WARNING – Variation in progress!


This is a follow up blog on my original blog on A Dummie’s Guide to Question Variation in Maths.

Variation theory is becoming increasingly popular in recent times, even though it has existed for quite some time. This is, amongst other things, down to the attempt to transfer pedagogy from the Far East, a rise in popularity of the idea around “mastery in maths”, popular books, Twitter and conferences. This is a good thing because aspects of variation theory, when applied correctly, lend themselves to increasing mathematical thought in students and making/discovering underlying mathematical connections. I was further inspired by Craig Barton’s #mathsconf14 session about how we can use an exercise to promote deeper mathematical relationships explicitly.

I just wanted to share some quick reflections based on my own experiences and, perhaps, induce some caution before people begin to jump on the bandwagon that will undoubtedly begin to sweep our nation’s class rooms.

  1. Vary less for more connections to be made – When designing question sets to include variation, keep to one focus – less is more. What do you want to vary that the students notice? What do you want to remain invariant? If you try to change too much from one question to the next then you will increase the chance of cognitive overload and, perhaps more importantly, will not be able to isolate the misconception in order to correct it. For example, if you want students to isolate identifying which angles are alternate angles on parallel lines, consider keeping the parallel lines and the transversal invariant but the two angles that are shown in the diagram vary from one question to the next. If you vary the orientation of the diagram AND the angles shown, there is too much going on and students are probably going to get lost – the power of the intelligent variation is lost. Similarly, when developing solving equations questions, changing the side of the unknown is desirable on it’s own…

10x = 20

5x = 20

20 = 5x

2 = 5x

2 = 0.5x

2 = 1/2x

You may think that question 2 and 3 are pointless side by side but how many times do NOVICE equation solvers get stuck because they are so used to be being presented with “standard” equations where the unknown is on the left hand side?

2. Model the intelligent practice mind set – if you want students to think like a mathematician then you must try to model this daily. What you present you promote  is a good way of thinking about this. If you present questions that are not connected then you are promoting that this topic has no connections. If you go through examples (that have not been carefully designed or sequenced) without explicitly making links between questions then how are students going to know *how* to do this in the future? My advice would be to pick one question and keep asking it over and over again – “What do you expect to happen in the next question?” “What’s the same?” “What’s different?” “What would a mathematician ask?” “Can you spot/complete the pattern?” “What is going on in general here? (Can you generalise?)” “Write me a question that could go in between question 4 and 5.” Eventually, you want to ask the question, “What am I going to ask you?” and you want to students to repeat the question you have been asking constantly for that period of time.

3. Provide time to create a connection making culture – Any time we change our pedagogy even slightly, it takes a lot of thought for us to break our old habits.  This is the same for students but on a larger scale. We see this in departments where the pedagogy from class room to class room does not have steel pillars of principles that are rooted in the day to day practice. We see this where expectations from one class room to another differ. If you want variation to work for your class, you have to create the culture. This will take time. Anne Watson claims you can change the culture of a class room in 6 weeks (about a half term). I would say this is probably the right amount of time before it becomes a habit for students to start trying to make connections. The way you praise students here is crucial. Praise students publicly and explicitly for attempting to display mathematical thought or trying to make connections. If the connection is not the correct one, praise their effort for doing so and then scaffold another question to help them make the connection. Use language like “interesting”, “links”, “connects”, “fascinating”, “structure”, “generalise”, “pattern” to show that maths is not just a set of boring isolated skills.

    4.    Multiple representations is not the same as varying procedures. Variation means change. To vary something can mean to represent it in a different way. My fear with this is that teachers will take this literally and think that showing a “variation of methods” to provide choice is desirable. While this may be the case when trying to show multiple representations of fractions (equal parts on number line, a quantity as a fraction of another quantity, equal parts of a circle, equal parts of shapes, equal parts of 3d objects etc.) it is not desirable to show different methods for solving equations throughout a department (flow chart method or “float n ping” compared to balancing method). Consistent use of a transferable procedure with the fewest limitations and that which does not contradict future learning, in my opinion, is better than varying procedures to increase student performance in the moment. Yes, students benefit from seeing different representations, using bars, counters, manipulatives but if there is more than one way to carry out a procedure, try to think of the best way as a department to do this and stick to it.

5. Don’t vary for the sake of it – There can be a danger that when we focus on one thing like variation it becomes a bit obsessive. There will be times when we simply haven’t had time to think about this in the detail that we want and, as a beginner, you may not feel confident writing intelligently varied questions on the board off the cuff. Writing a set of varied questions is not easy, especially at first, so don’t be too hard on yourself if you find a work sheet to provide students with practice that is not varied in the way you like. Students will still benefit from the practice but will possibly just not be able to make connections.

6. Try it yourself first – If you *do* feel confident, however, give it a go – live. Practice on your own first – take any topic and try to atomise it’s content. Here is my thought process… Take a standard question and you could think,

What is an important concept about this topic that I want to vary?

What am I going to alter from question one to two? From two to three?

Can I keep the question the same but write it is a different order?

In diagrams, can I keep the question the same but change the orientation?

Is there a pattern I want students to spot?

Can I miss some of the pattern out and ask them to complete it or generalise?

Can I include algebraic terms instead of numbers?

