Category Archives: Actual Maths

Actual Maths: Mathematical Literacy

The limits of my language mean the limits of my world.

Ludwig Wittgenstein

As we progress towards the first GCSE sitting of the new specification in Mathematics, it has become clear to me that as Mathematics teachers we now face two challenges:

  1. Ensuring students are confident with as many of the 250+ concepts as possible.
  2. Ensuring that they can interpret questions properly to choose the correct methods to answer the questions in the paper.

The latter is incredibly important now. It has struck me in recent months that in many of the questions, the Maths required is not that hard, particularly in the case of the ‘common’ Foundation and Higher tier questions. No, it’s knowing exactly where to start in tackling a question that’s the greater challenge.

With this in mind, on Friday as a department we looked at supporting students in developing close reading strategies to understand how to tackle a problem. We chose a 5 mark ‘classic’ functional problem, shown below.

1

We then followed these steps:

  1. The teacher reading the question aloud, fully.
  2. The teacher instructing the students to highlight any key words, statements and values as they read the question aloud.
  3. The ‘students’ (i.e. the rest of the department) then annotating the question with any things that they think they needed to do to tackle the problem – but not answering it.
  4. The teacher then leading the class through reading the question again, but highlighting key words, adding further annotations to the question text, diagram, etc, and establishing all the decisions that need to take place to successfully answer the question, but still not answering it.
  5. The ‘students’ then working out a solution to the question, taking into account all the required decisions in order to tackle the problem.
  6. As a group, going through the solution, linking the solution steps to the decision made.
  7. We then switched back into ‘teacher’ mode as a group, to see if there was anything we’d missed.

How many decisions do you think were made in tackling that problem? 5? 8? 10? I reckon it was more like 15-20 decisions, depending on a student’s fluency in timetables, estimation, use of formulae, choosing trapezia or rectangles and triangles in order to break down the floor area.

Here’s my annotations – it’s not a complete set, but you get the idea:

2

You may notice a couple of errors! That was me getting carried away… typical over-confident student.

My (amazing) AHoD, Mohammed Usman, made some great points. Don’t assume that what is obvious to you is obvious to the student – do they know if the fact that it’s a conservatory is important or not? Are they estimating to see if their answer is sensible? Do they recognise that if 4.5 packs are needed, that they need to buy 5 packs? Jenny Thompson rightly asked if you’d expect students to calculate the percentage discount on the pack price first, or as a discount on the whole cost – if the former, does that make the calculation more difficult? If the latter, will students forget to apply the discount, and make an incorrect judgement?

Carolyn Bate and I had a discussion about the final part of the question. Carolyn gets students to write a full summative sentence at the end of the method; I said that a simple ‘yes’ or ‘no’ would suffice with a clear method, but then could see Carolyn’s point given that students are being asked to give judgements based on assumptions in the new specification, and need practice in this regard. I also made clear that ultimately area is the multiplication of two dimensions – which the m2 in the question should trigger. All too often as an examiner I’ve seen students confuse the concepts of area and perimeter, and using the ‘m2‘ trigger should address that.

Conclusion

Some would say that students therefore need to improve their mathematical literacy, but I think it’s more important than that. I think we as teachers need to ensure we are clear on the number of decisions made in what might be a very simple question, and make sure we’re giving students the tools in order to make those them successfully – hence why as a department we’re placing real emphasis on close reading as part of our support for Y11 over the next few months, alongside our day-to-day teaching.

I’d like to thank Emma Steele and Carolyn Bate for leading the session, Head of School Jenny Thompson for bringing her English expertise and the rest of the Dixons Trinity Academy Maths Department for a) being brilliant as always and b) making the session really productive.

Actual Maths: 5 Questions Again

The detail is as important as the essential is. When it is inadequate, it destroys the whole outfit.

Christian Dior

In my previous post, I talked about how Craig Jeavons‘ idea of “if I only had five questions to get students to the objective I want them to achieve, what would they be?” looked to be thought process through which I can get students to explore conceptually challenging questions quickly in a lesson.

I was conscious, however, that I didn’t really go into detail regards what this actually looks like. So inspired by William Emeny’s follow up of Naveen Rivzi’s exploration of multiple-choice questioning, I offer up some examples.

Remember the idea is conceptual and procedural variation from question to question…

One step at a time

So, let’s a have a look at, say, expanding pairs of brackets. What do we actually want them to consider? Well, here’s a ‘minimally different examples’ style set of questions:

  1. (x + 1)(x + 2)
  2. (x + 1)(x – 2)
  3. (x – 1)(x – 2)
  4. (x + 1)2
  5. (x – 1)(x + 1)

Probably a little simplistic and obvious, but it’s pretty clear what’s happening each time. It has to be said in this case the numbers are irrelevant – it’s the procedures that are important.

Let’s take it up a notch. I set my Y8 students these five questions as part of an introduction to multiplying fractions recently:

Untitled

In this example, not only am I varying the concept – moving from unit to vulgar to algebraic fractions – I’m also varying the procedure, setting questions that might throw up a few misconceptions or ‘open goals’ – e.g. realising that the bottom right fraction will simplify to 2, or that some students might correctly multiply the numerators to get a2, but then incorrectly may think that they need to cancel both numerator and denominator by 2  to get a/9.

What was nice about this was that they were dealing with concepts in terms of the algebraic fractions that we would classically leave until much later in their secondary school Mathematics studies.

I replicated this methodology with dividing fractions, and it’s worked quite well – there’s still a little bit of tweaking to do around knowing when and how to simplify properly, but their conceptual understanding of multiplying fractions is pretty sound.

There’s some way to go with this. I am by no means perfect, and here’s some things I’ve learned so far:

  • The time you allow the students is crucial. For the multiplying fractions exercise I gave just 5 minutes maximum. My Y10 groups have been looking at area and volume scale factor problems – I’m not so harsh not to allow them at least 10-15 minutes on tackling 5 of those kinds of questions.
  • Don’t make the steps too big. There has to be some kind of connect from one question to another, and whilst it doesn’t have to be immediately obvious it shouldn’t require too much brain power. The goal is to make big conceptual and procedural leaps via all 5 questions, not between one question and the next.
  • Keep the entry point simple. Give students chance to get something right quickly, and you’ll hold their attention a little bit longer. If you are allowing enough lesson time to cover a single concept, you’ll be able to do this much easier than if you’re trying to cram too much into a lesson – but then you knew that anyway, didn’t you?

