Saturday, 29 September 2018

Time to revisit...Teaching for Mastery

In two weeks time on October 13th I will be delivering a session that shares the title of this blog. The blog is meant to act as a preview to the session.


“Mastery”. Some people see it as the latest buzz-word to be shunned until we wait for the next “big thing”. For others it is central to teaching. For some it is a confusing term with no clear idea of what it actually means. And I can sympathise with all of these views…

The idea of “mastery” has been around for a long time. People much more knowledgeable have written about its provenance, its history and its progress to the modern day. Neither this blog nor my mathsconf session will be trying to reinforce or reinterpret any of this. I will not be attempting to explain the structure of a mastery curriculum (which is not exclusive to a mathematics curriculum). Better men than me have already done this, not the least of which is the LaSalle CEO Mark McCourt (if you haven’t read his blogs on mastery then you must). Saying that, it is important to understand certain aspects of its structure to understand where I hope my session fits in.

One of the central aspects of a mastery curriculum is teaching in a way that all pupils can access from their starting point, and then carefully assessing their understanding throughout the teaching process. A second is the use of correctives where the initial teaching isn’t successful – having different ways of approaching concepts when the first way falls short. The biggest aim of my session is to try and showcase some of the ways that teachers can approach this. Starting with what I see as important ideas to consider when thinking about structuring learning, I then aim to share practical examples of approaches that could be used either as part of the initial teaching or as a corrective approach. For those that know me, it won’t be surprising to hear that much (but not all) of this focuses on the use of representations to reveal the underlying structure of an idea (given that my book “Visible Maths” is entirely concerned with the use of representations and manipulatives to reveal underlying structure).

As an example, but not one I am using in the session, consider the “rule” that one negative number divided by another negative number results in a positive answer. Consider -15 ÷ -3:

One way of representing this is to use double sided counters; these usually appear with a yellow side (positive) and a red side (negative). Two different coloured counters can also work, and in fact to model this calculation we only need to consider negatives so a single colour of counter will suffice. The image above shows -15, and now we have to think about how we divide that by -3. One way of thinking about division is to think about creating groups, so a possible way of looking at this calculation is, “Start with -15 and create groups of -3.” These groups can be seen below:

When we think about division like this, the result of the division is “How many groups can we create?”. We can see that this process creates 5 groups, which means that -15 ÷ -3 = 5.

  
Often this “rule” is taught as an arbitrary rule, without any attempt to show where it comes from. In many classrooms, one could be forgiven if kids believed that the only reason this is a “rule” of maths is because teachers says so. But this rule is a necessary rule – if division works in the way we know it does then the answer to -15 ÷ -3 cannot be anything but 5. I finish my session with a discussion around other “rules” of maths, how appropriate representations can show where these rules actually come from, and also discuss how we can manage the transition from using representations/manipulatives to the abstract calculations. Hopefully I have whetted your appetite to hear more about teaching approaches that can support mastery in mathematics, and I look forward to seeing you (whether in my session or not) in Birmingham. Don’t forget to join us for the pre-drinks and networking the night before as well!

Sunday, 23 September 2018

Relationships in teaching

The topic of relationships is one that undergoes perpetual discussion between people in education. Just the other day my line manager and I were talking about it. I am sure I saw a recent article appear on twitter about personality screening for teachers (although I now can't find it and the only ones appearing in my search are from 2012). Some teachers even profess to believe in a "teacher gene". Many talk about the "magic" of teachers (including this recent TES article).

Now I absolutely believe that relationships are an important, possibly the most important, part of working with young people in the classroom. But this belief raises some interesting questions:

1) How do you train people to build relationships?
2) To what extent do the systems in schools contribute to or detract from teachers' ability to form relationships? And if teachers struggle to form relationships how much of the responsibility lies with the teacher and how much with the school?
3) Can anyone learn to teach?
4) If there are certain people that are more predisposed to teach, because of some gene or some innate personality traits, then does it make economic sense to train people that don't have the predisposition as they are less likely to become "great" teachers?

