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Problematizing teaching and learning mathematics as “given” in STEM education

• Yeping Li 1 &
• Alan H. Schoenfeld 2  

International Journal of STEM Education volume  6 , Article number:  44 ( 2019 ) Cite this article

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Mathematics is fundamental for many professions, especially science, technology, and engineering. Yet, mathematics is often perceived as difficult and many students leave disciplines in science, technology, engineering, and mathematics (STEM) as a result, closing doors to scientific, engineering, and technological careers. In this editorial, we argue that how mathematics is traditionally viewed as “given” or “fixed” for students’ expected acquisition alienates many students and needs to be problematized. We propose an alternative approach to changes in mathematics education and show how the alternative also applies to STEM education.

Introduction

Mathematics is commonly perceived to be difficult (e.g., Fritz et al. 2019 ). Moreover, many believe “it is ok—not everyone can be good at math” (Rattan et al. 2012 ). With such perceptions, many students stop studying mathematics soon after it is no longer required of them. Giving up learning mathematics may seem acceptable to those who see mathematics as “optional,” but it is deeply problematic for society as a whole. Mathematics is a gateway to many scientific and technological fields. Leaving it limits students’ opportunities to learn a range of important subjects, thus limiting their future job opportunities and depriving society of a potential pool of quantitatively literate citizens. This situation needs to be changed, especially as we prepare students for the continuously increasing demand for quantitative and computational literacy over the twenty-first century (e.g., Committee on STEM Education 2018 ).

The goal of this editorial is to re-frame issues of change in mathematics education, with connections to science, technology, engineering, and mathematics (STEM) education. We are hardly the first to call for such changes; the history of mathematics and philosophy has seen ongoing changes in conceptualization of the discipline, and there have been numerous changes in the past century alone (Schoenfeld 2001 ). Yet changes in practice of how mathematics is viewed, taught, and learned have fallen far short of espoused aspirations. While there has been an increased focus on the processes and practices of mathematics (e.g., problem solving) over the past half century, the vast majority of the emphasis is still on what content should be presented to students. It is thus not surprising that significant progress has not been made.

We propose a two-fold reframing. The first shift is to re-emphasize the nature of mathematics—indeed, all of STEM—as a sense-making activity. Mathematics is typically conceptualized and presented as a body of content to be learned and processes to be engaged in, which can be seen in both the NCTM Standards volumes and the Common Core Standards. Alternatively, we believe that all of the mathematics studied in K-12 can be viewed as the codification of experiences of both making sense and sense making through various practices including problem solving, reasoning, communicating, and mathematical modeling, and that students can and should experience it that way. Indeed, much of the inductive part of mathematics has been lost, and the deductive part is often presented as rote procedures rather than a form of sense making. If we arrange for students to have the right experiences, the formal mathematics can serve to organize and systematize those experiences.

The second shift is suggested by the first, with specific attention to classroom instruction. Whether mathematics or STEM, the main focus of most instruction has been on the content and practices of the discipline, and what the teacher should do in order to make it accessible to students. Instead, we urge that the main focus should be on the student’s experience of the discipline – on the affordances the environment provides the student for disciplinary sense making. We will introduce the Teaching for Robust Understanding (TRU) Framework, which can be used to problematize instruction and guide needed reframing. The first dimension of TRU (The Discipline) focuses on the re-framing discussed above: is the content conceptualized as something rich and connected that can be experienced and codified in meaningful ways? The second dimension (Cognitive Demand) examines opportunities students have to do that kind of sense-making and codification. The third (Equitable Access to Content) examines who has such opportunities: is there equitable access to the core ideas? Dimension 4 (Agency, Ownership, and Identity) asks, do students encounter the discipline in ways that enable them to see themselves as sense makers, building both agency and positive disciplinary identities? Finally, dimension 5 (Formative Assessment) asks, does instruction routinely use formative assessment, allowing student thinking to become public so that instruction can be adjusted accordingly?

We begin with a historical background, briefly discussing different views regarding the nature of mathematics. We then problematize traditional approaches to mathematics teaching and learning. Finally, we discuss possible changes in the context of STEM education.

Knowing the background: the development of conceptions about the nature of mathematics

The scholarly understanding of the nature of mathematics has evolved over its long history (e.g., Devlin 2012 ; Dossey 1992 ). Explicit discussions regarding the nature of mathematics took place among Greek mathematicians from 500 BC to 300 AD (see, https://en.wikipedia.org/wiki/Greek_mathematics ). In contrast to the primarily utilitarian approaches that preceded them, the Greeks pioneered the study of mathematics for its own sake and pursued the development and use of generalized mathematical theories and proofs, especially in geometry and measurement (Boyer 1991 ). Different perspectives about the nature of mathematics were gradually developed during that time. Plato perceived the study of mathematics as pursuing the truth that exists in external world beyond people’s mind. Mathematics was treated as a body of knowledge, in the ideal forms, that exists on its own, which human’s mind may or may not sense. Aristotle, Plato’s student, believed that mathematicians constructed mathematical ideas as a result of the idealization of their experience with objects (Dossey 1992 ). In this perspective, Aristotle emphasized logical reasoning and empirical realization of mathematical objects that are accessible to the human senses. The two schools of thought that evolved from Plato’s and Aristotle’s contrasting conceptions of the nature of mathematics have had important implications for the ensuing development of mathematics as a discipline, and for mathematics education.

Several more schools of thought were developed as mathematicians tackled new problems in mathematics (Dossey 1992 ). Davis and Hersh ( 1980 ) provides an entertaining and informative account of these developments. Three major schools of thought in the early 1900s dealt with paradoxes in the real number system and the theory of sets: (1) logicism, as an outgrowth of the Platonic school, accepts the external existence of mathematics and emphasizes the form rather than the interpretation in a specific setting; (2) intuitionism, as influenced by Aristotle’s ideas, only accepts the mathematics to be developed from the natural numbers forward via “valid” patterns of mental reasoning (not empirical realization in Aristotle’s thought); and (3) formalism, also aligned with Aristotle’s ideas, builds mathematics upon the formal axiomatic structures to free mathematics from contradictions. These three schools of thought are similar in that they view the contents of mathematics as products , but they differ in whether products are viewed as pre-existing or created through experience. The development of these three schools of thought illustrates that the view of mathematics as products has its long history in mathematics.

With the gradual development of school mathematics since 1900s (Stanic and Kilpatrick 1992 ), the conception of the nature of mathematics has increasingly received attention from mathematics educators. Which notion of mathematics mathematics education adopts and uses has a direct and strong impact on the way of school mathematics being presented and approached in education. Although the history of school mathematics is relatively short in comparison with mathematics itself, we can find ample examples about the influence of different views of mathematics on curriculum and classroom instruction in the USA and other education systems (e.g., Dossey et al. 2016 ; Li and Lappan 2014 ; Li, Silver, and Li 2014 ; Stanic and Kilpatrick 1992 ). For instance, the “New Math” movement of 1950s and 1960s used the formalism school of thought as the core of reform efforts. The content was presented in a structural format, using the set theoretic language and conceptions. But the result was not a successful progression toward a school mathematics that is best for students and teachers (e.g., Kline 1973 ). Alternatively, Dossey ( 1992 ), in his review of the nature of mathematics, identified and selected scholars’ works and ideas applicable to both professional mathematicians and mathematics educators (e.g., Davis and Hersh 1980 ; Hersh 1986 ; Tymoczko 1986 ). Those scholars' ideas rested on what professional mathematicians do, not what mathematicians think about what mathematics is. Dossey ( 1992 ) specifically cited Hersh ( 1986 ) to emphasize mathematics is about ideas and should be accepted as a human activity, not strictly governed by any one school of thought.

Devlin ( 2000 ) argued that mathematics is not a single entity but has four different faces: (1) computation, formal reasoning, and problem solving; (2) a way of knowing; (3) a creative medium; and (4) applications. Further, he contended school mathematics typically focuses on the first face, makes some reference to the fourth face, but pays almost no attention to the other two faces. His conception of mathematics assembles ideas from the history of mathematics and observes mathematical activities occurring across different settings.

Our brief review shows that the nature of mathematics can be understood as having different faces, rather than being governed by any single school of thought. At the same time, the ideas of Plato and Aristotle continue to influence the ways that mathematicians, mathematics educators, and the general public perceive mathematics. Despite nearly a half century of process-oriented research (see below), let alone Pólya’s work on problem solving, mathematics is still perceived of largely as products —a body of knowledge as highlighted in the three schools (logicist, intuitionist, formalist) of thought, rather than ideas that call for active thinking and creation. The evolving conceptions about the nature of mathematics in history suggests there is room for us to decide how mathematics can be perceived, rather than being bounded by a pre-occupied notion of mathematics as “given” or “fixed.” Each and every learner can experience mathematics through different practices and “own” mathematics as a human activity.

Problematizing what is important for students to learn in and through mathematics

The evolving conceptions about the nature of mathematics suggest that choices exist when deciding what and how to teach and learn mathematics but they do not specify what and how to make the choice. Decisions require articulating options for conceptions of what is important for students to learn in and through mathematics and evaluating the advantages and drawbacks for the students for each option.

According to Stanic and Kilpatrick ( 1992 ), the history of school mathematics curricula presents two important and real changes over the years: one is at the turn of the twentieth century when school mathematics was reformed as a unified and applied curriculum to accommodate dramatically increased student populations from diverse backgrounds, and the other is the “New Math” movement of the 1950s and 1960s, intended to integrate modern mathematics into school curriculum. The perceived failure of the “New Math” movement led to the “Back to Basics” movement in the 1970s, followed by “Problem Solving” in the 1980s, and then the Curriculum Standards movement in the 1990s and after. The history shows school mathematics curricula have emphasized teaching and learning mathematical knowledge and skills, together with problem solving and some applications of mathematics, a picture that is consistent with what Devlin ( 2000 ) refers to as the 1st face and some reference to the 4th face of mathematics.

Therefore, although there have been reforms in mathematics curriculum and instruction, there are hardly real changes in how mathematics is conceptualized and presented in school education in the USA (Stanic and Kilpatrick 1992 ) and other education systems (e.g., Leung and Li 2010 ; Li and Lappan 2014 ). The dominant conception remains mathematics as products , frequently referring to a body of static knowledge and skills that need to be learned and acquired (Fisher 1990 ). This continues to be largely the case in practice, despite advances in conceptualization (see below).

It should be noted that conceptualizing mathematics as “a body of knowledge and skills” is not wrong, especially with such a long history of knowledge creation and accumulation in mathematics, but it is not adequate for school mathematics nowadays. The set of concepts and procedures, after years of development, exceeds what could be covered in any school curricula. Moreover, this body of knowledge and skills keeps growing, as the product of human intelligence and scholarship in mathematics. Devlin ( 2012 ) pointed out that school mathematics mainly covers what was developed in the Greek mathematics, plus just two further advances from the seventh century: calculus and probability theory. It is no wonder if someone questions the value of learning such a small set of knowledge and skills developed more than a thousand years ago. Meanwhile, this body of knowledge and skills are often abstract, static, and “foreign” to many students and teachers who learned to perceive mathematics as an external entity in existence (Plato’s notion) rather than Aristotelian emphasis on experimentation (Cooney 1987 ). It is thus not surprising for so many students and teachers to claim that mathematics is difficult (e.g., Fritz et al. 2019 ) and “it is ok—not everyone can be good at math” (Rattan et al. 2012 ).

What can be made meaningful should be critically important to those who want to (or need to) learn and teach mathematics. In fact, there is significant evidence that students often try to make sense of mathematics that is “presented” or “given” to them, although they made numerous errors that can be decoded to study their thinking (e.g., Ashlock 2010 ). Indeed, misconceptions are best thought of not as errors that need to be “fixed,” but as plausible abstractions on the basis of what students have learned—i.e., attempts at sense-making (Smith et al. 1993 ). Conceiving mathematics as about “ideas,” we can help students to play, own, experience, and think about some key ideas just like what they do in many other activities, such as game play (Gee 2005 ). Definitions of concepts and formal languages and procedures can be postponed until students are ready to consider why and how they are needed. Mathematics should be taken and accepted as a human activity (Dossey 1992 ), and developing students’ mathematical thinking (about ideas) should be emphasized in learning mathematics itself (Devlin 2012 ) and in STEM (Li et al. 2019a ).

