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## Visual Math Improves Math Performance

Mathematics educators have long known that engaging students in visual representations of mathematics is extremely helpful for their learning. When youcubed offered “ How Close to 100 ” as an activity for learning math facts with visual representations, teachers across the world were thrilled and responded with thousands of tweets showing students learning by playing the game.

Research shows the importance of visual thinking for mathematicians.

When researchers have considered the brains of mathematicians, and compared them with similarly high achieving academics who worked in non-mathematical fields, they find something fascinating (Amalric & Dehaene, 2016). We might assume that when people are thinking numerically, they are thinking “with language” and that high-level mathematical reasoning would draw from language processing areas of the brain. What the researchers found was that the brain activity that separated mathematicians from other academics came from visual areas of the brain – and this was true whatever the mathematical content. It was not only subjects like geometry or topology that caused activity in the visual brain areas, but algebra and calculations, with minimal use of any language areas. This led the researchers to offer the possible explanation that the mathematicians’ achievement in math came from early childhood experiences with “number and shape.”

Visual mathematics is an important part of mathematics for its own sake and new brain research tells us that visual mathematics even helps students learn numerical mathematics.

In a ground breaking new study Joonkoo Park & Elizabeth Brannon (2013), found that the most powerful learning occurs when we use different areas of the brain. When students work with symbols, such as numbers, they are using a different area of the brain than when they work with visual and spatial information, such as an array of dots. The researchers found that mathematics learning and performance was optimized when the two areas of the brain were communicating (Park & Brannon, 2013). (for math questions that encourage this use of visual and symbolic representations see our Tasks ). Additionally, they found that training students through visual representations improved students’ math performance significantly, even on numerical math, and that the visual training helped students more than numerical training.

## What is Visual Mathematics?

Each of these visuals highlights the mathematics inside the problem and helps students develop understanding of multiplication. Pictures help students see mathematical ideas, which aids understanding. Visual mathematics also facilitates higher-level thinking, enables communication and helps people see the creativity in mathematics.

Mathematics is a subject that allows for precise thinking, but when that precise thinking is combined with creativity, openness, visualization, and flexibility, the mathematics comes alive.

Teachers can create such mathematical excitement in classrooms with any mathematics question by asking students for the different ways they see and can solve the problems and by encouraging discussion of different ways of seeing problems.

For an example of visualizing algebra see here .

When we don’t ask students to think visually, we miss an incredible opportunity to increase students’ understanding and to enable important brain crossing.

This extract contains excerpts from Jo Boaler’s book Mathematical Mindsets , and her new, forthcoming book: Mathematical Diversity.

## References:

Amalric, M., & Dehaene, S. (2016) . Origins of the brain networks for advanced mathematics in expert mathematicians. Proceedings of the National Academy of Sciences, 113(18), 4909-4917

Park, J., & Brannon, E. (2013) . Training the approximate number system improves math proficiency. Association for Psychological Science , 1–7.

Happening May 13-14 at Stanford: Our ONLY 2024 workshop on Teaching Mathematics through Big Ideas in the Middle Years! Learn more here

## Initial Thoughts

Perspectives & resources, what is high-quality mathematics instruction and why is it important.

- Page 1: The Importance of High-Quality Mathematics Instruction
- Page 2: A Standards-Based Mathematics Curriculum
- Page 3: Evidence-Based Mathematics Practices

## What evidence-based mathematics practices can teachers employ?

- Page 4: Explicit, Systematic Instruction

## Page 5: Visual Representations

- Page 6: Schema Instruction
- Page 7: Metacognitive Strategies
- Page 8: Effective Classroom Practices
- Page 9: References & Additional Resources
- Page 10: Credits

## Research Shows

- Students who use accurate visual representations are six times more likely to correctly solve mathematics problems than are students who do not use them. However, students who use inaccurate visual representations are less likely to correctly solve mathematics problems than those who do not use visual representations at all. (Boonen, van Wesel, Jolles, & van der Schoot, 2014)
- Students with a learning disability (LD) often do not create accurate visual representations or use them strategically to solve problems. Teaching students to systematically use a visual representation to solve word problems has led to substantial improvements in math achievement for students with learning disabilities. (van Garderen, Scheuermann, & Jackson, 2012; van Garderen, Scheuermann, & Poch, 2014)
- Students who use visual representations to solve word problems are more likely to solve the problems accurately. This was equally true for students who had LD, were low-achieving, or were average-achieving. (Krawec, 2014)

Visual representations are flexible; they can be used across grade levels and types of math problems. They can be used by teachers to teach mathematics facts and by students to learn mathematics content. Visual representations can take a number of forms. Click on the links below to view some of the visual representations most commonly used by teachers and students.

## How does this practice align?

High-leverage practice (hlp).

- HLP15 : Provide scaffolded supports

## CCSSM: Standards for Mathematical Practice

- MP1 : Make sense of problems and persevere in solving them.

Number Lines

Definition : A straight line that shows the order of and the relation between numbers.

Common Uses : addition, subtraction, counting

Strip Diagrams

Definition : A bar divided into rectangles that accurately represent quantities noted in the problem.

Common Uses : addition, fractions, proportions, ratios

Definition : Simple drawings of concrete or real items (e.g., marbles, trucks).

Common Uses : counting, addition, subtraction, multiplication, division

Graphs/Charts

Definition : Drawings that depict information using lines, shapes, and colors.

Common Uses : comparing numbers, statistics, ratios, algebra

Graphic Organizers

Definition : Visual that assists students in remembering and organizing information, as well as depicting the relationships between ideas (e.g., word webs, tables, Venn diagrams).

Common Uses : algebra, geometry

Before they can solve problems, however, students must first know what type of visual representation to create and use for a given mathematics problem. Some students—specifically, high-achieving students, gifted students—do this automatically, whereas others need to be explicitly taught how. This is especially the case for students who struggle with mathematics and those with mathematics learning disabilities. Without explicit, systematic instruction on how to create and use visual representations, these students often create visual representations that are disorganized or contain incorrect or partial information. Consider the examples below.

## Elementary Example

Mrs. Aldridge ask her first-grade students to add 2 + 4 by drawing dots.

Notice that Talia gets the correct answer. However, because Colby draws his dots in haphazard fashion, he fails to count all of them and consequently arrives at the wrong solution.

## High School Example

Mr. Huang asks his students to solve the following word problem:

The flagpole needs to be replaced. The school would like to replace it with the same size pole. When Juan stands 11 feet from the base of the pole, the angle of elevation from Juan’s feet to the top of the pole is 70 degrees. How tall is the pole?

Compare the drawings below created by Brody and Zoe to represent this problem. Notice that Brody drew an accurate representation and applied the correct strategy. In contrast, Zoe drew a picture with partially correct information. The 11 is in the correct place, but the 70° is not. As a result of her inaccurate representation, Zoe is unable to move forward and solve the problem. However, given an accurate representation developed by someone else, Zoe is more likely to solve the problem correctly.

## Manipulatives

Some students will not be able to grasp mathematics skills and concepts using only the types of visual representations noted in the table above. Very young children and students who struggle with mathematics often require different types of visual representations known as manipulatives. These concrete, hands-on materials and objects—for example, an abacus or coins—help students to represent the mathematical idea they are trying to learn or the problem they are attempting to solve. Manipulatives can help students develop a conceptual understanding of mathematical topics. (For the purpose of this module, the term concrete objects refers to manipulatives and the term visual representations refers to schematic diagrams.)

It is important that the teacher make explicit the connection between the concrete object and the abstract concept being taught. The goal is for the student to eventually understand the concepts and procedures without the use of manipulatives. For secondary students who struggle with mathematics, teachers should show the abstract along with the concrete or visual representation and explicitly make the connection between them.

A move from concrete objects or visual representations to using abstract equations can be difficult for some students. One strategy teachers can use to help students systematically transition among concrete objects, visual representations, and abstract equations is the Concrete-Representational-Abstract (CRA) framework.

If you would like to learn more about this framework, click here.

## Concrete-Representational-Abstract Framework

- Concrete —Students interact and manipulate three-dimensional objects, for example algebra tiles or other algebra manipulatives with representations of variables and units.
- Representational — Students use two-dimensional drawings to represent problems. These pictures may be presented to them by the teacher, or through the curriculum used in the class, or students may draw their own representation of the problem.
- Abstract — Students solve problems with numbers, symbols, and words without any concrete or representational assistance.

CRA is effective across all age levels and can assist students in learning concepts, procedures, and applications. When implementing each component, teachers should use explicit, systematic instruction and continually monitor student work to assess their understanding, asking them questions about their thinking and providing clarification as needed. Concrete and representational activities must reflect the actual process of solving the problem so that students are able to generalize the process to solve an abstract equation. The illustration below highlights each of these components.

## For Your Information

One promising practice for moving secondary students with mathematics difficulties or disabilities from the use of manipulatives and visual representations to the abstract equation quickly is the CRA-I strategy . In this modified version of CRA, the teacher simultaneously presents the content using concrete objects, visual representations of the concrete objects, and the abstract equation. Studies have shown that this framework is effective for teaching algebra to this population of students (Strickland & Maccini, 2012; Strickland & Maccini, 2013; Strickland, 2017).

Kim Paulsen discusses the benefits of manipulatives and a number of things to keep in mind when using them (time: 2:35).

Kim Paulsen, EdD Associate Professor, Special Education Vanderbilt University

View Transcript

Transcript: Kim Paulsen, EdD

Manipulatives are a great way of helping kids understand conceptually. The use of manipulatives really helps students see that conceptually, and it clicks a little more with them. Some of the things, though, that we need to remember when we’re using manipulatives is that it is important to give students a little bit of free time when you’re using a new manipulative so that they can just explore with them. We need to have specific rules for how to use manipulatives, that they aren’t toys, that they really are learning materials, and how students pick them up, how they put them away, the right time to use them, and making sure that they’re not distracters while we’re actually doing the presentation part of the lesson. One of the important things is that we don’t want students to memorize the algorithm or the procedures while they’re using the manipulatives. It really is just to help them understand conceptually. That doesn’t mean that kids are automatically going to understand conceptually or be able to make that bridge between using the concrete manipulatives into them being able to solve the problems. For some kids, it is difficult to use the manipulatives. That’s not how they learn, and so we don’t want to force kids to have to use manipulatives if it’s not something that is helpful for them. So we have to remember that manipulatives are one way to think about teaching math.

I think part of the reason that some teachers don’t use them is because it takes a lot of time, it takes a lot of organization, and they also feel that students get too reliant on using manipulatives. One way to think about using manipulatives is that you do it a couple of lessons when you’re teaching a new concept, and then take those away so that students are able to do just the computation part of it. It is true we can’t walk around life with manipulatives in our hands. And I think one of the other reasons that a lot of schools or teachers don’t use manipulatives is because they’re very expensive. And so it’s very helpful if all of the teachers in the school can pool resources and have a manipulative room where teachers can go check out manipulatives so that it’s not so expensive. Teachers have to know how to use them, and that takes a lot of practice.

## Using Visual Representations in Mathematics

On this page:, drawing on technology tools, in the classroom, online resources for visual representations, introduction.

All students can benefit from using visual representations, although struggling students may require additional, focused support and practice. Visual representations are a powerful way for students to access abstract mathematical ideas. To be college and career ready, students need to be able to draw a situation, graph lists of data, or place numbers on a number line. Developing this strategy early during the elementary grades gives students tools for engaging with—and ways of thinking about—increasingly abstract concepts. Over time, they will work toward developing Common Core Standards for Mathematical Practice:

- CCSS.Math.Practice.MP2 (opens in a new window) Reason abstractly and quantitatively.
- CCSS.Math.Practice.MP4 (opens in a new window) Model with mathematics.
- CCSS.Math.Practice.MP5 (opens in a new window) Use appropriate tools strategically.

WAYS TO SUPPORT STUDENTS

Helping students choose the “right” visual representation often depends on content and context. In some contexts, there are multiple ways to represent the same idea. Show your students a variety of examples in order to demonstrate when (and why) they should choose each one (see UDL Checkpoint 2.5: Illustrate through multiple media (opens in a new window) ). Consider how you could use the following strategies to support your students:

- Check for understanding to determine a starting point. For example, you could ask the following questions: Why do you think that? How do you know that is correct? How does that picture represent the problem? Can you explain your answer? Is there another way you could do that?
- Ask students about features of the visual representation (including labels and scales, when appropriate).
- As students create visual representations, ask questions to ensure that they understand all the features of the representations. Prompt students to focus on the information the visual representations provide.
- When possible, include alternative visual representations and discuss the similarities and differences between them.
- Vary the shapes and orientations of representations so that students focus only on the important features as they learn about the objects and situations represented.
- Show your students a specific representation—a graph or a table—that is missing an important feature. Ask them to identify the missing feature.

New technologies are constantly expanding our ability to visualize data and explain mathematical concepts. For teachers looking to incorporate technology into the classroom, using virtual manipulatives (instead of physical ones) can be a good start. Students can begin with simple graphical representations of mathematical concepts and then work toward more complex modules that require them to create the data or work within a system of rules, like a game. Infographics (opens in a new window) —visualizations that are designed to communicate complex information effectively—have become increasingly popular. They can be used to “tell a story” with numbers, such as international democracy rankings (opens in a new window) or climate change impacts (opens in a new window) . Learning to create infographics gives students additional tools to communicate data and other quantitative information.

3D printing is a technology that, until recently, has been too expensive to make use of in a classroom. However, thanks to falling prices, they have now started to appear in high schools and it may not be long before elementary schools and middle schools also embrace this technology. 3D printing allows you to create solid, three-dimensional models from a digital design. You can explore what others have created (opens in a new window) to get a sense of what is possible. Imagine having students design and create their own mathematical models and manipulatives!

For more ideas on using technology to create visual representations, visit the Tech Matters blog (opens in a new window) or PowerUp’s Pinterest page (opens in a new window) . You can also check out the “ Virtual Manipulatives (opens in a new window) ” video, which supports students’ use of visual representations.

Geometry lends itself naturally to teaching with visual representations, as can be seen in Ms. Richardson’s Grade 6 class. So far, students have learned how to classify different quadrilaterals and triangles, and they are beginning to decompose polygons. They have also started using software (e.g., GeoGebra (opens in a new window) ) that can support their understanding by emphasizing the connections between mathematical language and visualization.

Ms. Richardson’s lesson objective is to have students decompose polygons into triangles, rectangles, and trapezoids. She will address two s Common Core State in this lesson:

- CCSS Math 6.G.1 (opens in a new window) Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
- CCSS Math MP4 (opens in a new window) Model with mathematics.

Ms. Richardson has students work on these standards within the context of a real-world example—a painting by the artist Sol LeWitt.

Sol LeWitt. Wall Drawing #1113. On a wall, a triangle within a rectangle, each with broken bands of color, 2003. Hirshhorn Museum and Sculpture Garden, Smithsonian Institution.

Students will build on their existing technology skills and create a model of this work, decomposing polygons and creating their own virtual LeWitt in the process. Ms. Richardson’s lesson plan is organized into three sections: a warm-up exercise to review concepts, the main learning task, and a closing discussion and assessment.

## Lesson plan

This article draws from the PowerUp WHAT WORKS (opens in a new window) website, particularly the Visual Representations Instructional Strategy Guide (opens in a new window) . PowerUp is a free, teacher-friendly website that requires no log-in or registration. The Instructional Strategy Guide on visual representations includes a brief overview with an accompanying slide show; a list of the relevant mathematics Common Core State Standards; evidence-based teaching strategies to differentiate instruction using technology; short videos; and links to resources that will help you use technology to support mathematics instruction. If you want to dig deeper into the research foundation behind best practices in the use of virtual manipulatives, take a look at our Tech Research Brief (opens in a new window) on the topic. If you are responsible for professional development, the PD Support Materials (opens in a new window) provide helpful ideas and materials for using the resources. Want more information? See PowerUp WHAT WORKS (opens in a new window) .

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## The Epistemology of Visual Thinking in Mathematics

Visual thinking is widespread in mathematical practice, and has diverse cognitive and epistemic purposes. This entry discusses potential roles of visual thinking in proving and in discovering, with some examples, and epistemic difficulties and limitations are considered. Also discussed is the bearing of epistemic uses of visual representations on the application of the a priori–a posteriori distinction to mathematical knowledge. A final section looks briefly at how visual means can aid comprehension and deepen understanding of proofs.

## 1. Introduction

2. historical background, 3.1 the reliability question, 3.2 visual means in non-formal proving, 3.3 a dispute: diagrams in proofs in analysis., 4.1 propositional discovery, 4.2 discovering a proof strategy, 4.3 discovering properties and kinds, 5. visual thinking and mental arithmetic, 6.1 evidential uses of visual experience, 6.2 an evidential use of visual experience in proving, 6.3 a non-evidential use of visual experience, 7. further uses of visual representations, 8. conclusion, other internet resources, related entries.

Visual thinking is a feature of mathematical practice across many subject areas and at many levels. It is so pervasive that the question naturally arises: does visual thinking in mathematics have any epistemically significant roles? A positive answer begets further questions. Can we rationally arrive at a belief with the generality and necessity characteristic of mathematical theorems by attending to specific diagrams or images? If visual thinking contributes to warrant for believing a mathematical conclusion, must the outcome be an empirical belief? How, if at all can visual thinking contribute to understanding abstract mathematical subject matter?

Visual thinking includes thinking with external visual representations (e.g., diagrams, symbol arrays, kinematic computer images) and thinking with internal visual imagery; often the two are used in combination, as when we are required to visually imagine a certain spatial transformation of an object represented by a diagram on paper or on screen. Almost always (and perhaps always) visual thinking in mathematics is used in conjunction with non-visual thinking. Possible epistemic roles include contributions to evidence, proof, discovery, understanding and grasp of concepts. The kinds and the uses of visual thinking in mathematics are numerous and diverse. This entry will deal with some of the topics in this area that have received attention and omit others. Among the omissions is the possible explanatory role of visual representations in mathematics. The topic of explanation within pure mathematics is tricky and best dealt with separately; for this an excellent starting place is the entry on explanation in mathematics (Mancosu 2011). Two other omissions are the development of logic diagrams (Euler, Venn, Pierce and Shin) and the nature and use of geometric diagrams in Euclid’s Elements , both of which are well treated in the entry diagrams (Shin et al. 2013). The focus here is on visual thinking generally, which includes thinking with symbol arrays as well as with diagrams; there will be no attempt here to formulate a criterion for distinguishing between symbolic and diagrammatic thinking. However, the use of visual thinking in proving and in various kinds of discovery will be covered in what follows. Discussions of some related questions and some studies of historical cases not considered here are to be found in the collection Diagrams in Mathematics: History and Philosophy (Mumma and Panza 2012).

“Mathematics can achieve nothing by concepts alone but hastens at once to intuition” wrote Kant (1781/9: A715/B743), before describing the geometrical construction in Euclid’s proof of the angle sum theorem (Euclid, Book 1, proposition 32). In a review of 1816 Gauss echoes Kant:

anybody who is acquainted with the essence of geometry knows that [the logical principles of identity and contradiction] are able to accomplish nothing by themselves, and that they put forth sterile blossoms unless the fertile living intuition of the object itself prevails everywhere. (Ewald 1996 [Vol. 1]: 300)

The word “intuition” here translates the German “ Anschauung ”, a word which applies to visual imagination and perception, though it also has more general uses.

By the late 19 th century a different view had emerged, at least in foundational areas. In a celebrated text giving the first rigorous axiomatization of projective geometry, Pasch wrote: “the theorem is only truly demonstrated if the proof is completely independent of the figure” (Pasch 1882), a view expressed also by Hilbert in writing on the foundations of geometry (Hilbert 1894). A negative attitude to visual thinking was not confined to geometry. Dedekind, for example, wrote of an overpowering feeling of dissatisfaction with appeal to geometric intuitions in basic infinitesimal analysis (Dedekind 1872, Introduction). The grounds were felt to be uncertain, the concepts employed vague and unclear. When such concepts were replaced by precisely defined alternatives without allusions to space, time or motion, our intuitive expectations turned out to be unreliable (Hahn 1933).