I have trained a fair few teachers in the last 10 years (teaching for more than that but have been involved in training for about that long) and one of the things I learned early on is that you shouldn't train people to teach like you. However good you may be, your job as a teacher trainer is to help that teacher teach as them, not teach like you. This has obvious challenges, particularly when you are have what might be termed a "robust" personality (as mine has been described) and you are working with someone who is a lot more unassuming in their personality. These people, being more introverted, can struggle to form relationships with pupils as they are more unsure of themselves in those myriad of interactions with all the different types of pupils they come across. But does that mean they can't be effective teachers? From my own experience I would say absolutely not, but it can be hard work to give them the confidence to let them be themselves in the classroom, particularly when the "effective" teachers they are often told to go and observe are those more extroverted individuals that seem to hold a classroom through the power of their presence alone.

I have heard many teachers and leaders say that "if you can't build relationships then no behaviour system can help you". Worse, I have heard many say "the best teachers don't need the behaviour system". I always wince when I hear this, because I always think "what are the behaviour systems for if not to support teachers in building and maintaining relationships?" I know from personal experience that when behaviour systems have been robust, and applied with a measure of consistency across the school, then relationship forming is easier. If kids know that it isn't personal, that when rewards and sanctions are applied they are done so within a clearly defined policy, then it makes it easier to work with them. For this to happen of course, all teachers need to be using them. If the best teachers are not using them, then anyone who does is automatically relegated to a lower status. This tends to be less experienced staff and staff newer to the school who haven't had chance to build those relationships yet, and so this can be a double whammy for those teachers. But is this the teacher's fault for not being "personable" enough to create those relationships, or the school's fault for not setting up their systems to support this in the best way possible?

The last two of my questions are perhaps the most contentious. As a teacher, I am committed to the idea that any of my pupils can learn what I want them to learn given enough time and good teaching. So my ideals should then naturally stretch to training teachers - any one who wants to teach should be able to learn to teach well given enough time and proper support. Of course the reality is slightly different from the ideal; every year that a teacher is not teaching well then kids suffer and the support given is often by other busy practitioners that have a whole lot more on their plate. But does this mean these teachers can't learn to teach, or that we simply don't have the time and tools to teach them? If there is such a thing as a "teacher gene" then does this mean that some could never be good teachers, or just that they will have to work harder to get there. But if that is the case, I wonder how demoralising that could be for those that are committed to teaching but are told time and time again that they don't quite have "it"?

This idea of the teacher gene, or better teachers having certainly personality traits, for me has to raise the question about training. Training and maintaining a teacher costs a huge amount of money, particularly in some subjects that attract large bursaries and golden hellos on top of the large pension contributions, CPD costs etc. And this is in a time when teachers are arguably not paid enough (the English starting salary is below the OECD average, as is the top salary attainable and the average hours are the seventh highest). This has to raise questions about value for money in terms of training people to teach that are unlikely to stay in the profession, or who will take a long time to reach a good level of practice if they do. As mentioned the idea of "personality screening" prior to training a teacher has been mooted before; it was actually part of a department for education white paper in 2010 (the same one that limited the number of attempts at the QTS literacy and numeracy tests) and was due to be introduced in September 2012 according to this Telegraph article, although I don't think it ever made it into practice. The idea was referenced again in this 2016 Guardian article - with a prediction that scenario-based questions could be used to ascertain details of prospective candidates character and thus screen for those that don't have the necessary "character". I must admit to being sceptical of this - I think that most people who would want to teach would be intelligent enough to identify suitable answers to these sorts of scenarios even if it they don't actually believe it. And of course saying you would act in a certain way when faced with a scenario is completely different to being able to actually pull this off in practice.

Overall I think it is clear that relationship building is important in education, and an important thing for teachers to be able to do. The big questions arise for the implications of this in terms of teacher training and school systems. And I think we should probably have a decent go at answering them before we say things like "Good teachers always have classes hanging on their every word", "You need a teacher 'presence'" (whatever that is) or "No system can help you if you can't form relationships".

