As I promised my first #mathsconf7 blog before the end of Monday, I thought I’d better crack on, which means turning pages of tiny scribbled notes on an old Osiris pad into something a little more coherent.
Much like @taylorda01, I’ve spent a lot of time in Leeds over the last decade (three years of university, followed by two years travelling up and down nearly every weekend, followed by a further five years living there and working in Yorkshire) and have never been to the Royal Armouries, but I concur with him that it worked well as a venue, and the unrelated event going on in the Leeds Dock area certainly added a bit of a party atmosphere every time I walked from New Dock Hall to the main Armouries building.
I wandered around the exhibition area for a bit, attempting to drink a coffee while picking up freebies from the various stalls. I was super-pleased to get my hands on one of @OxfordEdMaths pun pens to go with my badges from last time; I also got a fantastic free A Level poster. It was also great to see that @CambridgeMaths are still offering free, no-commitment evaluation copies of their new GCSE textbooks, so it’s worth checking them out at future conferences – I picked up a full set of GCSE AQA and Edexcel resources at #mathsconf3 last year.
Mark and Andrew’s opening address was brief, but set the tone for the day, with the only mention of the political climate being an incitement to get #mathsconf7 trending above #EUref. I don’t think we reached our goal of trending first worldwide, but we at least got to number 3, giving another indicator of how powerful these events are for bringing the maths teaching community together.
I thoroughly enjoyed the keynote from Mike Askew, which was a mini-workshop in itself. Mike set up the address by highlighting key recommendations from research into mathematics teaching, evaluating each before seguing into a discussion of problem-solving models. I particularly enjoyed the way he completed his list before presenting the level of evidence for each, reminding me that I have a duty to think critically about what I’m told and not accept things because “an expert” has recommended them.
The recommendations each offered significant food for thought, and although there was nothing new here, it’s nice to be reminded of important principles at the start of a day like this.
1. Space learning over time
I completely agree with this principle, and many teachers and planning documents subscribe to this notion. One point on this was a comment from Mike that seemed to imply that “current mastery ideas” and spaced learning are incompatible – I would disagree, and wholeheartedly believe that a workable scheme contains elements of both. I detest use of “mastery” as a buzzword; as many commentators have pointed out, “mastery” is the way that good mathematics teaching was done until introduction of the National Strategies, and isn’t new or revolutionary. For my money, a good scheme of work should allocate large blocks when introducing a big new concept, but that doesn’t mean that this concept will not be revisited in other contexts later on – i.e. when working on area of 2D shapes, include problems requiring pupils to multiply fractions and decimals as well as positive integers.
2. Interleaved examples and problems
This is something that I don’t feel most GCSE textbooks have got right. Most exercises are presented with a few examples at the top, then a large chunk of problems, requiring teachers to carefully guide pupils through the exercises and making independent learning quite difficult. The Edexcel series of A Level textbooks on the other hand have excellent worked examples with full explanations, and often include pointers in exercises referring pupils to the specific example that might be useful in that case – it would be good to see more of this earlier on in pupils’ education.
3. Combine graphics with verbal descriptions
4. Connect abstract and concrete representations
I jotted down a couple of key points from this linked pair of recommendations; firstly, Mike suggested that graphic representations should be accompanied by verbal descriptions rather than lots of written text, keeping the image clean and allowing for discussion. Secondly, Mike pointed to evidence that the thinking that understanding develops from concrete to abstract is flawed; young children are capable of engaging in abstract mathematical thinking (a point that tied in nicely with my third workshop choice, “A Golden Age?” from@dannytybrown), while adults and more advanced learners don’t get rid of concrete representations completely. I’ve seen evidence of this in my secondary practice (see this blog on completing the square), and Mike highlighted an example from his experiences as a university student when tackling higher-level concepts – despite being able to follow an abstract procedure, he had limited understanding compared with peers who could ground these concepts in concrete imagery.
5. Quizzes
Mike described the American process of setting “non-threatening” or low-stakes quizzes, both as a pre-assessment at the start of a unit and to re-expose pupils to concepts later on. Pre-assessments seem to be more a part of standard practice than they used to be, so it was interesting to see during Mike’s reveal of the evidence levels later on that there was low evidence that this actually increases pupils’ attainment. It was, however, interesting to see that quizzes as a means to re-expose pupils to content did have significant impact.
6. Ask deep explanatory questions
Mike used this as a springboard for his discussion about problem-solving and reasoning. Without deep questions, pupils only really experience a shallow, procedural version of mathematics. He highlighted the idea of posing a question at the end of the lesson and allowing pupils to “sleep on it”, with the suggestion that the brain will unconsciously continue to work on the problem.
