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## Undergraduate students' understanding of the contraposition equivalence rule in symbolic and verbal contexts

### Educational Studies in Mathematics (2004-03-01) 55: 133-162 , March 01, 2004

Literature suggests that the type of context wherein a task is placed relates to students' performance and solution strategies. In the particular domain of logical thinking, there is the belief that students have less difficulty reasoning in verbal than in logically equivalent symbolic tasks. Thus far, this belief has remained relatively unexplored in the domain of teaching and learning of mathematics, and has not been examined with respect to students' major field of study. In this study, we examined the performance of 95 senior undergraduate mathematics and education majors in *symbolic* and *verbal* tasks about the *contraposition equivalence rule*. The selection of two different groups of participants allowed for the examination of the hypothesis that students' major may influence the relation between their *performance* in tasks about contraposition and the *context*(symbolic/verbal) wherein this is placed. The selection of contraposition equivalence rule also addressed a gap in the body of research on undergraduate students' understanding of *proof by contraposition*. The analysis was based on written responses of all participants to specially developed tasks and on semi-structured interviews with 11 subjects. The findings showed different variations in the performance of each of the two groups in the two contexts. while education majors performed significantly better in the verbal than in the symbolic tasks, mathematics majors' performance showed only modest variations. The results call for both major- and context- specific considerations of students' understanding of logical principles, and reveal the complexity of the system of factors that influence students' logical thinking.

## PROSPECTIVE TEACHERS’ CHALLENGES IN TEACHING REASONING-AND-PROVING

### International Journal of Science and Mathematics Education (2013-12-01) 11: 1463-1490 , December 01, 2013

The activity of *reasoning-and-proving* is at the heart of mathematical sense making and is important for all students’ learning as early as the elementary grades. Yet, reasoning-and-proving tends to have a marginal place in elementary school classrooms. This situation can be partly attributed to the fact that many (prospective) elementary teachers have (1) weak mathematical (subject matter) knowledge about reasoning-and-proving and (2) counterproductive beliefs about its teaching. Following up on an intervention study that helped a group of prospective elementary teachers make significant progress in overcoming these two major obstacles to teaching reasoning-and-proving, we examined the challenges that three of them identified that they faced as they planned and taught lessons related to reasoning-and-proving in their mentor teachers’ classrooms. Our findings contribute to research knowledge about major factors (other than the well-known factors related to teachers’ mathematical knowledge and beliefs) that deserve attention by teacher education programs in preparing prospective teachers to teach reasoning-and-proving.

## Investigating the Guidance Offered to Teachers in Curriculum Materials: The Case of Proof in Mathematics

### International Journal of Science and Mathematics Education (2008-03-01) 6: 191-215 , March 01, 2008

Despite widespread agreement that *proof* should be central to all students’ mathematical experiences, many students demonstrate poor ability with it. The curriculum can play an important role in enhancing students’ proof capabilities: teachers’ decisions about what to implement in their classrooms, and how to implement it, are mediated through the curriculum materials they use. Yet, little research has focused on how proof is promoted in mathematics curriculum materials and, more specifically, on the guidance that curriculum materials offer to teachers to enact the proof opportunities designed in the curriculum. This paper presents an analytic approach that can be used in the examination of the guidance curriculum materials offer to teachers to implement in their classrooms the proof opportunities designed in the curriculum. Also, it presents findings obtained from application of this approach to an analysis of a popular US reform-based mathematics curriculum. Implications for curriculum design and research are discussed.

## Proof constructions and evaluations

### Educational Studies in Mathematics (2009-11-01) 72: 237-253 , November 01, 2009

In this article, we focus on a group of 39 prospective elementary (grades K-6) teachers who had rich experiences with proof, and we examine their ability to construct proofs and evaluate their own constructions. We claim that the combined “construction–evaluation” activity helps illuminate certain aspects of prospective teachers’ and presumably other individuals’ understanding of proof that tend to defy scrutiny when individuals are asked to evaluate given arguments. For example, some prospective teachers in our study provided empirical arguments to mathematical statements, while being aware that their constructions were invalid. Thus, although these constructions considered alone could have been taken as evidence of an empirical conception of proof, the additional consideration of prospective teachers’ evaluations of their own constructions overruled this interpretation and suggested a good understanding of the distinction between proofs and empirical arguments. We offer a possible account of our findings, and we discuss implications for research and instruction.

## Validation of Solutions of Construction Problems in Dynamic Geometry Environments

### International Journal of Computers for Mathematical Learning (2005-01-01) 10: 31-47 , January 01, 2005

This paper discusses issues concerning the validation of solutions of construction problems in Dynamic Geometry Environments (DGEs) as compared to classic paper-and-pencil Euclidean geometry settings. We begin by comparing the validation criteria usually associated with solutions of construction problems in the two geometry worlds – the ‘drag test’ in DGEs and the use of only straightedge and compass in classic Euclidean geometry. We then demonstrate that the *drag test criterion* may permit constructions created using measurement tools to be considered valid; however, these constructions prove inconsistent with classical geometry. This inconsistency raises the question of whether dragging is an adequate test of validity, and the issue of measurement versus straightedge-and-compass. Without claiming that the inconsistency between what counts as valid solution of a construction problem in the two geometry worlds is necessarily problematic, we examine what would constitute the analogue of the *straightedge-and-compass criterion* in the domain of DGEs. Discovery of this analogue would enrich our understanding of DGEs with a mathematical idea that has been the distinguishing feature of Euclidean geometry since its genesis. To advance our goal, we introduce the *compatibility criterion*, a new but not necessarily superior criterion to the drag test criterion of validation of solutions of construction problems in DGEs. The discussion of the two criteria anatomizes the complexity characteristic of the relationship between DGEs and the paper-and-pencil Euclidean geometry environment, advances our understanding of the notion of geometrical constructions in DGEs, and raises the issue of validation practice maintaining the pace of ever-changing software.

