“People are naturally curious, but they are not naturally good thinkers… Think of the to-be-learned material as answers, and take the time necessary to explain to the students the questions.”

“Memory is the residue of thought… the best barometer of every lesson plan is “Of what will it make the students think.” Daniel Willingham, Why Don’t Students Like School?

The two cognitive principles above have really begun to make me consider what it is that my students have to think about during my lessons during each moment and how I can design my instruction and activities to maximise the amount of time students spend thinking about what I want them to think about.

Variation theory lends itself well to increasing opportunity for student thought. It was originally thought up by Ference Marton (2005) and has been taken up and developed by great mathematical minds such as Anne Watson and John Mason.

Again, I confess that I am not at all an expert in this field. Indeed, I have only been playing around with these ideas for the last couple of months as I begin to develop my practice more around a mastery approach to teaching maths. This blog is more of a note of reflections to myself as I begin to understand variation in maths. Of course, this blog is also aimed at highly intelligent educators in mathematics who may or may not have knowledge about variation theory – not the dummies that the title suggests. I felt even more compelled to share some ideas after listening to Jane Jones lead HMI in maths’ podcast interview with Craig Barton – she refers to maths inspectors looking for examples of carefully selected questions rather than whole worksheets and how students make connections in maths.

Our instructional design, sequencing of examples, as well as the sequencing and variation of practice questions in an “exercise” is vital for student learning – I would say the most important parts of the planning the series of lessons as well as the careful planning of how you will carry out formative assessment and respond to learner’s needs.

The word “variation” implies “different” in other contexts. I am not talking about variety in terms of “pick ‘n’ mix” at a cinema! How “different” one example is from another is essential when designing sequences of instruction or a set of practice questions in maths. We all know that practise is important. We hear phrases like “practise makes perfect” or “practise makes permanent” but what if the practise we provide is so carefully thought out that something more than just practise occurs? What if we can manipulate an exercise so that practise is just one of the outcomes of the exercise – that connections can be made between questions, that students can begin to compare and contrast, make predictions and form expectations, develop multiplicative reasoning? Wouldn’t that be great?

“The central idea of teaching with variation is to highlight the essential features of the concepts through varying the non-essential features.”Gu, Hang and Marton. 2004

There are two types of variation that I am going to talk about with regards to maths:

Conceptual variation Procedural variation

Conceptual variation – all the different forms of the idea are considered.

There is always an opportunity for different representations of the same mathematical idea – particularly when introducing a concept or a transformation of one idea to another. Mark McCourt pointed out to me on Twitter that the big misconception with this idea is that concepts do not vary – they are invariant. The way we can represent them and explore them can be varied, however. Below summarises some view points from Gu et al. as well as Anne Watson and John Mason.

We must think carefully about what we are going to vary – is it going to make the students think about the essential features and make links between examples or representations. You many also want to show non-examples from the offset. This is just as powerful and will mean that students do not hold misconceptions in their head moving forward.

Simply displaying examples and non-examples on the board at the same time is a good way to make students think about defining terms and thinking about “why” things fall into categories. Asking the question, “what do you notice?” can make students think about features and then you can cold call on pupils to give their ideas. Alternatively, you can just tell students what you want them to notice – this is not a crime you know! 😉 The importance of “polygon-ness” (made that up) is that we have a 2d enclosed shape created by the joining of the ends of straight edges to enclose a single area – this can be clearly seen in the examples and non examples above. Of course the importance of fractions of shapes being shared into equal parts is the idea in the second set of examples.

You can develop this idea with most things to really develop the background knowledge of topics that are going to help students think critically later on. After all, we cannot think critically in the absence of background knowledge.

Do you show examples and non-examples of linear sequences, linear graphs, pictures of 1/2, prisms, quadrilaterals, the perpendicular height of triangle, pictures of radii, …? The list is endless.

Can you create activities that make students think about examples and non examples?

