Slice of advice 2019: What I have learned this year

I've been asked again this year to contribute to Craig Barton's end of year podcast: Slice of advice - what have I learned this year: http://www.mrbartonmaths.com/blog/slice-of-advice-2019-what-did-you-learn-this-year/

The main thing that I've learned this year is how few teachers are familiar with Richard Skemp's ideas on instrumental versus relational understanding. I meet a lot of teachers in my job and I will often have only one or two teachers in the room who have read it. These ideas have been fundamental to my development as a mathematics educator and I see it as the key concept that is most illuminating when considering ideas about teaching and learning in maths. It is also one of the main reasons why I am so passionate about the use of technology in maths.

Instrumental versus relational understanding

Skemp published his ideas in the 1970s but they are still available via the ATM website at: https://www.atm.org.uk/write/MediaUploads/Resources/Richard_Skemp.pdf

The paper contrasts two meanings to the word "understanding" that are used in relation to maths:

• Instrumental is just teaching the "how" of mathematical methods. Most commonly this will be when a method is presented to students, usually by working through an example, and then their activity is to be able to repeat this "recipe" for a number of different examples.
• Relational understanding emphasises the "why" of mathematics. Teaching for relational understanding focusses on the students being able to explain why mathematical methods work, especially by making connections, or relations, with other mathematical ideas.

For example, an instrumental approach to learning completing the square would be based on showing the students the mechanics of the method on particular examples and having them repeat this method themselves on some further examples. A relational approach would focus on relationship with the transformed graphs of quadratic equations and, in particular, the special cases of quadratic equations with repeated roots.

Instrumental understanding is shallower whereas relational understanding is longer lasting and more easily applied in solving novel problems.

Using technology to support relational understanding

There are a number of reasons why I believe technology, especially dynamic graphing software such as GeoGebra or Desmos, is so powerful in supporting relational understanding. The main two of these are:

• it constantly reinforces the link between algebraic, graphical and numerical representations of mathematical objects;
• it gives us the power to create many instances of a situation very quickly so that we can observe and generalise patterns.

An example: straight lines that intersect a parabola at one point

As an example I am going to think about the following problem:

What are the possible values of m and c so that the line y=mx+c has
a single repeated point of intersection with the curve y=x².

If you've not met this problem before then you might like to explore it in a graph-plotter yourself.

By putting this into a graph plotter it is fairly quick to find a number of different cases where the line cuts the curve at a single point (i.e. a tangent). These cases can be listed and observed algebraically, graphically and numerically. 

One way of solving this is based on using completing the square (I'm assuming that this is a task being explored by students before they've met calculus). Being able to observe that completing the square is applicable here is difficult for a student that has been taught the method instrumentally compared to a student who has a strong understanding of the relationship between completing the square and quadratic equations with repeated roots:

Skemp is still relevant

Although these ideas were first published over 40 years ago I think they are just as relevant today. There are increased demands for reasoning and problem solving at both GCSE and A level and teaching for relational understanding is key to supporting students with these. If you've not read Skemp before I'd suggest doing so and if you have I'd suggest a re-read.