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.

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.

Answers handed out promptly – ensures that you could discreetly check your own, and importantly, that everyone now has the same labels for each triangle

Linking to collecting like terms

Order through intuitive reasoning – no need to get bogged down in surds. Good application of telescoping sums in the extension.

A level extensions

To solve the question on the left…

• Alberto dived into trigonometry, using double angle formula for tangent.
• 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

‘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.

4 students collaborated on this. At times inefficient, but some great conversations.

Lovely annotations.

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

The challenges, from Don

Suggested scaffolding

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.

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.

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)

Excellent self-differentiation

This student worked far harder than usual – perhaps because he owns it?

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?

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? 
• 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.
  

  

  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.

 

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.

   

   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?

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:

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:

Focus.

The Task: Counting with different numbers of fingers.

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.

Vertical working to encourage collaboration

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.

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:

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 Origins of Counting

Rather than our usual Tuesday afternoon departmental meeting, the Maths Team left the school walls and decamped to ritzy Central London, for a meeting about the Origins of Counting, at the Royal Society. As always, the cultural disparities between Newham and the Mall were very apparent, from graffiti on walls to oil paintings of old venerable men on walls. An example of number sense in animals other than humans:

From Number Sense to Number Symbols

A talk by Francesco d’Errico, who claimed he would turn everyone in the audience into an archaeologist by the end of his 40 minutes. Many animals have number sense (hyenas can make comparative statements about the size of an enemy pack and use it to make a decision about whether to attack or retreat), but only humans have number symbols. How did this journey happen?

1. The journey was non-linear. Innovations in symbols happened, were lost, and then re-appeared later on, often in different cultures.
2. Homo Sapiens was not the first species to creat symbols. Neanderthals got there first, 80,000 years ago.
3. The appearance of number symbols was not due to genetic modification (since many animals have comparable number sense to humans, and neanderthals were able to make number symbols). Rather it was due to the plasticity of the brain, and to communication through culture. Symbols have only been developed in the last 5,000 years – nothing in evolutionary timescales.

Progression of number symbols. x-axis is time (larger x-value means further into the past), and blocks show when different techniques were used. Isolated dots on right show how practices were discovered and then forgotten and then discovered again. 

For a long time we have used symbols to externalise our memory. Outsourcing memory to google is not a new phenomenon.

A summary.

The Deep History of Number Words

By Mark Pagel

Excellent essential question

Language years: if a word is spoken in one language for 5 years and in another for 10 years, then that word has been spoken for 15 language years. “Two” (in its various forms) has been spoken for 148,000 language years. Ridiculous.

Evolution of words, like evolution of genes

Decay of words, like decay of isotopes

The more frequently a word is used, the less it changes. Number words change even slower than this regression would suggest.

The slow evolution of number words is not limited to Indo-European languages.

The only words that change slowly across all three language groups are number words. Number words are special!

Nice inforgraphic

Why do number words change so slowly, across languages and time? Possible answers:

• A link is built between the number words and the region of the brain links to numerosity. Numerosity is very stable, and so therefore the words are too
• “Two” is a very clearly defined concept, whereas other words (such as “sofa”) are not.
• If a word is to be used often, it must be short. There are not many short sounds that are still avaible (they are already words!), so there are no candidates for alternatives to number words.

Implications of Innate Numerosity-Processing Mechanism for Education

By Brian Butterworth, heavyweight academic and oganiser of the conference. He coined the term dyscalculia, and presented here an in-depth analysis of the current neuroscientific research into why some people do not have a strong in-built sense of numerosity.

The speed with which 3 year old children can count dots accurately predicts their numeracy skills at age 11

His rallying cry: dyscalculia is at least as important as dyslexia, and should be treated as such in schools.

Cedric Villani and Marcus du Sautoy in discussion

Two famous popularisers of maths end the conference with their ruminations on what we have learnt. Cedric is a great speaker, his hands wildly gesticulating and accurately representing the mathematical processes that his mouth is talking about. Marcus observes that, contrary to belief, mathematicians do not see the world in numbers – instead they are hypersensitive to structure. We should not be scared of showing children the big meaty ideas in maths, providing them with linguistic structure to tackle deep ideas.

Cedric speaks of this cartoon:

Thoughts

• We learnt nothing that could be immediately applied in the classroom as maths teachers. But, lifelong learners do not need to be immediately able to apply the new knowledge.
• We were frustrated by the lack of presentational skills by some of the speakers. Death by powerpoint is never excusable – intellectuals have a responsibility to communicate well.