This is my sixth year of teaching and I think it’s the first time I have taught equations properly to a KS3 class. I was almost there last year, and thought I was doing it well, but I now know there are several topics where I completely let the pupils down. This post is about how I could have been better-prepared earlier in my career, and avoided leaving later teachers with a mess to clean up.
Naveen Rizvi’s piece yesterday in the TES caused a stir that surprised me. Many people had a negative reaction beyond what I would have expected (I won’t link to them) and was followed by some negativity – or at least concern and alarmed questions – when Bodil subsequently shared an example of two pages from the booklets we give to pupils.
As I see it, these are some of the main barriers preventing pupils from achieving their potential in maths that CAN’T be dealt with by better resourcing:
|Possible solutions:Improved teacher pedagogy and understanding of how memories and connections are formed.
Improved teacher understanding of what fixed and growth mind-set actually is (not just a gimmick to console pupils when they underperform… my heart bleeds for Dweck).
|Possible solutions:Effective intervention and catch-up programmes in school (ideally supported at home).
|Possible solutions:School leadership foments a culture that challenges this (supported by classroom culture created by individual teachers), either through super-high expectations/tough love or alternative approach that challenges and changes issues that hold pupils back in school.||Possible solutions:Head of Department leads maths-focused CPD|
|Caveats:This is not easy. ITT doesn’t seem to cover this adequately, and it appears to be a relatively new part of most teachers’ pedagogy*, relatively complex to understand and highly complex to begin to incorporate into practice (particularly for the weakest pupils).
* This is, of course, excluding some very experienced and successful practitioners. In their case, it appears to be something they’ve come to understand intuitively and isn’t easily shared as it isn’t codified.
|Caveats:There are many programmes that appear to have high impact in closing the gap between pupils’ reading and chronological ages, or the gaps in their mathematical foundations. In particular, direct instruction programmes such as Connecting Maths Concepts (McGraw-Hill scripted direct instruction programme) and Lexia appear to be effective ‘off-the-shelf’ interventions (based on my own experience!).||Caveats:Really brave leadership on school culture, especially in challenging circumstances, is too rare (in my limited experience). Many bloggers have written about the gap between their school’s behaviour policy and the ‘real behaviour policy’ (teachers are left to defend their own classrooms, with little or no back up). In the best cases I’ve seen, there is total clarity about the positive, learning-focused culture the headmaster/mistress seeks to embed, and the behaviour policy serves this and is always upheld.||Caveats:This is incredibly time-consuming. Most HoDs simply don’t have the capacity to do this well. The number of conflicting interests they have makes this difficult: teaching as many of the critical/tricky classes as possible (as they are, hopefully, one of the strongest teachers), writing SOWs, managing staff shortage (it is maths, after all), retaining staff and keeping them happy, improving teaching quality. And, ideally, reading widely to prepare for new exam specs and maths education research…!|
However, there are more issues than this that are – I think – relatively neglected outside of the rarefied atmosphere of online edu-chat and conferences.
Barriers created in lessons:
- An capable but exhausted teacher who can’t prepare adequately for lessons (their department is under-resourced and teach a full and varied timetable)
- Confusion about what they should be covering to prepare for the end of Y11 (it is unclear what the pupils covered in Y7-9, or in how much detail; there is uncertainty about what should *actually* be taught when they see ‘averages, 1 week’ on the SOW… Does it mean calculating the mean, median, mode and range only, or complex questions where some values are missing and then one value is changed?).
- Painfully optimistic allocations of timing to teach topics (expressions – 1 week; fractions – 2 weeks), due to insufficient clarity about what should actually be taught.
- A gap between what they cover in lessons (superficial) and the rigour of the exam (increasingly higher, hopefully). A recent example of this was the GCSE question: Solve for a: 2a + a + a = 18. This question is beyond trivial, but many teachers had not prepared their class for the possibility that simplifying and solving could be used in the same problem.
- Unclear explanations, or rule-based explanations, that makes it difficult for pupils to use their knowledge flexibly or to ask useful questions (e.g. “change side, change sign” to solve linear equations because it seems quicker and easier, or convoluted steps to solve simultaneous equations).
- Inadequately scaffolded and varied practice in lessons that doesn’t prepare them for the variety of forms maths can take in the real world (or in exams…) (We all suffer from textbooks that escalate the difficulty of questions too quickly, so that your weakest pupils get only 2-3 questions practising questions in the form a+3=10 before they’re moved onto the other three operations).
- The practice gap (i.e. getting much less practice than pupils in other schools). Most textbooks DON’T HAVE ENOUGH QUESTIONS. At all. Most of the newest books boast how many more questions they have. It is not enough. If a pupil has only just begun to grasp a procedure, they need to do it many times to build their confidence and then begin very careful and gradual variations.
- Pupils forgetting that they have learned something (“I swear down they never taught us that”). This comes from haphazard, or no, continuous revision or interleaving (weaving old topics into current topics).
- Pupils doing what seems obvious to solve a problem, rather than what is mathematically correct (e.g. writing that 3/4 + 1/2 = 4/6). As above, an absence of revision and interleaving.
- Pupils knowing they’ve learned something, but muddle it (e.g. calculating the mean when asked to comment on the median). Also as above…
I am increasingly convinced that a good textbook would begin to address these ten problems. A good textbook:
- Offers interesting talks and prompts for pupils to have high-quality discussions in pairs and with the class. These can range from puzzles to problems that provoke cognitive dissonance (e.g. which is closer to 1/2, 1/3 or 1?)
- Offers worthwhile questions that allow pupils to use multiple strategies to solve a problem or to calculate (e.g. 4.5 x 24)
- Plans for revisiting old topics, particularly those that are high impact (directed numbers, fractions, equations, manipulation, mental maths, calculation) or easily confused (e.g. minimally different topics such as perimeter and area)
- Has carefully and thoughtfully sequenced content in the big picture (e.g. equations preceding graphs) and in the fine detail (e.g. breaking down directed numbers into the many strands of understanding and procedure that pupils need to grasp).
- Has identified key examples that a teacher might want to use with a class, covering the most important problem-types for a concept or procedure.
- Offers clear and highly accurate explanations of WHY something works.
- Has distilled clear steps to scaffold pupils’ work as they begin to tackle a new procedure.
- Offers memory devices to help pupils retain and recall concepts or steps (Chants for the 7 times tables, or mnemonics such a KFC for dividing fractions (Keep the first, Flip the other, Change to times, it’s no bother).
- Offers LOTS of practise at each level of difficulty in a procedure.
- Has lots of interleaving available, but sectioned off, so that the teacher can judge the level of complexity students should experience.
None of this replaces planning lessons. You still want to share enthusiasm, build excitement, anticipate common errors and misconceptions, explain clearly, model explicitly and unambiguously, check for understanding, grow their confidence in the face of setbacks, celebrate success, maintain pace and focus in a safe and happy environment and – of course – go back and refine the plan and resource after you’ve taught it. This all takes planning, deep thought about your classes and huge love of maths. I don’t understand how the existence of such a resource would compromise the idea that teachers tailor their teaching to their classes.
Sadly, such a resource doesn’t appear to exist. That’s why we’re making a textbook. Please get in touch, have a look, and help up improve it!