In some quarters this view turned into a general disdain for visual thinking in mathematics: “In the best books” Russell pronounced “there are no figures at all” (Russell 1901). Although this attitude was opposed by a few mathematicians, notably Klein (1893), others took it to heart. Landau, for example, wrote a calculus textbook without a single diagram (Landau 1934). But the predominant view was not so extreme: thinking in terms of figures was valued as a means of facilitating grasp of formulae and linguistic text, but only reasoning expressed by means of formulae and text could bear any epistemological weight.

By the late 20 th century the mood had swung back in favour of visualization: Mancosu (2005) provides an excellent survey. Some books advertise their defiance of anti-visual puritanism in their titles, for example Visual Geometry and Topology (Fomenko 1994) and Visual Complex Analysis (Needham 1997); mathematics educators turn their attention to pedagogical uses of visualization (Zimmerman and Cunningham 1991); the use of computer-generated imagery begins to bear fruit at research level (Hoffman 1987; Palais 1999), and diagrams find their way into research papers in abstract fields: see for example the papers on higher dimensional category theory by Joyal et al. (1996), Leinster (2004) and Lauda (2005, Other Internet Resources). But attitudes to the epistemology of visual thinking remain mixed. The discussion is mostly concerned with the role of diagrams in proofs.

## 3. Visual thinking and proof

In some cases, it is claimed, a picture alone is a proof (Brown 1999: ch. 3). But that view is rare. Even the editor of Proofs without Words: Exercises in Visual Thinking , writes “Of course, ‘proofs without words’ are not really proofs” (Nelsen 1993: vi). Expressions of the other extreme are rare but can be found:

[the diagram] has no proper place in the proof as such. For the proof is a syntactic object consisting only of sentences arranged in a finite and inspectable array. (Tennant 1986)

Between the extremes we find the view that, even if no picture alone is a proof, visual representations can have a non-superfluous role in reasoning that constitutes a proof. (This is not to deny that there may be another proof of the same conclusion which does not involve any visual representation.) Geometric diagrams, graphs and maps, all carry information. Taking valid deductive reasoning to be the reliable extraction of information from information already obtained, Barwise and Etchemendy (1996:4) pose the following question: Why cannot the representations composing a proof be visual as well as linguistic? The sole reason for denying this role to visual representations is the thought that, with the possible exception of very restricted cases, visual thinking is unreliable, hence cannot contribute to proof. Is that right?

Our concern here is thinking through the steps in a proof, either for the first time (a first successful attempt to construct a proof) or following a given proof. Clearly we want to distinguish between visual thinking which merely accompanies the process of thinking through the steps in a proof and visual thinking which is essential to the process. This is not always straightforward as a proof can be presented in different ways. How different can distinct presentations be and yet be presentations of the same proof? There is no context-invariant answer to this. Often mathematicians are happy to regard two presentations as presenting the same proof if the central idea is the same in both cases. But if one’s main concern is with what is involved in thinking through a proof, its central idea is not enough to individuate it: the overall structure, the sequence of steps and perhaps some other factors affecting the cognitive processes involved will be relevant.

Once individuation of proofs has been settled, we can distinguish between replaceable thinking and superfluous thinking, where these attributions are understood as relative to a given argument or proof. In the process of thinking through a proof, a given part of the thinking is replaceable if thinking of some other kind could stand in place of the given part in a process that would count as thinking through the same proof. A given part of the thinking is superfluous if its excision without replacement would be a process of thinking through the same proof. Superfluous thinking may be extremely valuable in facilitating grasp of the proof text and in enabling one to understand the idea underlying the proof steps; but it is not necessary for thinking through the proof.

It is uncontentious that the visual thinking involved in symbol manipulations, for example in following the “algebraic” steps of proofs of basic lemmas about groups, can be essential, that is neither superfluous nor replaceable. The worry is about thinking visually with diagrams, where “diagram” is used widely to include all non-symbolic visual representations. Let us agree that there can be superfluous diagrammatic thinking in thinking through a proof. This leaves several possibilities.

- (a) All diagrammatic thinking in a process of thinking through a proof is superfluous.
- (b) Not all diagrammatic thinking in a process of thinking through a proof is superfluous; but if not superfluous it will be replaceable by non-diagrammatic thinking.
- (c) Some diagrammatic thinking in a process of thinking through a proof is neither superfluous nor replaceable by non-diagrammatic thinking.

The negative view stated earlier that diagrams can have no role in proof entails claim (a). The idea behind (a) is that, because diagrammatic reasoning is unreliable, if a process of thinking through an argument contains some non-superfluous diagrammatic thinking, that process lacks the epistemic security to be a case of thinking through a proof.

This view, claim (a) in particular, is threatened by cases in which the reliability of the diagrammatic thinking is demonstrated non-visually. The clearest kind of example would be provided by a formal system which has diagrams in place of formulas among its syntactic objects, and types of inter-diagram transition for inference rules. Suppose you take in such a formal system and an interpretation of it, and then think through a proof of the system’s soundness with respect to that interpretation; suppose you then inspect a sequence of diagrams, checking along the way that it constitutes a derivation in the system; suppose finally that you recover the interpretation to reach a conclusion. (The order is unimportant: one can go through the derivation first and then follow the soundness proof.) That entire process would constitute thinking through a proof of the conclusion; and the diagrammatic thinking involved would not be superfluous.

Shin et al. (2013) report that formal diagrammatic systems of logic and geometry have been proven to be sound. People have indeed followed proofs in these systems. That is enough to refute claim (a), the claim that all diagrammatic thinking in thinking through a proof is superfluous. For a concrete example, Figure 1 presents a derivation of Euclid’s first theorem, that on any straight line segment an equilateral triangle is constructible, in a formal diagrammatic system of a part of Euclidean geometry (Miller 2001).

What about Tennant’s claim that a proof is “a syntactic object consisting only of sentences” as opposed to diagrams? A proof is never a syntactic object. A formal derivation on its own is a syntactic object but not a proof. Without an interpretation of the language of the formal system the end-formula of the derivation says nothing; and so nothing is proved. Without a demonstration of the system’s soundness with respect to the interpretation, one may lack sufficient reason to believe that all derivable conclusions are true. A formal derivation plus an interpretation and soundness proof can be a proof of the derived conclusion, but that whole package is not a syntactic object. Moreover, the part of the proof which really is a syntactic object, the formal derivation, need not consist solely of sentences; it can consist of diagrams.

With claim (a) disposed of, consider again claim (b) that, while not all diagrammatic thinking in a process of thinking through a proof is superfluous, all non-superfluous diagrammatic thinking will be replaceable by non-diagrammatic thinking in a process of thinking through that same proof. The visual thinking in following the proof of Euclid’s first theorem using Miller’s formal system consists in going through a sequence of diagrams and at each step seeing that the next diagram results from a permitted alteration of the previous diagram. It is clear that in a process that counts as thinking through this proof, the diagrammatic thinking is neither superfluous nor replaceable by non-diagrammatic thinking. That knocks out (b), leaving only (c): some thinking that involves a diagram in thinking through a proof is neither superfluous nor replaceable by non-diagrammatic thinking (without changing the proof).

Mathematical practice almost never proceeds by way of formal systems. Outside the context of formal diagrammatic systems, the use of diagrams is widely felt to be unreliable. A diagram can be unfaithful to the described construction: it may represent something with a property that is ruled out by the description, or without a property that is demanded by the description. This is exemplified by diagrams in the famous argument for the proposition that all triangles are isosceles: the meeting point of an angle bisector and the perpendicular bisector of the opposite side is represented as falling inside the triangle, when it has to be outside (Rouse Ball 1939; Maxwell 1959). Errors of this sort are comparatively rare, usually avoidable with a modicum of care, and not inherent in the nature of diagrams; so they do not warrant a general charge of unreliability.

The major sort of error is unwarranted generalisation. Typically diagrams (and other non-verbal visual representations) do not represent their objects as having a property that is actually ruled out by the intention or specification of the object to be represented. But diagrams very frequently do represent their objects as having properties that, though not ruled out by the specification, are not demanded by it. Verbal descriptions can be discrete, in that they supply no more information than is needed. But visual representations are typically indiscrete, in that they supply too much detail. This is often unavoidable, because for many properties or kinds \(F\), a visual representation cannot represent something as being \(F\) without representing it as being \(F\) in a particular way . Any diagram of a triangle, for instance, must represent it as having three acute angles or as having just two acute angles, even if neither property is required by the specification, as would be the case if the specification were “Let ABC be a triangle”. As a result there is a danger that in using a diagram to reason about an arbitrary instance of class \(K\), we will unwittingly rely on a feature represented in the diagram that is not common to all instances of the class \(K\). Thus the risk of unwarranted generalisation is a danger inherent in the use of many diagrams.

Indiscretion of diagrams is not confined to geometrical figures. The dot or pebble diagrams of ancient mathematics used to convince one of elementary truths of number theory necessarily display particular numbers of dots, though the truths are general. Here is an example, used to justify the formula for the \(n\) th triangular number, i.e., the sum of the first \(n\) positive integers.

The conclusion drawn is that the sum of integers from 1 to \(n\) is \((n \times n+1)/2\) for any positive integer \(n\), but the diagram presents the case for \(n = 6\). We can perhaps avoid representing a particular number of dots when we merely imagine a display of the relevant kind; or if a particular number is represented, our experience may not make us aware of the number—just as, when one imagines the sky on a starry night, for no particular number \(k\) are we aware that exactly \(k\) stars are represented. Even so, there is likely to be some extra specificity. For example, in imagining an array of dots of the form just illustrated, one is unlikely to imagine just two columns of three dots, the rectangular array for \(n = 2\). Typically the subject will be aware of imagining an array with more than two columns. This entails that an image is likely to have unintended exclusions. In this case it would exclude the three-by-two array. An image of a triangle representing all angles as acute would exclude triangles with an obtuse angle or a right angle. The danger is that the visual reasoning will not be valid for the cases that are unintentionally excluded by the visual representation, with the result that the step to the conclusion is an unwarranted generalisation.

What should we make of this? First, let us note that in a few cases the image or diagram will not be over-specific. When in geometry all instances of the relevant class are congruent to one another, for instance all circles or all squares, the image or diagram will not be over-specific for a generalisation about that class; so there will be no unintended exclusions and no danger of unwarranted generalisation. Here then are possibilities for reliable visual thinking in proving.

To get clear about the other cases, where there is a danger of over generalizing, it helps to look at generalisation in ordinary non-visual reasoning. Schematically put, in reasoning about things of kind \(K\), once we have shown that from certain premisses it follows that such-and-such a condition is true of arbitrary instance \(c\), we can validly infer from those same premisses that that condition is true of all \(K\)s, with the proviso that neither the condition nor any premiss mentions \(c\). The proviso is required, because if a premiss or the condition does mention \(c\), the reasoning may depend on a property of \(c\) that is not shared by all other \(K\)s and so the generalisation would be unsafe. For a trivial example consider a step from “\(x = c\)” to “\(\forall x [x = c]\)”.

A question we face is whether, in order to come to know the truth of a conclusion by following an argument involving generalisation on an arbitrary instance (a.k.a. universal generalisation, or universal quantifier introduction), the thinking must include a conscious, explicit check that the proviso is met. It is clearly not enough that the proviso is in fact met. For in that case it might just be the thinker’s good luck that the proviso is met; hence the thinker would not know that the generalisation is valid and so would not have genuinely thought through the proof at that step.

This leaves two options. The strict option is that without a conscious, explicit check one has not really thought through the proof. The relaxed option is that one can properly think through the proof without checking that the proviso is met, but only if one is sensitive to the potential error and would detect it in otherwise similar arguments. For then one is not just lucky that the proviso is met. Being sensitive in this context consists in being alert to dependence on features of the arbitrary instance not shared by all members of the class of generalisation, a state produced by a combination of past experience and current vigilance. Without compelling reason to prefer one of these options, decisions on what is to count as proving or following a proof must be conditional.

How does all this apply to generalizing from visual thinking about an arbitrary instance? Take the example of the visual route to the formula for triangular numbers using the diagram of Figure 2 . The diagram reveals that the formula holds for the 6 th triangular number. The generalisation to all triangular numbers is justified only if the visuo-spatial method used is applicable to the \(n\) th triangular number for all positive integers \(n\), that is, provided that the method used does not depend on a property not shared by all positive integers. A conscious, explicit check that this proviso is met requires making explicit the method exemplified for 6 and proving that the method is applicable for all positive integers in place of 6. (For a similar idea in the context of automating visual arguments, see Jamnik 2001). This is not done in practice when thinking visually, and so if we accept the strict option for thinking through a proof involving generalisation, we would have to accept that the visual route to the formula for triangular numbers does not amount to thinking through a proof of it; and the same would apply to the familiar visual routes to other general positive integer formulas, such as that \(n^2 =\) the sum of the first \(n\) odd numbers.

But what if the strict option for proving by generalisation on an arbitrary instance is too strict, and the relaxed option is right? When arriving at the formula in the visual way indicated, one does not pay attention to the fact that the visual display represents the situation for the 6 th triangular number; it is as if the mind had somehow extracted a general schema of visual reasoning from exposure to the particular case, and had then proceeded to reason schematically, converting a schematic result into a universal proposition. What is required, on the relaxed option, is sensitivity to the possibility that the schema is not applicable to all positive integers; one must be so alert to ways a schema of the given kind can fall short of universal applicability that if one had been presented with a schema that did fall short, one would have detected the failure.

In the example at hand, the schema of visual reasoning involves at the start taking a number \(k\) to be represented by a column of \(k\) dots, thence taking the triangular array of \(n\) columns to represent the sum of the first \(n\) positive integers, thence taking that array combined with an inverted copy to make a rectangular array of \(n\) columns of \(n+1\) dots. For a schema starting this way to be universally applicable, it must be possible, given any positive integer \(n\), for the sum of the first \(n\) positive integers to be represented in the form of a triangular array, so that combined with an inverted copy one gets a rectangular array. This actually fails at the extreme case: \(n = 1\). The formula \((n.(n + 1))/2\) holds for this case; but that is something we know by substituting “1” for the variable in the formula, not by the visual method indicated. That method cannot be applied to \(n = 1\), because a single dot does not form a triangular array, and combined with a copy it does not form a rectangular array. But we can check that the method works for all positive integers after the first, using visual reasoning to assure ourselves that it works for 2 and that if the method works for \(k\) it works for \(k+1\). Together with this reflective thinking, the visual thinking sketched earlier constitutes following a proof of the formula for the \(n\) th triangular number for all integers \(n > 1\), at least if the relaxed view of thinking through a proof is correct. Similar conclusions hold in the case of other “dot” arguments (Giaquinto 1993, 2007: ch. 8). So in some cases when the visual representation carries unwanted detail, the danger of over-generalisation in visual reasoning can be overcome.

But the fact that this is frequently missed by commentators suggests that the required sensitivity is often absent. Missing an untypical case is a common hazard in attempts at visual proving. A well-known example is the proof of Euler’s formula \(V - E + F = 2\) for polyhedra by “removing triangles” of a triangulated planar projection of a polyhedron. One is easily convinced by the thinking, but only because the polyhedra we normally think of are convex, while the exceptions are not convex. But it is also easy to miss a case which is not untypical or extreme when thinking visually. An example is Cauchy’s attempted proof (Cauchy 1813) of the claim that if a convex polygon is transformed into another polygon keeping all but one of the sides constant, then if some or all of the internal angles at the vertices increase, the remaining side increases, while if some or all of the internal angles at the vertices decrease, the remaining side decreases. The argument proceeds by considering what happens when one transforms a polygon by increasing (or decreasing) angles, angle by angle. But in a trapezoid, changing a single angle can turn a convex polygon into a concave polygon, and this invalidates the argument (Lyusternik 1963).

The frequency of such mistakes indicates that visual arguments (other than symbol manipulations) often lack the transparency required for proof. Even when a visual argument is in fact sound, its soundness may not be clear, in which case the argument is not a way of proving the truth of the conclusion, though it may be a way of discovering it. But this is consistent with the claim that visual non-symbolic thinking can be (and often is) part of a way of proving something.

An example from knot theory will substantiate the modal part of this claim. To present the example, we need some background information, which will be given with a minimum of technical detail.

A knot is a tame closed non-self-intersecting curve in Euclidean 3-space.

In other words, knots are just the tame curves in Euclidean 3-space which are homeomorphic to a circle. The word “tame” here stands for a property intended to rule out certain pathological cases, such as curves with infinitely nested knotting. There is more than one way of making this mathematically precise, but we have no need for these details. A knot has a specific geometric shape, size and axis-relative position. Now imagine it to be made of flexible yet unbreakable yarn that is stretchable and shrinkable, so that it can be smoothly transformed into other knots without cutting or gluing. Since our interest in a knot is the nature of its knottedness regardless of shape, size or axis-relative position, the real focus of interest is not just the knot but all its possible transforms. A way to think of this is to imagine a knot transforming continuously, so that every possible transform is realized at some time. Then the thing of central interest would be the object that persists over time in varying forms, with knots strictly so called being the things captured in each particular freeze frame. Mathematically, we represent the relevant entity as an equivalence class of knots.

Two knots are equivalent iff one can be smoothly deformed into the other by stretching, shrinking, twisting, flipping, repositioning or in any other way that does not involve cutting, gluing or passing one strand through another.

The relevant kind of deformation forbids eliminating a knotted part by shrinking it down to a point. Again there are mathematically precise definitions of knot-equivalence. Figure 3 gives diagrams of equivalent knots, instances of a trefoil.

Diagrams like these are not merely illustrations; they also have an operational role in knot theory. But not any picture of a knot will do for this purpose. We need to specify:

A knot diagram is a regular projection of a knot onto a plane which, when there is a crossing, tells us which strand passes over the other.

Regularity here is a combination of conditions. In particular, regularity entails that not more than two points of the strict knot project to the same point on the plane, and that two points of the strict knot project to the same point on the plane only where there is a crossing. For more on diagrams in knot theory see (De Toffoli and Giardino 2014).

A major task of knot theory is to find ways of telling whether two knot diagrams are diagrams of equivalent knots. In particular we will want to know if a given knot diagram represents a knot equivalent to an unknot , that is, a knot representable by a knot diagram without crossings.

One way of showing that a knot diagram represents a knot equivalent to an unknot is to show that the diagram can be transformed into one without crossings by a sequence of atomic moves, known as Reidemeister moves. The relevant background fact is Reidemeister’s theorem, which links the visualizable diagrammatic changes to the mathematically precise definition of knot equivalence: Two knots are equivalent if and only if there is a finite sequence of Reidemeister moves taking a knot diagram of one to a knot diagram of the other. Figure 4 illustrates. Each knot diagram is changed into the adjacent knot diagram by a Reidemeister move; hence the knot represented by the leftmost diagram is equivalent to the unknot.

In contrast to these, the knot presented by the left knot diagram of Figure 3 , a trefoil, may seem impossible to deform into an unknot. And in fact it is. To prove it, we can use a knot invariant known as colourability. An arc in a knot diagram is a maximal part between crossings (or the whole thing if there are no crossings). Colourability is this:

A knot diagram is colourable if and only if each of its arcs can be coloured one of three different colours so that (a) at least two colours are used and (b) at each crossing the three arcs are all coloured the same or all coloured differently.

The reference to colours here is inessential. Colourability is in fact a specific case of a kind of combinatorial property known as mod \(p\) labelling (for \(p\) an odd prime). Colourability is a knot invariant in the sense that if one diagram of a knot is colourable every diagram of that knot and of any equivalent knot is colourable. (By Reidemeister’s theorem this can be proved by showing that each Reidemeister move preserves colourability.) A standard diagram of an unknot, a diagram without crossings, is clearly not colourable because it has only one arc (the whole thing) and so two colours cannot be used. So in order to complete proving that the trefoil is not equivalent to an unknot, we only need prove that our trefoil diagram is colourable. This can be done visually. Colour each arc of the knot diagram one of the three colours red, green or blue so that no two arcs have the same colour (or visualize this). Then do a visual check of each crossing, to see that at each crossing the three meeting arcs are all coloured differently. That visual part of the proof is clearly non-superfluous and non-replaceable (without changing the proof). Moreover, the soundness of the argument is quite transparent. So here is a case of a non-formal, non-symbolic visual way of proving a mathematical truth.