Along with the shift from products to ideas in mathematics, scholars have already focused on how people work with ideas in mathematics. Elaborated in detail by Schoenfeld ( in press ), the revolution began with George Pólya (1887–1985) who had a fundamental interest in having students learn and understand content via problem solving. For Pólya, mathematics was about inquiry, sense making, and understanding how and why mathematical ideas (instead of content as products) fit together the way they do. The call for problem solving in the 1980s in the USA was (at least partially) inspired by Pólya’s ideas after a decade of “back to basics” in the 1970s. It has been recognized since that the practices of mathematics (including problem solving) are every bit as important as the content itself, and the two shouldn’t be separated. In the follow-up standards movement, the content and practices have been the “warp and weave” of the fabric doing mathematics, as articulated in Principles and Standards for School Standards (NCTM 2000 ). There were five content standards and five process standards (i.e., problem solving, reasoning, connecting, communicating, representing). It is widely acknowledged, also in the Common Core State Standards in the USA (CCSSI 2010 ), that both content and processes/practices are essential and they form the base for next steps.

Problematizing how mathematics is taught and learned, with connections to STEM education

How the ways that mathematics is often taught cause concerns.

Conceiving mathematics as a body of facts and procedures to be “mastered” has been long-standing in mathematics education practice, and it often results in students’ learning by rote memorization. For example, Schoenfeld ( 1988 ) provided a detailed account of the disasters of a “well-taught” mathematics course, documenting a 10th-grade geometry class taught by a confident and experienced teacher. The teacher taught and managed his class well, and his students also did well on standardized examinations, which focused on content and procedures. At the same time, however, Schoenfeld pointed out that the students developed counterproductive views of mathematics. Although the students developed some level of proficiency in content and procedures, they gained (or were reinforced in) the kinds of beliefs about mathematics as being fragmented and disconnected. Schoenfeld argued that the course led students to develop a robust and counterproductive set of beliefs about the nature of geometry.

Seeking possible origins about students’ counterproductive beliefs about mathematics from mathematics instruction motivated Schoenfeld’s study (Schoenfeld 1988 ). Such an intuitive motivation is also evident in other studies. Keitel ( 2006 ) compared the lessons of two teachers (T1 and T2) in Germany who taught their classes very differently. T1 regularly taught the class emphasizing routine individual practice and memorization of specific algebraic rules. T1 emphasized the importance of such practices for test taking, and the students followed his instruction. Even when T1 one day introduced a non-routine problem that connects algebra and geometry, the overwhelming emphasis on mastering routines and algorithms seemed to overshadow in dealing such a non-routine problem. In contrast, T2’s teaching emphasized students’ initiatives and collaboration, although T2 also used formal routine tasks. At the end, students in T2’s class reported positively about their experience, enjoyed working together, and appreciated the opportunities of thinking mathematically. Studies by Schoenfeld ( 1988 ) and Keitel ( 2006 ) indicate how students’ experience in mathematics classes influences their perceptions of mathematics and also imply the importance of learning about teachers’ perceptions of mathematics that likely guide their instructional practice (Cooney 1987 ).

Rattan et al. ( 2012 ) found that teachers with different perceptions of mathematics teach differently. Specifically, Rattan et al. looked at these teachers holding an entity (fixed) theory of mathematics intelligence (G1) versus incremental theory (G2). Through their studies, Rattan and colleagues found that G1 teachers more readily judged students to have low ability, comforted students for low mathematical ability, and used “kind” strategies (e.g., assigning less homework) unlikely to promote their engagement with the field than G2 teachers. Students who received comfort-oriented feedback perceived their teachers’ entity theory and low expectations and reported lowered motivation and expectations for their own performance. The results suggest how teachers’ inadequate perceptions of mathematics and beliefs about the nature of students’ mathematical intelligence contributed to low achievement, diminished self-esteem and viewed mathematics is only a set of static facts and procedures. Further, the results suggest that how mathematics is taught influences more than students’ proficiency with mathematics content in a class. Sun ( 2018 ) made a similar argument after synthesizing existing literature and analyzing classroom observation data.

Based on the 2012 US national survey of science and mathematics education conducted by Horizon Research, Banilower et al. ( 2013 ) reported that a vast majority of mathematics teachers, from 81% at the high school level to 90% at the elementary level, believe that students should be given definitions of new vocabulary at the beginning of instruction on a mathematical idea. Also, many teachers believe that they should explain an idea to students before having them consider evidence for it and that hands-on activities should be used primarily to reinforce ideas students have already learned. The report suggests many teachers emphasized pedagogical practices of “give” and “present,” perhaps influenced by conceptions of mathematics that are more Platonic than Aristotelian, similar to what was reported about teachers’ practices more than two decades ago (Cooney 1987 ).

How to change?

Given that the evidence demonstrates a compelling case for changing how mathematics is taught, we turn our attention to suggesting how to realize this transformation. Changing how mathematics is taught and learned is not a new endeavor for both mathematics educators and mathematicians (e.g., Li, Silver, and Li 2014 ; Schoenfeld in press ). For example, the “Moore Method,” developed and used by Robert Lee Moore (a famous topologist) in the early twentieth century, shifted instruction from teacher-centered lecturing to student-centered mathematical development (Coppin et al. 2009 ). In its purest form, students were presented with mathematical definitions and asked to develop and/or prove theorems from them after class, without reading mathematics books or using other resources. When students returned to the class, they were asked to prove a theorem. As a result, students developed the mathematics themselves, instead of the instructor presenting the proofs and doing the mathematics for students. The method has had its own success in producing many great mathematicians; however, the high-pressure environment also drowned many students who might have been successful otherwise (Schoenfeld in press ).

Although the “Moore Method” was used primarily in advanced mathematics courses at the post-secondary level, it illustrates how a different conception of mathematics led to a different instructional approach in which students developed mathematics. However, it might be the opposite end of a spectrum, in comparison to approaches that present mathematics to students in accommodating and easy-to-digest ways that can be as much easy as possible. Neither extreme is a good option for K-12 students. Again, it becomes important for us to consider options that can support the value of learning mathematics.

Our discussion in the previous section highlights the importance of taking mathematics as a human activity, ensuring it is meaningful to students, and developing students’ mathematical thinking about ideas, rather than simply absorbing a set of static and disconnected knowledge and skills. We call for a shift in teaching mathematics based on Platonic conceptions to approaches based on more of Aristotelian conceptions. In essence, Plato emphasized ideal forms of mathematical objects, perhaps inaccessible through people’s sense making efforts. As a result, learners lack ownership of the ideal forms of mathematical objects, because mathematical objects cannot and should not be created by human reasoning. In contrast, Aristotle emphasized that mathematical objects are developed through logic reasoning and empirical realization. In other words, mathematical objects exist only when they can be sensed and verified by people's efforts. This differs from Plato’s passive perspective, highlights human ownership of mathematical ideas and encourages people to make mathematics make sense, termed as making sense by McCallum ( 2018 ). Aristotelian conceptions view mathematics as objects that learners can actively develop and structure as mathematically meaningful, which is more in line with what research mathematicians do. McCallum ( 2018 ) argued that both sense-making and making-sense stances are needed for a complete view of mathematics and learning, recognizing that not attending to both stances carries risks. “Just as it is a risk of the sense-making stance that the mathematics gets ignored, it is a risk of the making-sense stance that the sense-maker gets ignored.” (McCallum 2018 ).

In addition, there is the issue of personal identity: if students come to avoid mathematics because they are uncomfortable with it (in fact, mathematics anxiety has become a widespread problem for all ages across the globe, see Luttenberger et al. 2018 ) then mathematics instruction has failed them, regardless of test scores.

In the following, we discuss sense-making and making-sense stances first with specific examples from mathematics. Then, we discuss connections to STEM education.

Sense making is much more than the acquisition of knowledge and skills

Sense making has long been emphasized in mathematics education community. William A. Brownell is a well-known, early 20 th century scholar who advocated the value of sense making in the learning of mathematics. For example, Brownell ( 1945 ) discussed how arithmetic can and should be taught and learned not only as procedures, but also as a meaningful system of thinking. He shared many examples like the following division,

Brownell suggested to ask questions: what does the 5 of 576 mean? Why must 57 be the first partial dividend? Do you actually divide 8 into 57, or into 57…’s? etc., instead of simply letting students memorize how to carry out the procedure. What Brownell advocated has been commonly accepted and emphasized in mathematics education nowadays as sense making (e.g., Schoenfeld 1992 ).

There can be different ways of sense making of the same computation. As an example, the sense making process for the above long division can come out with mental math as: I am looking to see how close I can get to 570 with multiples of 80; 7 multiples of 80 gives me 560, which is close. Of course, given base 10 notation, that’s the same as 8 multiples of 70, which is why the 7 goes over the 57. And when I subtract 560, there are 16 left over, so that’s another 2 8 s. Such a sense-making process also works, as finding the answer (quotient, k ) of 576 ÷ 8 is the same operation as to find k that satisfies 576 = k × 8. In mathematics, division and multiplication are alternate but equivalent ways of doing the same operation.

To help students build numerical reasoning and make sense of computations, many teachers use number talks in their classrooms for students to practice and share these mental math and computation strategies (e.g., Parrish 2011 ). In fact, new terms are being created and used in mathematics education about sense making, such as number sense (e.g., Sowder 1992 ) and symbol sense (Arcavi, 1994 ). Some instructional programs, such as Cognitively Guided Instruction (see, e.g., Carpenter et al., 1997 , 1998 ), make sense making the core of instructional activities. We argue that such activities should be more widely adopted.

Making sense makes the other side of mathematical practice visible, and values idea development and ownership

The making-sense stance, as termed by McCallum ( 2018 ), is not commonly practiced as it is pertinent to expert mathematician’s practices. Where sense making (as discussed previously) emphasizes the process of making sense of what is being learned, making sense emphasizes the process of making mathematics make sense. Making sense highlights the importance for students to experience mathematics through creating, designing, developing, and connecting mathematical ideas. As an example, for the above division computation, 8 \( \overline{\Big)576\ } \) , students may wonder why the division procedure is performed from left to right, which is different from the other operations (addition, subtraction, and multiplication) that are all performed from right to left. In fact, students can be encouraged to explore if the division can also be performed from right to left (i.e., starting from the one’s place). They may discover, with possible support from the teacher, that the division can be done in this way. However, once the division is moved to the high-value places, it will require the process to go back down to the low-value places for completion. In other words, the division process starting from the low-value place would require repeated processes of returning to the low-value places; as a result, it is inefficient. As mathematical procedure is designed to improve efficiency, the division procedure is thus set to be carried out from the high-value place to low-value place (i.e., from left to right). Students who work this out experience mathematics more deeply than the sense-making described by Brownell ( 1945 ).

There are plenty of making-sense opportunities in classroom instruction. For example, kindergarten children are often given opportunities to play with manipulatives like cube trains and snub cubes, to explore and learn about patterns, numbers, and measurement through various connections. The recording of such activities typically results in numerical expressions or operations of these connections. In addition, such activities can also serve as a context to encourage students to design and create a way of “recording” these connections directly with a drawing line next to the connected train cubes. Such a design activity will help students to develop the concept of a number line that includes the original/starting point, unit, and direction (i.e., making mathematics make sense), instead of introducing the number line to students as a mathematical concept being “given” years later.

Learning how to provide students with opportunities to develop mathematics may occur with experience. Huang et al. ( 2010 ) found that expert and novice teachers in China both valued students’ mastering of mathematical knowledge and skills and their development in mathematical thinking methods and abilities. However, novice teachers were particularly concerned about the effectiveness of their guidance, in contrast to expert teachers who emphasized the development of students’ mathematical thinking and higher-order thinking abilities and properly dealing with important and difficult content points. The results suggest that teachers’ perceptions and pedagogical practices can change and improve over time. However, it may be worth asking if support for teacher development would accelerate the process.

Connecting changes in mathematics and STEM education

Although it is commonly acknowledged that mathematics is foundational to STEM, mathematics is being related to STEM education at a distance in practice and also in scholarship development (English 2016 , see additional notes at the end of this editorial). Holding the conception of mathematics as products does not support integrating mathematics with other STEM disciplines, as mathematics can be perceived simply as a set of tools for these disciplines. At the same time, mathematics and science have often proceeded along parallel tracks, with mathematics focused on “problem solving” while science has focused on “inquiry.” To better connect mathematics and other disciplines in STEM, we should focus on ideas and thinking development in mathematics (Li et al. 2019a ), unifying instruction from the student perspective (the Teaching for Robust Understanding framework, discussed below).