Saturday, 12 May 2018

The Launch of the Midlands Knowledge Schools Hub

This week I have been privileged to spend quite a bit of time at St Martin's Catholic Academy in Stoke Golding, a village near the border of Leicestershire and Warwickshire. Eagle-eyed readers of education news and commentary will have come across this school before - it has been praised by people like Minister for State Nick Gibb and others for its rigorous focus on an academic curriculum for all pupils. This week St Martin's officially launched the Midlands Knowledge Schools Hub to support other schools in the midlands to make the journey towards a knowledge rich curriculum. As headteacher Clive Wright says "It would be easy to take pupils from an affluent area like Stoke Golding and just work with them, but that does not fit with our moral imperative as a Catholic school" (I paraphrase).

This week the Midlands Knowledge Schools Hub had two events to mark its inception, and official launch on Thursday, and then an inaugural conference today. I was pleased to be invited by Clive to attend the launch on Thursday, which featured inspiring speeches both by Nick Gibb (never thought I would hear myself say that!) and by Stuart Lock, Executive Principal of Advantage Schools. Nick spoke eloquently of the importance of knowledge, particularly the role of knowledge in tackling social justice and improving social mobility by making sure that young people from all backgrounds learn about "the best that has been thought and said". Stuart followed this with an equally impassioned speech about ensuring young people can "join the conversation of the educated citizens". This was certainly a fitting start to the Hub, but as is often true of the launch event it was more about ceremony than substance - the substance being left to today.

The conference today was attended by more than 180 people, a testament to the interest in this movement and what it has to offer. The day was mainly structured as a series of panel interviews/debates designed to explore the different facets of a knowledge rich school. The first debate examined the rationale for knowledge rich schools, and featured Helena Brothwell, Principal at Queen Elizabeth's Academy in Mansfield; Jon Brunskill, a teacher at an all-through school in London; Ros McCullen, Executive Principal of Midlands Academies Trust; and Robert Peal, Assistant Head in charge of Teaching and Learning at West London Free School. Much of this panel echoed the messages from the launch; the importance of a high level of academic knowledge to open opportunities for the future, as well as knowledge being the end in itself and that actually learning new things is something to enjoy. Jon summed it up nicely with the line "Lessons should be joyful, but what makes it joyful should be the content."

After a short break, the new panel consisting of the aforementioned Stuart Lock; along with Ben Newmark, a Head of Humanities in Rugby; Loraine Lynch-Kelly, Deputy Head at St Martins and co-founder of the Hub; and Alex Pethick, Deputy Head Teacher at West London Free Schools looked at "The How" of setting up a knowledge rich school. Props have to be given to Alex in particular for taking time out of her hen weekend to sit on the panel. It was Ben that set the scene here, talking about the orthodoxy of the last 20 years or more of education focusing on the teaching of 'transferable skills' and PLTs. Many teachers, Ben says, were led to feel guilty when actually teaching pupils, instead being praised for group work, discussion based tasks etc. and made to feel that they had to jump through hoops for lesson observations by doing things that actually had no impact on pupils' long term learning. Stuart spoke of the need to "play the long game" when it comes to curriculum, taking the time to work on it, refine it, and really have detailed discussions about "what" to teach as this would drive the "how" (i.e. the pedagogy and assessment). We were warned about treating knowledge rich curricula as a fad; if schools start saying "We use knowledge organisers so we are now a knowledge rich school" then this really important idea will simply go the way of other fads in education. Alex was able provide an invaluable primary perspective, talking about how to manage the teaching of a knowledge rich curriculum with primary staff who have to teach across the range of subjects, and Loraine added useful insights from St Martins own journey over the last 18 months or so.

