Visual and Kinesthetic Reasoning in Mathematics.

Walter Whiteley, Department of Mathematics, York University,

Graduate Programs in Mathematics, in Computer Science, and in Education.

Over the last few years, the strong connections between visual and kinesthetic reasoning in mathematics has been confirmed.   For many students, their critical breakthrough to conceptual understanding occurs in the context of kinesthetic experience with physical models.  At the recent ICME 10 conference in Copenhagen, one plenary, and several Regular Lectures, and some Topic Study Groups addressed this connection.  In spite of the 19^th century effort to ‘replace’ motion by set theory in the study of ‘limits’ in analysis, it is clear form observation of hand gestures, and from cognitive studies and brain images that this is the way both experts and novices process the concepts and ‘make sense’ out of the ideas.  

We will present some conclusions from this material as well as references for follow up.  Here are some points I hope to make in my presentation:

1. Recent studies of 3-D visualization also indicate that ‘scaffolding’ with physical objects lays the groundwork for later visual work using computer-based simulations of 3-D.   At the highest levels of mathematical practice, as well as applications of mathematics in other disciplines, good models, and such physically embedded visualizations remain an important tool for exploration and for communication.   

2. Feedback from other disciplines, such as stereo-chemistry, biochemistry and engineering reinforce the message that the current generation of students is not developing the necessary skills in 3-D reasoning, including spatial symmetries.  However, the ways these skills are used may not be, cognitively, the same way they are taught in Mathematics courses, nor are the supported by appropriate physical models, even in these applied areas.

3. As an initial consciousness raising step, it is important that students experiences  confirm that;

-         what they see is not what the student beside them sees, or what the teacher sees.

-         we can change what we see;

-         what we see has a major impact on how we think;

-         similar comments apply to kinesthetic experience and kinesthetic reasoning.

4. It is critical for teachers of mathematics at the post-secondary level to:

-         provide opportunities for students to explore with the kinesthetic sense;

-         observe the use of gesture by students (and themselves); 

-         encourage students to make full use of integrated kinesthetic and visual reasoning;

-         model for students the option of using visual and kinesthetic reasoning at the highest levels of mathematical work;

-         connect students to the models and visualizations which are used to support reasoning in applications of mathematics;

-         develop awareness among future teachers of mathematics that visual and kinesthetic reasoning are essential elements to the practice of mathematics, not optional add-ons.

5. A major gap in education in visual and kinesthetic reasoning is the development of widely shared conventions as well as support materials to be used over multiple courses to support effective use of visual and kinesthetic reasoning.  This is a community task, to which workshops such as this contribute in important ways.