At #MathsConf15, I was delighted to be asked to presented with Craig (@mrbartonmaths) and Ben (@mathsmrgordon). After #MathsConf14 in Kettering, in which Craig shared two series of examples based on expressing numbers as products of their prime factors and sharing in a ratio, I was thoroughly inspired, and went away and began writing my own sets of questions. Hence, minimallydifferent.wordpress.com was born!
Craig, unbeknownst to me at the time, was busy working on www.variationtheory.com, which I have subsequently been contributing to, and was finally launched at the weekend. I am so proud of the efforts that Craig, Ben and I have put into this site, and I genuinely believe it has the potential to be a real game changer in terms of Maths teaching. In this blog, I’m going to write up in details the things I shared at #MathsConf15, and hopefully discuss and respond to some of the criticisms that have surfaced in the last few days.
The topic I chose to talking about was solving one-step linear equations, where the ‘step’ to solve was either adding or subtracting. It had always seemed fairly obvious to me that in order for students to be able to solve complex linear equations, such as those with brackets, or unknowns on both sides, they would first need to be fluent with solving one-step equations. By fluent, I mean both that the procedural knowledge necessary was automatic and so not placing unnecessary demands on working memory, but also that pupils could understand and articulate the process of solving by balancing. This was key, for me: I didn’t want to teach my pupils a method that would not work at a later stage, and so I always emphasised the need to ‘balance’ from the beginning.
After having discussed some examples, I may have given my students the following sets of questions. Indeed, in my very earliest days of teaching, I would probably have provided them with these questions in the form of a card sort or something else ‘fun’. I thought this was great, and I was providing them with the opportunity to be successful (which would lead to motivation) and fluent (which would free up working memory later on).
This did not happen.
Reflecting back now, I would argue there are three main problems I found when using similar sets of questions with a class.
Firstly, pupils became bored incredibly quickly, and it isn’t hard to understand why! They would often do two or three, comment that they ‘got it now’, and I found it really hard to respond. Practice was necessary to ensure that they would retain the skill later and for it to become automatized, but this practice didn’t seem to be motivating pupils to reach that stage. Often, I’d comment ‘oh, just do every other question’ but there was absolutely no rationale behind that.
The second main problem was that pupils would forget how to solve simple one-step equations astonishingly quickly. Now, the more I have read and learnt, the more I am aware that there are multiple reasons for pupils forgetting, and multiple ways in which we can remedy this. One of those, however, is that this practice that I was insisting on was ‘mindless’; pupils were not being forced to think about how to solve an equation, the only thinking was ‘well, what is 19+6?’. This, I believe, was a problem.
The biggest problem I faced was that my pupils became completely unstuck when faced with ‘weird’ looking equations. The moment they encountered ‘7+x=2’ or even ‘2=x-7’, they would fall apart and all sorts of misconceptions would be exposed. The practice that I had been giving them was not intelligent, as each equation was identical in form. As such, I went away, did some thinking and some reading and began writing my own sets of questions, and eventually came up with this (which were shared at #MathsConf15).
Now, I am really pleased with this set of questions. Since using sequences such as these, I have noticed (and yes, this is anecdotal) an improvement in how my students are learning maths. I’m firstly going to run through the rationale behind the sequence, before discussing the effects using it has had (alongside some pitfalls).
The first equation is intentionally of the form ‘a+x=b’ rather than ‘x+a=b’. The reason behind this is I had encountered too many students perceiving the former type as something more difficult to solve, for the reason that they were less familiar with it. By introducing it from the very beginning, it normalises it and so students are less likely to panic when faced with it later.
The second and third equations are intentionally rearrangements of the first. This is something which is obvious to us as experts. It is something which may be obvious to some of our students. It is categorically not obvious to students who are learning this concept for the very first time, and so I wanted them to be exposed to this.
Following this, as students are becoming comfortable with subtracting to solve an equation, I’ve introduced adding to solve an equation. The fact that only one thing has changed means that pupils are able to attend to why the method to solve and the solution itself must also change, which is likely to be missed if the numbers are being changed significantly each time. This is followed with an equation where just one number changes. Will this make the solution bigger or smaller? Why? How do you know? Check!
I’ve then intentionally included two equations with the same solution. Again, the questioning that comes alongside is so important. Why do they have the same solution? Can you write me an equation in the same form with the same solution? How about an equation in a different form?
The inclusion of ‘-7=x-6’ is super important. I have taught high attaining students who would not be certain of the solution here, because the use of a negative will throw them. However, the fact that it follows directly from ‘7=x-6’ is important, as they are more likely to correctly reason ‘well, before I added 6 to 7, so now I need to add 6 to -7’. Next, I want them to notice that ‘-7=-6+x’ is simply a rearrangement. Again, this is obvious to an expert. It will not suddenly become obvious to a novice unless they are given the opportunity to see it, which a typical ‘solving equations’ exercise may not allow them.