After lunch, Ann Donaghy (Vice Principal at Nuneaton Academy) spoke about her personal journey to a knowledge based curriculum. To me, and many others, this was the highlight of the conference. Ann spoke with such passion about the growth in her practice, from her early years of being told to insert card sorts, group work, hot seating etc by those she trusted to know what worked in the classroom; and feeling like she had to ignore the advice and guidance of a maths teacher called Pete (you should always listen to maths teachers called Pete! 😉) when he tried to focus her attention on 'what' she was teaching and expecting kids to learn. This continued right up to taking her post as VP at Nuneaton Academy, with the early focus of their transformation of this once failing school on a well known teacher enhancement program, along with behaviour. Ann admitted that this did show some improvements, but the masses of triple marking, differentiated planning etc. led to a huge increase in teacher workload which started to burn teachers out. At the same time, whilst the school was a more pleasant environment, this didn't translate into much improved outcomes for pupils. Ann and the Principal, Simon Lomax, realised that they needed to change the focus of the school and its staff to the curriculum itself, and have started down the journey this year. Although too early to have externally validated data, Ann informs us that the signs look promising. Staggeringly, the latest tracking data shows disadvantaged pupils outperforming non-disadvantaged pupils across the board. Ann definitely had the lines of the day with quotes such as "we are no changing lives by delivering an unapologetically rigourous and academic curriculum" and "pupils shouldn't just have access to the same depth of knowledge as the best private schools in the country, they are entitled to it. It is their absolute right!" Ben Newmark had visited the school back in March and had these excellent words to say about it: https://bennewmark.wordpress.com/2018/03/21/a-building-shining-with-hope-my-visit-to-nuneaton-academy/. I had a brief stint of teaching at the Nuneaton Academy during some of its darkest times and I am really pleased that my old line manager Simon Lomax and his excellent team have managed to turn this school around and deliver the education that those pupils deserve.

The final session of the day was definitely the liveliest debate, with Lee Donaghy (husband of Ann and teacher in his own right), Jane Manzone (perhaps better known as @HeyMissSmith on twitter), Calvin Robinson and the indefatigable Andrew Old discussing whether behaviour or curriculum was the first thing to focus on. This was probably the first debate with real differences of opinion, particularly with Jane challenging the ideas that the high level of academic knowledge was the best pathway for some pupils and whether this would lead to increased social mobility for those pupils that will not make use of it once they leave school. Most panelists agreed that behaviour had to come before curriculum, however it was Andrew for me who hit the nail on the head. Andrew spoke about the necessary intertwining of behaviour with curriculum - with good learning behaviours being necessary for curriculum study, whilst the curriculum sets the behavioural expectations needed to access it. Whilst the topic was supposed to be behaviour and curriculum, the Chair Harry Yorke (Daily Telegraph political (and former education) correspondent) also prompted the panel to explore controversies such as the recent news for funding grammar expansion. I must admit to being dismayed that money is being made available for grammars to expand when people are working so hard to bring into all schools the sort of rigorous academic curriculum and extra-curricular enrichment that grammars are known for. The panel eventually returned to behaviour with a discussion on exclusions and the "zero-tolerance" behaviour policy. Again it was Andrew that stole the show; his commentary on the rise of exclusions being attributed to the changes that meant that schools didn't face the same obstructions to exclusions and his points about those reacting to the rise in knife crime by saying "they should never have been excluded from school" (because then it is only the teachers and their fellow pupils at risk!) having the audience in fits of laughter.

I found the day to be very useful and enjoyable. There is still plenty of the nitty-gritty of what different knowledge rich curricula actually look like both on paper and in the classroom, and I am hoping to visit some of the schools and trusts leading the way on this in the near future. In the meantime I am looking forward to the impact that the Midlands Knowledge Schools Hub can have on my local education landscape, and would like to offer my congratulations to Clive Wright and Loraine Lynch-Kelly along with all of their team at St Martins, well supported by people like Mark Lehain at Parents and Teachers for Excellence on a successful launch.