Emphasizing both sense making and making sense in mathematics education opens opportunities for connections with similar practices in other STEM disciplines. For example, sense making is very much emphasized in science education (Hogan 2019 ; Kapon 2017 ; Odden and Russ 2019 ), often combined with reflections in engineering (Kilgore et al. 2013 ; Turns et al. 2014 ), and also in the context of using technology (e.g., Antonietti and Cantoia 2000 ; Dick and Hollebrands 2011 ). Science is fundamentally about discovery and understanding of the natural world. This notion provides a natural link to mathematical modeling (e.g., Burkhardt 1981 ). Beyond that, in science education, sense making places a heavy focus on the construction and evaluation of explanation (Kapon 2017 ), and can even be defined as a process of constructing an explanation to resolve a perceived gap or conflict in knowledge (Odden and Russ 2019 ). Design and making play vital roles in engineering and technology education (Dym et al., 2005 ), as is student reflection on these experiences (e.g., Turns et al. 2014 ). Indeed, STEM disciplines share the same conceptual process of sense making as learners, individually or in a group, actively engage with the natural or man-made world, explore it, and then develop, test, refine, and use ideas together with specific explanation. If mathematics was conceived as an “empirical” discipline, connections with other STEM disciplines would be strengthened. In philosophical terms, Lakatos ( 1976 ) made similar claims Footnote 1 .

Similar to the emphasis on sense making placed in the Mathematics Curriculum Standards (e.g., NCTM, 1989 , 2000 ), Next Generation Science Standards (NGSS Lead States 2013 ) prompted a shift in science education away from simply knowing science content and procedures to practicing and using science, together with engineering, to make sense of the world and create the future. In a review, Fitzgerald and Palincsar ( 2019 ) concluded sense making is a productive lens for investigating and characterizing great teaching across multiple disciplines.

Mathematics has stronger linkages to creation and design than traditionally imagined. Therefore, its connections to engineering and technology could be much stronger. However, the deep-rooted conception of mathematics as products has traditionally discouraged students and teachers from considering and valuing design and design thinking (Li et al. 2019b ). Conceiving mathematics as making sense should help promote conceptual changes in mathematical practice to value idea generation and design activity. Connections generated from such a shift will support teaching and learning not only in individual STEM disciplines, but also in integrated STEM education.

At the same time, although STEM education as a commonly recognized field does not have a long history (Li 2014 , 2018a ), its rapid development can help introduce ideas for exploring how mathematics can be taught and learned. For example, the concept of projects is common in engineering professional practice, and the project-based learning (PjBL) as an instructional approach is a key component in some engineering programs (e.g., Berger 2016 ; de los Ríos et al. 2010 ; Mills and Treagust 2003 ). de los Ríos et al. ( 2010 ) highlighted three main advantages of PjBL: (1) development in technical, personal, and contextual competences; (2) students’ engagement with real problems from professional contexts; and (3) collaborative learning facilitated through the integration of teaching and research. Such advantages are important for students’ learning of mathematics and are aligned well with efforts to develop 21 st century skills, including problem solving, communication, collaboration, and critical thinking.

Design-based learning (DBL) is another instructional approach commonly used in engineering and technology fields. Gómez Puente et al. ( 2013 ) conducted a sampled review and concluded that DBL projects consist of open-ended, hands-on, authentic, and multidisciplinary design tasks. Teachers using DBL facilitate both the process for students to gain domain-specific knowledge and thinking activities to generate innovative solutions. Such features could be adapted for mathematics education, especially integrated STEM education, in concert with design and design thinking. In addition to a few examples discussed above about making sense in mathematics, there is a growing body of publications developed by and for mathematics teachers with specific examples of investigations, design projects, and instructional activities associated with STEM (Li et al. 2019b ).

A framework for helping students to gain important experiences in and through mathematics, as connected to other disciplines in STEM

For observing and evaluating classroom instruction in general and mathematics classroom instruction in specific, there are several widely used frameworks and rubrics available. However, a trial use of selected frameworks with sampled mathematics classroom instruction episodes suggested their disagreements on what counts as high-quality instruction, especially with aspects on disciplinary thinking being valued and relevant classroom practices (Schoenfeld et al. 2018 ). The results suggest the importance of choice making, when we consider a framework in discussing and evaluating teaching practices.

Our discussion above highlights the importance of shifting away from viewing mathematics simply as a set of static knowledge and skills, to focusing on ideas and thinking development in teaching and learning mathematics. Further discussion of several aspects of changes specifies the needs of developing and using practices associated with sense making, making sense, and connecting mathematics and STEM education for changes.

To support effective mathematics instruction, we propose the use of the Teaching for Robust Understanding (TRU) framework to help characterize powerful learning environments. With the conception of mathematics as “empirical” and a focus on students’ experience, then the focus of instruction should also be changed. We argue that shift is from instruction conceived as “what should the teacher do” to instruction conceived as “what mathematical experiences should students have in order for them to develop into powerful thinkers?” It is the shift in the frame of TRU that makes it so powerful and pertinent for all these proposed changes. Moreover, TRU only uses a small number of actionable dimensions after distilling the literature on teaching for robust or powerful understanding. That makes TRU a practical mechanism for problematizing instruction.

Figure 1 presents the TRU Math framework that identifies five key dimensions along which powerful classroom environments can be characterized: the mathematics; cognitive demand; equitable access; agency, ownership, and identity; and formative assessment. These five dimensions were distilled from an extensive literature review, thus capturing what the literature considers to be essential. They were tested against classroom videotapes and data on student performance, and the results indicated that classrooms that did well on the TRU dimensions produced students who did correspondingly well on tests of mathematical knowledge, thinking, and problem solving (e.g., Schoenfeld 2014 , 2019 ). In brief, the argument regarding the importance of the five dimensions of TRU Math is as follows. First, the quality of the mathematics discussed (dimension 1) is critical. What individual students learn is unlikely to be richer than what they experience in the classroom. Whether individual students’ understanding rises to the level of what is discussed/presented in the classroom depends on other factors, which are captured in the remaining four dimensions. For example, you surely have had the experience, at a lecture, of hearing beautiful content presented, and then not being able to do any of the assigned problems! The remaining four dimensions capture aspects needed to support the development of all students with respect to sense making, making sense, ownership, and feedback loop. Dimension 2: Cognitive Demand. Are students engaged in sense making and making sense? Are they engaged in “productive struggle”? Dimension 3: Equitable Access. Are all students fully engaged with the central content and practices of the domain so that every student can profit from it? Dimension 4: Agency, Ownership, and Identity. Do all students have opportunities to develop idea ownership and mathematical agency? Dimension 5: Formative Assessment. Are students encouraged and supported to share their thinking with a meaningful feedback loop for instructional adjustment and improvement?

The TRU Mathematics Framework: The five dimensions of powerful mathematics classrooms

The first key point about TRU is that students learn more in classrooms that are powerful along the five TRU dimensions. Second, the shift of attention from the teacher to the environment is fundamentally important. The key question is not “Is the teacher doing particular things to support learning?”; instead, it is, “Are students experiencing instruction so that it is conducive to their growth as mathematical thinkers and learners?” Third, the framework is not prescriptive; it respects teacher autonomy. There are many ways to be an excellent teacher. The question is, Does the learning environment created by the teacher provide each student rich opportunities along the five dimensions of the framework? Specifically, in describing the dimensions of powerful instruction, the framework serves to problematize instruction. Asking “how am I doing along each dimension; how can I improve?” can lead to richer instruction without prescribing or imposing a particular style or particular norms on teachers.

Extending to STEM education

Now, we suggest the following. If you teach biology, chemistry, physics, engineering, or any other STEM field, replace “mathematics” in Fig. 1 with your discipline. The first dimension is about rich content and practices in your field. And the remaining four dimensions are about necessary aspects of your students’ classroom engagement with the discipline. Practices associated with sense making, making sense, and STEM education are all be reflected in these five dimensions, with central attention on students’ experience in such classroom environments. Although the TRU framework was originally developed for characterizing effective mathematics classroom environments, it has been carefully framed in a way that is applicable to many different disciplines (Schoenfeld 2014 ). Our discussion above already specified why sense making, making sense, and specific instructional approaches like PjBL and DBL are shared across disciplines in STEM education. Thus, the TRU framework is applicable to other STEM disciplines. The natural analogue of the TRU framework for any field is given in Fig. 2 .

The domain-general version of the TRU framework

Both the San Francisco Unified School District and the Chicago Public Schools adopted the TRU Math framework and found results within mathematics sufficiently promising that they expanded their efforts to all subject areas for professional development and instruction, using the domain-general TRU framework. Work is still in its early stages. Current efforts might be best conceptualized as a laboratory for exploration rather than a promissory note for improvement across all different disciplines. It will take time to accumulate data to show effectiveness. For further information about the domain-general TRU framework and tools for professional development are available at the TRU framework website, https://truframework.org/

Finally, as a framework, TRU is not a set of specific tools or guidelines, although it can be used to guide their development. To help lead our discussion to something more practical, we can use the framework to check and identify aspects that are typically under-emphasized and move them to center stage in order to improve classroom instruction. Specifically, the following is a list of sample under-emphasized norms and practices that can be identified (Schoenfeld in press ).

Establishing a climate of inquiry, in which mathematics is experienced as a discipline of exploration and sense making.

Developing students’ ownership of ideas through the process of developing, sharing, refining, and using ideas; concepts and language can come later.

Focusing on big ideas, and not losing the forest for the trees.

Making student thinking central to classroom discourse.

Ensuring that classroom discourse is respectful and inviting.

Where to start? Begin by problematizing teaching and the nature of learning environments

Here we start by stipulating that STEM disciplines as practiced, are living, breathing fields of inquiry. Knowledge is important; ideas are important; practices are important. The list given above applied to all STEM disciplines, not just mathematics.

The issue, then, is developing teacher capacity to craft environments that have the properties described immediately above. Here we share some thoughts, and the topic itself can well be discussed extensively in another paper. To make changes in teaching, it should start with assessing and changing teaching practice itself (Hiebert and Morris 2012 ). Opening up teachers’ perceptions of teaching practices should not be done by telling teachers what to do!—the same rules of learning apply to teachers as they apply to students. Learning environments for teachers should offer teachers the same opportunities for rich engagement, challenge, equitable access, and ownership as we hope students will experience (Schoenfeld 2015 ). Working together with teachers to study and reflect on their teaching practices in light of the TRU framework, we can help teachers to find out what their students are experiencing and why changes are needed. The framework can also help guide teachers to learn what changes would be needed, and to try out changes to learn how their students’ learning may differ. It is this iterative and concrete process that can hopefully help shift participating teachers’ perceptions of mathematics. Many tools for problematizing teaching are available at the TRU web site (see https://truframework.org/ ). If teachers can work together with a focus on selected lessons like what teachers often do in China, the process would help form a school-based learning community that can contribute to not only participating teachers’ practice change but also their expertise improvement (Huang et al. 2011 ; Li and Huang 2013 ).

As reported before (Li 2018b ), publications in the International Journal of STEM Education show a mix of individual-disciplinary and multidisciplinary education in STEM over the past several years. Although one journal’s publications are limited in its scope of providing a picture about the scholarship development related to mathematics and STEM education, it can allow us to get a sense of related development.

If taking a closer look at the journal’s publications over the past three years from 2016 to 2018, we found that the number of articles published with a clear focus on mathematics is relatively small: three (out of 21) in 2016, six (out of 34) in 2017, and five (out of 56) in 2018. At the same time, we should point out that these publications from 2016 to 2018 seem to reflect a trend, over these three years, of moving toward issues that can go beyond mathematics itself, as what was noted before (Li 2018b ). Specifically, for these three articles published in 2016, they are all about mathematics education at either elementary school (Ding 2016 ; Zhao et al. 2016 ) or university levels (Schoenfeld et al. 2016 ). Out of the six published in 2017, three are on mathematics education (Hagman et al. 2017 ; Keller et al. 2017 ; Ulrich and Wilkins 2017 ) and the other three on either teacher professional development (Borko et al. 2017 ; Jacobs et al. 2017 ) or connection with engineering (Jehopio and Wesonga 2017 ). For the five published in 2018, two are on mathematics education (Beumann and Wegner 2018 ; Wilkins and Norton 2018 ) and the other three have close association with other disciplines in STEM (Blotnicky et al. 2018 ; Hayward and Laursen 2018 ; Nye et al. 2018 ). This trend likely reflects a growing interest, with close connection to mathematics, in both mathematics education community and a broader STEM education community of developing and sharing multidisciplinary and interdisciplinary scholarship.

Availability of data and materials

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Many thanks to Jeffrey E. Froyd for his careful review and detailed comments on an earlier version of this editorial that help improve its readability and clarity. Thanks also go to Marius Jung for his valuable feedback.

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EDITORIAL article

Editorial: psychology and mathematics education.