I particularly liked the inclusion of ‘-7+x=-7’ and ‘7+x=-7’. These were included as I had recently noticed some of my students struggling when faced with similar questions in isolation: ‘do I need to add 7? Or subtract 7? Wait, do the 7s cancel out or not? Will it be 0 or 14 or -14?’. I don’t want them to have these struggles, and so I want them to see them initially as part of a carefully designed sequence, and so when they are subsequently met in isolation, they will be more familiar.
Following this, I included two equations which are set as equal to 0. In the past, I have only ever explicitly taught equations as being equal to 0 when solving quadratic equations, and yet there is absolutely no reason for that. I don’t want them to think that the method for solving a linear equation might suddenly need to change because it is equal to 0; I want them to continue to use the same reasoning and the sequence to find the solution.
Finally, I included three equations which incorporate fractions. As I explained on the day, this sequence of question was written with a particular set of students in mind, and I knew that this would be appropriate for them. It may be equally appropriate to continue the sequence by incorporating decimals or powers, or by asking students to generate their own questions. This is something which has to be determined by the teacher in the best interests of their students. I’ve repeatedly said that there is not one perfect exercise which should always be used when introducing any topic; it will vary and be context specific.
In terms of the impact that sequences of questions like this have had with my classes, it is important to be honest and recognise that initially, it was limited. I had mistakenly thought that by presenting my students with these questions and little additional input from me, they would suddenly become amazingly confident and fluent mathematicians, and that they would make connections so everything would be wonderful. This did not happen.
The process of ‘reflect, expect, check’ is not something that came naturally to a lot of my students at first, and so I began to spend more time on modelling and making that process explicit. Even now, some of my students are far more at ease with this than others. For those that do struggle, encouraging them to retrospectively ‘expect’ has been really beneficial. ‘Okay, so we can see that this solution is 5, and this solution is also 5. How might you have been able to expect that based on the format of the questions?’
The real pitfall that I encountered when I first started using ‘minimally different questions’ was that I wanted to try and use it for everything. Obviously, this is not appropriate. In the same way that it would not be appropriate to run Maths lessons based solely on inquiry, or solely on group work, or solely using Increasingly Difficult Questions, or solely on low stakes quizzes, it isn’t appropriate to use minimally different questions at the expense of all else. Pupils should be (and are, in my lessons!) exposed to a variety of tasks and activities within the classroom, and the appropriateness will often be determined by the stage which pupils are at with their learning.
However, on a more positive note: I have observed countless benefits from introducing concepts in this way. One such benefit is that there is a real value and purpose to the practice which I am asking pupils to complete. I am not giving them random sets of practice questions and expecting them to complete an arbitrary selection of them; I am giving them a set of questions which matter and which all pupils deserve the opportunity to be exposed to. This will typically be followed up with a problem from www.openmiddle.com or www.mathsvenns.com or any number of places. The minimally different questions are only used in the very earliest stages of concept acquisition.
The main benefit that I feel I have observed when using such questions with my classes is how it is helping them to be more ‘mathematically minded’. This has been so much my experience that I was quite taken aback to discover that minimally different questions have been viewed as ‘changing procedures for the sake of it’ and as a hindrance to ‘thinking mathematically’. I have also seen it suggested that pupils need to first practise a procedure, and then be given the opportunity to think mathematically, but I would argue that this is setting up a false dichotomy.
In my experience (and again, yes, this is anecdotal), these sorts of questions have significantly enabled my pupils to think mathematically. If the questions were presented in isolation, with no follow up discussion and no other tasks, then the quality of mathematical thinking would almost certainly be low. However, this is not what is happening in my classroom. In my classroom, my pupils are more likely than ever to interject with ‘oh, miss, I think the answer is going to double because this part has doubled. Wait, no, that must be wrong because of [insert reason here] – I’ll have another look’. The process of ‘expect, reflect, check’ is having the biggest impact on their confidence and the way that they perceive Maths. They love it when they expect correctly and are able to articulate why. They have also benefited so much from being able to articulate why their expectations are incorrect. My students are thinking harder than ever about underlying structures and can express ideas such as ‘oh, wait, so this part of the questions had that effect on the answer’ better than ever before. They can create their own examples – included those which will have expected answers and those which have surprising and curious answers. To me, this is part of what it means to be mathematical. It is not sufficient, but it is a start.
- Minimally different questions (and any of the exercises from variationtheory.com) could be used badly.
- Any resource made by any teacher at any point could be used badly.
- I always think carefully about the questions my pupils are exposed to at the very beginning of learning about a concept – this will not always be perfect and some sequences of questions work better than others, but I am continuing to develop here.
- I want my students to think mathematically and reason and explain and conjecture and generalise.
- I have noticed that Minimally Different Questions has helped with this.
- I will continue to be reflective and think about how my practice can be improved and try new pedagogical approaches when the rationale behind them is sound.