Problem solving heuristics,affect, and discrete mathematics

It has been suggested that activities in discrete mathematics allow a kind of new beginning for students and teachers. Students who have been “turned off” by traditional school mathematics, and teachers who have long ago routinized their instruction, can find in the domain of discrete mathematics opportunities for mathematical discovery and interesting, nonroutine problem solving. Sometimes formerly low-achieving students demonstrate mathematical abilities their teachers did not know they had. To take maximum advantage of these possibilities, it is important to know what kinds of thinking during problem solving can be naturally evoked by discrete mathematical situations—so that in developing a curriculum, the objectives can include pathways to desired mathematical reasoning processes. This article discusses some of these ways of thinking, with special attention to the idea of “modeling the general on the particular.” Some comments are also offered about students' possible affective pathways and structures.     

If this is our mathematics, what are our stories?

This paper sets out to examine how narrative modes of thinking play a part in the claiming of mathematical territories as our own, in navigating mathematical landscapes and in conversing with the mathematical beings that inhabit them. We begin by exploring what constitutes the narrative mode, drawing principally on four characteristics identified by Bruner and considering how these characteristics manifest themselves in the activities of mathematicians. Using these characteristics, we then analyse a number of examples from our work with expressive technologies; we seek to identify the narrative in the interactions of the learners with different computational microworlds. By reflecting on the learners’ stories, we highlight how particular features, common across the microworlds—motion, colour, sound and the like—provided the basis for both the physical and psychological grounding of the behaviour of the mathematically constrained computational objects. In this way, students constructed and used narratives that involved situating mathematical activities in familiar contexts, whilst simultaneously expressing these activities in ways which—at least potentially—transcend the particularities of the story told.

Undergraduate students’ initial conceptions of factorials

Counting problems offer rich opportunities for students to engage in mathematical thinking, but they can be difficult for students to solve. In this paper, we present a study that examines student thinking about one concept within counting, factorials, which are a key aspect of many combinatorial ideas. In an effort to better understand students’ conceptions of factorials, we conducted interviews with 20 undergraduate students. We present a key distinction between computational versus combinatorial conceptions, and we explore three aspects of data that shed light on students’ conceptions (their initial characterizations, their definitions of 0!, and their responses to Likert-response questions). We present implications this may have for mathematics educators both within and separate from combinatorics.     

A case for combinatorics: A research commentary

In this commentary, we make a case for the explicit inclusion of combinatorial topics in mathematics curricula, where it is currently essentially absent. We suggest ways in which researchers might inform the field’s understanding of combinatorics and its potential role in curricula. We reflect on five decades of research that has been conducted since a call by Kapur (1970) for a greater focus on combinatorics in mathematics education. Specifically, we discuss the following five assertions: 1) Combinatorics is accessible, 2) Combinatorics problems provide opportunities for rich mathematical thinking, 3) Combinatorics fosters desirable mathematical practices, 4) Combinatorics can contribute positively to issues of equity in mathematics education, and 5) Combinatorics is a natural domain in which to examine and develop computational thinking and activity. Ultimately, we make a case for the valuable and unique ways in which combinatorics might effectively be leveraged within K-16 curricula.     

Values in preservice mathematics teachers’ discussions of the Body Mass Index - A critical perspective

This article explores the values that come to the fore when preservice mathematics teachers (PTs) ¹ engage in critical discussions about the role of mathematical models in society. The specific model that was discussed was the Body Mass Index (BMI) ². From the analysis of the PTs’ discussions of the BMI from a mathematical and societal point of view several mathematical and mathematics educational values were identified such as openness, rationalism, progress, reasoning, evaluating, and problematizing the instrumental understanding of mathematics. In addition, critical thinking about mathematics in society as emphasized in curricula in the three countries involved in the study, was identified with four categories of complementary pairs. Knowing the mathematical and mathematics educational values underpinning PTs’ discussions and their connection to critical thinking is important for successfully engaging with the role of mathematics in society.     