• 1 Institute of Mathematics and Computing, Ludwigsburg University of Education, Ludwigsburg, Germany
• 2 Institute of Mathematics, Ludwig Maximilian University, Munich, Germany
• 3 Institute of Psychology, University of Education Freiburg, Freiburg, Germany
• 4 Department of Psychology, University of Milano Bicocca, Milano, Italy
• 5 Department of Curriculum, Teaching and Learning at the University of Toronto, Toronto, ON, Canada

Editorial on the Research Topic Psychology and mathematics education

We use numbers and fractions every day, for example when we are doing our shopping or baking a cake. But mathematics is, of course, much more: it is the language of science, or, to use Galileo's words, “the book of Nature is written in mathematical language” ( Galileo, 1623 ) and some mathematical competencies beyond basic arithmetic are required in most professions. Basic mathematics, i.e., elementary arithmetic, elementary geometry and some elements of calculus, is taught in school, not just for everyday life, but as a tool for many different professions. In school, however, mathematics is either “loved” or “hated”, as Hersh and John-Steiner masterfully describe in their book “Loving and Hating Mathematics” ( Hersh and John-Steiner, 2010 ). Research in mathematics education has definitely contributed to reducing school students' hatred of mathematics and this reduction may be seen as one of its many goals.

In contrast with mathematics, the field of mathematics education is strongly interdisciplinary; the closest field to influence it directly is psychology. In fact, mathematics education is consistently shaped by both behavioral and cognitive perspectives, since so many factors—the power of visualizations, the effect of representation formats, but also factors like gender, self-efficacy, etc.—influence and sometimes determine students' performance.

Our aim for this Research Topic and for the collection of papers we are now publishing has thus been to illustrate the relevance of such various psychological perspectives for mathematics education using the contributions of colleagues from around the world. All the contributions we have collected address these interdisciplinary perspectives explicitly or implicitly.

We were surprised by the success of our Research Topic, which was perhaps triggered by the wide range of possible research directions addressed by its general title. The largest section of the papers presents empirical, original research carried out by a wide variety of specialists, with interesting results and suggestions for educators in the classroom. Other papers, which describe psychological features of mathematics education, review factual evidence and the relevant literature. Thus, the collection we present has a descriptive as well as a pragmatic and a prescriptive orientation. It consists of 39 contributions by 109 authors, including 29 original research articles, four brief research reports, three reviews, and three conceptual analyses.

Due to the large number of original research articles, we have decided to arrange them according to different sub-topics, which we list here:

1. Visualizations and representation formats

2. Reasoning, argumentation and biases in connection with mathematics

3. The influence of motor skills

4. Mathematics anxiety as a determining factor

5. Gender and its consequences for students' performance

6. Cultural differences, self-efficacy and consequences for mathematical development

7. Teachers' views, beliefs and culture in connection with teaching mathematics.

We now proceed to describe these sub-topics and cite examples from the corresponding papers.

1. Visualizations and representation formats—their fostering of mathematical intuitions

As we wrote in the overview for this Research Topic, a great inspiration for our endeavor was provided by the work of Herbert Simon with his concept of bounded rationality and its direct descendent, ecological rationality. The ecological rationality perspective allows a novel way of viewing typical aspects of mathematics education, pointing at its fundamental links with cognitive psychology.

We cite here one of Herbert Simon's most famous statements concerning the fundamental importance of representation for solving problems:

“Solving a problem simply means representing it so as to make the solution transparent.” ( Simon, 1996 , p. 132)

According to Simon, a solution becomes transparent if it emerges from the representation of the structure in terms of which the problem has been modeled. The structure results from an effort of adaptation between the endeavors to solve it and the conceptual constructions of the mind. Adopting the phrasing of Gigerenzer and his school, an adaptation is successful if it is ecologically rational (c.f. Gigerenzer et al., 1999 ). Ecological rationality is thus considered a fundamental characteristic of successful representations: it refers to behaviors and thought processes that are adaptive and goal-oriented in the context of the representational environment in which the organism is situated.

Constructing a representation that makes a problem easily solvable provides evolutionary advantages in terms of time resources to agents adopting it ( Martignon et al., 2020 ). The most popular example from mathematics is the representation of numbers based on the decimal number system. This example is so fundamental for Mathematics in every-day life, that it deserves a short digression: the positional representation, as is well known, made its way into the western world several centuries after its inception in India. Dysfunctional systems like Roman numerals had forced people in Europe to outsource their counting and computing: during the late Middle Ages and early Renaissance the Abakists did the numerical operations for businessmen, translating Roman numerals into strings of marbles and working with them on their abaci. Their results were then translated back into Roman numerals. Herbert Simon had wondered about the “theoretical” reasons that make the Hindu-Arabic number system so much more adaptive to our minds than Roman numerals. In fact, he wrote:

“We all believe that arithmetic has become easier since Arabic numerals and place notation replaced Roman numerals, although I know of no theoretical treatment that explains why” ( Simon, 1996 ) .

The studies of ecological rationality and simple heuristics by Gigerenzer and his school have provided good explanations (see Martignon et al., 2020 for a brief synthesis of the main arguments): the mind/brain, as has been empirically demonstrated, is akin to a sequential, lexicographic treatment of features for comparison tasks or classifications. In the case of two numbers in the decimal positional system their coefficients of powers of 10 are the features a comparison is based on. In fact, when having to compare, say 3.456 and 3.461, for instance, we first check that they are equally long. i.e., they have the same number of digits. Then we start checing each digit from left to right, stopping when we find a difference in digits and determine the largest number accordingly. This procedure can be described by a fast-and-fugal tree, as in the illustration at the center of Figure 1 . A fast-and-frugal tree is a simple lexicographic decision tool that proceeds in a sequential way. Its success is the consequence of the non-compensatory weights of its features, as, in the case of numbers, powers of 10 ( Figure 1 , Left).

Figure 1 . Aspects of the decimal representation system. On the left side, three powers of 10 are illustrated by means of cylinders of corresponding heights. They correspond to the weights of the simple non-compensatory linear model, which characterizes our number system. In the center of the figure, a simple, fast, and frugal tree describes the quick yet sequential decision procedure we use for number comparison. On the right side, the reader sees a famous illustration of Arithmetic represented through an allegory by Reisch in the Margarita Philosophica, who contemplates with a benevolent smile the “algarist” (image source: Typ 520.03.736, Houghton Library , Harvard University). The algarist performs computations based on the positional system, while the “abakist”, on the other side of the illustration, uses marbles for computation with his abakus. The work of abakists was fundamental during the beginnings of the Renaissance, when businessmen employed them to perform all their computations. The decimal positional system made arithmetic calculations simple, transparent, and thus accessible for everyone.

Obviously, the decimal positional system has other fundamental advantages for computing.

The illustration in the right panel of Figure 1 represents an illustration from the Margarita Philosophica (by Gregor Reisch) of 1503. Here the allegory of Arithmetica in the center, looks benevolently toward the “Algarist”, on the left side, who works with numbers in the positional system. On the right side, the Abakist, computes with marbles.

The relevance of ecological rationality for our Research Topic is made evident in several of the papers here. In fact, several authors address aspects of visualizations and representations in mathematical contexts which foster the adaptation of the mind to the mathematical contexts involved. Mathematics education treats representations of mathematical situations and entities as a fundamental aspect of didactics in the classroom. Of course, multiple representations of mathematical entities are possible: the advantages of juggling between them have been treated extensively by Dreher et al. (2016) . However, as work in this Research Topic demonstrates, for many types of problem, such as fractions and probabilities, different representations can be mathematically equivalent, yet be far from cognitively equivalent. That is, some representations are more adaptive and advantageous than others because they are aligned to the cognitive systems of human problem-solvers. Representing numbers geometrically has an ecological aspect, and the paper by Kempen et al. demonstrates empirically how figurative numbers enhance numerical understanding.

Approaching the realm of functions in mathematics, for instance, we recall that it would be cumbersome to think about them without the coordinate system introduced in the early seventeenth century by Descartes. Today, much progress has been achieved in dynamic visualizations of functions in coordinate systems. These visualizations allow us to go even further and to greatly improve the understanding of certain aspects of functions. In this connection, we mention the paper by Rolfes et al. in which the authors show that students can learn covariational aspects of the concept of function significantly better with dynamic visualizations than with static representations. However, as they also specify, there seems to be no significant difference in learning with linear or interactive dynamic visualizations.

A natural question related to representations seems to be whether self-generated drawings can always be of help. The answer seems to be that this is not always the case. Self-generated drawings are not always adaptive. In fact, only aspects of them, like their quality, may be of value. Krawitz et al. carried out a replication and elaboration study on the negative effect of self-generated drawings on the number of students' linear overgeneralisation and problem-solving performance. The drawing quality, but the not visual monitoring, affected the number of over generalizations. These results indicate the great importance of the quality of drawing as a strategy for problem-solving.

One mathematical field that has profited immensely from the search for adaptive representations is probability and statistics. Statistical situations concerning data sets profit hugely from dynamic representations. Hood et al. found that dynamic graphs in digital publications can potentially be used for communicating interactions (and other complex relationships) effectively. They are not, however, a panacea for people's challenges in understanding complicated data and more work is needed to take effective advantage of opportunities in digital data presentations.

Probabilistic inference has also profited greatly from adaptive representations: we recall here that it has been a milestone in the field of probabilistic reasoning to discover how certain representations and visualizations foster Bayesian reasoning ( Gigerenzer and Hoffrage, 1995 ) while others hamper it.

The paper by Eichler et al. is especially inspiring, quite in the spirit of ecological rationality, because of the empirically supported claim that people's strategies for solving Bayesian tasks are triggered by corresponding representations, exactly in the sense of Herbert Simon.

Another relevant finding in this field is that the characteristic of visualization making the nested-sets structure of a Bayesian situation transparent has a facilitating effect on people's Bayesian reasoning ( Macchi, 2003 ). Trees and double trees with nodes representing natural frequencies have been proven very effective in this context. A practical extension of the tree format with natural frequencies is the so-called “frequency net” proposed by Binder et al. . Here the disposition allows for sequential treatment in four directions but also provides a view of all relevant frequencies of conjunctions at a glance.

Another question that arises in the context of natural frequencies is whether younger children may exhibit forms of Bayesian reasoning when presented with simple formats of information. The paper by Till et al. presents an intervention study showing that primary school children at the age of nine to ten can understand probabilities and solve Bayesian problems using natural frequency formats. Hence, natural frequencies appear to be a suitable representation for grasping probabilities at an early stage and thus might support understanding more abstract contexts in higher grades.

It is a crucial result, that people may perform correct Bayesian inferences without recurring at all to numerical formats of information: Leuders et al. analyzed how people update their hypotheses based on uncertain evidence (e.g., teachers' updating their assumptions based on students' solutions), when they only have access to non-numerical information. They showed that people need strong support to apply a rational Bayesian strategy and otherwise resort to biased strategies for processing information–analogous to the strategies found in numerical settings.

The review by Neth et al. solves several well-known problems by representing them more transparently. Politely phrased as re-framing representational effects and suggesting a change in perspective, a more apt description of their achievement is to effectively put an end to academic debates and scientific practices that are sustained by obscure abstractions and idiosyncratic terminologies. Encouraged by editors and reviewers, the authors resisted the temptation to distribute their insights across several articles. The published product is long and detailed, but rewards its readers by seeing how numerous scientific puzzles and their solutions are alternative perspectives on a shared representational construct.

2. Reasoning, argumentation, and biases in connection with mathematics

Reasoning is the basis not only of mathematical thought, but of critical thinking in general. In an era of fake news and propaganda proliferated by social media, critical thinking acquires even greater relevance and has been declared one of the competencies of the 21st century by the OECD.

The work by Macchi et al. is represented in this Research Topic by two papers related to the improvement of logical and mathematical performance ( Bagassi et al. ; Bagassi and Macchi ) of children through a pragmatic approach, on the one hand, and on the possibility of facilitating problem-solving by viewing it as the overcoming of misunderstandings, on the other.

Another important field in the realm of reasoning deals with how humans cope with syllogisms. These used to be the basis of reasoning in the traditional approach to rigorous thinking. Syllogisms and how humans handle them have been a matter of research through the centuries. They are the essence of classical logic. But in heuristic decision-making, which definitely takes place when students approach problem-solving in mathematics, less “classical” logics may play an important role. In Chapter 3 of “The Science of the Artificial” Simon recommends the use of multiple logics and in Chapter 5, he explains how

“[m]ultiple logics may become necessary when approaching heuristic decision making” ( Simon, 1996 ) .

The paper by Vargas et al. introduces variations on syllogistic experimental tasks by (1) reshaping the pragmatics of the communication situations faced along the dimension of cooperative vs. adversarial attitudes and (2) rendering explicit the construction of counter-examples. It presents evidence on a significant switch in participants' performance and the strategies they employ while reasoning.