Where notions involving the infinite are in play, such as many involving limits, the use of diagrams is famously risky. For this reason it has been widely thought that, beyond some very simple cases, arguments in real and complex analysis in which diagrams have a non-superfluous role are not genuine proofs. Bolzano [1817] expressed this attitude with regard to the intermediate value theorem for the real numbers (IVT) before giving a purely analytic proof, arguing that spatial thinking could not be used to help justify the IVT. James Robert Brown (1999) takes issue with Bolzano on this point. The IVT is this:

If \(f\) is a real-valued function of a real variable continuous on the closed interval \([a, b]\) and \(f(a) < c < f(b)\), then for some \(x\) in \((a, b), f(x) = c\).

Brown focuses on the special case when \(c = 0\). As the IVT can be deduced easily from this special case using the theorem that the difference of two continuous functions is continuous, there is no loss of generality here. Alluding to a diagram like Figure 5, Brown (1999) writes

We have a continuous line running from below to above the \(x\)-axis. Clearly, it must cross that axis in doing so. (1999: 26)

Later he claims:

Using the picture alone, we can be certain of this result—if we can be certain of anything. (1999: 28)

Bolzano’s diagram-free proof of the IVT is an argument from what later became known as the Dedekind completeness of the real numbers: every non-empty set of reals bounded above (below) has a least upper bound (greatest lower bound). The value of Bolzano’s deduction of the IVT from the Dedekind completeness of the reals, according to Brown, is not that it proves the IVT but that it gives us confirmation of Dedekind completeness, just as an empirical hypothesis in empirical science gets confirmed by deducing some consequence of the hypothesis and observing those consequence to be true. This view assumes that we already know the IVT to be true by observing a diagram relevantly like Figure 5 .

That assumption is challenged by Giaquinto (2011). Once we distinguish graphical concepts from associated analytic concepts, the underlying argument from the diagram is essentially this.

- 1. Any function \(f\) which is \(\varepsilon\textrm{-}\delta\) continuous on \([a, b]\) with \(f (a) < 0 < f (b)\) has a visually continuous graphical curve from below the horizontal line representing the \(x\)-axis to above.
- 2. Any visually continuous graphical curve from below a horizontal line to above it meets the line at a crossing point.
- 3. Any function whose graphical curve meets the line representing the \(x\)-axis at a crossing point has a zero value.
- 4. So, any \(\varepsilon\textrm{-}\delta\) continuous function \(f\) on \([a, b]\) with \(f (a) < 0< f (b)\) has a zero value.

What is inferred from the diagram is premiss 2. Premisses 1 and 3 are assumptions linking analytical with graphical conditions. These linking assumptions are disputed. With regard to premiss 1 Giaquinto (2011) argues that there are functions on the reals which meet the antecedent condition but do not have graphical curves, such as continuous but nowhere differentiable functions and functions which oscillate with unbounded frequency e.g., \(f(x) = x \cdot\sin(1/x)\) for non-zero \(x\) in \([-1, 1]\) and \(f(0) = 0\).

With regard to premiss 3 it is argued that, under the standard conventions of graphical representation of functions in a Cartesian co-ordinate frame, the graphical curve for \(x^2 - 2\) in the rationals is the same as the graphical curve for \(x^2- 2\) in the reals. This is because every real is a limit point of rationals; so for every point \(P\) with one or both co-ordinates irrational, there are points arbitrarily close to \(P\) with both co-ordinates rational; so no gaps would appear if irrational points were removed from the curve for \(x^2- 2\) in the reals. But for \(x\) in the rational interval [0, 2] the function \(x^2- 2\) has no zero value, even though it has a graphical curve which visually crosses the line representing the \(x\)-axis. So one cannot read off the existence of a zero of \(x^2- 2\) on the reals from the diagram; one needs to appeal to some property of the reals which the rationals lack, such as Dedekind completeness.

This raises some obvious questions. Do any theorems of analysis have proofs in which diagrams have a non-superfluous role? Littlewood (1953: 54–5) thought so and gives an example which is examined in Giaquinto (1994). If so, can we demarcate this class of theorems by some mathematical feature of their content? Another question is whether there is a significantly broad class of functions on the reals for which we could prove an intermediate value theorem (i.e., restricted to that class).

If there are theorems of analysis provable with diagrams we do not yet have a mathematical demarcation criterion for them. A natural place to look would be O-minimal structures on the reals—this was brought to the author’s attention by Ethan Galebach. This is because of some remarkable theorems about such structures which exclude all the pathological (hence vision-defying) functions on the reals (Van den Dries 1998), such as continuous nowhere differentiable functions and “space-filling” curves i.e., continuous surjections \(f:(0, 1)\rightarrow(0, 1)^2\). Is the IVT for functions in an O-minimal structure on the reals provable by visual means? Certainly one objection to the visual argument for the unrestricted IVT does not apply when the restriction is in place. This is the objection that continuous nowhere differentiable functions, having no graphical curve, provide counterexamples to the premiss that any \(\varepsilon\textrm{-}\delta\) continuous function \(f\) on \([a, b]\) with \(f (a) < c < f (b)\) has a visually continuous graphical curve from below the horizontal line representing \(y = c\) to above. But the existence of continuous functions with no graphical curve is not the only objection to the visual argument, contrary to a claim of Azzouni (2013: 327). There are also counterexamples to the premiss that any function that does have a graphical curve which visibly crosses the line representing \(y = c\) takes \(c\) as a value, e.g., the function \(x^2 - 2\) on the rationals with \(c = 0\). So the question of a visual proof of the IVT restricted to functions in an O-minimal structure on the reals is still open at the time of writing.

## 4. Visual thinking and discovery

Though philosophical discussion of visual thinking in mathematics has concentrated on its role in proof, visual thinking may be more valuable for discovery than proof. Three kinds of discovery important in mathematical practice are these:

- (1) propositional discovery (discovering, of a proposition, that it is true),
- (2) discovering a proof strategy (or more loosely, getting the idea for a proof of a proposition), and
- (3) discovering a property or kind of mathematical entity.

In the following subsections visual discovery of these kinds will be discussed and illustrated.

To discover a truth, as that expression is being used here, is to come to believe it by one’s own lights (as opposed to reading it or being told) in a way that is reliable and involves no violation of epistemic rationality (given one’s epistemic state). One can discover a truth without being the first to discover it (in this context); it is enough that one comes to believe it in an independent, reliable and rational way. The difference between merely discovering a truth and proving it is a matter of transparency: for proving or following a proof the subject must be aware of the way in which the conclusion is reached and the soundness of that way; this is not required for discovery.

Sometimes one discovers something by means of visual thinking using background knowledge, resulting in a cogent argument from which one could construct a proof. A nice example is a visual argument that any knot diagram with a finite number of crossings can be turned into a diagram of an unknot by interchanging the over-strand and under-strand of some of its crossings (Adams 2001: 58–90). That argument is a bit too long to present accessibly here. For a short example, here is a way of discovering that the geometric mean of two positive numbers is less than or equal to their arithmetic mean (Eddy 1985) using Figure 6.

Two circles (with diameters \(a\) and \(b\)) meet at a single point. A line is drawn between their centres through their common point; its length is \((a + b)/2\), the sum of the two radii. This line is the hypotenuse of a right angled triangle with one other side of length \((a - b)/2\), the difference of the radii. Pythagoras’s theorem is used to infer that the remaining side of the right-angled triangle has length \(\sqrt{(ab)}\).Then visualizing what happens to the triangle when the diameter of the smaller circle varies between 0 and the diameter of the larger circle, one infers that \(0 < \sqrt{(ab)} < (a + b)/2\); then verifying symbolically that \(\sqrt{(ab)} = (a + b)/2\) when \(a = b\), one concludes that for positive \(a\) and \(b\), \(\sqrt{(ab)} \le (a + b)/2\).

This thinking does not constitute a case of proving or following a proof of the conclusion, because it involves a step which we cannot clearly tell is valid. This is the step of attempting to visually imagine what would happen when the smaller circle varies in diameter between 0 and the diameter of the larger circle and inferring from the resulting experience that the line joining the centres of the circles will always be longer than the horizontal line from the centre of the smaller circle to the vertical diameter of the larger circle. This step seems sound (does not lead us into error) and may be sound; but its soundness is opaque. If in fact it is sound, the whole thinking process is a reliable way of reaching the conclusion; so in the absence of factors that would make it irrational to trust the thinking, it would be a way of discovering the conclusion to be true.

In some cases visual thinking inclines one to believe something on the basis of assumptions suggested by the visual representation that remain to be justified given the subject’s current knowledge. In such cases there is always the danger that the subject takes the visual representation to show the correctness of the assumptions and ends up with an unwarranted belief. In such a case, even if the belief is true, the subject has not made a discovery, as the means of belief-acquisition is unreliable. Here is an example using Figure 7 (Montuchi and Page 1988).

Using this diagram one can come to think the following about the real numbers. When for a constant \(k\) the positive values of \(x\) and \(y\) are constrained to satisfy the equation \(x \cdot y = k\), the positive values of \(x\) and \(y\) for which \(x + y\) is minimal are \(x = \sqrt{k} = y\). (Let “#” denote this claim.)

Suppose that one knows the conventions for representing functions by graphs in a Cartesian co-ordinate system, knows also that the diagonal represents the function \(y = x\), and that a line segment with gradient –1 from \((0, b)\) to \((b, 0)\) represents the function \(x + y = b\). Then looking at the diagram may incline one to think that for no positive value of \(x\) does the value of \(y\) in the function \(x\cdot y = k\) fall below the value of \(y\) in \(x + y = 2\sqrt{k}\), and that these functions coincide just at the diagonal. From these beliefs the subject may (correctly) infer the conclusion #. But mere attention to the diagram cannot warrant believing that, for a given positive \(x\)-value, the \(y\)-value of \(x\cdot y = k\) never falls below the \(y\)-value of \(x + y = 2\sqrt{k}\) and that the functions coincide just at the diagonal; for the conventions of representation do not rule out that the curve of \(x\cdot y = k\) meets the curve of \(x + y = 2\sqrt{k}\) at two points extremely close to the diagonal, and that the former curve falls under the latter in between those two points. So the visual thinking is not in this case a means of discovering proposition #.

But it is useful because it provides the idea for a proof of the conclusion—one of the major benefits of visual thinking in mathematics. In brief: for each equation \((x\cdot y = k\); \(x + y = 2\sqrt{k})\) if \(x = y\), their common value is \(\sqrt{k}\). So the functions expressed by those equations meet at the diagonal. To show that, for a fixed positive \(x\)-value, the \(y\)-values of \(x\cdot y = k\) never fall below the \(y\)-values of \(x + y = 2\sqrt{k}\), it suffices to show that \(2\sqrt{k} - x \le k/x\). As a geometric mean is less than or equal to the corresponding arithmetic mean, \(\sqrt{[x \cdot (k/x)]} \le [x + (k/x)]/2\). So \(2\sqrt{k} \le x + (k/x)\). So \(2\sqrt{k} - x \le k/x\).

In this example, visual attention to, and reasoning about, the diagram is not part of a way of discovering the conclusion. But if it gave one the idea for the argument just given, it would be part of what led to a way of discovering the conclusion, and that is important.

Can visual thinking lead to discovery of an idea for a proof in more advanced contexts? Yes. Carter (2010) gives an example from free probability theory. The case is about certain permutations (those denoted by “\(p\)” with a circumflex in Carter 2010) on a finite set of natural numbers. Using specific kinds of diagram, easily seen properties of the diagrams lead one naturally to certain properties of the permutations (crossing and non-crossing, having neighbouring pairs), and to a certain operation (cancellation of neighbouring pairs). All of these have algebraic definitions, but the ideas defined were noticed by thinking in terms of the diagrams. For the relevant permutations \(\sigma\), \(\sigma(\sigma(n)) = n\); so a permutation can be represented by a set of lines joining dots. The permutations represented on the left and right in Figure 8 are non-crossing and crossing respectively, the former with neighbouring pairs \(\{2, 3\}\) and \(\{6, 7\}\).

A permutation \(\sigma\) of \(\{1, 2, \ldots, 2p\}\) is defined to have a crossing just when there are \(a\), \(b\), \(c\), \(d\) in \(\{1, 2, \ldots, 2p\}\) such that \(a < b < c < d\) and \(\sigma(a) = c\) and \(\sigma(b) = d\). The focus is on the proof of a theorem which employs this notion. (The theorem is that when a permutation of \(\{1, 2, \ldots, 2p\}\) of the relevant kind is non-crossing, there will be exactly \(p+1\) R-equivalence classes, where \(R\) is a certain equivalence relation on \(\{1, 2, \ldots, 2p\}\) defined in terms of the permutation.) Carter says that the proofs of some lemmas “rely on a visualization of the setup”, in that to grasp the correctness of one or more of the steps one needs to visualize the situation. There is also a nice example of some reasoning in terms of a diagram which gives the idea for a proof (“suggests a proof strategy”) for the lemma that every non-crossing permutation has a neighbouring pair. Reflection on a diagram such as Figure 9 does the work.

The reasoning is this. Suppose that \(\pi\) has no neighbouring pair. Choose \(j\) such that \(\pi(j) - j = a\) is minimal, that is, for all \(k, \pi(j) - j \le \pi(k) - k\). As \(\pi\) has no neighbouring pair, \(\pi(j+1) \ne j\). So either \(\pi(j+1)\) is less than \(j\) and we have a crossing, or by minimality of \(\pi(j) - j\), \(\pi(j+1)\) is greater than \(j+a\) and again we have a crossing. Carter reports that this disjunction was initially believed by thinking in term of the diagram, and the proof of the lemma given in the published paper is a non-diagrammatic “version” of that reasoning. In this case study, visual thinking is shown to contribute to discovery in several ways; in particular, by leading the mathematicians to notice crucial properties—the “definitions are based on the diagrams”—and in giving them the ideas for parts of the overall proof.

In this section I will illustrate and then discuss the use of visual thinking in discovering kinds of mathematical entity, by going through a few of the main steps leading to geometric group theory, a subject which really took off in the 1980s through the work of Mikhail Gromov. The material is set out nicely in greater depth in Starikova (2012).

Sometimes it can be fruitful to think of non-spatial entities, such as algebraic structures, in terms of a spatial representation. An example is the representation of a finitely generated group by a Cayley graph. Let \((G, \cdot)\) be a group and \(S\) a finite subset of \(G\). Let \(S^{-1}\) be the set of inverses of members of \(S\). Then \((G, \cdot)\) is generated by \(S\) if and only if every member of \(G\) is the product (with respect to \(\cdot\)) of members of \(S\cup S^{-1}\). In that case \((G, \cdot, S)\) is said to be a finitely generated group. Here are a couple of examples.

First consider the group \(S_{3}\) of permutations of 3 elements under composition. Letting \(\{a, b, c\}\) be the elements, all six permutations can be generated by \(\rf\) and \(\rr\) where

\(\rf\) (for “flip”) fixes a and swaps \(b\) with \(c\), i.e., it takes to \(\langle a, b, c\rangle\) to \(\langle a, c, b\rangle\), and

\(\rr\) (for “rotate”) takes \(\langle a, b, c\rangle\) to \(\langle c, a, b\rangle\).

The Cayley graph for \((S_{3}, \cdot, \{\rf, \rr\})\) is a graph whose vertices represent the members of \(S_{3}\) and two “colours” of directed edges, representing composition with \(\rf\) and composition with \(\rr\). Figure 10 illustrates: red directed edges represent composition with \(\rr\) and black edges represent composition with \(\rf\). So a red edge from a vertex \(\rv\) representing \(\rs\) in \(S_{3}\) ends at a vertex representing \(\rs\rr\) and a black edge from \(\rv\) ends at a vertex representing \(\rs\rf\). (Notation: “\(\rs\rr\)” abbreviates “\(\rs \cdot \rr\)” which here denotes “\(\rs\) followed by \(\rr\)”; same for “\(\rf\)” in place of “\(\rr\)”.) A black edge has arrowheads both ways because \(\rf\) is its own inverse, that is, flipping and flipping again takes you back to where you started. (Sometimes a pair of edges with arrows in opposite directions is used instead.) The symbol “\(\re\)” denotes the identity.

An example of a finitely generated group of infinite order is \((\mathbb{Z}, +, \{1\})\). We can get any integer by successively adding 1 or its additive inverse \(-1\). Since 3 added to the inverse of 2 is 1, and 2 added to the inverse of 3 is \(-1\), we can get any integer by adding members of \(\{2, 3\}\) and their inverses. Thus both \(\{1\}\) and \(\{2, 3\}\) are generating sets for \((\mathbb{Z}, +)\). Figure 11 illustrates part of the Cayley graph for \((\mathbb{Z}, +, \{2, 3\})\). The horizontal directed edges represent +2. The directed edges ascending or descending obliquely represent \(+3\).

Another example of a generated group of infinite order is \(F_2\), the free group generated by a pair of members. The first few iterations of its Cayley graph are shown in Figure 12, where \(\{a, b\}\) is the set of generators and a right horizontal move between adjacent vertices represents composition with \(a\), an upward vertical move represents composition with \(b\), and leftward and downward moves represent composition with the inverse of \(a\) and the inverse of \(b\) respectively. The central vertex represents the identity.

Thinking of generated groups in terms of their Cayley graphs makes it very natural to view them as metric spaces. A path is a sequence of consecutively adjacent edges, regardless of direction. For example in the Cayley graph for \((\mathbb{Z}, +, \{2, 3\})\) the edges from \(-2\) to 1, from 1 to \(-1\), from \(-1\) to 2 (in that order) constitute a path, representing the action, starting from \(-2\), of adding 3, then adding \(-2\), then adding 3. Taking each edge to have unit length, the metric \(d_S\) for a group \(G\) generated by a finite subset \(S\) of \(G\) is defined: for any \(g\), \(h \in G\), \(d_{S}(g, h) =\) the length of a shortest path from \(g\) to \(h\) in the Caley graph of \((G, \cdot, S)\). This is the word metric for this generated group.

Viewing a finitely generated group as a metric space allows us to consider its growth function \(\gamma(n)\) which is the cardinality of the “ball” of radius \(\le n\) centred on the identity (the number of members of the group whose distance from the identity is not greater than \(n\)). A growth function for a given group depends on the set of generators chosen, but when the group is infinite the asymptotic behaviour as \(n \rightarrow \infty\) of the growth functions is independent of the set of generators.

Noticing the possibility of defining a metric on generated groups did not require first viewing diagrams of their Cayley graphs. This is because a word in the generators is just a finite sequence of symbols for the generators or their inverses (we omit the symbol for the group operation), and so has an obvious length visually suggested by the written form of the word, namely the number of symbols in the sequence; and then it is natural to define the distance between group members \(g\) and \(h\) to be the length of a shortest word that gets one from \(g\) to \(h\) by right multiplication, that is, \(\textrm{min}\{\textrm{length}(w): w = g^{-1}h\}\).

However, viewing generated groups by means of their Cayley graphs was the necessary starting point for geometric group theory, which enables us to view finitely generated groups of infinite order not merely as graphs or metric spaces but as geometric entities. The main steps on this route will be sketched briefly here; for more detail see Starikova (2012) and the references therein. The visual key is to start thinking in terms of the “coarse geometry” of the Cayley graph of the generated group, by zooming out in visual imagination so far that the discrete nature of the graph is transformed into a traditional geometrical object. For example, the Cayley graph of a generated group of finite order such as \((S_{3}, \cdot, \{f, r\})\) illustrated in Figure 11 becomes a dot; the Cayley graph for \((\mathbb{Z}, +, \{2, 3\})\) illustrated in Figure 12 becomes an uninterrupted line infinite in both directions.