Conclusion

Writing 5 questions on a topic is easy to do. Writing 5 questions that build on a fundamental process and vary the parameters to make connections and develop problem solving skills is not so straightforward. Like any pedagogy it’s easy to understand but difficult to truly master. But it’s certainly something I’m going to pursue.

 

Actual Maths: 5 Questions

Without deviation from the norm, progress is not possible.

Frank Zappa

Everything changes

Recently at school I’ve been concerned with how we can expose students to more challenging questions to test their understanding of concepts, particularly in light of the new GCSE specifications. I worry that whilst my students can deal pretty well with procedural exercises, I don’t want to short change them when they face some of the types of questions that the Sample Assessment Materials have all frightened us with…

So I’ve been interested in how I quickly introduce conceptual and procedural variations on a theme. Rather than, say, getting students to practice multiplying lots of fractions (for example), can I keep the entry level to an exercise low, whilst manipulating the parameters slightly, but enough to develop students’ capacity to deal with any ‘spins’ on a problem.

I got 5 on it

Enter Craig Jeavons and his #mathsconf6 talk. Craig has been to Shanghai and witnessed first hand their take on conceptual and procedural variation, and has done well to bring light to these two ideas with the ‘Shanghai, pah’ response being fired his way.

I actually don’t care if an idea comes from Shanghai, New York or Wetwang – if it’s a bloody good idea I will take it.

Part of Craig’s talk was focused on the question “if I only had five questions to get students to the objective I want them to achieve, what would they be?”. We had a practice of this in the talk and I liked it so much I brought it to our department meeting the Monday immediately after for my staff to consider.

For me, the idea runs something like this: 5 questions that start off from a relatively straightforward opener, and then as you move from one question to the next, something about the concepts or procedures to solve the problem is adjusted – e.g. in the case of multiplying fractions: moving from unit fractions in the first question to vulgar fractions in the second; then throwing in a letter instead of a number as the numerator or denominator; then swapping in more letters until lo and behold you’ve got students multiplying algebraic fractions.

In relatively quick time, you can get students attempting some rather tricky problems; I’ve done something similar with my Y10s around volume of pyramids – start with a pyramid, then a cone, then given the volume and two dimensions calculate the other, and so on. In a short space of time they were looking at frustums.

If students finish the 5 questions in the time allocated to independent learning, then they’ll usually do a further practice exercise – and yes, this is going to be out of a textbook. But the minimum I expect is for them to get those 5 questions under their belt so they have exposure to a variety of possibilities. Once the time allocated to independent study is up, then we review those 5 questions in detail, using it as an opportunity for DIRT and for further points to be raised.

Early feedback from staff has been good – students of all abilities are getting real challenge now. The proof of the pudding is in the outcomes of assessments, but for now, the signs look positive.

Notes from #mathsconf6 – 10 things to take away

We like lists because we don’t want to die.

Umberto Eco

Hello all. Instead of writing a comprehensive report on the goings on at the latest #mathsconf in Peterborough this weekend, I’ve decided for reasons of efficiency, saving time and my sanity, to boil my findings down to the 10 things that had most of an impact on my thoughts. Hopefully you’ll find this just as useful, if not more so!

As Wikipedia showed, people are willing to share their knowledge and insight for the greater good, in their own time, without being paid.

I do not have any grudge against people who charge for sharing their time. Our profession has ever increasing demands on our time, and so whatever we can offer, and yet hundreds of people showed up on Saturday to learn about, share ideas on and celebrate the teaching of our subject. You cannot argue against this idea, and I want to personally thank everyone involved for all of their contributions – particularly the speakers and La Salle for putting everything together, but also people like Rob Smith (aka @rjs2122) who ran the tuck shop and the raffle without any expectation.

What can be seen as ‘traditional’ should carry no less weight for being so.

Andrew Taylor’s opening speech had much of what could be seen as ‘old school’ thought that unsurprisingly still holds consideration presently:

  • The Cockcroft report stating that we should not believe that it is fair for students to be entered for an assessment where they can only attempt a third of the Mathematics covered in the paper (he was talking about the CSE then, never mind the new GCSE).
  • The influence of SMP – developing and nuturing coherent approaches and pedagogy, not just resources and CPD; bringing users together; creating a curriculum model that drove assessment, not the other way round.
  • The example of leadership in Mathematics as a way to create opportunities for staff, understanding that good Maths teachers are rare and valuable, and that they should exercise that potential power.
  • The adage “whatever management tells you to do, don’t do it unless it helps students” – can this be argued with?
  • Nobody gets clever by sitting an exam: we have to understand the purpose, timing and assessment framework when putting together tests so that they are right and proper for the stage of a student’s learning – interestingly, Taylor stated that “teaching to the test is the right thing to do if the assessment is purposeful in it’s aims”. We can argue either way if the new GCSE is purposeful or not – but that is not a troubling thought taken out of present context.

There is still a place for intellectual approaches in the leadership and management of departments.

Ben Ward (aka @MrBenWard) and I took the stage in the first group of sessions to talk about data. Whilst I looked more at the practical opportunities to record, monitor and analyse data, Ben made connections to the wider running of a department and links to school leadership. One point that was made in our talk is that data in itself is not the end – it is the means to help make decisions, and widely, part of a set of tools in the case for forming arguments, i.e. the use of ethos, pathos and logos to form a rhetorical case.

  • Ethos in terms of appealing to authority – i.e. in the case of your teachers, using their professionalism to come to a conclusion of where to take a group of students forward to improve their understanding. In terms of management, demonstrating that you have a clear handle on the progress of students and what it is that needs to be focused on to make the greatest difference;
  • Pathos in terms of appealing to emotion – how do you get your staff to own and care about their data? By helping them see the bigger picture and how thought and planning on their part will not just impact on the life chances of those in their care, but also for them to see the contribution their efforts make on the greater performance of the department;
  • Logos in terms of appealing to the facts – gut feeling and hearsay is not enough. By providing objectivity to what are often subjective arguments, effort can be targeted more properly and in a more structure fashion.

Cynics may say that you should not have to persuade staff to take a course of action based on data – that staff should have the freedom to make their own decisions because they are professionals. I would retort with that idea that it is because they are professionals that they should use every scrap of evidence and support to help the make better judgements about their planning and interventions.

Some people think that the designing of questions and a marking policy is a complex process, but I can assure you it’s even more complex than that.

