This chapter describes how the authors have utilized digital graphics and Web 2.0 technologies to design an information technology environment, LessonSketch. In LessonSketch teachers can learn about mathematical practice in instruction through the transaction of representations of practice. The authors describe the main features of LessonSketch, its collection of lessons, and its authoring tools, and illustrate what teacher educators have done with them.
TopIntroduction And Background
The development and wide adoption of the Common Core State Standards for Mathematics (National Governors Association Center for Best Practices and Council of Chief State School Officers, 2010) has brought unprecedented attention to the importance of mathematical practice in classrooms. The eight Standards for Mathematical Practice (SMP) set expectations for what that mathematical practice needs to be like. As with the content standards but perhaps even more considering the novelty of the SMP, it is critical for policy makers and teacher developers to help teachers support students’ engagement in mathematical practice and recognize when the standards for mathematical practice are met. To facilitate this, the educational community needs a way to communicate about mathematical practice and in particular to illustrate what the SMP mean, what mathematical practice looks like, and what are the circumstances in which a particular Standard for Mathematical Practice could be met. In the following pages we suggest that technological tools, such as those we have developed in LessonSketch (https://www.lessonsketch.org/login.php), can be useful in the communication and learning of mathematical practice and its Standards.
LessonSketch is an online environment where teachers and teacher educators can create, share, and discuss classroom scenarios. At the center of LessonSketch is a graphic language for representing classroom interaction, and software tools that allow users to create comic strips displaying instructional scenarios. This graphic language is comprised of two-dimensional images of people and objects that populate classrooms (e.g., teachers, students, whiteboards, desks, textbooks), and tools for representing their interaction (e.g., speech bubbles, facial expressions, and arm positions). This graphic language provides an alternative to written or spoken language for communicating about classroom interaction. Inside LessonSketch, users can use this language to create comic strips displaying instructional scenarios that can then be shared and discussed.
LessonSketch is a useful tool that can help solve the challenge of how to make mathematical practice and its standards into a theme for discussion and learning. While the content Standards describe knowledge and skills that students should learn in school, the Standards for Mathematical Practice describe characteristics of the activities involved in creating and handling mathematical knowledge. Thus, while Mathematical Content Standards can be represented with lists of mathematical topics or problem types that students should be able to solve (e.g., The Geometry-Congruence 1 Standard says that high school geometry students should “Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc,” National Governors Association Center for Best Practices and Council of Chief State School Officers, 2010, p. 76), a richer representation may be needed to communicate what meeting the Standards for Mathematical Practice means. For example SMP 6 “Attend to precision”, is described in the Common Core State Standards as follows:
6 Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. (National Governors Association Center for Best Practices and Council of Chief State School Officers, 2010, p. 7)