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In school, at least in the US, we were taught to multiply by hand according to a standard algorithm. Most people find that algorithm difficult to use, and many children fail to learn it. We propose a new way to make sense of this difficulty: to treat explicit computation as perceptually supported physical and mental action. Based on recent work in neuroscience, we trace the flow of arithmetic information to emphasize demands on visual working memory and attention. We predict that algorithms that make moderate demands on memory and attention will work better than others that make stronger demands. We suggest that the judicious use of spatial schemas can reduce such cognitive demands. Experimental evidence from children in an inner-city school supports this claim. Our work suggests new ways to think about instruction. The goal should be to minimize demands that present obstacles and maximize instead what human eyes, bodies, and brains do well. (Contains 7 figures and 2 tables.)

Although children partition by repeatedly halving easily and spontaneously as early as the age of 4, multiplicative thinking is difficult and develops over a long period in school. Given the apparently multiplicative character of repeated halving and doubling, it is natural to ask what role they might play in the development of multiplicative thinking. We investigated this question by examining children's solutions to folding tasks, which involved predicting the number of equal parts created by a succession of given folds and determining a sequence of folds to create a given number of equal parts. Analyzing a combination of cross-sectional data and case studies from standardized clinical interviews, we found that children were most successful at coordinating folding sequences with multiplicative thinking when they used a conceptualization of doubling based upon recursion. This conceptualization tended to generate more sophisticated solutions. (Contains 1 figure and 2 tables.)

There is a need for further studies on students' learning of Differential Equations (DEs), especially in advanced undergraduate and graduate courses. Research on the mathematical education of engineers shows a conflict between students' demands for practical, contextualized pedagogies and the need for abstract reasoning and appropriate use of mathematical results. Few papers focus on engineering students' interpretation of theorems and their use as tools in argumentation and problem-solving. This paper takes a sociocultural stance on learning and employs dialogical inquiry – a methodology rooted in Bakhtinian theory, newly developed for collaborative inquiry and qualitative data analysis – to investigate the meanings that senior engineering students made while working on a task designed to evaluate their understanding of Existence and Uniqueness Theorems (EUTs) of solutions of DEs. We identified two important epistemological disconnections that explain the difficulties that some of our students faced in making meaning of solutions of DEs and the EUT. • There is a need for more studies on the learning of Differential Equations (DEs). • Few research papers focus on students' meaning-making of theoretical results in DEs. • Dialogical inquiry methodology was used to analyze students' meaning-making processes. • Two epistemological disconnections were found that explain students' difficulties. • The use of warrants in students' dialogue was important in their meaning-making. [ABSTRACT FROM AUTHOR]

Over the past few decades, Piaget's forms of abstraction have proved productive for developing explanatory models of student and teacher knowledge, yet the broader applicability of his abstraction forms to mathematics education remains an open question. In this paper, we adopt the Piagetian forms of abstraction to accomplish two interrelated goals. Firstly, we analyze instructional tasks to develop hypothetical accounts of the abstractions that might occur during students' engagement with them. Secondly, we draw on middle- and secondary-grades classroom data to discuss the abstractions that occurred during the implementation of those instructional tasks. Because this paper represents an initial attempt at extending the applicability of Piagetian forms of abstraction, we close with potential implications of such use and possible avenues for future research. Most notably, we highlight the complexities involved in supporting abstraction through curriculum and instruction. • We examine task materials and implementation. • We differentiate between forms of abstraction. • We identify abstraction's viability for such analysis. • We differentiate between student learning outcomes. • We suggest avenues for future research. [ABSTRACT FROM AUTHOR]

• The paper introduces three different settings of Example-Use: Spontaneous, Evoked, and Provided. • These constructs are used to organize and categorize selected research on exemplification in mathematics. • The spontaneous mode of example-use sheds light on students' thinking. • The evoked mode of example-use triggers students to deepen their understanding. • The provided example-use requires scaffolding of students' attention to relevant features of the example. The paper provides an overview of the broad field of research on the roles and use of examples in mathematical learning and thinking, specifically in concept learning and proving, and offers a way to organize and categorize selected research in this field. The main contribution of the paper is the distinction between three settings of example-use: spontaneous example-use, evoked example-use , and responsive example-use to a provided example. In the first two – the main source of examples comes from the student/learner, while in the third setting the main source is the teacher/researcher. The illustrations that accompany the description of each setting by research-based cases convey the affordances and challenges of each one. While these three settings are applicable to various kinds of learning, I highlight their affordances mainly for concept formation, and point to papers in this issue that provide a rich collection of cases that convey the place of these settings for conjecturing and proving. [ABSTRACT FROM AUTHOR]

