## Abstraction in Mathematics

Is there a reduction in the members of Mathematics fan club around the age group of 9 to 12? This is when in most curriculums students are introduced to algebra with its abstraction and the letter “x”. Abstraction or abstract mathematical reasoning can be described as the switch from concrete reasoning to more rule-based reasoning. This becomes significant for achieving success in high school mathematics.

How do we deal with this fear or misunderstanding? We need to remember that this abstraction builds on the fundamentals of numbers. So if a student is struggling with this, it might be the result of gaps in the understanding of numbers and operation on numbers. For instance – understanding the idea of distributive property of numbers can help understand expansion and factorization in algebra better.

Example

3(4+2)=3×4+3×2-Specification

3(x+y)=3×x+3×y-Generalization

28+36=4(7+9)-Specification

4x+12y=4(x+3y)-Generalization

These properties can help in moving from concrete thinking skills to logical, rule-based thinking skills.

A good understanding of place value helps in solving interesting problems and puzzles around digits and numbers. I will share one of such puzzle that I found in Alex Bellos book “Here’s looking at Euclid”.

- Choose a number in which the first and the last digits differ by at least two-for example 753.
- Now reverse the digits in this number i.e. 357.
- Subtract the smaller number from the larger, 753-357=396.
- Finally, add this number to its reverse, 396+693=1089
- Try this with different numbers and see what happens.

The proof for this is based on algebra. When I see a number puzzles like this, I want to get to “why it works” – the proof is always based on algebra/abstract mathematical reasoning. I have to say that I have become a greater fan of the subject once I started enjoying the abstraction.

And to get the students to overcome their fear – my suggestion is to have them read more mathematics, explore beyond the classroom syllabus and delve deeper into the reasoning.