Lofty Lipservice – Achieving Visions through Pythagoras.

This is the third chapter in Lofty Lipservice.

  1. First I focussed on the issues I have been struggling with
  2. Then the department spoke about our visions more broadly
  3. Now I will try and put one vision into practice.

My vision is that students direct and own their learning. The challenge is to plan a lesson that achieves this, while also teaching Pythagoras’ Theorem (all maths teachers are designing a Pythag lesson based on varying visions).

I chose this because, although my lessons are well-sequenced and carefully pitched at student level, the only choice normally available to a student is to skip a few questions if they find too easy. The lack of choice leads to lethargy and questions of why bother, in double lessons.

RF excellently reminded us of the difference between planning (thinking deeply about probing questions and how to ensure learning happens) and resourcing (finding worksheets, aligning your font, thinking about background colours). To that end, I will provide no slides – this blog is the lesson plan (Dan Meyer’s blog subtitle – “Less Helpful” –  starts to make sense)

Essential Question: What is the nicest triangle you can build?

Rationale: Pythagoras’ Theorem allows us to quickly find triangles where all sides have integer length and one angle is a right-angle (using Pythagorean triples). We are flipping the theorem – normally you think that if the triangle is right-angled then the theorem applies.

  • I wanted to be deliberately vague about what a nice triangle is. Maybe a student will decide that nice triangles have 1 (or 3) lines of symmetry. Or they will focus on integer angles rather than integer sides. Or they will search for a triangle with integer sides and angles. Student choice in interpreting the question.
  • I want to allow the students to use a variety of methods for answering their question. Some might use algebra to create an infinite number of Pythagorean triples. Some might use a compass to build a triangle if they have already chosen the side-lengths (and I would actively encourage this to link to constructions and motivate why we use them). Some might use an app to instantly create triangles given three bits of data. Some might decide they need a quick way of calculating the area of the triangle, and could be given Heron’s formula. I do not mind if their choice of question and choice of answer do not require Pythagoras – as long as some students need to use it then we can share our findings with each other. Student choice in answering the question.
  • There should be space for students to present and share their ideas. A video? A talk? A written-up paper? Student choice in presenting their answer.
  • Edit: MG reminds me that I should think more carefully about how to encourage productive student-definitions of the word “nice”. It is possible to provide them structures (peer-critiques, timelines) without foisting upon them my ideas of what a nice triangle is. 

Other possible routes of student-led enquiry using Pythagoras’ Theorem:

  • There are an infinite number of integer solutions to one equation, but no integer solutions to any one of an infinite number of equations. Can you convince yourself this is true? Why is this not a counterexample below? download
  • If I know two sides of a right-angled triangle, can I always find the third? Can you show me by building it explicitly?
  • How do I know that Pythagoras’ Theorem is true for any right-angled triangle? The bounty of delicious proofs that exist is excellent fodder for a sequence of lessons where each group is understanding and explaining a different proof.
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    Many options.
  • How far away is the horizon? I worked this out when walking along the Pembrokeshire coast-path, and Don has also thought about it.
  • Nrich’s excellent tilted squares problem
  • How many rational points (where both co-ordinates are rational) are there on the unit circle? (One of Nick’s undergraduate students asked this question – great application of Pythagorean triples)
  • How many right-angled triangles are there with hypotenuse 3125?
  • What about quadrilaterals? If I know three side lengths of a quadrilateral, how much information do I need about the angles before I can tell you the fourth length?

Side-note. Pythagoras’ Theorem is usually drawn with squares attached to each length. It is important to notice that any shape will do.

right_triangle_similar

 

P.S. Is it possible to have student-owned learning without the tasks becoming ridiculously open, or the teacher planning 24 different pathways in the lesson?

P.P.S Last year I started each class by asking the students to fill out a questionnaire asking about their interests outside of Maths. I learnt such fascinating facts as “M has never vomited, and her favourite film is Paranormal Activity” and “A has loads of pets and is proud of graduating into Yr9”. Have I ever used this knowledge when teaching? Do I want to base a lesson around Fast and Furious?

Next year, I will design a new questionnaire, one based on views of Maths. Why do you think you study Maths? Do you prefer proof or exploration? Open-ended tasks or structured worksheets? Links to other subjects or for its own sake? This information will be surely more useful.

