Mathematics Education

Lieven Verschaffel , ... Erik De Corte , in International Encyclopedia of the Social & Behavioral Sciences (Second Edition), 2015

Theoretical Views on Mathematics Education

One way to categorize the theoretical orientations of the research and analyses of the nature of mathematical cognition, how it is learnt, and how this learning can be optimally enhanced, is by referring to the broad paradigmatic traditions within psychology and education in general: the behaviorist/empiricist, the cognitive/rationalist, and the situative/pragmatist–sociohistoric tradition (Greeno et al., 1996 ). These general theories have largely influenced, and continue to influence, the research field of mathematics education.

The behaviorist/empiricist tradition includes associationism, behaviorism, and connectionism. According to this view, mathematical knowledge and skills are the result of an accumulation of acquired associations and skills. A classic example of this tradition is Thorndike, who applied his associationist view on learning (as differential strengthening of associations by reinforcement) to the learning of arithmetic. Although this tradition plays a minor role in contemporary research, its ideas still permeate into current instructional practices: for example, in drill-and-practice approaches. Conceptions of mathematics as providing correct answers to well-defined mathematics tasks, of learning mathematics as incremental (with errors to be avoided or immediately stamped out), and of mathematics teaching as the reinforcement of mathematically correct responses, remain prevalent in many current instructional practices and, as such, represent the legacy of this first tradition.

The cognitive/rationalist perspective involves research traditions such as Gestalt psychology, symbolic information processing, and constructivism. These traditions conceive mathematical knowledge as universal mental entities located in individual learners, such as cognitive schemes and procedural rules, and define learning mathematics as changes in these universal mental schemes and rules. Typical representatives of this tradition are Piaget, who theorized mathematical development in terms of the stepwise acquisition of increasingly complex logicomathematical structures; Wertheimer, who emphasized the insightful detection and exploitation of structure in mathematical thinking and learning; and information-processing theorists such as Simon, Anderson, and Siegler, who performed fine-grained analyses of the cognitive structures and processes involved in the acquisition and use of key mathematical concepts and skills. This perspective has also had a clear impact on the practice of mathematics education: performing detailed analyses of the concepts and skills needed to perform certain mathematical tasks, looking at mathematical learning processes rather than learning outcomes, paying ample attention to the role of prior knowledge, valuing conceptual understanding, and problem solving besides procedural fluency are reminiscent of this second perspective.

The situative/pragmatist–sociohistoric perspective entails traditions such as ethnography, anthropology, and situation theory. This third framework embraces all those theories that view mathematics learning as a reorganization of activity that accompanies the integration of an individual learner within a 'community of practice.' Having its origins in Vygotsky's work, this third tradition emerged strongly in the late 1980s in reaction to the then dominant cognitive view of mathematics learning as a highly individual and purely mentalistic process of knowledge and skill acquisition occurring in the learner. In contrast to this cognitivist view, the situated perspective stresses that mathematics learning is enacted essentially in interaction with social and cultural contexts and artifacts, and especially through participation in cultural activities and contexts. Whereas in the cognitivist view, mathematics is seen as a universal knowledge system, the situative/pragmatist–sociohistoric perspective assumes that mathematics differs according to the setting in which it has been developed and in which it is practiced. The influence of this perspective on mathematics education is reflected in attempts to implement elements of successful out-of-school learning settings into school mathematics, and increased attention to the sociocultural and affective side of mathematics learning and teaching.

However, research in mathematics education of the past decades is not merely, not even primarily, theoretically framed in terms of the three above-mentioned general traditions. Of the many theories that have been developed over the past decades and are being used within the field of mathematics education, the dominant ones are 'home-grown theories.' Such theories put the specificity and integrity of the domain at the center, with no hesitation, whatsoever, to borrow ideas and techniques from other disciplines. But they have little or no interest in the possible relevance of the theory outside the field of mathematics education. Therefore, since its emergence as a field in its own right, mathematics education has witnessed many such home-grown theories of mathematical cognition, learning, and teaching, such as Brownell's (1945) theory of what he called 'meaningful mathematics,' emphasizing the understanding of mathematical relationships and the ability to think quantitatively, Skemp's (1979) theoretical distinction between 'instrumental' and 'relational understanding' of mathematics, Van Hiele's (1986) theory of how children learn geometry, Pólya's (1945) analysis of the role of heuristic processes in mathematical problem solving, Freudenthal's (1983) view on mathematics (education) as a human activity, which formed the basis of the Dutch model of 'realistic mathematics education,' and Fischbein's (1987) theory of the intuitive sources of mathematical thinking – to mention just few 'giants' of the history of research in mathematics education. Although well versed in the theories and methods of psychology, most of them were quite critical of its limited application to the domain of mathematical cognition, learning, and teaching, and emphasized that (the psychology of) mathematics education had to develop its own questions, and theoretical and methodological perspectives (Verschaffel and Greer, 2013).