Ben Stafford gave a wonderful insight into how assessments are written and the level of detail that goes into getting a question right. I discovered some insights that I will take away when writing questions to test student understanding:

  • Ask questions that are unfamiliar – are you truly testing students if it’s completely obvious what they’re being asked to do?
  • Front load information, and ensure it’s laid out in a way that anything key doesn’t lack clarity.
  • Do give a clue to how students should answer.
  • Remember that in assessments parts of questions are independent of each other – if students can’t answer part a) of a question that should not prevent them from answering part b).
  • Avoid the need for students to assume a line of thinking. Set out the parameters of the question carefully, thinking about the language you use. Use as few words as possible, but enough for a student to understand what the question is asking.

There was also the argument regards reliability and validity. If you test enough students, you’ll get a statistically reliable result. But is it valid? Where you are giving marks for partially correct questions there has to be a level of structure in the mark scheme for validation to occur.

Even in an arena without politics, some people can still pursue an agenda.

I can appreciate why someone might ask about how one caters for EAL students in assessments. But Ben’s talk was not the right time, not was it the right stage or environment. There are lots of questions to be answered in that regard, but it was unfair to put Ben on the spot.

Don’t overthink things.

Well done to Megan Guinan for completing the Pringles Challenge! She passed on her know-how to her family – and look what happenedtwice!

Optimise the amount of questions you ask to get student to an objective, rather than bombarding them with practice.

I am a great fan of Craig Jeavons‘ work – practical, easy to follow ideas that can be implemented very quickly. As well as reiterating principles laid out by the likes of Bruno Reddy on working on the right foundations, building a culture for learning in your classroom, he also talked about how with whatever concepts you want to bring to your teaching – don’t go in all guns blazing. In relation to problem solving, ensure your students have good number sense, and an understanding of proportion. Incrementally develop problem solving as part of your day-to-day practice, and then don’t be afraid to start going ‘off piste’ or leaving things ‘open ended’.

I took one thing away from Craig’s talk that we are already looking at developing in our teaching practice in my department: the idea of “if I only had five questions to get students to the objective I want them to achieve, what would they be?”. I often worry with students that they can labour too long on the easier, rote elements of an exercise and not face challenge much earlier.

Through the use of minimally different examples, moving from one question to another with only one or two elements of procedural variation, you can actually get student from a low level of entry on an exercise to something quite complicated rather quickly. Tieing this in with proper review time that takes students through these five key questions, I do feel one can get students to be dealing with more contextual problems much quicker.

Some teachers and departmental leads are not responding to very public discussions and information sharing.

I was rubbing my hands at the thought of Eddie from OCR, Graham from Pearson and Andrew from AQA going toe-to-toe in front of a baying crowd. I’m only joking, but I was looking forward to some real insight cued up by some thoughtful questions. Unfortunately a large chunk of the talk was given up to answering the questions “what do the new grades look like” and “do you have an assessment package to support the grading of students”. I mean, COME ON. Have you been living under a rock? We’ve known the answers for a long time now. There are no, and have never been any ‘graded topics’. The grade descriptors given by the DfE are deliberately vague. As for assessment packages – the exam boards have been incredibly open around how they are as much in the dark as we are. They have sample assessment materials and supporting documentation to try and help guide us – but nobody truly has all the answers. All they can do is advise.

The decision to be made about GCSE tiers has been simplified somewhat.

I did baulk a little at the question “please can we have guidance on tiers?” – but gratefully we had something proper to take away from those who need to make such decisions. As I heard it – and I’m welcome to be corrected – the current fallout rate of students taking the higher tier, i.e. getting U – is 1% of entrants. Based on the standards of grading expected for the new GCSE, 10% of students currently getting a grade on higher spec would only get a U.

Also, bearing in mind that there will only be a sixth to a third of questions that students working at around grade 4 to 5 might confidently be able to answer, then asking these students to sit for 4 and a half hours (4 and a half hours!!!) with the confidence only to be able to accumulate at best about 25-30 marks per paper is perhaps placing undue stress on students. The system will still be gamed, don’t get me wrong, but the risks are even greater.

Even for an experienced dog like me, stay hungry.

It never fails to impress me how I can walk out of a conference absolutely buzzing with ideas. I am paying real attention to refreshing my teaching practice at present; all of the workshops that I attended helped my thinking about what I can do to get students a) retain concepts b) improve their problem solving skills and c) be prepared for exams.

I’d like to thank Ben Ward (aka @MrBenWard) for collaborating with me on our workshop on Getting The Most From Your Data – he’s a brilliant example of the modern, informed and proactive department leader, and I absolutely recommend that if you’re not already following him via Twitter then it’s about time you did.

Mark McCourt (aka @EmathsUK) has done a blinding job in launching and continuing the National Mathematics Conferences – I and a lot of people get a great deal out of each and every workshop and long may they continue. I’ll even let him off the fact that he doesn’t follow me on Twitter!

I hope the more concise approach to my notes from #mathsconf is just as enlightening and a little less demanding. Future posts will aim to be just as concise, but with more of a teaching and learning focus from now on. Watch this space!

Actual Maths: Notes on the National Maths Symposium, part 3

It took me a lifetime.

Picasso

This is the final part on a series on the National Maths Symposium held in Nottingham on the 21st of January. If you missed the first part, click here, and for the second part, click here. Otherwise, let’s get on with it…

Jeremy Hodgen – Mastery Approaches in Recent Decades

Prior to Jeremy’s presentation the symposium seemed to take a path in and out of mastery in terms of classroom experience and pedagogical thinking. The direction Jeremy’s presentation went was looking at a) where this thinking has come from and b) why it needs to deliver.

Parallel to the approaches taken by the previous speakers, Jeremy laid out some starting conditions. We were reminded of Bloom’s thoughts on teaching for mastery:

  • it uses formative assessment to determine understanding
  • quality feedback is deployed to close gaps where students are not moving towards mastery

In other words, teaching for mastery is adaptive. If students haven’t ‘got it’, then alternative approaches and additional time need to be given. If they have, then they need to go deeper into the concept.

We were then given the latest EEF toolkit evaluation of teaching for mastery. Their latest (beta) data implies that teaching for mastery improves progress by approximately 5 months, with 1-2 months further improvement for lower attainers.

Jeremy then laid out his standpoint on mastery, firstly in terms of why it needs to deliver. His research shows that the KS3 plateau in terms of student progress in Mathematics still continues – students make good progress generally in KS1 and KS2, and steady progress in KS4; KS3, however seems to be a period of latency.

Caution was asked for in terms of teaching concepts in small and fragmentary steps. Too much time on low level skills can impact negatively on lower attainers. Where high attainers confidently derive facts from underlying structures in problems, lower attainers focus too much on the low level skills and are unable to make the leap to higher order thinking.