Mike then discussed a few points around problem-solving and reasoning. This quote from Development of Maths Capabilities and Confidence in Primary School (DSCF-RR118) seemed to strike a chord with many people:
Mathematical reasoning, even more so than children’s knowledge of arithmetic, is important for children’s later achievement in mathematics
This is further elaborated in the document as follows:
Mathematical reasoning and knowledge of arithmetic (as assessed in year 4) make independent contributions to children’s achievement in mathematics in KS2 and 3. While both are important, mathematical reasoning is more important than knowledge of arithmetic for achievement in KS2 and 3.
Mike highlighted the implication that skills in arithmetic are not a predictor of problem-solving skills (and vice versa), then drew the distinction between direct and indirect objects of learning. The direct objects of learning are the topics we teach, and what pupils are most likely to identify when discussing their mathematics – “we did fractions this week”. The indirect objects (can’t be taught directly) break down into a list of three proficiencies:
- Fluency
- Problem-solving
- Reasoning
Mike then discussed the importance of ensuring that tasks planned to develop problem-solving capabilities don’t “decline” into procedural or fluency-based tasks and that teachers should be aware of “taking thinking away” from pupils by immediately diving in to help when pupils get stuck. He offered the suggestion that slow thinking is accompanied by emotional discomfort and tension, and that we’re almost hard-wired as teachers to remove elements of discomfort and tension from our classrooms – in removing this tension, we also remove the problem-solving element and the task declines to a more procedural activity. Pupils should be given long enough to think carefully about the problem and be allowed to struggle – but crucially, should not be allowed to think unsuccessfully for long enough that they get fed-up and disheartened with the task.
We should also think carefully about the sorts of problems we select for our pupils; this is what Mike called the ABC of problems.
1. Authentic problems
These are problems that pupils could conceivably encounter in their current circumstances – Mike offered the example of “plan a school disco”. The issue here is that authentic problems require authentic solutions, and realistically, we tend to “find a way around” such problems without engaging in much deep mathematical thinking.
2. Believable problems
These are problems that pupils could believably encounter at some point in their lives. Mike highlighted the issue with predicting such scenarios, and what’s believable or relevant for one pupil may not apply to others in the class; constructing such problems is very difficult.
3. Curious challenges
I loved the terminology used here, and immediately thought of Don Steward’s work. Rather than invent contexts and problems which may not be relevant to pupils, we look instead to provide them with challenges that hook them in – these may or may not be related to a “real-life” scenario. Mike offered a particularly excellent example that I’m looking forward to using with pupils:
Mike also offered a couple of rubrics for planning and developing lessons to enhance problem-solving skills. The first of these was around the teacher’s duties during such activities:
- Choose the tasks – we use our expertise to select tasks as above;
- Set them up – give the tasks a narrative or story;
- Orchestrate discussion of solutions – this is where the teacher should be most active during a problem-solving task, and focus should be on the mathematical content behind the problem, rather than just hearing voices. The teacher’s job is to look for the mathematically significant answers and draw these out.
The second plan offered was how to get pupils to engage with this discussion process. Mike cited a study looking at pupils’ reactions to discussion in upper primary – pupils thought it very important to have their say on the mathematics they had discovered, but frequently did not see the importance of listening to others. Suggestions included:
- Repeat and revoice – ask a pupil to repeat what another has said;
- Rephrase – ask a pupil to explain another’s answer in their own/different words;
- Build on – ask a pupil to add to a point raised by another;
- Agree/disagree – ask a pupil their opinion and give an explanation.
Following Mike’s address, we all engaged in the usual “speed-dating” activity to share resources. I’d brought along a simple idea I nicked from a colleague last year – a double-sided plastic mat with single integer values. It’s a gimmick, but an engaging one – I’ve used it for quick-fire questions and ask pupils to fold their mats to show the correct answer – the kids seem to love it! For example, you could ask pupils to fold and show you a square number:
Vicky shared a great idea for a hands-on approach to introducing angles in parallel lines; she described how the DT department has constructed a solid model of the parallel lines diagram from coloured Perspex, with two angle shapes tha slotted in. Pupils then used this to discover where angles were the same and developed rules from this.
Kate (@KateTarno) shared her department’s idea of attaching clear, raised Perspex to the tops of pupils’ desks. These then functioned like mini-whiteboard paint, and pupils were encouraged to use these for working out. She also explained how worksheets could be slotted underneath the Perspex, both saving on reprographics costs and providing an easy way for pupils to alter their work when working on diagrams.
Unfortunately I didn’t get my third date’s name and have subsequently lost the paper resource, but she shared a tangram-type template including most quadrilaterals, great for making 2D shape properties a little more interesting.
My workshop choices for the rest of the day were as follows:
- Avoiding Misleading Assumptions (Peter Mattock)
- 10 Things You Should Know About the New Maths A Levels (Christine Andrews, Andrew Taylor and Gary Wing)
- A Golden Age? (Danny Brown)
- More Interactive Problem-Solving Models (Matt Dunbar)
Individual blogs on each of these will be coming over the next week or so!