Monday, 30 April 2018

Concepts, Processes and Facts

Recently my department has been doing a lot of work towards our new Year 7 Scheme of Work. At some point (probably over the summer) I will get around to blogging about what we are doing and what has influenced it. Needless to say it is quite a departure from our current practice. The current scheme for Years 7 to 9 has been there since my predecessor was head of department. When I joined the school was only just adding Years 10 and 11 and so the development of schemes and materials rightly focused on those year groups. Now that we have seen a couple of years of GCSE through my attention has turned to Key Stage 3.

I have been doing a lot of development work with the team on some of the approaches and pedagogy behind our new scheme and one of the things we have talked about that I thought would be worth sharing is a recent session we did around the teaching of concepts, processes and facts. This is inspired, at least in part, by some of the excellent work that Kris Boulton has talked about in his mathsconf talks.

The session revolved around the idea that facts, processes and concepts are different forms of  knowledge, and will need to be approached in different ways. Although simplified, the major distinctions were:

Facts: Need to be taught explicitly and then tested repeatedly to support pupils retention in long term memory.

Processes: Need to be modelled, with each step broken down and explained, and then practised.

Concepts: Need to be illustrated and explored, allowing pupils to see the limits of the concept.

By way of example, we talked about this slide:


The line in quotation marks is taken from the National Curriculum in England as something expected of pupils in key stage 3. We looked at the distinct ideas that allowed pupils to reach the point where they could apply the properties indicated - namely they would need to know the particular given fact, they would need to be able to carry out the process of finding a missing angle on a straight line, and importantly they would need to understand what it meant for angles at a point to form a straight line. The point I made to staff is that very often we would provide pupils with the fact, then model and practice the process, and almost expect pupils to absorb the concept from the other two. Of course, this generally proves to be ineffective; we all know pupils that struggle to identify when angles form a straight line, particularly once they encounter diagrams that exhibit multiple properties. We also talked about the possibility that these were in the wrong order for teaching, and that a better order might actually look like this:

The logic being that once pupils are secure in the concept of what it means for angles at a point to form a straight line, then they learn the fact(s) associated with the concept, before carrying out the process of finding missing angles. This should support pupil learning a lot more because rather than learning a disparate and disconnected fact, they can connect the fact to the concept they have learned. There is copious research out there that suggests that connecting knowledge to other knowledge is important for pupil learning, and so approaching a topic in this way will make it easier for pupils to form those connections.

In terms of illustrating the concept we discussed different approaches - in this case I suggested that a series of examples and non-examples would allow pupils to form a strong understanding of the concept. These would be presented one at a time:
An important point here is the use of positive and negative examples that include diagrams pupils may not see until points in the future, for example the parallel line angle diagrams and circle theorem diagrams. It was pointed out that interior and exterior angles of different polygons would also be good examples to include here.

In our lesson design for the new scheme of work we will be focusing a great deal on the facts, processes and concepts we want pupils to learn, the most effective ways to teach/model/illustrate these and the best order to approach these in. I look forward to blogging about our work on this next year.

Wednesday, 18 April 2018

In defence of the Chartered College

Before I start properly I would like to make a couple of disclaimers:

1) I am writing a personal piece here. I am not writing on behalf of the Chartered College and nothing I write can be considered to be representative of the Chartered College or its members.

2) I am a Council member of the Chartered College.

My association with the Chartered College goes back to some way. I was an respondent to the original Princes Trust consultation back in about 2012 and attended the launch in London (I still have the document from that day somewhere with my response in it). After that I lost touch with it - I missed the crowd-funding situation (or I would have donated) and I didn't really see a lot about it until I saw the advert for new trustees later in 2016. I honestly thought the idea had petered out; it was ambitious at the time with the GTC so freshly in people's heads to say the least. I was so pleased when I saw it was still going that I immediately volunteered and was lucky enough to be one of the 7 selected to join the council. I have been a council member for a year and a half and I have felt privileged every moment.

Recently the Chartered College has been the subject of criticism. Some of this is not new, but two of the more recent ones I feel are unfounded and as a supporter of the Chartered College I wanted to redress this.

