Students’ perceived sociomathematical norms: The missing paradigm

This study proposes a framework for research which takes into account three aspects of sociomathematical norms: teachers’ endorsed norms, teachers’ and students’ enacted norms, and students’ perceived norms. We investigate these aspects of sociomathematical norms in two elementary school classrooms in relation to mathematically based and practically based explanations. Results indicate that even when the observed enacted norms are in agreement with the teachers’ endorsed norms, the students may not perceive these same norms. These results highlight the need to consider the students’ perspective when investigating sociomathematical norms.     

Bridging the gap between mathematical conjecture and proof through computer-supported cognitive conflicts

In many mathematical problems, students can feel that the universalityof a conjecture or a formula is validated by their experimentand experience. In contrast, students generally do not feelthat deductive explanations strengthen their conviction thata conjecture or a formula is true. In order to cope up withstudents’ conviction based only on empirical experienceand to create a need for deductive explanations, we developeda problem-solving activity with technology support intendedto cause cognitive conflict. In this article, we describe theprocess conducted for this activity that led students to contradictionsbetween conjectures and findings. The teacher could create familiarproblem-solving situations and use students’ naïveinductive approaches to make students think mathematically andestablish the necessity for proof via computer support.     

Secondary school students’ levels of understanding in computing exponents

The aim of this study is to describe and analyze students’ levels of understanding of exponents within the context of procedural and conceptual learning via the conceptual change and prototypes’ theory. The study was conducted with 202 secondary school students with the use of a questionnaire and semi-structured interviews. The results suggest that three levels of understanding can be identified. At the first level students’ interpretation of exponents is based upon exponents that symbolize natural numbers. At Level 2, students’ knowledge acquisition process is a process of enrichment of the existing conceptual structures. Students at this level are able to compute exponents with negative numbers by extending the application of prototype examples. Finally, at Level 3 students not only extend the prototype examples but also reorganize their thinking in order to compute and compare exponents with roots, a concept which is quite different from the concept of exponents with natural numbers.     

Students’ reflections on their learning experiences: lessons from a longitudinal study on the development of mathematical ideas and reasoning

This paper characterizes the views on mathematical learning of five high school students based on the students’ reflections on their mathematical experiences in a longitudinal study that focused on the development of mathematical ideas and reasoning in particular research conditions. The students’ views are presented according to five themes about learning which describe the students’ views on the nature of knowledge and what it means to know, source of knowledge, motivation to engage in learning, certainty in knowing, and how the students’ views vary with particular areas of mathematical activity. The study addresses the need for more research on epistemological beliefs of students below college age. In particular, the results provide evidence that challenge the existing assumption that, prior to college, students exhibit naïve epistemological beliefs.     

The role of operating premises and reasoning paths in upper elementary students’ problem solving

The validity of students’ reasoning is central to problem solving. However, equally important are the operating premises from which students’ reason about problems. These premises are based on students’ interpretations of the problem information. This paper describes various premises that 11- and 12-year-old students derived from the information in a particular problem, and the way in which these premises formed part of their reasoning during a lesson. The teacher’s identification of differences in students’ premises for reasoning in this problem shifted the emphasis in a class discussion from the reconciliation of the various problem solutions and a focus on a sole correct reasoning path, to the identification of the students’ premises and the appropriateness of their various reasoning paths. Problem information that can be interpreted ambiguously creates rich mathematical opportunities because students are required to articulate their assumptions, and, thereby identify the origin of their reasoning, and to evaluate the assumptions and reasoning of their peers.     

Promoting productive mathematical classroom discourse with diverse students

Productive mathematical classroom discourse allows students to concentrate on sense making and reasoning; it allows teachers to reflect on students’ understanding and to stimulate mathematical thinking. The focus of the paper is to describe, through classroom vignettes of two teachers, the importance of including all students in classroom discourse and its influence on students’ mathematical thinking. Each classroom vignette illustrates one of four themes that emerged from the classroom discourse: (a) valuing students’ ideas, (b) exploring students’ answers, (c) incorporating students’ background knowledge, and (d) encouraging student-to-student communication. Recommendations for further research on classroom discourse in diverse settings are offered.     

Toward a student-centred process of teaching arithmetic

This article describes a way toward a student-centred process of teaching arithmetic, where the content is harmonized with the students’ conceptual levels. At school start, one classroom teacher is guided in recurrent teaching development meetings in order to develop teaching based on the students’ prerequisites and to successively learn the students’ arithmetic. The students are assessed in interviews. Two special teachers participate and their current models of each student's arithmetic are tested when assessing the students. The students’ conceptual diversity and the consequent different content in teaching are shown. Further, the special teachers’ assessments and the class teacher's opinion of the new way of teaching are reported. A wide range both of the students’ conceptual levels and of the kinds of relevant problems was found. The special teachers manage their duties well and the classroom teacher has so far been satisfied with the new teaching process.     

