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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.”

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?

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:

- The students lack the knowledge to understand the Why at the stage it is being taught.
- Not knowing the Why does not inhibit their ability to do the How.
- The How is a mathematically sound method, not a trick that has no mathematical validity
- 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.

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.

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.

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## 1 Comment

Hi Nicole,

It’s great to see someone trying to crack this nut ie How best to teach stats.

I’m currently 3 years into a return to Uni as a career change. I started in a Maths & Stats major but have since moved into IT majoring in Data Analytics. I never did high level maths / physics at high school and I found that I really struggled in my Uni courses. Maths is a language and as such has a lot of terms that need understanding before you can gain an understanding of a concept – this is where I found I needed to spend hours and hours of extra study time to try and catch up. I would often read about a key concept, thought I’d understand it then go onto the next subject that needed this understanding only to find I needed to go back and do the first topic again – very frustrating!!

I still marked well (HDs / As) but I was only good at the procedures needed to do this … if you asked me today what I could use this knowledge for I’d have no idea.

For me, I’m a visual learner and I found the language of maths doesn’t draw pictures – you just need to ‘get’ what a term/concept means. I think this is where huge steps could be made in maths teaching ie describe the concepts in pictures and/or visual language. If you did this I think you’d find more young kids sticking with maths.

My [current] experience with my Uni lecturers is that they mostly only teach to the Aural/Verbal learners so they just don’t seem to get that there are students that need the material taught in a different way. It kind of says to me that those students that get through maths at school and go onto Uni lecturing are predominately one style of learner ie Aural/Verbal and continue that delivery in their classrooms?

As an aside, I found a short survey I did in one of my first Uni courses quite interesting. One question asked what style of learner we were and a high percentage (80-90% from memory) of students answered ‘Visual learner’.

So with all the above and to answer your question about the “… relationship between conceptual understanding and procedural fluency” – I think there should be many pathways that lead to both … at the same time.

I mean, some students may get the concept before they can do the procedure (ie “conceptual understanding” first) however others (like me) may need to do the procedure many times (“procedural fluency” first) before the concept ‘clicks’. I don’t think the teaching should be based on the content but rather the student learning styles and delivered in a synchronized way.

Easier said than done I’m sure but that’s my thoughts 🙂

Cheers