Assessing students' conceptions related to independence of events and determining probabilities from a sample space has been the focus of research in probability education for over 40 years. While we know a lot from past studies about predictable ways students may reason with well-known tasks, developing a diagnostic assessment that can be used by teachers to inform instruction demands the use of familiar and unfamiliar contexts. This paper presents the current work of a research team whose aim is to create a formative concept inventory with strong evidence of validity that uses a psychometric model to confidently predict whether a student exhibits one or more misconception across many items. We illustrate this process in this paper using a particular item with a context of a raffle aimed to measure whether a student reasons with misconceptions related to independence or equiprobability. The results of two aspects of the validity process: cognitive interviews to assess response processes on individual items, and a large-scale administration to examine internal structure of the concept inventory revealed difficulties in assessing students' reasoning about these key probability concepts and trends in the prevalence of misconceptions across grades. Results can provide guidance for others aiming to develop assessments in mathematics education and also support further possibilities for research into understanding students' reasoning about independence and sample space. • Details the validity process for constructing a probability concept inventory. • Discusses nuances in measuring whether a student reasons with misconceptions related to independence or equiprobability. • Cognitive interviews revealed interactions between item wording and students' reasoning that led to item modifications. • Large scale administration showed that proportion of students exhibiting a misconception decreased in higher grades. • Attending to item wording and using multiple methods of validation are encouraged for research in mathematics education. [ABSTRACT FROM AUTHOR]

To improve instruction, conduct communicable studies on student intuition, and provide an entry point for researchers interested in this area, the meaning of intuition in mathematics education needs to be addressed. This paper reviews 816 journal papers from top tier mathematics education journals to obtain an updated conceptualization of the intuition construct. Different research camps, such as a camp focused on instructional design that leverages learners' primary intuitions in the classroom and a camp concerned with sources of errors through the lens of intuitive rules theory, emerged. A critical discussion of theoretical choices for intuition research will be followed by several directions for future research on intuition specific to university mathematics education. [ABSTRACT FROM AUTHOR]

Despite the importance of graphical reasoning, graph construction and interpretation has been shown to be nontrivial. Paoletti et al. (2023) presented a framework that allows for a fine-grained analysis of students' graphical reasoning as they conceive of graphs as representing two covarying quantities. In this paper, we show how the framework can be used to not only characterize a student's graphing meanings and reasoning, but also to diagnose complexities in a student's development of such reasoning, and to design tasks that provide opportunities to resolve such complexities. We draw on data from a teaching experiment with a sixth-grade student in the U.S. to highlight how the framework allowed us to identify indications and contraindications of the student's engaging in reasoning compatible with the framework. Further, we describe how this analysis supported us in designing a task that was aligned with the framework and proved productive in supporting the student's learning. We conclude with a discussion of our findings and their implications for task design and future research. • We show how Paoletti et al. (2023) framework can support researchers, teachers, and curriculum designers to characterize a student's meanings and reasoning related to their understanding graphs as representing two covarying quantities. • We highlight how Paoletti et al.'s (2023) framework can also support researchers, teachers, and curriculum designers to diagnose complexities in a student's developing such reasoning and to design tasks that provide opportunities to resolve such complexities. • We use data from a teaching experiment with a middle-school student to highlight how the framework allowed us to identify indications and contraindications of the student's engaging in reasoning compatible with the framework. [ABSTRACT FROM AUTHOR]

