When is an exploration exploratory? A comparative analysis of geometry lessons

This paper presents a comparative analysis of two textbook lessons on the same topic from U.S. textbooks to learn how differently - designed “exploratory” lessons may structure content to enable or constrain student inquiry. One lesson, representative of a “reform - based” textbook, contains investigations of conditions of triangle congruence. The second is a “technology lab” on triangle congruence fro m a "traditional" textbook, the design of which is atypical for that textbook. Framing a lesson as a mathematical story, this analysis exposes three distinct ways that these lessons are different: (a) the proportion of the lesson in which mathematical questions remain unanswered, (b) the manner in which content unfolds to address each question, and (c) the way in which open mathematical questions overlap to increase the dynamically - changing number of questions that are pursued. This contrast of the two lessons illuminates how a lesson structure can prevent an "exploration" from being exploratory

Exposing the mathematical differences between enactments of the same written lesson

In this paper we respond to Huntley and Heck’s 2014 call for new conceptual frameworks that recognize mathematical differences between enactments of the same written lessons that stick “closely” to the textbook. We use a mathematical story framework to describe differences in the mathematical development of three enactments of the same algebra 1 lesson by three different experienced teachers. We find and document differences in how the lessons raise questions, sustain inquiry (or not), and progress toward resolution of the questions. These differences influence the overall mathematical and temporal structure of the enactments, which, in turn, affect the student experience and potentially affect student opportunities to learn

What do teachers attend to in curriculum materials?

In this paper, we describe an emerging methodology using eye tracking to explore teachers’ curricular attending as they interact with curriculum materials to design a lesson in order to learn what teachers pay attention to and how this attention shifts during planning. We propose affordances of this new method, remark on some of its limitations, and propose future directions

The changing expectations for the reading of geometric diagrams

Students studying geometry at the secondary level are expected to read diagrams in different ways than those in elementary school. In this paper, we present an analysis of the changes in diagrammatic expectations by comparing the geometric diagrams found in Grade 1 U.S. textbooks with those in U.S. high school geometry textbooks. This work included developing and using a coding scheme that recognizes dimensions of reading a diagram geometrically, including the type of object represented, use of deduction, use of mental redrawing, interpretation of markings, and the necessity of the diagram. The way in which elementary and secondary students are expected to interpret diagrams was shown to change along several of these dimensions, posing potential learning barriers for students. We end our paper with a discussion of what our results mean for the learning of geometry

The plot thickens: The aesthetic dimensions of a captivating mathematics lesson

We present an analysis of a sixth-grade mathematics lesson in which an aesthetically-rich moment of mathematical surprise, inspired by a decontextualized integer addition problem, spurred students to ask mathematical questions and actively sustain inquiry into the lesson’s central ideas. In order to understand how the unfolding mathematical content enabled this moment, we interpret the lesson as a mathematical story. Using this narrative framework, we describe the aesthetic dimensions of the story including its plot, density, coherence, and rhythm, and connect them to the unfolding mathematical content. This analysis demonstrates how these aesthetic elements of a lesson can be recognized and how they help explain the students’ productive engagement. This framework offers a potential tool for researchers and practitioners who seek to understand, design, and enact captivating mathematical experiences.Accepted manuscrip

The Role of Sequence in the Experience of Mathematical Beauty

In this article, I analyze the aesthetic dimensions of a sequence of mathematical events found in an unusual first grade lesson in order to demonstrate how sequencing may affect an individual’s experience of mathematical beauty. By approaching aesthetic as a sense or felt quality of an experience in context (Sinclair, 2001, 2011), this analysis explains how sequence can affect the way mathematical objects or actions are experienced by an individual. Thus, rather than questioning whether or in what ways a set of mathematical objects are beautiful or not, this paper addresses under what conditions is the mathematics in play beautiful. It is argued that with a better understanding of the temporal dimension of mathematical beauty, educational experiences with mathematics can be designed to captivate attention and nurture interest and positive disposition by students toward mathematics

Shaping mathematics into compelling stories: A curriculum design heuristic

This article describes a mathematics curriculum design heuristic that was developed and used in the design of CPM Educational Program textbooks. It introduces a metaphor of mathematics curriculum as a narrative story, which allows for the design of mathematical experiences that emotionally moves students and teachers and compels them to engage. Specifically, it explains how curiosity and surprise can be intentionally designed within a mathematical sequence of curricular elements, such as tasks. Background that spurred the development of the mathematical story framework is offered and examples from designing lessons for Algebra and middle school are provided