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.     

Counter-examples for refinement of conjectures and proofs in primary school mathematics

The purpose of this study is to explore how primary school students reexamine their conjectures and proofs when they confront counter-examples to the conjectures they have proved. In the case study, a pair of Japanese fifth graders thought that they had proved their primitive conjecture with manipulative objects (that is, they constructed an action proof), and then the author presented a counter-example to them. Confronting the counter-example functioned as a driving force for them to refine their conjectures and proofs. They understood the reason why their conjecture was false through their analysis of its proof and therefore could modify their primitive conjecture. They also identified the part of the proof which was applicable to the counter-example. This identification and their action proof were essential for their invention of a more comprehensive conjecture.     

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.     

The student experience of mathematical proof at university level

While proofs are central to university level mathematics courses, research indicates that some students may complete their degrees with an incomplete picture of what constitutes a proof and how proofs are developed. The paper sets out to review what is known of the student experience of mathematical proof at university level. In particular, some evidence is presented of the conceptions of mathematical proof that recent mathematics graduates bring to their postgraduate course to teach high school mathematics. Such evidence suggests that while the least well-qualified graduates may have the poorest grasp of mathematical proof, the most highly qualified may not necessarily have the richest form of subject matter knowledge needed for the most effective teaching. Some indication of the likely causes of this incomplete student perspective on proof are presented.     

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.     

Schülerprobleme beim Lösen von geometrischen Beweisaufgaben —eine Interviewstudie—

In this article we report on an interview study involving ten grade 8 students. These interviews served as a qualitative supplement for a large-scale quantitative study on proof and argumentation (N=659). During videotaped interviews the students were asked to solve geometrical proof problems. The results indicate that students’ difficulties with proof and logical argumentation may be explained by insufficient knowledge of facts, deficits in their methodological knoledge about mathematical proofs, and a lack of knowledge with respect to developing and implementing a proof strategy. Low-achieving students show difficulties with respect to all these three aspects, whereas high-achieving students’ difficulties are mainly based on deficits of developing an adequate and correct proof strategy.     

On high-school students' use of graphic calculators in mathematics

When students are working with hand held technology, such as graphic calculators, we usually only see the outcomes of their activities in the form of a contribution to a written solution of a mathematical problem. It is more difficult to capture their process of thinking or actions as they use the technology to solve the problem. In this paper we report on two case studies that follow the progress of students as they solve mathematical problems. We use software that works in the background of the graphic calculator capturing the students' keystrokes as they use the calculator. The aim of the research studies described in this paper was to provide insights into the working styles of these students. Through a detailed analysis of their graphic calculator keystrokes, interviews and associated written solutions we will discuss the effectiveness of their solution strategies and the efficiency of their use of the technology and identify some barriers to the use of graphic calculators in mathematical problem solving.     

Mathematics and Science Teachers' Use of and Confidence in Empirical Reasoning: Implications for STEM Teacher Preparation

The recent trend to unite mathematically related disciplines (science, technology, engineering, and mathematics) under the broader umbrella of STEM education has advantages. In this new educational context of integration, however, STEM teachers need to be able to distinguish between sufficient proof and reasoning across different disciplines, particularly between the status of inductive and deductive modes of reasoning in mathematics. Through a specific set of mathematical conjectures, researchers explored differences between mathematics (n = 24) and science (n = 23) teachers' reasoning schemes, as well as the confidence they had in their justifications. Results from the study indicate differences between the two groups in terms of their levels of mathematical proof, as well as correlational trends that inform their confidence across these levels. Implications particularly for teacher training and preparation within the context of an integrated STEM education model are discussed.     

Explication as a lens for the formalization of mathematical theory through guided reinvention

Realistic Mathematics Education supports students’ formalization of their mathematical activity through guided reinvention. To operationalize “formalization” in a proof-oriented instructional context, I adapt Sjogren's (2010) claim that formal proof explicates (Carnap, 1950) informal proof. Explication means replacing unscientific or informal concepts with scientific ones. I use Carnap's criteria for successful explication – similarity, exactness, and fruitfulness – to demonstrate how the elements of mathematical theory – definitions, axioms, theorems, proofs – can each explicate their less formal correlates. This lens supports an express goal of the instructional project, which is to help students coordinate semantic (informal) and syntactic (formal) mathematical activity. I demonstrate the analytical value of the explication lens by applying it to examples of students’ mathematical activity drawn from a design experiment in undergraduate, neutral axiomatic geometry. I analyze the chains of meanings (Thompson, 2013) that emerged when formal elements were presented readymade alongside those emerging from guided reinvention.     

The Beverton–Holt dynamic equation

The Cushing–Henson conjectures on time scales are presented and verified. The central part of these conjectures asserts that based on a model using the dynamic Beverton–Holt equation, a periodic environment is deleterious for the population. The proof technique is as follows. First, the Beverton–Holt equation is identified as a logistic dynamic equation. The usual substitution transforms this equation into a linear equation. Then the proof is completed using a recently established dynamic version of the generalized Jensen inequality.     

Using components of mathematical ability for initial development and identification of mathematically promising students

Kruteskii's work on the mathematical abilities of school children is a seminal work on the nature of mathematical ability. However, the task of developing methods for the practical application of his work is still a significant problem in mathematics education. The authors have developed a practical application of Kruteskii's approach to the important problem of initially developing components of mathematical ability in student and thereafter identifying mathematically promising students. Examples of problems that were designed to develop ability to generalize, flexibility and reversibility of mental processes are presented. A practical guide for determining the level of development of components of mathematical abilities in individual students, in terms of specified observables, is presented as a set of structured reference tables. The authors set out a practical application protocol that combines use of the tables and sets of specially developed problems for initial development of mathematical abilities prior to identification of mathematically promising students in the general classroom. A significant motivation for this work is the desire to avoid time-consuming and resource intensive practices such as interviews and summer schools which therefore have been used successfully because these practices are now out of reach for all but very wealthy countries or highly ideologically driven systems. On the other hand, special examinations heavily depend on the level of preparedness of the students for the particular examination, and therefore some students with high abilities but with fewer opportunities to prepare could be overlooked.     

