Teaching Surds – Part 2

In part 1 (available here), I outline how I taught surds for the very first time. This is part 2.

The second time I taught surds, I had put significantly more thought into my planning. I had thought about common misconceptions and mistakes that I had seen students make previously, and so I had considered how I would address these. I had discussed the topic with more experienced colleagues, and had drawn upon their expertise. I had decided on a logical progression through the content which needed to be covered, and so I was feeling pretty pleased about how this would go.

In the weeks leading up to introducing surds to my new class, I had consistently included questions in their Do Now to ensure that they were fluent with their square numbers. For those that weren’t, I had explained to them that they would find our next topic significantly easier to access if they were able to recall these without even having to think, and provided them with a homework task on our online platform to help with this.

The first lesson arrived. By the end of this lesson, I wanted all pupils to be able to explain what a surd was, as well as being able to simplify surds. This felt achievable. I began by showing them an isosceles right-angled triangle, where both legs had length 1. Asking how we could determine the length of the hypotenuse, a significant proportion recalled that we could use Pythagoras’ Theorem. A couple of pupils correctly talked me through the procedure, which I modelled on the whiteboard, and we concluded that the hypotenuse was of length root 2. One pupil commented ‘okay, so the answer is 1.414213562 then, so 1.41 to 2 decimal places’.

This obviously led nicely into me discussing what surds were and why they were useful – we spoke about the story of Hippasus, which they were pleasingly interested in (or possibly, just humouring me), and then we looked at a couple of surds and non-surds. (root 3 is a surd,  root 4 is not a surd). I then presented them with root 12 , we discussed if it was/wasn’t a surd, and when I felt convinced that all pupils could articulate why it was a surd, I explained and modelled the process of simplifying.

After this, we went through a few more examples as a class of how to simplify surds, and pupils completed some questions on mini-whiteboards, which meant I knew exactly who would initially quire some additional input from me. Following that, pupils did some practice of the skill. I had given them a set of questions (and I genuinely can’t remember which questions – either a physical textbook or an online worksheet) and they got on fairly well with this. In fact, the majority of our lessons on surds seemed to go quite well.

Surds are a higher GCSE topic, and they are something that a lot of pupils find conceptually difficult, but through assessing my groups’ performance in class (by using mini-whiteboards and multiple choice questions and just looking at their written work), I was so impressed with how they were doing. I felt quite happy with my surds planning and delivery this time round – pupils had made great progress, I reasoned – bring on the next topic!

As you may well have realised, things were not quite this straightforward. I try to regularly incorporate topics from previous lessons/weeks/terms into current lessons as much as possible, to ensure that pupils’ learning is retained and not forgotten over time (still got some work to do on improving this). Now, a curious thing was happening: whenever they came across a surds version, they would start manipulating them in all sorts of weird (for which you can read incorrect) ways.

For example, I once presented them with this question:

Simplify root 20.

All pupils had excelled with this in class time, but when exposed to it later, and out of context, less than half of all pupils obtained the right answer. Several had written that the answer was root 5, which I was baffled by. When I queried this, I encountered this response: ‘well, that’s how you simplify, isn’t it? You just keep on dividing by 2’. Ah.

Evidently, there was still some work to be done. Even more commonly, pupils would simplify  root 20 to 5 root 4  – this is not an uncommon misconception, but still an alarming one. These errors that were being made by pupils were serious, as they suggest that the fundamental concept of what a surd is and the basic rules for manipulating surds had possibly never been understood by my pupils. Looking back now, I think that my careful modelling and examples and practice time had resulted in a class who could replicate a procedure, but that was it.

I am a huge advocate for procedural fluency in mathematics, on the basis that when key procedures are automatized, working memory can handle complicated parts of a problem with ease, and I would have argued that my initial series of lessons on surds with that class was about developing procedural fluency. The main objection to that, though, is that it had failed. My pupils were not procedurally fluent.