The word metric of a generated group is discrete: the values are always in \(N\). How is this visuo-spatial association of a discrete metric space with a continuous geometrical object achieved mathematically? By quasi-isometry. While an isometry from one metric space to another is a distance preserving map, a quasi-isometry is a map which preserves distances to within fixed linear bounds. Precisely put, a map \(f\) from \((S, d)\) to \((S', d')\) is a quasi-isometry iff for some real constants \(L > 0\) and \(K \ge 0\) and all \(x\), \(y\) in \(S\) \[ d(x, y)/L - K \le d'(f(x), f(y)) \le L \cdot d(x, y) + K. \]

The spaces \((S, d)\) and \((S', d')\) are quasi - isometric spaces iff the quasi-isometry \(f\) is also quasi-surjective, in the sense that there is a real constant \(M \ge 0\) such that every point of \(S'\) is no further than \(M\) away from some point in the image of \(f\).

For example, \((\mathbb{Z}, d)\) is quasi-isometric to \((\mathbb{R}, d)\) where \(d(x, y) = |y - x|\), because the inclusion map \(\iota\) from \(\mathbb{Z}\) to \(\mathbb{R}\), \(\iota(n) = n\), is an isometry hence a quasi-isometry with \(L = 1\) and \(K = 0\), and each point in \(\mathbb{R}\) is no further than \(1/2\) away from an integer (in \(\mathbb{R}\)). Also, it is easy to see that for any real number \(x\), if \(g(x) =\) the nearest integer to \(x\) (or the greatest integer less than \(x\) if it is midway between integers) then \(g\) is a quasi-isometry from \(\mathbb{R}\) to \(\mathbb{Z}\) with \(L = 1\) and \(K =\frac{1}{2}\);.

The relation between metric spaces of being quasi-isometric is an equivalence relation. Also, if \(S\) and \(T\) are generating sets of a group \((G, \cdot)\), the Cayley graphs of \((G, \cdot, S)\) and \((G, \cdot, T)\) with their word metrics are quasi-isometric spaces. This means that properties of a generated group which are quasi-isometric invariants will be independent of the choice of generating set, and therefore informative about the group itself.

Moreover, it is easy to show that the Cayley graph of a generated group with word metric is quasi-isometric to a geodesic space. [ 1 ] A triangle with vertices \(x\), \(y\), \(z\) in this space is the union of three geodesic segments, between \(x\) and \(y\), between \(y\) and \(z\), and between \(z\) and \(x\). This is the gateway for the application of Gromov’s insights, some of which can be grasped with the help of visual geometric thinking.

Here are some indications. Recall the Poincaré open disc model of hyperbolic geometry: geodesics are diameters or arcs of circles orthogonal to the boundary, with unit distance represented by ever shorter Euclidean distances as one moves from the centre towards the boundary. (The boundary is not part of the model). All triangles have angle sum \(< \pi\) ( Figure 13, left ), and there is a global constant δ such that all triangles are δ-thin in the following sense:

A triangle \(T\) is δ- thin if and only if any point on one side of \(T\) lies within δ of some point on one of the other two sides.

This condition is equivalent to the condition that each side of \(T\) lies within the union of the δ-neighbourhoods of the other two sides, as illustrated in Figure 13 , right. There is no constant δ such that all triangles in a Euclidean plane are δ-thin, because for any δ there are triangles large enough that the midpoint of a longest side lies further than δ from all points on the other two sides.

Figure 13 [ 2 ]

The definition of thin triangles is sufficiently general to apply to any geodesic space and allows a generalisation of the concept of hyperbolicity beyond its original context:

- A geodesic space is hyperbolic iff for some δ all its triangles are δ-thin.
- A group is hyperbolic iff it has a Cayley graph quasi-isometric to a hyperbolic geodesic space.

The class of hyperbolic groups is large and includes important subkinds, such as finite groups, free groups and the fundamental groups of surfaces of genus \(\ge 2\). Some striking theorems have been proved for them. For example, for every hyperbolic group the word problem is solvable, and every hyperbolic group has a finite presentation. So we can reasonably conclude that the discovery of this mathematical kind, the hyperbolic groups, has been fruitful.

How important was visual thinking to the discoveries leading to geometric group theory? Visual thinking was needed to discover Cayley graphs as a means of representing finitely generated groups. This is not the triviality it might seem: Cayley graphs must be distinguished from the diagrams we use to present them visually. A Cayley graph is a mathematical representation of a generated group, not a visual representation. It consists of the following components: a set \(V\) (“vertices”), a set \(E\) of ordered pairs of members of \(V\) (“directed edges”) and a partition of \(E\) into distinguished subsets, (“colours”, each one for representing right multiplication by a particular generator). The Cayley graph of a generated group of infinite order cannot be fully represented by a diagram given the usual conventions of representation for diagrams of graphs, and distinct diagrams may visually represent the same Cayley graph: both diagrams in Figure 14 can be labelled so that under the usual conventions they represent the Cayley graph of \((S_{3}, \cdot, \{f, r\})\), already illustrated by Figure 10 . So the Cayley graph cannot be a diagram.

Diagrams of Cayley graphs were important in prompting mathematicians to think in terms of the coarse-grained geometry of the graphs, in that this idea arises just when one thinks in terms of “zooming out” visually. Gromov (1993) makes the point in a passage quoted in Starikova (2012:138)

This space [a Cayley graph with the word metric] may appear boring and uneventful to a geometer’s eye since it is discrete and the traditional (e.g., topological and infinitesimal) machinery does not run in [the group] Γ. To regain the geometric perspective one has to change one’s position and move the observation point far away from Γ. Then the metric in Γ seen from the distance \(d\) becomes the original distance divided by \(d\) and for \(d \rightarrow \infty\) the points in Γ coalesce into a connected continuous solid unity which occupies the visual horizon without any gaps and holes and fills our geometer’s heart with joy.

In saying that one has to move the observation point far away from Γ so that the points coalesce into a unity which occupies the visual horizon, he makes clear that visual imagination is involved in a crucial step on the road to geometric group theory. Visual thinking is again involved in discovering hyperbolicity as a property of general geodesic spaces from thinking about the Poincaré disk model of hyperbolic geometry. It is hard to see how this property would have been discovered without the use of visual resources.

While there is no reason to think that mental arithmetic (mental calculation in the integers and rational numbers) typically involves much visual thinking, there is strong evidence of substantial visual processing in the mental arithmetic of highly trained abacus users.

In earlier times an abacus would be a rectangular board or table surface marked with lines or grooves along which pebbles or counters could be moved. The oldest surviving abacus, the Salamis abacus, dated around 300 BCE, is a white marble slab, with markings designed for monetary calculation (Fernandes 2015, Other Internet Resources). These were superseded by rectangular frames within which wires or rods parallel to the short sides are fixed, with moveable holed beads on them. There are several kinds of modern abacus — the Chinese suanpan, the Russian schoty and the Japanese soroban for example — each kind with variations. Evidence for visual processing in mental arithmetic comes from studies with well trained users of the soroban, an example of which is shown in Figure 15.

Each column of beads represents a power of 10, increasing to the left. The horizontal bar, sometimes called the reckoning bar , separates the beads on each column into one bead of value 5 above and four beads of value 1 below. The number represented in a column is determined by the beads which are not separated from the reckoning bar. A column on which all beads are separated by a gap from the bar represents zero. For example, the number 6059 is represented on a portion of a schematic soroban in Figure 16.

On some sorobans there is a mark on the reckoning bar at every third column; if a user chooses one of these as a unit column, the marks will help the user keep track of which columns represent which powers of ten. Calculations are made by using forefinger and thumb to move beads according to procedures for the standard four numerical operations and for extraction of square and cube roots (Bernazzani 2005,Other Internet Resources). Despite the fact that the soroban has a decimal place representation of numbers, the soroban procedures are not ‘translations’ of the procedures normally taught for the standard operations using arabic numerals. For example, multidigit addition on a soroban starts by adding highest powers of ten and proceeds rightwards to lower powers, instead of starting with units thence proceeding leftwards to tens, hundreds and so on.

People trained to use a soroban often learn to do mental arithmetic by visualizing an abacus and imagining moving beads on it in accordance with the procedures learned for arithmetical calculations (Frank and Barner 2012). Mental abacus (MA), as this kind of mental arithmetic is known, compares favourably with other kinds of mental calculation for speed and accuracy (Kojima 1954) and MA users are often found among the medallists in the Mental Calculation World Cup.

Although visual and manual motor imagery is likely to occur, cognitive scientists have probed the question whether the actual processes of MA calculation consist in or involve imagining performing operations on a physical abacus. Brain imaging studies provide one source of evidence bearing on this question. Comparing well-trained abacus calculators with matched controls, evidence has been found that MA involves neural resources of visuospatial working memory with a form of abacus which does not depend on the modality (visual or auditory) of the numerical inputs (Chen et al. 2006). Another imaging study found that, compared to controls without abacus training, subjects with long term MA training from a young age had enhanced brain white matter related to motor and visuospatial processes (Hu et al. 2011).

Behavioural studies provide more evidence. Tests on expert and intermediate level abacus users strongly suggest that MA calculators mentally manipulate an abacus representation so that it passes through the same states that an actual abacus would pass through in solving an addition problem. Without using an actual abacus MA calculators were able to answer correctly questions about intermediates states unique to the abacus-based solution of a problem; moreover, their response times were a monotonic function of the position of the probed state in the sequence of states of the abacus process for solving the problem (Stigler 1984). On top of the ‘intermediate states’ evidence, there is ‘error type’ evidence. Mental addition tests comparing abacus users with American subjects revealed that abacus users made errors of a kind which the Americans did not make, but which were predictable from the distribution of errors in physical abacus addition (Stigler 1984).

Another study found evidence that when a sequence of numbers is presented auditorily (as a verbal whole “three thousand five hundred and forty seven” or as a digit sequence “Three, five, four, seven”) abacus experts encode it into an imaged abacus display, while non-experts encode it verbally (Hishitani 1990).

Further evidence comes from behavioural interference studies. In these studies subjects have to perform mental calculations, with and without a task of some other kind to be performed during the calculation, with the aim of seeing which kinds of task interfere with calculation as measured by differences of reaction time and error rate. An early study found that a linguistic task interfered weakly with MA performance (unless the linguistic task was to answer a mathematical question), while motor and visual tasks interfered relatively strongly. These findings suggested to the paper’s authors that MA representations are not linguistic in nature but rely on visual mechanisms and, for intermediate practitioners, on motor mechanisms as well (Hatano et al. 1977).

These studies provide impressive evidence that MA does involve mental manipulation of a visualized abacus. However, limits of the known capacities for perceiving or representing pluralities of objects seem to pose a problem. We have a parallel individuation system for keeping track of up to four objects simultaneously and an approximate number system (ANS) which allows us to gauge roughly the cardinality of a set of things, with an error which increases with the size of the set. The parallel individuation system has a limit of three or four objects and the ANS represents cardinalities greater than four only approximately. Yet mental abacus users would need to hold in mind with precision abacus representations involving a much larger number of beads than four (and the way in which those beads are distributed on the abacus). For example, the number 439 requires a precise distribution of twelve beads. Frank and Barner (2012) address this problem. In some circumstances we can perceive a plurality of objects as a single entity, a set, and simultaneously perceive those objects as individuals. There is evidence that we can keep track of up to three such sets in parallel and simultaneously make reliable estimates of the cardinalities of the sets (if not more than four). If the sets themselves can be easily perceived as (a) divided into disjoint subsets, e.g. columns of beads on an abacus, and (b) structured in a familiar way, e.g. as a distribution of four beads below a reckoning bar and one above, we have the resources for recognising a three-digit number from its abacus representation. The findings of (Frank and Barner 2012) suggest that this is what happens in MA: a mental abacus is represented in visuospatial working memory by splitting it into a series of columns each of which is stored as a unit with its own detailed substructure.

These cognitive investigations confirm the self-reports of mental abacus users that they calculate mentally by visualizing operating on an abacus as they would operate on a physical abacus. (See the 20-second movie Brief interview with mental abacus user , at the Stanford Language and Cognition Lab, for one such self-report.) There is good evidence that MA often involves processes linked to motor cognition in addition to active visual imagination. Intermediate abacus users often make hand movements, without necessarily attending to those movements during MA calculation, as shown in the second of the three short movies just mentioned. Experiments to test the possible role of motor processes in MA resulted in findings which led the authors to conclude that premotor processes involved in the planning of hand movements were involved in MA (Brooks et al. 2018).

## 6. A priori and a posteriori roles of visual experience

In coming to know a mathematical truth visual experience can play a merely “enabling” role. For example, visual experience may have been a factor in a person’s getting certain concepts involved in a mathematical proposition, thus enabling her to understand the proposition, without giving her reason to believe it. Or the visual experience of reading an argument in a text book may enable one to find out just what the argument is, without helping her tell that the argument is sound. In earlier sections visual experience has been presented as having roles in proof and propositional discovery that are not merely enabling. On the face of it this raises a puzzle: mathematics, as opposed to its application to natural phenomena, has traditionally been thought to be an a priori science; but if visual experience plays a role in acquiring mathematical knowledge which is not merely enabling, the result would surely be a posteriori knowledge, not a priori knowledge. Setting aside knowledge acquired by testimony (reading or hearing that such-&-such is the case), there remain plenty of cases where sensory experience seems to play an evidential role in coming to know some mathematical fact.

A plausible example of the evidential use of sensory experience is the case of a child coming to know that \(5 + 3 = 8\) by counting on her fingers. While there may be an important \(a\) priori element in the child’s appreciation that she can reliably generalise from the result of her counting experiment, getting that result by counting is an a posteriori route to it. For another example, consider the question: how many vertices does a cube have? With the background knowledge that cubes do not vary in shape and that material cubes do not differ from geometrical cubes in number of vertices (where a “vertex” of a material cube is a corner), one can find the answer by visually inspecting a material cube. Or if one does not have a material cube to hand, one can visually imagine a cube, and by attending to its top and bottom faces extract the information that the vertices of the cube are exactly the vertices of these two quadrangular faces. When one gets the answer by inspecting a material cube, the visual experience contributes to one’s grounds for believing the answer and that contribution is part of what makes the belief state knowledge. So the role of the visual experience is evidential; hence the resulting knowledge is not a priori . When one gets the answer by visually imagining a cube, one is drawing on the accumulated cognitive effects of past experiences of seeing material cubes to bring to mind what a cube looks like; so the experience of visual imagining has an indirectly evidential role in this case.

Do such examples show that mathematics is not an a priori science? Yes, if an a priori science is understood to be one whose knowable truths are all knowable only in an a priori way, without use of sense experience as evidence. No, if an a priori science is one whose knowable truths are all knowable in an a priori way, allowing that some may be knowable also in an a posteriori way.

Many cases of proving something (or following a proof of it) involve making, or imagining making, changes in a symbol array. A standard presentation of the proof of left-cancellation in group theory provides an example. “Left-cancellation” is the claim that for any members \(a\), \(b\), \(c\) of a group with operation \(\cdot\) and identity element \(\mathbf{e}\), if \(a \cdot b = a \cdot c\), then \(b = c\). Here is (the core of) a proof of it:

Suppose that one comes to know left-cancellation by following this sequence of steps. Is this an a priori way of getting this knowledge? Although following a mathematical proof is thought to be a paradigmatically a priori way of getting knowledge, attention to the role of visual experience here throws this into doubt. The case for claiming that the visual experience has an evidential role is as follows.

The visual experience reveals not only what the steps of the argument are but also that they are valid, thereby contributing to our grounds for accepting the argument and believing its conclusion. Consider, for example, the step from the second equation to the third. The relevant background knowledge, apart from the logic of identity, is that a group operation is associative. This fact is usually represented in the form of an equation that simply relocates brackets in an obvious way:

We see that relocating the brackets in accord with this format, the left-hand term of the second equation is transformed into the left-hand term of the third equation, and the same for the right-hand terms. So the visual experience plays an evidential role in our recognising as valid the step from the second equation to the third. Hence this quite standard route to knowledge of left-cancellation turns out to be a posteriori , even though it is a clear case of following a proof.

Against this, one may argue that the description just given of what is going on in following the proof is not strictly correct, as follows. Exactly the same proof can be expressed in natural language, using “the composition of \(x\) with \(y\)” for “\(x \cdot y\)”, but the result would be hard to take in. Or the proof can be presented using a different notational convention, one which forces a quite different expression of associativity. For example, we can use the Polish convention of putting the operation symbol before the operands: instead of “\(x \cdot y\)” we put “\(\cdot x y\)”. In that case associativity would be expressed in the following way, without brackets:

The equations of the proof would then need to be re-symbolised; but what is expressed by each equation after re-symbolisation and the steps from one to the next would be exactly as before. So we would be following the very same proof, step by step. But we would not be using visual experiences involved to notice the relocation of brackets this time. This suggests that the role of the different visual experiences involved in following the argument in its different guises is merely to give us access to the common reasoning: the role of the experience is merely enabling. On this account the visual experience does not strictly and literally enable us to see that any of the steps are valid; rather, recognition of (or sensitivity to) the validity of the steps results from cognitive processing at a more abstract level.

Which of these rival views is correct? Does our visual experience in following the argument presented with brackets (1) reveal to us the validity of some of the steps, given the relevant background knowledge ? Or (2) merely give us access to the argument? The core of the argument against view (1) is this:

Seeing the relocation of brackets is not essential to following the argument.

So seeing merely gives access to the argument; it does not reveal any step to be valid.

The step to this conclusion is faulty. How one follows a proof may, and in this case does, depend on how it is presented, and different ways of following a proof may be different ways of coming to know its conclusion. While seeing the relocation of brackets is not essential to all ways of following this argument, it is essential to the normal way of following the argument when it is symbolically presented with brackets in the way given above.

Associativity, expressed without symbols, is this: When the binary group operation is applied twice in succession on an ordered triple of operands \(\langle a, b, c\rangle\), it makes no difference whether the first application is to the initial two operands or the final two operands. While this is the content of associativity, for ease of processing associativity is almost always expressed as a symbol-manipulation rule. Visual perception is used to tell in particular cases whether the rule thus expressed is correctly implemented, in the context of prior knowledge that the rule is correct. What is going on here is a familiar division of labour in mathematical thinking. We first establish the soundness of a rule of symbol-manipulation (in terms of the governing semantic conventions—in this case the matter is trivial); then we check visually that the rule is correctly implemented. Processing at a more abstract, semantic level is often harder than processing at a purely syntactic level; it is for this reason that we often resort to symbol-manipulation techniques as proxy for reasoning directly with meanings to solve a problem. (What is six eighths divided by three fifths, without using any symbolic technique?) When we do use symbol-manipulation in proving or following a proof, visual experience is required to discern that the moves conform to permitted patterns and thus contributes to our grounds for accepting the argument. Then the way of coming to know the conclusion has an a posteriori element.

Must a use of visual experience in knowledge acquisition be evidential , if the visual experience is not merely enabling? Here is an example which supports a negative answer. Imagine a square or look at a drawing of one. Each of its four sides has a midpoint. Now visualize the “inner” square whose sides run between the midpoints of adjacent sides of the original square (Figure 17, left). By visualizing this figure, it should be clear that the original square is composed precisely of the inner square plus four corner triangles, each side of the inner square being the base of a corner triangle. One can now visualize the corner triangles folding over, with creases along the sides of the inner square. The starting and end states of the imagery transformation can be represented by the left and right diagrams of Figure 17.

Visualizing the folding-over within the remembered frame of the original square results in an image of the original square divided into square quarters, its quadrants, and the sides of the inner square seem to be diagonals of the quadrants. Many people conclude that the corner triangles can be arranged to cover the inner square exactly, without any gap or overlap. Thence they infer that the area of the original square is twice the size of the inner square. Let us assume that the propositions concerned are about Euclidean figures. Our concern is with the visual route to the following:

The parts of a square beyond its inner square (formed by joining midpoints of adjacent sides of the original square) can be arranged to fit the inner square exactly, without overlap or gap, without change of size or shape.

The experience of visualizing the corner triangles folding over can lead one to this belief. But it cannot provide good evidence for it. This is because visual experience (of sight or imagination) has limited acuity and so does not enable us to discriminate between a situation in which the outer triangles fit the inner square exactly and a situation in which they fit inexactly but well enough for the mismatch to escape visual detection. (This contrasts with the case of discovering the number of vertices of a cube by seeing or visualizing one.) Even though visualizing the square, the inner square and then visualizing the corner triangles folding over is constrained by the results of earlier perceptual experience of scenes with relevant similarities, we cannot draw from it reliable information about exact equality of areas, because perception itself is not reliable about exact equalities (or exact proportions) of continuous magnitudes.