Many believe teacher leaders can play a central role in meeting the needs of students in schools, but it is presumptuous to think that teachers intuitively know how to lead their colleagues or schools without any focused support. This paper describes how one middle school mathematics teacher learned to enact leadership through an informal role as a teacher leader. The results of this autoethnographic study also reveal how the teacher’s view of leadership changed during the year of the study. As that leadership identity transitioned, the researcher found that adopting a lead-by-example and lead-learner stance supported her work with colleagues. A presentation of findings related factors that supported and hindered the transition from mathematics teacher to that of teacher leader are shared. Factors that supported teacher leadership included maintaining a disposition of continuous learning, developing a community of practice with colleagues, and developing a systems view of leading. The experiences and factors that hindered the process of becoming a teacher leader included confusion about one’s leadership role, navigating the middle ground between colleague and leader, and the lack of communication with administration. The findings in this paper suggest several courses of action for supporting emergent teacher leaders. [ABSTRACT FROM AUTHOR]

This paper reports a mixed-methods, causal-comparative study of two instructional approaches in a differential equations course for engineering undergraduates. One classroom emphasized decontextualized techniques to solve canonical differential equations. The other classroom emphasized modeling principles to derive and interpret canonical differential equations as models of real world phenomena. Both classrooms were lecture based. This paper presents a brief review of both texts and compares a lesson on the same topic from each classroom in order to illustrate what it means to emphasize modeling principles. An ANCOVA was used to compare student performance on common final exam items while controlling for prior mathematics achievement. The main effect for instructional approach was statistically significant suggesting that a modeling perspective for teaching differential equations aided student learning. This quasi-experimental study corroborates previously reported recommendations for improving teaching and learning of differential equations. [ABSTRACT FROM AUTHOR]

© 2015 Elsevier Inc. This paper uses a sociocultural conceptual framework to provide an integrated view of academic literacy in mathematics for English Learners. The proposed definition of academic literacy in mathematics includes three integrated components: mathematical proficiency, mathematical practices, and mathematical discourse. The paper uses an analysis of a classroom discussion to illustrate how the three components of academic literacy in mathematics are intertwined, how academic literacy in mathematics is situated, and how participants engaged in academic literacy in mathematics use hybrid resources. The paper closes by describing the implications of this integrated view of academic literacy in mathematics for mathematics instruction for English Learners, arguing that it is important that the three components not be separated when designing instruction in general, and it is essential that mathematics instruction for English Learners address these three components simultaneously.

Analogical reasoning is an important mathematical process for undergraduate students. However, it is unclear how students understand analogies that are presented to them, and more importantly, how students understand and create their own analogies. In this paper, we present a case study of four students as they reason analogically about several structures in abstract algebra. In particular, we expand on the notion of structure sense to include a wider range of structures in advanced mathematics and attend to each students' analogical structure sense associated with each structure. Findings suggest that although students may possess a strong structure sense for group-theoretic structures, it is not necessarily the case that they possess a comparatively strong analogical sense of structure for ring-theoretic structures. In addition, those students with weaker senses of structure for group-theoretic structures are still able express productive reasoning about ring-theoretic analogies. Implications for future research and instructional practice are discussed. • Sense of structure can entail a wider variety of structures, such as sub-structure and quotient structure. • Students can successfully construct analogical structures in ring theory by analogy with structures in group theory. • Sense of analogical structure for ring-theoretic concepts is not necessarily coordinated with sense of group structure. • Students may reason about non-normative concepts in ring theory by analogy with structures in group theory. [ABSTRACT FROM AUTHOR]

Despite its almost four-decade history, research remains in the early stages of understanding the phenomenon of mathematical problem posing. In particular, the quality of problems created by beginning posers has been recognized as a persistent challenge. In this conceptual paper, I endorse an approach that identifies posing and solving as co-emergent components of steps learners take in a mathematically problematic situation. I further argue that some of the initial responses to the situation may constitute productive ingredients for creating problems that are personally meaningful and interesting to the learners. Drawing on the literature, I offer two principles through which the process of transitioning from initial responses to fully-fledged problems can be supported: making the didactical contract of the problem-posing activity transparent and immersing learners in socio-mathematical settings that are conducive to "good" problems. The approach is illustrated with fragments from a workshop for in-service teachers. The concluding discussion focuses on how the presented approach addresses some common issues in problem posing research. • Problem posing studies are subject to particular conventions. • Implicit problem posing is embedded in learners' mathematical activity. • Initial responses to a problematic situation can be useful to pose problems. • A problematic situation and a socio-mathematical setting are important. [ABSTRACT FROM AUTHOR]