The question on how to foster argumentation skills deals with the design of adequate learning environments and can be influenced, as Sommerhoff et al. , show, by whether a sequential or a concurrent instructional approach is used in the classroom. Their paper highlights that sequential and concurrent approaches are both effective in supporting the resources underlying mathematical argumentation and proof skills; however, the concurrent approach can have slightly better effects on mathematical argumentation skills, especially in the case of weaker students.

The paper with most views so far is the one by Bruckmaier et al. on the cognitive illusions studied by Kahneman and Tversky, which released a flurry of fundamental investigations on human reasoning. They provide a unified framework for the basic treatment of the classical teasers analyzed by the school associated with Tversky and Kahneman.

Playing games may sharpen reasoning and lead to concept formation. Özel et al. report having worked with children from 8 to 10 years old, who played different versions of a code-breaking game in guided game-based instruction. After this process, a post-test showed that children were remarkably sensitive to key principles in their mathematical reasoning when dealing with information. This adds to evidence that game-based instruction can be a powerful tool for making mathematics moreintuitive.

Through playing games the phenomenon of “help-seeking”, which is so relevant in the context of learning and problem-solving, can be efficiently analyzed, as Taylor et al. . They show that help-seeking is not correlated with a real need for help. The important paper by Jonsson et al. is devoted to how mathematical understanding can be fostered by creativity and cognitive proficiency.

Pursuing another line, in their paper, Reinhold et al. empirically analyzed the biases—the natural number bias in particular—that make working with fractions difficult, especially for low-achieving students.

The paper by Yang et al. analyzes how concept formation, and understanding of categories fosters analytical and mathematical competencies.

3. The influence of individual learner characteristics

3.1. the influence of motor skills.

The paper by Fischer et al. on the surprising effect of fine motor skills to mathematical insight is particularly relevant. This is the only paper in the whole Research Topic collection that treats connections between movement and mathematical competency.

3.2. Mathematics anxiety as a determining factor

Three papers in our collection treat mathematics anxiety in connection with mathematics in the context of school ( Maldonado Moscoso et al. ; Primi et al. ) Among the psychological factors that trigger impairments in mathematics, mathematics anxiety has been suggested to play a key role. It has been defined as feelings of apprehension and increased physiological reactivity when individuals have to manipulate numbers, solve mathematical problems or when they are exposed to an evaluative situation connected with mathematics. Mathematics anxiety involves psychological arousal, negative cognitions, escape and/or avoidance behaviors and, when the individual cannot avoid the situation, performance deficits. It is described as a multidimensional construct that is related to, but distinct from, other forms of anxiety, such as trait, social or test anxiety. Mathematics anxiety has been shown to hinder mathematics performance. This phenomenon is very common not just among school students. Adults suffer from it as well.

The paper by Primi et al. describes a new scale for measuring it already in young children.

The paper by Moliner et al. describes a how peer tutoring among school students can become a factor thast reduces math anxiety.

3.3. Gender and its consequences on students' performance

In the paper by Uclés et al. on “ Gender Differences in Visuospatial Abilities and Complex Mathematical Problem Solving ” the authors provide empirical evidence that students with the ability to solve complex mathematical problems exhibit stronger spatial skills. It also shows that boys and girls present similar spatial abilities, and that there is no significant interaction between the ability to solve complex problems and gender.

3.4. Cultural differences, self-efficacy, and consequences for mathematical development

The role of self-efficacy for mathematical development has become ever more evident since the discoveries of Bandura in the second half of the 20th century (Bandura, 1997). The paper by Siefer et al. reveals that including written data (notes) and non-verbal data (gestures and actions) leads to a more accurate analysis of self-explanations than an analysis solely based on verbal data. This influence is even stronger for the categorization of self-explanations as “adequate” or “inadequate”.

In the paper by Siefer et al. the authors explore the potential multi-dimensionality of self-efficacy focused on three task characteristics:

(1) the representational format,

(2) embedding in a real-life context,

(3) the required operation.

The paper highlights the fact that even within a specific content domain students' self-efficacy can and should be considered a multi-dimensional construct.

The paper by Salle describes how self-explanation, gestures and notes trigger self-assurance and self-efficacy.

The paper by Zakariya addresses causal relationships between the previous and current mathematics performance of undergraduate students.

Pursuing another line, in their paper Meng et al. deal with the thought-provoking topic of the influence of specific cultural phenomena in connection with self-efficacy. In fact, it treats cultural aspects which are apparently more specific to upbringing in China, and shows that these aspects have an influence on factors analogous to self-efficacy when dealing with mathematical tasks.

Findings by Wang and Sperling revealed that those interventions grounded in metacognition-oriented theories and those interventions that targeted multiple strategies including cognitive, metacognitive, and motivational, tended to yield effective increases in both mathematics achievement and self-regulated learning.

4. Teachers' views, beliefs, and culture in connection with teaching mathematics

The role of teachers, their training and their views in the discussion on mathematics education is treated by two papers in the collection.

The paper by Patterson et al. shows the positive effects of special units of teacher training on the performance of students. The findings indicate that students profit from their teacher's participation in special training interventions.

The paper by Tanas et al. addresses how the views of teachers on technology and their perceived ease of technology use affects their use of technology in the mathematics classroom. These new tools provide education with many new opportunities, but their application often meets with a variety of difficulties. Many of those difficulties are general and appear across different areas of technology use. The paper confirms that perceived usefulness has a stronger direct impact on technology use and that user friendly technology increases use.

Besides research articles, our Research Topic contains two research reports, one systematic review, three conceptual analyses, two review articles and ends with an opinion.

5. Further contributions: research reports, systematic review, conceptual analyses, review article and opinion

5.1. research reports.

Sturm et al. report on an empirical study on how the attitudes and beliefs of young students correlate with their problem-solving performance. They also claim that this correlation can be affected by student participation in a training programme.

In his brief research report Rolfes treats the interpretations of pictorial charts involving differences in areas and differences in volumes, as understood by readers of popular reports. His claim is that readers do not seem to interpret two-dimensional pictures of three-dimensional objects spatially.

5.2. Systematic review

The paper Wang et al. analyzes findings on social-cognitive self-regulated learning and discuss implications for good practices in the classroom.

5.3. Conceptual analyses

In Bertram the author examines future directions in research on digital games in mathematics and computer science education. She highlights the importance of a sound psychological foundation for the development of learning games and the need for interdisciplinary research projects and randomized controlled experimental designs to evaluate the effectiveness of games and game features.

The analysis of Kramer on iconic mathematics is extremely pertinent to the realm of ecological rationality in the context of mathematics education. It reminds us of the necessity of reviewing representations of mathematical entities and processes that produce features that are appealing to the mind/body and thus become easy to grasp.

The analysis by Kurdoglu et al. describes one step in the conceptual view on uncertainty and is therefore relevant for the realm of decision-making, having connotations (implications?) that are meaningful for the teaching of probability. It deals with complete uncertainty, a situation that goes beyond mathematically structured scenarios. Under very high levels of uncertainty, decision-makers rely on heuristics to no avail. Kurdoglu et al. posit that eristic reasoning (i.e. self-serving inferences for hedonic pursuits), rather than heuristic reasoning, is adaptive when uncertainty is extreme. They explain how decision-makers can benefit from heuristic vs. eristic reasoning under different levels of uncertainty. Although the authors establish no immediate connection with mathematics in the classroom, their approach is novel and clearly relevant.

5.4. Review article

In Barrocas et al. the authors mostly review the large collection of their own discoveries concerning finger-counting as related to later arithmetic abilities. Their report fits in perfectly with the intention of the Research Topic.

5.5. Opinion

As a final perspective from the Research Topic, we cite here the paper by Simplicio et al. Here the authors insist that results from research should find their way into classrooms, but they see the need for more integration of different perspectives and fruitful collaborations between researchers of different disciplines and educators. Only then, they claim, are there real chances of transferring results from basic research into educational practice. Yet, they also point out that, as has been said by Minshall (2009) , “knowledge transfer is a ‘contact sport'; it works best when people meet to exchange ideas, … and spot new opportunities”.

We definitely agree with their statements and conclude the description of our Research Topic with the hope that more steps toward the integration of research on the psychology and even on the neuroscience of mathematics acquisition are soon taken at all levels of research and implementation.

Author contributions

LMar prepared the draft of the editorial, which was revised and completed by KB and KL. KB and KL had previously sorted all papers in the Research Topic and classified them in meaningful categories, as listed in the editorial. LMac and GH fully agreed with the editorial in its present form.

Acknowledgments

We want to thank Dr. Peter Dines for editing some of the texts published in this Research Topic. We also thank Stela Moldovan for her help with figures in some of the articles in this collection.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher's note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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PubMed Abstract | Google Scholar

Keywords: mathematics, psychology, pedagogy, classroom, learning

Citation: Martignon L, Binder K, Loibl K, Macchi L and Hanna G (2023) Editorial: Psychology and mathematics education. Front. Psychol. 14:1243419. doi: 10.3389/fpsyg.2023.1243419

Received: 20 June 2023; Accepted: 27 June 2023; Published: 27 July 2023.

Edited and reviewed by: Bernhard Hommel , University Hospital Carl Gustav Carus, Germany

Copyright © 2023 Martignon, Binder, Loibl, Macchi and Hanna. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY) . The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Laura Martignon, martignon@ph-ludwigsburg.de ; Karin Binder, karin.binder@lmu.de ; Katharina Loibl, katharina.loibl@ph-freiburg.de ; Laura Macchi, laura.macchi@unimib.it ; Gila Hanna, gila.hanna@utoronto.ca

Disclaimer: All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.

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The transition from school to university in mathematics education research: new trends and ideas from a systematic literature review

Pietro di martino.

1 Department of Mathematics, Università Di Pisa, Largo B. Pontecorvo 5, 56125 Pisa, Italy

Francesca Gregorio

2 Université de Paris, Univ. Paris Est Creteil, CY Cergy Paris Université, Univ. Lille, UNIROUEN, LDAR, F-75013 Paris, France

3 Haute École Pédagogique du Canton de Vaud, Lausanne, Switzerland

Paola Iannone

4 Department of Mathematics Education, Loughborough University, Loughborough, LE11 3TU UK

Associated Data

The datasets analysed during the current study are available from the correspondent author upon reasonable request. These datasets were derived from the following public domain resources: Google Scholar and ERIC.

Investigating the transition between educational levels is one of the main themes for the future of mathematics education. In particular, the transition from secondary school to STEM degrees is problematic for the widespread students’ difficulties and significant for the implications that it has on students’ futures. Knowing and understanding the past is key to imagine the future of a research field. For this reason, this paper reports a systematic review of the literature on the secondary-tertiary transition in Mathematics Education from 2008 to 2021. We constructed two corpuses: one from the proceedings of three international conferences in mathematics education (PME, ICME, and INDRUM) and the other from peer reviewed research papers and book chapters returned by the databases ERIC and Google Scholar. A clear evolution in perspectives since 2008 emerges from the analysis of the two corpuses: the research focus changed from a purely cognitive to a more holistic one, including socio-cultural and — to a lesser extent — affective issues. To this end, a variety of research methods were used, and specific theoretical models were developed in the considered papers. The analysis also highlights a worrisome trend of underrepresentation: very little research comes from large geographical areas such as South America or Africa. We argue that this gap in representation is problematic as research on secondary tertiary transition concerns also consideration of socio-cultural and contextual factors.

Introduction

The study of the transition from secondary school to university mathematics — also referred to as the secondary-tertiary transition (STT) — has a long tradition in mathematics education research. Seminal studies were conducted by Tall since the 1980s, investigating significant cognitive discontinuities in STT (Tall, 1991 ; Tall & Vinner, 1981 ). Tall’s studies inspired the development of research specifically focused on students’ difficulties in STT. At the end of the past millennium, De Guzman et al. ( 1998 ) summarised the main findings of this field of research and identified three categories related to students’ difficulties in STT: epistemological-cognitive, sociological-cultural, and didactical.

A quarter of a century later, students’ difficulties in STT are still a significant issue for modern society. The high dropout of undergraduate students in STEM subjects reported in several countries (Rach & Heinze, 2017 ) is problematic for at least two reasons. The first relates to the need of advanced mathematics competencies for economies to flourish (Adkins & Noyes, 2016 ). The second is linked to equity, given the opportunities afforded by STEM degrees (science, technology, engineering, and mathematics) in terms of social mobility and future earnings (see Higher Education Statistics Agency, 2018 , for the UK context). Indeed, the European Mathematical Society recently conducted a survey amongst mathematicians to collect information to devise national and international actions that may help reduce students’ dropout during the STT (Koichu & Pinto, 2019 ). The authors underline that the STT is a complex and multi-faceted process, provoking frustration in first-year students, as well as in lecturers.

