A cohort leaves

Yr11 students left school today for the final time, after their last exam. I recevied a few lovely cards.

A reminder to myself that students appreciate the work the teacher puts in, even if it is not evident at the time. A reminder to stay true to my principles – convincing students that Maths is beautiful and constantly remaining positive.

To Trig or not to Trig?

Mixed ability class. I made a call about who should continue to study Pythagoras, and who should move on to learning about Trigonometry.

In the unit assessment, one glorious student (who I had decided should stay on Pythagoras), flew through the Trig questions, doing better than a lot of the students who had learnt it in class. He had gone home and asked his cousin/youtube to teach him.

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Outstanding example of the dangers of overly rigid differentiation. What should I have done differently?

Mike Ollerton’s Problems

RF, DSE and AG went to a session on a Saturday morninng by Mike Ollerton, to get stuck into some problems. Here is one, that DSE and AG presented back to the team.

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Answers handed out promptly – ensures that you could discreetly check your own, and importantly, that everyone now has the same labels for each triangle
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Linking to collecting like terms
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Order through intuitive reasoning – no need to get bogged down in surds. Good application of telescoping sums in the extension.
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A level extensions

To solve the question on the left…

  • Alberto dived into trigonometry, using double angle formula for tangent.IMG_6082.JPG
  • I dived into analytic geometry (after first “cheating” and finding the answer on geogebra), working out the equation of each line, the intersection of the lines, and then using the shoelace formula to work out the area

There surely must be a simpler way to work out the area, but nobody could find it yet.

 

An excellently stretchy task – plenty of further questions:

  • How many triangles in a 4 by 4 grid?
  • What about an m by n grid?
  • Explore areas? Angles?

Sphinx: other example of stretchy problem

Sticks and Shadows

Paul Lockhart’s excellent book Measurement (a constructive follow-up to his infamous diatribe about the start of maths education, A Mathematician’s Lament), includes this gem:

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I adapted for mixed-ability Yr9 class. Some students were thinking deeply about the shape of the curve, others were practicing measuring angles and lengths and plotting coordinates.

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From formal to concrete
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Multiple ways to measure the shadow…

 

Geogebra app proved the most popular, since the least brainpower required to use it. Maybe only give this to the students once they have struggled with a physical metre-ruler or the unit circle first?

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Lovely conversations about how a shadow can have a negative length, and why we would want to think of the metre-ruler sticking straight into the ground and still casting a shadow.

Some students found four squares representing 30 degrees a challenge. Which graph is correct?

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Spot the difference.
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Student explanation

‘Tis but a Scratch

Inspired by this Monty Python sketch, I asked students to find the area of a triangle, if they know the lengths of the three sides. Gradually I chopped off a mathematical limb, removing ipads, calculators and compasses, before only a brain and a pencil remained.

Some great and unexpected learning:

  • The compass was too small to draw an arc of 13cm. So we enlarged the shape by a factor of 0.5, and thought hard about how that would affect area
  • We thought about how to accurately drop a perpendicular from a point to a line, when measuring the perpendicular height of a triangle
  • We wanted to work out the square root of 21 x 8 x 7 x 6, but without using a calculator. Prime factors to the rescue!

I had given the class 5 minutes to try and find a way of calculating the area of the triangle, using only a pencil and a brain. Some wildly inventive methods, that involved inventing an adapted version of Pythagoras got students close to the correct area. Great creativity, but lacking in precision.

The start of understanding the proof:

Reflections:

  •  If a student knows every constituent skill then they can understand the proof. Whole is greater than sum of the parts – the multiple steps are difficult to keep in your working memory
  • Students have well-trained instinct to expand brackets. Sometimes this isn’t helpful!
  • One student spent a long time trying to prove that s = 1/2(a + b + c). The distinction between something you can define and something you have to prove.
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4 students collaborated on this. At times inefficient, but some great conversations.
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Lovely annotations.
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Outstanding clarity and thought

Lofty Lipservice: Reaching for Student Choice

This is the fourth 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. I planned a lesson, to put my vision of student-choice into practice
  4. Now I will reflect

Last term I developed a lesson on Pythagoras’ Theorem, focussing on providing students with the choice of which question to answer, and how to answer it. I have been working, across Yrs 7,8 and 9, to give more productive freedom to students.