In this conceptual paper, we explore a framework of reasoning practices for decision-making in context. We extend prior work related to socioscientific issues (SSI) (Sadler et al., 2007) to consider the applicability of this framework to sociomathematical issues, specifically using the context of fairness in political (re)districting. We illustrate the usefulness of the SSI framework for sociomathematical issues drawing on student work and reflections from two undergraduate courses. We conclude by suggesting adjustments to reasoning practices of the SSI framework that might uniquely reflect the nature of sociomathematical reasoning. We discuss implications of our findings for conceptualizing reasoning practices that include mathematical perspectives within social and political contexts and for mathematics education generally. • Reasoning about a societal issue consists of intertwining personal, practical, contextual, and mathematical meaning. • A framework based on Sadler, Barab & Scott's (2007) analysis of reasoning practices regarding socioscientific issues is useful for analyzing emergent student reasoning in the context of redistricting, a sociomathematical issue. • Students reason in multiple, sophisticated ways about the fairness of specific maps and redistricting processes. [ABSTRACT FROM AUTHOR]

This paper reports the results from a set of exploratory teaching interviews in which students constructed individualized algebraic expressions (called personal expressions) to describe their meanings for partial sums. Our analysis focused on one student, Emily, who constructed two distinct personal expressions for partial sums, one novel and one based on her image of summation notation. Emily created her novel expression to denote the process of generating the summands to compute the value of a partial sum. Emily adopted summation notation to describe the value of the partial sum. After reflecting on her inscription for a series' general term of summation, Emily constructed a single expression to describe either the process of computing an arbitrary partial sum or the value of the sum itself. Using Emily's story, we propose three categories for students' coordination of their meanings for partial sums with a corresponding representation. ● The term personal expression is defined from the constructivist perspective. ● A case study of one student's expressions for arbitrary partial sums is reported. ● The student constructed two distinct expressions to denote a process and a concept. ● The student reconciled her expressions into one symbol conveying process and concept. ● The students' cognitive steps necessary to facilitate this transition are described. [ABSTRACT FROM AUTHOR]

Recognizing and describing children's mathematical thinking in humanizing ways, especially when students engage in confusion, productive struggle, and mistakes, is a complex and challenging process. This paper describes an exploratory, mixed-methods study about how elementary teacher candidates (TCs) describe children's thinking as a right to exercise and to value their humanity when learning mathematics. The study analyzed transcripts from 64 TCs' summative assessments, which consisted of mock parent-teacher conferences (MPTC). Findings suggest that TCs described children's confusion, productive struggle, and mistakes (RotL 1 and 2) as: a teacher's observation, an opportunity for students to correct or clarify their thinking, an opportunity for teachers to adjust instruction or provide support, and as a normal part of the learning process. More importantly, some TCs reassured children that learners have fundamental rights when learning mathematics, especially when feeling confused and claiming a mistake. Implications for research and teacher education are discussed. Keywords: Elementary, teacher education, mathematics, mixed methods, rehumanizing, Torres' rights of the learner • Topic modeling and sentiment analysis helps to quantify and describe valence that arises from large amounts of text. • 32 TCs identified children's confusion, generalized struggle, and mistakes (Torres' RotL 1 and 2) in their MPTC across 50 instances. • Some TCs described children's confusion and mistakes as rights to exercise, a potential rehumanizing stance. [ABSTRACT FROM AUTHOR]

Children with mathematics difficulties often show challenges in additive reasoning and problem solving and may experience challenges with more advanced mathematics. To address these challenges, we implemented a computer-based intervention to promote children's conceptual understanding of numbers, counting, and additive strategies. The purpose of this exploratory case study was to provide an in-depth analysis of the progression of counting and composite unit development as well as strategies for solving addition and subtraction problems by two third-graders with mathematics difficulties in a computer-based learning environment. We analyzed video-recorded sessions, field notes, and pre- and post-assessments. Our findings show that participating students negotiated solving addition and subtraction problems through inventing more advanced and efficient additive double-counting strategies. In addition, it seems that the growth of their counting through the computer-based intervention was generalized to solving additive problems on a paper-and-pencil test. • More advanced and efficient counting strategies appear after engaging in computer-based instruction. • Advanced counting strategies that participants developed include additive double counting by composite units. • In the computer-based learning environment, participants developed composite units of tens and fives. • The challenges to counting acts may contribute to the participants' growth. • The growth of counting seems to be generalized to the traditional paper-and-pencil tasks. [ABSTRACT FROM AUTHOR]

