## 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.
6. Use vectors as the aspirin, to fix the headache.

Resources

• 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

• 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?

Resources

• 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.

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.

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?

## 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.

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

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.