ANALYZING THE DIAGRAMMATIC REGISTER IN GEOMETRY TEXTBOOKS: TOWARD A SEMIOTIC ARCHITECTURE

Diagrams are key resources for students when reasoning in geometry. Over the course of the 20th century, diagrams in geometry textbooks have evolved from austere collections of strokes and letters to become diverse arrays of symbols, labels, and differently styled visual parts. Diagrams are thus multisemiotic texts that present meanings to students across a range of communication systems. We propose a scheme for analyzing how geometric diagrams function as resources for mathematical communication in terms of four semiotic systems: type, position, prominence, and attributes. The semiotic architecture we propose draws on research in systemic functional linguistics (Halliday, 2004; O’Halloran, 2005) and suggests a framework for analyzing how geometry diagrams function as mathematical texts.http://deepblue.lib.umich.edu/bitstream/2027.42/91288/1/DiagrammaticRegisterJKDPH.pdf-

INSTRUCTIONAL SITUATIONS AND STUDENTS’ OPPORTUNITIES TO REASON IN THE HIGH SCHOOL GEOMETRY CLASS

We outline a theory of instructional exchanges and characterize a handful of instructional situations in high school geometry that frame some of these exchanges. In each of those instructional situations we inspect the possible role of reasoning and proof, drawing from data collected in intact classrooms as well as in instructional interventions.This manuscript is part of the final report of the NSF grant CAREER 0133619 “Reasoning in high school geometry classrooms: Understanding the practical logic underlying the teacher’s work” to the first author.All opinions are those of the authors and do not represent the views of the National Science Foundation.http://deepblue.lib.umich.edu/bitstream/2027.42/78372/1/Instructional_Situations_in_Geometry.pd

Prospective Teachers’ Interactive Visualization and Affect in Mathematical Problem-Solving

Research on technology-assisted teaching and learning has identified several families of factors that contribute to the effective integration of such tools. Focusing on one such family, affective factors, this article reports on a qualitative study of 30 prospective secondary school mathematics teachers designed to acquire insight into the affect associated with the visualization of geometric loci using GeoGebra. Affect as a representational system was the approach adopted to gain insight into how the use of dynamic geometry applications impacted students’ affective pathways. The data suggests that affect is related to motivation through goals and self-concept. Basic instrumental knowledge and the application of modeling to generate interactive images, along with the use of analogical visualization, played a role in local affect and prospective teachers’ use of visualization

Approaching Euclidean proofs through explorations with manipulative and digital artifacts

The combined use of origami and dynamic geometry software has recently appeared in mathematics education to enrich students’ geometric thinking. The objective of this research is to study the roles played by the interaction of two artifacts, paper folding and GeoGebra, in a construction-proving problem as well as its generalization in the Euclidean geometry context. For this, we designed and implemented two mathematical tasks with 52 secondary education students (15–16 years old, 10th grade) during the COVID-19 emergency lockdown period in Italy. The tasks involved four phases: constructing, exploring, conjecturing, and proving. This article presents an epistemic analysis of the tasks and a cognitive analysis of the answers given by one of the students. The theoretical tools of the onto-semiotic approach supported these analyses. Cognitive analysis allows us to confront the intended meanings of the task and the meanings actually employed by a student, thus drawing specific conclusions about the roles of such artifacts in written arguments and give an interpretation of their combined use in mathematics education

Cognitive Conditions of Diagrammatic Reasoning

Forthcoming in Semiotica (ISSN: 0037-1998), published by Walter de Gruyter & Co.In the first part of this paper, I delineate Peirce's general concept of diagrammatic reasoning from other usages of the term that focus either on diagrammatic systems as developed in logic and AI or on reasoning with mental models. The main function of Peirce's form of diagrammatic reasoning is to facilitate individual or social thinking processes in situations that are too complex to be coped with exclusively by internal cognitive means. I provide a diagrammatic definition of diagrammatic reasoning that emphasizes the construction of, and experimentation with, external representations based on the rules and conventions of a chosen representation system. The second part starts with a summary of empirical research regarding cognitive effects of working with diagrams and a critique of approaches that use 'mental models' to explain those effects. The main focus of this section is, however, to elaborate the idea that diagrammatic reasoning should be conceptualized as a case of 'distributed cognition.' Using the mathematics lesson described by Plato in his Meno, I analyze those cognitive conditions of diagrammatic reasoning that are relevant in this case

The Study of Taiwanese Students' Experiences with Geometric Calculation with Number (GCN) and Their Performance on GCN and Geometric Proof (GP).