By using the right tools and models in our teaching, we can help students move up from reliance on these low level skills and start to derive more concrete understanding of concepts. As if to underline this point, Hodgen shared the finding that in his research, only 5% of 14 year olds know how to determine if 2n or n+2 is larger, and when this is the case. It was only students who scored significantly highly on the algebra test used during the research that managed to articulate an correct answer.

So, how do we address this problem? Going back to Bloom’s two required criteria when teaching for mastery, firstly formative assessment:

  • Clarify understanding and share learning intentions;
  • Allow effective classroom discussion;
  • Use feedback that allows learning to move forward;
  • Activate learners as being resources for one another;
  • Active learners as being owners of their own learning.

Jeremy then went further into detail in regards to feedback, after giving the caveat that 38% of studies showed that the feedback given by teachers had a negative effect on student learning. It is important, therefore, that we establish what good feedback looks like:

  • It is specific, accurate and clear;
  • It informs students when they’re right, and why;
  • It is used sparingly and is meaningful;
  • It makes clear how students can improve
  • Praise and marks are given separately.

Additionally, when we listen to learners, rather than being evaluative (waiting for the right answer) we should be interpretive, asking why students have responded in the way they have, and demonstrating real interest in what they think.

To conclude the talk, we were reminded of one or two ‘classic’ resources for teaching for mastery – in particular, the School Mathematics Project – the 1970s textbooks, especially. Ultimately, what ever we chose to do in our classrooms, mastery balances learning that tests the conceptual, the procedural and the development of knowledge.

In terms of a closing piece, Jeremy Hodgen’s talk neatly distilled much of what the previous speakers shared. But what was most stark was how he succinctly clarified how much students need teachers to act!

Conclusion

The notion of mastery has spread like wildfire through the mathematics teaching sphere. As soon as OFSTED and the new National Curriculum mentioned ‘mastery’ in their documentation, then there was a rush for schools to claim that they were teaching a ‘mastery curriculum. Although, to paraphrase Mark McCourt, they’re not. Mastery is often proclaimed as ‘teaching less, better’, but what does the word ‘better’ actually entail?

Though the speakers shared viewpoints that might have appeared different, in actuality there were a number of overlapping, and therefore central principles to what teaching for mastery should truly consider:

  1. Mastery is asymptotic: it is never the case that one achieves mastery, at any level of perceived difficulty, as John Mason’s clever twists on ordering decimals and counting showed. Whatever your students study, whilst they may have been able to achieve 100% on a exercise based on a topic, there are always ways of setting problems that extend and deepen thinking.
  2. The journey towards mastery is an interplay of conceptual and procedural understanding. Knowledge develops by testing and challenging students’ confidence in these two strands, whatever is being taught. As the layers of knowledge develop, we must ensure that students are not settling into pattern-based thinking: conjecture what students think they understand and then vary the problem sets to keep them in the zone of proximal development.
  3. Culture is everything. Firstly by setting and reinforcing routines, expectations and a shared vision for success in the classroom. Secondly by taking advantage of the space that this affords, teachers can then properly offer opportunities that establish students’ viewpoints on what is being studied, and establish if these viewpoints have true appreciation of the concepts in focus.
  4. We need to have a range of tools and models for teaching concepts; whether this taking the path through the concrete-pictorial-abstract methodology or otherwise. This affords teachers the ability to provide levels of depth in teaching, but also an alternative model where students have shown to have gaps in understanding after initial study of a topic.
  5. Formative assessment is the true arbiter of how teachers dictate the pace and difficulty of teaching and learning in the classroom. It is not enough to take a list of learning intentions and plough through them; keeping a constant view of, to quote Robert Wilne, students’ attainment hitherto and setting out a path from thereon makes more sense if students are to make real progress.
  6. Avoid granular approaches. Learning concepts should build incrementally, yes, but as Tony Gardiner pointed out, the curriculum is a web of concepts, each their own a ‘mini-web’ of ideas, methods, and procedures. If we teach topics in isolation from one another then we will not engender proper conceptual and procedural understanding in students.
  7. An element of flair and creativity is essential. The path to mastery is not achieved through the deployment of a textbook exercise, but through intelligent practice, subtly changing parameters within an exercise and using a variety of models to test understanding.
  8. Well over 30 years after the publication of the Cockcroft Report, we are still harking back to the answers to the then identified issues in Mathematics education that the document offered. We still hark back to them because a) they’re still happening and b) we have an assessment system that is a straight-jacket that doesn’t allow a proper response. Pretty much every speaker discussed the need to properly develop knowledge, skills and understanding as the three stages of the journey towards mastery; three stages that Cockcroft himself spoke of all that time ago.
  9. Interestingly for me, the talk of knowledge, skills and understanding strikes a similar chord of what I have also being researching, the concept of the trivium: grammar (knowledge), logic (skills) and rhetoric (understanding); ideas that have been around for an incredibly long time! So perhaps it’s true, mastery is nothing new after all.

As you can probably tell, the Symposium had a significant impact on my thinking. I feel that my department has a good handle on what mastery is truly about, and I know that we have the skillset, flair and creativity required to take students to a point where they are able to become great mathematicians whilst achieving great qualifications. I am adamant that the two are not mutually exclusive, and it continues to be my job to ensure that that is the case.

Actual Maths: Notes on the National Maths Symposium, part 2

“It is time to reverse this prejudice against conscious effort and to see the powers we gain through practice and discipline as eminently inspiring and even miraculous.”

Robert Greene

This is the second of a three part series. If you wish to read the first part, click here. Otherwise, read on…

John Mason – The Mystery of Mastery

Mention the names ‘Mason’ and ‘Watson’ in Mathematics teaching circles and a reverential air suddenly eminates. John Mason and his wife Anne Watson have worked for years on researching the best ways to engage and develop student thinking whilst studying Maths, particularly through questioning and the creation of tasks that push students into their individual zones of proximal development (hello Vygotsky). In fact if I had to choose two texts that teachers starting out in our rarified field should read, then Mason and Watson’s Questions and Prompts for Mathematical Thinking and Mason (et al)’s Thinking Mathematically would probably be my top candidates.

You can probably understand my excitement, therefore, when John Mason took the stage. His opening remarks betrayed principles that correlate with mine own: everything he proposed is merely conjecture (very Socratic!) and that professional development should be fundamentally phenomenological, i.e. a ‘lived experience’.

