In speaking to some teachers about mathematics learning there is sometimes a bit of misunderstanding about what constitutes mathematical understanding. The most common one is that if they can get it right, and they can regularly get it right then that shows understanding. But it is my argument that being able to apply a procedure is very different to understanding the procedure, it is easy to follow a procedure, but not so easy to know why it exists and why it works.
In workshops I run with teachers from pre-school through to secondary I like to use the following problem to explain the difference.
When three numbers are added in pairs the sums are 22, 39 and 45. What are the three numbers?
What I like about the problem is that it is simple enough that anyone from any level of schooling can understand it, but it is also complex for all those people. It also lends itself to a range of different ways to attack it.
When I present the task to teachers in workshops I split the room in half, one half have a go at it themselves with no prompts, hints or scaffolds, whilst the other half are given a very clear proceedure to follow. This proceedure is as follows
Typically I give each group about 10 mins to work on the problem. When I give teachers the method described I get responses as shown below. They normally work though a few of these in the 10 mins.
When I talk to the other group of teachers they have made some progress on the problem, but don't have an answer by the end of the time they are given. So if we were only looking at understanding as being able to get the answer then the group with the procedure would be seen to be more proficient. I give both groups the opportunity to talk about how their group approached the problem and after this I ask them one more question which is
Why do we add the sums together and divide by 2 in the first step?
This question throws them, initially they think it is an average until they realise that there are three numbers and we are dividing by 2. But a really interesting thing happens, the group that didn't have that procedure seems to pick up on why that first step exists and the group who did have the procedure doesn't pick up on it at all. The group with procedure where so used to following it without having to think about it that they could not see the reason for that step. The group who didn't have the procedure, and also didn't get the answer had to think much more deeply about the question, and in struggling with it they were able to develop a deeper understanding of the underlying framework.
In looking at the procedure I chose to answer the problem that first step is not intuitive and I choose that method particularly for that purpose. It is a very simple way of solving it and it takes some shortcuts, both of which are common practices for teaching mathematics in many classrooms. But in my experience, the easiest method and taking shortcuts also leads to a lack of understanding. Realising that you add the sums and divided by two because in adding the sums you have added each number twice, so dividing by 2 gives you the sum of the three numbers is an important pattern that they could have made for themselves had they not been first given the procedure.
A way of solving this problem I have found online and in textbooks is in the way below
Although this does not take any shortcuts, I also think that it tries to make it to "mathsy", it makes it more complicated than it really needs to be, it complicates the thinking to a point which the method of solving it does not closely resemble the original problem. Although some students really understand and can make sense of this method, for many in the past this has just become like the first way, a method to follow rather than something to understand.
For those who approach the question more intuitively and without the procedure I at times give them the following diagram to help them make sense of the question, normally though without the blue dot.
with this image it is clearer why the dividing by 2 is important, but when given this image most do not solve it that way. They look at of the boxes and notice that it is common to 2 sums, 45 and 39. They reason that the difference between the sums 45-39 = 6 is also the difference between the other two boxes. and the method they follow generally follows one of the two ways below.
Both of these methods make more sense intuitively of how to solve the problem and they don't bypass seeing the connections in the problem and developing a deep understanding of how it works. Hopefully as they progress though solving one or more extra of these problems they will be able to progress from the method on the left to the one on the right. Although the endpoint is the same the difference is in the connections they have made along the way and this gets to the heart of the difference between fluency and understanding.
Senior Leader of Pedagogical Innovation and Mathematics Coordinator in Regional South Australia.
Opinions in this blog are my own and do not necessarily represent the views of my employer.