 

Lofty Lipservice: Maths as a Vehicle

Based on my previous post where I struggled to reconcile my beliefs about Maths with the things that happen day to day in schools, the department spent a morning fleshing out our thoughts

  1. Read article in silence, and reflect independently
  2. Harkness Debate. Essential Question: “What is the purpose of Maths at School 21, and how does it inform our pedagogy?” Some really outstanding insights, I learnt a lot.
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    Record of conversation.

    A quick summary:

  • Debs: We all feel this contradiction between the system and our values. It is useful to attack this head-on.
  • Karenann: Any conversation about making maths optional, or splitting maths into beauty and numeracy, should be treated very carefully. If we allow students to drop maths early we are running the risk of closing doors and widening social inequalities.
  • Rosie: “Maths is a vehicle for deeper skills”. It is not the only subject in which you can learn deeper skills. It is merely something that we happen to be passionate about. As long as the teacher is passionate, then the students will be too.
  • Alberto: “It is likely that learning Maths massively helped me in later life. Even if I cannot pinpoint how or why, I do not want to deprive my children of this chance”
  • Karenann: “There is a distinction between content and process. We as teachers should ensure content is covered, but students should see lessons as process.”  Students should not enter the maths classroom thinking that they will learn how to add fractions. They should focus on broader skills, such as logical thinking, spotting patterns, explaining reasoning, collaborating, constructively challenging. Teachers should ensure that the necessary content is covered, but the students do not necessarily need to be aware of this.
  • Maths is difficult. This should be celebrated, as a way of making brains bigger. Not apologised for.

3. Brainstorm possible beliefs, and the resulting actions.

4. Sort these in a two-way table. Which beliefs do I believe in but never put into practice? Which beliefs do I disagree with but accidentally (or otherwise) put into practice?

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My highlights:

  • Lessons should be student-led. Students should decide the broad direction of study, and should focus on skills such as collaboration, spark, pattern spotting. It is the teacher’s creative task to weave in the necessary content through this. Maths is the vehicle, not the driver. (I do not yet do this)
  • Maths is not a unique subject. Student should not be taken out of other subjects for Maths Intervention – our curriculum should be good enough for all.
  • I think that maths should be hard, but often plan to remove as many obstacles for the students as possible. Re-find this balance!
  • I think that students should never work through a drill-and-kill worksheet, but set them often (either for behaviour management or because I haven’t had time to plan properly). How to resolve this?

Thank you to everyone for thinking so deeply and effectively on these important issues!

 

Next steps: plan a lesson on Pythagoras that encapsulates a belief that I hold but rarely use.

What if I had 12 fingers?

A Yr4 teacher is currently finding it difficult to stretch the most mathematically able students in her class – how to provide depth of understanding rather than just front-loading them with content they will learn anyway in secondary school.

I spent an afternoon with four students, trying to put into practice the vision that you can choose any topic and make it as difficult as you like. We thought about place value.

Warm-up:

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Some interesting misconceptions being born here, that crop up later in secondary school:

  • A must be equal to 1, because it is the first letter in the alphabet.
  • The value of A must be the same in question 1 and question 2

Some absolutely outstanding vocalisation of thought process here:

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Focus.

The Task: Counting with different numbers of fingers.

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We used this table to try and write the same number in different bases. This forced us to think really hard about place value, and the patterns that are common across different number systems.

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Vertical working to encourage collaboration
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Some outstanding thinking shown here

Reflections

  • We should have used physical representations initially (the classic secondary maths teacher’s mistake…). Here are 15 blocks. I can either group them into one 10 and five 1s, or into one 8 and 7 ones… (Thanks Margherita for this observation!)
  • In the next session we might look at Happy Numbers, explore this calculator that converts between bases, explore Kaprekar’s Constant in different bases, the 1089 trick in different bases…
  • Margherita was struck by the attitude of the students – the joy at being given a difficult problem. “This is HARD!” they crowed delightedly. Contrast to the “This is HARD!” some of the secondary students moan lethargically. How can we cultivate this love of challenge, and ensure that it doesn’t get lost as students become older?
  • There was one student in particular that Lisa (the Yr4 teacher) found amazingly strong at Maths. We did not know who he/she was, and would have been unable to pick him out from the group at the end of the session. Why does the class teacher’s assessment of ability not match ours? Our session concentrated on doing simple things in unusual settings and being able to talk through the process – maybe he is doing different problems in class?

