The Development of Mathematical Discussions: An Investigation in a Fifth-Grade Classroom

Promoting discussion and argumentation of mathematical ideas among students are aspects of the vision for communication in recent school mathematics reform efforts. Having rich mathematical discussions, however, can present a variety of classroom challenges. Many factors influence classroom discussions and need to be addressed in ways that will assist teachers in creating more inquiry-based mathematics classrooms. The study presented here examined the development of mathematical discussions in a fifth-grade classroom over the course of a school year. Various aspects of the participants' interactions, teacher's pedagogy, and the classroom microculture were investigated. One major result is the evolution of student participation from nonactive listening to active listening and use of others' ideas to develop new conjectures. These changes were paralleled by changes in the teacher's role in the classroom and the nature of her questions, in particular.     

Possible Student Justification of Proofs

The National Council of Teachers of Mathematics calls for an increased emphasis on proof and reasoning in school mathematics curricula. Given such an emphasis, mathematics teachers must be prepared to structure curricular experiences so that students develop an appreciation for both the value of proof and for those strategies that will assist them in developing proving skills. Such an outcome is more likely when the teacher feels secure in his/her own understanding of the concept of “mathematical proof” and understands the ways in which students progress as they take on increasingly more complex mathematical justifications. In this article, a model of mathematical proof, based on Balacheff's Taxonomy of Mathematical Proof, outlining the levels through which students might progress as they develop proving skills is discussed. Specifically, examples of the various ways in which students operating at different levels of skill sophistication could approach three different mathematical proof tasks are presented. By considering proofs under this model, teachers are apt to gain a better understanding of each student's entry skill level and so effectively guide him/her toward successively more sophisticated skill development.     

Game Design as an Interactive Learning Environment for Fostering Students'' and Teachers'' Mathematical Inquiry

Many learning environments, computer-based or not, have been developed for either students or teachers alone to engage them in mathematical inquiry. While some headway has been made in both directions, few efforts have concentrated on creating learning environments that bring both teachers and students together in their teaching and learning. In the following paper, we propose game design as such a learning environment for students and teachers to build on and challenge their existing understandings of mathematics, engage in relevant and meaningful learning contexts, and develop connections among their mathematical ideas and their real world contexts. To examine the potential of this approach, we conducted and analyzed two studies: Study I focused on a team of four elementary school students designing games to teach fractions to younger students, Study II focused on teams of pre-service teachers engaged in the same task. We analyzed the various games designed by the different teams to understand how teachers and students conceptualize the task of creating virtual game learning environment for others, in which ways they integrate their understanding of fractions and develop notions about students' thinking in fractions, and how conceptual design tools can provide a common platform to develop meaningful fraction contexts. In our analysis, we found that most teachers and students, when left to their own devices, create instructional games to teach fractions that incorporate little of their knowledge. We found that when we provided teachers and students with conceptual design tools such as game screens and design directives that facilitated an integration of content and game context, the games as well as teachers' and students' thinking increased in their sophistication. In the discussion, we elaborate on how the design activities helped to integrate rarely used informal knowledge of students and teachers, how the conceptual design tools improved the instructional design process, and how students and teachers benefit in their mathematical inquiry from each others' perspectives. In the outlook, we discuss features for computational design learning environments. This revised version was published online in August 2006 with corrections to the Cover Date.     

Alice Bache Gould: mathematician in search of war work, 1918

Alliances between American mathematics and the military were not well defined when the United States entered World War I in April of 1917. Although academic mathematicians – like other American faculty members and students – were among the strongest supporters of the cause, it took time to identify how and where to utilize their technical training in the war effort. This paper investigates the case of Alice Bach Gould, one of E H Moore's mathematics graduate students at the University of Chicago, and her efforts to support US military endeavors with her scientific expertise. Although Gould's substantial research accomplishments in Spanish archives eventually surpassed her mathematical achievements, her quest for war work in 1917 nonetheless illustrates the difficulty of contributing mathematical training to patriotic service in the United States during World War I.     