## Seeking research-grounded solutions to problems of practice: classroom-based interventions in mathematics education

### ZDM (2013-05-01) 45: 333-341 , May 01, 2013

Research on classroom-based interventions in mathematics education has two core aims: (a) to improve classroom practice by engineering ways to act upon problems of practice; and (b) to deepen theoretical understanding of classroom phenomena that relate to these problems. Although there are notable examples of classroom-based intervention studies in mathematics education research since at least the 1930s, the number of such studies is small and acutely disproportionate to the number of studies that have documented problems of classroom practice for which solutions are sorely needed. In this paper we first make a case for the importance of research on classroom-based interventions and identify three important features of this research, which we then use to review the papers in this special issue. We also consider the issue of ‘scaling up’ promising classroom-based interventions in mathematics education, and we discuss a major obstacle that most such interventions find on the way to scaling up. This obstacle relates to their long duration, which means that possible adoption of these interventions would require practitioners to do major reorganizations of the mathematics curricula they follow in order to accommodate the time demands of the interventions. We argue that it is important, and conjecture that it is possible, to design interventions of short duration in mathematics education to alleviate major problems of classroom practice. Such interventions would be more amenable to scaling up, for they would allow more control over confounding variables and would make more practicable their incorporation into existing curriculum structures.

## Impacting prospective teachers’ beliefs about mathematics

### ZDM (2013-05-01) 45: 393-407 , May 01, 2013

This study investigated: (1) the changes in the beliefs about mathematics held by 25 prospective elementary teachers as they went through a university mathematics course that aimed, among other things, to promote a problem-solving view about mathematics; and (2) the possible factors that accounted for the observed changes. The course incorporated specific features that prior research suggested reflect successful mechanisms for belief change (e.g., cognitive conflict). The data included students’ reflections, and responses to prompts and interview questions. Analysis of the data revealed the following major trends: (1) a movement towards a problem-solving view from the more traditional Platonist and instrumentalist views; and (2) no change in students’ initial views. Activities creating cognitive conflict, as well as the implementation of instruction valuing group collaboration and explanations, appear to have played important roles in the process of belief change. The findings have implications for research on teacher beliefs and teacher education.

## Level, strength, and facet-specific self-efficacy in mathematics test performance

### ZDM (2017-06-01) 49: 379-395 , June 01, 2017

Students’ self-efficacy expectations (SEE) in mathematics are associated with their engagement and learning experiences. Going beyond previous operationalisations of SEE we propose a new instrument that takes into account not only *facet-specificity* (expectations related to particular competences or skills) and *strength* (confidence of the expectations), but also *level* (perceived task difficulty) of these expectations as proposed by Bandura (Self-efficacy: The exercise of control, W. H. Freeman & Co, New York, 1997; Self-efficacy beliefs of adolescents, Information Age Publishing, Greenwich, 2006). In particular, we included level-specific items referring to perceived difficulty on a subsequent national test in mathematics. In total 756 Norwegian grade 5, 8, and 9 students completed the “Self-Efficacy Gradations of Difficulty Questionnaire.” We fitted plausible multitrait-multimethod models using structural equation models. The best fitting model included three factors representing levels of perceived difficulty, and a-priori specified correlated uniquenesses representing four facets. The facets related to problem solving or students’ self-regulation skills during the test in order to accomplish the following: (1) complete a certain number of problems, (2) solve tasks of a certain challenge, (3) concentrate, and (4) not give up for a certain amount of time. The results indicated that three correlated constructs representing levels of SEE are associated with scores on national tests in mathematics, and that the strongest association is between national test scores and medium level SEE. Taking level (difficulty) into account broadens our understanding of the self-efficacy construct, and allows investigation into differential relationships between SEE and performance.

## Preservice teachers’ knowledge of proof by mathematical induction

### Journal of Mathematics Teacher Education (2007-06-01) 10: 145-166 , June 01, 2007

There is a growing effort to make *proof* central to all students’ mathematical experiences across all grades. Success in this goal depends highly on teachers’ knowledge of proof, but limited research has examined this knowledge. This paper contributes to this domain of research by investigating preservice elementary and secondary school mathematics teachers’ knowledge of proof by mathematical induction. This research can inform the *knowledge about preservice teachers* that mathematics teacher educators need in order to effectively teach proof to preservice teachers. Our analysis is based on written responses of 95 participants to specially developed tasks and on semi-structured interviews with 11 of them. The findings show that preservice teachers from both groups have difficulties that center around: (1) the essence of the base step of the induction method; (2) the meaning associated with the inductive step in proving the implication *P*(*k*) ⇒ *P*(*k* + 1) for an arbitrary *k* in the domain of discourse of *P*(*n*); and (3) the possibility of the truth set of a sentence in a statement proved by mathematical induction to include values outside its domain of discourse. The difficulties about the base and inductive steps are more salient among preservice elementary than secondary school teachers, but the difficulties about whether proofs by induction should be as encompassing as they could be are equally important for both groups. Implications for mathematics teacher education and future research are discussed in light of these findings.