Most educational and philosophical thought about mathematics focuses on the logical structure of the subject and considers mathematicians and students to be people whose primary practices are verifying statements within this structure. We claim that acts of discernment – careful choices driven by aesthetic considerations – are as important as acts of verification in mathematical work. This paper offers a conceptualization of this “aesthetically guided choice” that differentiates between three interrelated acts of discernment: nominating ideas, arranging ideas, and balancing ideas. We argue that aesthetically guided choice should be supported in school, and that such acts are notably lacking from the “Standards for Mathematical Practice” in the Common Core State Standards. This paper is the development of a theory at heart, built around rich descriptions of a mathematics class with middle school students, in which young thinkers were engaged in aesthetic choice. It includes analyses of the history of mathematics and studies of mathematical work to support claims about the nature of mathematics and to interpret the work of young mathematics learners from the perspective of mathematics as a discipline. [ABSTRACT FROM AUTHOR]

The number of students who leave U.S. schools mathematically underprepared has prompted widespread concern. Low achieving students, many of whom have been turned off mathematics, are often placed in low tracks and given remedial, skills-oriented work. This study examines a different approach wherein heterogeneous groups of students were given responsibility and agency and asked to engage in a range of mathematical practices collaboratively. The teaching intervention, which was introduced in the first paper, took place as part of a summer class on algebra, and it gave students the opportunity to participate with mathematics in changed ways. This paper will report evidence that the vast majority responded with increased engagement, achievement, and enjoyment. The students chose collaboration and agency as critical to their improved relationships with mathematics. [ABSTRACT FROM AUTHOR]

This paper discusses the process of proving from a novel theoretical perspective, imported from cognitive psychology research. This perspective highlights the role of hypothetical thinking, mental representations and working memory capacity in proving, in particular the effortful mechanism of cognitive decoupling : problem solvers need to form in their working memory two closely related models of the problem situation – the so-called primary and secondary representations – and to keep the two models decoupled , that is, keep the first fixed while performing various transformations on the second, while constantly struggling to protect the primary representation from being “contaminated” by the secondary one. We first illustrate the framework by analyzing a common scenario of introducing complex numbers to college-level students. The main part of the paper consists of re-analyzing, from the perspective of cognitive decoupling, previously published data of students searching for a non-trivial proof of a theorem in geometry. We suggest alternative (or additional) explanations for some well-documented phenomena, such as the appearance of cycles in repeated proving attempts, and the use of multiple drawings. [ABSTRACT FROM AUTHOR]

Students make sense of mathematical ideas using a variety of representations including physical models, pictures, diagrams, spoken words, and mathematical symbols. As students’ understanding of mathematical ideas becomes more general and abstract, there is a need to express these ideas using mathematical notation. This paper describes students’ movement from model building and personal notations to elegant use of mathematical symbols that show their understanding of advanced counting ideas. Specifically, this paper shows how earlier ideas from investigations of specific combinatorics problems (questions about making pizzas with different toppings and using cubes to build towers) are retrieved and built upon using the formal mathematical register to explain the meaning of Pascal's Identity, the addition rule of Pascal's Triangle. This analysis also shows the power of shared communication in mathematical problem solving. [ABSTRACT FROM AUTHOR]

Covariation and covariational reasoning have become key themes in mathematics education research. In this paper, I build on the construct of covariation by considering cases where more than two variables relate to each other, in what I term "multivariation." This paper provides two major contributions. The first is a conceptual analysis describing distinct theoretical multivariation structures: independent, dependent, and nested. The second is an empirical study that explored students' mental actions in reasoning about multivariation. The results of this study showed that while students' multivariational reasoning was, as expected, connected to covariational reasoning, there were many extensions and new mental actions used by the students. These included different types of recognizing, reducing multivariation into isolated covariations, switching variables and constants, different types of coordinating multiple simultaneous changes, and continuous multivariation consisting of imagining an entire system evolving. • Examines situations with more than two changing variables, called "multivariation". • A conceptual analysis is provided of distinct multivariational structures. • An empirical study is related that explores students' multivariational reasoning. • Several multivariational reasoning mental actions are identified and described. • Some mental actions are related to covariational reasoning, others are entirely new [ABSTRACT FROM AUTHOR]

