Learning together about Algorithms

My class and I are learning about algorithms, together. I was nervous because:

  • I don’t know much about algorithms
  • I had always found algorithms a fairly dry subject.

In fact, the process has been a beautiful example of student enthusiasm and knowledge motivating the teacher, of us learning together.

Pre-work:

  • Teach yourself to code
  • Teach yourself to solve a rubik’s cube (Every student can do this, and the craze is slowly spreading across the year-group. I cannot solve it still.)

Intro sessions, to understand the journey from words, to pseudo-code/flowchart, to formal python code.

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“What is the point of pseudo-code?”. Half an hour later: “Ahhh, now I see…”
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How to socialise in the corridor…
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Rubik to Computer
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Rubik to flowchart

Collaboration with Computer Scientists:

  • We have joined up with the Computer Science class. We send them algorithms, they make them sexy and efficient. They send us algorithms, we prove they always work.
  • Example: the above pseudo-code for corridor greetings became this code. A joke death-predictor became this
  • A student who studies both maths and Computing went away and revised his knowledge of matrices to find the inverse of a matrix

We debated, inspired by the moral arguments of Cathy O’Neil in Weapons of Math Destruction. The Wright brothers used to swap sides when they couldn’t agree, to empathise with the other point of view. In that vein, the computer scientists had to argue that “Algorithms do more harm than good”.

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An algorithm for working out which side won
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Creativity with Desmos

Task: collect as many blue dots as possible, as few reds as possible.

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Learning: inequalities in the plane.

Outcomes were very surprising. The assumption was that students would fiddle around with straight lines. Instead, they leapt ahead to bizarre curves, often using trial-and-error, and the power of online graph-sketching.

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What even is this!?
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Some great transformation of functions
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The equation of an ellipse has never been taught…
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Successfully turned the task into something very boring and easy. (Again, the student had only ever seen circles centred at the origin before)
  • Is it okay that these curves would never have been dreamt up if the students didn’t have access to technology?
  • Would there have been more profitable learning if I had restricted the game to straight lines?

 

Big Problems, Motivation

Debs, Dudds and I spent a half-term’s lesson study focussing on the following question:

How does introducing content through big problems affect motivation?

Some initial clarifying points:

  • Dudds took “big problems” to mean real-world applications on a large scale. He instinctively viewed the question through his engineering lens. Debs and I had originally thought of “big problems” as anything mathematically meaty. Neither is superior, but it is interesting to note that language is confusing.
  • We chose to focus on motivation, rather than achievement. The desire to learn is a pre-requisite for sustainable learning. Throughout the study Debs, as class teacher, understandably was concerned about achievement – “But look at their marks on the vector assessment, maybe this means our approach is rubbish?”. This was not relevant for our specific question however.

We planned three lessons.

Lesson 1: What is the point of vectors?

  1. Draw the situation.
  2. Plot on Geogebra, to draw many more, very quickly.
  3. Spot patterns
  4. Attempt to prove, using the tools you currently have.
  5. HEADACHE
  6. Use vectors as the aspirin, to fix the headache.

Resources

Comments

  • This open style of lesson can sometimes be a bit fluffy. We went through each specific idea that the students needed to learn about vectors, and worked out which would need to be taught separately. The lesson might look free and flexible, but we have thought carefully about sequencing content beforehand.
  • Matilda guessed that the ratio of exam:problem-solving in lesson was 20:1. Yusrah thought it was 5:1. Why the discrepancy?
  • Students are nervous that in open problem-solving lessons, they never actually get to an answer. No sense of satisfaction, which lowers motivation.
  • Students quick to teach themselves Geogebra, especially its new browser version.
  • The weaker students went for the harder problem, and floundered. The stronger students went for the more manageable version, and achieved something. Being able to assess the difficulty of tasks is an important skill.

 

Lesson 2: Vectors solve another headache

Resources

Comments

  • Debs enjoyed this lesson a lot more – she felt that the class were beginning to slot into the routine – draw, geogebra, proof. This is despite the fact that the mathematical concepts underlying the proof are much more difficult.
  • Reflection task as starter – excellent calming device after hectic lunch.
  • Slide 7 caused controversy. Was it removing any student independence or creativity (Debs) or providing necessary support for a difficult task (me)?