Promoting uncertainty to support preservice teachers’ reasoning about the tangent relationship

Including opportunities for students to experience uncertainty in solving mathematical tasks can prompt learners to resolve the uncertainty, leading to mathematical understanding. In this article, we examine how preservice secondary mathematics teachers’ thinking about a trigonometric relationship was impacted by a series of tasks that prompted uncertainty. Using dynamic geometry software, we asked preservice teachers to compare angle measures of lines on a coordinate grid to their slope values, beginning by investigating lines whose angle measures were in a near-linear relationship to their slopes. After encountering and resolving the uncertainty of the exact relationship between the values, preservice teachers connected what they learned to the tangent relationship and demonstrated new ways of thinking that entail quantitative and covariational reasoning about this trigonometric relationship. We argue that strategically using uncertainty can be an effective way of promoting preservice teachers’ reasoning about the tangent relationship.     

The catwalk task: Reflections and synthesis: Part 2

In this article we recount our experiences with a series of encounters with the catwalk task and reflect on the professional growth that these opportunities afforded. First, we individually reflect on our own mathematical work on the catwalk task. Second, we reflect on our experiences working with a group of community college students on the catwalk task and our interpretations of their mathematical thinking. In so doing we also detail a number of innovative and novel student-generated representations of the catwalk photos. Finally, we each individually reflect on the entire experience with the catwalk problem, as mathematics learners, as teachers, and as professionals.     

Proofs as bearers of mathematical knowledge

Yehuda Rav’s inspiring paper “Why do we prove theorems?” published in Philosophia Mathematica (1999, 7, pp. 5–41) has interesting implications for mathematics education. We examine Rav’s central ideas on proof—that proofs convey important elements of mathematics such as strategies and methods, that it is “proofs rather than theorems that are the bearers of mathematical knowledge”and thus that proofs should be the primary focus of mathematical interest—and then discuss their significance for mathematics education in general and for the teaching of proof in particular.     

Investigating students’ perceptions of graduate learning outcomes in mathematics

ABSTRACT

The purpose of this study is to explore the perceptions mathematics students have of the knowledge and skills they develop throughout their programme of study. It addresses current concerns about the employability of mathematics graduates by contributing much needed insight into how degree programmes are developing broader learning outcomes for students majoring in mathematics. Specifically, the study asked students who were close to completing a mathematics major (n = 144) to indicate the extent to which opportunities to develop mathematical knowledge along with more transferable skills (communication to experts and non-experts, writing, working in teams and thinking ethically) were included and assessed in their major. Their perceptions were compared to the importance they assign to each of these outcomes, their own assessment of improvement during the programme and their confidence in applying these outcomes. Overall, the findings reveal a pattern of high levels of students’ agreement that these outcomes are important, but evidence a startling gap when compared to students’ perceptions of the extent to which many of these – communication, writing, teamwork and ethical thinking – are actually included and assessed in the curriculum, and their confidence in using such learning.     

An Exploratory Study of College Students' Views of Mathematical Thinking in a Historical Approach Calculus Course

Heuristic training alone is not enough for developing one's mathematical thinking. One missing component is a mathematical point of view. This study reports findings regarding outcomes of a historical approach calculus course to foster Taiwanese college students' views of mathematical thinking. This study consisted of 3 stages. During the initial phase, 44 engineering majors' views on mathematical thinking were tabulated by an open-ended questionnaire, and 9 randomly selected students were invited to participate in follow-up interviews. Students then received an 18-week historical approach calculus course in which mathematical concepts were problematized to challenge their intuition-based empirical beliefs about doing mathematics. Near the end of the semester, all participants answered the identical questionnaire, and we interviewed the same students to pinpoint any shifts in their views on mathematical thinking. We found that participants were more likely to value logical sense, creativity, and imagination in doing mathematics. Further, students were leaning toward a conservative attitude toward certainty of mathematical knowledge. Participants' focus seemingly shifted from mathematics as a product to mathematics as a process.     

