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.     

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.     

Conceptual metaphors for isomorphism and homomorphism: Instructors’ descriptions for themselves and when teaching

Group isomorphism and homomorphism are topics central to abstract algebra, yet research on instructors’ views of these concepts is limited. Based on interviews from two instructors as well as classroom video from eight class periods, this paper examines the language used to discuss isomorphism and homomorphism. Language used by instructors in interviews and classroom settings are identified and classified into four main categories: formal definition, mapping, sameness, and combinations of sameness and mapping language. How the two instructors drew on language classified into those four categories in the interview and instruction settings are examined for isomorphism and homomorphism. Similarities and differences between the interview and instruction contexts reveal the wide variety of ways of understanding isomorphism and homomorphism as well as a research need to examine mathematicians’ content knowledge in more than one context.     

Teaching methods comparison in a large calculus class

We report findings from a classroom experiment in which each of two sections of the same Calculus 1 course at a North American research-focused university were subject to an “intervention” week, each for a different topic, during which a less-experienced instructor encouraged a much higher level of student engagement, promoted active learning (answering “clicker” questions, small-group discussions, worksheets) during a significant portion of class time and built on assigned pre-class tasks. The lesson content and analysis of the assessments were informed by existing research on student learning of mathematics and student interviews, though the interventions and assessments were also intended to be compatible with typical course practices in an attempt to appeal to practitioners less familiar with the literature. Our study provides an example of active learning pedagogy (including materials and assessment used) for students at this level of mathematics in a classroom of over one hundred students, and we report improved student performance—on conceptual items in particular—with a switching replication in that each section outperformed the other on the topic for which it received the intervention.     

Problem Posing at All Levels in the Calculus Classroom

This article explores the use of problem posing in the calculus classroom using investigative projects. Specially, four examples of student work are examined, each one differing in originality of problem posed. By allowing students to explore actual questions that they have about calculus, coming from their own work or class discussion, or questions arising from studying supplementary material, all students can successfully engage in problem posing.     

Teachers’ responses to student mathematical thinking in Chinese elementary mathematics classrooms

Recent research on teachers’ use of student mathematical thinking (SMT) and recommendations for effective mathematics instruction claim that how teachers respond to SMT has great impact on student mathematical learning in the classroom. This study examined some Chinese mathematics teachers’ responses to student in-the-moment mathematical thinking that emerged during whole class discussion. The findings of this study revealed that the majority of Chinese elementary mathematics teachers in the data involved the whole group of students to make sense of in-the-moment SMT. They either invited students to digest SMT involved in the instance or provided an extension of the instance to further develop student mathematical understanding.     

Using Prediction to Promote Mathematical Understanding and Reasoning

Research has shown that prediction has the potential to promote the teaching and learning of mathematics because it can be used to enhance students' thinking and reasoning at all grade levels in various topics. This article addresses the effectiveness of using prediction on students' understanding and reasoning of mathematical concepts in a middle school algebra context. In the treatment classroom, prediction questions were utilized at the launch of each algebra lesson, and in the control classroom such questions were not used. Both classrooms were taught by the same teacher and used the same curriculum. After completing each of the linear and exponential units, the two classrooms were compared in terms of their mathematical understanding and reasoning through unit assessments. Overall, the treatment classroom outperformed the control classroom on the unit assessments. This result supports that prediction is a valid construct with respect to enhanced conceptual understanding and mathematical reasoning.     

Implementing inquiry-oriented curriculum: From the mathematicians’ perspective

As part of an effort to scale up an instructional innovation in abstract algebra, several mathematicians have implemented an inquiry-oriented, group theory curriculum. Three of those mathematicians (co-authors here) also participated in iterative rounds of interviews designed to document and investigate their experiences as they worked to implement this curriculum. Analyses of these interviews uncovered three themes that were important to these mathematicians: coverage, goals for student learning, and the role of the teacher. Here, by drawing on interview data, classroom data, and first person commentaries, we will present and discuss each teacher's perspective on these three themes.     