The first criticism has surrounded the review process of articles for the Chartered College journal, Impact - specifically an article written by Greg Ashman. Now I want to take this opportunity to publicly state I have absolutely nothing but the highest respect for Greg. I read his blog whenever I can (the man is so prolific it can be impossible to catch everything!) and I think he speaks a huge amount of sense on a lot of issues. I read the article in question and I thought there were some interesting points raised, and I think it could be a useful read to spark debate. The article can be found on Greg's blog here for the interested reader. I do respect the opinion of the reviewers and the people at the College that put the journal together, and they decided that the tone of the article wasn't in keeping with the style of the journal. There have however been some implications of a bias, and that the article was blocked because people found the content of the article unpalatable. I was not involved in the review process in anyway, and nor am I part of the committee that oversees the journal on behalf of the council, but I know those people. I have worked with them, talked to them, shared hopes and dreams for the Chartered College with them, and I can categorically state that there is no bias in them. These people are teachers, as I am. They spend their weeks in the classroom or in schools working to educate young people. They are people like Natalie Scott (@nataliehscott), Jemma Watson (@thefinelytuned ) and Aimee Tinkler (@aimeetinkler). There is no agenda behind us and no wish to exclude from debate, and it makes me unhappy to think that people might believe that of us.

Some will believe I am being naive at this point, and if so fine, but I would rather believe that these people are doing what they think are the right things and for the right reasons than look for hidden motives behind these decisions.

The second recent criticism has been around the launch of the Fellowship. There have been a couple of comments about this. The first is the idea that affiliates and not just members can nominate and be nominated for Fellowship. I personally don't take issue with this. I think we have to recognise that, while the contribution of teachers to our schools is immeasurable, it is not the only contribution. There are many people I can think of that have added immense value to my career as an educator but that no longer work in the classroom. I would personally nominate Professors Anne Watson and John Mason immediately; their contribution to maths education has been incalculable and they are two of the most passionate and dedicated people I have ever met when it comes to trying to ensure that our young people develop a deep understanding and appreciation of mathematics. They are just two of about 10 people I could immediately think of that would be worthy of the honour.

Another criticism around Fellowship is the idea that "As a Founding Fellow, you will be encouraged to support members and other teachers to engage with and promote the use of evidence." People out there are seeing this as asking people to pay to do extra work. This saddens me greatly. As a teacher with 12 years experience I see it as my moral and professional duty to support other teachers and support them in finding approaches to teaching that can help them overcome problems they may be facing. Granted I don't need to pay the Fellowship fee to do this, but I believe that if we are going to approach the standards of other professions then the Fellowship as a mark of someone who has the experience, skills and knowledge of our profession is an important milestone. For me, those people who deserve to be fellows would see the opportunities to support other members as a positive. As teachers I believe we must be an outward facing group if we are to solve the problems that currently plague our daily work.

A further criticism of Fellowship are some of the additional benefits, such as the Fellows roundtable and the reservation of certain Council positions as Fellows only. I am not going to go into the details but I can tell you as a member of the Constitutional Committee that we considered this very carefully. In the end it was concluded that to be effective in these roles one would need to meet the criteria for Fellowship, and that it would need someone committed enough to the ideals and ethos of the Chartered College that they would seek out that sort of role. It is hard to imagine how a teacher of less experience or less passion could effectively lead the body that holds the Royal Charter for our profession. That isn't to say that the views of newer teachers aren't important, in fact they are crucial to ensuring that the College is representative of the views of all its members. This is why there are council member positions open to all. But those positions that are required to drive the College forward, to ensure that the governance of this body is robust, are those that need to be filled by people that have the experience, knowledge and skills developed over time to fulfill that need.

The final criticism I will address is the funding. It is no secret that the College are currently funded by the government, to the tune of £5 million. This naturally raises questions about independence - how can a body be funded by government be able to criticise policy and practice? People may not believe this but I can honestly say that it doesn't really feature in our discussions. Hard as it may be to believe, but for all their flaws the DfE recognise that a strong, well-connected and informed teaching profession is a positive thing. There are people there that care as passionately about young people as we teachers do. As for criticism of policy and practice, we have always been clear that we want to work with and not against. For the profession to lead the way we cannot be a group that shouts and screams when things happen that we don't like. That is not to say there is not a time for anger, nor a time for action, but always first should be an effort to reach out, to work with, to influence by being a calm and well-informed voice.