I would love to continue working with Primary Students, and cannot wait for the Middle School next year.

Blogs: Why, How, What?

Why?

  • To work out what you want to say, and say it well. (Jeffrey)
  • To curate professional portfolio of your practice
  • To help formalise being a reflective practitioner
  • To share expertise and ideas with other people
  • To inform your performance management
  • For your CV

How? 

In a 45 minute session, focus on deep thinking rather than presentation/organisation. If you are not confident with writing an actual blog post then either:

  • Ask someone for help (or look at the tips we gave last time)
  • Write your post as an email to yourself, in a google doc, in a note on your ipad. Worry about building the actual blog another time – the thinking is the important thing.

We will share with a partner our thoughts, at the end of the session.

What?

Stuck for something to write about?

  • Put the flesh on Jeffrey’s story-skeleton –  “Basically, you’ll reach a point in teaching when…”, transformative anectode, “Have you ever… ?”, “Picture the scene: You… “,  convert abstract idea to tangible metaphor. Humans are addicted to stories. Patronise your reader. “So the next time you find yourself…”
  • Reflect on your favourite lesson of the last few weeks – why was it a success?
  • Reflect on your performance management targets for the year – how are you achieving them?
  • Share your snappy findings from this term’s CPD modules?

Examples of blogs 

Rekindling Beauty

After a few weeks of mocks and many lessons of focussed revision, it was high time in my Yr11 class for a double lesson of open-ended problem solving. Enough time silently in rows, get rowdy at the whiteboards please.

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Problem-solving helps you become a better mathematician AND pass exams

 We struggled with an excellent problem from Underground Maths, designed as a transition between GCSE and A level mathematics thinking.

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Stage 1
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Stage 2

What went well?

  • So much surprising maths covered here. The difference between proving an identity and solving an equation. The equation of a circle. The meaning of a variable. The properties of a quadrilateral. How and when to shift from using a compass to using algebraic variables. (The temptation to correct methods that I hadn’t thought of had to be quashed, to explore surprising links with other topics.)

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    Lots of excellent thinking here
  • Outstanding buzz in the room – stealing from other whiteboards, keen to solve the problem, naturally on-task. See video
  • The shift from constructing a circle given a triangle (very tangible) to finding a triangle given a circle (quite abstract) was effective – well done Underground Maths team!
  • I was surprised by who made the best insights into the problems. Not the same as those smashing out the top marks in the exams…
  • Extensions into special types of Pythagorean triples and beyond into Fermat’s Last Theorem (Numberphile video)

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Example of student work (one of the weaker students in the class, or so I thought…)

 Some questions

  • Students flagging towards the end of the afternoon – how do I keep up the pace without providing too much structure?
  • This process took 100 minutes. How to teach students to do a miniature version of this when in an exam, and without any collaboration?
  • How could we have used technology better? A useful visualisation here, which we did not make much use of

Apples and Kings

Four lessons in the morning in Stratford, and then into central London for two conferences.

Apple Teacher

In a luxury hotel just off Trafalgar Square, we were served posh cakes and snazzy presentations from a slick Apple Team. At times the focus on design was almost laughable – we spent forty minutes learning how to remove the background from an image. All style, no substance.


“Learning how to tell compelling stories with data is a skill that we all need in the age of fake news” we were told. This seems questionable. Surely a more important skill is to critically evaluate data, to work out if it can be trusted? Twisting messy data into neat stories is a problem, not a solution.

Some things to take home:

  • “Browse each other’s learning”. Explicitly encourage students to observe other students, learn from each other. This could be through whiteboards, walking around the room, or ensuring that all work is publicly viewable online.
  • The ipad is a tool for getting out of the classroom. You can explore the universe by app, facetime an expert on the other side of the world, ask a large group of people a question on Twitter… (Is there a danger that what was once a field trip now becomes an exercise on Google Earth?)

Walk over Westminster Bridge in blazing sunshine to Lambeth, to the King’s Maths School, wedged tightly in between blocks of flats. A different world. For…

Further Maths Forum

A workshop, led by Michael Davies (head of Westminster Maths Dept for 30 years and writer of STEP questions). Great contrast to previous conference – chalk and blackboard, with occasional computer graphing. Clear delivery of talk, but no question of no substance here.