An analytical comparison of students’ reasoning in the context of Inquiry-Oriented Instruction: The case of span and linear independence

In this report we analyze differences in reasoning about span and linear independence by comparing written work of 126 linear algebra students whose instructors received support to implement a particular inquiry-oriented (IO) instructional approach compared to 129 students whose instructors did not receive that support. Our analysis of students’ responses to open-ended questions indicated that IO students’ concept images of span and linear independence were more aligned with the formal concept definition than the concept images of Non-IO students. Additionally, IO students exhibited more coordinated conceptual understandings and used deductive reasoning at higher rates than Non-IO students. We provide illustrative examples of systematic differences in how students from the two groups reasoned about span and linear independence.     

Asymptotic probability of majority inversion under a general binomial voting model

We obtain limits on the probability of majority inversion when the number of voters tends to infinity, for a binomial voting model specific to each state with different population sizes, and for different voting quotas in the two stages of the voting procedure. The case of weighted votes at the second stage is also discussed. For an important special case where the limit cannot be determined, we provide an exact expression for the inversion probability, but only for unweighted votes.     

Graphing in Groups: Learning About Lines in a Collaborative Classroom Network Environment

This article presents a design experiment in which we explore new structures for classroom collaboration supported by a classroom network of handheld graphing calculators. We describe a design for small group investigations of linear functions and present findings from its implementation in three high school algebra classrooms. Our coding of the problem-solving efforts of six student focus pairs in this environment over the course of several class sessions indicates that these students tended to move from exploratory and visual to more analytic means of establishing lines of a specified slope. As they adopted these analytic approaches, they were also more likely to enact their strategies jointly. In closer examination of emerging analytic strategies in episodes selected from the work of one of the pairs, we argue that the processes by which these students discovered the need for coordinated action on their respective points, and came to establish mathematical meaning for the relations between their coordinate locations as slope, were overlapping and intertwined.     

Assessing the Use of Graphing Calculators in College Algebra: Reflecting on Dimensions of Teaching and Learning

This article considers various aspects of teaching and learning stemming from the integration of graphing calculator use in a semester-long college algebra course. The project examined four class sections, in which two instructors each taught one section using graphing calculators and one section using a traditional approach. Achievement and attitude data showed no significant differences for treatment or instructor; a significant difference in achievement was found for gender. Students in the calculator sections responded to an open-ended questionnaire about their use of the calculator and were generally supportive of the technology. Students were in agreement about specific topics for which the technology was most useful. Overall, findings indicated that the technology use had a positive impact on various dimensions of student learning.     

Mathematics teaching practices with technology that support conceptual understanding for Latino/a students

We analyze how three seventh grade mathematics teachers from a majority Latino/a, linguistically diverse region of Texas taught the same lesson on interpreting graphs of motion as part of the Scaling Up SimCalc study (Roschelle et al., 2010). The students of two of the teachers made strong learning gains as measured by a curriculum-aligned assessment, while the students of the third teacher were less successful. To investigate these different outcomes, we compare the teaching practices in each classroom, focusing on the teachers’ use of class time and instructional format, their use of mathematical discourse practices in whole-class discussions, and their responses to student contributions. We show that the more successful teachers allowed time for students to use the curriculum and software and discuss it with peers, that they used formal mathematical discourse along with less formal language, and that they responded to student errors using higher-level moves. We conclude by discussing implications for teachers and mathematics educators, with special attention to issues related to the mathematics education of Latinos/as.     

Mobile-phone-based classroom response systems: Students’ perceptions of engagement and learning in a large undergraduate course

Using a Classroom Response System (CRS) has been associated with positive educational outcomes, by fostering student engagement and by allowing immediate feedback to both students and instructors. This study examined a low-cost CRS (VotApedia) in a large first-year class, where students responded to questions using their mobile phones. This study explored whether the use of VotApedia retained the advantages of other CRS, overcame some of the challenges of other CRS, and whether new challenges were introduced by using VotApedia. These issues were studied within three themes: students’ perceptions of using VotApedia; the impact of VotApedia on their engagement; and the impact of VotApedia on their learning. Data were collected from an online survey, focus groups and student feedback on teaching and course content. The results indicated that using VotApedia retains the pedagogical advantages of other CRS, while overcoming some of the challenges presented by using other CRS, without introducing any new challenges.     