A geometry teacher's use of a metaphor in relation to a prototypical image to help students remember a set of theorems

This article asks the following: How does a teacher use a metaphor in relation to a prototypical image to help students remember a set of theorems? This question is analyzed through the case of a geometry teacher. The analysis uses Duval's work on the apprehension of diagrams to investigate how the teacher used a metaphor to remind students about the heuristics involved when applying a set of theorems during a problem-based lesson. The findings show that the teacher used the metaphor to help students recall the apprehensions of diagrams when applying several theorems. The metaphor was instrumental for mediating students’ work on a problem and the proof of a new theorem. The findings suggest that teachers’ use of metaphors in relation to prototypical images may facilitate how they organize students’ knowledge for later retrieval.     

A model of students’ combinatorial thinking

Combinatorial topics have become increasingly prevalent in K-12 and undergraduate curricula, yet research on combinatorics education indicates that students face difficulties when solving counting problems. The research community has not yet addressed students’ ways of thinking at a level that facilitates deeper understanding of how students conceptualize counting problems. To this end, a model of students’ combinatorial thinking was empirically and theoretically developed; it represents a conceptual analysis of students’ thinking related to counting and has been refined through analyzing students’ counting activity. In this paper, the model is presented, and relationships between formulas/expressions, counting processes, and sets of outcomes are elaborated. Additionally, the usefulness and potential explanatory power of the model are demonstrated through examining data both from a study the author conducted, and from existing literature on combinatorics education.     

Students’ whole number multiplicative concepts: A critical constructive resource for fraction composition schemes

This article reports on the activity of two pairs of sixth grade students who participated in an 8-month teaching experiment that investigated the students’ construction of fraction composition schemes. A fraction composition scheme consists of the operations and concepts used to determine, for example, the size of 1/3 of 1/5 of a whole in relation to the whole. Students’ whole number multiplicative concepts were found to be critical constructive resources for students’ fraction composition schemes. Specifically, the interiorization of two levels of units, a particular multiplicative concept, was found to be necessary for the construction of a unit fraction composition scheme, while the interiorization of three levels of units was necessary for the construction of a general fraction composition scheme. These findings contribute to previous research on students’ construction of fraction multiplication that has emphasized partitioning and conceptualizing quantitative units. Implications of the findings for teaching are considered.     

Flexibility and coordination among acts of visualization and analysis in a pattern generalization activity

This study aims at exploring processes of flexibility and coordination among acts of visualization and analysis in students’ attempt to reach a general formula for a three-dimensional pattern generalizing task.The investigation draws on a case-study analysis of two 15-year-old girls working together on a task in which they are asked to calculate the number of blocks in a three-dimensional tower of different heights. The students’ activity was video- and audio-taped, fully transcribed and lasted for 50 min.The analysis discloses several instances of how the students were linking acts of visualization and analysis to reach a general formula. However, regarding flexibility, we found that it was more natural for the students to change visual format than to change analytical position and direction in their attempts to generalize the three-dimensional pattern of the task in a closed formula.     

Exploring college students’ mental representations of inferential statistics

We report a case study that explored how three college students mentally represented the knowledge they held of inferential statistics, how this knowledge was connected, and how it was applied in two problem solving situations. A concept map task and two problem categorization tasks were used along with interviews to gather the data. We found that the students’ representations were based on incomplete statistical understanding. Although they grasped various concepts and inferential tests, the students rarely linked key concepts together or to tests nor did they accurately apply that knowledge to categorize word problems. We suggest that one reason the students had difficulty applying their knowledge is that it was not sufficiently integrated. In addition, we found that varying the instruction for the categorization task elicited different mental representations. One instruction was particularly effective in revealing students’ partial understandings. This finding suggests that modifying the task format as we have done could be a useful diagnostic tool.     

Why do U.S. and Chinese students think differently in mathematical problem solving?: Impact of early algebra learning and teachers’ beliefs

This paper reports two studies that examined the impact of early algebra learning and teachers’ beliefs on U.S. and Chinese students’ thinking. The first study examined the extent to which U.S. and Chinese students’ selection of solution strategies and representations is related to their opportunity to learn algebra. The second study examined the impact of teachers’ beliefs on their students’ thinking through analyzing U.S. and Chinese teachers’ scoring of student responses. The results of the first study showed that, for the U.S. sample, students who have formally learned algebraic concepts are as likely to use visual representations as those who have not formally learned algebraic concepts in their problem solving. For the Chinese sample, students rarely used visual representations whether or not they had formally learned algebraic concepts. The findings of the second study clearly showed that U.S. and Chinese teachers view students’ responses involving concrete strategies and visual representations differently. Moreover, although both U.S. and Chinese teachers value responses involving more generalized strategies and symbolic representations equally high, Chinese teachers expect 6th graders to use the generalized strategies to solve problems while U.S. teachers do not. The research reported in this paper contributed to our understanding of the differences between U.S. and Chinese students’ mathematical thinking. This research also established the feasibility of using teachers’ scoring of student responses as an alternative and effective way of examining teachers’ beliefs.     

