Pre-service mathematics teachers’ concept images of radian

This study investigates pre-service mathematics teachers’ concept images of radian and possible sources of such images. A multiple-case study was conducted for this study. Forty-two pre-service mathematics teachers completed a questionnaire, which aims to assess their understanding of radian. Six of them were selected for individual interviews on the basis of theoretical sampling. The data indicated that participants’ concept images of radian were dominated by their concept images of degree. As the data in this study suggested, pre-service mathematics teachers were reluctant to accept trigonometric functions with the inputs of real numbers but rather they use value in degrees. More interestingly, they have two distinct images of π : π as an angle in radian and π as an irrational number.     

Preservice teachers’ pictorial strategies for a multistep multiplicative fraction problem

Previous research has documented that preservice teachers (PSTs) struggle with understanding fraction concepts and operations, and misconceptions often stem from their understanding of the referent whole. This study expands research on PSTs’ understanding of wholes by investigating pictorial strategies that 85 PSTs constructed for a multistep fraction task in a multiplicative context. The results show that many PSTs were able to construct valid pictorial strategies, and the strategies were widely diverse with respect to how they made sense of an unknown referent whole of a fraction in multiple steps, how they represented the wholes in their drawings, in which order they did multiple steps, and which type of model they used (area or set). Based on their wide range of pictorial strategies, we discuss potential benefits of PSTs’ construction of their own representations for a word problem in developing problem solving skills.     

Student teachers’ concept definitions of area and their understanding about two-dimensionality of area

We examine what kind of concept definitions of area a group of Finnish primary and lower secondary student teachers (N = 82) use, and how the quality of the definitions is associated with the participants’ success in seven exercises involving area. We are especially interested in how the understanding of the two-dimensionality of area appears in the participants’ responses. Only six student teachers were able to give a mathematically precise and correct definition of area. Altogether 26 participants defined it as ‘the size of a figure’ and 20 respondents required that a figure must be bounded. Further, 22 of them associated area with a formula or an example and eight respondents gave an incorrect or nonsensical definition. On average, student teachers master rather well the area formulae of a circle and a rectangle but already the relationship between the surface area of a cube and its volume is less commonly perceived. Most student teachers associate the area of an irregular domain with the method of exhaustion but clearly fewer of them acknowledge the difference between the area and an approximation of it. Surprisingly, there is only a weak Spearman correlation between the participants’ scores in the test exercises and the qualitatively ordered categories of concept definitions.     

Are self-constructed and student-generated arguments acceptable proofs? Pre-service secondary mathematics teachers’ evaluations

This study examined 14 pre-service secondary mathematics teachers’ productions and their evaluations of self-constructed and student-generated arguments in the domains of algebra, geometry, and number theory. Pre-service secondary mathematics teachers’ (PSMTs) evaluations of their own arguments indicate if they considered self-productions as proofs from a learner perspective. Similarly, PSMTs’ evaluations of student-generated arguments indicate if they decided given students’ productions could be counted as proofs from a teacher perspective. Our results show that the majority of PSMTs suspected that their invalid productions did not qualify as proofs. Furthermore, the PSMTs who were confident with their work and claimed that they had constructed a proof were more likely to make a correct judgment on four of the six student-generated arguments. We discuss implications of these findings for supporting PSMTs’ learning of proof and future research on the construction-evaluation activity.     

Prospective elementary teachers’ claiming in responses to false generalizations

When faced with a false generalization and a counterexample, what types of claims do prospective K-8 teachers make, and what factors influence the type and prudence of their claims relative to the data, observations, and arguments reported? This article addresses that question. Responses to refutation tasks and cognitive interviews were used to explore claiming. It was found that prospective K-8 teachers’ claiming can be influenced by knowledge of argumentation; knowledge and use of the mathematical practice of exception barring; perceptions of the task; use of natural language; knowledge of, use of, and skill with the mathematics register; and abilities to technically handle data or conceptual insights. A distinction between technical handlings for developing claims and technical handlings for supporting claims was made. It was found that prudent claims can arise from arguer-developed representations that afford conceptual insights, even when searching for support for a different claim.     

School mathematics curriculum materials for teachers’ learning: future elementary teachers’ interactions with curriculum materials in a mathematics course in the United States

This report describes ways that five preservice teachers in the United States viewed and interacted with the rhetorical components (Valverde et al. in According to the book: using TIMSS to investigate the translation of policy into practice through the world of textbooks, Kluwer, 2002) of the innovative school mathematics curriculum materials used in a mathematics course for future elementary teachers. The preservice teachers’ comments reflected general agreement that the innovative curriculum materials contained fewer narrative elements and worked examples, as well as more (and different) exercises and question sets and activity elements, than the mathematics textbooks to which the teachers were accustomed. However, variation emerged when considering the ways in which the teachers interacted with the materials for their learning of mathematics. Whereas some teachers accepted and even embraced changes to the teaching–learning process that accompanied use of the curriculum materials, other teachers experienced discomfort and frustration at times. Nonetheless, each teacher considered that use of the curriculum materials improved her mathematical understandings in significant ways. Implications of these results for mathematics teacher education are discussed.     