We grappled with Taylor and Maclaurin Series.

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This function’s Maclaurin series has radius of convergence 1, unless a is less than 1 (because of singularities in the complex plane)
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Every derivative of this function at 0 is 0, so its Maclaurin series is the constant function y = 0. This is an example of a non-analytic function (where analytic unhelpfully means its taylor series converges to the function)

Pedagogical points about this Maths:

  • It is surprising that complicated functions can be approximated by polynomials. No reason why this should be true. Why should the behaviour at x = 0 tell you the behaviour for any other value of x? If I know exactly how you behave today, can I predict what you will do for the rest of your life?
  • Each time I make a better approximation by adding more terms in polynomial, the earlier terms do not change. Again, not obvious. This makes approximation by this iterative process practical.
  • An identity is when coefficients of each power of x is the same on each side. 1 + x + x^2 + …. = 1/1-x is not an identity, it is just pointwise true when you evaluate at any valid point of x.
  • 2+ 3 + 5 is just the sum of the numbers in the set (2,3,5). In an infinite sum you are not doing this – consider the alternating harmonic series as counterexample. Infinite sums are properly weird
  • If gradient and curvature are both 0, not necessarily a point of inflection. (What if the third derivative is 0 too and locally the graph looks like a quartic?)

 

General points:

  • Think carefully about the example sheets you give students. Constantly interweave previous learning, and make the questions “desirably difficult” – challenging enough for deep thought. Too often we focus on quick wins.
  • For A level you need good technical skills (to be able to do a page of algebraic manipulation, for example). Start this early – don’t allow any coasting through the beginning of the course and relish wading through the working.

Fun with Further Maths (these are the problems we worked on)

I asked Michael for any tips to ease the transition to difficult A Level Mathematics for students who struggled with GCSE. “I don’t have that problem, all my students got an A* and are very strong”. Understandable, but frustratingly unhelpful.

Underground Maths

An engaging day thinking about rich A level resources, with Underground Maths. The website is crammed full of rich tasks with plenty of teacher support. A useful reference.

We did this Starter activity. A many-ways problem – either very algebraic or very graphical.

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My pre-work. I enjoyed putting into practice the techniques I ask my students to use.

A problem that doesn’t look hard, but actually is challenging. How does this affect motivation? Would I prefer a problem that looks really scary but actually is pretty simple?

This would be a great problem to throw at students before they know how to differentiate – if this is the headache then what is the aspirin?

Mathematical Mindsets

We discussed things that we would like our students to think about mathematics, using these cards. Some key thing that I want my students to think:

  • It is good to get stuck. (Teachers need to model this). Also, it is okay to get stuck-in!
  • There are no right answers in maths.
  • Problem-solving is the most important thing in Maths.
  • Good problem-solvers do well in exams
  • Understanding > memory (within reason)
  • If you pass an exam using death by practice paper then you might get a good grade but you will not do well in life. You are merely delaying your failure.

Get stuck-in. Get stuck.

Timing a problem

There seem to be three times to use problems:

  1. Before you know the techniques. Motivation for learning techniques
  2. Just after you have learnt the technique. Application of fresh knowledge
  3. A long time after you have learnt the technique. To encourage connections, recall, re-deriving knowledge.

From GCSE to A level

There is an excellent transition problems bundle, using GCSE knowledge but requiring deeper reasoning.

  • Between the lines (why is there an actual answer for this!)
  • A tangent is
  • A perfect fit (with an excellent extention question: how many pythagorean triples make a triangle with inscribed circle of radius 6? We found 6 integer solutions. What about a circle of radius 7? We found 3 integer solutions. What about a circle of radius n? Lovely opportunity to use excel or python)
  • Name that graph
  • Quadratic solving sorter

You cannot do this task without knowing what an asymptote is. This video clip shows students struggling to use the equation of a circle to find an asymptote (thus showing a deep misconception), without any teacher intervention. When do we leave students to struggle and when do we step in? Does the teacher need to know about every mistake students make?

 

What is 0 the power 0? You need to know the answer to properly understand this problem. Outstanding problem generation.

 

We believe that there should not be a right method or a right answer. This problem fails this criteria.

