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.

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.

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!