About this site

Introduction

In Chapters 6 and 7 of my book, “How I wish I’d taught maths: Lessons learned from research, conversations with experts, and 12 years of mistakes“, I described my process for introducing concepts to students, and helping them develop their understanding. The cornerstone of this process was the use of example-problem pairs with Silent Teacher, followed by intelligent, varied practice.

Since writing the book, I have become a little obsessed with this process, and in particularly the nature of this intelligent varied practice. So, I decided to put together this website in the hope it will be of some use to both teachers and students wanting to know more.

 

Variation Theory

My approach to intelligent, varied practice is built upon my understanding of variation theory. The following two quotes capture the key point perfectly:

When certain aspects of a phenomenon vary when its other aspects are kept constant, those aspects vary are discerned (Lo, Chik & Pang, 2006)

If learners think that mathematical examples are fairly random, or come mysteriously from the teacher, then they will not have the opportunity to experience the expectation, confirmation and confidence-building which come from perceiving variations and then learning that their perceptions are relevant mathematically (Watson and Mason, 2006)

By holding constant as much as possible and varying one element, we can direct students’ attention to that element that has varied. Any change (or lack of) in the answer may then be attributed to the change in the element. Moreover, because each question or example is related to the one that preceded it, students are able to form expectations as to what the answer will be. I call this process reflect, expect, check. This can lead to significant moments of revelation and discussion when these expectations are not realised, compelling students to think more deeply about the processes invovled, instead of just cruising through an exercise on autopilot.

I am aware that there is far, far more to Variation Theory than simply the nature of the examples and exercises I give my students. But for me, this focus has had a significant effect on the complexity of questions my students can answer, how much we get through in lessons, how much they retain, and the class discussions that we have. In short, it has been a revelation. For discussion of the other aspects of Variation Theory, then I recommend listening to my interview with Anne Watson and John Mason, and checking out the book that Anne edited: Variation in mathematics: A collection of writings from ATM Mathematics Teaching, a snippet of which can be found below:

For me, the activities and sequences of questions on this website are predominantly targeted at that Consolidation phase.

 

Non-intelligently varied practice

To better understand the nature of practice, consider the following sequence of questions on sharing in a ratio:

Notice how the answer to question 1 does not enable the student to form any expectations about the answer to question 2. They are separate entities. There is certainly plenty of practice, but there is also too much variation – and that is a wasted opportunity.

Intelligently varied practice

Now consider this sequence of questions:

By asking our students to pause before attempting each question, reflect on what is the same and what is different form the preceding question, and hence form a prediction of what they expect the answer to be, and then check that expectation, we are allowing our students to think deeper about the mathematical structure and processes involved.

If a student’s expectation about the answer turns out to be correct when they check it, then brilliant as they get a nice confidence booster. And if it is not realised, then even better as they get a cognitive shock. For example, in the sequence of questions above, a student may expect the answer to the third question to be double that of the second. But when they check, they find the answers are the same. This is likely to throw the student off auto-pilot and force them to question why, thus thinking hard in way that simply does not happen with less structured sequences of questions.

 

Four types of activity

It is still early days for me in my voyage into the wonderful world of Variation Theory, and this website will house my efforts. There will be four types of activity, reflecting my four main uses of the principles of variation theory. In time I will also record a short video to accompany each post so you can see what this might look like in action in my classroom. These videos might also be useful to share with students to watch in their own time.

The key principle that covers all four types of activities is what I call: reflect, expect, check

Before answering question I want students to pause and reflect for a moment, consider what has changed and what has stayed the same from the example or question before. To ensure they are doing this, I may ask them to put their pens down. I then ask them to form an expectation as to what they expect the answer to the current question to be based on the previous answer. Finally I ask them to check their answer, either by carrying out the procedure they have just been taught (in the case of Examples) or through discussion or teacher demonstration. For me, reflect, expect, check is one of the keys to learning.

 

1. Examples
I present students with an example that I first model in absolute silence so my students can concentrate as much as possible on what is happening on the board (what I call Silent Teacher). I then annotate and narrate over the top of my example to give clarity to the process. Then my students try a mathematically similar example on mini whiteboards, and I would choose a couple of attempts to utilise the technique of show-call. This process, which goes under the name example-problem pairs, is described in detail in my book. Then comes the intelligent, varied practice. Students can use the techniques from the example-problem pair to solve each question, but via careful variation their attention is drawn towards the critical features of questions. The teacher can help by explicitly focussing on the key features that have changed during whole-class discussion. This enables students to form exceptions of answers, considering the deeper structure of the processes invovled. When students have finished, they can be challenged to continue the sequence of questions, justifying their choices.

