The Development of Mathematical Thinking Curriculum

The content and methodology of the VITAL: Early Childhood Mathematics Education resource are based on a series of mathematics education courses taught by Prof. Herbert Ginsburg (Co-PI) at Teachers College, Columbia University, and by Prof. Rochelle Kaplan (Investigator) at William Paterson University. These courses employ learning activities using brief video clips (or "cases") to help students (prospective and practicing teachers) in undergraduate and graduate courses analyze the development of young children's mathematical thinking and learning, and critically examine early mathematics instruction.

The "Development of Mathematical Thinking" Curriculum

Prof. Herbert Ginsburg's course at Teachers College covers topics ranging from children's informal understanding of basic mathematical concepts to the implications of developmental psychology for formal pedagogy and curriculum. The course is based on the premise that teachers can be more effective if they understand and can build upon the mathematics that children have already learned from an early age, with or without limited instruction. Students in the course are expected to learn theories of developmental psychology as they pertain to mathematics, and specific techniques adapted from psychology research.

Approach and Topics

Students spend the first part of the course examining the developmental features of children's mathematical knowledge in the context of a series of discrete video-based "cases." Later, students focus on how certain aspects of mathematics instruction, including pedagogical theory and curriculum design, can and should be informed by an understanding of children's mathematical thinking and learning. Topics include the following:

• Concepts in Infants and Little Children
• Counting, Cardinal Numbers, Shape and Space
• Everyday Mathematics
• Transition to Symbols
• Assessment, Clinical Interview
• Number Facts
• Calculation and Procedures
• Understanding, Constructivism and Manipulatives
• Preschool Curriculum
• Pedagogy
• Textbooks
• Educational Media

Course Structure

Each week students prepare for class by completing readings, watching videos, and completing assignments all within the VITAL environment. Completing assignments in VITAL in advance of class not only prepares students to engage more deeply in the lecture, it helps Prof. Ginsburg anticipate their questions. By highlighting comments selected from the week's assignments, he can connect his lecture more closely with students' concerns and also motivate them with the knowledge that their interpretations may be recognized publicly in class.

During class, Professor Ginsburg uses video clips of children solving mathematical problems to illustrate important concepts. Both in and out of class, he encourages students to think critically about their observations and hypotheses and to use evidence from the videos and readings to support their ideas.

Course Assignments

Students complete three types of assignments during the semester: multimedia essays, video lessons, and a final project for which they create and try out a mathematical activity and an interview, and reflect on the outcome in an extended multimedia essay.

Multimedia Essays

Multimedia essays are designed to help students focus on specific topics by requiring an extended analysis of the course materials, combining students' writing with "quoted" excerpts of digital video that are embedded as links within the essay. There are several videos and readings assigned in each week of the course, along with a guiding question to help students make connections between the readings and videos, which illustrate key concepts described in the readings. Students are asked to watch the videos carefully and excerpt the most relevant moments in order to support the thesis proposed in their essay.

Video Lessons

Video lessons are intended to introduce students to the clinical interview as a technique for uncovering children's thinking. The video lessons are presented as an interactive series of screens that include brief video clips and questions that guide students to focus on children's behavior and language as well as the researchers' style. After watching a clip, students are prompted to comment on what they observed, to make suggestions for follow-up questions, and to critique the researchers' technique. Some video lessons make use of expert commentaries to show a diversity of interpretation. During the term, students must complete several video lessons in preparation for the interview that they will conduct for their final project.

Final Project

The culminating activity for the course is an independent research project with a report written in VITAL. Students must design a mathematical activity and videotape themselves carrying it out with a child and interviewing the child afterwards. Their video is added to the VITAL Digital Library, and students use clips from their videos in the context of an extended essay that addresses the background of their activity, their methodology, and the results of their investigation. The project gives students an opportunity to integrate the various strands of the course: selecting a particular psychological topic to investigate, developing an instructional activity based on theory, implementing the activity, using clinical interviewing techniques to assess the child's understanding, and completing an analysis in VITAL using the video as evidence to support the student's original hypothesis.