 

Modelling

MathConf9

To a school in Bristol to share and learn with plenty of nerdy Maths teachers. Who knew – not all teachers are under 35! School 21 is a bit of a bubble…

We did speed-dating with people around us, to share ideas and make friends. Here is my favourite:

  • Type a random 3 digit number into your calculator
  • Divide it by 7. Hands up who got a whole number?
  • Divide it by 11. Hands up who got a whole number?
  • Divide it by 13. Hands up who got a whole number?
  • Now, get you 3 digit number and type it in twice (for example becomes 456456). Divide this by 7, then 11, then 13. Hands up who got a whole number?! What number did you get?!

7 x 11 x 13 is 1,001. How could you extend this problem? 101 is prime, 10,001 factorises to 73 x 137, 100,001 factorises to 11 x 9091. Hmm.

 

Talk 1: From Abacus to Zero

By Ed Southall (who shares excellent problems on twitter)

 

We explored trivia about the etymology of mathematical language. Fun, and sometimes useful to hang meaning on, but not much maths. Here are some titbits:

  • Calculus derives from a word for pebbles (maths used to be done by counting pebbles). You can have calculus on your teeth – a term used in dentistry to describe tiny pebbles of plaque.
  • Which words connected to one?
    • Atone
    • Reunion
    • Alone
    • Only
    • None
    • Condone
    • Onion
      • All, apart from condone. Onion means “the big one”. Pearls used to be called onions.
  • If only eleven and twelve were oneteen and twoteen are rules for naming numbers would be almost sane. We are lucky – in Danish the word for 54 translates as “two-and-a-half-of-twenty-and-four”
  • Factor and factory have common root – they both build products (two puns in that sentence).
  • Linear and lingerie have common root – they both derive from the word for thread.
  • Average is an abbreviation of “averie damage”.  When a boat gets broken, how do you work out how much each investor has to pay? You share the total equally between the number of investors, obviously.

Enjoyable, but unsure of how I will use this to motivate students who drag their feet when entering the maths classroom.

 

Talk 2: Two A-Level Topics

Geogebra links:

 

Two friends who did their PHDs together are now writing a textbook for A level, chock-full of proof and links to applications, both at university and in industry. We explored geogebra apps to visualise differentiation from first principles and Newton-Raphson Method for solving messy equations (most equations are horribly messy).

 

Why do we care about Newton-Raphson method? Because it is used in Formula 1 car design, in Google searches, in weather forecasting… This argument left me dissatisfied – either show me explicitly how it is used (although this is presumably horribly complicated) or I won’t believe you.

 

One thing that got me excited was thinking about when the Newton Raphson method would fail (for example if you get stuck in a loop) – for me that is more mathematically profound than tenuous links to the weather.

 

Talk 3: Algebra Tiles!

Mark McCourt, Chief Excec of La Salle Education, ran an excellent session on the use of algebra tiles, from early primary right up to A level. Key thesis: we jump straight to abstract representations of mathematical concepts far too quickly in a topic, at far too young an age.

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Don’t just focus on abstract.

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Claim: they are all the same question.

Bus-stop notation tells you exactly what you are doing. Given the area of a rectangle and its height, what is its length?

Bus-stop division is exactly as hard as dividing algebraic polynomials – you just shift from base 10 to base x. If you understand this deeply enough using algebra tiles then all will be well.

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Bus-stop division with Dienes blocks generalises to algebraic division
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The red side of a tile represent a negative number. To get rid of a positive number you have to add its negative version to both sides.

 

Lofty Lipservice – Are we lying to ourselves?

Edit: @SolveMyMaths has written an excellent blog on this

Edit: a talk by Conrad Wolfram – thanks to RF for sharing

Edit: Vsauce’s compelling argument that exploring Maths is a function of our curiosity – the thing that sets us apart from Neandertals. See video, 17.45 in, or start earlier to learn about the Ross-Littlewood Paradox.

Noble Thoughts

In a maths meeting this week we spoke about why people should learn maths, and what key things we should ensure are covered in the lessons. Things that came up were:

  • To teach core wellbeing principles, such as how to fail and how to work with others (present throughout all schools)
  • An appreciation of truth and beauty (maybe in sixth form)
  • Learning to problem-solve generally, for use in 21st century jobs (maybe in the middle school)
  • Understanding links between maths and other subjects, between maths and things in the real world (maybe in secondary school)

 

Different teachers placed emphasis on these differently, but broadly we agree that these four are important. We think noble thoughts, but we do do noble deeds?

