Mathematics Majors' Perceptions of Conviction,Validity, and Proof

In this paper, 28 mathematics majors who completed a transition-to-proof course were given 10 mathematical arguments. For each argument, they were asked to judge how convincing they found the argument and whether they thought the argument constituted a mathematical proof. The key findings from this data were (a) most participants did not find the empirical argument in the study to be convincing or to meet the standards of proof, (b) the majority of participants found a diagrammatic argument to be both convincing and a proof, (c) participants evaluated deductive arguments not by their form but by their content, but (d) participants often judged invalid deductive arguments to be convincing proofs because they did not recognize their logical flaws. These findings suggest improving undergraduates' comprehension of mathematical arguments does not depend on making undergraduates aware of the limitations of empirical arguments but instead on improving the ways in which they process the arguments that they read.     

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)

     

Exploring informal mathematical products of low achievers at the secondary school level

This article examines the notion of informal mathematical products, in the specific context of teaching mathematics to low achieving students at the secondary school level. The complex and relative nature of this notion is illustrated and some of its characteristics are suggested. These include the use of ad-hoc strategies, mental calculations, idiosyncratic ideas, everyday rather than mathematical language, non-symbolic explanations, visual justifications and common-sense based reasoning. The main argument raised in the article concerns the challenge of valuing informal mathematical products, created by low achievers, and using them within the mathematics classroom as means for advancing such students. The data draws from several research and design projects conducted in Israel since 1991. Selected examples of students’ products, gathered from low-track mathematics classrooms involved in these projects, are presented and analyzed. The analyses highlight various features of such products, and portray the possible gains of teaching approaches that legitimize, and build onwards from, informal products of low achievers.     

Proof in School Mathematics: Insights from Psychological Research into Students' Ability for Deductive Reasoning

There are currently increased efforts to make proof central to school mathematics throughout the grades. Yet, realizing this goal is challenging because it requires that students master several abilities. In this article we focus on one such ability, namely, the ability for deductive reasoning, and we review psychological research to enhance what is currently known in mathematics education research about this ability in the context of proof and to identify important directions for future research. We first offer a conceptualization of proof, which we use to delineate our focus on deductive reasoning. We then review psychological research on the development of students' ability for deductive reasoning to see what can be said about the ages at which students become able to engage in certain forms of deductive reasoning. Finally, we review two psychological theories of deductive reasoning to offer insights into cognitively guided ways to enhance students' ability for deductive reasoning in the context of proof.     

Leveraging interdiscursivity to support elementary students in bridging the empirical-deductive gap: the case of parity

Research has recognized deductive reasoning as challenging but not impossible for young mathematics learners. In this paper, we present a learning environment developed to assist elementary-school students to bridge the empirical-deductive gap in the context of parity of numbers. Using the commognitive framework, we construe the empirical-deductive gap as part of a broader divide between two discourses that abide by different rules of a “mathematical game”: a discourse on specific numbers and a discourse on numeric patterns. Interdiscursivity is leveraged as a mechanism for instructional design, where students’ familiar routines with specific numbers are teased out and advanced to make sense in the new discourse. We mobilize this mechanism to create opportunities for students to play an active role in recognizing issues with empirical reasoning and generating deductive arguments to establish the validity of universal statements. The environment is illustrated with a small group of 8-year-olds who learned to justify deductively that “odd + odd = even”.     

Proof in School Mathematics: Insights from Psychological Research into Students' Ability for Deductive Reasoning

There are currently increased efforts to make proof central to school mathematics throughout the grades. Yet, realizing this goal is challenging because it requires that students master several abilities. In this article we focus on one such ability, namely, the ability for deductive reasoning, and we review psychological research to enhance what is currently known in mathematics education research about this ability in the context of proof and to identify important directions for future research. We first offer a conceptualization of proof, which we use to delineate our focus on deductive reasoning. We then review psychological research on the development of students' ability for deductive reasoning to see what can be said about the ages at which students become able to engage in certain forms of deductive reasoning. Finally, we review two psychological theories of deductive reasoning to offer insights into cognitively guided ways to enhance students' ability for deductive reasoning in the context of proof.     

Integrating learning theories and application-based modules in teaching linear algebra

The research team of The Linear Algebra Project developed and implemented a curriculum and a pedagogy for parallel courses in (a) linear algebra and (b) learning theory as applied to the study of mathematics with an emphasis on linear algebra. The purpose of the ongoing research, partially funded by the National Science Foundation, is to investigate how the parallel study of learning theories and advanced mathematics influences the development of thinking of individuals in both domains. The researchers found that the particular synergy afforded by the parallel study of math and learning theory promoted, in some students, a rich understanding of both domains and that had a mutually reinforcing effect. Furthermore, there is evidence that the deeper insights will contribute to more effective instruction by those who become high school math teachers and, consequently, better learning by their students. The courses developed were appropriate for mathematics majors, pre-service secondary mathematics teachers, and practicing mathematics teachers. The learning seminar focused most heavily on constructivist theories, although it also examined socio-cultural and historical perspectives. A particular theory, Action-Process-Object-Schema (APOS) [10], was emphasized and examined through the lens of studying linear algebra. APOS has been used in a variety of studies focusing on student understanding of undergraduate mathematics. The linear algebra courses include the standard set of undergraduate topics. This paper reports the results of the learning theory seminar and its effects on students who were simultaneously enrolled in linear algebra and students who had previously completed linear algebra and outlines how prior research has influenced the future direction of the project.     