He started out his thoughts on mastery by answering two questions: a) what is a concept? and b) what is a procedure? John stated that

  • Concepts provide access to relevant actions
  • Procedures are a sequence of actions, organised by underlying concepts

In other words, concepts and procedures are interdependent, and mastery is the journey of layering concepts on procedures on concepts on procedures and so on. Most importantly, this journey never has a final end point: mastery is asymptotic (hello Mr Fitzpatrick!).

John then proceeded to punch through we delegates’ (assumed) mastery of ideas such as ordering decimals, counting, functions, and area/perimeter. He did this via a similar approach to Tony Gardiner’s earlier examples: start from simple principles, vary the initial conditions, run the exercise again, constantly challenging the human brain’s need to find pattern and run with it (to paraphrase John’s words), all the while paying attention to how these problems are seen.

John then moved into discussing his belief in the need for ‘didactic tactics’ – when students have completed a task, they shouldn’t put it down, but think how it can be extended? By varying conditions of a problem testing a skill, we are eventually diverting students attention away from the fact they are carrying out the skill, to the point where it becomes second nature – ergo, mastery. Mastery is focus on the goal, rather than on the process (which is automatic).

John closed by stating there are strategies for deepening appreciation and comprehension of concepts by:

  • Enriching example spaces and methods of example construction;
  • Refining personal narratives;
  • Extending connections between pervasive mathematics themes (doing and undoing)

Ultimately, we were told (and in full agreement) that there are many ways to gain procedural fluency, moving along the path from cognition, to affect, and on to awareness.

I have to be honest I could have sat for hours listening to John and trying out his activities. The measure, honesty, practicality and rationalism that his ideas and methods demonstrate are quite someting to behold. There is no sell, there is no overconfidence – as he says, it is all conjecture, and if it works, it works. There is a great deal of thought that he and his wife Anne put into how we can test the true mastery of concepts and I want to know more. So I will be adding more of their writings to my reading list in due course, and my department have a lot to look forward to in terms of trying some of the ideas out in meetings, and hopefully their classrooms, in the not too distant future.

Robert Wilne – What does ‘mastery’ mean?

After Tony Gardiner and John Mason’s academic (but very important) viewpoints on mastery, it was now the turn of Robert Wilne to take on the baton, framing his findings very much in terms of what is going on in classrooms as we speak.

Robert started out by defining mastery in very simple terms: masterry is achieved if students

  • demonstrate solid conceptual understanding
  • are fluent in their methods
  • can apply their mathematics to a range of applications

all whilst making connections to other facts and ideas. If at this point you’ve read the Cockcroft Report and thought ‘knowledge, skills and understanding’ then you will not be alone!

Robert stated that mastery does not refer to what the teacher does but what the pupil gets out of it. The teacher should work to ensure that all students are confident, secure, flexible and connected in their mathematical understanding. Robert was anxious to emphasise the word ‘all’: all students can perform well mathematically, given the opportunity; “don’t reify ability into character… only talk about attainment hithero”.

Robert has been heavily involved in the NCETM Shanghai project, and shared his findings, which were extremely relevant to the mastery concept. Lessons in Shanghai:

  • have less chopping and changing of concepts;
  • work through conceptual steps slowly;
  • rapidly move towards increasing abstraction;
  • have more questioning and focus on intelligent practice;
  • provide interventions quickly to close gaps (same or next day);
  • provide more time for pupils to discuss and improve on concepts.

All students made progress at broadly the same pace, and teaching, whilst not brilliant, was consistently good as a result of building lessons around these principles. Study was done through problems that had a richness and sophistication that did not require differentiation.

Teaching for mastery, Robert therefore conjectured, was based on four things:

  • provision of good models and representations of concepts and procedures;
  • offering procedural and conceptual variation in teaching and practice;
    • in other words, predicting likely misconceptions by raising and resolving them;
  • provision of intelligent practice through increase creativity.

Example thereof Robert shared in great detail, and it would be remiss of me to try and summarise these at this stage, but I am sure that Robert and his colleagues at the NCETM would be happy to share them.

Importantly, (developing on an analogy on kite-flying that the Shanghai teachers proffered) as learning progress, find out what the students have seen, and then establish if they’ve seen the right things. Make connections through intelligent practice, which is practicing the thinking process through more creative means.

Robert then moved on to how we can manifest these ideas in our own classrooms, based on what the Shanghai teachers suggested:

  • Start with the topic, establish your objectives and how they’ll be achieved through the specific focus of your lesson;
  • Have a quality textbook that tests procedural and conceptual variation;
  • Use department meetings to develop ways of implementing intelligent practice;
  • Use the knowledge of colleagues both in ‘real life’ and social media;
  • Try things out and discuss the impact!

Now some people reading this will say, well duh, we knew this already, and did it need trips back and forth from Shanghai in order to confirm our thoughts? I have to say I have been skeptical about the whole Shanghai affair, however I do feel there are subtleties that have been shown in much of the teaching and learning that goes on which I do not think in this country we appreciate: particularly how conceptual and procedural variation play out in the classroom. I was glad to hear from Steve McCormack that work continues apace in transmitting and testing the impact of practice taken from the Shanghai project and I do hope that the subtleties Robert and his team have found do show up in teaching in this country, particularly in our quest for students working towards true mastery.

If Tony Gardiner and John Mason focused mainly on the ‘coalface’ actions of enabling the journey towards mastery then it is clear that Robert, along with Bruno Reddy, laid out how mastery can be facilitated for in terms of environment, culture, teacher development and planning. Both approaches are two sides of the same coin, and as you will see in the next part of my notes, it was down to Jeremy Hodgen to further tie these viewpoints together. Don’t change that dial…

Actual Maths: Notes on the National Maths Symposium, part 1

Man’s reach should exceed his grasp, or what’s a heaven for?

Robert Browning

As I come to the end of formalising my principles on leadership and the systems that I believe make a Mathematics department effective, I have begun to turn my attention back to the classroom. As a department, we follow a ‘mastery curriculum’, spending more time on topics to facilitate understanding of topics, rather than – to paraphrase one of the symposium’s speakers, Tony Gardiner – continuing to “nibble”, coming back to topics and not getting students understanding to a satisfactory level.

What is all too readily apparent, however, is that whilst one has a framework that facilitates teaching for mastery, what does that actually look like in the classroom? How can I take the (already excellent) subject knowledge and teaching skill of my department, and ensure all learners’ learning opportunities are optimal? Also, now that everyone and their mother is claiming they offer a mastery curriculum (thanks to OFSTED fears), in many shapes and forms, can we bring the notion of mastery back to what is should really be about? What *is* it really about? My attendance of the National Maths Symposium (hosted by La Salle Education) had the goal of answering those questions.

Mark McCourt – Is Mastery New?

