It seems like math teachers do not care enough if you know how to get an answer. They care more that you can prove that you know how to get an answer. Here is an example to illustrate the difference:
I was helping an 11-year-old with his math homework. The assignment was to find the greatest common factor (gcf) for eleven pairs of numbers. He is really good at this. In fact, he figured out the answers to all eleven problems completely in his head, in less than one minute. This should be a good thing. This means he "knows" this concept. On to the next concept! Right? No, wait... what does that say on the assignment? Something about "showing your work..." Yes, that's right, even though this child did absolutely no work to get these answers, he has to write down some work. But not just some work. A lot of work.
So for the pair of numbers (60, 18), the gcf is 6. My little friend got this answer immediately upon looking at the numbers, as anyone who knows the concept would. But instead of writing down "6" and moving on, he has to write all this:
6 3 6 10
^ ^ ^
2 3 2 3 2 5
18 = 2*3*3
60 = 2*2*3*5
Both numbers have one 2 and one 3 in common, so the gcf = 2*3 = 6
An assignment that was completed in less than one minute has turned into an hour-long session of unnecessary writing. He has to write this process over and over again, eleven times, to prove that he knows the steps. Even though he did not use these steps to get the answer in the first place! I don't know a better way to make a kid feel like math is boring and hard and requires tedious processes.
I ask: WHY? Why do teachers do this? It would take one minute of a teacher's time, sitting with this child, to realize he knows this concept. Instead, so much of his precious childhood is eaten up filling out paperwork like this, night after night.
This is a child who is brilliant in math. He often arrives at answers to complicated problems in his head, without writing down any work, and without being taught any steps. Sometimes it takes me ten minutes, after he has already told me the right answer, to explain the steps he needs to write down for his teacher's benefit.
On tests, he often loses for not showing work. I have already shared some of my feelings about tests in general (Part I and Part II). But here is another great example of TESTS (Teaching Everyone Some Terrible Stuff). Now this mathematically gifted child is bringing home test scores in the "C" range, even if most of his answers are correct. This is harmful to both his confidence and his interest in math. How can he ever believe he is good at math if he brings home such low grades? And this will only get worse as he gets older.
I can remember being upset about this when I was a student, but fortunately it did not completely turn me off from my favorite subject. And I have both a Bachelor's and a Master's degree in math to prove it.
This practice is ruining math for our children. Can we please make it stop?
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