Section 27.4 Closing Ideas
In this section, we took a concept that is traditionally very confusing for students and presented it as a single image that most people find simple and intuitive. We are once again leveraging both your intuition and experience with numbers to create a framework that is easy for you to understand and use. But this really isn't about the calculations. It's about the idea of the calculation. We are in the process of exploring just how far we an go with addition and subtraction using the number line. What we have shown in this section is that the number line is a more powerful idea than the addition and subtraction algorithms.
And that is the general situation in mathematics. Very often, a good idea will take you further than just applying formulas or prescribed algorithms. Developing a depth of mathematical thinking can help you solve problems that you've never seen before by relating it back to things that you have. These calculations with negative numbers are extremely accessible and understandable without needing to do a lot of extra work because we were able to relate it back to the number line, which we have had plenty of experience with over the last few sections. This is far better than simply having you memorize a complex set of rules for adding and subtracting in columns.
Whenever possible, you should try to focus on ideas rather than rules. You will be able to go so much further if you do.