Though the visual experience could not provide good evidence for the belief, it is possible that we erroneously use the experience evidentially in reaching the belief. But it is also possible, when reaching the belief in the way described, that we do not take the experience to provide evidence. A non-evidential use is more likely, if when one arrives at the belief in this way one feels fairly certain of it, while aware that visual perception and imagination have limited acuity and so cannot provide evidence for a claim of exact fit.

But what could the role of the visualizing experience possibly be, if it were neither merely enabling nor evidential? One suggestion is that we already have relevant beliefs and belief-forming dispositions, and the visualizing experience could serve to bring to mind the beliefs and to activate the belief-forming dispositions (Giaquinto 2007). These beliefs and dispositions will have resulted from prior possession of cognitive resources, some subject-specific such as concepts of geometrical figures, some subject-general such as symmetry perception about perceptually salient vertical and horizontal axes. A relevant prior belief in this case might be that a square is symmetric about a diagonal. A relevant disposition might be the disposition to believe that the quadrants of a square are congruent squares upon seeing or visualizing a square with a horizontal base plus the vertical and horizontal line segments joining midpoints of its opposite sides. (These dispositions differ from ordinary perceptual dispositions to believe what we see in that they are not cancelled when we mistrust the accuracy of the visual experience.)

The question whether the resulting belief would be knowledge depends on whether the belief-forming dispositions are reliable (truth-conducive) and the pre-existing belief states are states of knowledge. As these conditions can be met without any violation of epistemic rationality, the visualizing route described incompletely here can be a route to knowledge. In that case we would have an example of a use of visual experience which is integral to a way of knowing a truth, which is not merely enabling and yet not evidential. A fuller account and discussion is given in chapters 3 and 4 of Giaquinto (2007).

There are other significant uses of visual representations in mathematics. This final section briefly presents a couple of them.

Although the use of diagrams in arguments in analysis faces special dangers (as noted in 3.3 ), the use of diagrams to illustrate symbolically presented operations can be very helpful. Consider, for example, this pair of operations \(\{ f(x) + k, f(x + k) \}\). Grasping them and the difference between them can be aided by a visual illustration; similarly for the sets \(\{ f(x + k), f(x - k) \}\), \(\{ |f(x)|, f(|x|) \}\), \(\{ f(x)^{-1}, f^{-1}(x), f(x^{-1}) \}\). While generalization on the basis of a visual illustration is unreliable, we can use them as checks against calculation errors and overgeneralization. The same holds for properties. Consider for example, functions for which \(f(-x) = f(x)\), known as even functions, and functions for which \(f(-x) = -f(x)\), the odd functions: it can be helpful to have in mind the images of graphs of \(y = x^2\) and \(y = x^{3}\) as instances of evenness and oddness, to remind one that even functions are symmetrical about the \(y\)-axis and odd functions have rotation symmetry by \(\pi\) about the origin. They can serve as a reminder and check against over-generalisation: any general claim true of all odd functions, for example, must be true of \(y = x^{3}\) in particular.

The utility of visual representations in real and complex analysis is not confined to such simple cases. Visual representations can help us grasp what motivates certain definitions and arguments, and thereby deepen our understanding. Abundant confirmation of this claim can be gathered from working through the text Visual Complex Analysis (Needham 1997). Some mathematical subjects have natural visual representations, which then give rise to a domain of mathematical entities in their own right. This is true of geometry but is also true of subjects which become algebraic in nature very quickly, such as graph theory, knot theory and braid theory. Techniques of computer graphics now enable us to use moving images. For an example of the power of kinematic visual representations to provide and increase understanding of a subject, see the first two “chapters” of the online introduction to braid theory by Ester Dalvit (2012, Other Internet Resources).

With regard to proofs, a minimal kind of understanding consists in understanding each line (proposition or formula) and grasping the validity of each step to a new line from earlier lines. But we can have that stepwise grasp of proof without any idea of why it proceeds by those steps. One has a more advanced (or deeper) kind of understanding when one has the minimal understanding and a grasp of the motivating idea(s) and strategy of the proof. The point is sharply expressed by Weyl (1995 [1932]: 453), quoted in (Tappenden 2005:150)

We are not very pleased when we are forced to accept a mathematical truth by virtue of a complicated chain of formal conclusions and computations, which we traverse blindly, link by link, feeling our way by touch. We want first an overview of the aim and the road; we want to understand the idea of the proof, the deeper context.

Occasionally the author of a proof gives readers the desired understanding by adding commentary. But this is not always needed, as the idea of a proof is sometimes revealed in the presentation of the proof itself. Often this is done by using visual representations. An example is Fisk’s proof of Chvátal’s “art gallery” theorem. This theorem is the answer to a combinatorial problem in geometry. Put concretely, the problem is this. Let the \(n\) walls of a single-floored gallery make a polygon. What is the smallest number of stationary guards needed to ensure that every point of the gallery wall can be seen by a guard? If the polygon is convex (all interior angles < 180°), one guard will suffice, as guards may rotate. But if the polygon is not convex, as in Figure 18, one guard may not be enough.

Chvátal’s theorem gives the answer: for a gallery with \(n\) walls, \(\llcorner n/3\lrcorner\) guards suffice, where \(\llcorner n/3\lrcorner\) is the greatest integer \(\le n/3\). (If this does not sound to you sufficiently like a mathematical theorem, it can be restated as follows: Let \(S\) be a subset of the Euclidean plane. For a subset \(B\) of \(S\) let us say that \(B\) supervises \(S\) iff for each \(x \in S\) there is a \(y \in B\) such that the segment \(xy\) lies within \(S\). Then the smallest number \(f(n)\) such that every set bounded by a simple \(n\)-gon is supervised by a set of \(f(n)\) points is at most \(\llcorner n/3.\lrcorner\)

Here is Steve Fisk’s proof. A short induction shows that every polygon can be triangulated, i.e., non-crossing edges between non-adjacent vertices (“diagonals”) can be added so that the polygon is entirely composed of non-overlapping triangles. So take any \(n\)-sided polygon with a fixed triangulation. Think of it as a graph, a set of vertices and connected edges, as in Figure 19.

The first part of the proof shows that the graph is 3-colourable, i.e., every vertex can be coloured with one of just three colours (red, white and blue, say) so that no edge connects vertices of the same colour.

The argument proceeds by induction on \(n \ge 3\), the number of vertices.

For \(n = 3\) it is trivial. Assume it holds for all \(k\), where \(3 \le k < n\).

Let triangulated polygon \(G\) have \(n\) vertices. Let \(u\) and \(v\) be any two vertices connected by diagonal edge \(uv\). The diagonal \(uv\) splits \(G\) into two smaller graphs, both containing \(uv\). Give \(u\) and \(v\) different colours, say red and white, as in Figure 20.

By the inductive assumption, we may colour each of the smaller graphs with the three colours so that no edge joins vertices of the same colour, keeping fixed the colours of \(u\) and \(v\). Pasting together the two smaller graphs as coloured gives us a 3-colouring of the whole graph.

What remains is to show that \(\llcorner n/3\lrcorner\) or fewer guards can be placed on vertices so that every triangle is in the view of a guard. Let \(b\), \(r\) and \(w\) be the number of vertices coloured blue, red and white respectively. Let \(b\) be minimal in \(\{b, r, w\}\). Then \(b \le r\) and \(b \le w\). Then \(2b \le r + w\). So \(3b \le b + r + w = n\). So \(b \le n/3\) and so \(b \le \llcorner n/3\lrcorner\). Place a guard on each blue vertex. Done.

The central idea of this proof, or the proof strategy, is clear. While the actual diagrams produced here are superfluous to the proof, some visualizing enables us to grasp the central idea.

Thinking which involves the use of seen or visualized images, which may be static or moving, is widespread in mathematical practice. Such visual thinking may constitute a non-superfluous and non-replaceable part of thinking through a specific proof. But there is a real danger of over-generalisation when using images, which we need to guard against, and in some contexts, such as real and complex analysis, the apparent soundness of a diagrammatic inference is liable to be illusory.

Even when visual thinking does not contribute to proving a mathematical truth, it may enable one to discover a truth, where to discover a truth is to come to believe it in an independent, reliable and rational way. Visual thinking can also play a large role in discovering a central idea for a proof or a proof-strategy; and in discovering a kind of mathematical entity or a mathematical property.

The (non-superfluous) use of visual thinking in coming to know a mathematical truth does in some cases introduce an a posteriori element into the way one comes to know it, resulting in a posteriori mathematical knowledge. This is not as revolutionary as it may sound as a truth knowable a posteriori may also be knowable a priori . More interesting is the possibility that one can acquire some mathematical knowledge in a way in which visual thinking is essential but does not contribute evidence; in this case the role of the visual thinking may be to activate one’s prior cognitive resources. This opens the possibility that non-superfluous visual thinking may result in a priori knowledge of a mathematical truth.

Visual thinking may contribute to understanding in more than one way. Visual illustrations may be extremely useful in providing examples and non-examples of analytic concepts, thus helping to sharpen our grasp of those concepts. Also, visual thinking accompanying a proof may deepen our understanding of the proof, giving us an awareness of the direction of the proof so that, as Hermann Weyl put it, we are not forced to traverse the steps blindly, link by link, feeling our way by touch.

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How to cite this entry . Preview the PDF version of this entry at the Friends of the SEP Society . Look up topics and thinkers related to this entry at the Internet Philosophy Ontology Project (InPhO). Enhanced bibliography for this entry at PhilPapers , with links to its database.

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## The role of visual representations in the learning of mathematics

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- A. Arcavi 1

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Visualization, as both the product and the process of creation, interpretation and reflection upon pictures and images, is gaining increased visibility in mathematics and mathematics education. This paper is an attempt to define visualization and to analyze, exemplify and reflect upon the many different and rich roles it can and should play in the learning and the doing of mathematics. At the same time, the limitations and possible sources of difficulties visualization may pose for students and teachers are considered.

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Arcavi, A. The role of visual representations in the learning of mathematics. Educational Studies in Mathematics 52 , 215–241 (2003). https://doi.org/10.1023/A:1024312321077

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March 18, 2021, by Rupert Knight

## Using visual models to solve problems and explore relationships in Mathematics: beyond concrete, pictorial, abstract – Part 1

This two-part blog series by Marc North explores some thinking and strategies for using representations in Mathematics lessons. Part 1 unpicks some of the key theoretical ideas around the use of representations and models and foregrounds how representations can be used to both solve problems and explore mathematical relationships. Part 2 will illustrate these theoretical ideas practically via a classroom based Maths activity.

Our brains don’t like abstract ideas!

There is little doubt that visual models are a key part of the learning and teaching of mathematics. One of the reasons for this is that while much of school mathematics involves abstract concepts that can be generalized across a range of topics and problems, our brains actually don’t like abstract ideas! Instead, many of us prefer to learn and think through concrete examples and deliberately look for concrete and practical examples to help to explain abstract concepts. See, for example, the Learning Scientists site .

Visual models provide a useful tool for ‘concretising’ complex and abstract ideas. The human brain responds positively to information packaged in creative and visual ways, which is why throughout our daily lives we are constantly bombarded with visual imagery and stimuli. It is often easier to remember information presented in picture form than as a string of words, and visual models provide succinct and organised summaries of information. Visual models can also demonstrate relationships between different items and, when shown dynamically, can show how those relationships change and evolve.

Multiple Representations in Teaching for Mastery

The emphasis on a ‘Teaching for Mastery’ approach in both Primary and Secondary schools has ushered in a clear priority for using representations to model and illustrate mathematical ideas and problems. As indicated in Figure 1, the ‘Big Idea’ of representation and structure provides students with access to mathematical concepts and supports them to visualise patterns and make connections both within and between concepts.

In the Teaching for Mastery approach, the Concrete-Pictorial-Abstract (CPA) framework provides the main approach that structures how teachers are encouraged to work with different representations. The CPA approach is a multi-sensory teaching model that introduces abstract concepts in a concrete and tangible way, by moving from concrete materials, to pictorial representations, to abstract symbols and problems.

Figure 2: Concrete-Pictorial-Abstract approach (Caroll, Pikul, Foust & Grodziak)

The Concrete dimension is the ‘doing’ stage, where children use physical resources (e.g. manipulatives) to model problems. This stage can also involve the use of concrete situations that are linked to real-life contexts. The Pictorial dimension is the ‘seeing’ stage, where pictures are used to model the concrete resources and the problems. The ‘abstract’ stage is the ‘symbolic’ stage, where abstract mathematical symbols are used to model problems. Many teachers have adopted this approach enthusiastically, operationalizing this practice in various ways:

Origins of the CPA approach – Bruner’s Representation Modes

The theoretical origins of this approach stem from Jerome Bruner’s wor k on different representation modes. As a social-constructivist, Bruner argues that children’s problem-solving skills are developed through inquiry and discovery, and also that to support deep learning subject matter should be represented and experienced by children in terms of how they will view and experience the world. This is facilitated by using different representation modes that model the stages of our learning and that reflect the ways in which humans store and encode knowledge and information in memory. The enactive stage (‘based on action’) (from birth to one year old) involves the encoding and storage of information through direct manipulation of objects – for example, think of a baby playing with a rattle. At this early stage, there is no clear internal representation of the object by the individual. The iconic stage (‘based on images’) (from one to six years old) involves an internal representation of external objects visually in the form of a mental image or icon – for example, a child being able to draw a picture of a tree without actually having a tree in front of them. The symbolic stage (‘based on language’) (seven years and up) is when information is able to be stored in the form of a code or symbol – for example, being able to describe a tree in words or through writing.

Potential challenges with the CPA approach

Although the CPA approach is based on Bruner’s work, there are some important distinctions – which also give rise to some potential challenges with this approach.

First, the CPA framework has adopted a theory that describes children’s development and learning through various age ranges into a sequence for instruction for children at all ages. This has resulted in some teachers using this approach in a strictly hierarchical way, always starting with the concrete and progressing to the abstract – and treating the abstract as the ultimate goal of the learning experience in Mathematics. Those students who are not able to demonstrate mastery of the abstract are then deemed to have a lower level of understanding (or no understanding), despite potentially still being able to demonstrate deep understanding wile engaging with concrete and pictorial representations. Some also use the CPA model as a differentiation tool, with lower-attaining students presented with tasks containing mainly concrete and pictorial representations and higher-attaining students encouraged to engage more quickly with abstract elements. Although Bruner’s representation modes are hierarchical in the sense that they map out children’s learning stages through various age ranges, by age 7 years the expectation is that ALL children are capable of creating and storing knowledge at a symbolic level. As such, learning experiences should offer all children opportunities to experience their learning through actions, images, and more formal symbolic means. The sequence in which different representations of knowledge are explored should be determined by the sequence that will lead to the most in-depth understanding of a concept. This could mean working symbolically first, then drawing a picture, then working concretely (e.g. by building a model), or engaging backwards and forwards with each representation mode concurrently while developing and refining understanding – which is precisely what architects and engineers do.

Second, it is problematic to associate the ‘abstract’ stage in the CPA framework exclusively with abstract knowledge and to think that it is only through engagement with formal mathematical symbols and calculations that abstract knowledge is developed. Abstract mathematical structures can also be engaged and represented through enacted activities and pictures or icons. For example, uni-fix cubes are commonly treated as an abstract representation of a concrete object (like apples). Similarly, a bar model inherently contains a degree of abstractness because it shows a standardized or generic model of a unique scenario.

There is a risk, then, that some teachers may not recognise how much abstractness their teaching and resources contain and may wonder why some students continue to be confused despite access to different representations. Encountering formal mathematical structures in enactive or iconic forms does not automatically reduce the degree of abstractness; rather, it merely presents these structures via a medium other than symbols, notation and language.

Although there are numerous classroom resources available that use the CPA approach (for example, see here ), what is less common are resources that help teachers understand which models are the most effective for illustrating certain concepts, why this is, how to build links between different models to support deep relational understanding, and how to use models to compare and contrast different methods. So, while some teachers are using a large number of different representations, they are not always able to give students insight into the decisions behind why specific models are prioritised over others, which makes it difficult for students to know which models to choose when working independently. Also, while models are most commonly used to describe problems and then aid with the solving of those problems, less common is the use of models to compare and contrast different ways of working and to explore mathematical relationships and structures.

It seems important to consider two agenda:

1. the importance of deliberateness when choosing and using models; 2. and, using models to compare and contrast different ways of working and to explore mathematical relationships and structures.

The discussion below draws out some key ideas that have framed these agendas.

Different purposes for mathematics models

While the CPA and Enactive-Iconic-Symbolic frameworks set out different types of models and representations, it is also helpful to think about the different purposes that these can serve. The Realistic Mathematics Education (RME) approach provides some useful thinking around this. RME, as explained here , was developed in the Netherlands as a specific approach to the teaching of subject Mathematics. This approach has also been used in the United Kingdom with GCSE-resit Mathematics students, as shown here . Three key features of RME are helpful for this discussion:

1. Use of realistic contexts 2. Different purposes for models 3. The ‘progressive formalisation of models’ principle

The first key feature is engagement with abstract mathematics contents in realistic contexts (1), where ‘realistic’ refers to contexts that students can imagine and relate to. The contexts provide an anchor in which to ground understanding of abstract contents, a reference point for structuring thinking about abstract ideas. In part, this reflects some similarity with the ‘concrete’ dimension of the CPA framework.

A second feature of RME is different purposes for working with models (2). ‘Models of’ mathematics are models developed to represent a scenario or problem, with the model bearing a close connection to the problem situation at hand – for example, using a picture of a pizza to represent a fraction of a whole. When (or if) these models are developed and generalised to represent, describe and investigate mathematical structures and relationships over a range of problem situations and even content topics, the model becomes a ‘Model for’ exploring and understanding mathematics. Arrays, bar models and double number-lines are example of models that can be used in this way to describe and investigate mathematical methods, structures and relationships across a range of problem types and situations. ‘Models for’ are powerful precisely because they allow us to investigate mathematical relationships and explore different ways of working.

From a RME perspective, when using representations, it is essential to choose models and representations that can easily be developed from models of a specific local situation to models for describing more general and abstract relationships. This progressive formalisation of models (3) helps students navigate a learning trajectory to abstract concepts and equips them with a small number of models that have applicability over a range of problem and content types.

A key distinction here with the CPA approach is that, from a RME perspective, there is no expectation for students to work through a hierarchy from concrete experiences and pictures to symbolic representations. Instead, the move is from experiences and representations that are bound to local situations towards experiences and representations that are generalizable across a range of situations. The focus is less on the format of the representation and more on how the representation can be molded and developed to explore mathematical relationship and structure.

What does this mean for classroom practice?

The discussion above has attempted to highlight the importance of thinking about both the formats of the representations we use in our teaching AND the purpose of those representations. Using different types of representations that are blended into a deliberate sequence (like the CPA sequence) is helpful for supporting students to think about mathematical concepts from different perspectives – like different pieces of a puzzle, with each piece giving some unique information about the whole picture. However it is also important to think about what we use representation for so that students don’t believe that the only purpose for different representations is to solve problems. A much richer understanding is that representations, when carefully chosen, allow us to explore mathematical relationships, to see connections between mathematical concepts, and – in so doing – to develop a deeper understanding.

Looking ahead to part 2

Part 2 illustrates these theoretical ideas practically via a classroom based Maths activity that explores the relationship between different methods for solving a problem involving a conversion rate (from miles to kilometers). Without giving away too many clues, the key questions that Part 2 explores are:

How are each of these different methods linked, what’s the same and what’s different about them, are there methods that haven’t been considered yet, and what is the most effective ‘model for’ exploring the similarities and differences between them?

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I found this a really helpful analysis Dr North. Thank you very much.

