– all angles 60°
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.
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.
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.
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.
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.
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
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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.
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March 18, 2021, by Rupert Knight
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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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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Reflections on visualization in mathematics and in mathematics education, author information, authors and affiliations.
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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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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.
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.
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.
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 ).
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.
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.
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).
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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ME carried out the introductory literature review, the analysis of the first case study, and drafted the manuscript. SE carried out the analysis of the third case study and contributed towards the “Conclusions” section of the manuscript. TM carried out the second case study. All authors read and approved the final manuscript.
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Evagorou, M., Erduran, S. & Mäntylä, T. The role of visual representations in scientific practices: from conceptual understanding and knowledge generation to ‘seeing’ how science works. IJ STEM Ed 2 , 11 (2015). https://doi.org/10.1186/s40594-015-0024-x
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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.
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 .
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:
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.
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:
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!
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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Published 2021
When the recepient of one of the most prestigious prizes in mathematics recently credited his success to playing with a Rubik's cube, it reinforced the importance of encouraging students to think visually.
Fields medallist and number theorist Manjul Bhargava explained to our colleagues at Plus how exploring his Rubik's cube led to his award-winning mathematical breakthrough:
"I was in my dorm room getting ready to go to bed, I had these Rubik's cubes in my room and I was just looking at one. I just have this memory of thinking, "What happens if you put numbers on the corners of this Rubik's cube? Instead of just thinking of it as a simple cube, think of it with numbers on the corners". I put numbers on the corners of that cube and I did some manipulations, and I saw three quadratic forms coming out, three quadratic expressions coming out. I decided not to go to sleep, that I'd figure out what these three quadratic expressions [were], how were they related."
There's a student-friendly version of the interview with Plus, and a teacher version .
A willingness to work flexibly and explore different approaches is deemed a highly desirable quality by some of our top ranking university mathematics departments (e.g. Mathematics at Cambridge: Guidance on admission selection ). Adopting a visual approach can enable students to spot patterns which might not be immediately apparent as they progress through a number curriculum packed with primes, patterns, and finite and infinite series. Nevertheless, many of the solutions we receive at NRICH come from students who mainly rely on algebraic approaches, rather than drawing on the power of visual imagery to convince their readers and justify their thinking.
Working visually not only enables students to grasp increasingly complex ideas, it has other benefits too. Visuals can also provide valuable opportunities to inspire students with the beauty of some of those images. Consider for example the power of a Sierpinski Triange for engaging students and prompting their mathematical curiosity:
Visual representations also offer a powerful approach for supporting students to recall key mathematical ideas, as noted by Lynn Steen (joint editor of 'Mathematics Magazine'):
"The various relationships embedded on [ sic ] a good diagram represent real mathematics awaiting recognition and verbalization. So as a device to help students learn and remember mathematics, proofs without words are often more accurate than (mis-remembered) proofs with words."
We would argue that the value of encouraging students to visualise is not in question. However, we do need to consider its overall role in the reasoning process and whether a visual image is acceptable as a proof. Consider the following conjecture: 'The sums of two odd number is always even'. Although this is a very general statement, many students understandably begin exploring it by playing with different number pairs to see if it holds. Although this approach, which mathematics educator John Mason refers to as 'specialising', enables students to try out a few examples to convince themselves about the results, to reach the level of a proof they need to go further and generalise their ideas. Their image need not be overly complex, but it does need to capture the underlying mathematics. Indeed, simplicity is often the key. After all, "successful visual representations tend to be spartan in their detail." (Borwein & Jorgensen, 2001). Consider this visual proof published by Rick Mabry (1999):
This image does appear very convincing and we would welcome more students to submit solutions which include a visual representation. However, perhaps we also need to consider the meaning of proof to determine the role of visual representations. The authors of Proof Without Words and Beyond (Doyle et al., 2014) have suggested that there are two key perspectives to consider relating to visual proofs. Firstly, the authors noted those who believe that "a proof must be expressed so as to 'explicitly draw logical connections between mathematical propositions". Since many visual representions illustrate a specific case rather than a generalisation, as is also the case when specialising with numbers, then it could be argued that visual representations are perhaps well suited for engaging and convincing others when working with large data sets, but often insufficient to satisfy the demands of a rigorous proof. Alternatively, the authors noted the alternative viewpoint that visual representations "can be far more rapidly and deeply convincing than traditional, propositional mathematical argumentation, and are therefore (in such cases) perfectly acceptable, even occasionally preferable proofs.'" Either way, it appears that visual representations are becoming increasingly important vehicle for engaging with mathematical ideas.