Noble Deeds?

In a typical lesson, students will be given carefully and excellently planned tasks, controlled by the teacher, to progress through a topic (To clarify, I wish I could plan lessons as deeply thought of as this one, here is one by me that I am less pleased with). The focus will be on doing a rule, and possibly applying it to worded problems. Assessments to check understanding are skills-based, with minimal problem-solving (at least in my lessons, despite my best efforts). The majority of projects link maths to art, sometimes tenuously (for example I am now running a project where the students are using their geometry to design a gate).

An example. This week in Yr7 we have been adding fractions (very thoughtful planning by Alberto). I have not had to add fractions since I was at school myself. If I needed to, I would probably use a calculator anyway. Why do we teach adding fractions? (Edit: there might be better examples of topics that are hard to justify. For example, polynomial equations, volume of cone, negative indices…)

  1. Because it is on the exam. We are meant to be re-inventing education, this is simply not good enough.
  2. Because it might be useful to some students in their future careers. So is gardening, or coding, but we don’t make everyone learn those.
  3. Because it is a building block of more difficult maths. It might be a building block to more difficult, but even more irrelevant maths (trigonometry, probability of picking two red balls from a bag…). 
  4. Because it offers a glimpse into the structure of maths. Not the way we teach it (here is a method, now practice it).

I currently cannot give a satisfactory answer for why students should be able to add fractions (please tell me I am wrong!). Differentiating between real and fake news, understanding how an infographic might be persuading you to think a certain thing – these are far more obviously useful.

Learning how to stay healthy (by 2030 a third of adults in the UK will be obese) is surely more important than finding the length of the hypotenuse. Currently we cannot teach children either.

Conclusion

Our teaching practice is at a mismatch to our vision. Either we change our vision, or we change our practice. If we are to change our practice, we should do it one step at a time, making each other accountable for small achievable differences. (This is not yet a satisfactory conclusion).

 

 

Postscript

By the way, there are plenty of other models out there for what every lesson should include. We should walk before we can run (by implementing our vision), but for interest they are:

 

Don Steward says every lesson should include a glimpse of infinity – a chance to generalise beyond specific examples.

Mathwithbaddrawings says every lesson should involve the following process:

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Rosie and I sketched another process:

  1. Do the rules
  2. Apply the rules
  3. Understand the rules
  4. Break the rules

 

Dan Meyer says that the headache (a difficult and novel problem) should be presented first, before giving the aspirin (a new mathematical technique).

The Maths Diary

Every student in my Yr11 class is making a Maths Diary. In it they will summarise everything they need to know about Mathematics, to help them become better Mathematicians, and to strive to get an excellent grade in their summer exams.

Crudely – it worked for me, so hopefully it will work for them (there are obvious dangers in this approach).

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Notes on 3-dimensional manifolds from my masters. Added notes in margins, colour, quick reminders. Clear but not precious.
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On the inside of each book…

Why summarising?

The act of condensing notes requires the mathematician to identify the essential from ther superfluous.

Why a book?

Cue cards and folders tend to lead to disorganised piles of notes. A book is a self-contained treasure-trove of information.

Some observations

List of every algebraic skill the students need to be able to do. Student plans which topics to focus on and which to ignore, before diving in. Topic list is not high quality (stole from the web)
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Time devoted in lessons and homework to giving each other feedback
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Excellent awareness of what makes a question difficult.
The Drafting Process
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Beautiful presentation, but is there much here beyond facts? Is that a problem?
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Great tips, but hard to scan. Does that matter?

The pros

  • For students who enjoy neat presentation of work, this is a comforting dream.
  • Every mini-test or lesson that students do now ends with a few minutes transferring key ideas in quick note-form back into the diary (key notes jotted in back of book for reference). This is the central store of all knowledge.

The cons

  • What about the student who finds it hard to formally explain their understanding and might benefit more from reams of practice? Is the art of summarising important enough to warrant a short-term loss in their performance?
  • What if the book is lost? (We have already mourned the loss of one diary)
  • It takes a lot of lesson time to maintain and keep momentum