Worse than this, was that I had always intended for them to be procedurally fluent alongside having a deeper understanding of the concepts at hand. Conceptual understanding, after all, had to be our ultimate goal. I had really tried hard to communicate the concept of a surd clearly, and whenever I was modelling a process, I was full of questions to try and promote this conceptual understanding. John, why do I need to find a factor that is also a square number? Ally, which of these is a surd? Which isn’t a surd? How do you know? Convince me. Convince me. Convince me.

Despite this, it hadn’t worked. This is more difficult to admit than with the first time I taught surds. This time round I had thought much more carefully about how best to communicate relevant mathematical ideas, and yet it had still resulted in pupils thinking that to simplify a surd, you can just divide by 2. Admittedly, some of the class had continue to demonstrate a good level of performance over time (and hence, I feel reasonably confident in asserting that learning had taken place), but this is not a good enough outcome – I wanted all of my pupils to be at this stage.

As such, I went away, did some reading and some hard thinking and some more discussion with colleagues about what had and what not worked.

Part 3 will be the story of what I do now, and why I think it is the most successful way (spoiler: minimally different questions may be mentioned).

Teaching Surds – Part 1

How I first taught surds

During my NQT year, I taught a top set Year 9 class. During our first few weeks, I became quite frustrated with the fact that whatever I tried to teach them, they had seen before. Multiplying decimals? Check. Product of prime factors? Check. Fractional and negative indices? Check. Although I was able to challenge and extend their existing understanding, I was desperate for the opportunity to teach them something new. As such, I was delighted to see ‘Surds’ (simplifying, multiplying, dividing, expanding brackets, rationalising the denominator) appear on our SOW – finally, here was my chance!

Disappointingly, it was a disaster.

For some context, I had spent my training year in that same school and things had generally gone reasonably well. For more context, a big focus on that school was on students’ independence and learning for themselves and using technology to support this as much as possible. A lesson in which a topic was explained carefully and precisely by the teacher, before giving students the opportunity to practise independently would not have been well regarded (or, at least, this was a widely held perception at the time).

It just so happened that my first lesson on Surds with this Year 9 class was going to be observed by my NQT professional mentor, and as you would expect, I wanted to impress. I went to Google, typed in any combination of ‘surds’ ‘introduction’ ‘fun’ ‘investigation’ ‘discovery’ and ‘lesson’, and came across a carousel activity. This seemed perfect. It would minimise teacher talk and allow the students to be really active in their learning. I downloaded it, made a few small tweaks and printed it, confident in the fact that I was doing the exact right thing. (I should add: this is not intended as a criticism of this resource – more that I used it incredibly poorly).

The day of the lesson arrived. I was astonishingly well-resourced: I had a booklet made for each student, and hint cards ready to hand out if anyone was struggling. I had put my students into carefully considered groups of 3. I had a timer on the board, to provide students with a visual reminder of how long they had to complete each stage of the carousel. My learning objectives with differentiated three-ways (‘All, Most, Some’). I had an exit ticket all prepared, and lots of mini-whiteboards on hand. This could only go well.

However. In addition to being astonishingly well-resourced, I was astonishingly badly planned. I had not considered what prior knowledge my students would need to access ‘surds’, and I hadn’t thought about how to assess if that prior knowledge was in place. I hadn’t thought of the sequencing of the content. I hadn’t thought about common misconceptions, and if it would be better to tackle them explicitly or respond to them as and when they occurred. I hadn’t thought about how best to draw links between surds and other areas of the curriculum. Given this, it is unsurprising things did not develop as I had hoped.

In the lesson itself, I began by telling my class ‘today, we’re going to do things a bit differently. I’m not going to tell you anything about surds, but you’re going to use the information I’ve given out to learn about them for yourself.’ There would have been five different stages to the carousel activity (‘what is a surd?’ ‘simplifying surds’ ‘multiplying and dividing surds’ ‘expanding brackets’ ‘rationalising the denominator’). Each group was a given a stage of the carousel, and they had approximately 10 minutes to read the notes and examples, before trying some practice questions. I was constantly flitting between groups and handing out hint cards where appropriate.

Very quickly, students went off task. Although I wanted them to be able to discuss the maths, and so I knew the class wouldn’t be silent, I knew a significant proportion were not focused on what they were being asked to do. I continue to move around the classroom. Students were telling me ‘I don’t get it, I don’t understand’ – but, I reasoned, this was because they were not engaging in the way that they should be. Students weren’t even bothering to read the hint cards. If they actually read the hint cards, they would understand! How could they not?