Pre- and in-service teachers’ perceived value of an experimental real analysis course for teachers

Prospective secondary mathematics teachers are usually required to complete several university advanced mathematics courses before being certified to teach secondary mathematics. However, teachers usually do not find these courses to be valuable for their teaching. We designed an experimental real analysis course with the goal of making real analysis content useful and relevant to teaching. Our approach was to ground the real analysis content in pedagogical situations that problematized a secondary mathematics topic, where the nuances of teaching secondary mathematics could be informed by the real analysis that was covered. The experimental course was implemented in a graduate teacher education programme with 32 pre- and in-service teachers (PISTs). After the course, we conducted focus group interviews with 20 of these PISTs to get feedback on how the course was valuable to their teaching practice. Many PISTs found the course to be valuable for teaching secondary mathematics, as well as for their understanding of secondary mathematics and real analysis.     

Addressing prospective elementary teachers’ mathematics subject matter knowledge through action research

There is international dissatisfaction regarding the standard of mathematics subject matter knowledge (MSMK) evident among both qualified and prospective elementary teachers. Ireland is no exception. Following increasing anecdotal evidence of prospective elementary teachers in one Irish College of Education (provider of initial teacher education programme) demonstrating weaknesses in this regard, this study sought to examine and address the issue through two cycles of action research. The examination of the nature of prospective teachers’ MSMK (as well as related beliefs in the main study) informed the design and implementation of an intervention to address the issue. A mixed method approach was taken throughout. In both cycles, Shapiro's criteria were used as a conceptual framework for the evaluation of the initiative. This paper focuses on the perceived and actual effects of the intervention on participants’ MSMK. As well as its contribution at a local and national level, the study provides an Irish perspective on approaches taken to address the phenomenon internationally.     

The roles of tools and models in a prospective elementary teachers’ developing understanding of multidigit multiplication

It is important for prospective elementary teachers to understand multidigit multiplication deeply; however, the development of such understanding presents challenges. We document the development of a prospective elementary teacher’s reasoning about multidigit multiplication during a Number and Operations course. We present evidence of profound progress in Valerie’s understanding of multidigit multiplication, and we highlight the roles of particular tools and models in her developing reasoning. In this way, we contribute an illuminating case study that can inform the work of mathematics teacher educators. We discuss specific instructional implications that derive from this case.     

Future mathematics teachers’ professional knowledge of elementary mathematics from an advanced standpoint

This paper reports a joint research project by researchers from three countries on an international comparative study that examines the professional knowledge of prospective mathematics teachers in elementary mathematics from an advanced standpoint. For this study, mathematical problems on various topics of elementary mathematical content were developed. Using this instrument, the mathematical knowledge of future teachers from Germany, Hong Kong, China (Hangzhou) and South Korea was measured empirically. The paper presents the design of the study, and also results are discussed. The results show that there are systematic differences among the participating countries; for example, the Korean future teachers outperform their counterparts in other countries. A more detailed analysis of the results suggests that the future teachers often do not seem to be able to link school and university knowledge systematically and cannot achieve the crucial “advanced standpoint” from the teacher training programme.     

Prospective and current secondary mathematics teachers’ criteria for evaluating mathematical cognitive technologies

As technology becomes more ubiquitous in the mathematics classroom, teachers are being asked to incorporate it into their lessons more than ever before. The amount of resources available online is staggering and teachers need to be able to analyse and identify resources that would be most appropriate and effective with their students. This study examines the criteria prospective and current secondary mathematics teachers use and value most when evaluating mathematical cognitive technologies (MCTs). Results indicate all groups of participants developed criteria focused on how well an MCT represents the mathematics, student interaction and engagement with the MCT, and whether the MCT was user-friendly. However, none of their criteria focused on how well an MCT would reflect students’ solution strategies or illuminate their thinking. In addition, there were some differences between the criteria created by participants with and without teaching experience, specifically the types of supports available in an MCT. Implications for mathematics teacher educators are discussed.     

A tiling proof of Euler’s Pentagonal Number Theorem and generalizations

The Ramanujan Journal - In two papers, Little and Sellers introduced an exciting new combinatorial method for proving partition identities which is not directly bijective. Instead, they consider...     

Pre-service science teachers’ perceptions of mathematics courses in a science teacher education programme1

Most science departments offer compulsory mathematics courses to their students with the expectation that students can apply their experience from the mathematics courses to other fields of study, including science. The current study first aims to investigate the views of pre-service science teachers of science-teaching preparation degrees and their expectations regarding the difficulty level of mathematics courses in science-teaching education programmes. Second, the study investigates changes and the reasons behind the changes in their interest regarding mathematics after completing these courses. Third, the current study seeks to reveal undergraduate science teachers’ opinions regarding the contribution of undergraduate mathematics courses to their professional development. Being qualitative in nature, this study was a case study. According to the results, almost all of the students considered that undergraduate mathematics courses were ‘difficult’ because of the complex and intensive content of the courses and their poor background mathematical knowledge. Moreover, the majority of science undergraduates mentioned that mathematics would contribute to their professional development as a science teacher. On the other hand, they declared a negative change in their attitude towards mathematics after completing the mathematics courses due to continuous failure at mathematics and their teachers’ lack of knowledge in terms of teaching mathematics.     