In the course of development, tensions and perspectival differences between mathematics educators on the one hand and psychologists, educational scientists, and mathematicians on the other hand, have continued. The books edited by Sierpinska and Kilpatrick (1999) and Sriraman and English (2010), for instance, make clear that the field has far from settled into a phase of 'normal science' (in the Kuhnian sense). But rather it has entered a period of considerable complexity and diversity in theoretical perspectives, experimental methodologies, and reflections, not only on the fundamental questions of what mathematics is and how it can be learnt and taught, but also of the equally fundamental question what is mathematics (education) education for?

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780080970868920488

The Development of Early Childhood Mathematics Education

Crystal Day-Hess , Douglas H. Clements , in Advances in Child Development and Behavior, 2017

3 Conclusion

Early childhood mathematics education has received increasing attention from researchers, practitioner, and policy makers. Over the next several years, the DREME Network will conduct research and development activities that will have implications for interventions, from the state and district levels, to professional development, families, and teaching. As an immediate contribution, we report on the initial work of one of the projects, a comprehensive review of the instructional activities in six research-based curricula. These data will provide researchers, curriculum developers and specialists, and other educators with new information and new perspectives when creating, evaluating, and choosing curricula and instructional activities in early mathematics.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/S0065240717300137

Epistemic States of Convincement. A Conceptualization from the Practice of Mathematicians and Neurobiology

Mirela Rigo-Lemini , Benjamín Martínez-Navarro , in Understanding Emotions in Mathematical Thinking and Learning, 2017

Abstract

Some Mathematics Education research deals with convincement, certainty, security and doubt; nevertheless, those studies often do not precisely define the terms. The resulting polysemy may hinder agreement among experts and the progress of research. This chapter aims to conceptualize those internal states (in this work called "epistemic states of convincement"), in order to gain a deeper understanding and provide a justified account of their related didactical phenomena. Based on a process of dialectic triangulation of empirical data taken from the professional practice of mathematics and from notions taken from various fields, the first part of the chapter shows that epistemic states are associated with beliefs, the reasons that support said beliefs and the motives a person has for believing. It moreover shows that there are different epistemic states and that they are expressed through certain behavioral patterns and bodily expressions. It also follows that these states hold a certain relationship with the cognitive world, but also with the affective realm. In order to delve deeper into these studies, the second part of the chapter uses resorts the model suggested by Damasio for emotions and feelings. From this perspective, epistemic states are conceived as certain types of emotions and feelings; in addition to being consistent with the notion derived from the first part, this new interpretation allows us to deepen our understanding of how these epistemic states arise and change during the practice of mathematics. It offers the possibility, for instance, of suggesting a potential route for the basis and evolution of states of certainty during deductive type tests, as explained in the second part of the document. The last part of the chapter provides some didactic considerations. Specifically, it suggests that connections between propositions of mathematical content and the epistemic states are re-trainable. A re-training mechanism consists of, to the extent this is feasible, the teacher abstaining from encouraging certain types of grounds (such as those that are based on authority), which are contrary to those driving mathematics; another consists of, under the guidance of the teacher, students engaging in conscious evaluation of the connections they have made between beliefs and epistemic states, as well as the connections between grounds and epistemic states.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780128022184000042

Evaluation of Mathematics Education Programs

J. Cai , in International Encyclopedia of Education (Third Edition), 2010

Formative and Summative Evaluations in Mathematics Education

The evaluation of mathematics education programs is an integral part of mathematics education. Evaluation is the process of judging the value or worth of something. Like in any evaluation, the critical words in the evaluation of mathematics education programs are value or worth. When we evaluate a mathematics education program, we engage in a process designed to provide information to help make a judgment about the program. In evaluating such programs, the worth or value is usually related to students' learning.

Sometimes in mathematics education, evaluation has been used interchangeably with assessment and testing, without a clear distinction (National Council of Teachers of Mathematics, 1989). In fact, assessment is a broad term defined as a process for obtaining information that is used to make decisions about students, curricular programs, and policy (National Council of Teachers of Mathematics, 1995). Thus, assessment can provide information to evaluate a mathematics education program, but is not always involved in the judgment of worth. A test is defined as an instrument or systematic procedure for observing and describing one or more characteristics of a student. In other words, a test is a special form of assessment using a formal instrument, but an assessment does not have to involve a test because we can assess a student's learning through informal observation. Test scores are usually used to judge how well students perform or how effectively teachers teach. Please note that testing itself is neutral and does not involve any value judgment. As a result, a test is a powerful tool for evaluating mathematics education programs.

