As a follow up to my last post, take the example of ‘cancelling,’ a common technique in algebra. A student may be asked to simplify the fraction:
Instead of actually calculating 13 times 20, and then dividing by 20, a clever student may instead ‘cancel’ the 20 on the top and the bottom giving a final answer of 13. This can also be done with more complicated fractions. For example:
Since , we can cancel the 25 on the top and bottom to leave us with the much simpler expression:
However, my students often do not understand the limits of this cancelling technique. For example, they may think that in the fraction
the 3’s can be cancelled and reduce the fraction to
This leads to a whole mess of confusions.
Instead, we will start with the statement ‘you can always cancel any number on the top and bottom of a fraction.’
Next, a few choice examples make this rule seem plausible:
We can check that each example is valid by comparing the decimal representations on a calculator. Sure enough, the initial and final fractions are the same. The students now think that the rule is true. While they have not proved it, emotionally, they are sold on the technique. They believe it.
It is at this moment we create a shocking counterexample:
Everyone knows that 2 is not equal to 4. But our method has produced just that. Thus, our method is incorrect. The statement ‘you can always cancel any number on the top and bottom of a fraction’ must be false.
Of course you can sometimes cancel numbers and get the correct answer. But math is not about sometimes. If we state a general rule, it should hold all the time. By exhibiting a counterexample where the rule fails, and doing so with an emotional shock, hopefully students will not attempt to follow the misguided rule.