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## The role of visual representations in scientific practices: from conceptual understanding and knowledge generation to ‘seeing’ how science works

- Maria Evagorou 1 ,
- Sibel Erduran 2 &
- Terhi Mäntylä 3

International Journal of STEM Education volume 2 , Article number: 11 ( 2015 ) Cite this article

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The use of visual representations (i.e., photographs, diagrams, models) has been part of science, and their use makes it possible for scientists to interact with and represent complex phenomena, not observable in other ways. Despite a wealth of research in science education on visual representations, the emphasis of such research has mainly been on the conceptual understanding when using visual representations and less on visual representations as epistemic objects. In this paper, we argue that by positioning visual representations as epistemic objects of scientific practices, science education can bring a renewed focus on how visualization contributes to knowledge formation in science from the learners’ perspective.

This is a theoretical paper, and in order to argue about the role of visualization, we first present a case study, that of the discovery of the structure of DNA that highlights the epistemic components of visual information in science. The second case study focuses on Faraday’s use of the lines of magnetic force. Faraday is known of his exploratory, creative, and yet systemic way of experimenting, and the visual reasoning leading to theoretical development was an inherent part of the experimentation. Third, we trace a contemporary account from science focusing on the experimental practices and how reproducibility of experimental procedures can be reinforced through video data.

## Conclusions

Our conclusions suggest that in teaching science, the emphasis in visualization should shift from cognitive understanding—using the products of science to understand the content—to engaging in the processes of visualization. Furthermore, we suggest that is it essential to design curriculum materials and learning environments that create a social and epistemic context and invite students to engage in the practice of visualization as evidence, reasoning, experimental procedure, or a means of communication and reflect on these practices. Implications for teacher education include the need for teacher professional development programs to problematize the use of visual representations as epistemic objects that are part of scientific practices.

During the last decades, research and reform documents in science education across the world have been calling for an emphasis not only on the content but also on the processes of science (Bybee 2014 ; Eurydice 2012 ; Duschl and Bybee 2014 ; Osborne 2014 ; Schwartz et al. 2012 ), in order to make science accessible to the students and enable them to understand the epistemic foundation of science. Scientific practices, part of the process of science, are the cognitive and discursive activities that are targeted in science education to develop epistemic understanding and appreciation of the nature of science (Duschl et al. 2008 ) and have been the emphasis of recent reform documents in science education across the world (Achieve 2013 ; Eurydice 2012 ). With the term scientific practices, we refer to the processes that take place during scientific discoveries and include among others: asking questions, developing and using models, engaging in arguments, and constructing and communicating explanations (National Research Council 2012 ). The emphasis on scientific practices aims to move the teaching of science from knowledge to the understanding of the processes and the epistemic aspects of science. Additionally, by placing an emphasis on engaging students in scientific practices, we aim to help students acquire scientific knowledge in meaningful contexts that resemble the reality of scientific discoveries.

Despite a wealth of research in science education on visual representations, the emphasis of such research has mainly been on the conceptual understanding when using visual representations and less on visual representations as epistemic objects. In this paper, we argue that by positioning visual representations as epistemic objects, science education can bring a renewed focus on how visualization contributes to knowledge formation in science from the learners’ perspective. Specifically, the use of visual representations (i.e., photographs, diagrams, tables, charts) has been part of science and over the years has evolved with the new technologies (i.e., from drawings to advanced digital images and three dimensional models). Visualization makes it possible for scientists to interact with complex phenomena (Richards 2003 ), and they might convey important evidence not observable in other ways. Visual representations as a tool to support cognitive understanding in science have been studied extensively (i.e., Gilbert 2010 ; Wu and Shah 2004 ). Studies in science education have explored the use of images in science textbooks (i.e., Dimopoulos et al. 2003 ; Bungum 2008 ), students’ representations or models when doing science (i.e., Gilbert et al. 2008 ; Dori et al. 2003 ; Lehrer and Schauble 2012 ; Schwarz et al. 2009 ), and students’ images of science and scientists (i.e., Chambers 1983 ). Therefore, studies in the field of science education have been using the term visualization as “the formation of an internal representation from an external representation” (Gilbert et al. 2008 , p. 4) or as a tool for conceptual understanding for students.

In this paper, we do not refer to visualization as mental image, model, or presentation only (Gilbert et al. 2008 ; Philips et al. 2010 ) but instead focus on visual representations or visualization as epistemic objects. Specifically, we refer to visualization as a process for knowledge production and growth in science. In this respect, modeling is an aspect of visualization, but what we are focusing on with visualization is not on the use of model as a tool for cognitive understanding (Gilbert 2010 ; Wu and Shah 2004 ) but the on the process of modeling as a scientific practice which includes the construction and use of models, the use of other representations, the communication in the groups with the use of the visual representation, and the appreciation of the difficulties that the science phase in this process. Therefore, the purpose of this paper is to present through the history of science how visualization can be considered not only as a cognitive tool in science education but also as an epistemic object that can potentially support students to understand aspects of the nature of science.

## Scientific practices and science education

According to the New Generation Science Standards (Achieve 2013 ), scientific practices refer to: asking questions and defining problems; developing and using models; planning and carrying out investigations; analyzing and interpreting data; using mathematical and computational thinking; constructing explanations and designing solutions; engaging in argument from evidence; and obtaining, evaluating, and communicating information. A significant aspect of scientific practices is that science learning is more than just about learning facts, concepts, theories, and laws. A fuller appreciation of science necessitates the understanding of the science relative to its epistemological grounding and the process that are involved in the production of knowledge (Hogan and Maglienti 2001 ; Wickman 2004 ).

The New Generation Science Standards is, among other changes, shifting away from science inquiry and towards the inclusion of scientific practices (Duschl and Bybee 2014 ; Osborne 2014 ). By comparing the abilities to do scientific inquiry (National Research Council 2000 ) with the set of scientific practices, it is evident that the latter is about engaging in the processes of doing science and experiencing in that way science in a more authentic way. Engaging in scientific practices according to Osborne ( 2014 ) “presents a more authentic picture of the endeavor that is science” (p.183) and also helps the students to develop a deeper understanding of the epistemic aspects of science. Furthermore, as Bybee ( 2014 ) argues, by engaging students in scientific practices, we involve them in an understanding of the nature of science and an understanding on the nature of scientific knowledge.

Science as a practice and scientific practices as a term emerged by the philosopher of science, Kuhn (Osborne 2014 ), refers to the processes in which the scientists engage during knowledge production and communication. The work that is followed by historians, philosophers, and sociologists of science (Latour 2011 ; Longino 2002 ; Nersessian 2008 ) revealed the scientific practices in which the scientists engage in and include among others theory development and specific ways of talking, modeling, and communicating the outcomes of science.

## Visualization as an epistemic object

Schematic, pictorial symbols in the design of scientific instruments and analysis of the perceptual and functional information that is being stored in those images have been areas of investigation in philosophy of scientific experimentation (Gooding et al. 1993 ). The nature of visual perception, the relationship between thought and vision, and the role of reproducibility as a norm for experimental research form a central aspect of this domain of research in philosophy of science. For instance, Rothbart ( 1997 ) has argued that visualizations are commonplace in the theoretical sciences even if every scientific theory may not be defined by visualized models.

Visual representations (i.e., photographs, diagrams, tables, charts, models) have been used in science over the years to enable scientists to interact with complex phenomena (Richards 2003 ) and might convey important evidence not observable in other ways (Barber et al. 2006 ). Some authors (e.g., Ruivenkamp and Rip 2010 ) have argued that visualization is as a core activity of some scientific communities of practice (e.g., nanotechnology) while others (e.g., Lynch and Edgerton 1988 ) have differentiated the role of particular visualization techniques (e.g., of digital image processing in astronomy). Visualization in science includes the complex process through which scientists develop or produce imagery, schemes, and graphical representation, and therefore, what is of importance in this process is not only the result but also the methodology employed by the scientists, namely, how this result was produced. Visual representations in science may refer to objects that are believed to have some kind of material or physical existence but equally might refer to purely mental, conceptual, and abstract constructs (Pauwels 2006 ). More specifically, visual representations can be found for: (a) phenomena that are not observable with the eye (i.e., microscopic or macroscopic); (b) phenomena that do not exist as visual representations but can be translated as such (i.e., sound); and (c) in experimental settings to provide visual data representations (i.e., graphs presenting velocity of moving objects). Additionally, since science is not only about replicating reality but also about making it more understandable to people (either to the public or other scientists), visual representations are not only about reproducing the nature but also about: (a) functioning in helping solving a problem, (b) filling gaps in our knowledge, and (c) facilitating knowledge building or transfer (Lynch 2006 ).

Using or developing visual representations in the scientific practice can range from a straightforward to a complicated situation. More specifically, scientists can observe a phenomenon (i.e., mitosis) and represent it visually using a picture or diagram, which is quite straightforward. But they can also use a variety of complicated techniques (i.e., crystallography in the case of DNA studies) that are either available or need to be developed or refined in order to acquire the visual information that can be used in the process of theory development (i.e., Latour and Woolgar 1979 ). Furthermore, some visual representations need decoding, and the scientists need to learn how to read these images (i.e., radiologists); therefore, using visual representations in the process of science requires learning a new language that is specific to the medium/methods that is used (i.e., understanding an X-ray picture is different from understanding an MRI scan) and then communicating that language to other scientists and the public.

There are much intent and purposes of visual representations in scientific practices, as for example to make a diagnosis, compare, describe, and preserve for future study, verify and explore new territory, generate new data (Pauwels 2006 ), or present new methodologies. According to Latour and Woolgar ( 1979 ) and Knorr Cetina ( 1999 ), visual representations can be used either as primary data (i.e., image from a microscope). or can be used to help in concept development (i.e., models of DNA used by Watson and Crick), to uncover relationships and to make the abstract more concrete (graphs of sound waves). Therefore, visual representations and visual practices, in all forms, are an important aspect of the scientific practices in developing, clarifying, and transmitting scientific knowledge (Pauwels 2006 ).

## Methods and Results: Merging Visualization and scientific practices in science

In this paper, we present three case studies that embody the working practices of scientists in an effort to present visualization as a scientific practice and present our argument about how visualization is a complex process that could include among others modeling and use of representation but is not only limited to that. The first case study explores the role of visualization in the construction of knowledge about the structure of DNA, using visuals as evidence. The second case study focuses on Faraday’s use of the lines of magnetic force and the visual reasoning leading to the theoretical development that was an inherent part of the experimentation. The third case study focuses on the current practices of scientists in the context of a peer-reviewed journal called the Journal of Visualized Experiments where the methodology is communicated through videotaped procedures. The three case studies represent the research interests of the three authors of this paper and were chosen to present how visualization as a practice can be involved in all stages of doing science, from hypothesizing and evaluating evidence (case study 1) to experimenting and reasoning (case study 2) to communicating the findings and methodology with the research community (case study 3), and represent in this way the three functions of visualization as presented by Lynch ( 2006 ). Furthermore, the last case study showcases how the development of visualization technologies has contributed to the communication of findings and methodologies in science and present in that way an aspect of current scientific practices. In all three cases, our approach is guided by the observation that the visual information is an integral part of scientific practices at the least and furthermore that they are particularly central in the scientific practices of science.

## Case study 1: use visual representations as evidence in the discovery of DNA

The focus of the first case study is the discovery of the structure of DNA. The DNA was first isolated in 1869 by Friedrich Miescher, and by the late 1940s, it was known that it contained phosphate, sugar, and four nitrogen-containing chemical bases. However, no one had figured the structure of the DNA until Watson and Crick presented their model of DNA in 1953. Other than the social aspects of the discovery of the DNA, another important aspect was the role of visual evidence that led to knowledge development in the area. More specifically, by studying the personal accounts of Watson ( 1968 ) and Crick ( 1988 ) about the discovery of the structure of the DNA, the following main ideas regarding the role of visual representations in the production of knowledge can be identified: (a) The use of visual representations was an important part of knowledge growth and was often dependent upon the discovery of new technologies (i.e., better microscopes or better techniques in crystallography that would provide better visual representations as evidence of the helical structure of the DNA); and (b) Models (three-dimensional) were used as a way to represent the visual images (X-ray images) and connect them to the evidence provided by other sources to see whether the theory can be supported. Therefore, the model of DNA was built based on the combination of visual evidence and experimental data.

An example showcasing the importance of visual representations in the process of knowledge production in this case is provided by Watson, in his book The Double Helix (1968):

…since the middle of the summer Rosy [Rosalind Franklin] had had evidence for a new three-dimensional form of DNA. It occurred when the DNA 2molecules were surrounded by a large amount of water. When I asked what the pattern was like, Maurice went into the adjacent room to pick up a print of the new form they called the “B” structure. The instant I saw the picture, my mouth fell open and my pulse began to race. The pattern was unbelievably simpler than those previously obtained (A form). Moreover, the black cross of reflections which dominated the picture could arise only from a helical structure. With the A form the argument for the helix was never straightforward, and considerable ambiguity existed as to exactly which type of helical symmetry was present. With the B form however, mere inspection of its X-ray picture gave several of the vital helical parameters. (p. 167-169)

As suggested by Watson’s personal account of the discovery of the DNA, the photo taken by Rosalind Franklin (Fig. 1 ) convinced him that the DNA molecule must consist of two chains arranged in a paired helix, which resembles a spiral staircase or ladder, and on March 7, 1953, Watson and Crick finished and presented their model of the structure of DNA (Watson and Berry 2004 ; Watson 1968 ) which was based on the visual information provided by the X-ray image and their knowledge of chemistry.

X-ray chrystallography of DNA

In analyzing the visualization practice in this case study, we observe the following instances that highlight how the visual information played a role:

Asking questions and defining problems: The real world in the model of science can at some points only be observed through visual representations or representations, i.e., if we are using DNA as an example, the structure of DNA was only observable through the crystallography images produced by Rosalind Franklin in the laboratory. There was no other way to observe the structure of DNA, therefore the real world.

Analyzing and interpreting data: The images that resulted from crystallography as well as their interpretations served as the data for the scientists studying the structure of DNA.

Experimenting: The data in the form of visual information were used to predict the possible structure of the DNA.

Modeling: Based on the prediction, an actual three-dimensional model was prepared by Watson and Crick. The first model did not fit with the real world (refuted by Rosalind Franklin and her research group from King’s College) and Watson and Crick had to go through the same process again to find better visual evidence (better crystallography images) and create an improved visual model.

Example excerpts from Watson’s biography provide further evidence for how visualization practices were applied in the context of the discovery of DNA (Table 1 ).

In summary, by examining the history of the discovery of DNA, we showcased how visual data is used as scientific evidence in science, identifying in that way an aspect of the nature of science that is still unexplored in the history of science and an aspect that has been ignored in the teaching of science. Visual representations are used in many ways: as images, as models, as evidence to support or rebut a model, and as interpretations of reality.

## Case study 2: applying visual reasoning in knowledge production, the example of the lines of magnetic force

The focus of this case study is on Faraday’s use of the lines of magnetic force. Faraday is known of his exploratory, creative, and yet systemic way of experimenting, and the visual reasoning leading to theoretical development was an inherent part of this experimentation (Gooding 2006 ). Faraday’s articles or notebooks do not include mathematical formulations; instead, they include images and illustrations from experimental devices and setups to the recapping of his theoretical ideas (Nersessian 2008 ). According to Gooding ( 2006 ), “Faraday’s visual method was designed not to copy apparent features of the world, but to analyse and replicate them” (2006, p. 46).

The lines of force played a central role in Faraday’s research on electricity and magnetism and in the development of his “field theory” (Faraday 1852a ; Nersessian 1984 ). Before Faraday, the experiments with iron filings around magnets were known and the term “magnetic curves” was used for the iron filing patterns and also for the geometrical constructs derived from the mathematical theory of magnetism (Gooding et al. 1993 ). However, Faraday used the lines of force for explaining his experimental observations and in constructing the theory of forces in magnetism and electricity. Examples of Faraday’s different illustrations of lines of magnetic force are given in Fig. 2 . Faraday gave the following experiment-based definition for the lines of magnetic forces:

a Iron filing pattern in case of bar magnet drawn by Faraday (Faraday 1852b , Plate IX, p. 158, Fig. 1), b Faraday’s drawing of lines of magnetic force in case of cylinder magnet, where the experimental procedure, knife blade showing the direction of lines, is combined into drawing (Faraday, 1855, vol. 1, plate 1)

A line of magnetic force may be defined as that line which is described by a very small magnetic needle, when it is so moved in either direction correspondent to its length, that the needle is constantly a tangent to the line of motion; or it is that line along which, if a transverse wire be moved in either direction, there is no tendency to the formation of any current in the wire, whilst if moved in any other direction there is such a tendency; or it is that line which coincides with the direction of the magnecrystallic axis of a crystal of bismuth, which is carried in either direction along it. The direction of these lines about and amongst magnets and electric currents, is easily represented and understood, in a general manner, by the ordinary use of iron filings. (Faraday 1852a , p. 25 (3071))

The definition describes the connection between the experiments and the visual representation of the results. Initially, the lines of force were just geometric representations, but later, Faraday treated them as physical objects (Nersessian 1984 ; Pocovi and Finlay 2002 ):

I have sometimes used the term lines of force so vaguely, as to leave the reader doubtful whether I intended it as a merely representative idea of the forces, or as the description of the path along which the power was continuously exerted. … wherever the expression line of force is taken simply to represent the disposition of forces, it shall have the fullness of that meaning; but that wherever it may seem to represent the idea of the physical mode of transmission of the force, it expresses in that respect the opinion to which I incline at present. The opinion may be erroneous, and yet all that relates or refers to the disposition of the force will remain the same. (Faraday, 1852a , p. 55-56 (3075))

He also felt that the lines of force had greater explanatory power than the dominant theory of action-at-a-distance:

Now it appears to me that these lines may be employed with great advantage to represent nature, condition, direction and comparative amount of the magnetic forces; and that in many cases they have, to the physical reasoned at least, a superiority over that method which represents the forces as concentrated in centres of action… (Faraday, 1852a , p. 26 (3074))

For giving some insight to Faraday’s visual reasoning as an epistemic practice, the following examples of Faraday’s studies of the lines of magnetic force (Faraday 1852a , 1852b ) are presented:

(a) Asking questions and defining problems: The iron filing patterns formed the empirical basis for the visual model: 2D visualization of lines of magnetic force as presented in Fig. 2 . According to Faraday, these iron filing patterns were suitable for illustrating the direction and form of the magnetic lines of force (emphasis added):

It must be well understood that these forms give no indication by their appearance of the relative strength of the magnetic force at different places, inasmuch as the appearance of the lines depends greatly upon the quantity of filings and the amount of tapping; but the direction and forms of these lines are well given, and these indicate, in a considerable degree, the direction in which the forces increase and diminish . (Faraday 1852b , p.158 (3237))

Despite being static and two dimensional on paper, the lines of magnetic force were dynamical (Nersessian 1992 , 2008 ) and three dimensional for Faraday (see Fig. 2 b). For instance, Faraday described the lines of force “expanding”, “bending,” and “being cut” (Nersessian 1992 ). In Fig. 2 b, Faraday has summarized his experiment (bar magnet and knife blade) and its results (lines of force) in one picture.

(b) Analyzing and interpreting data: The model was so powerful for Faraday that he ended up thinking them as physical objects (e.g., Nersessian 1984 ), i.e., making interpretations of the way forces act. Of course, he made a lot of experiments for showing the physical existence of the lines of force, but he did not succeed in it (Nersessian 1984 ). The following quote illuminates Faraday’s use of the lines of force in different situations:

The study of these lines has, at different times, been greatly influential in leading me to various results, which I think prove their utility as well as fertility. Thus, the law of magneto-electric induction; the earth’s inductive action; the relation of magnetism and light; diamagnetic action and its law, and magnetocrystallic action, are the cases of this kind… (Faraday 1852a , p. 55 (3174))

(c) Experimenting: In Faraday's case, he used a lot of exploratory experiments; in case of lines of magnetic force, he used, e.g., iron filings, magnetic needles, or current carrying wires (see the quote above). The magnetic field is not directly observable and the representation of lines of force was a visual model, which includes the direction, form, and magnitude of field.