How would the inclusion of a negative in the next question change student thought?

Can I include something in the next question that will surprise students?

Is there a way I can combine the previous two questions to help students find the answer the third question?

How can I aid students multiplicative reasoning?

How can I aid students understanding of place value and multiplying and dividing by powers of 10?

Below is a slide show of some that I have created recently from different topics to give you a flavour, inspired by Craig Barton. Notice how some things remain invariant and some vary – providing opportunity for discussion. The inequality on a number line and mean from frequency tables ones should be completed in columns.

This slideshow requires JavaScript.

7. Be aware of the curse of knowledge – As a mathematician, when designing these questions, we are designing them from an expert point of view. Experts and novices think very differently because of the way our schemas are arranged. Novices think of halving a number by dividing by 2 and probably separate dividing an odd number and even number as two separate things. We as experts have a much wider idea of halving in terms of dividing by 0.5, different representations of what a half looks like, that doubling and halving are inverses etc. This is a really basic example but my warning is to ask yourself, “have my students already got the fluent skills they need in order to make these connections?” This is why fluency is so important. What happens if you design a question set where the solution to the first one is 66 and the solution to the second is 198? We know that this has trebled, but will the students in front of you? Try to choose numbers that are better going to help scaffold these connections, particularly with lower attaining pupils to begin with. If you want them to make connections it has to be based on things they are quite fluent with. Don’t ask them to spot relationships between length and area scale factors if they don’t know their square numbers. There are certain topics that feed into everything. This blog from @maths_master Will Emeny talks about the topics that students need to be fluent with that feed into everything else.

8. Don’t differentiate intelligent practice – There is still a mentality in many schools, often driven by SLT that all lessons should be differentiated by outcomes and resource. This is a myth and is a large workload driver and stress inducing concept. “If my lesson is not differentiated 3 ways then I am not accounting for my students and I am a poor teacher compared to those that do.” This is a nonsense in my opinion and I will not subscribe to it. High challenge for all, then differentiate down to scaffold complex tasks. Keep the class together in the initial skill acquisition phase. This fits in with a lot of the research that has been done of effective instruction by Barak Rosenshine. I hope we do not see things appear on websites that fall under the category of “red, amber, green varied practice on BIDMAS” – this is missing the point. It is extremely arrogant to think that because a student has higher prior attainment on paper that they are not going to benefit from seeing connections between intelligently varied examples with simple numbers. There is no added benefit to differentiating these questions, especially when you use them as a sequence of examples. On a practical level that would mean making connections between three different sets of questions to three different sets of students. NONSENSICAL! Differentiation by resource has it’s place in some lessons, some of the time at certain points in a topic – but it certainly shouldn’t be the norm every lesson so that some monitoring adult can tick the differentiation box on their lesson observation checklist.

9. Deliberate department culture – if you are going to uptake this, ensure that you do not start thinking, “Ah, I am going to have a variation lesson today.”

Variation is not a strategy it is a principle. If you build your example sequences and practice questions or questions to assess on mini-whiteboards using these principles then a non-specialist needs to know about it. Staff are not going to part take in variation if they know that the Assistant Head teacher (Science teacher) comes into the lesson and sees no differentiation. Due to the ridiculous high stakes nature of observations in the current climate – staff will always revert back to their safety net – the school policy. Each department should have their own unique policy on such things. We differentiate in maths like this because… We differentiate like this in English because…

A statement, shared with senior leaders, might go something like this…

“You will not see differentiation in certain parts of the lesson because we feel that intelligently varying questions for the whole class allows connections to be made that wouldn’t otherwise. It allows us to keep the class together in the initial skills acquisition phase, forming deeper conceptual understanding, interleaving key building blocks in maths with other topics to embed them into long term memory and allows us to train students to think mathematically – a shared language that students can use for our communal departmental curriculum. We believe that students can learn everything they need to learn and more from one carefully designed set of questions.”

10. No silver bullet – As with all things in education that get a lot of air time, there will be cynics who believe this is just another fad that will come and go. I agree that there needs to be caution thrown to the idea this is the thing that is going to improve maths results across the country (the whole points of this blog). It is not. In isolation, nothing is sufficient. Curriculum design, formative assessment, responsive teaching, considerations to cognitive science and behaviour management are all still obviously vital. While it may be true that a lesson without variation can still be a good lesson,  a lesson that includes quality variation will better promote opportunities for mathematical thinking. While it may be true that teachers will try, and sometimes fail, to produce a quality set of varied questions, there will be some that master it and hopefully share their ideas online like @fortyninecubed has been doing. I would argue, from personal experience over the last 8 months of using these principles that your pupils become better independent thinkers, pattern spotters, generalisers, questioners and more interested in the subject that we hold so dear to our hearts.

Further reading

@fortyninecubed Jess Prior has an excellent blog/website dedicated to sharing such examples

Anne Watson and John Mason’s paper – Seeing an exercise as a single mathematical object

Naveen Rizvi’s blog – Resourcing: Applying Variation Theory

Register with the NCETM and view this article

Craig Barton’s book “How I Wish I’d Taught Maths” has an excellent chapter on Intelligent Practice.

Kris Boulton’s My Best Planning Series has some carefully chosen intelligent examples for solving simultaneous equations that are minimally varied.

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