The co-construction of arguments by middle-school students

This paper highlights the value of student collaboration in doing mathematics, demonstrates how urban, middle-school students, working together, co-constructed justifications for their solutions, and shows that certain conditions are associated with the promotion of a culture of reasoning. It is documented that students collaboratively built arguments that took the form of proof, challenged each others’ arguments, and justified these arguments in small groups and whole class discussions. In producing their mathematical justifications, students included the input of others. Finally, the way in which students, by expanding on the arguments of others, also used alternative forms of reasoning which in many cases led to even more refined arguments is discussed.     

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.     

Undergraduate Engineering Majors' Beliefs About Mathematics

This study examined the mathematics beliefs of college students in 10 undergraduate mathematics classes at a large engineering school in the Midwest. The beliefs of 254 engineering majors were measured by the Indiana Mathematics Belief Scales and compared to the beliefs of elementary education majors and remedial college mathematics students obtained from earlier studies using the same instrument. The results were interpreted in terms of the students' daily attitudes towards their mathematics classes and corresponding academic and demographic parameters. The study showed that in many respects, the beliefs of the engineering majors were not that different from the other populations. The correlations among beliefs for the engineering group tended to be higher although there were relatively few significant correlations between belief and background variables. Attitude data were collected across a full semester for the engineering majors. The relatively modest day-to-day variation in those attitudes suggests that they are based on deeply seated beliefs.     

“Playing the game” of story problems: Coordinating situation-based reasoning with algebraic representation

This study critically examines a key justification used by educational stakeholders for placing mathematics in context –the idea that contextualization provides students with access to mathematical ideas. We present interviews of 24 ninth grade students from a low-performing urban school solving algebra story problems, some of which were personalized to their experiences. Using a situated cognition framework, we discuss how students use informal strategies and situational knowledge when solving story problems, as well how they engage in non-coordinative reasoning where situation-based reasoning is disconnected from symbol-based reasoning and other problem-solving actions. Results suggest that if contextualization is going to provide students with access to algebraic ideas, supports need to be put in place for students to make connections between formal algebraic representation, informal arithmetic-based reasoning, and situational knowledge.     

Measuring early mathematics knowledge via number skills and task types

Early number skills are a critical aspect of early mathematics development. However, the constructs that comprise early number skills differ across assessments, and previous studies have proposed various models of early mathematics skills comprised of formal and informal tasks. This study explored the factor structure of a researcher-developed measure of mathematics administered to a large, geographically diverse sample of kindergarten students at risk for mathematics difficulty (n = 580) in a randomized control trial. Consistent with previous research, factors representing early number skills and task types emerged. Importantly though, the best fitting model was one in which both skill types (e.g., number identification, magnitude comparison) and task types (i.e., informal and formal) were modeled. The inclusion of task type as a factor in early mathematics assessment has many potentially important ramifications. Recommendations for attending to task types when assessing early number skills, and implications for instruction and measurement are discussed.     

Levels of arithmetic reasoning in solving an open-ended problem

This paper presents the results of an experimental teaching carried out on 12-year-old students. An open-ended task was given to them and they had not been taught the algorithmic process leading to the solution. The formal solution to the problem refers to a system of two linear equations with two unknown quantities. In this mathematical activity, students worked cooperatively. They discussed their discoveries in groups of four and then presented their answers to the whole class developing a rich communication. This study describes the characteristic arguments that represent certain different forms of reasoning that emerged during the process of justifying the solutions of the problem. The findings of this research show that within an environment conducive to creativity, which encourages collaboration, exploration and sharing ideas, students can be engaged in developing multiple mathematical strategies, posing new questions, creating informal proofs, showing beauty and elegance and bringing out that problem solving is a powerful way of learning mathematics.     

ON THE RELIABILITY OF MEDIUM LOGIC(ML)

Medium logic (ML) is set up for the common theoritical foundation of the classical mathematics and fuzzy mathematics. It has been formalized as a new theory of logic. See note (1) and note (2). As mathematical logic the ML's construct reches the study of the informal deductive inference by means of studying the formal inference in it, and it demands the formal inference of ML reliable reflected the deductive inference. For this reason, this paper deals with its reliability. The result shows that the formal inference of M L consists with deductive inference, and that ML reliably reflects the deductive inference.     

The Argument from Silence

The argument from silence is a pattern of reasoning in which the failure of a known source to mention a particular fact or event is used as the ground of an inference, usually to the conclusion that the supposed fact is untrue or the supposed event did not actually happen. Such arguments are widely used in historical work, but they are also widely contested. This paper surveys some inadequate attempts to model this sort of argument, offers a new analysis using a Bayesian probabilistic framework that isolates the most problematic step in such arguments, illustrates a key problem besetting many uses of the argument, diagnoses the attraction of the argument in terms of a known human cognitive bias affecting the critical step, and suggests a standard that must be met in order for any argument from silence to have more than a very weak influence on historical reasoning.     

Reinventing solutions to systems of linear differential equations: A case of emergent models involving analytic expressions

An enduring challenge in mathematics education is to create learning environments in which students generate, refine, and extend their intuitive and informal ways of reasoning to more sophisticated and formal ways of reasoning. Pressing concerns for research, therefore, are to detail students’ progressively sophisticated ways of reasoning and instructional design heuristics that can facilitate this process. In this article we analyze the case of student reasoning with analytic expressions as they reinvent solutions to systems of two differential equations. The significance of this work is twofold: it includes an elaboration of the Realistic Mathematics Education instructional design heuristic of emergent models to the undergraduate setting in which symbolic expressions play a prominent role, and it offers teachers insight into student thinking by highlighting qualitatively different ways that students reason proportionally in relation to this instructional design heuristic.