Struggling with the Curriculum

In CPD we have been thinking about how to build subject curricula – when should they be individualised and when should all subjects follow same structure. Here are some disorganised mullings, to come back to in the future.

 

Transcendental Concepts

The question: If you were to  distil Mathematics into a handful of concepts – never-ending ideas that Mathematics is about, what would they be? Things that 4 year olds and researchers both grapple with.

Why? To use as the spine of a curriculum, from 4 to 18.

 Questions about the question: Why only within Maths? Why not in the sciences or across all academic pursuits? Why is it necessarily true that  all mathematicians of all ages and flavours are struggling with the same core concepts? If it isn’t true, is it still a helpful approximation? (Further Maths has no direct relation to everyday citizenship. Yr1 Maths definitely does).

 Currently, secondary Maths curricula are split into strands – Number, Geometry, Algebra, Data. This fizzles out at A level.

 A first draft:

  1. How many numbers are there?
  2. When are models useful?
  3. Does Maths change?
  4. What is the best way to represent relationships?

Why is this different to the strands? The fact that they are questions emphasise curiosity, rather than learning from the canon. There is a greater emphasis on process rather than content. Something that could come out of this that would be genuinely new would be a deep connection between maths and history/philosophy (this is the Hinterland of maths). Currently students are not aware that once upon a time, the concept of “-5” never existed. Understanding the history of maths encourages an understanding of the progression of mathematical concepts.

A draft of how you might split up school (elementary, primary, middle, secondary, a level) by these questions:

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Key dilemma 1: this requires significant reworking. Primary buy in to external curriculum, maths mastery, focussing on solid ability on key skills. Why would they want to change if they are happy with current practice? Should they be happy with current practice?

Key dilemma 2: is the purpose of Maths proximal (stepping-stone) or ultimate (worthy in its own right)? Is it different for different ages/students? Should we split into Maths and Numeracy?

Key dilemma 3: the curriculum from 30 years is saying the same things that I am trying to say, but more eloquently. Why are we making no progress on this? (See detailed notes on this at the bottom of post)

 

The portfolio of problem-solving is a key way to achieve our vision of students valuing more than just exam-style questions.

 

(Note, earlier draft by maths department is here)

 

1985 Curriculum

Disappointingly relevant to today’s discussions about Mathematics – surely we should have progressed in 30 years?

These aims are “indispensable but it is recognised that there may be others which teachers will wish to add.” Which are missing from School 21?

Aim Juicy Quote Six21
Communication.  “The main reason for teaching mathematics is its importance in the analysis and communication of information and ideas. The mere manipulation of numerical or algebraic symbols is of secondary importance.” There is a mediumly strong emphasis on interpreting the answer – does it make sense? What does it mean?
Tools “It is not the mathematics itself but the result obtained which is the important thing” Mechanics and Stats are an obvious example. Not enough effort to use Pure Maths as a tool, but I would argue this is not an issue.
Relationships “In school mathematics this fundamental feature of mathematics might not be appreciated by pupils as they become preoccupied with trying to master the details” Active effort to find problems that link material we have covered together, and to bridge gap between pure and applied (for example Fundamental Theorem of Calculus)
Fascination “Much will depend on the enthusiasm of the teachers” Youtube is an outstanding tool for this.
Imagination “Opportunities need to be given to pupils to use their expertise to find their own ways through problems and investigations in which the strategies are not immediately obvious” Prob-squads hits this squarely on the head.
Systematic Work “The aim that pupils should learn to work in a systematic way does not clash with the aim that they should learn to show imagination” We could make this more of a priority. Dedicating time to writing up solutions, or modelling good habits (either teacher-led or sharing excellent student work)
Independent work “Pupils will have developed well mathematically when they are asking and answering their own questions” Question generation is a specific focus of prob-squads. Another meaning of independent work – the ability to overcome obstacles without outside help and to adapt methods appropriately, could be further improved.
Collaborative work “Working together on a common task where all the pupils in the group make a contribution” Whiteboards are an excellent visual reminder to students and teachers to work collaboratively. A simple concrete way to improve teaching is to put whiteboards in every maths classroom in the country.
Depth “Because of the commonly held view that ‘many pupils cannot concentrate for any length of time’, many textbooks are planned to provide a rapidly changing experience” Each half-term has an overarching theme – for example 6 weeks of integration. We are able to spend time on problems, on consolidation, on links to previous work. Students haven’t shown evidence of boredom or restlessness.
Confidence “At present many teachers feel that they are under pressure … to push pupils to cover mathematical content for which they are not ready” We as teachers are not thinking about this sufficiently. Sixth form classes are mixed ability.
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Two Podcasts…

