Learning mathematics through algorithmic and creative reasoning

There are extensive concerns pertaining to the idea that students do not develop sufficient mathematical competence. This problem is at least partially related to the teaching of procedure-based learning. Although better teaching methods are proposed, there are very limited research insights as to why some methods work better than others, and the conditions under which these methods are applied. The present paper evaluates a model based on students’ own creation of knowledge, denoted creative mathematically founded reasoning (CMR), and compare this to a procedure-based model of teaching that is similar to what is commonly found in schools, denoted algorithmic reasoning (AR). In the present study, CMR was found to outperform AR. It was also found cognitive proficiency was significantly associated to test task performance. However the analysis also showed that the effect was more pronounced for the AR group.     

Locally logical mathematics: An emerging teacher honoring both students and mathematics

This case study explores the mathematics engagement and teaching practice of a beginning secondary school teacher. The focus is on the mathematical opportunities available to her students (the classroom mathematics) and how they relate to the teacher's personal capacity and tendencies for mathematical engagement (her personal mathematics). We use a mathematical process-and-action approach to analyze mathematical engagement and then employ the teaching triad—mathematical challenge, sensitivity to students, and management of learning—to situate mathematical engagement within the larger context of teaching practice. The article develops the construct of locally logical mathematics to underscore the cogency of mathematical engagement in the classroom as part of a coherent mathematical system that is embedded within a teaching practice. Contributions of the study include the process-and-action approach, especially in tandem with the teaching triad, as a tool to understand nuances of mathematical engagement and differences in demand between written and implemented tasks.     

Game and decision theory in mathematics education: epistemological, cognitive and didactical perspectives

In the 1950s, game and decision theoretic modeling emerged—based on applications in the social sciences—both as a domain of mathematics and interdisciplinary fields. Mathematics educators, such as Hans Georg Steiner, utilized game theoretical modeling to demonstrate processes of mathematization of real world situations that required only elementary intuitive understanding of sets and operations. When dealing with n-person games or voting bodies, even students of the 11th and 12th grade became involved in what Steiner called the evolution of mathematics from situations, building of mathematical models of given realities, mathematization, local organization and axiomatization. Thus, the students could participate in processes of epistemological evolutions in the small scale. This paper introduces and discusses the epistemological, cognitive and didactical aspects of the process and the roles these activities can play in the learning and understanding of mathematics and mathematical modeling. It is suggested that a project oriented study of game and decision theory can develop situational literacy, which can be of interest for both mathematics education and general education.     

Educative experiences in a games context: Supporting emerging reasoning in elementary school mathematics

Reasoning as a process supports students’ success in mathematics, yet reports on its development in elementary school are scarce. An action research project with grade 5 and 6 students investigated how growth in reasoning occurred within abstract strategy games. Reasoning within the board game context was framed by Dewey’s conceptualization of experience which emphasizes the importance of students’ active participation and reflection. Through characteristics of interaction and continuity, students analyzed moves, generalized toward strategies, and convincingly justified effective approaches through accepted structures of reasoning. Elaborating on reasoning as a process, results show that students can grow in their capability to reason through multiple experiences of developing convincing arguments in an authentic context.     

Generalization and extension of a problem from an American mathematics competition

ABSTRACT

The purpose of these notes is to generalize and extend a challenging geometry problem from a mathematics competition. The notes also contain solution sketches pertaining to the problems discussed.     

Geometry students’ hedged statements and their self-regulation of mathematics

Statements conveying a degree of certainty or doubt, in the form of hedging, have been linked with logical inference in students’ talk (Rowland, 2000). Considering the current emphasis on increasing student autonomy for effective mathematical discourse, I posit a relationship between hedging and student autonomy. In the current study, high school Geometry students’ frequency of producing hedged mathematical statements were correlated with their perceived mathematical autonomy to determine if a relationship existed. Results found a strong and statistically significant correlation, providing support for a connection between students’ hedging and their perceived autonomy. However, additional analysis revealed that perceptions of mathematical competence and social relatedness were also influential to hedging. Implications of these results are discussed.     

Preservice teachers learning to teach proof through classroom implementation: Successes and challenges

Proof and reasoning are central to learning mathematics with understanding. Yet proof is seen as challenging to teach and to learn. In a capstone course for preservice teachers, we developed instructional modules that guided prospective secondary mathematics teachers (PSTs) through a cycle of learning about the logical aspects of proof, then planning and implementing lessons in secondary classrooms that integrate these aspects with traditional mathematics curriculum in the United States. In this paper we highlight our framework on mathematical knowledge for teaching proof and focus on some of the logical aspects of proof that are seen as particularly challenging (four proof themes). We analyze 60 lesson plans, video recordings of a subset of 13 enacted lessons, and the PSTs’ self- reported data to shed light on how the PSTs planned and enacted lessons that integrate these proof themes. The results provide insights into successes and challenges the PSTs encountered in this process and illustrate potential pathways for preparing PSTs to enact reasoning and proof in secondary classrooms. We also highlight the design principles for supporting the development of PSTs’ mathematical knowledge for teaching proof.     

Flexible thinking and met-befores: Impact on learning mathematics

In this paper we study the difficulties resulting from changes in meaning of the minus sign, from an operation on numbers, to a sign designating a negative number, to the additive inverse of an algebraic symbol on students in two-year colleges and universities. Analysis of the development of algebra reveals that these successive meanings that the student has met before often become problematic, leading to a fragile knowledge structure that lacks flexibility and leads to confusion and long-term disaffection. The problematic aspects that arise from changes in meaning of the minus sign are identified and the iconic function machine is utilized as a supportive strategy, along with formative assessment to encourage teachers and learners to seek more flexible and effective ways of making sense of increasingly sophisticated mathematics.     