I believe that teachers deserve to be a well-connected and authoritative body when it comes to the practice and standards of those who choose the profession. I think this is essential to us being universally considered a profession. I don't think we are there yet, but I believe that signs are hopeful. Above all, I believe that the Chartered College has the potential to be a force for good in this regard. Mistakes have and will happen along the way; we are human and not immune to them. And that is really the key - we are human. We are teachers, like many others, and we are trying. I hope that others will see that and lend us their support. And I hope that those who doubt us will either join and become part of the influence, or at least give us the benefit of that doubt while we keep trying.

Friday, 6 April 2018

The Power of Interpretations

Probably the most celebrated mathematics of recent years (1994 seems a long time ago now, but it really isn't) is probably Andrew Wiles proof of Fermat's Last Theorem. In fact, it seems likely that anyone will be able to say something similar to this until such time as the Riemann Hypothesis is proved. What people forget is that Wiles' proof was not actually directly of Fermat's Last Theorem. Wiles' proof was concerning two (at the time) separate branches of mathematics, elliptic curves and modular forms. The real wonder in Wiles' proof is that it suddenly showed that elliptic curves, which had been worked on in one way or another since the time of the Roman Empire were linked to modular forms, an invention only a hundred or so years old at the time. The fact that this also proved Fermat's Last Theorem was almost incidental (although it clearly was the focus of the media coverage) - for mathematicians the power was that Wiles' proof had allowed them to reinterpret problems in one area of maths as an equivalent problem in the connected area. Problems that had gone unsolved in modular forms could be reinterpreted as a problem in elliptic curves, and all the understanding from that area could be brought to bear on solving it.

I first read this story about 7 or 8 years ago in Simon Singh's book, "Fermat's Last Theorem", and it resonated with one I had heard at university as a undergrad*. Both spoke of a key message in mathematics; that often a different interpretation of a problem can drastically change our ability to solve it. Indeed, Wiles' own proof was only possible because he viewed the problem in a different way to others, and was therefore able to approach it in a way that no one else had tried to (it is worth noting it still took him 8 years!).

I think this message is as important in the mathematics classroom as in the realms of professional mathematics. Many of the different problems we ask pupils to work with in the classroom have different ways of thinking about the underlying mathematics, and if those pupils don't have flexibility in their ways of thinking about the underlying mathematics then clearly they are going to find this difficult. Consider, for example, the three questions below:

1) Cans of pop are sold in packs of 6. If Russell buys 24 cans altogether, how many packs does he buy?

2) Russell walks 24 miles in 6 hours. What is Russell's average speed?

3) The two rectangles shown below are similar. What is the scale factor of the enlargement from the smaller rectangle to the larger rectangle?

All of these problems are solved using the calculation 24 ÷ 6 but importantly the way we interpret the division is different in all three cases. The first question is a classic interpretation of division as grouping; take a dividend and create groups of a certain size (the divisor). In this case take 24 and create groups of 6. The answer to the division (the quotient) is the number of groups created. A key part of this is that the starting number and the number in each group are unit consistent - we start with 24 cans and create groups of 6 cans. However, the quotient is not in the same unit; the answer is 4 packs, not 4 cans.

Contrast this with the second question. In the second question we are not creating groups of 6 miles. Rather we are sharing the 24 miles into 6 shares, with each share being a single hour. The answer now is 4 miles - the dividend and the quotient have the same unit, but the dividend does not. This is clearly a different way of thinking about the division.