Lesson 3: What is the point of algebraic proof?

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The planning process was an utter joy.

Resources

Comments

  • Easy wins as students ran to the board to fill in the table of results
  • How to encourage students to write things down, regardless of whether they are correct or not? Too many students only willing to write it down once they have checked it definitely works.
  • Searching tasks reward the resilient, rather than the “talented”. Is this good?
  • “You plus them and then times by two” when pattern-spotting. Informal language as student chatter to each other.
  • It took 25 minute to properly set up the problem, before the students moved on to the main theorems. This is a lot of time, but it was time well spent.
  • Nobody asked “what’s the point?” because they were all inherently excited about the hunt for solutions.
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Student work. Not yet comfortable moving to algebra. Big numbers as alternative. Maybe this is okay.

 

Final conclusions

  • Introducing problem-solving norms takes weeks. Introducing a single problem takes half a lesson. Be patient. Pace is not the only thing.
  • We would like to continue to work together. Collaboration is a joy.
  • If the teacher cannot work out how to scaffold the task, then it is too difficult (see Lesson 2)
  • Students are motivated with early easy wins, by tangible beginnings, by common structures.

Melting Brains with Infinite Series

My papa sparked a desire to understand wacky series, when he told a 9-year-old me that the alternating harmonic series can sum to whatever you wish it to. Addition commutes, except if you do it lots and lots?!

A happy two hours exploring this with FM class, using Excel without fully trusting Excel.

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The students’ emotions at the end – thoroughly angry at Maths. Mission Accomplished

Best Student Quotes:

  • “If the alternating harmonic series can converge to anything, does that mean 1 – 1 + 1 – 1 … also can?”
  • “1 + 1/4 + 1/9 + 1/16 + … is an infinite sequence of squares. Lined up they are infinitely long, but do they have an infinite area? Oo, isn’t that like Vsauce?”
  • “Eventually the percentage of numbers with a 9 in them is basically 100”
  • “I jump to the right more and more, and then end up … on the left?!” yells W as she hops around the room angrily (-1/12).
  • “Don’t worry sir, it doesn’t take too long to add up the first 50,000 terms”
  • “I feel dumb in Further Maths”. Obvious Warning: if it is too hard, people will drop out. How to ensure students know that this kind of thing is not on the exam, without constantly referencing exams?

Further links:

 

 

Thing Explainer

With five of my Yr12 FM gang, into the heart of London for a talk on how to explain science and maths simply to a public audience.

Reddish tinge of sunset

 

We were listening to Randal Munroe (of xkcd fame), and Marcus du Sautoy.

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We looked around at the thousand or so people in the audience. “There is not a single Asian person I can see” exclaimed Ashwin. “They are old and white” said Cody. Very different to Stratford station, where we began our journey.

 

Some points from the talk:

  • 500 years ago it was possible for one person to hold in their brain almost all of the scientific knowledge we had. Now, no one person could properly explain how an iphone works.
  • Marcus was given “A Mathematician’s Apology” by a maths teacher, gave him the key to a secret magical garden. He writes to show the magical garden to as many people as possible
  • You sometimes use sophisticated language to convince yourself and others that you do in fact deserve to be here, you are clever enough.
  • Marcus used the first few rows of the audience to show Cantor’s diagonal proof – each row was an infinite decimal, each person a digit
  • Humour is a way of connecting speaker/audience (we are the same!) and to diffuse tension
  • Teacher: “What is the biggest number?” Student: “5246?” T: “What about 5247?” S: “Ah, I was so close!”
  • The hardest problem that humans tackle is working out how they come across to other people.
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Beautiful room!

Inventing Triangles



The advantage of multiple classes is you can refine, compare and contrast. Loving it.