(d) Modeling: There is no denying that the lines of magnetic force are visual by nature. Faraday’s views of lines of force developed gradually during the years, and he applied and developed them in different contexts such as electromagnetic, electrostatic, and magnetic induction (Nersessian 1984 ). An example of Faraday’s explanation of the effect of the wire b’s position to experiment is given in Fig. 3 . In Fig. 3 , few magnetic lines of force are drawn, and in the quote below, Faraday is explaining the effect using these magnetic lines of force (emphasis added):

Picture of an experiment with different arrangements of wires ( a , b’ , b” ), magnet, and galvanometer. Note the lines of force drawn around the magnet. (Faraday 1852a , p. 34)

It will be evident by inspection of Fig. 3 , that, however the wires are carried away, the general result will, according to the assumed principles of action, be the same; for if a be the axial wire, and b’, b”, b”’ the equatorial wire, represented in three different positions, whatever magnetic lines of force pass across the latter wire in one position, will also pass it in the other, or in any other position which can be given to it. The distance of the wire at the place of intersection with the lines of force, has been shown, by the experiments (3093.), to be unimportant. (Faraday 1852a , p. 34 (3099))

In summary, by examining the history of Faraday’s use of lines of force, we showed how visual imagery and reasoning played an important part in Faraday’s construction and representation of his “field theory”. As Gooding has stated, “many of Faraday’s sketches are far more that depictions of observation, they are tools for reasoning with and about phenomena” (2006, p. 59).

## Case study 3: visualizing scientific methods, the case of a journal

The focus of the third case study is the Journal of Visualized Experiments (JoVE) , a peer-reviewed publication indexed in PubMed. The journal devoted to the publication of biological, medical, chemical, and physical research in a video format. The journal describes its history as follows:

JoVE was established as a new tool in life science publication and communication, with participation of scientists from leading research institutions. JoVE takes advantage of video technology to capture and transmit the multiple facets and intricacies of life science research. Visualization greatly facilitates the understanding and efficient reproduction of both basic and complex experimental techniques, thereby addressing two of the biggest challenges faced by today's life science research community: i) low transparency and poor reproducibility of biological experiments and ii) time and labor-intensive nature of learning new experimental techniques. ( http://www.jove.com/ )

By examining the journal content, we generate a set of categories that can be considered as indicators of relevance and significance in terms of epistemic practices of science that have relevance for science education. For example, the quote above illustrates how scientists view some norms of scientific practice including the norms of “transparency” and “reproducibility” of experimental methods and results, and how the visual format of the journal facilitates the implementation of these norms. “Reproducibility” can be considered as an epistemic criterion that sits at the heart of what counts as an experimental procedure in science:

Investigating what should be reproducible and by whom leads to different types of experimental reproducibility, which can be observed to play different roles in experimental practice. A successful application of the strategy of reproducing an experiment is an achievement that may depend on certain isiosyncratic aspects of a local situation. Yet a purely local experiment that cannot be carried out by other experimenters and in other experimental contexts will, in the end be unproductive in science. (Sarkar and Pfeifer 2006 , p.270)

We now turn to an article on “Elevated Plus Maze for Mice” that is available for free on the journal website ( http://www.jove.com/video/1088/elevated-plus-maze-for-mice ). The purpose of this experiment was to investigate anxiety levels in mice through behavioral analysis. The journal article consists of a 9-min video accompanied by text. The video illustrates the handling of the mice in soundproof location with less light, worksheets with characteristics of mice, computer software, apparatus, resources, setting up the computer software, and the video recording of mouse behavior on the computer. The authors describe the apparatus that is used in the experiment and state how procedural differences exist between research groups that lead to difficulties in the interpretation of results:

The apparatus consists of open arms and closed arms, crossed in the middle perpendicularly to each other, and a center area. Mice are given access to all of the arms and are allowed to move freely between them. The number of entries into the open arms and the time spent in the open arms are used as indices of open space-induced anxiety in mice. Unfortunately, the procedural differences that exist between laboratories make it difficult to duplicate and compare results among laboratories.

The authors’ emphasis on the particularity of procedural context echoes in the observations of some philosophers of science:

It is not just the knowledge of experimental objects and phenomena but also their actual existence and occurrence that prove to be dependent on specific, productive interventions by the experimenters” (Sarkar and Pfeifer 2006 , pp. 270-271)

The inclusion of a video of the experimental procedure specifies what the apparatus looks like (Fig. 4 ) and how the behavior of the mice is captured through video recording that feeds into a computer (Fig. 5 ). Subsequently, a computer software which captures different variables such as the distance traveled, the number of entries, and the time spent on each arm of the apparatus. Here, there is visual information at different levels of representation ranging from reconfiguration of raw video data to representations that analyze the data around the variables in question (Fig. 6 ). The practice of levels of visual representations is not particular to the biological sciences. For instance, they are commonplace in nanotechnological practices:

Visual illustration of apparatus

Video processing of experimental set-up

Computer software for video input and variable recording

In the visualization processes, instruments are needed that can register the nanoscale and provide raw data, which needs to be transformed into images. Some Imaging Techniques have software incorporated already where this transformation automatically takes place, providing raw images. Raw data must be translated through the use of Graphic Software and software is also used for the further manipulation of images to highlight what is of interest to capture the (inferred) phenomena -- and to capture the reader. There are two levels of choice: Scientists have to choose which imaging technique and embedded software to use for the job at hand, and they will then have to follow the structure of the software. Within such software, there are explicit choices for the scientists, e.g. about colour coding, and ways of sharpening images. (Ruivenkamp and Rip 2010 , pp.14–15)

On the text that accompanies the video, the authors highlight the role of visualization in their experiment:

Visualization of the protocol will promote better understanding of the details of the entire experimental procedure, allowing for standardization of the protocols used in different laboratories and comparisons of the behavioral phenotypes of various strains of mutant mice assessed using this test.

The software that takes the video data and transforms it into various representations allows the researchers to collect data on mouse behavior more reliably. For instance, the distance traveled across the arms of the apparatus or the time spent on each arm would have been difficult to observe and record precisely. A further aspect to note is how the visualization of the experiment facilitates control of bias. The authors illustrate how the olfactory bias between experimental procedures carried on mice in sequence is avoided by cleaning the equipment.

Our discussion highlights the role of visualization in science, particularly with respect to presenting visualization as part of the scientific practices. We have used case studies from the history of science highlighting a scientist’s account of how visualization played a role in the discovery of DNA and the magnetic field and from a contemporary illustration of a science journal’s practices in incorporating visualization as a way to communicate new findings and methodologies. Our implicit aim in drawing from these case studies was the need to align science education with scientific practices, particularly in terms of how visual representations, stable or dynamic, can engage students in the processes of science and not only to be used as tools for cognitive development in science. Our approach was guided by the notion of “knowledge-as-practice” as advanced by Knorr Cetina ( 1999 ) who studied scientists and characterized their knowledge as practice, a characterization which shifts focus away from ideas inside scientists’ minds to practices that are cultural and deeply contextualized within fields of science. She suggests that people working together can be examined as epistemic cultures whose collective knowledge exists as practice.

It is important to stress, however, that visual representations are not used in isolation, but are supported by other types of evidence as well, or other theories (i.e., in order to understand the helical form of DNA, or the structure, chemistry knowledge was needed). More importantly, this finding can also have implications when teaching science as argument (e.g., Erduran and Jimenez-Aleixandre 2008 ), since the verbal evidence used in the science classroom to maintain an argument could be supported by visual evidence (either a model, representation, image, graph, etc.). For example, in a group of students discussing the outcomes of an introduced species in an ecosystem, pictures of the species and the ecosystem over time, and videos showing the changes in the ecosystem, and the special characteristics of the different species could serve as visual evidence to help the students support their arguments (Evagorou et al. 2012 ). Therefore, an important implication for the teaching of science is the use of visual representations as evidence in the science curriculum as part of knowledge production. Even though studies in the area of science education have focused on the use of models and modeling as a way to support students in the learning of science (Dori et al. 2003 ; Lehrer and Schauble 2012 ; Mendonça and Justi 2013 ; Papaevripidou et al. 2007 ) or on the use of images (i.e., Korfiatis et al. 2003 ), with the term using visuals as evidence, we refer to the collection of all forms of visuals and the processes involved.

Another aspect that was identified through the case studies is that of the visual reasoning (an integral part of Faraday’s investigations). Both the verbalization and visualization were part of the process of generating new knowledge (Gooding 2006 ). Even today, most of the textbooks use the lines of force (or just field lines) as a geometrical representation of field, and the number of field lines is connected to the quantity of flux. Often, the textbooks use the same kind of visual imagery than in what is used by scientists. However, when using images, only certain aspects or features of the phenomena or data are captured or highlighted, and often in tacit ways. Especially in textbooks, the process of producing the image is not presented and instead only the product—image—is left. This could easily lead to an idea of images (i.e., photos, graphs, visual model) being just representations of knowledge and, in the worse case, misinterpreted representations of knowledge as the results of Pocovi and Finlay ( 2002 ) in case of electric field lines show. In order to avoid this, the teachers should be able to explain how the images are produced (what features of phenomena or data the images captures, on what ground the features are chosen to that image, and what features are omitted); in this way, the role of visualization in knowledge production can be made “visible” to students by engaging them in the process of visualization.

The implication of these norms for science teaching and learning is numerous. The classroom contexts can model the generation, sharing and evaluation of evidence, and experimental procedures carried out by students, thereby promoting not only some contemporary cultural norms in scientific practice but also enabling the learning of criteria, standards, and heuristics that scientists use in making decisions on scientific methods. As we have demonstrated with the three case studies, visual representations are part of the process of knowledge growth and communication in science, as demonstrated with two examples from the history of science and an example from current scientific practices. Additionally, visual information, especially with the use of technology is a part of students’ everyday lives. Therefore, we suggest making use of students’ knowledge and technological skills (i.e., how to produce their own videos showing their experimental method or how to identify or provide appropriate visual evidence for a given topic), in order to teach them the aspects of the nature of science that are often neglected both in the history of science and the design of curriculum. Specifically, what we suggest in this paper is that students should actively engage in visualization processes in order to appreciate the diverse nature of doing science and engage in authentic scientific practices.

However, as a word of caution, we need to distinguish the products and processes involved in visualization practices in science:

If one considers scientific representations and the ways in which they can foster or thwart our understanding, it is clear that a mere object approach, which would devote all attention to the representation as a free-standing product of scientific labor, is inadequate. What is needed is a process approach: each visual representation should be linked with its context of production (Pauwels 2006 , p.21).

The aforementioned suggests that the emphasis in visualization should shift from cognitive understanding—using the products of science to understand the content—to engaging in the processes of visualization. Therefore, an implication for the teaching of science includes designing curriculum materials and learning environments that create a social and epistemic context and invite students to engage in the practice of visualization as evidence, reasoning, experimental procedure, or a means of communication (as presented in the three case studies) and reflect on these practices (Ryu et al. 2015 ).

Finally, a question that arises from including visualization in science education, as well as from including scientific practices in science education is whether teachers themselves are prepared to include them as part of their teaching (Bybee 2014 ). Teacher preparation programs and teacher education have been critiqued, studied, and rethought since the time they emerged (Cochran-Smith 2004 ). Despite the years of history in teacher training and teacher education, the debate about initial teacher training and its content still pertains in our community and in policy circles (Cochran-Smith 2004 ; Conway et al. 2009 ). In the last decades, the debate has shifted from a behavioral view of learning and teaching to a learning problem—focusing on that way not only on teachers’ knowledge, skills, and beliefs but also on making the connection of the aforementioned with how and if pupils learn (Cochran-Smith 2004 ). The Science Education in Europe report recommended that “Good quality teachers, with up-to-date knowledge and skills, are the foundation of any system of formal science education” (Osborne and Dillon 2008 , p.9).

However, questions such as what should be the emphasis on pre-service and in-service science teacher training, especially with the new emphasis on scientific practices, still remain unanswered. As Bybee ( 2014 ) argues, starting from the new emphasis on scientific practices in the NGSS, we should consider teacher preparation programs “that would provide undergraduates opportunities to learn the science content and practices in contexts that would be aligned with their future work as teachers” (p.218). Therefore, engaging pre- and in-service teachers in visualization as a scientific practice should be one of the purposes of teacher preparation programs.

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## Best Practices for Teaching Math to Unique Learners: Visual Representations

## Kelley Spainhour

Special Education Consultant

But what about middle school and high school teachers? And what about teachers of students with learning disabilities—at any grade level?

To many, the use of visuals is an elementary school practice. Counting on fingers, modeling with stick figures, and coloring in bar graph worksheets seem like appropriate strategies for young students who are learning foundational math concepts. In older grade levels, visual representations are often highlighted in the first or second instructional example per the guidance of the textbook, but they are rarely emphasized as a critical part of the lesson.

## The Importance of Visual Representations

Research suggests that visual representations are critical to the success of math students at all ages and ability levels, especially those with learning disabilities. There’s an important caveat to these research findings, though. Simply adding a picture or a diagram to a word problem for additional context is far less helpful than explicitly teaching students how to use a particular visual representation to solve the problem. Explicit instruction with visual representations includes plenty of opportunities for students to practice, get feedback, and ask questions. These include pictures and drawings , diagrams, graphs, physical gesturing, and manipulatives. As students become more adept at using visuals, the cognitive demand for solving the problem lessens. In other words, it gets easier for them to solve math problems.

Students respond better to visuals that the teacher prescribes than to those they select on their own. This may be surprising to some, especially as much of the current conversation in education involves student-centered learning. Yes, it is important to empower students to make meaning of their own learning. But the role of the teacher cannot be understated. Students depend on their teacher to guide them through their problem-solving processes by offering models, correction, suggestion, and encouragement along the way. For example, students typically do not intuitively know how to use fraction tiles to compare fractions with unlike denominators. They need their teacher to set up structured opportunities for them to explore fraction comparison with these manipulatives.

Unique Learning System is an excellent example of a curriculum that emphasizes the importance and relevance of visuals throughout its math lessons. It is designed to meet the needs of learners with diverse communication and cognitive needs and pairs math vocabulary, math symbols, and pictorial representations in every lesson. Teachers can supplement the visual representations as needed to scaffold up or down .

## Using Visuals to Teach Perimeter

A middle school math special education teacher is planning her upcoming geometry unit, and she knows from experience that many students confuse perimeter and area, especially when the two problem types are mixed throughout a worksheet (e.g., a problem on finding the perimeter of a rectangle followed by a problem on finding the area of a square). In the past, she typically has started this unit by teaching perimeter. She shows students how to calculate perimeter by drawing a 2D shape on the board and labeling the sides of the shape with their measurements. She then models how to add those measurements to calculate the total distance around the shape. Students write “total distance around a shape = perimeter” in their notebooks.

To incorporate more visuals into her teaching, she might try these strategies:

- Begin with a kinesthetic activity that introduces students to the concept of distance in a tangible way. Offer standard and nonstandard tools to measure the distance around the classroom. Use anything from yardsticks or measuring tapes to shoes or chairs. Students will be immediately engaged as they are able to move about, work together, and use tools that are familiar.
- Provide explicit instruction on calculating perimeter by following the steps described above. After this, teach students to identify the problem as a “perimeter problem” and label the problem with a capital letter, P. This will become important when students are asked to distinguish between perimeter problems and area problems later in the unit.
- Next, model for students how to use a highlighter to trace the perimeter of the 2-D shape. This simple but powerful visual representation of “distance around” reinforces the concept of perimeter and reminds students that they must account for and label all sides of the shape.

## Using Pictures to Support Context

Teachers know that pictures are a helpful accommodation for English language learners (ELLs) and struggling readers. These kinds of pictures serve a different purpose than those that visually represent the mathematics of the problem.

For example, a word problem about finding the slope of a ski hill may include a picture of a person skiing and bundled up with winter clothes. This picture adds helpful context for an ELL student who may not be familiar with skiing or snowy weather. It doesn’t help students understand the mathematics of the problem, but it does serve as an entry point. These kinds of pictures are encouraged when used along with visuals that represent the mathematics of the problem.

A visual representation of the mathematics of this problem may be a sketch of the ski hill with lines drawn to represent the height of the hill and the distance from the base to the finish line. Students would then learn to transfer the diagram to a coordinate plane, identify the x- and y- coordinates, and apply the formulas for determining slope.

## Visual Representations and Standards for Mathematical Practice

The Common Core State Standards Initiative has published standards for teaching mathematical content and standards for teaching mathematical practice . Some states follow Common Core and some do not. However, the practice standards are relevant and useful for math teachers across the country. They stand apart from the content standards and describe the broad habits of mathematically proficient students. For example, the first of the eight standards is to make sense of problems and persevere in solving them . The standards are the same from kindergarten through grade 12, and regardless of the math content you teach, you can apply one or more of them.

The use of visual representations can be connected to any of the eight practice standards, but they are strongly emphasized in these three:

- Math.Practice.MP2—Reason abstractly and quantitatively
- Math.Practice.MP4— Model with mathematics
- Math.Practice.MP5—Use appropriate tools strategically

If you use the standards for mathematical practice to shape your philosophy of math instruction and develop strong math habits in your budding mathematicians, this is an important connection to make. Explicitly teaching students how to model and use tools appropriately helps them learn to transfer known skills to novel problems, which can be especially challenging for your unique learners. When students learn to apply visual representations to solve math problems, their math confidence grows. They might even get excited about the challenge of a new problem and learn to love math!

## About the Author

Kelley Spainhour is a special education professional with a decade of teaching and leadership experience. She is passionate about the unique needs of children with medical needs and enjoys collaborating in multidisciplinary contexts. Kelley currently serves as a special education consultant and writer.

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## Visual Representation in Mathematics

Print Resource

by Ian Matheson and Nancy Hutchinson

Mathematics is a subject that deals with abstract ideas in order to solve problems. Information can be represented with numbers, words, and other types of symbols. Representing information with symbols can be a difficult practice for students with learning disabilities (LDs) to understand, and eventually begin to use themselves.

Information is often represented visually in mathematics as a method of organizing, extending, or replacing other methods of presentation. Visual representation in mathematics involves creating and forming models that reflect mathematical information (van Garderen & Montague, 2003).

## LDs and Problem-Solving

Although there are a number of problem solving strategies that students use in mathematics, good problem solvers usually construct a representation of the problem to help them comprehend it (van Garderen & Montague, 2003). Students with LDs can have an especially challenging experience solving problems in math, and research suggests that their use of visual representation strategies differs from their typically-achieving peers in:

- frequency of use (Montague, 1997),
- type of visual representations used (van Garderen & Montague, 2003), and
- quality of visual representations (van Garderen, Scheuermann, & Jackson, 2012b).

Creating a visual representation to solve a problem in mathematics is a process that involves:

- processing the information in the problem,
- selecting important information, and
- identifying the goal of the problem.

Students with LDs struggle with visual representation in mathematics because they typically have difficulties processing information (Swanson, Lussier, & Orosco, 2013).

Visual representation is an important skill because higher-level math and science courses increasingly draw on visualization and spatial reasoning skills to solve problems (Zhang, Ding, Stegall, & Mo, 2012). Additionally, it is simply another strategy that students can use when they are thinking of the best way to answer a problem in mathematics.

## The Importance of Explicit Instruction

Perhaps the most consistent message in the literature about visual representation in mathematics is that it needs to be explicitly taught to students. Representing information visually is not a skill that comes naturally to students, and so it must be taught and practiced.

When first introducing a new skill to students, it is important to model the skill in order for them to see how it is used followed by opportunity for students to try it themselves.

Click here to access the article Explicit Instruction: A Teaching Strategy in Reading, Writing, and Mathematics for Students with Learning Disabilities .

The concrete-representational-abstract (CRA) approach to instructing students is a method of explicit instruction that is supported by research as being effective for students with LDs (Doabler, Fien, Nelson-Walker, & Baker, 2012; Mancl, Miller, & Kennedy, 2012). The concrete level involves the use of objects to represent mathematical information (e.g., counters, cubes); the representational level involves the drawing of pictures to represent the objects that were used in the previous level; and the abstract level replaces pictures of objects with mathematical symbols and numbers.

Click here to access the article Concrete-Representational-Abstract: And Instructional Strategy for Math .

