# The art of teaching anti-functional maths

Here is a scene that has played out many times in my classroom over the years, with students of various ages:

Ellie: “Mr Barton, when will we actually need trigonometry, you know, like in real life?”

Me: “Well, Ellie, one day you might become an architect or a civil engineer, and then trigonometry will be extremely useful to you”

Matthew: “How about Vectors?”

Me: “Well, Matthew, you might be the captain of a ship…”

Kiel: “Are you honestly telling me I will ever need to do long division problems like this?”

Me: “Of course, Kiel. Imagine, you go out for a meal, and there are six of you, and you are trying to work out how to split the bill… (Kiel glances down at his mobile)… and for some reason everyone has forgotten their phones and the waitress refuses to lend you her calculator”

Jenny: “How about Pythagoras?”

Me: “Jen, let’s say you have a ladder, leaning against a wall, and you are desperate to know how high it reaches…”

Josie: “And adding fractions?”

Me: “Hmmm…”

The realisation I reached a few years ago was that the vast majority of maths topics that students cover during their time at school will never be of any use to them whatsoever in their day to day lives. Of course, this is not unique to maths – rarely have I found myself explaining the properties of igneous rocks or reciting the neon gases, apart from Tuesday evening at the Black Bull Quiz. And the truth of the matter – that we need to learn these topics as they will be appearing on the exam – is so blunt in its honestly that it can remove any lingering enjoyment and love for the subject that a student may have.

The antidote to this apparent perceived lack of relevance of mathematics in students’ lives goes by many names: Functional Maths, Real Life Maths, Real World Applications, to name a few. All of them are based on the (entirely reasonable) assumption that if you can make the mathematics students are doing in the classroom link directly to something the can relate to, then engagement and achievement will follow. Sounds a flawless plan, right?

But there is a fundamental problem. Well two, actually.

Firstly, as we have seen from the questions above, not all of school mathematics lends itself well to functional, real-life applications. Take the adding of fractions. Name me one job, or indeed one real life situation, where that is going to come into play. How about angle properties on parallel lines? Fractional laws of indices? Multiplying two negative numbers? So, the examples you end up coming up with are so tenuous and contrived, that students see right through them, and engagement levels plummet.

The second problem is equally dangerous. Any “real life” situation that can be modelled mathematically tends to involve a level of maths far above what students could be expected to do at school level. For example, I once attempted to do a vectors lesson based free kicks taken by Beckham, Toure, etc. I thought I was on to a real winner here. It took me ages to put together, and I had YouTube clips, and everything. But, of course, I had simplified the situation so much that it had lost all resemblance of reality. Students were, quite rightly, asking where was the wind, what about the shot power, the pressure, the goal keeper. Was I really claiming that Beckham worked out 3a – 2b on a piece of paper before curling the ball in? And before I knew it, my poor, fragile little model collapsed to reveal a bog-standard lesson on vectors. No-one was fooled. No-one was happy.

Fortunately, I believe there is a solution to this problem. And it comes in the most unlikeliest of guises. It is in problems, challenges and activities that are so far removed from any situation that could ever be called real life, relevant, or even useful. They do not claim to be functional. The do not claim to impart students with skills that they will use in their every-day lives. They don’t even claim to help students pass exams. But they do all that, and more.

The first branch of these activities, I like to call Anti-Functional Maths. They can be summarised as activities that involve a situation that is often bizarre and crucially has no disenable practical use to anyone. Students are challenged to solve it.

Many of my favourite types of this activity fall under the umbrella of “3 Act Math” from the USA. High school teachers, Andrew Stadel has a brilliant activity that starts with a video of him throwing scrunched up A4 paper balls into a bin. You see about 8 go in, and then the video suddenly stops. The question is: how many paper balls would fill up the bin?

A completely pointless question, but one which has engaged every single group of students (and teachers!) I have done this activity with. Before you know it, students are reaching for paper, rulers and calculators, they are scrunching up balls, working out averages, discussing strategy, and best of all demanding to know what the volume of a cylinder is, how to work out the volume of a sphere, and even what grams per square metre rating the A4 paper Andrew used has!

In other similar activities students are trying to figure out how many Post-it notes will cover a filing cabinet, and how many cups with stack high enough to reach the ceiling. All arbitrary, contrived situations, that do not profess to be any more than that, but which stimulate the students and get them willingly performing a whole host of high level mathematics.

I have collated an index of my favourite 3 Act Maths activities on my blog here: http://www.mrbartonmaths.com/blog/links-best-maths-websites-world/#rich

I call the second branch “Low Barrier, High Ceiling” tasks. They are activities that any students can access and make progress within the first 20 seconds, and yet there is enough maths in there to keep the most able challenged and asking questions for days.

One of my favourite example of this was found on Mike Ollerton’s excellent blog, and I have done it with classes of all ages and abilities:

*“I have chosen 3 numbers, and I have added the 1 ^{st} and 2^{nd}, the 2^{nd} and 3^{rd}, and the 1^{st} and 3^{rd}. And the three totals I have ended up with are: 17, 21 and 18. Can you figure out what my original three numbers are?”*

Simple instructions, no worksheet or fancy equipment required, just a pen and some paper. And once students have figured out the numbers, then the fun really begins as you can ask them the following:

- Make up three numbers for your partner, give them the three totals and see if they can figure out your numbers
- Can you come up with a strategy to find the numbers?
- How can you tell from the totals if two of the numbers are the same?
- What about if the numbers are in a sequence?
- What happens if one of the numbers is a negative/decimal/fraction/surd?
- If three random numbers are chosen as totals, can you always find three numbers that would create them? Can you prove it?
- How about if you change the rules and subtract numbers? Or multiply them? Or if you have four numbers?

Several of the questions above came from the students themselves. The task lasted a good three lessons, and the students still talk about it now. I was able to different effectively with my questions, meaning students were both challenged and supported. They were engaged, focussed, doing all types of mathematics, as well as hypothesising, communicating, generalising, problem solving, and a whole host of wider skills.

And at no stage in the lesson did any student ask: “when we will ever need this in real life, sir?”

*There are a whole host of activities like this around, from both sides of the Atlantic. I have collated my favourite on my “Links to the Best Websites in the World” page here: **http://www.mrbartonmaths.com/blog/links-best-maths-websites-world/#rich*

*Craig Barton is an Advanced Skills Maths Teacher at Thornleigh Saleisan College, Bolton. He is also the TES Maths Adviser, and creator of mrbartonmaths.com and diagnosticquestions.com*