Now turn to the answers

In the course material I have been given to teach from, and in mathematics more generally, the answers appear on the next page, or sometimes even the same page, as the questions. This can present a real problem. Students will wrangle the calculator in ways they don't quite understand, but if they can get it to display the answer that is given they move on to the next question. They may not have understood the topic; the point of the problems is to understand through doing, so if that understanding is not full the student should repeat or try a similar problem.

For example: I answer a question and look in the answers. I have not got the answer given. I try the question a slightly different way and still get an incorrect answer^#. I try a third time and finally I get the same answer as in the book. Have I really understood the technique I am applying? Which application will I use in the exam? Still, I have the correct answer, so I move on happily.

Here is an example, the type of question most commonly given on the coursework. I am given a formula:
y = A * exp(k * t)
I am told the values of y at two values of t (e.g. "when t=1, y=63.2") and asked to find the values of the constants A and k. I convert to logarithmic form and make a pair of simultaneous equations. Solving these give the values of A and k (let's say 40 and 0.33). Now the formula is
y = 40 * exp(0.33t)
For the second part of the question I am asked to predict the value of y when t is some greater value; t=7, say. To do this, I put t=7 into the equation.

The crucial point to notice here is that I can check whether my values for A and k are correct by putting the values I have found into the formula and asking what is the value when t=1. Since I have been told the value at t=1 is y=63.2, say, I can calculate my value at t=1 and see if they match. If they match for both the points given in the question, I have the correct values for A and k. Using my values to predict the value at t=7, say, I am now completely confident I have the answer correct. There is no need to check the answers in the back. In this mathematics the truth is absolute.

I have tried to stress to the students the power of working the answer back into the question like this. I did this in my A-Level Mathematics and came out of the exam confident I had an A because I knew I had answered so many of the questions correctly. (In Further Mathematics and at degree level it is not so absolute, but at this level of mathematics as a toolbox it is the case). Given the question we can find the answer; given the answer we can find the question; etc. After all, this is how I set the questions in the first place, picking an answer arbitrarily and working back from there to form the question.

Two points come from this line of thought:

1. There might be benefit in getting students to set their own questions, since they will see the symbiosis of question and answer and this could help them in solving this sort of problem.
2. It might be interesting to set questions which are back to front. In some ways we do this already, the second part of the question above (predicting the value at some future time t) is really just finding values to set up the question with. However, to bring this home the answer could be given and we ask "what was the question?"

These points are not useful in this case at the moment, since the coursework is due in only a few days. Still, a student aware of this process should not fail to get 100% in the coursework I have set, since all the questions can be derived back from the answers, and I have tried to express this to the few students I have met.

And of course, this process of proving the answer correct is far more realistic. There are no answers in the real world.


# of course, there is always the possibility that the answer given is not correct and this has happened in the course notes. It is most gratifying when a student notices an answer that is incorrect, since this shows real deep understanding and confidence (and represents a valuable life lesson: we are all only human and prone to err), but it can really destroy the confidence of a student who has not yet fully built up their understanding.