Mathematics teachers’ support and retention: using Maslow's hierarchy to understand teachers’ needs

As part of a larger study, four mathematics teachers from diverse backgrounds and teaching situations report their ideas on teacher stress, mathematics teacher retention, and their feelings about the needs of mathematics teachers, as well as other information crucial to retaining quality teachers. The responses from the participants were used to develop a hierarchy of teachers’ needs that resembles Maslow's hierarchy, which can be used to better support teachers in various stages of their careers. The interviews revealed both non content-specific and content-specific needs within the hierarchy. The responses show that teachers found different schools foster different stress levels and that as teachers they used a number of resources for reducing stress. Other mathematics-specific ideas are also discussed such as the amount of content and pedagogy courses required for certification.     

The relationship between preservice elementary teachers’ solutions to a pattern generalization problem and difficulties they anticipate in teaching it

To explore the relationship between elementary preservice teachers’ (PTs’) solutions to a pattern generalization problem and the difficulties they expected to encounter when teaching the same problem to students, we administered a task-based questionnaire to 154 participants at a large Southwestern university in the US. Employing inductive content analysis, we identified possible links between PTs' solutions and their anticipated difficulties. PTs who solved the problem using figurative reasoning tended to anticipate difficulties related to pedagogical moves to support students’ mathematical understanding. In contrast, PTs who solved the problem using algebraic formulations were likely to anticipate difficulties related to teaching algebraic knowledge and supporting procedural fluency. Also, only PTs who solved the problem using figurative reasoning anticipated difficulties associated with eliciting and evaluating student thinking, whereas PTs who used formulas to solve the problem expected difficulties related to their own self-efficacy and confidence. We discuss three implications for mathematics teacher education.     

Secondary teachers’ meanings for function notation in the United States and South Korea

This study investigates what teachers in U.S. reveal about their meanings for function notation in their written responses to the Mathematical Meanings for Teaching secondary mathematics (MMTsm) items, with particular attention to how productive those meanings would be if conveyed to students in a classroom setting. We then report South Korean teachers’ responses to see whether the meanings U.S. teachers demonstrated are shared with South Korean teachers. The results show that many U.S. teachers use function notation to name rules instead of to represent relationships. The data from South Korean teachers indicates that the problematic meanings in U.S. teachers’ responses are shared with a minority of South Korean teachers. The results suggest a need for attention to ideas regarding function notation in teacher education for pre-service teachers and professional development programs for in-service teachers.     

Secondary mathematics teachers’ meanings for measure,slope, and rate of change

This article reports an investigation of 251 high school mathematics teachers’ meanings for slope, measurement, and rate of change. The data was collected with a validated written instrument designed to diagnose teachers' mathematical meanings. Most teachers conveyed primarily additive and formulaic meanings for slope and rate of change on written items. Few teachers conveyed that a rate of change compares the relative sizes of changes in two quantities. Teachers’ weak measurement schemes were associated with limited meanings for rate of change. Overall, the data suggests that rate of change should be a topic of targeted professional development.     

The challenges facing initial teacher education: Irish prospective elementary teachers’ mathematics subject matter knowledge

Given the acknowledged relationship between teachers’ knowledge, their teaching and pupil learning, teachers’ mathematics subject matter knowledge (MSMK) has received increased attention internationally. As children's early mathematics experiences have been recognized as a critical stage, elementary teachers’ MSMK has become a focal point among researchers and policy makers alike. International research findings have uncovered that in many cases, there is a mismatch between what is perceived to be an appropriate MSMK for teaching elementary mathematics and that demonstrated by many qualified and prospective elementary teachers. Following repeated incidences of weak MSMK during interactions with prospective elementary teachers in one Irish College of Education (provider of initial teacher education programme for elementary teachers), this study sought to examine and address the issue purposefully through two cycles of action research. This article focuses on the data collected prospective teachers’ MSMK in the initial stage (reconnaissance) of these cycles, i.e. pre-test findings. While considerable differences were evident among the pre-test population, the findings suggest that prior to the intervention stage many participating prospective teachers; regardless of previous mathematics achievements or the level of mathematics study; demonstrate weaknesses and gaps in their ‘common’ MSMK. Particular difficulties were evident in relation to pre-test items requiring knowledge of rational numbers, conceptual understanding or problem solving. These findings highlight the inadequacy of previous mathematics achievements and indeed minimum entry requirements as predictors of MSMK for teaching. As well as its contribution at a local and national level, the findings provide an Irish perspective on this international issue.