## Resilience in the classroom

Recently I’ve written a lot of posts around puzzles and problems that people have set me to solve. This is something I find fun to do and I have enjoyed solving them. I have also enjoyed sharing my thought processes and proofs here, including the dead ends I went down.

The ideas involved in doing this have led me to think a lot about which, if any, of my students would have been able to solve each of the problems.

I started thinking about my A level classed, I know that the vast majority would have given them a go. One or two might have not started because they didn’t know where to start, but the rest would have given it a try. I’m fairly certain that at least 5 of them, given enough time, would have solved most of the puzzles. I intend to test this hypothesis after the holidays.

I then thought about my year 11 class. They are a top set and I have been trying to build a resilience in them this year. When I first took the class over in September one of the students complained to my colleague, her science teacher, that I had “given a worksheet and not even told us how to do it.” It had been a starter task designed to double check the class were familiar with and able to use Pythagoras’s Theorem. My colleague then asked her “could you not do it then?” to which she had replied “I could actually, it was Pythagoras’s Theorem.” He then asked, “so why did you want him to tell you what to do?” She had no answer.

This is quite common in schools, it’s an idea a lot of pupils have. That they should be explictly told what to do for each type if question. But I think that if we are to create the mathematicians of the future we need to be building a resilience into pupils. We need to equip them with the skills and knowledge required to solve the problems, and allow them to select whichever bit of it they want/need to solve it.

I think the puzzles I’ve discussed recently are good examples of tasks that do this. Some of them have the added bonus of being solvable multiple ways, often given rise to a “low barrier, high ceiling” task that can be set to 11-18 year olds and be solvable, yet challenging, to all.

This one which started the chain is a lovely puzzle based around algebraic fractions. It is solvable to a clever yr7 student who just has a basic knowledge of fractions, or via simultaneous equations and complex numbers, which is how most people with an advanced knowledge went about it.

This triangle puzzle was particularly nice, it had a lovely solution which requires a knowledge of the sine rule for the area of a triangle and knowledge of the sine curve, or a much simpler visual one which only requires a primary school level understanding of triangles!

This triangle puzzle is the best example of one with multiple solutions. I used Heron’s Formula (which no one else seems to have heard of!) But it is equally solvable using accurate drawing, similar and congruent triangles rules and/or Trigonometry (including Pythagoras’s Theorem).

These problems are great, and will build resilience, but the two most recent ones are the ones which illustrate this best.

While discussing constructing a proof with my brother he was asking how algebraic proof worked. He has two A levels in maths (at A no less) but he stopped studying maths a decade ago so is a little rusty. His questions, though, made me think about my classes. I know my year 13 pupils can construct proofs, but I’m not sure about all of those pupils who are younger. I am going to ensure I build more opportunities for this into my lessons.

In solving this problem I noticed a pattern in the numbers, I expressed this pattern algebraically and manipulated the algebra to prove the pattern was true for all numbers. This is what mathematicians have done for centuries and how theorems are born. And this is a skill I need to instil in my students.

In the root of the problem I discussed solving a problem which involved searching for integer solutions of an equation in two variables. It was fun, and again I was asked, “how on earth did you work that out? I wouldn’t know where to start.” Well I didn’t know where to start either, I just tried things until I got something that was right. This is what the best of my students do. This post gives some great examples of this. It needs to be more though. As maths teachers we need to make sure our students are willing to do this, if they don’t know what to do to apply things they know until they get an answer. I have been using this approach with my pupils. I won’t help them unless they have tried something first.

The way the maths exams have been previously has made this spoonfeeding possible and far too common. In the last couple if years the exam boards have thrown a few curveballs, which has meant that students have had to apply their knowledge in different ways to the past papers. I think this is the way forward, and hope the new GCSE and A Level exams address this.

Reblogged this on The Echo Chamber.

I’m not sure the “you haven’t told us how to do it” thing is about resilience. If a school has low expectations regarding the retention of knowledge, students will not expect to be asked to remember something they haven’t been taught that day. If kids are shocked at being asked to do something without an explanation it can simply be that there is not enough emphasis on knowledge and recall in their lessons (generally, not specifically in maths). I’ve encountered kids who would react like this to things they had been taught the day before, and not because they cannot recall it (or look it up in their books) if they tried, but because they were not used to being expected to recall *anything* they had learnt since primary school.

Good points, I have managed to train the class in question to be able to recall knowledge, but they really should have been used to it. The younger year groups are much better, so perhaps that signals a shift in culture.

Resilience is definitely something we need to build though, too many pupils are not prepared to try things out and if they try something that doesn’t work give up. I’ve been working a lot with my classes on this and have seen a vast improvement.