Visual representations enable students to access problems, and learn and recall their mathematics. They can also lead to mathematical breakthroughs, such as with Bhargava, and inspire others to begin to understand the awe and wonder of mathematics, whether through the beauty of Sierpinski's triangle or other powerful images. As computers enable visual representations to play an increasingly key role in communicating mathematical ideas, we need to ensure that our students frequently engage with visual representations, use them to support their mathematical thinking and enable them to provide convincing arguments. We look forward to receiving and publishing many more student submissions which feature visual representations in the coming months.
The NRICH secondary team
February 2021
Borwein, P. and Jorgensen, L. (2001). Visible structures in number theory. Retrieved from https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Borwein897-910.pdf
Doyle, T., Kutler, L., Miller, R., and Schueller, A. (nd). Proof without words and beyond. Retrieved from https://www.maa.org/press/periodicals/convergence/proofs-without-words-and-beyond
Mabry, R. (1999). Proof Without Words: (1/4)+(1/4)2+(1/4)3+⋯=1/3,"(1/4)+(1/4)2+(1/4)3+⋯=1/3. Mathematics Magazine, 72(1), 63.
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Home / Blog / Math / The Importance of Visual Learning in Math
There are so many different kinds of learning styles that it’s impractical for math educators to rely on just one. In fact, research shows that there are 70+ learning styles with visual, aural, verbal, and kinesthetic (aka VARK) approaches as the main sensory approaches. Interestingly, studies show that 65% of the population are visual learners. This is why it is important for educators who want their students, especially children, to better engage with math content to understand the needs of visual learners.
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Visual learning is the interaction and processing of knowledge through graphics. This includes drawings, images, charts, graphs, maps, and other types of visual aids. To engage visual learners, educators can use chalkboards, whiteboards, screens, projectors, and other mediums for displaying visual information to help students learn and remember what they see.
While visual learners can learn through other means as well, they learn best through a “show me” approach. For example, they are typically better at watching someone fix a flat tire than reading instructions in a printed manual. Even directions, when spoken, can be difficult for a visual learner to understand when compared to looking at a map.
So, how does this translate into the math class? Consider teaching addition. For a child who is a visual learner, showing a diagram with three apples in one hand and three in the other allows the child to visually count each apple until they reach the number six. This can be a more effective way for them to learn than a teacher verbally explaining the concept of three plus three equals six.
In a scientific paper released in the Journal of Applied & Computational Mathematics, researchers showed that “mathematical thinking is grounded in visual processing.” According to the lead researcher, Dr. Jo Boaler, a professor at the Stanford Graduate School of Education, when you try to solve a mathematical problem, brain activity occurs in several regions of the brain. Two of these regions are the ventral and dorsal pathways, which are visual pathways. And this activity happens to both children and adults.
The takeaway is that even if we don’t do it actively, on some level, our brains will subconsciously render the numbers we see into images. This allows us to understand space and quantity much better. So, when children are learning math, their brains are working under the hood trying to visualize the operations.
Another study observed activity in the brain region associated with finger representation, aka counting on your fingers, and perception when we look at mathematical calculations. Many kids naturally use their fingers as visual aids when performing arithmetic. What the study showed was that our brain does this type of “finger counting” anyway by representing the calculations as fingers without us even realizing it.
All this means is that visuals can transform the math class for many students. A visual provides a shift in perspective for some students, allowing them to understand math problems differently. While one student can hear numbers and math problems being said out loud and grasp the underlying math concepts, their classmate might need to look at the same problems visually before they can truly understand the “why” behind the problem. These visual learners then translate these visual representations, such as the six apples, back into numbers in their brain, putting them on par with their fellow non-visual peers.
There are more reasons why visual learning is important in math, even if a student is not a visual learner. Here are a few of them.