The third question is again different from the other two. This time we are not creating groups of 6, or sharing 24 into 6 shares. Rather we are comparing the 6 and the 24 to see how many times bigger 24 is than 6. There are two main types of comparison like this: additive comparison, where we examine how much bigger one value is than the other (this is an interpretation of subtraction often called the "difference" between two numbers) and multiplicative comparison (which is the type used in question 3) where the values are compared to see how many times bigger one value is than the other. This has many different names, probably the most common being a "scale factor".

My point in writing this blog is that if pupils don't understand these different interpretations of division, then they won't be able to answer all of these questions. I regularly encounter (and I am sure other teachers do as well) pupils that do not realise that the way to answer the third question is by using a division. Their concept of division is deficient and although they are capable of answering 24 ÷ 6, they are unable to see that this calculation is the one required. 

Of course, I am not just talking about division here; many concepts in mathematics have multiple interpretations and the more ways of thinking about a concept pupils have the more likely they are to recognise a concept in a certain problem, and therefore be able to work towards solving it.

In my upcoming book (provisionally titled "Representations in Mathematics") I talk about different interpretations of some fundamental mathematical concepts, and how we can use representations to highlight and unite the underlying structures behind them. However, whether you choose to read my book or not I would urge all maths teachers out there to consider the following when approaching teaching new concepts:
  • What are the different ways of thinking about this concept and how it can work?
  • How can I make explicit to pupils the different interpretations of this concept and ensure they are comfortable with each?
  • When solving different problems, how can I ensure my pupils understand the particular interpretation this problem type requires? Note I am not saying that this needs to be explicitly taught to pupils, teachers may choose to ask pupils to explore a number of the possible interpretations to try and make sense of them.
  • Do my pupils have the necessary understanding of this concept to interpret this problem type in the correct way?
Making the different ways of thinking about a concept clear to pupils is going to be increasingly important, not just for pupils' examinations where they will be faced with unfamiliar contexts, but also for beyond school. The evidence is now increasingly convincing that skills such as problem solving and critical thinking are domain specific skills; they depend on strong knowledge of the area to which these skills are expected to be applied. But importantly these skills still need to be developed, and for me part of this strong subject knowledge is a flexibility in interpreting the central concepts of that subject.

* The story in question also appears in the introduction to my upcoming book, so I cannot reproduce it here. I am hoping that now I have finished writing the book that I can get back to blogging more regularly again!

Thursday, 8 March 2018

What is x? Explaining the meaning of algebraic notation.

Recently I have been teaching linear equation solving with Year 7. We have explored various interpretations; function machines and their inverses, balancing, inverse operations, blank box etc. But one thing I did before any of this was explicitly define the role that x plays.

It is well known that pupils can struggle with algebra when it is introduced immediately as an abstract concept. Much has been made about the use of representations to support the teaching of algebra - indeed I dedicate a good proportion of my upcoming book to exploring different ways of representing algebraic expressions and equations, and how these representations can aid pupils in understanding how algebra is manipulated. However, one thing the representations struggle with is communicating the nature of the letter itself.

I once saw an excellent use of Geogebra to create a dynamic visual representation of completing the square that would allow the value of x to be varied, the squares shrinking and growing really hammered home the idea of x being a variable in the expression. However the static representations that we often use cannot convey the same meaning. So before I started working with them I decided to introduce some of the basic interpretations of letters in mathematics; in particular viewing them as variables, as parameters and unknowns.

To be fair, I didn't stress parameter too much, just a vague definition about them having a particular meaning for a value that doesn't change, like b for the base of a triangle or A for the area of a particular shape. But we did talk a lot about the difference between an unknown and a variable, and then we revisited the ideas as we went through the different lessons. At the beginning of each lesson I would ask the class what role x was playing in an equation, and that meant I went through the entire topic without once being asked what x is - which could be a first for the introductory teaching of algebra!

It seems like it really helped the pupils to understand something about the different roles that the letters can play, and in particular the role that they played in the equations we were working with. I would certainly recommend teachers introducing algebra by giving clarity over the roles that letters play in mathematics so that pupils have a sense of what they are working with before they are asked to manipulate them.