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Mathematical Thinking: How to Develop it in the Classroom

Masami Isoda and Shigeo Katagiri
World Scientific
Publication Date: 
Number of Pages: 
Monographs on Lesson Study for Teaching Mathematics and Sciences 1
[Reviewed by
Tom Schulte
, on

Hatsumon is a Japanese word for "posing a problem.” Teachers know that the way in which the problem is posed influences students' learning significantly. A teacher's considered hatsumon will orient a student's thinking so that the answer and learning appears to the student to be from within ε a discovery — as compared to a more occidental approach of step-by-step recipes intended to funnel the student toward a solution that becomes understanding after practice. The authors here offer a two-part work composed of first a largely high level view of the hatsumon approach, followed and buttressed by a final part of low-level, detailed classroom examples.

Since developing a higher level of mathematical reasoning is a major aim of mathematics education, it is important for the teacher to have some idea of what “thinking” is and specifically how mathematical reasoning can be expressed. This work seeks to go in that direction by providing a taxonomy of thinking variants (analogical, integrative, developmental, etc.) and describing how each facet of the thought manifold can be touched. While anything as complex as the thinking process will be described differently by different experts, there is a fulfilling sense of completeness to the authors’ analysis of thinking modes and how each is covered in such techniques as their template questions and approaches.

Regardless of the psychological aspects of their work, the authors offer to Western grade school teachers a refreshing, inspiring set of Asian approaches, such as parallel number lines to introduce multiplication and mapping work rate problems to calculating the areas of rectangles. More unexpected directions are fully explored in about one hundred seventy examples of thought-provoking learning activities (games, lessons, and blackboard presentations), all based on tables of numbers with various properties (squares made of odd whole numbers, zero to hundred, etc.). These examples are detailed in their goals, age applicability (based on the Japanese curriculum system), desired outcome (type of mathematical thinking cultivated), and preparation requirements.

While largely focused on elementary and high school students, the book offers approaches applicable to all levels of mathematics education even when the specific examples do not transfer directly.

Tom Schulte is a community college mathematics instructor in Michigan who read this book in time stolen away from lesson plans and wrote the review while proctoring an algebra exam.

  • Introductory Chapter: Problem Solving Approach to Develop Mathematical Thinking
  • Mathematical Thinking: Theory of Teaching Mathematics to Develop Children Who Learn Mathematics for Themselves:
    • Mathematical Thinking as the Aim of Education
    • The Importance of Cultivating Mathematical Thinking
    • The Mindset and Mathematical Thinking
    • Mathematical Methods
    • Mathematical Ideas
    • Mathematical Attitude
    • Questioning to Enhance Mathematical Thinking
  • Appendix for the List of Questions for Mathematical Thinking
  • Developing Mathematical Thinking with Number Tables: How to Teach Mathematical Thinking from the Viewpoint of Assessment:
    • Example 1: Sugoroku: Go Forward Ten Spaces If You Win, or One If You Lose
    • Example 2: Arrangements of Numbers on the Number Table
    • Example 3: Extension of Number Arrangements
    • Example 4: Number Arrangements: Sums of Two Numbers
    • Example 5: When You Draw a Square on a Number Table, What Are the Sum of the Numbers at the Vertices, the Sum of the Numbers Along the Perimeter, and the Grand Total of All the Numbers?
    • Example 6: Where Do Two Numbers Add up to 99?
    • Example 7: The Arrangement of Multiples
    • Example 8: How to Find Common Multiples
    • Example 9: The Arrangement of Numbers on an Extended Calendar
    • Example 10: Development of the Arrangement of Numbers in the Extended Calendar
    • Example 11: Sums of Two Numbers in an Odd Number Table
    • Example 12 When You Draw a Square on an Odd Number Table, What Are the Sum of the Numbers at the Vertices and the Grand Total of All the Numbers?