For an example of Examples see Product of Prime Factors or Factorising into double brackets

 

2. Rules
Instead of attempting to explain a concept, rule or definition, I present students will a selection of carefully varied examples, and silently indicate which ones belong and which ones do not. Again, the examples are chosen to draw students’ attention to the key features involved in the concept, rule or definition. They are encouraged to pause and reflect. This should enable them to form a more holistic understanding of the concept that makes sense to them, as opposed to trying to make sense of an abstract, difficult explanation from me. After this process, I present students with a few questions to complete independently to see if they can apply what they have learned, followed by a group discussion.

For an example of Rules see Ready to be added? or In standard form?

 

3. Patterns
I like to harness my students’ natural ability to spot patterns by presenting carefully varied sequences of examples in the hope that they are able to spot what is going on and fill in the gaps. I answer the first question in silence and then pause. I then answer the second. After each answer I invite my students to attempt to complete the rest of the pattern, making corrections where needed and reflecting on why those corrections are needed. Many of these sequences of examples have what I call a gap of understanding – that is, a break in the pattern followed by two further examples for students to complete. This is based on a principle I learned from Anne Watson and John Mason to ensure students go across the grain as opposed to just carrying on a pattern of numbers without reflecting as to where the pattern comes from in the first place. After this process, I present students with a few questions to complete independently to see if they can apply what they have learned, followed by a group discussion. Much like the gap of understanding, this final stage is crucial, as I need my students to be able to answer questions in isolation – in other words when they are not part of a carefully varied sequence.

For an example of Patterns see Adding negative numbers or Expanding single brackets

 

4. Demonstrations
Here I make use of the wonderful world of technology, including dynamic geometry and graphing packages such as Desmos and Geogebra. Whereas in the past I would use interactive demonstrations, I now harness the principles and power of Variation Theory by always having two versions on the screen at the same time. This allows me to keep everything the same, change one thing, and my students’ attention is more easily drawn towards the change and its effect than if only one example is on screen at any one time. It is a small change, but one that has made a huge difference.

For an example of Demonstrations see Angles on a straight line or Sketching y = mx + c

None of this is rocket science – nor indeed complex mathematics pedagogy 🙂 But using this approach regularly when introducing my students to a concept has significantly improved their initial understanding, along with the ability to retain and transfer the acquired knowledge. This approach can then be followed by the use of SSDD problems to help our students develop into the problem solvers we all want them to be.

 

A note on pedagogy

It is important to note that an intelligently varied sequence of examples or practice questions will not magically make students understand something better. Whilst the structure and sequencing of the questions, together with the explicit processes of reflect, expect, check certainly makes it far more likely that students’ attention will be focused upon the thing you want them to focus on, there is certainly no guarantee of this. These examples and questions need to be accompanied by a pedagogy that supports them.  Anne Watson describes this point beautifully:

This is important for the way that variation theory is seen  within the mathematics education field, because a superficial look at it might suggest that the theory as applied to design is purely cognitive and concerned only with  what is presented to learners in order for them to construct meaning individually  through inductive reasoning. The addition of teachers’ gestures and speech to the mix indicates a need to think also about attention and the disposition of learners to  discern what is intended.

For me it is the teacher pausing when answering a question, and the way they explicitly indicate which feature of the previous example they are focusing on when considering the new answer. It is the carefully planned discussion that follows the completion of an exercise, avoiding unnecessarily vague questions such as “what did you notice?” in favour of more pertinent ones like “how did the answer to the previous question help you answer this one?”, or “which answer surprised you the most?”. It is these teacher actions that give intelligently varied sequences of examples and questions their power.

And in terms of creating and presenting the activities, this image by Ben Gordon sums it up better than I ever could:

 

FAQs

When I discuss these ideas of variation with teachers, I get asked a number of questions:

How do I get my students to reflect and form an expectation instead of just ploughing through the questions?
This can be tricky, and of course there is no way of ensuring every student will do. Modelling this process is key. I always begin by ensuring just two questions are visible, and students have their pens down. I then say “okay, so let’s reflect – what is the same and what is different about this second question?”. I insist on at least 10 seconds of silence, and then a paired discussion, then I take some suggestions. “Okay, now based on our reflections, what do we expect the answer to this question to be?”. More silence, and then more discussion. When it comes to setting the students off on their own, I have found – at least initially – asking students to put to their pens to convince me that they are reflecting and expecting is one way to stop them ploughing through questions on auto-pilot. More importantly, informing students exactly why the process of forming that expectation is so important can also help them buy into it. Many students like the boost in confidence that a correct expectation brings, and also enjoy the challenge of trying to figure out why an incorrect one didn’t work out. But, like anything, it will not work equally as well for every student. But I am convinced it does work for the vast majority, and that the positive effects can be huge.