  • In Year 7, freedom in ways of working
  • In Year 8, freedom in interacting with objects in the classroom
  • In Year 9, freedom in developing an individual question

In a (dream) mixed-ability class, students should be working on self-generated questions, in their own way. The teacher should be ensuring the questions are rigorous, the groups productive, and the methods mathematically useful.

Year 7: Ways of working

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The challenges, from Don
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Suggested scaffolding
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Great teamwork from students who would normally be more distracted

The class had a healthy competitive atmosphere. Students were practising substitution, trial and improvement, negative numbers, collaboration.

Year 8: Interacting with objects

Finding the curvy length around a circle is difficult. We scoured the classroom for circles, measured their diameter and curvy length, and tried to spot patterns. Students enjoyed thinking creatively, searching for doorknobs, clocks, watches, stools, buttons on their iPads… Useful to emphasise that Pi is, by definition, the ratio between circumference and diameter.

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Recording ratios, sum and products…

 

 

Year 9: Developing individual questions

I told the students that a nice triangle, for me, is one with integer side lengths and one 90 degree angle. With-holding Pythagoras’ Theorem, I asked them to search for as many “nice” triangles as possible.

Next, students developed their own definition of nice.

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An attempt to ensure that “nice” has a productive definition

Some student definitions include:

  • Whole-number area
  • Whole-number perimeter (even if individual side-lengths are not whole numbers…)
  • “Any normal simple triangle” (the result of too much freedom is ridiculous definitions)
  • One side is a multiple of the others
  • “Area” (some students find it difficult to ask well-posed questions)
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Excellent self-differentiation
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This student worked far harder than usual – perhaps because he owns it?
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A productive definition

Further questions

  • When everyone is answering a different question it is difficult to provide meaningful targets for quantity (and quality?) of work. A student with a very restricted definition might not find as many triangles as someone who is only searching for triangles with one line of symmetry.
  • I have a fine-tuned notion of what makes a good question. How to train students in this more effectively?
  • Searching for circles is tangible. What about topics that do not lend themselves to physical objects like this?

How many right-angles?

After a week of exploring angle sums in polygons, both through interior and exterior angles, we attacked an excellent problem by Don:

I have a shape with 100 sides. What is the maximum number of right-angles the shape can have?

Fascinating to see different reactions to the problem:

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Go straight for the jugular: try and draw a 100-sided shape
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A similar (but digital) attempt
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Overly keen pattern spotting. (The odds/evens pattern breaks for n = 11,12)

One group came up with a hierarchy for solving the problem, that I would struggle to improve on:

  1. Guess
  2. Draw it out
  3. Understand the pattern and apply it

Students worked in trios:

  • Geogebra master (in charge of using technology to quickly sketch ideas)
  • Pattern spotter (in charge of generalising any results)
  • Organiser (ensuring good communication within the group)

 

Learning Together

Over the last 9 months I have been penpalling with James, a student studying for a Masters in Maths (he loves everything from the history of geometry to application of ODEs to engineering, but specialises in Number Theory), while also a prisoner at HMP Grendon. In order to go to Grendon you have to volunteer. Prisoners have the power to ask fellow inmates to leave, if they become too antisocial. Grendon is specifically a therapy prison (art, physco-drama, poetry, music, gym…).
Today I was lucky enough to get the day off school to go celebrate his graduation from Learning Together. Learning Together is a collaboration between the prison and Cambridge University. Students, from both institutions, meet once a week for 9 weeks to discuss, listen to lectures, write essays. They can study Criminology or Literary Criticism.

Note plenty of time for mingling and reflecting
Here are some observations from the day:

  • The room was so supportive of friends as they took to the stage to share their experiences of learning. Learning as messy, difficult, rewarding. No unkindness ever hinted at, such ,warmth and love from everyone. 
  • I watched as groups of learners, from university and prison, bantered away, completely at ease with their friends. When we used to go in for a day’s singing workshop this level of collaboration was never quite reached.
  • Learning or Togetherness – which is the more important? Learning about the academic definition of Legitimacy or being open to people from seemingly different worlds to yours?
  • Ruth and Amy, co-founders of the scheme, are an incredible team. Great vision, drive, humour. They believe completely in what they are doing, so humbling and great to see! (FOFO – “full on or f*&^ off”)
  •  By the end of the day I was unable to accurately play the game “University or Prison?”. Nor did I want to. Everyone was a learner.
  • Ruth spoke about the close relationship between brilliance and brokenness. In order to reach academic brilliance you must first become aware of and accept the ways in which you are broken, and the ways in which our society is broken.