Learning about teaching for understanding through the study of tutoring

In this study, we conducted a fine-grained analysis of an expert tutor's (Nancy Mack) tutorial actions as she attempted, successfully, to help students learn fractions with understanding. Our analysis revealed that, as Mack tutored students in two different research studies, she took two types of tutorial actions previously unrecorded in the literature. By analyzing her actions using a methodology involving production rules, we suggest how her content knowledge, pedagogical content knowledge, and her knowledge of her students were interrelated and how they impacted on her instructional decisions and teaching actions. We also provide an example of how using production rules can be useful to discern some of the complexities involved in teaching and tutoring.     

The Development of Mathematical Discussions: An Investigation in a Fifth-Grade Classroom

Promoting discussion and argumentation of mathematical ideas among students are aspects of the vision for communication in recent school mathematics reform efforts. Having rich mathematical discussions, however, can present a variety of classroom challenges. Many factors influence classroom discussions and need to be addressed in ways that will assist teachers in creating more inquiry-based mathematics classrooms. The study presented here examined the development of mathematical discussions in a fifth-grade classroom over the course of a school year. Various aspects of the participants' interactions, teacher's pedagogy, and the classroom microculture were investigated. One major result is the evolution of student participation from nonactive listening to active listening and use of others' ideas to develop new conjectures. These changes were paralleled by changes in the teacher's role in the classroom and the nature of her questions, in particular.     

Student Difficulties with Accessing and Using Mathematical Knowledge

This article examines the issue of why students fail to activate and use mathematical knowledge during problem solving when it is known that they possess the required knowledge. This issue is explored by analyzing problem-solving attempts of a high-achieving student and a low-achieving student in the domain of plane geometry. On the basis of these data and other literature, three major sources of mathematical knowledge-access difficulties are identified that might be considered by classroom teachers, including a student's (1) dispositional state, (2) management of the problem-solving process, and (3) state of organization of his or her mathematical knowledge. It is argued that teaching practices that place emphasis on careful management of problem-solving activity could help students activate and extend the use of mathematical knowledge acquired in lesson activities.     

A different perspective of the teaching philosophy of RL Moore

Dr RL Moore was undoubtedly one of the finest mathematics teachers ever. He developed a unique teaching method designed to teach his students to think like mathematicians. His method was not designed to convey any particular mathematical knowledge. Instead, it was designed to teach his students to think. Today, his method has been modified to focus on using student participation toward the goal of the conveyance of mathematical knowledge rather than on Dr Moore's goal of teaching students to think. This article proposes that undergraduates would be better served if they took at least one course using Dr Moore's original method and his original goal.     

Comparing German and Taiwanese secondary school students’ knowledge in solving mathematical modelling tasks requiring their assumptions

Mathematization is critical in providing students with challenges for solving modelling tasks. Inadequate assumptions in a modelling task lead to an inadequate situational model, and to an inadequate mathematical model for the problem situation. However, the role of assumptions in solving modelling problems has been investigated only rarely. In this study, we intentionally designed two types of assumptions in two modelling tasks, namely, one task that requires non-numerical assumptions only and another that requires both non-numerical and numerical assumptions. Moreover, conceptual knowledge and procedural knowledge are also two factors influencing students’ modelling performance. However, current studies comparing modelling performance between Western and non-Western students do not consider the differences in students’ knowledge. This gap in research intrigued us and prompted us to investigate whether Taiwanese students can still perform better than German students if students’ mathematical knowledge in solving modelling tasks is differentiated. The results of our study showed that the Taiwanese students had significantly higher mathematical knowledge than did the German students with regard to either conceptual knowledge or procedural knowledge. However, if students of both countries were on the same level of mathematical knowledge, the German students were found to have higher modelling performance compared to the Taiwanese students in solving the same modelling tasks, whether such tasks required non-numerical assumptions only, or both non-numerical and numerical assumptions. This study provides evidence that making assumptions is a strength of German students compared to Taiwanese students. Our findings imply that Western mathematics education may be more effective in improving students’ ability to solve holistic modelling problems.     

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.     

A sense-making approach to proof: Strategies of students in traditional and problem-based number theory courses

This paper reports the results of an exploratory study of the perceptions of and approaches to mathematical proof of undergraduates enrolled in lecture-based and problem-based “transition to proof” courses. While the students in the lecture-based course demonstrated conceptions of proof that reflect those reported in the research literature as insufficient and typical of undergraduates, the students in the problem-based course were found to hold conceptions of and approach the construction of proofs in ways that demonstrated efforts to make sense of mathematical ideas. This sense-making manifested itself in the ways in which students employed initial strategies, notation, prior knowledge and experiences, and concrete examples in the proof construction process. These differences were also seen when students were asked to determine the validity of completed proofs. These results suggest that such a problem-based course may provide opportunities for students to develop conceptions of proof that are more meaningful and robust than does a traditional lecture-based course.     