What do I do if my students simply do not / cannot expect an answer?
I get this question a lot, and I have spent some time considering it. Let’s take the 4th question in the sequence of ratio questions above: Share £40 in the ratio 4 : 6. A student may look at that question, compare it to the previous one (Share £20 in the ratio 4 : 6) and either fail to spot the connection, or simply not be able to form an expectation. Is this a big problem? Well, not really. The student can still answer the question using the procedure you have just taught them, so they are not at any disadvantage compared to if this was a randomly chosen, unconnected question. But, having got their answer, they can then be encouraged to reflect upon it. The answer to this question is £16 : £24. The answer to the previous question was £8 : £12. The student can be prompted to ask themselves the question: “could I have expected this answer before I attempted the question?” This kind of reflection can be incredibly powerful. So, there is nothing wrong at all with reflection coming after students have worked out the answer. If if they still do not spot what is going on, they are no worse off than if they had attempted an unconnected question, and they have a third chance to understand it during any discussions that follow.

Where is the differentiation?
Ah, the eternal quest for perfect differentiation that has plagued my teaching for years. A huge mistake I feel I made for the vast majority of my teaching career was to attempt to differentiate too early on in the process. I made assumptions that certain students would understand something, and hence set them off on more challenging work when in fact it turned out that the fundamentals were not in place. Likewise, I was too quick to provide help and support without challenging students to actually have a go. For me, the best form of differentiation is the time it takes students to answer questions. Some will spot patterns, identify relationships and understand the deep structure quicker than others, and for those students I will challenge them to explain this to others, or offer up a challenge in the form of a UKMT question. But I want these students to start with the exact same questions as everyone else, because I believe the questions I have chosen are important for all. Likewise, I believe that presenting questions and examples in this way gives a far greater number of students an a opportunity to succeed than the randomly chosen questions as I used to use. So I will withhold support longer than I used to do, because I believe students can succeed. But if students are still struggling, then of course I will be there to help them.

My students are not getting enough practice of the procedure as they are working out answers using the previous answer! What do I do?
I never underestimate the value of practice, and the ability to fluently carry-out a key procedure, and this use of intelligently sequenced questions should never get in the way of that. So, you can “force” students to practice the key procedure during the Check phase by asking to see evidence of it. A quick holding up of mini-whiteboards usually does the trick. Or, I love the idea of taking a set of questions from the site, and building in columns for expecting and checking on a worksheet, like this one from Jenny Beech:

Additionally there will always be those students who get so good at the “expecting” that they do not feel the need to Check, and resent you for asking them to do so. When force fails, I find a really effective technique is to make use of the hyper-correction effect. The child who simply expects but does not check is likely to be on the receiving end of a significant cognitive shock when they find out – either through a quick glance at their partner’s work, or when the answers are projected up, or during the class discussion – that their expectation was false. Showing students the importance of the Check phase beats telling them importance any day of the week.

Won’t my students only be able to answer questions that form parts of a sequence?
Maybe. But is that a bad thing? Take the ratio sequence above, and the question “share £20 in the ratio 7:1”. Now, if that question was presented in isolation initially, a student may say “I can’t do that because you can’t divide 20 by 8”. However, when part of a logical sequence of questions, the student may reason that the answer is likely to be half of the answer to “share £40 in the ration 7:1”. Now, if they are faced with a similar question in the future and can reason that they may double (or treble, quadruple, etc) the amount to get one that does divide nicely, and then do the inverse to their answer, then they are thinking like a mathematician, and I am delighted! Of course, the only way to know if that is the case is to ensure you do present them with a question such as that in isolation in the future, which brings me to the next question…

How do I know my students have really understood it?
In short, you don’t. Or, you don’t yet at least. Just like anything we teach our students, we can only get a better sense if they have truly understood something if we ask them on various occasions in the future. The sequence of intelligently varied questions is just the first step. Presenting a single question in isolation as part of a starter, a low-stakes quiz or a mixed topic homework will give you and your students a sense of whether they have understood and retained the knowledge you hoped they would have done. And this needs to happen again, and again, an again.

