Category Archives: Mastery

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.

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.