In Taiwan, students have considerable experience with tasks requiring geometric calculations with number (GCN) prior to their study of geometric proof (GP). This dissertation examined closely the opportunities provided to Taiwanese students’ experiences with GCN and their performance on GCN and GP. Three sequential studies were conducted, corresponding roughly to key aspects of the Mathematical Tasks Framework (MTF); namely, GCN tasks as found in instructional/curricular materials, GCN as enacted by teachers and students, and student performance on paired GCN and GP. Study One found that GCN used by one Taiwanese mathematics teacher were drawn not only from the textbooks but also from other sources (e.g., tests) and the tasks varied with respect to cognitive complexity, with the tasks additionally included being generally more demanding than those found in the textbooks. The high demand GCN appeared to afford opportunities for Taiwanese students to master the types of knowledge, the reasoning and the problem-solving skills that are essential not only for proficiency with GCN but also for GP. Study Two showed how the teacher sustained the cognitive demand levels by making the diagrams more complex and using gestural moves to scaffold students' visualization of the diagrams so that they could sustain their work on the tasks. Through scaffolded experiences with GCN containing complex diagrams, the teacher appeared to nurture students’ competence in constructing and reasoning about geometric relationships in ways that are likely to support their later work with GP. Study Three presented the analysis of Taiwanese students’ performance on matched pairs of GCN and GP, which require the same diagrams and geometric properties to obtain solutions. The findings strongly support the hypothesis that students’ prior experiences with GCN can support their competence in constructing GP. Taken together the three studies sketch a plausible pathway through which Taiwanese students might gain high levels of proficiency in creating GP through their experiences with GCN. In addition, the use of a sequence of three studies that examine different aspects of students’ experiences with mathematical tasks appears to have utility as a model for other research that seeks to understand cross-national differences in mathematics performance.Ph.D.EducationUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/78852/1/huiyuhsu_1.pd

Making mathematics on paper : constructing representations of stories about related linear functions

This dissertation takes up the problem of applied quantitative inference as a central question for cognitive science, asking what must happen during problem solving for people to obtain a meaningful and effective representation of the problem. The core of the dissertation reports exploratory empirical studies that seek to answer the descriptive question of how quantitative inferences are generated, pursued, and evaluated by problem solvers with different mathematical backgrounds. These are framed against a controversy, described in Chapter 2, over the theoretical and empirical validity of current cognitive science accounts of problems, solutions, knowledge, and competent human activity outside of laboratory or school settings.Chapter 3 describes a written protocol study of algebra story problem solving among advanced undergraduates in computer science. A relatively open-ended interpretive framework for "problem-solving episodes" is developed and applied to their written solution attempts. The resulting description of problem-solving activities gives a surprising image of competence among an important occupational target for standard mathematics instruction.Chapter 4 follows these results into detailed verbal problem-solving interviews with algebra students and teachers. These provide a comparison across settings and levels of competence for the same set of problems. The results corroborate similar generative activities outside the standard formalism of algebra across levels of competence. Notable among these nonalgebraic problem-solving activities are "model-based reasoning tactics," in which people reason about quantitative relations in terms of the dimensional structure of functional relations described in the problem. These tactics support different activities within surrounding solution attempts and usually describe "states" in the problem's situational structure.Chapter 5 holds these activities accountable to local combinations of notation and quantity, reinterpreting results for model-based reasoning in an ecological analysis of material designs for constructing and evaluating quantitative inferences. This analysis brings forward important relations between what material designs afford problem solvers and the complexity of episodic structure observed in their solution attempts. The dissertation closes with a reappraisal of the relationship between knowledge, person, and setting and, I will argue, puts us on a more promising track for a descriptively adequate theoretical account of constructing mathematical representations that support applied quantitative inference