The difficulties that students encounter when moving from school mathematics to the mathematics in a STEM degree still represent a strange story, not fully explained, that involves and overwhelms even students considered excellent in mathematics during their school experience (Di Martino & Gregorio, 2019 ). Understanding this complex and apparently unexplained phenomenon requires many theoretical approaches and, as Schoenfeld ( 2000 ) notices, ‘findings are rarely definitive; they are usually suggestive. Evidence is not on the order of proof but is cumulative’ (p. 648). This cumulative nature of findings suggests the relevance of literature reviews on selected topics: they allow researchers to identify trends and provide insights for possible developments of the research field (Pan, 2016 ).

An influential review of research on STT was conducted by Gueudet in 2008. Gueudet highlights three types of transitions involving individual, social, and institutional factors: transition in ways of thinking, transition to proof and the technical language of mathematics, and institutional transition related to changes in the didactical contract. Gueudet’s review also confirms that up to that time, the tertiary transition in mathematics was mainly studied through cognitive and epistemological lenses, even though socio-cultural and affective issues had since assumed an important role in mathematics education (Lerman, 2000 ). Gueudet ( 2008 ) concludes that there is ‘the need for further research, and for teaching designs grounded in their findings’ (p. 252). Her call was heard by the mathematics education research community, and research on STT has since adopted new theoretical perspectives, discussed new results, and highlighted new lines of interest. Indeed, the transitions to higher education are one of the themes emerging from a recent international survey of researchers in mathematics education answering the question: ‘on what themes should research in mathematics education focus in the coming decade?’ (Bakker et al., 2021 , p. 2).

Recently, Bergsten and Jablonka ( 2019 ) traced the development of studies on STT in terms of the theoretical approaches taken. However, a systematic review of the research on STT in the last 15 years is still missing. To fill this gap, we analyse the research on STT in mathematics education since 2008.

We developed our systematic review to answer the following research questions related to the period 2008–2021:

RQ1: Which are the methods and the specific theoretical frameworks used to approach STT?

RQ2: What are the main themes of STT research emerging from an analysis of published research?

RQ3: What is the geographical distribution of STT research? Do we know enough about diverse cultural context coming from a wide range of countries?

Methodology

The development of a systematic review involves several aspects: in this section, we briefly describe the underlying choices of our review and their reasons, as well as its constraints and the possibility of reproducibility.

The development of a literature review is a systematic process (Pan, 2016 ), involving three steps needed to identify the corpus of interest (Green et al., 2006 ): selection of the topic, definition of the sources of information, and definition of the selection criteria employed (Table ​ (Table1 1 ).

The three steps for a literature review (Green et al., 2006 )

The relevance of the topic and the reasons for the review (Step 1) were discussed in the introduction of this paper.

The description of Steps 2 and 3 is of crucial importance for a systematic literature review: it allows other scholars to replicate the selection of the corpus by following the given criteria. Related to these two steps, we differentiate between the two searches we conducted: one focused on conference proceedings, the other on refereed journal papers and book chapters. For both searches, the starting year was 2008 and the ending was October 2021. However, the database and, in part, the automatic search criteria were different between the two searches.

According to our focus, we included in this review studies describing the STT to university mathematics of students enrolled in STEM degrees. These studies may involve pre-service secondary mathematics teachers in those countries where they are taught mathematics in the same lectures of students enrolled in a STEM degree. However, we consider the STT for pre-service teachers (including teachers for primary and lower secondary education) as a special case of transition to tertiary mathematics deserving, in our view, a separate discussion, not addressed in this paper.

We developed a specific search for conference proceedings since international conferences are likely to be a forum to share ideas which are then developed in journal papers or book chapters, and results emerging from conference proceedings often anticipate trends in the development of research themes.

The first choice we made was related to which conference’s proceedings to include in the review. We decided to include two of the most popular (by number of participants) international conferences in mathematics education: the International Congress on Mathematical Education (ICME) and the Conference of the International Group for the Psychology of Mathematics Education (PME). For the likely relevance to our focus, we also included the conferences organised by the International Network for Didactic Research in University Mathematics (INDRUM).

Since our third research question concerned the representation of geographical regions in mathematics education research, one of the main reasons for the selection of ICME, PME, and INDRUM was that the international program committees of these conferences are opened to members coming from all over the world. Using this criterion, we did not include in our search other potentially relevant conferences such as CERME (Congress of the European society for Research in Mathematics Education), RUME (the Research on Undergraduate Mathematics Education of the Mathematical Association of America), PME-NA (North American Chapter of the International Group for the Psychology of Mathematics Education), and other PME regional conferences.

The database for the construction of this first corpus therefore consisted of the published proceedings of ICME, PME, and INDRUM since 2008. We identified all the contributions including the key term ‘transition’ in the title, abstract, or text (automatic search criterion). We then considered only contributions where the occurrence was related to the secondary-tertiary transition and with STT as the significant focus of the paper (not mentioned as a secondary topic).

For the construction of the second corpus (which includes journal papers and book chapters), we first considered what search criteria to use, as this is a crucial step for every literature review. Gusenbauer and Haddaway ( 2020 ) recognised three main quality criteria for literature searches: completeness (identifying all the most significant resources about the topic), transparency, and reproducibility. Through the analysis of the systematic search qualities of 28 academic search engines, the authors suggested to conduct literature reviews using at least two different search engines, according to their quality with respect the three criteria mentioned above. Regarding Google Scholar, Gusenbauer and Haddaway ( 2020 ) observe that this search engine has several limitations in terms of transparency and reproducibility of searches; however, ‘it is considered a suitable supplementary source of evidence (including on grey literature)’ (p. 196) for systematic literature reviews.

Based on the above considerations, we used two databases: ERIC (Education Resources Information Center, https://eric.ed.gov ), the online database of education research promoted by the US Department of Education, and Google Scholar ( https://scholar.google.com ). The use of Google Scholar in our review appeared important to bring to the fore contributions by researchers who may not have access to the main journals and conferences of our community (for example due to the high cost of conference fees and travel). We also followed Haddaway et al.’s ( 2015 ) advice to focus on the first 200 to 300 results returned by Google Scholar for systematic reviews as supplementary source of evidence.

Since the databases were not specific to mathematics education, we used a strict automatic search criterion. We used ‘transition AND mathematics AND school AND university’ as search terms. We observed a posteriori that the inclusion of the word ‘university’ in the Google Scholar search was not useful: Google Scholar searches the entire document — unlike ERIC which searches only the title, the abstract, and the keywords — and therefore the word university featured in all documents where the names of authors and their institutions were included. At the end of this first phase, following the recommendation of Haddaway et al. ( 2015 ), we considered the corpus produced by the union of the 252 results produced from ERIC and the first 300 results by relevance from Google Scholar taking care to eliminate repetition of entries between the two sets of results since, as expected, there was a significant overlap.

The selection criteria used to include a study identified using the automatic search criteria were the same for both databases. Assuming the definition of STT given in the Encyclopaedia of Mathematics Education as ‘the process experienced by students leaving secondary school and entering different kinds of postsecondary institutions: universities, engineering schools, etc.’ (Gueudet & Thomas, 2020 , p. 762), we selected the contributions describing a transitional process , considering a ‘before’ and an ‘after’ this process. This means we excluded studies where the focus was solely on one of the stages of this transition (e.g., students’ experiences with proof at the start of a mathematics degree) and did not consider the ‘before’ and ‘after’ of the STT. Because of this choice, papers focusing on the design of transition to proof courses or university preparatory classes were not included in our review.

A margin of subjectivity in the application of the selection criteria clearly exists in this process. However, the first two authors developed an investigator triangulation to balance individual biases (Mok & Clarke, 2015 ). The two authors applied independently the selection criteria, sharing their outcome. They then discussed the cases of disagreement until a full agreement for the selection of the final corpuses was reached.

The result of this process was the two final corpuses: the first one including 59 reports presented in the three selected conferences and the second one including 55 papers from peer reviewed journals and books.

Once we selected the corpus, aligned with the research questions of the systematic review, each paper was categorised according to: year of publication, type of study (theoretical, empirical, didactical design) and — if empirical — methods (quantitative, qualitative, mixed), instruments, sample, context 1 (country where the research was developed, how many and which schools, university, or texts were involved in the research), theoretical framework(s), research question(s), and focus (cognitive, affective, socio-cultural). This classification allowed for multiple labels: for example, a paper could be classified with cognitive and socio-cultural foci if explicit references to both cognitive and socio-cultural issues were made in the theoretical framework or in the research question(s).

In addition to the described classification, we also noted the main findings of the study for each paper.

The recognition of the main research themes was based on a coding process related to the research questions aimed to recognise frequencies and patterns (Cohen et al., 2007 ). Initially, a very specific label was assigned to each research question of the paper considered, then — following Miles et al. ( 2003 ) — the labels evolved during the process to describe wider categories capable to bring together several similar focuses (an example of this evolution is summarised in Fig.  1 ).

Example of the coding process for the research questions

Results and discussion

Stt in conference proceedings from 2008 to 2021.

As previously described, we considered in our review three international conferences: ICME, INDRUM, and PME.

ICME congresses

ICME is a large international congress held every 4 years involving researchers worldwide. Four ICME congresses took place since 2008: the proceedings of ICME-11 (Monterrey, Mexico 2008) and of ICME-14 (Shanghai, China, 2021, online conference) are not currently available, while those of ICME-12 (Seoul, Korea, 2012) and ICME-13 (Hamburg, Germany, 2016) are published as open access Springer books. According to the spirit and nature of the activities of the Topic Study Groups (TSG), the existing proceedings consist of abstracts of the discussions developed in the sessions rather than a collection of research reports. For this reason, the discussion which follows is short and mainly descriptive.

The evolution of the presence of STT in the discussion of the ICME community can be recognised by the analysis of the proceedings, scientific programs, and books related to developed activities. In what follows, we detail the evolution of research on STT within these four ICME congresses.

Although one of the TSGs of ICME-11 was about university mathematics, STT was not mentioned in the description of its scientific program.

The situation was different 4 years later at ICME-12. The transition ‘problem’ was explicitly mentioned in the call for papers and in the description of TSG 2: ‘Mathematics education at the tertiary level and access to tertiary level’. Within this TSG, we found five contributions on STT by authors from five different continents (Brazil, Canada, South Korea, South Africa, Sweden). Four out of these five contributions analysed students’ difficulties with specific mathematics topic (matrices, axiomatic method, calculus) with a predominantly cognitive approach. There was also a survey team dedicated to STT, and the outcomes of this work were findings of an international survey of 79 mathematics lecturers from 21 countries. Findings pointed to ‘a multi-faceted web of cognitive, curricular and pedagogical issues’ (Thomas et al., 2015 , p. 278) related to STT, also depending on the institutional context. Despite the evident differences between institutions, and therefore the emergence of socio-cultural aspects, the lens of this survey is still strongly cognitive, without specific reference to affective issue. However, the final mention of the emerging interest for how students experience their first encounters with advanced mathematical topics represented a first step towards the inclusion of affective issues in the discussion.

At ICME-13, STT was a recurrent topic: seven contributions presented in TSG2: ‘Mathematics education at the tertiary level’ were focused on STT. In addition, transition issues were the topic of a discussion group and of the plenary panel. The latter inspired an ICME monograph where the main results of the research and ‘the blind spots that remain unquestioned’ (Gueudet et al., 2016 , p. 19) were discussed. The summary of the results included considerations about institutional differences (for example in class size or equipment) and pedagogical and cognitive issues, but no mention of affect was made.

At ICME-14, four contributions in TSG2 were focused on STT. In particular, Pinto, Gamlieli, and Koichu discussed the results of an international survey conducted among 310 tertiary mathematics lecturers from 30 countries. This confirms the potential of ICME Congresses to foster discussion between researchers from different contexts. The wide geographical reach of the ICME Congresses is particularly relevant for research on STT considering that international comparisons are still rare elsewhere in our field.

To conclude, the diachronic analysis of the proceedings and scientific programmes of these four ICME editions shows an increasing interest for STT. However, while a lack of attention to affective issues clearly emerges, the discussion about socio-cultural issues is promoted through the involvement of researchers from a variety of countries, with a considerable participation of researchers from countries currently underrepresented in mathematics education.

INDRUM conferences

The growing interest for undergraduate mathematics education in our field is also evidenced by the foundation of the International Network for Didactic Research in University Mathematics (INDRUM) in 2015. Since 2016, this network holds a biannual conference with open access to online proceedings. To date, the following conferences have been held: INDRUM 2016 (Montpellier, France, Nardi et al., 2016 ), INDRUM 2018 (Kristiansand, Norway, Durand-Guerrier et al., 2018 ), and INDRUM 2020 (virtually held in Bizerte, Tunisia, Hausberger et al., 2020 ). The book Research and Development in University Mathematics Education (Durand-Guerrier et al., 2021 ) provides a detailed overview of the discussion on topics in the 2016 and 2018 INDRUM conferences. Focusing on STT and according to our criteria, we identified 18 contributions relevant to our review (Table ​ (Table2 2 ).

