### ADAPTIVE INSTRUCTION MARKET RESEARCH

## What is Mathematical Thinking & Why is It Important?

*By Dr. Kaye Stacey, Mathematics Educator*

### Review Summary by Diya Jaishankar

*Grade X Student, Market Research Intern*, *Adaptive Instruction*

*Diya Jaishankar is a tenth grade student in Bangalore. She is curious and observant and enjoys playing the piano, watching crime shows, solving puzzles and chatting with her friends.*

This is a summary of a presentation made by Mathematics educator and Emeritus professor, Kaye Stacey at University of Melbourne. The presentation explores the subject of Mathematical thinking and its importance. These topics are illustrated by means of two examples, which serve to demonstrate how mathematical thinking is used by both teachers and students of Mathematics.

Mathematical thinking is a goal of education and an efficient method of learning and teaching Mathematics. Mathematical Literacy is one of the core concepts involved in Mathematical Thinking and is the ability to use Mathematics for problem solving in daily life and in higher studies. It also involves various skills such as reasoning, modelling, and making connections. The skills required for Mathematical thinking can be divided into two types: technical skills, which include an in-depth knowledge of mathematics, reasoning, and self-learning (heuristic strategies), and non-technical skills, such as communication, organization, teamwork, and the presence of an inquisitive nature. The gist of the presentation is given below.

### 1. Requirements to improve Problem-solving

- Experience – of solving unique problems in the classroom
- Reflection – on lessons learned from the experience
- Strategies – which are effective and heuristic in nature, along with the development of mathematical and analytical thinking

### 2. Processes involved in Mathematical Thinking

- Generalising – Looking for patterns and relationships; making connections
- Specialising – Looking at several different individual examples to collect information
- Conjecturing – Predicting patterns and results
- Convincing – Discovering and explaining reasons behind observations to oneself and to others

### 3. Example 1: The Flash Mind Reader

- Instructions: Choose a number from 0-99. Add the digits. Then subtract from the original number. The crystal ball displays the symbol you are thinking of
- Specialising – by trying out the mind reader on several distinct numbers
- Generalising – by discovering that within certain ranges of numbers, the resultant number is the same (ex: 80 to 89 results in 72)
- Conjecturing – any number selected results in a magic number (9,18…63,81) which are multiples of 9. Each multiple of 9 corresponds to the same symbol seen on the ball
- Convincing – Brute force method – proving conjecture by performing due process on a given number and obtaining same result (predicted symbol) on the crystal ball.
- Algebraic method – Proving by algebra that all possible resultant numbers are multiples of 9 having the same symbol
- Teaching potential – Depending on the level of mathematical knowledge of the student, it enhances their ability to recognize patterns and prove conjectures, and develops a problem-solving attitude in students

### 4. Mathematics and Pedagogy

- The combination of both leads to efficient teaching and allows mathematical thinking
- It helps to analyse study material, achieve student comprehension, and understand students’ thought processes

### 5. Example 2: To teach students Area and Perimeter

- Instruction: to draw a rectangle of given area (20 cm
^{2}) and cut out – encourages investigative tasks by the student - Teacher uses geometrical knowledge and mathematical definitions to show to a student that a square is a rectangle
- Uses logical reasoning to recognize reason behind a measurement error made by a student and allows student to make the correction utilizing heuristic strategies
- Uses reasoning to explain how formula/counting of squares gives the area
- With interactive Q&A session, teacher ensures that students are aware of the limited circumstances of application of this formula to area of a rectangle preventing overgeneralization
- Allows student exploration by asking them to find varying rectangles of the same area
- Lays emphasis on generalization as students discover that all lengths and widths were factors of 20
- Students repeat the process with rectangle of area 16 cm
^{2 }and determine varying perimeters of these different rectangles, and lastly find the perimeters of different shapes whose area is 12 cm^{2} - Teacher thus practices many concepts and steps (such as geometry, numbers, area, and perimeter) using the mathematical thinking processes aforementioned

### Conclusion

To conclude, this presentation clearly explains the meaning of mathematical thinking and demonstrates its usage in a classroom setting by both teachers and students using comprehensible examples. The presentation also details how mathematical thinking can be used by mathematics students at any level, be it elementary or high school students. The provision of examples of utilization of mathematical thinking techniques well achieves the goal of convincing the reader of the need for and importance of mathematical thinking.