Is there a danger that there is simply too much going on in a sequence of questions?
This is a great question that Dan Pearcy asked me after one of my workshops. And I think the answer is yes. Take the ratio sequence of questions above, for example. Above and beyond the procedure needed to share a quantity in a given ratio, there are a lot of concepts and big ideas within this sequence. What happens when the quantity being shared changes? What happens when the number of parts doubles? What happens when one of the parts stays the same but the other changes? And just after students tackle one big idea, they are hit with another! And so, there is the definite danger that considering these questions when forming expectations, and then carrying out a procedure that they are not all that confident and secure with, will simply be too much for some students, potentially leading to cognitive overload and little learning taking place. So, what is the solution? Well, I think it requires a judgement call from us teachers utilising the knowledge we have about our class. I would (and indeed, have!) used the above sequence of ratio questions with a bright, confident Year 7 class who love a challenge and who whizzed through the example-problem pair. However, with a Year 9 class who lack confidence – perhaps who have met and struggled with ratio before – I would adapt the sequence of questions. I would include more examples between each big idea. So, 3 questions where just the quantity doubles/halves instead of 1, followed by a few questions where the ratio doubles/halves. I am acutely aware both of the need to instill confidence and a sense of success in my students, together with not throwing too much at them at the same time so they have the best chance to take stuff in and learn from the experience.

Does this mean I can’t teach students *why* something works?
Not at all! Take the Example category of activities. If I am teaching sharing in a ratio or finding the mean, I am not dividing into an Example-problem Pair with Silent Teacher cold. I may introduce the concept using manipulatives, analogies, bar models, animations, dynamic geometry, stories – whatever I think will convey the concept best. It is only when I am at the point that I want to get students practicing a procedure that I begin the process described above. But – and I know this does not go down well with everyone – I am much more prepared to teach the How before the Why these days. So, if I can get students feeling successful carrying out a procedure, and I can get them asking questions during the discussion phase when they describe how they formed their expectations and how some were not realised, then this leads nicely into a discussion of the why at a time when students are in a better position to understand it. Likewise, Rules, Patterns and Demonstrations serve the same purpose. Students begin to form their own conceptions that get tried and tested throughout the activity – and then they can be laid out during the class discussion that follows.

This looks like death by worksheet! Won’t my students get bored?
On the face of it, some of these collections of questions look dull as anything. I mean, just look at the set of ratio questions above. But appearances can be deceptive. I believe there is so much good maths and potential for such amazing discussions lurking within these apparently dull looking questions. Students will only start to appreciate this when they engage with the reflect, expect, check process – and this might take a while. Likewise, anyone who visits your lesson and sees these kinds of questions on the wall might just need explaining the subtleties of what you are trying to do.

I tried this and my lesson was a disaster! What am I doing wrong?
It did the first time I tried it. My students did not have a clue what was going on. They were shouting stuff out, asking what was the point – it was terrible. New things take time to embed. If students are not used to reflecting and forming expectations, it can take a while. My advice is to stick at it. Try it once, leave it a couple of lessons, and then try again. After the third time, reflect on what worked and what you might need to tweak whilst keeping the fundamentals the same. Maybe try it with another class. Convince a colleague to try it as well and discuss your experiences. Find out what works for you. As a rule of thumb, I like to give every new thing I try in lessons at least 3 weeks.

Are you saying I should do this every lesson?
Not at all. Whilst I firmly believe that using the principles of intelligent variation via the four activity types on this website can contribute to the learning and development of the vast majority of mathematical topics, there is clearly the need for much more. I see students’ mathematical experiences as a diet. Intelligent sequences of questions are a key part of that diet, but so too are less structured problems, inquiries and student generated examples. As I discuss in my book, I still make use of rich tasks and investigations, but I am just more choosy when I do so, preferring to ensure students’ knowledge and understanding of a concept is at the point where they can really benefit from and enjoy such activities. Likewise, I am a huge fan of the principle of Purposeful Practice, which is my preferred way to have students practice and revisit concepts after initial instruction (I discuss Purposeful Practice in Chapter 10 of my book, and in my podcast interview with Colin Foster). I believe that carefully chosen examples and practice are the most effective way of helping students to understand new concepts and procedures, but they are certainly not the be all and end all.

You do realise that there is a lot more to Variation Theory than just this, right?
Yes I do. Very much so. This is just my interpretation of a specific aspect of the principles of variation. I know not everyone will like it, but I do. And I genuinely believe it has had a significant impact on the confidence, understanding, retention and work-rate of my students.

 

Comments from teachers

If you used any of the activities on this site in your lessons, sent me a tweet (@mrbartonmaths) and I will collect together comments here.

 

Get involved!

Few things in life would make me happier than if this grew to be the world’s largest collection of intelligently varied maths questions. With that in mind, it would be amazing if you could share your efforts. You can find out exactly how to get invovled here.

 

I hope you and your students find this site useful.
Craig

My book

How I wish I’d taught maths, is available directly from my publisher, John Catt Educational Ltd, from Amazon, and from most good (and evil) bookshops.