INDRUM proceedings: the complete list of selected publications

We first notice a clear prevalence (78%) of contributions by European researchers, probably related to the origin of the INDRUM group. Within this European prevalence, there is a strong presence of French authors who account for 39% of the reports on STT. The French influence on the INDRUM community is not only evident in terms of participation: it is also evident from the theoretical frameworks used in the research reports included in the proceedings of these conferences.

Three out of the four theoretical papers on STT used the anthropological theory of the didactic (ADT) (Chevallard, 1992 ) for identifying praxeologies in STT. The fourth, the paper by Artigue ( 2016 ), discussed the challenges of the research in mathematics education at the tertiary level. As for the empirical papers, four reported and interpreted students’ difficulties in STT using the lens of the anthropological theory of didactic. The common hypothesis of these studies was that several phenomena in STT can be interpreted as institutional issues, determined by the strong discontinuity between school and university praxeologies. This approach was extensively described in Gueudet and Pepin ( 2017 ).

Using this framework, Winsløw ( 2008 ) introduced a model with repeated cycles of two transitions to university mathematical praxeologies: the first is related to the need for students to extend their praxeologies considering the role of theory in mathematics. The second transition takes place when the elements of the theory become objects; students need to work with autonomously: in this case, the emergence of new objects can require further transitions.

Other important theoretical frameworks developed by French researchers and adopted in papers included in the INDRUM proceedings for studying STT are the theory of didactical situations (Brousseau, 2002 ) — that Bloch and Gibel ( 2016 ) used for developing a tool for modelling students’ reasoning processes — and the Instrumental Approach (Rabardel, 2002 ), used by Gueudet and Pepin ( 2016 ) for theorising the use of technology in STT.

The INDRUM papers pay great attention to cognitive and epistemological issues in STT; several of the authors’ theoretical approaches fall ‘under the umbrella term advanced mathematical thinking’ (Hochmuth et al., 2021 , p. 195) and are often situated within APOS (action-process-object-schema) theory (Dubinsky, 1991 ). However, the use of the ATD also shows some consideration of institutional and social perspectives, but only one paper (Gueudet & Pepin, 2016 ) analysed and compared two case studies from different national contexts (France and UK).

Only two papers in our INDRUM sample have an affective focus. Quéré ( 2016 ) analysed the different forms of autonomy required in the move from secondary to tertiary education in mathematics, also discussing the role of theories of success and expectations in students’ difficulties in STT. Geisler and Rolka ( 2018b ) discussed the relationship between students’ success in mathematics in their first university year and some affective variables (self-concept, interest, view of mathematics, basic needs, self-efficacy).

PME conferences

PME conferences have annual frequency; therefore, we analysed 13 editions: from Mexico 2008 to the edition virtually held in Thailand in 2021. 2 PME includes several types of contributions and activities: research report, short oral, working group, discussion group, poster, and plenary. In the following, we will focus on the 26 selected research reports (Table ​ (Table3 3 ).

PME proceedings: the complete list of selected publications

Table ​ Table3 3 illustrates both a small but steadily growing presence of empirical research reports on STT since 2009, and the absence of theoretical reports. The latter may also result from the strict length limitation of the contributions. The analysis of representation by country returns a clear picture: 23 out of 26 reports (88%) were developed in a European country by European researchers (one of these papers discusses a comparison between France and Brazil, also involving a Brazilian author). These data highlight an issue of underrepresentation in the PME conferences concerning the discussion about STT.

Moreover, 16 (62%) out of 26 reports are by researchers from German universities. This regional predominance is also reflected in the research methodologies: the quantitative approach is prevalent in these empirical studies (62.5% versus 29% for the qualitative approach and 8% for the mixed one), and samples are usually large, notwithstanding the presence of three interesting case studies. Several reports focus on the identification of significant correlations between academic success/failure and other variables. This aim is related to a clear definition of the variables involved, to the development of instruments for measuring these variables and the students’ success or failure, and to the adoption of statistical models (the more frequently adopted being the Rasch model).

Considering the 7-year period 2008–2014, we found 9 research reports about STT measuring students’ preparedness in mathematics and students’ learning strategies at the beginning of their university experience. These studies aim to determine whether and to what extent mathematics dropout can be predicted by the analysis of cognitive performance at the beginning of university (Halverscheid & Pustelnik, 2013 ). In first period, two exceptions are represented by the paper by Di Martino and Maracci ( 2009 ), stressing the need to go beyond a purely cognitive approach also in research about STT, and the paper by Dias et al. ( 2010 ), that, within the ATD framework, developed a socio-cultural comparison between Brazil and France regarding STT, considering differences in educational systems and in educational cultures.

In the subsequent 7-year period (2015–2021), we found 17 research reports about STT. The clear difference in the number of reports between the period 2008–2014 and the period 2015–2021 highlights the growing interest for STT in the mathematics education community. The analysis of the frameworks and the research questions used in the research reports in the period 2015–2021 shows a greater consideration for affective and socio-cultural constructs in STT research. Ufer ( 2015 ) analysed the relationship between students’ motivations to choose a mathematical programme and their success, Jeschke et al. ( 2016 ) attempted to measure students’ ‘academic buoyancy’ and its role in early dropout, and Kouvela et al. ( 2017 ) studied the students’ identities as mathematics learners and the influence that messages given by their lecturers have on the development of these identities.

The 2018 edition of PME in Umeå deserves special attention because of its variety of approaches to STT. Bampili et al. ( 2018 ) analysed how social and institutional issues shape the development of a new identity for first year mathematics students by adopting the theoretical framework of Communities of Practice (Wenger, 1998 ). Meehan et al. ( 2018 ) studied how STT affects high-achieving students’ ‘sense of belonging to math’, that is related ‘to whether one feels a member of a mathematical community and feels valued and accepted by that community’ (p. 371). Di Martino and Gregorio ( 2018 ) introduced and analysed the so called ‘first-time phenomenon’, that is the cognitive and emotional reactions of (successful) students to the first experience of failure in mathematics. Geisler and Rolka ( 2018b ) discussed the relationship between affective variables (such as self-concept and beliefs about the nature of mathematics) and academic procrastination.

Therefore, the diachronic analysis of the research reports on STT in the 13 PME conferences shows a clear evolution toward a more holistic view of STT that increasingly includes socio-cultural and affective considerations.

STT in journal papers and book chapters

Following the selection criteria described in the method section, we obtained a corpus of 55 papers (Table ​ (Table4). 4 ). As hypothesised, we found several papers extending the ideas discussed by the authors in the international conferences presented in the previous section; therefore, the analysis of this corpus is particularly significant to gain a clear picture of the state of the art of STT research.

Journals and books: the complete list of selected publications

The data about the university affiliation of the authors and the geographical contexts where the studies were developed underlines again an issue of representation: the largest number of contributions comes from authors working in European universities reporting studies developed in Europe (63%). Research concerning other regions, such as South America, Asia, Africa, is almost completely absent in the identified corpus.

Concerning the analysis of the methods in the empirical studies, we found a balance between qualitative (43%) and quantitative (37%) approaches (20% used a mixed approach), as well as a variety of instruments and targets (summarised in Table ​ Table5 5 ).

Targets and instruments

In particular, the analysis of National School Standards or textbooks allows us to compare different contexts, leading to the awareness that the systems in school and university are culturally embedded (Frank & Thompson, 2021 ). To this aim, Vollstedt et al. ( 2014 ) elaborated a framework for analysing and comparing mathematics textbooks.

The analysis of the research methods highlights two related issues: most data are collected through online surveys (but the work by Geisler and Rolka ( 2021 ) represents a recent exception) and the sample is almost exclusively a sample of volunteers. This latter aspect, recurrent in the mathematics education research, appears to be particularly relevant in STT research since it involves adults, asking them to report an event often perceived as a personal failure.

Theoretical models

The 55 selected papers include a great variety of theoretical frameworks: several of these theoretical approaches are related to Tall’s ideas of advanced mathematical thinking and the three worlds of mathematics (e.g., Deeken et al., 2020 ; Hong et al., 2009 ), others are based on the Anthropological Theory of the Didactic (e.g., Hausberger, 2018 ), and others are within the theory of commognition by Sfard ( 2008 ). In the latter case, the focus is on the shift of mathematical discourses between secondary school and university (Thoma & Nardi, 2018 ).

Recently, more authors have adopted frameworks related to affective constructs not originally developed in the field of STT: Hernandez-Martinez et al. ( 2011 ) focused on mathematical identity adopting the perspective of Sfard and Prusak ( 2005 ); Ufer et al. ( 2017 ) conceptualised interest within the person-object theory of interest by Krapp ( 2002 ); Dibbs ( 2019 ) used engagement theory (Fredricks et al., 2004 ) to study students’ affective reactions to failure in a calculus course; Geisler ( 2021 ) analysed the role of attitude in the dropout from university mathematics within the three-dimensional model of attitude (Di Martino & Zan, 2010 ).

In 2008, Clark and Lovric observed that ‘perhaps the most notable feature of the existing body of research on transition is the absence of a theoretical model’ (p. 25). In a special issue of the Mathematics Education Research Journal dedicated to transitions, two specific (and influential) models were discussed: the three worlds of mathematics by Tall ( 2008 ) and the rite of passage by Clark and Lovric ( 2008 ).

The three worlds of Mathematics is a theory about the development of mathematical thinking that Tall presented at PME in Bergen (Tall, 2004 ). In this theory, the development of mathematical thinking is described as the development of perceptions of three different but interrelated worlds. In 2008, Tall used this theory to focus on the changes in thinking involved in the STT and on the individual development needed for this transition. According to this theoretical model, there are three fundamental mental structures that shape long-term learning and mathematical thinking: recognition of patterns, repetition of sequences of actions, and language. In this cognitive and epistemological perspective, Tall ( 2008 ) identified ‘three worlds of mathematics that develop in sophistication in quite different ways’ (p. 7): conceptual embodiment, proceptual symbolism (APOS theory is included in this world), and axiomatic formalism. Tall explained how, in his view, the blending of embodiment and symbolism gives a more accurate way of developing sophistication in mathematical thinking. In the final version of his theoretical model, Tall ( 2013 ) included an affective dimension recognising the role of emotions in the interpretation of previous experiences with mathematics, and therefore in the individual development of mathematical thinking.

Clark and Lovric ( 2008 ) adapted a well-established anthropological theory to the study of STT: that of rites of passage. This model recognised three stages in STT (Fig.  2 ).

The three stages of the rite of passage

The liminal stage includes the period from the last part of high school to the first part of university. It is characterised by an unavoidable crisis, known mathematical routines are challenged, and first year students need to find their place in a new mathematics community. This model appeared initially strongly influenced by the cognitive perspective in STT; Clark and Lovric ( 2008 ) discussed only the cognitive shock of the passage from informal to formal language and reasoning in mathematics. A year later, Clark and Lovric ( 2009 ) recognised that the rite of passage inevitably leads to the emergence of affective reactions: the initial reaction of euphoria, the feelings of inadequacy during the crisis, and the recovery after the resolution of the crisis. These affective reactions are particularly strong for those students who were successful in secondary school: this is the case of most of the first-year students in mathematics (Di Martino & Gregorio, 2019 ).

The rite of passage model has two main implications. First, dealing with a significant crisis is a necessary step for a successful passage: instead of avoiding the crisis — for example making the new context like the old one — the crisis needs to be understood to offer students support to overcome it. As Thomas and Klymchuk ( 2012 ) observed: ‘the existence of demanding aspects of transition that are difficult to control is not in itself a good enough reason to ignore those that can be managed to produce a better experience for students’ (p. 285). An unsuccessful rite of passage may result in a never completed incorporation stage, and students’ dropout can be interpreted in this perspective. Second, the rite of passage model stresses the fact that socio-cultural aspects cannot be disregarded in the analysis of STT: the old and new communities through which the rite of passage is accomplished are context specific.

Main themes of research

Through the coding process, we identified, inductively, four main research themes. We describe them in turn below.