For many, visuals are more memorable than text or auditory learning. Since the information in visuals is presented in bite-sized chunks, it is easier to process and store. So, for example, when online education platforms break down complex topics into short videos, it allows students to digest the topic in short, manageable sessions instead of reading a long chapter in text format.
With visual learning, students become more creative in their problem-solving. And they carry on this lesson in other aspects of their daily lives outside the classroom.
For example, think of the last time you misplaced something around the house. You might have tried to tackle the problem visually by closing your eyes and imagining where the item might be. As you visualize in your mind the steps you took the last time you remember having, let’s say, your keys, you’re experiencing visual problem solving.
When visuals are made to be a part of the process, learning can be fun. This can help students stay engaged longer, which is important in this day and age where attention spans are at an all-time low. Studies have shown that the average attention span is currently eight seconds. It’s easy to lose a student’s attention if the lesson doesn’t start with attention-grabbing and relatable visuals.
Learning math through visualization gives students a deeper understanding of mathematical concepts and their practical applications. Visuals are an important aspect of math education, and this applies to everyone⏤not just visual learners. This is not to say that visual learning should replace all other types of learning in the classroom; rather, as we’ve seen, it should be used to increase engagement, understanding, and retention.
As a global education platform, we at BYJU’S FutureSchool understand the importance of visual learning in math. Our research-based approach to visual math combines one-to-one attention with a hands-on approach to teaching children aged 6 to 14. Even our group classes are intimate, consisting of only four students per teacher. This ensures that everyone gets the attention and guidance they need.
BYJU’S FutureSchool’s interactive platform was built to be engaging and intuitive. Our math mission is to foster students’ confidence in math. We use visuals to drive concepts home, rather than simply asking students to memorize formulas without knowing why. Our storytelling approach to teaching enables students to visualize what they learn. We also celebrate mistakes as learning opportunities. This helps to mitigate the fear of failure while also encouraging students to embrace taking chances.
Byju's futureschool.
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Mathematical representation is one of the fundamental abilities used in mathematics to build the connection between abstract ideas and logical thinking to understand and solve mathematics problems. The level of student representation ability will affect the way students convey the idea of solving mathematics problems. This shows how important this ability is in mathematics. But in fact, the ability of mathematical representation is felt to be less comprehensively mastered. This is indicated by the large number of students having difficulty conveying their ideas in solving mathematics problems. This paper describes students' abilities in mathematical representation to solve math problems. The indicators for mathematical representation ability in this research are classified as verbal representation (written text), visual representation (picture, diagram, graph, or table), and symbolise representation (mathematical statement, numerical/algebra symbol, mathematical notation). The subject of this research was 8-grade students of one of the junior secondary schools at Bukittinggi. The results showed that 22 out of 34 students have difficulty in conveying their ideas to solve math problems, while the rest feel quite good about it.
COMMENTS
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 ...
As teachers, complete the mathematical tasks yourselves, a key component of Math for All, to explore and understand the various physical models, visual representations, and symbolic processes that can be developed in support of the mathematical content. Value the concrete models and pictures of the mathematical content in a similar way to how ...
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.
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 ...
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 ...
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.
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 ...
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 ...
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.
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 accurate ...
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 ...
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 ...
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
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 ...
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 ...
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.
Visual representations enable students to access problems, and learn and recall their mathematics. They can also lead to mathematical breakthroughs, such as with Bhargava, and inspire others to begin to understand the awe and wonder of mathematics, whether through the beauty of Sierpinski's triangle or other powerful images.
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 use of multiple visual representations is common practice in science, technology, engineering, and mathematics (STEM) domains (NRC 2006). Consequently, educational practice guides emphasize the importance of using multiple visual representations (NCTM 2000, 2006; NRC 1996, 2002, 2006), and instruction relies on visual representations in science
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 Learning is Engaging. When visuals are made to be a part of the process, learning can be fun. This can help students stay engaged longer, which is important in this day and age where attention spans are at an all-time low. Studies have shown that the average attention span is currently eight seconds.
Mathematical representation is one of the fundamental abilities used in mathematics to build the connection between abstract ideas and logical thinking to understand and solve mathematics problems. The level of student representation ability will affect the way students convey the idea of solving mathematics problems. This shows how important this ability is in mathematics. But in fact, the ...