A few students (who were female, very conscientious and extremely highly-attaining) were working really well and persevering. ‘Fantastic!’, I thought. This shows that this is an appropriate task, but it is the behaviours of the others which means they are not learning as well as they could be. These students would want to ask questions: ‘miss, with expanding brackets, I don’t understand which terms we multiply?’, but I felt unable to stop and help deal with misconceptions because I needed desperately to re-engage the pupil who had just thrown a pen on the other side of the classroom.

The lesson ended. I felt fairly certain that only a minority of my pupils had learnt anything whatsoever in that hour. Despite this, the feedback from my observer (a non-Maths specialist) was overwhelmingly positive. He remarked that giving them the opportunity to learn independently was ‘exactly what our students need’ and that it was a case of persevering with similar tasks to build up their resilience. Yes, the students were noisy, but that was a sign that they were engaging well in the task. However, when I collected in books, it was very apparent that despite this ‘engagement’, learning (or, to be more precise, performance) had been minimal.

In particular: the students who had completed the carousel activity in the order outlined earlier had completed far more of the practice questions than those who had been given the ‘expanding brackets’ or ‘rationalise the denominator’ questions first. This is entirely unsurprising. In order to begin attempting rationalising the denominator, pupils will already need to be fluent with simplification of surds and multiplying surds; they had not been given the opportunity to develop this fluency. I regret so much about how I delivered this lesson, but particularly the fact that I had not thought about this sequencing, and consequently, disadvantaged multiple students.

In our next lesson, having reflected upon just why the first had been so unsuccessful, I made an attempt to go ‘back to basics’ with surds. I would introduce what surds were, and demonstrate the process of simplifying them. This still did not go well. The majority of pupils in this class (who were typically highly motivated, and would normally display a positive attitude towards learning maths) had decided that surds were difficult, and their mind-set was very much ‘I can’t do surds.’ I tried my best to work around this: I would make everything as simple as possible, by breaking down each step in any procedure. I checked that they could quickly recall all the square numbers up to 225. I told them the story of the Pythagoreans and root 2. Ultimately, it was still a real challenge. I tried countless strategies over the next few weeks, but that initial exposure had left them believing that surds were ‘basically impossible’.

We moved on with the SOW, and I was lucky enough to have some spare at the end of that academic year, in which I retaught surds. Luckily, this was for more successful! So successful, in fact, that one student commented to another (who had joined midway through the school year), ‘honestly, Miss made us do these before and we all hated them but I’m not sure why’.

More to follow in Part 2!

 

Maths Conf 15 – Reflect, Expect, Check and Minimally Different Questions

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).

solvingA

solvingB

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).

minimally solving one step

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.

Impact

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.

Final summary:

  • 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.

 

 

#MathsConf15

On Saturday 23rd June, I attended my second ever #MathsConf, in Manchester. After attending #MathsConf14 in Kettering, I was completely hooked! For anyone reading who hasn’t yet been, I cannot recommend them more strongly – Mark (@EMathsUK) and the team @LaSalleEd do such a fantastic job of putting the day together, and it is by far and away the most effective CPD I’ve attended. With that in mind, I’m going to share some reflections on the day itself. (This is quite long – you have been warned).

Session 1:

I attended Jo Morgan’s (@mathsjem) session on looking at Indices in Depth. While I’ve read Jo’s posts before on her Topics in Depth series, I’d never seen her speak and I am just so glad I did! She began by explaining the rationale behind the project, which is that due to time constraints and other pressures, teachers typically cover topics at a superficial level, which Jo is keen to change.

She began by discussing how she used to teach the three laws of indices in one lesson, and how this was – on a surface level – successful (it even led to her being offered a job!). However, this meant that her students couldn’t answer more challenging problems, and her A Level classes still had gaps in their knowledge. This really resonated with me. Only two weeks previously, I had planned a ‘revision lesson’ for my top set Year 10s, in which I rushed through a few difficult examples, on the basis ‘oh, you’ll have done this in Year 9 – you should be able to apply that knowledge here’ and it was fairly disastrous. As such, I was really keen to think more carefully about how I introduce this topic.