Exploring affordances and limitations of between-group movement in elementary mathematics classrooms

In this exploratory study, we examine how between-group movement, as an autonomy-promoting practice, might incentivize or disincentivize sixth-grade students’ engagement in two mathematical practices: (1) making sense of problems and persevering in solving them; and (2) constructing viable arguments and critiquing the reasoning of others. Between-group movement refers to a pedagogical strategy wherein teachers allow groups to physically move within the classroom while problem-solving to discuss strategies, ask for help, or check their work with other groups. Exploring both the affordances and limitations of between-group movement, we found that between-group movement supported groups to construct viable justifications, among other sense-making mathematical practices. However, we also found that some groups over-scaffolded during between-group conversations which disincentivized meaningful engagement in mathematical practices. Furthermore, between-group movement revealed some equity concerns in relation to status-based privileges. The findings imply that between-group movement can be a constructive pedagogical practice under specific conditions.     

Probing for reasons: Presentations, questions, phases

This paper reports on a research study based on data from experimental teaching. Undergraduate dance majors were invited, through real-world problem tasks that raised central conceptual issues, to invent major ideas of calculus. This study focuses on work and thinking by these students, as they sought to build key ideas, representations and compelling lines of reasoning. Speiser and Walter's psychological and logical perspectives (see Speiser, Walter, & Sullivan, 2007) provide opportunities to focus not just on the students’ thinking, but perhaps most especially, through detailed examination of important choices, on their exercise of agency as learners. Close analysis of student data through these lenses triggered the development of two new analytic categories—logic of agency and logic of proof. The analysis presented here treats students as active shapers of their own experience and understanding, whose choices open opportunities for continued growth and learning, not just for themselves but also for each other.     

Measuring interaction quality in mathematics instruction: How differences in operationalizations matter methodologically

Quality of interaction can enhance or constrain students’ mathematical learning opportunities. However, quantitative video studies have measured the quality of interaction with very heterogeneous conceptualizations and operationalizations. This project sought to disentangle typical methodological choices to assess interaction quality in six quality dimensions, each of them in task-based, move-based, and practice-based operationalizations. The empirical part of the study compared different conceptualizations with their corresponding operationalizations and used them to code video data from middle school students (n = 210) organized into 49 small groups who worked on the same curriculum materials. The analysis revealed that different conceptualizations and operationalizations led to substantially different findings, so their distinction turned out to be of high methodological relevance. These results highlight the importance of making methodological choices explicit and call for a stronger academic discourse on how to conceptualize and operationalize interaction quality in video studies.     

Teaching routines and student-centered mathematics instruction: The essential role of conferring to understand student thinking and reasoning

We compare two lessons with respect to how a teacher centers student mathematical thinking to move instruction forward through enactment of five mathematically productive teaching routines: Conferring To Understand Student Thinking and Reasoning, Structuring Mathematical Student Talk, Working With Selected and Sequenced Student Math Ideas, Working with Public Records of Students’ Mathematical Thinking, and Orchestrating Mathematical Discussion. Findings show that the lessons differ in the enactment of teaching routines, especially Conferring to Understand Student Thinking and Reasoning which resulted in a difference in student-centeredness of the instruction. This difference highlights whose mathematics was being centralized in the classroom and whether the focus was on correct answers and procedures or on students’ mathematical thinking and justifying.     

Towards an integrated theory of mathematics conceptual learning and instructional design: The Learning Through Activity theoretical framework

We discuss the theoretical framework of the Learning Through Activity research program. The framework includes an elaboration of the construct of mathematical concept, an elaboration of Piaget’s reflective abstraction for the purpose of mathematics pedagogy, further development of a distinction between two stages of conceptual learning, and a typology of different reverse concepts. The framework also involves instructional design principles built on those constructs, including steps for the design of task sequences, development of guided reinvention, and ways of promoting reversibility of concepts. This article represents both a synthesis of prior work and additions to it.     

Computing devices,mathematics education and mathematics: Sexton’s omnimetre in its time

Material objects can tell us much about mathematical practice. In 1899, Albert Sexton, a Philadelphia mechanical engineer, received the John Scott Medal of the Franklin Institute for his invention of the omnimetre. This inexpensive circular slide rule was one of a host of computing devices that became common in the United States around 1900. It is inscribed “NUMERI MUNDUM REGUNT”. In part because of instruments such as the omnimetre, numbers increasingly ruled the practical world of the late 19th and early 20th century. This changed not only engineering, but mathematics education and mathematical work.     

Logical form,mathematical practice,and Frege's Begriffsschrift

For over a century we have been reading Frege's Begriffsschrift notation as a variant of standard notation. But Frege's notation can also be read differently, in a way enabling us to understand how reasoning in Begriffsschrift is at once continuous with and a significant advance beyond earlier mathematical practices of reasoning within systems of signs. It is this second reading that I outline here, beginning with two preliminary claims. First, I show that one does not reason in specially devised systems of signs of mathematics as one reasons in natural language; the signs are not abbreviations of words. Then I argue that even given a system of signs within which to reason in mathematics, there are two ways one can read expressions involving those signs, either mathematically or mechanically. These two lessons are then applied to a reading of Frege's proof of Theorem 133 in Part III of his 1879 logic, a proof that Frege claims is at once strictly deductive and ampliative, a real extension of our knowledge. In closing, I clarify what this might mean, and how it might be possible.