I finally met James! Former professional wrestler, tapestry-artist, number theorist, beard-nurturer. Quiet, kind, fascinating. In a room full of bustling conversation we, the nerds, sat in a corner and worked through some geometry problems, getting confused about scale factors and applications of Pythagoras. 

Our scribbles – we made friends through a problem…

We spoke about education in prisons. Despite the lip-service, there is only funding in prisons for English and Maths to Level 2 (equivalent of a C at GCSE). If you want to o beyond this, you can study through distance-learning with NEC for A levels, or with the Open University for degrees. Four phone conversations with a tutor, and many lonely nights wading through textbooks. 
46% of prisoners have literacy that is below that expected of an 11 year old (three times the proportion in the general population). For Maths, 52-65% (depending on sources) have numeracy that is below that expected of an 11 year old (shockingly for the general population it is still 49%). 80% of prisoners reject education (I couldn’t find the equivalent stat for the general population, or what this statement really means). 

Learning Togehr: Maths?

James and I would like to set up a scheme, similar to the ones for Criminology/Literary Criticism, but for Maths. Should we focus on numerical competency or instil a deep love of the subject? James spoke passionately about this, taking the words out of my mouth – “Give them the love and they will go away and learn the nuts and bolts as a conesequence”. Prisons provide basic numeracy education, lets give something that only we can provide. (Compare this to the excellent One to One Maths Charity, where prisoners teach each other basic numeracy). One idea would be to organise an intense 1 week summer school, during the 2 week slot in the summer when therapy sessions do not run.
Learning Together is so successful because it brings two groups of people together, who would not normally meet. Who would our second group be, given we were thinking of doing this in the summer holidays? Students about to start their first year of uni? Students at local adult education colleges? Old peoples’ homes? Staff at the prison?
James taught me an excellent phrase – it is “quicker to plait fog” than to use the prison computers. No graphing software to be used here then… He taught me about partition theory and the maths of juggling (originally developed for its own sake, and now with applications to computing).

We spoke of primes…

A levels – opening doors or enlarging brains?

I am thinking of which type of exam to work towards in the new sixth form. In simplistic terms, the one that students probably will do better at, or the one that is more mathematically genuine. Here are some thoughts:

  • Secondary school teachers often are frustrated by the crazy grade inflation they perceive in students Yr6 SATS marks. “No way is ____ a Level 5” we mutter. The accusation is not of cheating, but of coaching to the test. It is unhelpful in secondary school, giving an inaccurate picture of the students and providing unrealistic GCSE targets. If we want accurate marks from primary school, then surely we should give accurate marks from secondary school.
  • Inaccurate grades are not only unhelpful for teachers/universities, they are unhelpful for the student. If a university course requires an A grade, what it really requires is a student capable of A-grade level thinking (if such a thing exists…).
  • I asked my mum (a research scientist) about this. She cannot see any conflict – “Of course you should take the intellectually honest and challenging route”. Set high expectations and students will meet them.
  • I asked MK (an American) about this. In America standardised tests are… standardised. No choice of exam board. Individual teachers/schools don’t have to make this horrible choice, between honesty and what they might perceive as helping their students by opening doors with good grades.

In summary – if a student does not deserve an A grade (whatever that might mean), then playing the system to get them an A grade is delaying failure. Choose the more challenging course, make brains bigger, dream big!