Mark opened the session with a brief introducory address, reprising his research piece on the history of the idea as well as what the various historical proponents of mastery actually agreed it looked like:

  • Diagnostic pre-assessment with pre-teaching
  • High quality group based initial instruction
  • Progress monitoring through regular formative assessment
  • High quality corrective instruction (that uses different models and metaphors to the original)
  • Second parallel formative assessment
  • Enrichment or extension activities

It is hard to argue that these ideas are other than good, principled teaching. I also agreed with Mark’s point that true mastery relies on the interplay of mathematical language along with concrete, pictorial and abstract forms. Here we were setting the tone for what the rest of the symposium was to do – determine how those principles should play out in the classroom.

Tony Gardiner – Mastery: Confusion and Contradiction? Or Coherence?

I have encountered Tony in conferences before but his talk today was a tour de force. Though his presentational style may not be lining him up for a TED talk soon, his message was formidable.

Tony first set out the concepts of assimilation and accomodation (via Piaget):

  • Assimilation is understanding a concept in terms of what you already know. Politicians and policy makers are assimilating “mastery” as a term to up their game, cherry picking some ideas and fitting them into existing frameworks;
  • Accomodation is getting a jolt in your thinking, a discontinuity which changes your understanding: we can accomodate “mastery” principles by setting up the necessary support structures and training needed make that jolt happen.

I agreed with Tony’s point that the word “mastery” (in terms of Mathematics teaching) “is used so unthinkingly… that [it] risks passing directly from obscurity to meaninglessness without any intervening period of coherence”. This correlates with the question I posed above. It’s almost like the scene in the eponymous film: “I’m Spartacus” – when enough people say it, who truly knowns what mastery is?

Tony then shared 5 problems that highlight students’ journey towards mastery (more of that later), that we were encouraged to have a go at. All five underpinned an interesting conceit – requiring simple techniques, but multi-step, and open-ended, leading to more abstract thought.

I was particularly pleased with myself when one of the problems led me to demonstrating a formal algebraic proof. But I digress…

Mastery, therefore, requires something more than providing a framework and style in teaching. It requires a didactism based on technical detail that allows students to rapidly move beyond the employment of procedure and see how techniques in Maths fit within the “webs of ideas” that make up the subject.

To quote Tony: “each “basic technique” then brings with it a “web of related ideas”, a “mini-universe of methods and problems” that must be integrated into the teaching sequence”. In other words, it is not simply enough to teach addition: problem sets must allow a student to see the concept in a variety of contexts before one can say that said student is moving towards mastery. This is done through sequencing ‘webs’ in a workable order, exploring it and it’s variations in  unit, establishing key methods, trialled to address misconceptions.

There was also one very interesting subtext to Tony’s talk: much of the problem of students being unable to move along the path to mastery is an Anglo-Saxon one. Too many right-thinking governments and institutions have failed to bridge the gap between the expectation of classical rigour of simple techniques (good grades) and the need to produce true problem solvers (good mathematicians). I have touched on this point before, and I hope that under the cover of the sloganeering that appears to be going on, a real attempt to bridge this divide truly happens.

Bruno Reddy – Culture for mastery

Bruno, to many people, is Mr Mastery, thanks to the ripples we are all still comprehending as a result of his and his colleagues’ work at King Solomon Academy. Today he spoke about the importance of the right culture in the classroom, creating an environment for mastery to happen.

In his mind, the best Maths teachers work at the intersection of good questioning, good modelling, and creating a good culture for learning. Culture is a multiplier of the other two – if the culture is right then the effect is appreciably better; if it isn’t, then the effect is diminshed.

Interestingly, it is a result of my previous experience that I often worry about ploughing through content before getting culture right. I am now rectifying that situation. 

So what is culture, in Bruno’s mind? Starting from your values and language, and expressing them through your actions:

  • Routineering – how do your lessons play out on a daily basis?
  • Visioneering – how do you inspire your students to show their best every lesson?
  • Expectations setting – what do you want to see happen in terms of your students actions?

Bruno demonstrated first hand how this looks in his classrooms, and he showed how by through getting the culture right each day, every day, wonderous things can occur.

Whilst Bruno did not delve too much into explicit Mathematics teaching (apart from Rolling Numbers), there is something profound about the culture that his and similar schools of thought has in terms of impact on Mathematics teaching. Such steps cut through the pre-determined alienation that many students feel about Mathematics, creating a safe space to enjoy learning instead of what can be often hostile environments to try and teach in. Even though I am very familiar with Bruno’s work, it is always great to have a refresher on the methods he employs and their effectiveness. I again am taking stock!

In the next part I aim to summarise the thoughts of John Mason and Robert Wilne, whilst in the third, those of Jeremy Hodgen and some conclusions and other points to take away. Stay tuned.

Actual Maths: A Canon of Mathematical Thought, for Students

I find television very educating. Every time somebody turns on the set, I go into the other room and read a book.

Groucho Marx

This week, I had the pleasure of being in attendance at a talk by Elfi Pallis, the author of Oxbridge Entrance: The Real Rules. The aim of the talk was to share how state schools can support students in applying for and entering studies at Oxford or Cambridge.

What became readily apparent was that it isn’t enough for students to be academically brilliant. ‘Oxbridge’ expects a balance of both academic excellence and cultural capital, and it was made clear that it is important that we invest that cultural capital in students from early in their secondary education. Also, it is not just the case that we invest this cultural capital as a broad stroke: instead we should look for subject specific reading that students can pick up and start to think more deeply about the broader impact and development of that subject, as they begin to move from GCSE to A-level, then from A-Level to higher education.

So, this got me thinking: is there a ‘canon’ of Mathematics books for the more general reader at a level that students at Secondary and Sixth Form level could access? I started to formulate a rough list, but I’m interested to see what readers can come up with. Here’s a few of my choices, but I’ll publish an updated post once people have had chance to make their own suggestions:

Introducing Mathematics – Sardar, Ravetz, et al

This is the bluffer’s guide, to be honest, a whistle-stop history of Mathematics and schools of thought from the days of Pythagoras, Euclid and Diophantus to relatively recent developments such as Andrew Wiles’ solution of Fermat Last Theorem. In a comic book, idosyncratic style, it’s no bigger than a Mills and Boon special but packed with snippets of the best historical and cultural touchstones in Mathematics.

Fermat’s Last Theorem – Singh

Talking of Wiles, Simon Singh probably wrote the perfect example of a book on rather heavy Mathematics for the general reader. When the book starts going into modular forms and elliptic curves one might need to slow down a little, but it’s not too exacting. Written as part history, part biography, part thiller, but blended together to tell the story of one man’s determination solve one of pillar problems of our subject.