Generally speaking, every evaluation process involves an object to be evaluated, a scale of value, and a way of collecting information so that the object can be placed on the scale of value for judgment (Eisner, 1994; Fitzpatrick et al., 2004). The ultimate goal of program evaluation is to improve students' learning of mathematics. To reach this goal, both formative and summative evaluations are needed.

An evaluation is defined as formative if the primary goal is to provide information for program improvement. For example, if the goal of evaluating teaching of mathematics is to improve teaching, then it is formative evaluation. Formative evaluation determines if a mathematics education program is delivered appropriately and then determines ways for improvements if they are needed. In the formative evaluation of teaching, for example, the evaluation provides a way to understand teachers' current way of teaching and then creates a plan to enhance their professional growth (Fitzpatrick et al., 2004; NCTM, 1991).

An evaluation is defined as summative if the primary goal is to determine the effectiveness of a mathematics education program. It looks at the results. It is common to speak of short-term and long-term outcomes for a mathematics education program. For example, we can evaluate if a particular instructional intervention is effective within one year (short-term outcome) or in four or more years (long-term outcome).

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780080448947016316

Evidence for Cognitive Science Principles that Impact Learning in Mathematics

Julie L. Booth , ... Jodi L. Davenport , in Acquisition of Complex Arithmetic Skills and Higher-Order Mathematics Concepts, 2017

Abstract

Numerous issues with mathematics education in the United States have led to repeated calls for instruction to align more fully with evidence-based practices. The field of cognitive science has identified and tested a number of principles for improving learning, but many of these principles have not yet been used to their fullest to improve mathematics learning in US classrooms. In this chapter, we describe eight principles that may have particular promise for mathematics education: abstract and concrete representations, analogical comparison, feedback, error reflection, scaffolding, distributed practice, interleaved practice, and worked examples. For each principle, we review laboratory and classroom evidence related to benefits for mathematics learning and identify priorities for future research.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780128050866000138

Special Needs in Mathematics Classrooms

Melissa Rodd , in Understanding Emotions in Mathematical Thinking and Learning, 2017

Subtle Interplay: Example 2

A more visual mathematics education example of subtle interplay, discussed above, can be found in Finesilver's research. The picture ( Fig. 3) shows a mathematical artefact "cocreated" (ibid, p. 71) by student and teacher with the student empowered to abstract the mathematics (here, of multiplication) to the extent she can manage in that particular lesson. The teacher-student set-up involved student agency, shared attention, and patience (similar to the parable of the stationary donkey; see Pedagogical Principles for a "Math-Care Environment" section). When I discussed this chapter at our informal weekly seminar, some of my colleagues were concerned about my using the image of Fig. 3 because "it was not correct" (e.g., bottles are missing from a box in the van). On reflection, this "error" that the teacher need not notice and the child has left behind, is part of the collusion concerning existence, analogous to that in maternal care, and is a good example to illustrate a shared view in a math-care environment.

Fig. 3. Finesilver's visual subtle interplay (Finesilver, 2014, p. 71).

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780128022184000091

Mathematics Teacher Education

K. Krainer , S. Llinares , in International Encyclopedia of Education (Third Edition), 2010

Mathematics Teacher Education: A Practical and Emerging Field of Research

In the last 20 years, researchers in mathematics education have increased attention on mathematics teacher education (MTE). This shift is reflected in the emergence of international handbooks. In 1996, the first International Handbook of Mathematics Education (Bishop et al., 1996) was published. The first International Handbook of Mathematics Teacher Education (Wood et al., 2008) has been published in 2008. Till about 1990, MTE was mainly a field of practice (mostly presenting success stories, more or less grounded on the basis of theory and evidence.). Since then it also increasingly became a field of research. In 1998, the Journal of Mathematics Teacher Education (JMTE) was launched. This journal marked the beginning of a new era since this was the first international journal focusing genuinely on research in MTE. It was only 4   years earlier that the founding editor of JMTE stressed that "as a profession we had just begun to recognize the significance of conceptual orientations for guiding research on teacher education (Cooney, 1994)." Simultaneously, numerous books and articles on the problems and progress in describing and interpreting learning processes by mathematics teachers were also published (see e.g., Jaworski et al., 1999; Krainer et al., 1999; Lin and Cooney, 2001; Peter-Koop et al., 2003; Sowder, 2007).

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780080448947006801

Discrete and Continuous Presentation of Quantities in Science and Mathematics Education

Ruth Stavy , Reuven Babai , in Continuous Issues in Numerical Cognition, 2016

Abstract

Continuous quantities, part of everyday science and mathematics education, are difficult for many students. Difficulties may stem from the interference of salient irrelevant quantities, which are automatically/intuitively processed. We focus on the effect of mode of presentation on students' ability to overcome this interference.

We describe several studies: comparison of perimeters, comparison of ratios, and comparison of areas and numbers. In each we used discrete and continuous modes of presentation.

