Thoughts on Negatives

2024-07-13

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One of my favorite mathematical tools is the number line. It's an amazingly effective tool to help students understand quantities in a tangible way. Seeing spatial representations of numbers not only helps to manifest examples of magnitude and step, but it helps lead naturally into the concept of the Cartesian coordinate system.

Typically a number line is presented to students as a horizontal line. The point 0 is often set in the middle of the line, and numbers to the right of 0 are positive while numbers to the left are negative. Students are then told to think of positive numbers as being "to the right", and vice versa for negative numbers.

This is a handy way to think about positive and negative numbers when dealing with addition and subtraction. However, it is an oversimplification, and I believe it is the root of a common misconception in multiplication and division. It's easy to reason that multiplying two positive numbers (to the right of 0) should lead to another positive number to the right of 0, but students naturally think that multiplying two negative numbers (to the left of 0) will lead to another negative number to the left of 0. This mistake is ubiquitous among young math learners, and I suspect it's the moment when many students begin to feel that math doesn't make sense.

There is a much more useful way to think about negative numbers and how they interact with positive numbers. Instead of negatives always pointing to the left, negatives simply point in the opposite direction of whichever direction you're facing on the number line, starting with the positive direction.

As an example, consider the addition problems 8 + -2 and 8 - -2. In the first understanding, the negative sign in front of 2 tells us to move two units to the left. This is true in the first problem, where we are adding a negative number, but it is not true in the second problem, where the negatives cancel out and we move two units to the right. Our second understanding tells us to face the opposite direction that we normally would for a subtraction problem: facing to the right (positive) instead of to the left. Thus we move two units to the right.

Similarly, consider the multiplication problem -3 × -5. Our first understanding tells us that since -3 and -5 are to the left of 0, their product should also be to the left--an incorrect result. Our second understanding tells us to start with the positive direction, turn around for the negative sign in front of 3, and turn around again for the negative sign in front of 5. We end facing the positive direction again, and we know our answer will be positive.

Many tools in mathematics suffer from problems like this. They are introduced early in mathematical curriculum, and to help explain them to young minds, they are simplified too far. This can lead to many problems down the road, and I think we need to be more careful how we summarize explanations of math concepts.

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[Last updated: 2024-07-13]