• The mathematical gap between secondary school and university. This gap is described in terms of students’ thinking (Godfrey & Thomas, 2008 ), approach and content (Brandell et al., 2008 ), acceptance criteria for justification (Selden, 2012 ; Sommerhoff & Ufer, 2019 ) and for legitimate mathematical activity (Jablonka et al., 2017 ), teaching style and assessment (Thomas & Klymchuk, 2012 ), didactic contract (Pepin, 2014b ) and messages students receive (Kouvela et al., 2018 ), identity (Jooganah & Williams, 2016 ), and high school calculus outcomes and university calculus requirements (Ghedamsi & Lecorre, 2021 ). These studies are mainly conducted within a socio-cultural perspective, adopting the three worlds of mathematics framework (Tall, 2008 ). In recent years, the approach to STT has become more holistic, including new viewpoints. For example, some studies consider the social and discursive perspective related to the analysis of commognitive conflicts (Thoma & Nardi, 2018 ); others describe the gap between school and university mathematical experiences through the description of the perception of the main actors involved: students (e.g., Hernandez-Martinez et al., 2011 ; O’Shea and Breen, 2021 ), schoolteachers (Hong et al., 2009 ), lecturers (Klymchuk et al., 2011 ; Deeken et al., 2020 ), or compare these different perceptions (Corriveau, 2017 ).
• The potential of technology to facilitate the STT. Two issues emerge within this category: the effective use of CAS (Computer Algebra Systems) in STT (Hong & Thomas, 2015 ; Varsavsky, 2012 ) and the analysis of the potential of ICT (Information and Communication Technology) to facilitate the STT (Bardelle & Di Martino, 2012 ; Daza et al., 2013 ). It is interesting to observe that these kinds of studies are limited to the period 2012–2015: the recent events related to the pandemic could (should?) generate new interest towards this line of research in STT (Chan et al., 2021 ).
• The factors correlated with academic success. The studies in this category consider a wide range of cognitive, social, and affective factors and are mostly quantitative, involving large samples. Their explicit goal is to highlight significant correlations between academic success and other factors, such as students’ attitudes (Geisler, 2021 ), standardised test results (Culpepper et al., 2010 ), attended secondary school (Adamuti-Trache et al., 2013 ), prior knowledge (Rach & Ufer, 2020 ), students’ learning prerequisites (Rach & Heinze, 2017 ), interest (Kosiol et al., 2019 ), and students’ beliefs (Geisler & Rolka, 2021 ).
• Failure in STT. This category represents studies of failure to transition: they are mostly qualitative, based on the collection of narratives, often framed as case studies (Hernandez-Martinez, 2016 ). The shared assumption of these studies is that the difficulties in STT are inevitable — according to Clark and Lovric ( 2009 ) even essential — and there is a thin line between success and failure. The description and interpretation of the failure is considered a significant key to understand, prevent, and overcome the students’ difficulties in STT. The definition of failure in STT varies in these studies: there is a local meaning, i.e., negative results in some first-year university exam (Dibbs, 2019 ) and a global meaning, i.e., students who leave university studies (Di Martino & Gregorio, 2019 ).

These four themes of research are highly specific to mathematics. This is evident for the first two themes which deal with subject-related issues. However, it is also the case for the latter two themes. Regarding the third theme, the factors considered are mathematics-related: for example, students’ beliefs and attitudes reveal students’ perceptions about the nature of the mathematics they are encountering at university. The fourth theme, that of failure/success in STT, is related to the students’ vision of the nature of mathematics. A change in one’s mathematical theory of failure/success is a change in one’s vision of mathematics. The mathematics epistemology conveyed by the mathematical culture in the new university environment affects students’ mathematical theories of success through the introduction of new mathematical symbols, knowledge, customs, and requirements for success. Therefore, all the elements of difficulty, not only the cognitive ones, but also the affective and sociocultural ones, are highly specific to mathematics.

Conclusion and directions for research

First, a meta-reflection about literature searches based on fully automated search engines such as Google Scholar. Such searches not only present an issue of non-reproducibility (Gusenbauer & Haddaway, 2020 ) but also present an issue of control (or rather of poor control). These searches are affected by external and contextual factors. The list produced is based on an unknown ranking algorithm (Beel & Gipp, 2009 ) and the profiling process and geolocation of who develops the search also play a role. The introduction of external selection criteria not fully controlled by researchers is a significant dilemma for the scientific community.

Notwithstanding this significant issue, the academic search engines are an essential resource to develop systematic literature reviews. We believe that adopting the recommendations of Haddaway et al. ( 2015 ) — for example using a primary and a supplementary search engine, adopting a suitable search query, and considering a large sample of the papers — it is possible to obtain a representative picture of the state of the art in the field investigated, also highlighting the so called ‘grey literature’ (Haddaway et al., 2015 ).

Our review of the research literature on STT from 2008 to 2021 highlights the variety of theoretical frameworks in use, the growing awareness of the complexity of phenomena at play in STT and the consequent adoption of a more holistic approach to STT that goes beyond a purely cognitive interpretation, including socio-cultural and affective issues. The first studies focused on the cognitive and epistemological obstacles of the shift toward advanced mathematical thinking have been integrated and complemented by studies considering social, cultural, and affective issues in the last decade. On the other hand, Artigue ( 2021 ) recently underlined how ‘the socio-cultural turn... has not yet impacted research’ (p. 9) at university level and we argue that the ‘affective turn’ has also not been completely fulfilled in STT research.

The research results obtained considering social, cultural, and affective issues have already some clear implications for the teaching and learning of mathematics at secondary school and at university level. Since the seminal work of Tall ( 1991 ), we know much about the difference between tertiary mathematics and secondary school mathematics: mathematics is understood and presented in a different (advanced) way at the university level. The more recent research on the STT highlights how tertiary transition involves other changes than the purely cognitive and other actors. Many difficulties the learners experience in the passage from school mathematics to university mathematics appear to be related to a sudden change in their mathematics identity (Hernandez-Martinez et al., 2011 ). Many successful students develop a different view of mathematics in their passage to university, often perceiving that their ability in mathematics is suddenly reduced and, consequently, developing very strong negative emotions in their university experience (Geisler & Rolka, 2021 ). This affective phenomenon is however related to an epistemological aspect: the meaning of being a successful student in mathematics. The individuals’ theory of success in mathematics are rooted and consolidated during the school experience (Di Martino & Zan, 2010 ), going often into crisis during the tertiary transition. Whether the development of the students’ theories of success during STT is unavoidable and related to the epistemology of the advanced mathematics encountered at university is an open question at the boundary between epistemological and didactical issues.

Concerning the second research question we posed, we identified four main themes of the STT research: the mathematical gap between secondary school and university, the potential of technology to facilitate STT, and the factors correlated to academic success and to academic failure in STT. These strands of research produced significant outcomes. However, we believe that more research is needed to improve our understanding of the relationship between cognitive, affective, social, and cultural aspects of the STT. We encourage studies that consider the dynamic nature of STT — that consider STT as a process — and bring into play the cultural context in which this crisis takes place. In this perspective, comparative studies between cultural or institutional contexts are still rare: an interesting exception is represented by the study of Deeken et al. ( 2020 ) that, through the Delphi method, described which students’ mathematical abilities are considered minimal prerequisites by university mathematics lecturers. Studies involving and comparing several institutions or countries are rare (Di Martino et al., 2022 , is one such example), but much needed, since they can help to understand the role of contextual factors and the generalisability of studies conducted in a specific context. Further studies in this direction would surely represent a valuable addition to the current body of research.

From a methodological point of view, the analysis of existing research highlights on the one hand the use of several different instruments, approaches, and samples. On the other hand, it points to two significant issues. First, most data are collected through online surveys: this choice is ‘not neutral’. Psychologist have discussed the impact of computer versus paper–pencil survey in collecting self-reports (Bates & Cox, 2008 ) and, in mathematics education, recent publications address the impact in students’ performance in online versus pen and paper tasks (Lemmo, 2021 ).

Second, the participants of these studies are almost exclusively volunteers, and we need to consider and discuss the limitations related to this. The effects of a sample of volunteers appear to be particularly significant in the analysis of failure in STT, where students who drop out or are about to do so are often the focus of research. The volunteer sampling in the STT may create a bias towards higher achieving students (Vollstedt et al., 2014 ).

In our overview, we included papers such as Doukhan ( 2020 ), Griese ( 2017 ), Jablonka et al. ( 2017 ), and Hong and Thomas ( 2015 ). Those papers focus on STT concerning non-specialist students: i.e., students who study mathematics as part of their degree but are not enrolled in a degree course in mathematics. The study of the differences in the mathematical transition to the various STEM degrees — the differences in the mathematical transition between specialist and non-specialist students — appears to be a significant perspective for further research.

These differences can be related to the different mathematical identities of the first-year students, to the different degrees of discontinuity of mathematical contents (for example, calculus) as they are presented at school and as they are presented in the different degrees, as well as to the different mathematical practices and requests in the different degrees. In our view, this research should also involve epistemological, socio-cultural, and affective aspects.

Finally, the result of our systematic review confirms a need of our community: that of learning more about STT in the areas of the world that are not represented in our final corpus. This is not only an important issue of equity and participation, but — considering the role of cultural, affective, and social factors in the STT — our limited knowledge of many educational contexts signifies a limited understanding of the STT, an understanding too culturally bound. We firmly believe that the mathematics education community should make all possible efforts to support researchers from underrepresented educational contexts interested in developing research about STT.

Open access funding provided by Università di Pisa within the CRUI-CARE Agreement.

Data availability

Declarations.

The authors declare no competing interests.

1 For the theoretical studies, we reported and considered the author’s nationality.

2 The PME conference was not held in 2020, due to the Covid-19 pandemic.

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Contributor Information

Pietro Di Martino, Email: [email protected] .

Francesca Gregorio, Email: [email protected] .

Paola Iannone, Email: [email protected] .

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A Project-Based Guide to Undergraduate Research in Mathematics

Starting and Sustaining Accessible Undergraduate Research

• © 2020
• Pamela E. Harris 0 ,
• Erik Insko 1 ,
• Aaron Wootton 2

Mathematics, Williams College, Williamstown, USA

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Mathematics, Florida Gulf Coast University, Fort Myers, USA

Mathematics, university of portland, portland, usa.

• Provides students with all the tools and information they will need to pursue undergraduate research within concise and accessible chapters
• Guides faculty through creating sustainable research programs by offering helpful hints and tips
• Contains a unique chapter on pursuing research in mathematics education

Part of the book series: Foundations for Undergraduate Research in Mathematics (FURM)

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Table of contents(11 chapters)

Front matter, folding words around trees: models inspired by rna.

• Elizabeth Drellich, Heather C. Smith

Phylogenetic Networks

• Elizabeth Gross, Colby Long, Joseph Rusinko
• Tropical Geometry
• Ralph Morrison

Chip-Firing Games and Critical Groups

• Darren Glass, Nathan Kaplan

Counting Tilings by Taking Walks in a Graph

• Steve Butler, Jason Ekstrand, Steven Osborne

Beyond Coins, Stamps, and Chicken McNuggets: An Invitation to Numerical Semigroups

• Scott Chapman, Rebecca Garcia, Christopher O’Neill

Lateral Movement in Undergraduate Research: Case Studies in Number Theory

• Stephan Ramon Garcia

Projects in (t, r) Broadcast Domination

• Pamela E. Harris, Erik Insko, Katie Johnson

Squigonometry: Trigonometry in the p-Norm

• William E. Wood, Robert D. Poodiack

Researching in Undergraduate Mathematics Education: Possible Directions for Both Undergraduate Students and Faculty

• Milos Savic

Undergraduate Research in Mathematical Epidemiology

• Selenne Bañuelos, Mathew Bush, Marco V. Martinez, Alicia Prieto-Langarica
• Undergraduate research in mathematics
• Phylogenetic networks
• Numerical Semigroups
• Squigonometry
• Mathematical Epidemiology
• combinatorics

Pamela E. Harris

Aaron Wootton

Book Title : A Project-Based Guide to Undergraduate Research in Mathematics

Book Subtitle : Starting and Sustaining Accessible Undergraduate Research

Editors : Pamela E. Harris, Erik Insko, Aaron Wootton

Series Title : Foundations for Undergraduate Research in Mathematics

DOI : https://doi.org/10.1007/978-3-030-37853-0

Publisher : Birkhäuser Cham

eBook Packages : Mathematics and Statistics , Mathematics and Statistics (R0)

Copyright Information : Springer Nature Switzerland AG 2020

Hardcover ISBN : 978-3-030-37852-3 Published: 18 April 2020

Softcover ISBN : 978-3-030-37855-4 Published: 18 April 2021

eBook ISBN : 978-3-030-37853-0 Published: 17 April 2020

Series ISSN : 2520-1212

Series E-ISSN : 2520-1220

Edition Number : 1

Number of Pages : XI, 324

Number of Illustrations : 61 b/w illustrations, 81 illustrations in colour

Topics : Combinatorics

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