Teaching students maths so they learn

Introduction

These web pages look at the sort of pictures, or “mental models” that students construct in their heads when they are learning mathematics.

These web pages focus on the types of mental models constructed as students learn mathematics. Clearly, this is only one component of student learning. They do not discuss other learning theory, nor the neuroscience, nor the psychology relating to student learning, all of which are also important considerations in helping students to learn mathematics effectively.

Part 1 of these web pages looks at the concept of a mental model, and compares the mental models of students to those of experts. Part 2 looks at what can be done to assist students to develop high quality mental models in mathematics.

The information in these web pages has been sourced from the literature on learning mathematics. As the body of literature on constructing mental models in university students studying mathematics is currently small, some work studying mental models in college physics students and children studying mathematics has also been included.

Contents

• Part 1: Mental models in experts and novices
  • What is a mental model?
  • We are different from most of our students
  • Student maths mental models
  • Doing maths without good mental models
  • Six differences between mental models in experts and novices
    • 1. Memorized procedures
    • 2. Choice of problem features
    • 3. Level of detail
    • 4. Contradictory models
    • 5. Model adaptation
    • 6. Model choice
• Part 2: Teaching students how to develop good mental models
  • Building mental models is hard
  • Conceptual models
  • From concrete … to conceptual … to mental model
  • Fixing misconceptions
• Part 3 What to do?
• References