OCR (B) Exams

In conjunction with MEI, OCR have devised a new spec. Features I like:

  • The large data set is comparative information about countries over time. Great for highlighting social inequalities and getting students debating.
  • The third paper includes mathematical comprehension – read a new bit of Maths and then answer questions based on it. Excellent for encouraging mathematical independence.
  • Problem-solving questions abound, with lots of interpretations (that are not hoop-jumping but ensure deep understanding)

 

Edit: KS says “Surely we can teach deep problem-solving in lessons while also using a less mathematically challenging exam board?” A response: our systems should nudge everyone to always keep our mathematical integrity. GCSE interventions often lose sight of the beauty of Maths, because they can. If GCSE exam questions all required deep problem-solving then rote-learning how to rationalise a denominator would become less tempting.

 

Learning from Kings

Karenann and I were excellently hosted by Dan, headteacher at Kings College Maths School. Every student studies Maths, Further Maths and Physics at A level.

  • Every lesson is designed around the principle “Don’t tell the students what to do”. We teach through discussion and questioning. We only break this principle in interventions. Interventions are timetabled in, happen after assessments, and are there to quickly support students.
  • While every lesson will involve problem-solving, we also dedicate a session a week specifically to problem-solving. PHD students come in and help run them, we give far less guidance, and focus on encouraging students to fail, to keep on failing, to tenaciously strive for a solution. No lesson objectives, no rush.
  • Students are assessed by their teachers on core study skills – collaboration, communication, independence and organisation. Interventions (for example go to board-games club to improve collaboration) are put in place.
  • Teachers have a weekly planning meeting, to skill-up those who are new to the Maths, and to ensure teaching is consistent and high-quality.

How do we support the transition from GCSE to A level?

  • It is not true that all our students, even though we are a specialist Maths school, are ready for A level. They might lack study skills (see above), might think of maths as a subject where they can easily find “the right answer”, or might have some subject-gaps.
  • We start with a topic that is new and impressive, but that also will enable the basics (algebraic manipulation) to be covered. Complex numbers works well – requiring expanding brackets, collecting like terms, while also being something that none of the students will have seen before. Recapping completing the square pales in comparison.
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    Deliberately hide snazzy methods (multiplying by complex conjugate) at first to encourage excellent algebraic manipulation
  • Students find mechanics particularly hard. This is due to a difficulty separating intuition about forces from the formal modelling. For example, reaction force is equal and opposite to the weight of an object lying on the floor, but this is nothing to do with Newton’s Third Law of Motion.
  • Early interventions are key

Lesson Observation: Yr13 Mechanics

Essential Question: “When I push a block, will it slide or topple?”

  • 30 minutes of teacher-led exploration of the question. 20 minutes of applying the broad techniques to unfamiliar contexts. The students could have been given the question and nothing else. However, in this lesson the content had to be covered quickly, so more teacher-leading was necessary. Dan guided us through, with constant questioning and time to reflect, talk to each other, and predict. Pacy but still involving students and ensuring we all thought deeply.
  • Repeated links to intuition and the physical example (we all had blocks to play with).  “What do you think it might depend on?” , “Intuitvely, should it depend on mu?”, “Translate this into english please”,  “I feel that…”, “Think about the point when it is just starting to happen”. Conscious effort to hone and improve intuition.
  • The most difficult part about the lesson was the logical structure.
    • If I assume that the block slides, then…
    • If I assume that the block topples, then..
  • Teacher quickly assesses work on whiteboards. Nothing really written down formally, no huge emphasis on taking good notes. Focus is on the group collectively thinking deeply together.

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Counterintuitively, whether the object will slide or topple is not dependent on height of block or mass of block.

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Futher problems
Lesson Observation: Yr12 Matrices.

Lesson focus: to use matrices to solve simultaneous equations.

  • Excellently clear link between prior knowledge and new method. Students unconvinced for the need for matrices to solve equations that they already have a method for. A possible opportunity for technology here: matrices can solve simultaneous equations in three variables, and an app can split out the inverse (useful if you don’t know how to invert big matrices yet).
  • Teacher completes problems on board while students complete on A3 “mini”-whiteboards. Deliberately supportive culture for a class that finds maths hard.
  • Usefully uncovered misconceptions – you can divide by a matrix, and matrices commute.

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Example of student work

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Two geometric interpretations of M= a. How are they linked? I don’t yet know. 
Thank you so much to Dan and Kings Maths School for hosting our visit! We are excited to think how to use some of the exciting things we saw here next year at Six21.