When I discuss these ideas of 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 following sequence of ratio questions:
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. Likewise – and depending on the specific nature of the activity – students could be encouraged to consider if the answer with be bigger, smaller or the same as the previous one. Then, having got their answer by carrying out the procedure, 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.
Of course, sometimes it will be impossible/unhelpful to epxect a certain answer. This is the case when two questions appear connected on the surface, but actually there is no quick way to jump from one to the other. For example the equations:
7 = x – 11
-7 = x – 11
The student who works out 18 for the first, may expect -18 for the second and experience a cognitive shock when the answer does not come out this way.
Discussions based around these Transition Points are just as powerful as when an expectation is possible
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, continue the sequence of questions, 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 students.
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. Some students may take a few questions to get into the swing of it, and then be able to reflect and expect. If I do not give them the chance to do this – as would be the case with an non-intelligently varied sequence of questions – I am putting a ceiling on what they can achieve.
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:
Or this one from Danielle Moosajee:
Additionally there will 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. I find a really effective technique is to make use of the hypercorrection 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.
Do I need to discuss every Transition Point?
I don;’t think this is possible. That is why I advise choosing two to discuss with each class, and have individual or small-group conversations about other Transition Points whilst interacting with students as they work through the sequence of questions. I find that the most appropriate way to differentiate effectively, offering support and challenge when needed.
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.