An Exploratory Study of College Students' Views of Mathematical Thinking in a Historical Approach Calculus Course

Heuristic training alone is not enough for developing one's mathematical thinking. One missing component is a mathematical point of view. This study reports findings regarding outcomes of a historical approach calculus course to foster Taiwanese college students' views of mathematical thinking. This study consisted of 3 stages. During the initial phase, 44 engineering majors' views on mathematical thinking were tabulated by an open-ended questionnaire, and 9 randomly selected students were invited to participate in follow-up interviews. Students then received an 18-week historical approach calculus course in which mathematical concepts were problematized to challenge their intuition-based empirical beliefs about doing mathematics. Near the end of the semester, all participants answered the identical questionnaire, and we interviewed the same students to pinpoint any shifts in their views on mathematical thinking. We found that participants were more likely to value logical sense, creativity, and imagination in doing mathematics. Further, students were leaning toward a conservative attitude toward certainty of mathematical knowledge. Participants' focus seemingly shifted from mathematics as a product to mathematics as a process.     

Asking mathematical questions mathematically

It is proposed that the style and format of the questions used by lecturers and tutors profoundly influence students' conceptions of what mathematics is about and how it is conducted. By looking at reasons for asking questions, and becoming aware of different types of questions which mathematicians typically ask themselves, we can enrich students' experience of mathematics. Drawing on recent work by Watson and Mason stimulated by the ideas of Zygfryd Dyrszlag, the paper proposes that mathematical themes, powers, heuristics and activities generate a mathematical discourse which is not always represented in the questions students are asked, and that the real purpose of questions is to provoke students into construal, into constructing their own stories which constitute meaning and understanding, and which equip them for the future. The use of questions of whatever type depends on both scaffolding and fading their use with and in front of students, so that students internalize the questions into their own learning and doing of mathematics. The framework directed—prompted—spontaneous is proposed as an alternative to scaffolding—fading for informing interactions with students.     

Advanced Mathematical Thinking

In this article we propose the following definition for advanced mathematical thinking: Thinking that requires deductive and rigorous reasoning about mathematical notions that are not entirely accessible to us through our five senses. We argue that this definition is not necessarily tied to a particular kind of educational experience; nor is it tied to a particular level of mathematics. We also give examples to illustrate the distinction we make between advanced mathematical thinking and elementary mathematical thinking. In particular, we discuss which kind of thinking may be required depending on the size of a mathematical problem, including problems involving infinity, and the types of models that are available.     

Measurement as relational,intensive and analogical: Towards a minor mathematics

Minor mathematics refers to the mathematical practices that are often erased by state-sanctioned curricular images of mathematics. We use the idea of a minor mathematics to explore alternative measurement practices. We argue that minor measurement practices have been buried by a ‘major’ settler mathematics, a process of erasure that distributes ‘sensibility’ and formulates conditions of mathematics dis/ability. We emphasize how measuring involves the making and mixing of analogies, and that this involves attending to intensive relationships rather than extensive properties. Our philosophical and historical approach moves from the archeological origins of human measurement activity, to pivotal developments in modern mathematics, to configurations of curriculum. We argue that the project of proliferating multiple mathematics is required in order to disturb narrow (and perhaps white, western, male) images of mathematics—and to open up opportunities for a more pluralist and inclusive school mathematics.     

The role of examples in the learning of mathematics and in everyday thought processes

The purpose of this paper is to present a view of three central conceptual activities in the learning of mathematics: concept formation, conjecture formation and conjecture verification. These activities also take place in everyday thinking, in which the role of examples is crucial. Contrary to mathematics, in everyday thinking examples are, very often, the only tool by which we can form concepts and conjectures, and verify them. Thus, relying on examples in these activities in everyday thought processes becomes immediate and natural. In mathematics, however, we form concepts by means of definitions and verify conjectures by mathematical proofs. Thus, mathematics imposes on students certain ways of thinking, which are counterintuitive and not spontaneous. In other words, mathematical thinking requires a kind of inhibition from the learners. The question is to what extent this goal can be achieved. It is quite clear that some people can achieve it. It is also quite clear that many people cannot achieve it. The crucial question is what percentage of the population is interested in achieving it or, moreover, what percentage of the population really cares about it.     