How To Solve It – Polya

it is one thing to do Mathematics, it is another to think like a mathematician. Polya’s treatise captures the essence of how to solve mathematical problems, taking a rather strategic approach of gathering information and planning an ‘attack’ on a given question. It is clearly aimed at anyone of any level of ability who wishes to start taking Mathematics seriously.

1089 And All That – Acheson

A curious little book that summarises some of the classic concepts and curiosities in Mathematics: infinite series, three-body motion, e, π, calculus, etc. There’s a breathless pace to the book, but it’s packed full of knowledge and a great way of sparking inspiration in terms of fields for students to delve further into.

It Must Be Beautiful – Farmelo

I have to admit I have lost my copy of this book! Essentially it shows how some of the most revolutionary concepts in Science are underpinned by wonderfully elegant equations, such as Dirac’s equation, Schroedinger’s wave equation and Drake’s equation to estimate the number of extraterrestrial civilisations. The 11 essays in the book are wonderfully written, and absolutely convincing of the power of Mathematics in making a difference to the world.

The Numbers Game – Anderson, Sally

This book is, on the face of it, about football, but it shows how the use of statistics can essentially determine strategies to take, how seemingly random behaviour can be boiled down to a set of rather simple variables and relationships. People may think that this is an odd choice, but I also think it’s important for students to see how even in an apparently unrelated field, mathematical analysis can help provide an organisation or individual an edge over one’s competitors.

The Penguin Dictionary of Curious and Interesting Numbers – Wells

Quite literally a list of numbers, but with a twist. Many numbers in themselves have stories behind them, from -1 all the way up to Graham’s Number (with some jumps and skips in between). Another book that is a curious blend of history, theory and inspiration, moving from Coprime, Sociable and Lucas-Carmichael numbers (what?) in easy to follow steps.

Alex’s Adventures in Numberland and Alex Through The Looking Glass – Bellos

Vying with Simon Singh for the best writer of Mathematics books for the general reader, both of these books by Alex Bellos are notable for their breadth, depth and sheer accessibility. Like Singh, Bellos is adept at pulling together ideas behind mathematical concepts through the power of storytelling. I particularly like his tale about Claude Shannon teaming up with a dodgy Mafia type, going to Las Vegas and taking the casinos to the cleaners. Gambling is bad, kids.

I think ultimately that where possible, books go quite deep into concepts. Huge tomes covering the history of Mathematics are great, but in some respects it’s also important for a switch to be flicked in the reader’s mind that motivates further study.

I look forward to hearing people’s suggestions, and I’ll summarise them in a few weeks’ time.

N.B. I haven’t linked these to any bookseller in particular, simply because various sellers sell these books for various prices. That said, AbeBooks is good to trawl through the web to get a decent deal on used copies.

Actual Maths – Have I solved the grade descriptor problem?

The text has disappeared under the interpretation.

Nietzsche

I, like most heads of department, have scratched their heads at the government-issued grade descriptors for the new GCSE Mathematics specifications. I mean, look at them:

To achieve grade 8, candidates will be able to:

  • perform procedures accurately
  • interpret and communicate complex information accurately
  • make deductions and inferences and draw conclusions
  • construct substantial chains of reasoning, including convincing arguments and formal proofs
  • generate efficient strategies to solve complex mathematical and non-mathematical problems by translating them into a series of mathematical processes
  • make and use connections, which may not be immediately obvious, between different parts of mathematics
  • interpret results in the context of the given problem
  • critically evaluate methods, arguments, results and the assumptions made

To achieve grade 5, candidates will be able to:

  • perform routine single- and multi-step procedures effectively by recalling, applying and interpreting notation, terminology, facts, definitions and formulae
  • interpret and communicate information effectively
  • make deductions, inferences and draw conclusions
  • construct chains of reasoning, including arguments
  • generate strategies to solve mathematical and non-mathematical problems by translating them into mathematical processes, realising connections between different parts of mathematics
  • interpret results in the context of the given problem
  • evaluate methods and results

To achieve grade 2, candidates will be able to:

  • recall and use notation, terminology, facts and definitions; perform routine procedures, including some multi-step procedures
  • interpret and communicate basic information; make deductions and use reasoning to obtain results
  • solve problems by translating simple mathematical and non-mathematical problems into mathematical processes
  • provide basic evaluation of methods or results
  • interpret results in the context of the given problem

Did someone go crazy with a thesaurus? Clearly they’re designed to stop people assigning grades to topics, which any Mathematics teacher worth their salt will tell you that this is a good thing, even if we’re still expected to report grades as we cover content. Indeed, as Mel Muldowney rightly stated on her wonderful blog: “the practice of categorising topics for GCSE is just plain harmful”.

However. However. Ultimately a Head of Department needs to know that a) their students are being taught correctly in terms of content and standard and b) their students are as well prepared for an exam as possible (not, I hope you note, by teaching to the test – I think we’re all in agreement that this was pretty bloody hard in the first place and looking at the new SAMs nigh on impossible). Now if you look at the updated SAMs you will see that topics that would normally be considered as ‘straight forward’ are now being pitched at any point on the difficulty scale, for example, this question:

AQA-83002H-SQP-2015 Question 21. Thanks to AQA.

AQA-83002H-SQP-2015 Question 21. Thanks to AQA.

Because the price isn’t given, and the amounts have to be calculated, this goes from a classic better value question to something more detailed. Will students realise that the price doesn’t matter (and therefore using £1 is a good option?). There’s lots to consider.

My worry, therefore, is being able to translate the grade descriptors into something more concrete. So, I have looked for examples, including appealing to the exam boards for more details (who in the main were very helpful, and thanks also to Mel again for her continued support):

Including typos! Authentic!

Including typos! Authentic!

However I still wasn’t satisfied with what was put forward, and so decided to press on. Then, lo and behold, I had an epiphany, prompted by a tweet that Bruno Reddy had posted regards “Questions and Prompts for Mathematical Thinking” (written by Anne Watson and John Mason, published by the Association of Teachers of Mathematics). The book is a masterpiece, a structured approach to designing questions that will develop a range of mathematical skills. Watson and Mason are legends of the Mathematics teaching game, and their knowledge comes through in the breadth of different examples they offer.