The first key point that I took away from this session was Jo’s carefully considered examples and non-examples – rather than just showing pupils ‘this is how/why the rule works’, we also need to show them ‘how/why the rule does not work’, so that they can build up a more complete understanding. The other point that I found fascinating was the use of language surrounding this topic – I know from conversations with other attendees that I’m not the only one who has been referring to the ‘power’ incorrectly. More generally, I’m aware that I rarely (if ever) spend sufficient time in lessons looking at mathematical key terms and defining these effectively, so this is something that I want to continue to work on.

Session 2:

I attended Peter Mattock’s (@MrMattock) session on Revisiting Measuring. Pete is someone I’ve met previously and conversed with on Twitter, so I was absolutely delighted to get to see him speak. Now, I will be the first to admit that I’ve never really considered ‘measuring’ as a topic – it’s always seemed something that should have been covered in sufficient depth at a primary level. While I’ve explicitly taught ‘how to use a protractor’ and ‘how to use a ruler’, when necessary, I’ve not taken the time to think about what it actually means to measure.

We looked at how measurement is a tool of comparison – comparing something to a unit, and how we define what this unit is. I was so fascinated to learn about how various units have been defined and re-defined over time. I had absolutely no idea that a metre had once been defined as ‘one ten-millionth of the distance from the equator to the North Pole’, but now I am desperate to share this with my students.

My main takeaway was in terms of measuring/calculating areas. I’ve always introduced area from the perspective of ‘counting squares’, before moving onto formulae for areas of rectangles/triangles etc. The problem I’ve encountered with this is that the moment a formula is used, students seem to forget how this formula relates to the ‘counting squares’ approach – and so it becomes just another procedure to remember. Pete suggested creating ‘area rulers’ and using these as a way to promote a deeper understanding – I’m so excited to try this in my classroom.

 

Plenary session:

The plenary session of MathsConf15 was run by Simon Singh (@SLsingh), which I was so excited about! His book ‘Fermat’s Last Theorem’ was one of the first things that really made me interested in Maths, and I recently gave a talk on it for an enrichment session as school. In his session, he covered all sorts of interesting ideas – we were played a clip of Led Zeppelin before experiencing the same clip in reverse, which initially sounded like nonsense. Before playing the clip for a second time, he explained we would now hear some words about Satan, which, inevitably, we did. We had been tricked by the power of suggestion – a really useful reminder to remain critical and evidence-focused!

He also played us some clips from the documentary he made for the BBC about Andrew Wiles and Fermat’s Last Theorem. I’ve watched this before, but it is such a powerful film. I definitely want to spend more time with my students looking at just how beautiful and powerful Maths can be, and this provides a really nice starting point for those discussions. The documentary is still available online at:

Simon also discussed some of his most recent book ‘The Simpsons and their Mathematical Secrets’ with us, before sharing his latest project: The Parallel Project. This project is aimed at Year 7/8 students, and provides them with weekly challenges and puzzles, to stretch them beyond typical ‘school maths’. This is definitely something that I want to investigate further, as the puzzles are excellent quality and may be the catalyst some of my students need to fall in love with Maths!

 

Session 3:

Session 3 was on problem-solving with Clare (@abcdmaths). It was a packed session, with people sat on the floor. Clare (like me!) was making her Maths Conf debut – I think she did fantastically. We were encouraged to think about various tasks and if we would consider them as ‘problem-solving’ or not. This was a really interesting discussion – it goes without saying that we want our students to be excellent problem-solvers, but it’s really hard to identify what that actually means! I had an interesting discussion with Ashton (@ashtonC94), who suggested that a ‘problem’ is anything which students have not been taught explicitly.

For example, finding the radius of a cylinder when given its volume and height could be considered a ‘problem’, in that a student may not find it immediately obvious what they have to do. However, if a class have been taught a procedure to find the radius of a cylinder, and then complete 10 questions on it, then those would not constitute ‘problems’. In short – there isn’t one definition of what a problem is, as it depends on the student’s prior knowledge and all sorts of other things. Yes, there is a case that it must be a ‘non-routine’ question, but perhaps not if those ‘non-routine’ questions are taught directly.

