Why is -3-(-9) equal to +6?

Have you ever asked yourself:

‘Why is the result of -3-(-7) positive?’

Or, if you are a teacher of mathematics, has one of your students ever raised this or similar questions about integer numbers such as:

‘Why is negative times negative positive?’

As a response to the first question, have you ever been taught that this is the ‘rule’:

‘When you subtract a negative number, you change the sign of the number being subtracted and then apply the rules of addition.'

In the second scenario, where a student raises this question, how would you respond? Would you respond to the question by reminding the student of the ‘rule’.

Whilst mastering this ‘rule’ would help the student to solve similar integer questions, as a mathematics educator (Vesife Hatisaru, ECU School of Education) and a mathematician (Steven Richardson, ECU School of Science), we think that ‘knowing’ this rule, and applying it successfully to solve similar questions, would add little to the student’s conceptual learning. That is, the development of student’s understanding of key mathematical concepts, or big ideas, is more powerful than simply learning rules or facts. ???

In a joint paper published with Mathematics Teaching (https://www.atm.org.uk/Mathematics-Teaching-Journal-Archive/177718), we offer our insights into how we might teach mathematical content so as to develop conceptual understanding.

We use examples that many school students may ask: ‘Why does, for example, -4-(-7) result in a positive number?’ Our approach in responding to this question, in other words the big idea that we employ, is applicable to other mathematical concepts including factorisation examples such as:

In essence, our joint effort provides at least two contributions to practise. Firstly, by presenting the mathematical background behind ‘rules’, we contribute to the efforts to enhance conceptual learning in students. And, secondly, our approach has the potential to prepare students for algebra learning.

The full article is available on the following link, and we welcome any of your feedback:

https://www.researchgate.net/publication/370593185_A_structural_perspective_to_mathematics_teaching_Why_does_the_difference_of_-3_and_-7_yield_4