Which mathematics should we teach engineering students? An empirically grounded case for a broad notion of mathematical thinking

While many engineering educators have proposed changes to theway that mathematics is taught to engineers, the focus has oftenbeen on mathematical content knowledge. Work from the mathematicseducation community suggests that it may be beneficial to considera broader notion of mathematics: mathematical thinking. Schoenfeldidentifies five aspects of mathematical thinking: the mathematicscontent knowledge we want engineering students to learn as wellas problem-solving strategies, use of resources, attitudes andpractices. If we further consider the social and material resourcesavailable to students and the mathematical practices studentsengage in, we have a more complete understanding of the breadthof mathematics and mathematical thinking necessary for engineeringpractice. This article further discusses each of these aspectsof mathematical thinking and offers examples of mathematicalthinking practices based in the authors' previous empiricalstudies of engineering students' and practitioners' uses ofmathematics. The article also offers insights to inform theteaching of mathematics to engineering students.     

What does it mean to critique? An exploration of the types of critiques promoted in seventh-grade mathematics curriculum and assessment materials

While research in mathematics education has shown that mathematics assessments are highly consequential, traditional assessments often lag behind advancements in instructional methods. One such advancement is the promotion of mathematical habits of mind such as students' abilities to critique others' reasoning. This study explored the use of student work embedded in seventh-grade curriculum-based mathematics assessment tasks as a mechanism for critiquing others' thinking. The researcher investigated the prevalence and nature of student work in assessment tasks as compared to textbook tasks from five seventh-grade, Common Core State Standards for Mathematics-aligned curriculum series. The text analyses findings revealed that while there were multiple critique types in student work across both assessment and textbook tasks, there were substantial differences between students' opportunities to make sense of someone else's mathematical thinking in curriculum-based assessments as compared to the student textbooks. These findings reinforced prior curriculum and assessment research that found assessment often lags behind instructional methods.     

Mathematics Learning and Participation as Racialized Forms of Experience: African American Parents Speak on the Struggle for Mathematics Literacy

This article draws on 3 ethnographic and participant observation studies of African American parents and adults from 3 northern California communities. Although studies have shown that African American parents hold the same folk theories about mathematics as other parents, stressing it as an important school subject, few studies have sought to directly examine their beliefs about constraints and opportunities associated with mathematics learning for both themselves and their children. I argue that, as they situate the struggle for mathematical literacy within the larger contexts of African American, political, socioeconomic, and educational struggle, these parents help reveal that mathematics learning and participation can be conceptualized as racialized forms of experience. As they attempt to become doers of mathematics and advocates for their children's mathematics learning, discriminatory experiences have continued to subjugate some of these parents, whereas others—as demonstrated in their oppositional voices and behaviors—resisted their continued subjugation based on a belief that mathematics knowledge, beyond its role in schools, can be used to change the conditions of their lives. The characterization of mathematics learning as racialized experience put forth in this article contrasts with culture-free and situated perspectives of mathematics learning often found in the literature. As a result of their experiences with oppression in this society, the concept of race has historically played a major role in the lives of African Americans. Although race has dubious value as a scientific classification system, it has had real consequences for the life experiences and life opportunities of African Americans in the United States. Race is a socially constructed concept which is [a] defining characteristic for African American group membership. (Sellers, Smith, Shelton, Rowley, & Chavous, 1998, p. 18)     

Developing shift problems to foster geometrical proof and understanding

Meaningful learning of formal mathematics in regular classrooms remains a problem in mathematics education. Research shows that instructional approaches in which students work collaboratively on tasks that are tailored to problem solving and reflection can improve students’ learning in experimental classrooms. However, these sequences involve often carefully constructed reinvention route, which do not fit the needs of teachers and students working from conventional curriculum materials. To help to narrow this gap, we developed an intervention—‘shift problem lessons’. The aim of this article is to discuss the design of shift problems and to analyze learning processes occurring when students are working on the tasks. Specifically, we discuss three paradigmatic episodes based on data from a teaching experiment in geometrical proof. The episodes show that is possible to create a micro-learning ecology where regular students are seriously involved in mathematical discussions, ground their mathematical understanding and strengthen their relational framework.