The session has left me with a lot to think about and reflect on; I definitely want to do some more reading around how much of problem-solving is domain specific and how much is fairly general. Is there anything similar between solving a tricky circle theory problem or a simultaneous equations problem? I’m not sure yet, and will continue to think about this.

 

Session 4:

So, this was the session in which I made my #mathsconf debut! I was incredibly honoured to have been asked to present by Craig (@mrbartonmaths) alongside Ben (@mathsmrgordon). In it, Craig discussed how thinking about variation/examples has had a significant effect on his teaching, and Ben and I both presented an exercise which we had written and used in our classes. This culminated in the reveal of www.variationtheory.com, which I am so delighted to have been able to reveal after months of secrecy! I’m not going to go into much detail here, as I want to write a separate blog post concerning the exercise I shared, and hopefully respond to some of the criticisms that have surfaced.

 

Overall, I had the most fantastic time at #mathsconf15 – I learnt so much, and as before, left with new ideas buzzing around my head and things to take back to the classroom. A huge thank-you to Mark McCourt and La Salle Education for organising the whole conference so well – the whole day runs incredibly smoothly and it is just a dream to attend. If you’ve not been before, please look at attending! I promise you will have the best time.

 

 

Learning Maths – let’s not pretend that it’s easy.

Within any Key Stage 3 or GCSE Maths classroom, I am an expert. I am certainly not saying that I have the best subject knowledge (more on this to come!) in the world, or even in my department, but I don’t have to struggle to solve equations or calculate the mean from a frequency table, and I can detect immediately whether a problem will require the use of trigonometry or Pythagoras’ Theorem. Obviously, with the advent of the new GCSE, some of the more unusual problems may occasionally cause me to pause and consider the most efficient method, but even then, my expertise means that I can come a solution with relative ease. This surely, is important, for a teacher. I’ve never been particularly convinced by the claims that ‘as long as you’re one step ahead of the students, you’ll be fine’. I don’t just want to be one step ahead – I want to be multiple steps ahead so that I can challenge and probe and I know exactly where the learning is headed.

My expertise, as is the case for any teacher, has been developed after years of practice. Now, obviously I was reasonably lucky at school in that I enjoyed Maths and it came relatively easily, but the cause of my expertise is simply the sheer quantity of Maths that I have done. Thinking back to my own years at school, although I could replicate procedures/solve problems/explain concepts to others, I was far less of an expert than I am now. When I reflect back, while I never struggled with factorising quadratic expressions, for example, the procedure was not automatic in the same way that it is for me now. More importantly, when I reflect more deeply, I remember that learning Maths was difficult.

Yes, I enjoyed Maths, and yes, I was successful at Maths, but the process of learning new concepts/methods is something that I would have found difficult – because it is difficult! This is something that I am trying to work on conveying to my students at the moment. In the past, I would have prefaced any introduction to a topic with how successful they were going to be and how easy they would find it. ‘Angles in parallel lines? You are going to love this! This is so straightforward; I promise you are all going to be great at this!’ Regardless of the topic, they would hear a similar introduction. On occasion, I might begin with ‘today, we’re going to look at Circle Theorems. Now, people may say this is hard, but all of you are going to find this super easy and be amazing at it – let’s go!’.

This was obviously done with the best of intentions – it seemed important to promote the idea that all students would experience high levels of success and it would boost their self-esteem from the beginning. Except, it didn’t. And it doesn’t. Even with the best teacher explanation and resources and careful scaffolding in place, learning is (or at least, should be!) hard work. Every time that I told a student that they were going to find something easy and they didn’t (which was inevitable – most of the students that I teach are novice learners of Maths), they would blame themselves. ‘If this is so easy, and I still can’t get it, then it must be me that’s the problem’.