Investigating the transformation of a secondary teacher’s knowledge of trigonometric functions

Shulman (1987) defined pedagogical content knowledge as the knowledge required to transform subject-matter knowledge into curricular material and pedagogical representations. This paper presents the results of an exploratory case study that examined a secondary teacher’s knowledge of sine and cosine values in both clinical and professional settings to discern the characteristics of mathematical schemes that facilitate their transformation into learning artifacts and experiences for students. My analysis revealed that the teacher’s knowledge of sine and cosine values consisted of uncoordinated quantitative and arithmetic schemes and that he was cognizant only of the behavioral proficiencies these schemes enable, not the mental actions and conceptual operations they entail. Based on these findings, I hypothesize that the extent to which a teacher is consciously aware of the mental activity that comprises their mathematical conceptions influences their capacity to transform their mathematical knowledge into curricular material and pedagogical representations to effectively support students’ conceptual learning.     

The effectiveness of resources created by students as partners in explaining the relevance of mathematics in engineering education

First-year engineering students often struggle to see the relevance of theoretical mathematical concepts for their future studies and professional careers. This is an issue, as students who do not see relevance in fundamental parts of their studies may disengage from these parts and focus their efforts on other subjects they think will be more useful to them. In this study, we surveyed engineering students enrolled in a first-year mathematics subject on their perceptions of the relevance of the individual mathematical topics taught. Surveys were administered at the start of semester when some of these topics were unknown to them, and again at the end of semester when students had not only studied all these topics but also watched a set of animated videos. These videos had been produced by higher-year students to explain where they had seen applications of the mathematical concepts presented in the first year. We notice differences between the perceived relevance of topics for future study and for professional careers, with relevance to study rated higher than relevance to careers. We also find that the animations are seen as helpful in understanding the relevance of first-year mathematics. The majority of students indicated that lecturers with students as partners should work collaboratively to produce future videos.     

数学建模课内容和教学方法的探讨

拟就以下内容进行了探讨 .(i)该课程究竟应该讲什么内容、怎样讲 ,才能使学生在较短的时间内 ,掌握数学建模的基本知识和基本方法 ;(ii)该课程怎样与数学实验更好地结合起来 ,以培养学生的动手能力 ;(iii)该课程应采用什么样的教学手段和教学方法 ,才能加大课堂信息量 ,加强直观性和趣味性等 .我们的解决方法是 :(i)以介绍建立数学模型为主 ,按数学知识内容的不同来选取数学模型的典型案例 ,通过案例介绍 ,使学生学会怎样建立模型 .(ii)适当介绍数学软件包 ,让学生掌握运用软件包来求解模型能力 .(iii)做大作业 ,教员给出题目 ,学生自己收集资料、讨论、上机求解 ,最后写出报告 .(iv)开展多媒体教学 ,对主要的教学内容进行模块化教学 ,将建模分成 1 4个专题 ,做成 1 4个多媒体课件     

Toward the problem-centered classroom: trends in mathematical problem solving in Japan

In this paper, I summarize the influence of mathematical problem solving on mathematics education in Japan. During the 1980–1990s, many studies had been conducted under the title of problem solving, and, therefore, even until now, the curriculum, textbook, evaluation and teaching have been changing. Considering these, it is possible to identify several influences. They include that mathematical problem solving helped to (1) enable the deepening and widening of our knowledge of the students’ processes of thinking and learning mathematics, (2) stimulate our efforts to develop materials and effective ways of organizing lessons with problem solving, and (3) provide a powerful means of assessing students’ thinking and attitude. Before 1980, we had a history of both research and practice, based on the importance of mathematical thinking. This culture of mathematical thinking in Japanese mathematics education is the foundation of these influences.     