Interactionist perspective on negotiation processes of students’ different understandings during small group work on linear algebra

Student group work represents a central learning setting within mathematics programs at the university level. In this study, a theoretical perspective on collaboration is adopted in which the differences between students’ interpretations of a mathematical concept are seen as an opportunity for individual restructuring processes. This so-called interactionist perspective is applied to student group work on linear algebra. The concepts of linear algebra at the university level are characterized by a versatility of different modes of expression and interpretation. For students of linear algebra, the flexible transitions between the different interpretations of linear algebra concepts usually pose a challenge. This study focuses on how students negotiate their different interpretations during group work on linear algebra and how transitions between interpretations might be stimulated or hindered. Video recordings of eight student groups working on a task that required flexible transition between interpretations of homomorphisms were sampled. The recordings were analyzed from an interactionist perspective, focusing on interaction situations in which the participating students expressed and negotiated different interpretations of homomorphisms. The analyses of students’ interactions highlight a phenomenon whereby differences in students’ interpretations remain implicit in group discussions, which constitutes an obstacle to the negotiation process.     

Voting power in a system of compound majority rule

In 1973, the United States Supreme Court ruled that a water district's voting scheme that apportioned votes on the basis of the assessed valuation of acreage in the district was constitutional. Among the justifications for the constitutionality of this scheme was the concurrent requirement that legislation be approved by a majority of voters as well as by a majority of weighted votes. However, analysis of this voting scheme in game-theoretic terms indicates that this justification is only partial: when two sets of winning coalitions must form simultaneously in order to pass legislation, the voting power of each voter in the combined system equals the mean of the voting power afforded each voter in each simple system. The results can be generalized to three or more concurrent requirements.     

Evaluating correct classification probability for weighted voting classifiers with plurality voting

Weighted voting classifiers (WVCs) consist of N units that each provide individual classification decisions. The entire system output is based on tallying the weighted votes for each decision and choosing the winning one (plurality voting) or one which has the total weight of supporting votes greater than some specified threshold (threshold voting). Each individual unit may abstain from voting. The entire system may also abstain from voting if no decision is ultimately winning. Existing methods of evaluating the correct classification probability (CCP) of WVCs can be applied to limited special cases of these systems (threshold voting) and impose some restrictions on their parameters. In this paper a method is suggested which allows the CCP of WVCs with both plurality and threshold voting to be exactly evaluated without imposing constraints on unit weights. The method is based on using the modified universal generating function technique.     

An experimental study of voting rules and polls in three-candidate elections

We report the results of elections conducted in a laboratory setting, modelled on a threecandidate example due to Borda. By paying subjects conditionally on election outcomes, we create electorates with (publicly) known preferences. We compare the results of experiments with and without non-binding pre-election polls under plurality rule, approval voting, and Borda rule. We also refer to a theory of voting “equilibria,” which makes sharp predictions concerning individual voter behavior and election outcomes. We find that Condorcet losers occasionally win regardless of the voting rule or presence of polls. Duverger's law (which asserts the predominance of two candidates) appears to hold under plurality rule, but close three-way races often arise under approval voting and Borda rule. Voters appear to poll and vote strategically. In elections, voters usually cast votes that are consistent with some strategic equilibrium. By the end of an election series, most votes are consistent with a single equilibrium, although that equilibrium varies by experimental group and voting rule.     

Children’s conceptions on the structure of an array: Using quick images as a gateway to multiplicative ideas

In this paper we report the conceptions about arrays that came to the fore as one class of second-grade students participated in whole classroom discussions and activities focused on the structure of arrays presented as a Quick Images routine. Before the intervention, students were not introduced to formal multiplication but had completed a unit on arrays. A constant comparative method was used to identify numeric and spatial structuring strategies that allowed for students’ conceptions about the structure of the array to emerge. Results indicated that not all students automatically use arrays as a composite of rows. We found that the use of Quick Images with larger arrays and non-arrays within the whole classroom discussion was successful at eliciting and directing students’ attention towards the spatial features of an array, including seeing an array as made of a composite of rows (or columns).