In recent months, I have started teaching myself Further Maths A Level – I only studied Maths at A Level and then a non-Maths related subject at university, so the vast majority of the content has been brand new to me. I am learning about matrices and complex numbers and sequences and all sorts of wonderful mathematics, and this is fascinating and I am so excited about everything that I am learning. It’s been really hard work, as you might imagine. Although I can now find the roots of polynomial equations with relative ease, trying to learn it was really tough! To add some context, my partner is currently completing his PhD in Bayesian statistics – so his mathematical expertise is far greater than my own. When I asked him for help recently on a particularly challenging problem, his immediate response was ‘oh, well obviously you need to…’. To him, it was obvious, because he had acquired that expertise. To me, a relative novice, it was far from obvious, and now I felt frustrated in the fact that I hadn’t known how to do it.

I felt similarly frustrated when a colleague recently emailed around a difficult problem. I had sat down with my pen and paper, and was just getting stuck in, when another colleague emailed around the solution. Although I still went on to obtain the correct answer, I’d felt far less satisfied by achieving it than I would have otherwise.

This has been a massive revelation to me in terms of how I relate to my students in the classroom. Firstly, although I will still introduce topics to my students by telling them how much I love/they will love/how fascinating it is, I am far more honest about the fact that it will be hard, hard work. They will need to focus entirely and want to achieve and be prepared to make mistakes – and then, after time, they will find it easy and forgotten that they ever had reason to struggle. I am also promoting the fact that hard work, when you are successful at it, is far more enjoyable than not having to think at all.

 I am also far more thoughtful about how I give students answers to problems – in the past, I would stop the class at particular points to read out answers to ‘the questions you should have completed so far’. Imagine that you are the student working on question 5 – it’s a difficult question involving multiple properties of angles in polygons, but you’re confident you know what you need to be doing. You’re working through, quite happily, when all of a sudden you are interrupted: ‘Question 5 is 25°’. There’s little point in continuing with that question now – the satisfaction that you would have experienced when you had reached the solution was taken away from you. As a result of this, I now will typically have answers to problems/exercises available to all students – these will be on the back of a worksheet/printed separately, and this means that they can check their work when they are ready. I don’t know if I’ve been particularly lucky with the students that I am teaching currently, but the vast majority don’t want to cheat.

Despite our difference statuses as novice and expert, we achieve that same ‘YES!’ feeling when we make it to the end – and that ‘YES’ feeling is all the more worthwhile when it has taken considerable effort to get there. Yes, Maths can be easy and straightforward – but learning it often isn’t.

 

Using Minimally Different Problems in the Classroom: Sequences and the Nth Term.

Since I started teaching, one of my favourite topics to teach has been ‘finding the nth term of linear sequences’. There’s a few reasons for this, including that it doesn’t take too long to come up with examples, and there’s loads of resources on this already available. Primarily, though, I love the fact that every single student in my nth term lessons has been successful. Regardless of their age, prior attainment, or their perceptions of maths, they’ve all managed to find the nth term of simple increasing sequences. After all, the procedure is straightforward and can be quickly replicated.

The way I’ve introduced this topic has varied over time – at points, I’ve spent significantly more time explaining what the nth term is and why it is useful, and at others, I’ve gone through a couple of silent examples before providing students with the opportunity to practice. Regardless of the teacher exposition section of the lesson, the practice questions would have looked something like this:

Find the nth term of the following sequences:

nth term

And typically, all my students would have been successful with such questions. Obviously, some would require more teacher support, and some would have been quickly moved on to decreasing sequences and fractional sequences, but all would have been well. However, although I mentioned that my students were initially successful, it remained that the vast majority (including high-attaining, well-motivated students) could not remember how to find the nth term when tested on it at any point in the future.

I have lost count of the number of students, who, when asked to find the nth term of:

3, 8, 13, 18

in a test will write down ‘n+5’ or ‘n+3’. I have also lost count of the number of students who will simple continue the sequence, writing down ’23, 28, 33′ in a desperate attempt to gain marks. Evidently, something was going wrong. Now, in part this is due to the distinction between performance and learning. As stated by John Mason, ‘teaching takes place in time, but learning takes place over time’ (Griffin, 1989). Arguably, the primary reason for my students failing to remember was that they were not given adequate opportunities to regularly revisit this skill, and making full use of retrieval practice to ensure that finding the nth term was something that had been learnt, rather than just being performed.  Without doubt, this was part of the problem, but with reflection, I think something else was going on.