The findings clearly demonstrate that changing the mode of presentation affects performance. It is suggested that changing the presentation mode changes the perceptual information, leading to a change in salience level. Such a change could encourage the use of appropriate solution strategies and thus improve the ability to overcome the interference. We discuss the importance of these findings to the understanding of reasoning processes related to quantities and their practical implications in science and mathematics education.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780128016374000135

Mathematics, Teaching of

Steven R. Williams , in Encyclopedia of Applied Psychology, 2004

2 Current Views of Mathematical Understanding

In many ways, the foundational issue in mathematics education is the nature of mathematics itself. The decisions we make about teaching mathematics and the kinds of understanding we value in students both depend on what we believe to be most fundamental about mathematics. Mathematics has traditionally been seen as a collection of facts and procedures to be mastered and later applied to appropriate situations. Certainly, learning mathematics does involve mastery of some basic information—multiplication facts, procedures for solving equations, and so forth. Many early applications of psychology to mathematics learning viewed mathematics in just this way. However, it is increasingly recognized that mathematics is more than a collection of facts and procedures. Knowing mathematics involves having the conceptual understanding and habits of mind that enable one to use mathematics powerfully in daily life. Conceptual understanding of mathematics involves understanding how the facts and procedures fit together and when they apply. It involves knowing why procedures work and how to use them to solve problems. In short, it is connected knowledge that provides an organized framework for understanding. Mathematical habits of mind include the ability to reason logically with mathematics ideas, to justify procedures and answers, and to explain mathematical ideas to others—in short, to engage in the practices of mathematics. The acquisition of all these skills, concepts, and habits of mind is what is meant by learning mathematics with understanding. When this becomes our instructional goal, it significantly affects the psychology of teaching mathematics.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B0126574103002440

Appraising Emotion in Mathematical Knowledge

Inés M. Gómez-Chacón , in Understanding Emotions in Mathematical Thinking and Learning, 2017

Epilog: Open Questions and Unresolved Issues

This section lists important issues around affect in mathematics education that have not yet been suitably solved by research (this author's included). While many have been broadly discussed in the area of affective science, no clear answers have been forthcoming.

Experience vs expression. The existence of a one-to-one relationship between emotional experience and emotional expression is perhaps the most widespread and potentially problematic assumption.

While it may be tempting to think of emotional experience and expression as two sides of the same coin, recent research suggests that such a conclusion is tenuous at best. Physiological and bodily channels obviously convey information on a person's emotional state. Nonetheless, the exact value of such channels as emotional indicators has yet to be determined.

Intercomponent coherence in emotion. Another assumption made by many researchers is that emotional experience is expressed as a synchronized response involving peripheral physiology, facial expression, and instrumental behavior. The corollary to that assumption is that the signals detected are more intense when several channels are observed. With the exception of prototypically intense expressions of emotional experience, however, research findings show low correlations among the various components of a given emotional episode (Barrett, 2006). The identification of cases where emotional components are synchronized as opposed to those where they are only weakly related is a matter outstanding more effective detection.

Affective structures. Labels and categories of emotions based on two- or three-dimensional models are of crucial importance to researchers. The identification of a suitable level of representation can favor progress. Here two significant structures in a person's affective dimension, local and global affect-cognitive structure, are proposed given their effectiveness in the research conducted. In natural learning contexts, motivation, cognition, and emotion can be conceptualized and worked with empirically at three or more levels of specificity to explore how the self transitions at those levels.

The interrelationships among cultural, social, and personal systems in students' emotions around mathematical thinking. The data showed the existence of interrelationships among cultural, social, and personal systems in students' emotions around mathematical thinking. The study covered these students' social identity and what mathematics and the learning process meant to them. The findings suggest that new approaches to the affective dimension of mathematics in lifelong learning are possible, at least for groups similar to the one studied (branded with a negative identity). The traits comprising these students' identity in their context can be equated to a network of meanings relevant to that network which will emerge in mathematics learning. Such meanings provide insight into the pursuit of a fuller understanding of persons' global affect, their affective view of and reaction to mathematics and mathematics learning, the way they build their belief systems, and their awareness of that process.

Awareness of the importance of social processes in the explanation of emotion is growing steadily. The aim in this chapter (and in this author's research) is to stress the role of culture and social processes on the one hand and the variety of positions adopted by individuals on the other. Social and cultural factors cannot be construed as a "uniform static package." A dialog emerges around the self. The case studies showed that students' behavior reveals different strategies for identifying and negotiating their social identity and consequently different emotional responses.

The emotional dimension can be studied from the standpoint of that dialog by combining perspectives and methods that seek to describe, with a full wealth of symbols, the meanings of affect in the social reality in which it arises. The frameworks for interpreting emotion have traditionally run the risk of centering more on an individual's cognitive events than on issues around arguments and conflicts in "mathematical practice."

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780128022184000029