Odds and evens: mathematical reasoning and informal proof among high school students

Ten high school algebra students were asked to judge simple statements about combining odd and even numbers, stating whether they were true or false. They were also asked to give justifications or explanations for their decisions. All of the students initially reasoned inductively or empirically, appealing to specific cases and justifying their answers with additional examples. On being prompted for any further explanations, seven of the students attempted to formulate some type of non-empirical rationale. However, only three students were able to create fairly coherent arguments, none of which used standard algebraic notation. Instead, two of these original, idiosyncratic arguments were based on visual representations of odd and even numbers, and the third consisted of an informal and partial argument by cases.

Intuition comes to us much earlier and with much less outside influence than formal arguments … Therefore, I think that in teaching high school age youngsters we should emphasize intuitive insight more than, and long before, deductive reasoning. —Polya, George (1981, pp. 2–128)

     

Instructional identities of geometry students

We inspect the hypothesis that geometry students may be oriented toward how they expect that the teacher will evaluate them as students or otherwise oriented to how they expect that their work will give them opportunities to do mathematics. The results reported here are based on a mixed-methods analysis of twenty-two interviews with high school geometry students. In these interviews students respond to three different tasks that presented students with an opportunity to do a proof. Students’ responses are coded according to a scheme based on the hypothesis above. Interviews are also coded using a quantitative linguistic ratio that gauges how prominent the teacher was in the students’ opinions about the viability of these proof tasks. These scores were used in a cluster analysis that yielded three student profiles that we characterize using composite profiles. These profiles highlight the different ways that students can experience proof in the geometry classroom.     

“Then you can take half … almost” — Elementary students learning bar graphs and pie charts in a computer-based context

Forty Swedish elementary students, 7-12 years of age and working in pairs, constructed a series of bar graphs and pie charts using a graphing application software as an instructional tool under the guidance of the researcher. After successive withdrawal of help, each pair drew a small number of graphic displays manually at the end of the data collection period. Evidence is provided that children's engagement with the graphing application software enhanced their understanding of essential graphical ideas and that even the youngest students appropriated and talked insightfully about a number of critical aspects of graphing. The students’ gradual mastering of different aspects of graphing is argued to be movements within their “zones of proximal development” towards a more competent use of graphs.     

Teachers’ mathematical activity in inquiry-oriented instruction

This work investigates the relationship between teachers’ mathematical activity and the mathematical activity of their students. By analyzing the classroom video data of mathematicians implementing an inquiry-oriented abstract algebra curriculum I was able to identify a variety of ways in which teachers engaged in mathematical activity in response to the mathematical activity of their students. Further, my analysis considered the interactions between teachers’ mathematical activity and the mathematical activity of their students. This analysis suggests that teachers’ mathematical activity can play a significant role in supporting students’ mathematical development, in that it has the potential to both support students’ mathematical activity and influence the mathematical discourse of the classroom community.     

Are beliefs believable? An investigation of college students’ epistemological beliefs and behavior in mathematics

College students’ epistemological belief in their academic performance of mathematics has been documented and is receiving increased attention. However, to what extent and in what ways problem solvers’ beliefs about the nature of mathematical knowledge and thinking impact their performances and behavior is not clear and deserves further investigation. The present study investigated how Taiwanese college students espousing unlike epistemological beliefs in mathematics performed differently within different contexts, and in what contexts these college students’ epistemological beliefs were consistent with their performances and behavior. Results yielded from the survey of students’ performances on standardized tests, semi-open problems, and their behaviors on pattern-finding tasks, suggest mixed consequences. It appears that beliefs played a more reliable role within the well-structured context but lost its credibility in non-standardized tasks.     

Learning trajectories in teacher education: Supporting teachers’ understandings of students’ mathematical thinking

Recent work by researchers has focused on synthesizing and elaborating knowledge of students’ thinking on particular concepts as core progressions called learning trajectories. Although useful at the level of curriculum development, assessment design, and the articulation of standards, evidence is only beginning to emerge to suggest how learning trajectories can be utilized in teacher education. Our paper reports on two studies investigating practicing and prospective elementary teachers’ uses of a learning trajectory to make sense of students’ thinking about a foundational idea of rational number reasoning. Findings suggest that a mathematics learning trajectory supports teachers in creating models of students’ thinking and in restructuring teachers’ own understandings of mathematics and students’ reasoning.     

The mathematical beliefs and behavior of high school students: Insights from a longitudinal study

There is a documented need for more research on the mathematical beliefs of students below college. In particular, there is a need for more studies on how the mathematical beliefs of these students impact their mathematical behavior in challenging mathematical tasks. This study examines the beliefs on mathematical learning of five high school students and the students’ mathematical behavior in a challenging probability task. The students were participants in an after-school, classroom-based, longitudinal study on students’ development of mathematical ideas funded by the United States National Science Foundation. The results show that particular educational experiences can alter results from previous studies on the mathematical beliefs and behavior of students below college, some of which have been used to justify non-reform pedagogical approaches in mathematics classrooms. Implications for classroom practice and ideas for future research are discussed.