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.     

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.     

Integrating algebra and proof in high school mathematics: An exploratory study

Frequently, in the US students’ work with proofs is largely concentrated to the domain of high school geometry, thus providing students with a distorted image of what proof entails, which is at odds with the central role that proof plays in mathematics. Despite the centrality of proof in mathematics, there is a lack of studies addressing how to integrate proof into other mathematical domains. In this paper, we discuss a teaching experiment designed to integrate algebra and proof in the high school curriculum. Algebraic proof was envisioned as the vehicle that would provide high school students the opportunity to learn not only about proof in a context other than geometry, but also about aspects of algebra. Results from the experiment indicate that students meaningfully learned about aspects of both algebra and proof in that they produced algebraic proofs involving multiple variables, based on conjectures they themselves generated.     

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.     

Authority and whole-class proving in high school geometry: The case of Ms. Finley

Students’ experiences with proving in schools often lead them to see proof as a static product rather than a negotiated process that can help students justify and make sense of mathematical ideas. We investigated how authority manifested in whole-class proving episodes within Ms. Finley’s high school geometry classroom. We designed a coding scheme that helped us identify the proving actions and interactions that occurred during whole-class proving and how Ms. Finley and her students contributed to those processes. By considering the authority over proof initiation, proof construction, and proof validation, the episodes illustrate how whole-class proving interactions might relate to students’ potential development (or maintenance) of authoritative proof schemes. In particular, the authority of the teacher and textbook limited students’ opportunities to engage collectively in proving and sometimes allowed invalid arguments to be accepted in the public discourse. We offer suggestions for research and practice with respect to authority and proof instruction.     

Examining preservice teachers' responses to area conservation tasks

The purpose of this work was to explore how elementary preservice teachers responded to area conservation tasks. We administered written pre-assessments, followed by semi-structured interviews with 23 preservice teachers, asking them to respond to and reason with area conservation tasks. Findings highlighted several interesting preservice teachers' struggles when assessing area conservation tasks. In many cases, preservice teachers exhibited struggles similar to students, especially with regards to the justification of their area conservation claims. We provide recommendations to assist preservice teachers in their development of mathematical content knowledge in their teacher education programs, so that in the future they may better plan area lessons that promote procedural fluency from conceptual understanding in area measurement.     

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.     

Studying teachers’ mathematical argumentation in the context of refuting students’ invalid claims

This study investigates teachers’ argumentation aiming to convince students about the invalidity of their mathematical claims in the context of calculus. 18 secondary school mathematics teachers were given three hypothetical scenarios of a student's proof that included an invalid algebraic claim. The teachers were asked to identify possible mistakes and explain how they would refute the student's invalid claims. Two of them were also interviewed. The data were analysed in terms of the content and structure of argumentation and the types of counterexamples the teachers generated. The findings show that teachers used two main approaches to refute students’ invalid claims, the use of theory and the use of counterexamples. The role of these approaches in the argumentation process was analysed by Toulmin's model and three types of reasoning emerged that indicate the structure of argumentation in the case of refutation. Concerning the counterexamples, the study shows that few teachers use them in their argumentation and in general they underestimate their value as a proof method.     

Exploring the Link Between Preservice Teachers' Conception of Proof and the Use of Dynamic Geometry Software

This case study investigated how secondary preservice mathematics teachers perceive the need for and the benefits of formal proof when given geometric tasks in the context of dynamic geometry software. Results indicate that preservice teachers are concerned that after using dynamic software high school students will not see the need for proofs. The participants stated that multiple examples are not equivalent to a proof but, nonetheless, questioned the value of formal proof for high school students. Finally, preservice teachers found the greatest value of geometric software to be in helping students understand key relationships within a problem or theorem. Participants also tended to study a problem more deeply with the software than without it.     

Conceptual blending: Student reasoning when proving “conditional implies conditional” statements

Conceptual blending describes how humans condense information, combining it in novel ways. The blending process may create global insight or new detailed connections, but it may also result in a loss of information, causing confusion. In this paper, we describe the proof writing process of a group of four students in a university geometry course proving a statement of the form conditional implies conditional, i.e., (p → q) ⇒ (r → s). We use blending theory to provide insight into three diverse questions relevant for proof writing: (1) Where do key ideas for proofs come from?, (2) How do students structure their proofs and combine those structures with their more intuitive ideas?, and (3) How are students reasoning when they fail to keep track of the implication structure of the statements that they are using? We also use blending theory to describe the evolution of the students’ proof writing process through four episodes each described by a primary blend.     

Understanding kindergarten teachers’ perspectives of teaching basic geometric shapes: a phenomenographic research

Geometry is one of the disciplines children involve within early years of their lives. However, there is not much information about geometry education in Turkish kindergarten classes. The current study aims to examine teachers’ perspectives on teaching geometry in kindergarten classes. The researchers inquired about teachers’ in-class experiences in geometry and asked a series of questions such as “what are the benchmarks in your kindergarten class?”; “what kind of tools and materials you use to teach geometry in your class?”; “what shape do you teach first in your kindergarten class?”; “what do you expect to hear when you asked your students ‘what is square’?”; “how do you teach rectangular?”. The study utilized one of the qualitative research methods, namely phenomenography, to collect the data and analyze the data. The study involved with eight kindergarten teachers who work in different schools in central Kutahya, Turkey. The researchers collected data by conducting face-to-face half-structured interviews. The findings of this phenomenographic research showed that kindergarten teachers have some difficulties in teaching geometry and have lack of knowledge and skills in teaching geometry in kindergarten classes.     