There has been significant growth in the literature base exploring questions of teaching in undergraduate mathematics. In this paper we synthesize the literature on the teaching of proof-based undergraduate mathematics, drawing on 104 published reports from a range of countries and research traditions. We primarily differentiate the papers into those which explore lecture-based pedagogy and student-centered pedagogy. For each type of instruction, we focus on three categories of findings from the literature: description of instruction, instructor beliefs and rationales, and the relationship between instruction and students (cognitive, participatory, affective, and equity oriented). Much is known about the enactment of lecture-based teaching, including instructors' cognitive and affective goals. The student-centered literature focuses on tensions and challenges implementing curricula with a greater focus on participatory goals. Overall, there are few studies that attempt to link instructors' classroom activity and students' learning. Similarly, attention to equity is relatively lacking in the extant research. • In the lecture setting, we have a rich understanding of what mathematicians do and why, but only limited work on its influence on student activity and outcomes. • In the student-centered setting, we have evidence of affective and participation benefits to students, but more work is needed on the teachers' role. • We propose a framework by which to conduct research on teaching in advanced mathematics in both lecture-based and student-centered formats. [ABSTRACT FROM AUTHOR]

This paper addresses the empirical phenomenon of adaptive instruction by a lesson case on covariational reasoning. Two author groups have investigated this lesson from distinct perspectives, one by emphasizing sign use and the other by emphasizing collective argumentation. How their approaches mutually complement each other guides the research this paper is presenting. I conduct a networking of theories approach as it offers strategies to relate the two frameworks, their enacted analyses and their results. The aim is to capture a synthesis of the two analyses to gain a more comprehensive understanding of the complexity of adaptive teaching. The result is a two-layer-model that distinguishes "two ways of being" in the behaviour of the teacher, s equentially , fostering students' conceptualizing of covariation while simultaneously , composing a structure of collective argumentation independent of the sequential order. Theoretically, this two-layer-model may promote new ways to network theories and practically, to guide teacher education. [ABSTRACT FROM AUTHOR]

Research in mathematics and science education reveals a disconnect for students as they attempt to apply their mathematical knowledge to science and engineering. With this conclusion in mind, this paper investigates a particular calculus topic that is used frequently in science and engineering: the definite integral. The results of this study demonstrate that certain conceptualizations of the definite integral, including the area under a curve and the values of an anti-derivative, are limited in their ability to help students make sense of contextualized integrals. In contrast, the Riemann sum-based “adding up pieces” conception of the definite integral (renamed in this paper as the “multiplicatively-based summation” conception) is helpful and useful in making sense of a variety of applied integral expressions and equations. Implications for curriculum and instruction are discussed. [ABSTRACT FROM AUTHOR]

The general context of this paper is the power of intuitive thinking, and how it can help or hinder analytical thinking. The research literature in cognitive psychology teems with tasks where intuitive thinking leads subjects to “non-normative” answers, including tasks for which they have all the knowledge necessary for the normative answer. The best explanation to date for such phenomena is dual-process theory, which stipulates the activation of a quick automatic intuitive process ( System 1 ), together with the failure of the heavy, lazy, and computationally expensive analytical process ( System 2 ) to intervene and correct the intuitive response. In an earlier paper, we have documented a clash between intuitive and analytical thinking concerning functions, which we have termed the changing-the-input phenomenon. The discovery of the changing-the-input phenomenon, however, left us with a puzzle: Why has this phenomenon concerning functions – a purely mathematical concept – been observed in computer science classes but not in mathematics ones? The purpose of the present paper is to address this puzzle. More generally we ask, under what conditions the changing-the-input phenomenon will or will not be manifested? Still more generally, in learning about functions, when is the intuitive scaffolding of functions via actions-on-tangible-objects helpful, and when does it get in the way of deeper understanding? [ABSTRACT FROM AUTHOR]