Understanding maths, or writing down the answers

Bipods have two legs, and tripods have three.   At least two of each land on earth, and they have 23 legs.  How many of each are there?

Well, I thought of solving it with equations.  But there are two unknowns and only one equation.  So I borrowed two primary maths books from the excellent Educational Resource Collection (for trainee teachers - hope they also got the ones they wanted!) at work.   Starting with a guess, was the suggestion.  So we did, and with a bit of use of the 2 and 3 times tables, we arrived at an answer.

There is more than one answer, so we tried to find another.   But we did not understand how we had got to the first answer, so we could go not go any further.

I was not sure what to do next.  I am a Primary PGCE drop out.   So, I recommended we (well, my son) ask his teacher.

The next night, my son wrote down his own fraction problems.  He wanted to do them himself, and take them to school to show his teacher.  An admirable application of what he had learned. 

Some were easily solved using the techniques he had learned at school.

One was 5/10 of 48.  He drew 10 boxes and proceeded to share 48 dots between them.  He ended up with some boxes with 6 and some with 5 dots and was not sure what to do next.   I asked how else you could write 5/10?    1/2, he said.   He tried again, with two boxes, and worked it out.  

Then the next was 6/9 of 50.  The same problem with the boxes, and with no solution in sight, he was very upset that he would have to take the answers to school with blank spaces in.  One of his friends had written out some questions and had no blank spaces, he said.   Maybe, I said, he had not come up with any questions that he had not done at school.  Remember, these were questions my son had written himself, and however many times I told him not to worry, because he had come up against some maths he did not know, he was most upset.   Having the answers was more important than knowing how you got there, and more important than saying "I don't know" and realising that the methods you know don't work, even if you don't know why.

So, when devising assessments or assignments, I need to remember - make them so people have to understand a process, as well as having an answer.