An “inverse” relationship between mathematics identities and classroom practices among early career elementary teachers: The impact of accountability

This qualitative case study guided by portraiture examines the relationships between three early career elementary teachers’ beliefs about themselves in relation to mathematics (mathematics identities) and their classroom practices. Through autobiographical inquiry, reflective practice, classroom observations, interviews, and artifacts, findings show that all three second grade teachers appeared to have an “inverse” relationship between their mathematics identities and their classroom practices. In this relationship, as negative as they felt about themselves with regards to mathematics, they expended that much more effort to ensure that their students would have positive experiences with it and not be stigmatized by it as they had been. Accountability to schools, students, and parents, to increase student achievement appeared to play an important role in this relationship. Implications for preservice teacher education, inservice professional development, and research on beliefs and practices are discussed.     

A classification of teacher interventions in mathematics teaching

One of the best-known quotes in pedagogical literature is Maria Montessoris “Help me to do it myself.” This citation can be applied to many open questions. For example, how to help students working autonomously on cognitively demanding tasks is not only an unanswered question in didactical literature, but there has also been relatively little research done in this area. This article reflects upon qualitatively oriented studies from the German research project DISUM and selected literature about “teacher interventions”. Based on this, we propose, from a mathematically didactic point of view, a multi-dimensional framework, which allows us to identify central aspects of teacher interventions.     

Likelihood and sample size: The understandings of students and their teachers

The aim of this study was to consider the match of student statistical understanding and teacher pedagogical content knowledge in relation to sample size and likelihood. Students were given two contexts within which to compare the likelihood of events for different sample sizes. Teachers were presented with one of the contexts and asked what their students would do and how they would remediate incorrect responses. The data also provided the opportunity for a detailed hierarchical analysis of students’ and teachers’ understandings. Analysis of student solutions revealed a wide range of reasoning, some of which was apparently unfamiliar to teachers.     

Beliefs and beyond: hows and whys in the teaching of proof

In this paper, we report on a study aimed at describing the way secondary school teachers treat proof and at understanding which factors may influence such a treatment. This study is part of a wider project on proof carried out for many years. In our theoretical framework, we combine references from research on proof with those from research on teachers in relation to their beliefs. The study was carried out through interviews with secondary school teachers aimed at learning how they describe their work with proof in the classroom, and to elicit beliefs and other factors that shape this work. Through the interviews we were able to detect reasons behind teachers’ choices in planning their work in the classroom. In the present paper, we concentrate on four cases that, among other factors, offer elements suitable to unravel the problem of inconsistencies using the construct of leading beliefs, i.e., beliefs (whose nature may vary from teacher to teacher) that seem to drive the way each teacher treats proof.     

Integrating Algebra and Proof in High School: Students' Work with Multiple Variables and a Single Parameter in a Proof Context

In the United States, researchers argue that proof is largely concentrated in the domain of high school geometry, thus providing students a distorted image of what proof entails, which is at odds with the central role that proof plays in mathematics. Despite the centrality of proof, there is a lack of studies addressing how to integrate proof into other mathematical domains. In this article, we discuss a teaching experiment designed to integrate algebra and proof in the high school curriculum. Algebraic proof was envisioned as the vehicle that would provide high school students the opportunity to learn not only about proof in a context other than geometry but also about aspects of algebra. Results from the experiment indicate that students meaningfully learned about aspects of both algebra and proof in that they produced algebraic proofs involving multiple variables and a single parameter, based on conjectures they themselves generated.     

Transformations in the Visual Representation of a Figural Pattern

Multiple representations of a given mathematical object/concept are one of the biggest difficulties encountered by students. The aim of this study is to investigate the impact of the use of visual representations in teaching and learning algebra. In this paper, we analyze the transformations from and to visual representations that were performed by 18 students (aged between 10 and 13) in a task designed to explore a figural pattern. The data were collected from an audio recording of the class, the students’ work, and the teacher’s notes about each lesson. The results confirm that visual representations are important. However, visual treatments of any kind of representation are decisive, since they give students other possibilities for seeing and understanding tasks, continuity and flexibility in their activities, and the ability to make conversions between representations. The creative realization of visual treatments is necessary, and the teacher has a significant role in helping students to learn how to do this.     

Peer interactions in a computer lab: reflections on results of a case study involving web-based dynamic geometry sketches

A case study, originally set up to identify and describe some benefits and limitations of using dynamic web-based geometry sketches, provided an opportunity to examine peer interactions in a lab. Since classes were held in a computer lab, teachers and pairs faced the challenges of working and communicating in a lab environment.Research has shown that particular teacher interventions provide motivation for the consideration of new ideas, and help uncover misunderstandings that may interfere with student progress [Towers, J. (1999). In what ways do teachers interventions interact with and occasion the growth of students’ mathematical understanding. Doctoral Dissertation, University of British Columbia, Unpublished]. Examples of student discourse presented here suggest that certain peer interactions act in similar ways—helping propel students towards new understanding. On the other hand, they also show that some peer interactions, although superficially similar to teacher interventions, may hamper student progress.