This kind of lesson requires, from the teacher:

  • Mathematical agility, to be able to quickly check if a triangle is worth exploring, or if a conjecture is worth exploring
  • General knowledge, to link in unexpected ways (see below)
  • Love of chaos – got no idea where we were going to travel

One group decided the outer diagonals of the triangle would be 1,3,5,7,9… and then used a Pascal-like rule to construct the rest. They abstracted to just odds and evens, and started noticed nested patterns of triangles. 


Another group started with rt(2) as the diagonal, and multiplied rather than added. They too found the nested triangles…


I got very excited at the parallels with Sierpinski’s triangle. The whole class crowded around, awed into silence, to compare and contrast the two. (One made by continuing down an infinite number of rows, the other chops a finite thing infinitely many times).


Later that afternoon I noticed precisely the same pattern in a textbook. After yelling about being the Neil Armstrong of triangles, boldly going to new mathematical lands, I was disappointed to see that someone got there before the kids. Does it matter? Not really. It does matter that I was ignorant of it existing during the lesson – the excitement would have been less genuine.

My Grandkids’ Lung Cancer

“But the project finished two terms ago!”, Rochelle complained after legging it from her home to meet Cameron and I outside the school gates one weekday evening.
That isn’t how truly authentic projects work – they don’t neatly fit into term-schedules. We were off to summarise our report on the air pollution that would be caused by a proposed set of concrete factories, slap-bang in the olympic park.
In two hours of dense, tightly protocoled discussion, Cameron was the only speaker to get a laugh. Finishing his planned speech, he paused, “I just have two more things I want to say”. I couldn’t decide whether to leap in and move things on, but he spoke, eloquently and from the heart, about intergenerational responsibility. “When you are old we are going to look after you” was the punchline that got the chuckle.
Rochelle, finishing our presentation with the line “Do you really want to be responsible for giving my grandkids lung cancer?” (Which we hastily had drafted while in the waiting outside, scribbling up against the wall), cut through the technical jargon (Section 10.3, 2011 Planning Act, B9 usage) to create a phrase that the room remembered. Several people came up at the end to shake her hand and reference that line. 
The committee voted unanimously against the building of the concrete factories. (My) summary of their position – we accept that all cities require building materials, but we are shocked by the poor quality and thoughtlessness of this specific application. The chairman, in his summary, specifically praised the students – “to hear the two youngsters speak was powerful”. 


“Most adults would have struggled doing what you just did” said Terry Paul, local councillor, to Rochelle


The other protestors went to the pub to celebrate. The kids declined the offer of a beer, and instead went home safe in the knowledge that their learning had truly changed something . “Don’t let anyone tell you that you cannot make a difference to the world” one of the local residents joyously yelled as he shook Cameron’s hand. 

A Tentative Lesson Structure

I have been naturally falling into a rhythm in my Further Maths classes. This structure seems to work well when students are already on board with Maths (and therefore follow you through headaches and open exploration), and when they are meeting lots of new mathematical structures for the first time.

 

 

Table to clarify thinking, and then a record of the lesson below.

 

 

Obviously, any attempt to codify what all good lessons looks like is ridiculous.

Chapter

In general

In particular – Matrices

Why?

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Provide students with a headache to wrestle with

Ask students to solve increasingly complex simultaneous equations

Before providing the aspirin, you need to give the headache. Dan Meyer

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Give students key information that it would not be reasonable for them to discover themselves

Lecture on how to multiply matrices (link back to multiplying vectors, which in turn links back to multiplying numbers)

Minimise teacher-talk while not wasting time on wishy washy discovery-based learning

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Ask students what questions they have. Important to do individually first – no cross-contamination

“What do you want to know about matrices?’

Now that they have a basic idea, questions start flowing. (Without teacher-input, the blank page is too daunting)

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Explore! In groups of three or four students?

Structured comparison of matrices and numbers.

Students need to learn how to behave like a mathematician – conjecture, root around in the darkness, play, prove.

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Big reveal: you now have the aspirin for your headache

After playing around with matrices and working out what the equivalent of division is, you can solve big systems of simultaneous equations

Provide an ending to the story, reminder of why this new bit of Maths is useful.

A record of the lesson

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Headache time
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Initial questions
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Explore time