Thanks Craig!!

Anne Mason and John Watson

  • Watch this video by Polya
  • Anne taught a 100%-coursework GCSE. Wow!
  • The danger of group-tasks is that students self-select to do tasks that they can already do. Nobody climbs up the ladder.
  • There is no such thing as rich tasks. Only rich pedagogy.
  • Two types of exercise: with the grain (easy but without structure) and against the grain (more challenging, reveals structure with careful variation). LInks to John’s two-way tables
  • Some good questions: “What would a mathematician ask?”, “What do you expect me to say next?”
  • Asking “what would you tell yourself when you first started teaching” is pointless – I would have ignored the advice that my future self now believes in.

Andrew Blair (and his amazing website)

  • The novice/expert dichotomy is meaningless. “First master the basics, and then problem-solve” is also meaningless.
  • Focus on student thinking, not teacher actions.
  • A purpose of inquiry maths is for students to learn how to learn. More than just motivation, it is about providing general skills (as universities say, students come to them without any research skills or independence)
  •  The art of teaching: to be on the line between what is known and what is unknown, for every student in the class.

  • When debating with teachers about whether studen-led exploration or teacher-led direct instruction is the best way to learn, you are really debating about much deeper philosophies of education. What is the purpose of education? What is Mathematics? Agree on these first, before thinking about the classroom.

Prob-Squad 2 Write-up

See reflection on lesson here.

Thoughts:

  • Avoid simply narrating what happened in the exploration. Move from description to evaluation.
  • Avoid relying on technology to be your brain (e.g. graph sketching on desmos)
  • Freedom is a difficult responsibility, some groups spent about half the time deciding what they would explore. It is okay to fail at this!

Students becoming very comfortable in highlighting where they don’t fully understand the maths. Excellent!

Plenty of great interplay between algebra, words and diagrams when explaining:

Meta-cognition improving:

What a great conjecture!

A few more pics:

Swans and a Cello (Music and Maths)

Night before school starts again, back to the Barbican for a sumptuous programme: Sibelius 5 (The Swan Hymn), Elgar’s Cello Concerto (an elegy to WW1), and a world premiere by Giguere.

Conducted by Susanna Malkki, the exception that proves the rule of conducting = patriarchy.

Choice quotes from students:

  • “I didn’t know classical music was still being made”
  • “If that violinist wasn’t white he could be my grandfather!”

Thoughts on parallels between Music and Maths? (Danger area – the analogies are often really forced and unhelpful). Focus here on processes, rather than things like “You can use your knowledge of lowest common multiple to create poly-rhythms”

  • Both are being invented by professionals all the time. (Symphonies aren’t only composed by dead white men. Maths is not something that has always existed).
  • You can appreciate the genius of pre-existing discoveries (by playing great music, or by understanding for yourself great proofs)
  • Improvisation – riffing on a simple intial idea (short phrase in jazz, or geometric prompt in a prob-squad) without really knowing or needing to know where you are going
  • Interplay between practice of skills and seeking of art. See Lockhart’s Lament for more on this

Check out the Proof and Composition collaboration! Deep link or gimmick?