As stated earlier, the procedure for finding the nth term is relatively straightforward – once students have performed two or three similar questions, they are likely to be able to perform the method, in the moment, with little thought as to what they are doing. To quote Willingham (2010), ‘memory is the residue of thought’ – that is, we remember what we think about. The danger, therefore, with students practising the same style of question over and over and over is that they stop thinking; the moment they stop thinking, the chance of committing the procedure to memory is greatly reduced.

I was now unsure how to deal with this with relation to my own practice – to have students move on to a new procedure/concept after only three questions on finding the nth term seemed deeply counter-intutitive, but to have students mindlessly answering procedural questions was not yielding positive long-term results. In recent months, as a result of reading Craig Barton’s (2018) ‘How I Wish I’d Taught Maths’ and in an attempt to address this, I have started writing sequences of ‘minimally different’ questions, applying what I have learnt about variation theory to my own practice.

For example, if I now wanted students to find the nth term of linear sequences  (in particular, when they are learning this procedure for the first time), I would present them with the following:

nth term 2

At first glance, nothing much appears to have changed from the set of questions earlier, and yet on closer examination, each has been chosen for a specific reason. Question (a) is straightforward, and then (b) is related to (a) in that each term in the sequence is 1 lower. (c) is related to (b) for the same reason, and yet this time, students are forced to think about the ‘+/-0′ –  does it need to be there? Question (d) is again very similar, in that the common difference is constant and yet the starting point is 3 lower. With question (e), the starting point remains the same, and yet the common difference changes – how will this affect the nth term? With questions (g) and (f), sequence (g) has been doubled – will that double the entire nth term or just part of it? With question (j), perhaps not all students will reach this point, but some will – either way, this provides an excellent introduction to decreasing linear sequences.

I have only used this set of questions once so far, with my Year 8s, just before Easter. As such, I cannot possibly yet reveal the effects on their long-term learning. Indeed, if I do not give them the chance to practise and retrieve this skill in future, then it is still highly likely it will be forgotten. However, in the lesson itself, something felt very different to how my nth term lessons have felt in the past. Pupils, on the whole, were significantly more interested in trying to predict what the nth term would be, and were hugely delighted when they predicted correctly.  With question (f), several students initially predicted that ‘the second part of the nth term would be the same, because they’ve got the same starting point’, and were astonished when this turned out not to be the case. The fact that their predictions were incorrect caused them to refocus and check their work independently, and then they began to reason as to why. When students are given the opportunity to predict and reflect upon their answers, then they are likely to ask deeper questions about the procedure or concept in question, which will hopefully lead to greater learning over time.

I have used ‘minimally different’ questions with all of my classes now, across a variety of topics and I have been astonished each time by the initial reaction of my students, and so I would really recommend trialling this. However, this needs to be done with caution. If you are creating a set of questions yourself, you need to constantly be thinking ‘what do I want them to notice here? What do I want my students to be thinking about?’. As you create, work out the answers simultaneously – I keep (despite being reasonably proficient at all GCSE Maths!) being surprised by what I’m noticing about how one change in a question can completely change an answer – make a note of that to discuss with your students. If you have found a set of questions from elsewhere, work them out yourself beforehand, regardless of whether the answers are included. Most importantly of all, you will need to model with your students exactly what is expected of them, as this will almost certainly not come naturally at first. If you give your students some carefully chosen practice questions, but don’t encourage them to predict and reflect, then the effort that has gone into the creation of these questions has been wasted. If you don’t stop the class at appropriate intervals and discuss their predictions and reflections, then the questions will have little impact above any other set of randomly chosen questions, as students will begin to mindlessly work through the exercise.

In short, carefully chosen, intelligently varied exercises will only have a positive impact on initial student learning if they are combined with discussion and forcing students to predict and reflect upon their answers. They will only have an impact on long-term student learning if students are given the opportunity to regularly revisit the skill or concept in question.