Conceptual vs procedural when teaching maths and stats

April 2008, Salt Lake City. It was my first NCTM conference and I was awed by the number of dedicated teachers of mathematics in one place. I had soaked in a pre-conference series about teaching statistics and my head was full of revolutionary ideas. I can’t remember the workshop I was attending but I declared that I saw no point in teaching students to calculate standard deviations by hand – and that I never did. The response was awesome! There was just about a stand-up battle between teachers who agreed with me and those who would defend to the death their belief in hand-calculations as the road to understanding.

The thing we all agree on is that we would like our students to understand what they are doing. What we probably disagree on, is what it means to understand. My favourite quote about understanding is from Moore and Cobb, that “Mathematical understanding is not the only understanding.”

All understanding is limited

The thing is, that all understanding has limitations. Take for example the normal distribution. I understand how the normal distribution is a good model for a lot of natural, manufacturing and human processes. I understand that as n gets bigger, the binomial distribution approaches the normal distribution. I understand how to use the normal distribution to model processes, and answer questions. Could I derive the formula for the normal distribution? Um… Nope! I might possibly recognise the Gaussian formula if I saw it, and I think it is pretty cool as it has pi and e in it. So do I understand the normal distribution? I think so. Certainly I understand it well enough to use it sensibly and know when it is less likely to be useful.

I know that there are statistics educators who do not believe people should perform regression analysis unless they understand the assumptions of regression and can perform residual analysis to check for violated assumptions. They have a point. But until people understand what regression actually is, they are not going to understand the finer points. And the key to understanding in statistics, for many people, is doing! And the question remains, where do you put the boundary on what constitutes understanding?

“How I wish I’d taught maths: lessons learned from research, conversations with experts, and 12 years of mistakes”

My current reading is a book by Craig Barton, “How I wish I’d taught maths: lessons learned from research, conversations with experts, and 12 years of mistakes”. I have also started listening to Craig Barton’s podcasts. The style of writing is engaging, and I love the reference to research. But some of it is doing my head in. I’m hoping by the time I get to the end of the book I will have an idea of how to implement the ideas without boring my class and myself rigid. To be fair, the author acknowledges this: “Now, I know what you are thinking – ‘God, your lessons must be so boring…’” But so much of what he says rings true with my experience teaching maths and stats.

This, however, I just love. Barton is talking about the question of which comes first, the How or the Why. It is in the chapter about the perennial question about the ordering of conceptual understanding and procedural fluency. As Barton puts it, “Both procedural fluency and conceptual understanding are clearly desirably and arguably completely useless without each other. But what is not so obvious is which we as teacher should try to help our students develop first.” In the following quote, the Why is a shorthand for conceptual understanding, and the How for procedural fluency.

“When assessing whether or not I should teach the How [procedures] before the Why [concepts] I need the following criteria to be met:

1. The students lack the knowledge to understand the Why at the stage it is being taught.
2. Not knowing the Why does not inhibit their ability to do the How.
3. The How is a mathematically sound method, not a trick that has no mathematical validity
4. The How needs to be a durable method that can be built upon.”

I like this set of criteria and find it helpful for examining my own choices about the order of teaching statistical analysis.

Procedures and concepts for line-fitting

Let’s take a look at line-fitting. It is helpful for students to see that you can fit a line to a set of points on an x y scatter plot. When we fit the line we can see that we are trying to get it so that all of the points are about the same distance from the line. We can get most computer programs, such as Excel, Google sheets or statistical packages to fit a line for us. Generally it will use least squares to do so. I see no need for students to understand how least-squares works before using a computer program to fit a line for them. There are other types of understanding, and the visual representation works. What is more important to me is for them to have a conceptual understanding of the interpretation of the line. The best way to develop this is by exposure to multiple lines in multiple contexts so that they can generalise their understanding. Computers enable us to experience multiple contexts without the burden of fitting the line by hand.

Students at early high-school level can understand what it means to have a line fitted on a graph. Not knowing precisely how it happens does not limit their ability to interpret the fitted line. The method is sound – every statistician uses computer programs to fit lines. No one fits lines from first principles. And the method is durable. They will never have to unlearn fitting a line using a computer.

Understanding builds on procedural fluency

Recently I taught for two weeks a class of Year 10 girls who have yet to discover their mathematical capability. Almost all of them told me they hated maths. I got them analysing data using a statistical program that drew graphs and found the summary statistics. They were successful at using the program and interpreting the output, drawing sensible conclusions about the data. Imagine my surprise to find out they didn’t know how to calculate a mean or a median. Actually, I’m quite pleased I didn’t know that ahead of time, because they were successful at doing some meaningful analysis, when I might have been stuck teaching them something procedural and never got to the meaning.

There is so much more to think about concepts and procedures in maths and in stats. I hesitate to say I have the answers. But hopefully I have some good questions. I’d love to hear any answers or questions you have about the relationship between conceptual understanding and procedural fluency.