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.     

On the importance of mathematical play

A number of existing theories and proposals for the meaning and characteristics of ‘play’ are considered before the authors suggest six characteristics of mathematical play, including the idea that it is not confined to childhood. Previous studies provide evidence for relating play to cognitive gain while the place of mathematical play in research activities is illustrated by describing a mathematician's approach to a number investigation from the classroom-The Six Circles. The problem-solving process for the Six Circles and observations of students solving calculator and integration problems are analysed in relation to theories of play and cognitive gain and also considered from the perspective of the students' experience. Piaget's theory for the assimilation and accommodation of new information and Davis's view of play as ‘space to support learning’ are reflected in the authors' rationale for suggesting that open questions and mathematical play provide opportunities for students to develop their own conjectures, with no threat of failure, and provide a foundation for mathematical learning. Some difficulties of implementing a ‘play’ approach in the classroom are discussed and further research questions proposed.     

Classroom sociomathematical norms for proof presentation in undergraduate in abstract algebra

This paper is a case study of the teaching of an undergraduate abstract algebra course with a particular focus on the manner in which the students presented proofs and the class engaged in a subsequent discussion of those proofs that included validating the work. This study describes norms for classroom work that include a set of norms that the presenter of a proof was responsible for enacting, including only using previously agreed upon results, as well as a separate set that the audience was to enact related to developing their understanding of the presented proof and validating the work. The study suggests that the students developed a sense of communal and individual responsibility for contributing to growing the body of mathematical knowledge known by the class, with an implied responsibility for knowing the already developed mathematics. Moreover, the behaviors that norms prompted the students to engage were those that literature suggests leads to increased comprehension of proofs.     

High school students’ perceptions of geometrical proofs proving and refuting geometrical claims of the ‘for every … ’ and ‘there exists’ type

Proving and refuting mathematical claims constitute a significant element in the development of deductive thinking. These issues are mainly studied during geometry lessons and very little (if at all) in lessons of other mathematical disciplines. This study deals with high school students’ perceptions of proofs in the geometry. The study explores whether students know when to use a deductive proof and when an example is sufficient for proving or refuting geometrical claims. The findings indicate that in cases of simple claims, the students corroborate them by using a deductive proof. However, when the claim is more complex, the students tend to present both a proof and an example. Moreover, they are unsure whether using an example can constitute a method for proving a mathematical claim, believing that in mathematics everything must be proven. They believe that examples are used merely for illustration purposes rather than as a means of convincing. The research conclusions support the need for deepening and developing the students’ distinction between cases where examples are insufficient and cases where an example is sufficient for proving a claim.     

When Two Circles Determine a Triangle. Discovering and Proving a Geometrical Condition in a Computer Environment

Visualization of mathematical relationships enables students to formulate conjectures as well as to search for mathematical arguments to support these conjectures. In this project students are asked to discover the sufficient and necessary condition so that two circles form the circumscribed and inscribed circle of a triangle and investigate how this condition effects the type of triangle in general and its perimeter in particular. Its open-ended form of the task is a departure from the usual phrasing of textbook’s exercises “show that…”.     

Investigating undergraduate students’ proof schemes and perspectives about combinatorial proof

Combinatorics is an area of mathematics with accessible, rich problems and applications in a variety of fields. Combinatorial proof is an important topic within combinatorics that has received relatively little attention within the mathematics education community, and there is much to investigate about how students reason about and engage with combinatorial proof. In this paper, we use Harel and Sowder’s (1998) proof schemes to investigate ways that students may characterize combinatorial proofs as different from other types of proof. We gave five upper-division mathematics students combinatorial-proof tasks and asked them to reflect on their activity and combinatorial proof more generally. We found that the students used several of Harel and Sowder’s proof schemes to characterize combinatorial proof, and we discuss whether and how other proof schemes may emerge for students engaging in combinatorial proof. We conclude by discussing implications and avenues for future research.     

Distinguishing Features of Mechanical and Human Problem-Solving

Over the years, research in mathematical problem-solving has examined expert/novice problem-solving performance on various types of problems and subjects. In particular, DeFranco examined two groups of Ph.D. mathematicians as they solved four mathematics problems and found that although all were content experts, only one group were problem-solving experts. Based on this study, this article posits the notion that one distinguishing feature between experts and novices is that experts tend to look for special features of a problem and use algorithms only as a “fail-safe” system while novices act like “machines” relying on algorithms to solve the problems. Why? The article explores the idea that novice problem solvers learned to solve problems the way they learned proof, that is, in a formal, abstract and mechanizable way. Beliefs about proof and the culture in which it is practiced help frame a mathematician's view of the discipline and ultimately impacts classroom practice. The authors believe that current classroom instruction tends to create a culture that fosters algorithmic proficiency and a “machine-like” approach to the learning of mathematics and problem-solving. Further, they argue that mathematicians need to be aware of the distinction between knowing a proof is true and explaining why it is true. When these distinctions are appreciated and practiced during classroom instruction, then and only then will students begin to acquire the mathematical knowledge to become better problem solvers.