Anyway, the epiphany was this, a rubric for questions in the middle of the book:

Copyright Association of Teachers of Mathematics, 1998

Now, if some of you are reading this and going ‘well, duh’, I’m not going to apologise. This is not me trying to pass off old thinking as something revolutionary – in fact I’ve had the book for some time. No, what this did was provide a key to unlocking my need to provide concrete examples of posing content-based questions that test the skills outlined in the new GCSE grade descriptors. There are 48 different ways of structuring a question based on the rubric above, which I think you’ll agree allows for testing depth of understanding on a topic at any level. This therefore prepares students for the threat of being questioned on a topic that is classically procedural in a way that doesn’t quite required a straightforward method.

Anyway, I had a go at filling in a rubic for the N1 section of the new AQA GCSE specification. Some questions are straight-up lifted from other sources, some are my own. I’m not a great content creator if I’m honest, so any critiques are welcome.

If you click here, you’ll be able to access a copy.

Now I am not saying that we can then start to grade questions – not at all – but it does make it easier to judge whether a student understands the syllabus at a level approximating a grade 2, 5, 8 or somewhere in between. Hopefully then we can move away from the developing practice of publishers (including one who I have long had a lot of time for but have let me down big time recently) assigning grades to topics, pegged to the old GCSE grading system.

I’ll be honest, it’s not easy – I think I’ll get quicker at it as I go through. Bearing in mind there are 97 different content statements, however, it is a huge project, and someone with more time than a SLT member/Head of Department/Father of a toddler would make a huge number of teachers very happy if they were willing to take up the baton.

One final point – these questions should not take the place of deliberate, independent practice. Instead, these should be used to identify if students have truly mastered a concept. In fact, that’s what they should be called. Concept mastery questions.

Now, I’ve got my reservations about the Association of Teachers of Mathematics but if they were to be the ones who did take up this baton and run with it then a) they’d be helping out a huge number of very grateful teachers and b) therefore would see their subscription rates skyrocket. Otherwise, I’ll see I can plod on through…

Thoughts, as always – are welcome.

 

 

 

Actual Maths: Thoughts on #mathsconf5, part 5

If history were taught in the form of stories, it would never be forgotten.

Rudyard Kipling

This is the fifth (fifth!) in a multi-part series on the 5th National Mathematics Conference. If you like your blog posts in linear order, then start here. I really recommend that you do.

Kris Boulton – The Stories of Mathematics

I have a lot of time for Kris Boulton. Kris’ ideas and work have long been an influence on mine own, and over the past year or so as I’ve got to know him better, his well considered and deeply-researched ideas continue to augur my own motivation to improve my own work.

The last workshop of Kris’ that I attended was on the assessment of mastery and it is still having reprecussions as I continue to develop the practice of my department. However like me Kris has an appreciation of the interweaving of strands that make up Mathematics teaching – not just the subject, or the pedagogy, but also the stories behind how our subject came to be.

An excellent point that Kris made at the beginning is that we often forget how much hard work goes into the development of Mathematics, the creation of new methods, the proof of theorems (and how they themselves came about. We often take for granted things like Pythagoras’ Theorem, the Fundamental Theorem of Arithmetic, the Cartesian coordinate system.

Kris himself had put a lot of work into this talk, about 40 hours of watching and reading, getting an understanding of not just the impact on our subject but on civilisations too. Major historical figures and their contributions were also considered. Kris produced a timeline of Mathematics – you can find it here. It’s an incredible piece of work in itself.

Kris wisely decided to choose a sample of the events along the line, placing the spotlight on paradigm shifts, rather than incremental developments. So we were taken to a time where we found out what people did before they could count, how the number of things we can remember is generally limited to four, and thus how counting, tallying and one-to-one pairings – ideas at the foundations of all Mathematics – came to be.

We were then taken through a journey of a combination of classic stories, scientific truths and debunking myths. I particularly liked the focus on linguistic development of terms like Trigonometry, and how an understanding of the etymology of the jargon we use is a great way of demystifying such terms.

I also think that there is something else to consider here. Mathematics, ultimately, is a branch of Philsophy. Early in human civilisations, when people asked why the sun rose, the rain fell, the wind blew, etc, those in power would retort that it was the will of Zeus, Gilgamesh, Ra, etc. Only when people began to realise that this was not good enough, and that a true understanding of the nature of the world would allow them to make it a better place, did true progress come to be, i.e. it was the beginning of Philosophy that started to answer such questions.

Thales of Miletus, of whom Kris spoke in his talk, was the first ‘name’ in Philosophy, the first individual recognised for his philosophical approach. He gains recognition because of being able to measure the height of the Great Pyramids, one of the first occasions where something considered only in the scope of the efforts of the Gods brought under the yoke of human thinking. Pythagoras, as a student of Thales, extended that idea by touring the known world and linking the knowledge of three of the great ancient civilisations (Greek, Babylonian and Egpytian) and began to codify the discipline of Mathematics as an answer to philosophy’s great questions – over 100 years before Socrates developed his methods and well over 200 years before Euclid created the Elements.

I believe there is a case that there should be a greater element of the history of Mathematics within the programmes of study at GCSE. Certainly the philosophical elements like Platonic ideals (which led to geometry as a formal discipline), the Vedic appreciation of the void that let to the concept of zero as a proper mathematical concept (alien to the likes of Aristotle), Leibniz’s understanding of binary numbers as the creation of something out of nothing.

Ultimately I feel that Mathematics is the language of nature, and that language was formed by the philosophical questions that have been asked from the days of Thales. What power there would be if we could answer the question ‘what’s the point?’ with a bit of historical/philosophical storytelling? Also, from a purely functional point of view, there is apparent evidence that teaching Philosophy has a positive impact on the progress students make in Mathematics – why not explore that at a deeper level?

Kris didn’t have time to go through his whole plan, which was a shame considering it was an engaging and lucid presentation of findings that weaved together into an appreciation of the breadth, scope and constant evolution of the subject we teach everyday. It was worthy of a longer, more encompassing keynote, a double-session at least. I hope Kris produces a sequel or series to follow up this start.

In the interim, I know Kris recommends the book Zero – The Biography of a Dangerous Idea by Charles Seife, and just finishing reading it I concur with this. If the history of Mathematics is something you’re interested in, particularly the development of ‘big’ ideas, I can happily recommend everyone reads Euclid’s Window by Leonard Mlodinow and Marcus Du Sautoy’s Music of the Primes. Sometimes we are too concerned with the process of teaching our subject, losing the reasoning of how Mathematics came about and what the ultimate opportunities of having mathematical skill mean. It’s a healthy process to stop, reflect on what we’re all here for, and thing about how we can translate this to our teaching. The stories of Mathematics are not a bad place to start.