Coordinating Theories of Learning to Account for Second-Grade Children's Arithmetical Understandings

This article coordinates social constructivism and socioculturalism orientations to explain 2nd-grade children's reasoning with 2-digit quantities. From a social constructivist position, we illustrate how the classroom teacher and the students constituted what counted as an acceptable mathematical explanation. As children offered informal and conventional ways of interpreting problem situations, they were expected to reason with quantities in sensible ways. From a sociocultural position, we explain how the teacher's and students' contributions were situated within the mathematical ways of knowing constituted by the community at large. Particular children's contributions were clarified in terms of the ways in which they participated in socially organized activities. By coordinating these lenses, we argue the local classroom mathematical practices constrained and enabled the mathematical practices of the wider society.     

Enabling the Practice of Mathematics Teachers in Context: Toward a New Equity Research Agenda

In this article, I address the need for a more clearly articulated research agenda around equity issues by proposing a working definition of equity and a focal point for research. More specifically, I assert that rather than pitting them against each other, we must coordinate (a) efforts to get marginalized students to master what currently counts as “dominant” mathematics with (b) efforts to develop a critical perspective among all students about knowledge and society in ways that ultimately facilitate (c) a positive relationship between mathematics, people, and equity on the planet. I make this argument partly by reviewing the literature on (school) contexts that engage marginalized students in mathematics. Then, I argue that the place that holds the most promise for addressing equity is a research agenda that emphasizes enabling the practice of teachers and that draws more heavily on design-based and action research, thereby redefining what the practice of mathematics means along the way. Specific research questions are offered.     

Creative and algorithmic mathematical reasoning: effects of transfer-appropriate processing and effortful struggle

Two separate studies, Jonsson et al. (J. Math Behav. 2014;36: 20–32) and Karlsson Wirebring et al. (Trends Neurosci Educ. 2015;4(1–2):6–14), showed that learning mathematics using creative mathematical reasoning and constructing their own solution methods can be more efficient than if students use algorithmic reasoning and are given the solution procedures. It was argued that effortful struggle was the key that explained this difference. It was also argued that the results could not be explained by the effects of transfer-appropriate processing, although this was not empirically investigated. This study evaluated the hypotheses of transfer-appropriate processing and effortful struggle in relation to the specific characteristics associated with algorithmic reasoning task and creative mathematical reasoning task. In a between-subjects design, upper-secondary students were matched according to their working memory capacity.

The main finding was that the superior performance associated with practicing creative mathematical reasoning was mainly supported by effortful struggle, however, there was also an effect of transfer-appropriate processing. It is argued that students need to struggle with important mathematics that in turn facilitates the construction of knowledge. It is further argued that the way we construct mathematical tasks have consequences for how much effort students allocate to their task-solving attempt.     

Princess Elizabeth of Bohemia and Descartes’ letters (1650–1665)

After Descartes’ death in 1650, Princess Elizabeth generously shared with others several letters she had received from the philosopher, which contained philosophically as well as mathematically exciting material. In this article I place the transmission of these copies in context, revealing that Elizabeth steadily became an intellectually inspiring figure, attracting international attention. In the 1650s she stayed at Heidelberg where she discussed Cartesian philosophy with professors and students alike, including the professor of philosophy and mathematics Johann von Leuneschlos. In the mid-1660s, an initiative was taken from the English side of the Channel (Pell, More) to obtain Descartes’ mathematical letters to Elizabeth that had not yet been published. One letter of Elizabeth herself on this very subject has been preserved. The letter, addressed to Theodore Haak, will be published here for the first time. It is of special interest, because the princess supplies a general outline of her solution to the mathematical problem Descartes gave her to solve in 1643. It substantiates the hypothesis regarding Elizabeth’s solution earlier proposed by Henk Bos.     

New challenges in the 2011 revised middle school curriculum of South Korea: mathematical process and mathematical attitude

The ‘future-oriented middle school mathematics curriculum focused on creativity and personality’ was revised in August of 2011 with the aim of nurturing students’ mathematical creativity and sound personalities. The curriculum emphasizes: contextual learning from which students can grasp mathematical concepts and make connections with their everyday lives; manipulation activities through which students may attain an intuitive idea of what they are learning and enhance their creativity; and reasoning to justify mathematical results based on their knowledge and experience. Since students will not be able to engage in the intended mathematical process with the study-load imposed by the current curriculum, the newly revised curriculum modifies or deletes some parts of the contents that have been traditionally taught mechanically. This paper provides a detailed overview of the main points of the revised curriculum.