For example:

This really helps students to separate the essential features of what it is that makes a term a like term to another. Of course you could fill this in yourself and ask students “what do you notice?” and carefully include all of the common misconceptions that you want students to focus on – i.e. where to include negative terms, fractional terms, terms that don’t contain variables, terms like ab and ba, terms that contain the same co-efficient but not the same variable etc. These all form good questions to ask pupils.

Can you think of an activity for the next topic you teach that includes this kind of activity?

You may also use conceptual variation to develop the same mathematical idea in different contexts or problem areas to broaden the scope of what the student connects the idea to:

How many times do you introduce the idea of fractions with 3D objects as shown above?

Procedural Variation

If you look at experts in the field of fitness and muscle building, they often advise “compound exercises.” What this means is that you exercise a large muscle group but other smaller muscle groups also reep the benefit. This is how I think about procedures that are carefully varied – they provide the practice of the main skill but also benefit other areas such as reasoning and making connections.

Often referred to by the NCETM as “intelligent practice” this involves carefully constructed and sequenced question sets. The idea is to”minimally vary” non-essential features of the question. Only one aspect varies each time. Some of the reasons for doing this are:

It reduces cognitive load* since the questions are only changing each time in a small way Students can make connections between questions Students can begin to form expectations and make predictions Still develops fluency of skills through the practice gained

For example,

Consider the question sets below:

What you’ll notice is that both questions sets contain the same questions but are ordered differently. Once completed, each student will have gained the same level of practice in terms of difficulty. The reason that set B would be classed as more intelligent practice is that question 1 and question 2 are related. As a teacher, you could say, “complete the first three questions – once you complete them, what do you notice? Why is this? How does link to the next set of questions?”

qOther way to present the problems:

Variation links

Above is what I have called a variation link – these are the same 3 questions as the questions shown above but the lay out means it is easy to make links between the top question and the two below. Students then have to complete the bottom with either an example that is minimally different to those in the diagram or how about a non-example using some of the numbers used. The example-problem pair layout means that they can then complete their own link diagram using similar ideas but different numbers on the right hand side.

Around the clock variation

Here is an example of minimally different questions. The idea is that you base your questions around the example in the middle. You start at 12 o’clock and work clockwise around the clock. Each time you can ask your students, “what is the same? Why is this? What is different? Why is this? What links can you make between each questions and that in the middle? What about make links between questions around the clock? Are there any links between questions that are directly after one another clockwise? What about anti-clockwise? What about opposite each other?”

Forming expectations

(x+4)(x+1)=x^2 + 5x +4

(x+4)(x+2)=x^2 + 6x +8

(x+4)(x+3)=x^2 + 7x +12

By varying the second bracket and keeping the first bracket the same you can ask students to spot patterns and form expectations. These kind of questions can help us to form our question and answer sessions and make students think of the underlying principles. What do you want them to focus on? Here the focus is on the effect of altering the integers on the LHS of the identity on the constant and co-efficient on the RHS.

(x+3)(x+2)=x^2+5x+6 (x+4)(x+5)=x^2+9x+20

(x-3)(x-2)=x^2 – 5x +6 (x-4)(x-5)=x^2-9x+20

——————————————————————-

(x+3)(x-2) = x^2+x-6 (x+5)(x-2) = x^2+3x-10

(x-3)(x+2)=x^2-x -6 (x-5)(x+2) = x^2-3x-10

We can ask questions like “what is the same?” “What’s different?” “What would happen if I change _____?” “What do you expect to happen if I just simply swapped the order of the brackets?” “How can I form a double bracket where the constant is -12?” “How can I form a double bracket where the co-efficient of x in the middle is 0?” “What do you notice about the order the terms on the RHS are written in?”

Again, I will repeat that I am a novice when it comes to variation theory. I would advise you to have a look into it and begin to introduce it into your instructional sequences and practice activities.

Remember, we want the student to think about essential features of content. Memory is the residue of thought – how can we vary our questions intelligently to make the main thing the main thing?

Anne Watson and John Mason have written a nice summary here for ATM:

http://www.pmtheta.com/uploads/4/7/7/8/47787337/variation_2007.pdf