Scaffolding is another approach to teaching visual representation to students with LDs that is supported by research (van Garderen, Scheuermann, & Jackson, 2012a). Scaffolding involves the use of temporary supports during the learning process as needed by the student. Providing a student with an incomplete diagram and having them finish it and then use it to solve a problem is an example of a scaffolding technique.

## Types of Visual Representation

When you are talking about visual representation in mathematics, you may be talking about representing information on a page with a diagram or chart , or representing information in your head with an image . Fortunately, researchers have focussed on helping students improve their visual representation both externally (e.g., van Garderen, 2007) and internally (e.g., Zhang et al., 2012). Developing both external and internal visual representation strategies is important for students as both help support student learning in mathematics for different types of problems.

In addition to the distinction between internal and external visual representations, researchers have also outlined differences in visual imagery based on the purpose. Pictorial imagery is used for representing the visual appearance of objects or information. Schematic imagery is used for representing the spatial relationships between objects or information. While both can be used to help students learn and solve problems in mathematics, schematic imagery is more effective as a method for supporting problem solving. Students with LDs are more likely to use pictorial imagery when solving problems in math (van Garderen & Montague, 2003).

## External Visual Representation

Diagrams and graphic organizers are two types of external visual representations that are used in mathematics, and both are supported by research for use with students with LDs. This is a type of visual representation that can be modelled to students as it is something that can be seen on a page or on the board at the front of the class.

Diagrams are visual displays that use the important information in mathematical problems. They are typically used to demonstrate how the important information is related , and can be used to organize information as well as to compute the answer to a problem. A common type of diagram might be a drawing that a student creates to represent the objects within a word problem.

Individuals with LDs may have a poorer understanding of what a diagram is, as well as when and how to use it (van Garderen et al., 2012b). Diagrams are effective for students with LDs as they can help highlight essential information and leave out information that is not necessary for solving a problem. This can simplify the problem-solving process (Kolloffel et al., 2009).

Distinctions can be made between pictorial diagrams and schematic diagrams . An example of a pictorial diagram would be a drawing of the important objects within a word problem, while a schematic diagram would be a drawing that includes the spatial relationships between the objects. As mentioned earlier, schematic diagrams are more useful for students and typically result in more success with problem solving (van Garderen & Montague, 2003).

## Tree Diagrams

A tree diagram is a method of representing independent events or conditions related to an action, and it is often used to teach students probability theory.

This type of diagram might be used to teach students about probability through coin flipping or through drawing from a deck of cards. It is considered a powerful method of teaching probability in math, and is a great example of a visual representation that is a diagram (Kolloffel, Eysink, de Jong, & Wilhelm, 2009).

## Number Lines

A number line is another type of diagram that is being used increasingly by mathematicians (Gersten et al., 2009). A number line consists of a straight line that has equally spaced numbers along it on points, and can be easily drawn by students for use when solving problems.

Number lines are often used for the teaching of integers , as well as for simple addition and subtraction problems, as they provide students with a visual that they can touch to keep track of their place .

## Graphic Organizers

A graphic organizer is another type of external visual representation that is often used in mathematics. There are many types of graphic organizers and each have situations that they are best used for. While graphic organizers are often thought of simply as organizational tools , they can be used to make rapid inferences to solve different types of problems.

Research supports the idea that students with language disorders may benefit from learning and instruction using nonverbal information such as a graphic organizer (Ives, 2007). Additionally, the use of graphic organizers to support learning has been found to improve the comprehension of facts and text for students with LDs at all ages (Dexter & Hughes, 2011), as well as enhancing conceptual understanding in mathematics (Ives, 2007).

Graphic organizers may be a great support for students with LDs because they take some of the organizational pressure off these individuals who may have difficulty sorting through information and seeing the relationships between different mathematical objects or concepts (Ives, 2007).

Four main types of graphic organizers can be used in mathematics:

## 1. Semantic Maps

A semantic map is one type of graphic organizer that can be used to support learning in mathematics. This type of graphic organizer is mainly used to relate conceptual information , and could be used to support conceptual learning in mathematics.

One example might be to use a semantic map to help young students who are learning to classify shapes into different categories. While shapes might be the main heading, students might organize shapes into further sub-heading such as round, symmetrical, right-angle, etc.

## 2. Semantic Feature Analysis

A semantic feature analysis is another type of graphic organizer. This graphic organizer is characterized by a matrix format , where features or characteristics of objects or concepts are displayed. A semantic feature analysis might be used to compare shapes in geometry, where comparisons could be made between number of sides, vertices, types of angles, etc.

## 3. Syntactic/Semantic Feature Analysis

A syntactic/semantic feature analysis is similar to the semantic feature analysis, but sentences are added in to help students identify specific features about each object.

An example sentence that might follow the matrix is “ A ________ has the most sides of all of the shapes we have looked at. ”

## 4. Visual Displays

## Selecting the Appropriate Graphic Organizer

Though each type of graphic organizer can be used for learning mathematics by individuals with LDs, the differences in their design suggest that they may be best used in specific situations.

- Semantic maps and Semantic feature analyses are considered to be better for recalling facts though they are more difficult to understand and to learn how to use (Dexter & Hughes, 2011).
- Syntactic/semantic feature analyses and visual displays are considered to be more efficient for making computations to solve problems, and for recalling the information within these types of graphic organizers (Dexter & Hughes, 2011).

The advantages to each type of graphic organizer suggest that initial instruction of a mathematical concept may be best with more complicated graphic organizers, and that independent review and studying could be done with less complicated graphic organizers to improve recall of information for students with LDs (Dexter & Hughes, 2011).

## Explicit Instruction of External Visual Representation

It is important to remember that explicit instruction is necessary for both diagrams and graphic organizers. This instruction should highlight the purpose of each type of external visual representation, as well as when and how to use them . Both types of external visual representations can be easily modelled for students as educators can physically construct them and explain their thinking as they do so in front of students.

In a study conducted by van Garderen (2007), the researcher examined the effectiveness of a three-phase instructional strategy for teaching students with LDs to use diagrams in mathematics:

- The first phase involved explicit instruction about what diagrams are, as well as how and when they are used.
- The second phase connected the use of diagrams to one-step word problems , where students created diagrams that represented the information that they knew and the information that they did not know.
- The third phase was focused on two-step word problems which have more than one unknown piece of information, and students used diagrams to determine the ultimate goal of the problem, as well as the secondary pieces of information that they would need to find in order to compute the ultimate goal.

Teaching students with LDs to use diagrams with this sequence of explicit instruction resulted in improved performance , satisfaction of the students, and students were also more likely to use diagrams with other types of problems.

## Internal Visual Representation

While external visual representation can be easy to model and teach explicitly to students, internal visual representation is not as easy to show students as it is a mental exercise. A visual schematic representation involves the creation or recall of visual imagery to represent information.

Students are often asked to visualize the problem in order to better understand it and solve it. This can be a difficult task for students and it should not be assumed that this is a skill that all students already possess .

To create a mental picture when solving a word problem in math, students must combine information from the problem with their prior knowledge of the topic. While students cannot see the mental images that their teachers create, it is still possible to walk students through the process of creating the mental image as a verbal model , or even to draw images of what they are seeing in their head to make it more explicit.

Click here to access the article Verbalization in Math Problem-Solving .

A group of researchers (Krawec, Huang, Montague, Kressler, Melia de Alba, 2012) developed an intervention to support students with LDs as they solve problems in mathematics. The intervention was aimed at explicitly instructing students about the cognitive processes that proficient problem solvers use in math, including visualization . The intervention was delivered by teachers who were trained to follow a sequence of instruction that included teaching visualizing to the students. The intervention involved the teachers “ thinking aloud ” as they progressed through the stages of the problem solving process. Students who were a part of the intervention reported using more strategies when solving problems in math, including the strategy of visualizing the problem (Krawec et al., 2012).

## Visual-Chunking

One strategy that teachers can use to support their students with LDs in creating internal visual representations is known as visual-chunking representation . Chunking is the practice of combining bits of information that are related in some way in order to reduce the overall amount of information for easier processing. For students with LDs, a reduction in the amount of information to be processed can make exercises such as math problem solving much easier.

A group of researchers examined one method of visual-chunking for students with difficulties in math, where students were working with geometric shapes and transformations (Zhang et al., 2012). One group received series of nets of geometric shapes, while another group received the same nets, though sections had been shaded or “chunked” in an effort to see if it made a difference in their transformations. The group that received the visual-chunking support performed better than the other group, and found the exercise to be easier when provided with the visual-chunking support (Zhang et al., 2012).

## Visual Schematic Representations

Visual schematic representations have been shown to be effective for individuals with specific difficulties in math, which can include LDs (Swanson et al., 2013). Instructing students about how to create internal visual representations can be difficult as it does not easily lend itself to explicit instructional techniques . Despite this challenge, teachers can still support the development of this skill by creating diagrams of their mental images as well as by thinking aloud as they are visualizing while problem solving.

The use of visual representation during instruction and learning tends to be an effective practice across a number of subjects , including mathematics (Gersten et al., 2009). While using visual representation alone as a teaching method does produce significant learning improvements for students in mathematics, these improvements are even greater when other teaching methods are used as well (Gersten et al., 2009).

Having students represent mathematical information verbally and in written form along with visual representation is encouraged. For students with LD, both receiving instruction and solving problems in a number of ways will help support their deeper understanding of concepts and operations in mathematics (Suh & Moyer, 2007).

The importance of using explicit instruction to teach students how to make visual representations cannot be overstated. The CRA method is an example of an effective sequence of explicitly instructing students with LD to use visual representation as a step towards the use of mathematical symbols exclusively (Mancl et al., 2012).

There are many types of diagrams (Kolloffel et al., 2009) and graphic organizers (Dexter & Hughes, 2011) that can be effectively used support students with LD in mathematics. While internal visual representation can be difficult to model, strategies do exist that can support students with LD as they develop this skill (Zhang et al., 2012). Educators are encouraged to use a combination of external and internal visual representation strategies in their instruction to students in the interest of helping students develop both types of skills.

## Related Resources on the LD@school Website

Click here to access the article Mind Maps .

Click here to access the article LDs in Mathematics: Evidence-Based Interventions, Strategies, and Resources .

Dexter, D. D., & Hughes, C. A. (2011). Graphic organizers and students with learning disabilities: A meta-analysis. Learning Disability Quarterly, 34 , 51-72.

Doabler, C. T., Fien, H., Nelson-Walker, N. J., & Baker, S. K. (2012). Evaluating three elementary mathematics programs for presence of eight research-based instructional design principles. Learning Disability Quarterly, 35 , 200-211.

Gersten, R., Chard, D. J., Jayanthi, M., Baker, S. K., Morphy, P., & Flojo, J. (2009). Mathematics instruction for students with learning disabilities: A meta-analysis of instructional components. Review of Educational Research, 79 , 1202-1242.

Ives, B. (2007). Graphics organizers applied to secondary algebra instruction for students with learning disorders. Learning Disabilities Research & Practice, 22 , 110-118.

Kolloffel, B., Eysink, T. H., de Jong, T., & Wilhelm, P. (2009). The effects of representational format on learning combinatorics from an interactive computer simulation. Instructional Science, 37 , 503-517.

Krawec, J., Huang, J., Montague, M., Kressler, B., Melia de Alba, A. (2012). The effects of cognitive strategy instruction on knowledge of math problem-solving processes of middle school students with learning disabilities. Learning Disabilities Quarterly, 36 , 80-92.

Mancl, D. B., Miller, S. P., & Kennedy, M. (2012). Using the concrete-representational-abstract sequence with integrated strategy instruction to teach subtraction with regrouping to students with learning disabilities. Learning Disabilities Research & Practice, 27 , 152-166.

Montague, M. (1997). Cognitive strategy instruction in mathematics for students with learning disabilities. Journal of Learning Disabilities, 30 , 164–177.

Suh, J., & Moyer, P. S. (2007). Developing students’ representational fluency using virtual and physical algebra balances. Journal of Computers in Mathematics and Science Teaching, 26 , 155-173.

Swanson, H. L., Lussier, C., & Orosco, M. (2013). Effects of cognitive strategy interventions and cognitive moderators on word problem solving in children at risk for problem solving difficulties. Learning Disabilities Research & Practice, 28 , 170-183.

van Garderen, D. (2007). Teaching students with LD to use diagrams to solve mathematical word problems. Journal of Learning Disabilities, 40 , 540-553.

van Garderen, D., & Montague, M. (2003). Visual-spatial representation, mathematical problem solving, and students of varying abilities. Learning Disabilities Research & Practice, 18 , 246-254.

van Garderen, D., Scheuermann, A., & Jackson, C. (2012a). Developing representational ability in mathematics for students with learning disabilities: A content analysis of grades 6 and 7 textbooks. Learning Disability Quarterly, 35 , 24-38.

van Garderen, D., Scheuermann, A., & Jackson, C. (2012b). Examining how students with diverse abilities use diagrams to solve mathematics word problems. Learning Disabilities Quarterly, 36 , 145-160.

Zhang, D., Ding, Y., Stegall, J., & Mo, L. (2012). The effect of visual-chunking-representation accommodation on geometry testing for students with math disabilities. Learning Disabilities Research & Practice, 27 , 167-177.

Searches were conducted of the literature for content appropriate for this topic that was published in scientific journals and other academic sources. The search included online database searches (ERIC, PsycINFO, Queen’s Summons, and Google Scholar). The gathered materials were checked for relevance by analysing data in this hierarchical order: (a) titles; (b) abstracts; (c) method; and (d) entire text.

Relevant journals’ archives were also hand-searched between issues from 2010 and the most recent issues. These journals included Learning Disability Quarterly, Journal of Learning Disabilities, and Learning Disabilities Research & Practice.

Nancy L. Hutchinson is a professor of Cognitive Studies in the Faculty of Education at Queen’s University. Her research has focused on teaching students with learning disabilities (e.g., math and career development) and on enhancing workplace learning and co-operative education for students with disabilities and those at risk of dropping out of school. In the past five years, in addition to her research on transition out of school, Nancy has worked with a collaborative research group involving researchers from Ontario, Quebec, and Nova Scotia on transition into school of children with severe disabilities. She teaches courses on inclusive education in the preservice teacher education program as well as doctoral seminars on social cognition and master’s courses on topics including learning disabilities, inclusion, and qualitative research. She has published six editions of a textbook on teaching students with disabilities in the regular classroom and two editions of a companion casebook.

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Thinking about many possible visual representations is the first step in creating a good one for students. The Progressions published in tandem with the Common Core State Standards for mathematics are one resource for finding specific visual models based on grade level and standard. In my fifth-grade example, what I constructed was a sequenced ...

Visual mathematics is an important part of mathematics for its own sake and new brain research tells us that visual mathematics even helps students learn numerical mathematics. In a ground breaking new study Joonkoo Park & Elizabeth Brannon (2013), found that the most powerful learning occurs when we use different areas of the brain.

What evidence-based mathematics practices can teachers employ? Page 5: Visual Representations. Yet another evidence-based strategy to help students learn abstract mathematics concepts and solve problems is the use of visual representations.More than simply a picture or detailed illustration, a visual representation—often referred to as a schematic representation or schematic diagram—is an ...

Research shows that the use of visual representations may lead to positive gains in math achievement. Visual representations help students develop a deeper understanding of the problems they are working with, making them more efective problem solvers. Visual representations such as manipulatives, number lines, pictorial representations, and ...

Multimedia-based Representations . In mathematics and mathematics education multimedia-based representations play an important role (Ollesch et al., 2017). The use of the medium affects brain function by stimulating the function of some of its parts. The processes

Visual representations are a powerful way for students to access abstract mathematical ideas. To be college and career ready, students need to be able to draw a situation, graph lists of data, or place numbers on a number line. Developing this strategy early during the elementary grades gives students tools for engaging with—and ways of ...

Visualization in mathematics learning is not new. Because mathematics involves the use of signs such as symbols and diagrams to represent abstract notions, there is a spatial aspect involved, that is, visualization is implicated in its representation. However, in contrast with the millennia in which mathematics has existed as a discipline ...

The kinds and the uses of visual thinking in mathematics are numerous and diverse. This entry will deal with some of the topics in this area that have received attention and omit others. Among the omissions is the possible explanatory role of visual representations in mathematics.

The first one considered the importance of visual representations in science and its recent debate in education. It was already shown by philosophers of the Wiener Kreis that visual representation could serve for a better understanding and dissemination of knowledge to the broader public. As knowledge can be condensed in different non-verbal ...

Visualization, as both the product and the process of creation, interpretation and reflection upon pictures and images, is gaining increased visibility in mathematics and mathematics education. This paper is an attempt to define visualization and to analyze, exemplify and reflect upon the many different and rich roles it can and should play in the learning and the doing of mathematics. At the ...

This is because students can offload the intrinsic demands of a word problem to the representation. Another potential benefit of visual representations is that they can help teach students how to discern important connections between variables, quantities, and relational terms in word problems. Previouschapter in book.

This two-part blog series by Marc North explores some thinking and strategies for using representations in Mathematics lessons. Part 1 unpicks some of the key theoretical ideas around the use of representations and models and foregrounds how representations can be used to both solve problems and explore mathematical relationships. Part 2 will illustrate these theoretical ...

mathematical knowledge, stressing even more the relevance of visual representation in mathematics education. More directly related with our work, there are a number of studies that stress the importance of the pictorial representations in solving geometrical tasks and discuss some difficulties involved in the representation of geometrical figures.

The use of visual representations (i.e., photographs, diagrams, models) has been part of science, and their use makes it possible for scientists to interact with and represent complex phenomena, not observable in other ways. Despite a wealth of research in science education on visual representations, the emphasis of such research has mainly been on the conceptual understanding when using ...

A visual image, as the product of visualisation, is defined as "a mental construct depicting visual or spatial information" (Presmeg, 2006, p.207). Visual representations, on the other hand, are the tools (diagrams or spatial arrangements) that may support visualisation (Duval, 2014). As a process, visualisation influences our thinking, but ...

Education researchers have long studied the role and importance of visual representation strategy in mathematics learning and problem solving (Arcavi, 2003; Debrenti, 2015; Goldin, 1998;Stylianou ...

on. using. visual representations of mathematical ideas in the classroom. We then provide the design of the project in terms of the training for teachers and also in terms of measuring the outcomes of the project. The results of the project are then presented, and then the conclusions that are drawn from the project.

Representation is an important element for teaching and learning mathematics since utilization of multiple modes of representation would enhance teaching and learning mathematics. Representation is a sign or combination of signs, characters, diagram, objects, pictures, or graphs, which can be utilized in teaching and learning mathematics.

The use of visual representations can be connected to any of the eight practice standards, but they are strongly emphasized in these three: Math.Practice.MP2—Reason abstractly and quantitatively. Math.Practice.MP4— Model with mathematics. Math.Practice.MP5—Use appropriate tools strategically.

General problem-solving skills are of central importance in school mathematics achievement. Word problems play an important role not just in mathematical education, but in general education as well. ... Visual representation often helps in understanding a problem. Using visual representations leads to a better understanding and to improving ...

DOI: 10.4172/2168-9679.1000325 Corpus ID: 63269571; Seeing as Understanding: The Importance of Visual Mathematics for our Brain and Learning @article{Boaler2016SeeingAU, title={Seeing as Understanding: The Importance of Visual Mathematics for our Brain and Learning}, author={Jo Boaler and Lang Chen and Cathy M. Williams and Montserrat Cordero}, journal={Journal of Applied and Computational ...

Visual representation is an important skill because higher-level math and science courses increasingly draw on visualization and spatial reasoning skills to solve problems (Zhang, Ding, Stegall, & Mo, 2012). Additionally, it is simply another strategy that students can use when they are thinking of the best way to answer a problem in mathematics.

The use of visual representation is o ne method that can be utiliz ed to teach mathematics. I t is. any method for producing pictures, diagrams, or animations to convey a point. Since the ...

The exploration of higher-order and multicomponent extensions of the nonlinear Schrödinger equation holds significant importance across diverse applications, notably within the field of optics. Among these equations, the integrable Sasa-Satsuma equation stands out for its captivating soliton solutions. The Sasa-Satsuma equation serves as a mathematical representation for describing the ...