The View From Mathematics

I have a lot of respect for Professor Sylvia Serfaty. Not only is she a brilliant and esteemed mathematician, but she recently brought two of my favourite things together when she said this:

“You enjoy solving a problem if you have difficulty solving it. The fun is in the struggle with a problem that resists. It’s the same kind of pleasure as with hiking: You hike uphill and it’s tough and you sweat, and at the end of the day the reward is the beautiful view.”

There is a certain exhilaration you feel when – after carrying a heavy pack on your back for kilometres on end through mud, up hills, feeling that gross sweat trickle down your back, and running out of things to say to your hiking partners – you arrive at your destination. There you are in the middle of dense bushland, with not a roof, road or electricity wire in sight. Instead, stretched out around is unending greenery and the vastness of the sky above. You are in a patch of the world that very, very few people will ever get to see. And yes, you can be proud in knowing that you worked hard to get there.

And this natural beauty can be compared to maths?!

Just like hiking, there is much in the journey of problem solving that is hard work and will challenge you. Mathematicians – and I use that term broadly, to describe educators, academics, students and those who are in some way engaged in the field – take joy in getting to the destination. Problem solving is not like relying on your GPS to get somewhere, where each step you are told what to do next. “At the roundabout take the third exit. In five hundred metres, use the second from the left lane to turn left. You have arrived at your destination.” Nope. Why would we bother with mathematics if it was that mundanely easy? It’s hard and mathematicians knowingly struggle. Serfaty took nearly 18 years to solve one problem. She’s also not the first to show such extreme mathematical persistence (e.g. Andrew Wiles‘ momentous  journey with Fermat’s Last Theorem).

On solving a problem, mathematicians reach a point of (sometimes momentary) finality. There is perspective on the method used to get there- what was effective, what held them back, how they failed, but then learned from it. And, just like the hiker’s view, there is immense satisfaction that comes with overcoming your own personal limitations to arrive somewhere new.

(As inspired by Serfaty and Ben Orlin)

Amazing Things Young People Do

Who says that reading is under-rated? This 4 year-old has read 1,000 books. No kidding.

Pop quiz: What has 15 year-old Toby Thorpe from Huonville High, Hobart done?

a) Created a two-year renewable energy plan for his community.
b) Met the President of Kazakhstan, Nursultan Nazarbayev
c) Won \$US100, 000.
d) Met Abu Dhabi’s Crown Prince, Sheikh Mohamed Bin Zayed.

Both of these stories blow me away. Four year-old, Daliyah, and Toby have both achieved incredible things at such a young age. Their feats are of the kind that we look to in admiration, unsure of whether we could achieve what they’ve done.

The useful takeaways from their stories though, is not that we should feel inadequate for not having done similar things. That response feeds into unhelpful beliefs and attitudes, such as the sense that “I just wasn’t born to be able to do X”. We don’t need to look at success for the purposes of drawing comparison, but we can still celebrate the achievements of others and take joy in what they have done.

We can also learn from what these two stories tell us about supporting young people to pursue their passions. Daliyah’s parents entered her in a reading program where the bar was set extraordinarily high. Toby’s teachers supported his participation in a prestigious international competition. In both cases, the adults involved could have stopped and not provided the opportunity out of concern for Daliyah or Toby facing failure. It’s easy to think “What’s the point of my child/student doing this? The chances of them succeeding are incredibly slim”. But if opportunities are never provided in the first place, then there’s no way of knowing what potential height can be reached. Opportunity combined with passion form a powerful couple. As these two young people have shown us, passion is a great motivator that pushes us to exceed expectations.

Daliyah’s and Toby’s stories have left me wondering about how we can harness the passion/opportunity combo for more young people. Wouldn’t it be wonderful if one of the goals of our education system was to support every individual to uncover and pursue something that got them excited and deeply curious, without the pressure of having a defined end-point (e.g. curriculum outcome) or standardised metric for success (e.g. report mark)?

Reinvigorating Maths Class

Maths anxiety cripples students’ ability to participate and progress in class. Forty percent of students begin Year 7 already behind in mathematics. The number of students going on to complete advanced maths in Years 11 and 12 and at university is decreasing.

Yet a projected 75% of future jobs will rely on STEM skills, notably mathematics.

What’s being done to address this chasm between present problems and future need? Check out this article I wrote recently for Teach for Australia’s Stories Blog about a maths program that is successfully turning around student learning outcomes.

The Maths Anxiety Kid

Anxiety, by definition, is an irrational fear of something. Now I’ve heard the term ‘maths anxiety’ bandied about and know that according to empirical research this phenomenon exists – though for some reason you never hear about ‘English anxiety’ or ‘geography anxiety’*.

Whatever the reason for its existence, this week the maths anxiety of some of my students smacked me in the face.

Here’s the situation: mid-year exam coming up, students are given summary sheet templates to organise their notes and have some class time to do so. In theory, a perfect opportunity to revise a semester’s worth of work, practice questions and put together notes that can provide guidance during the exam.

For the Maths Anxiety Kid, instead this means leaving all books closed and pens out of sight, before finally writing a couple of words in their book, shortly followed by tearing out the page. The Maths Anxiety Kid will also engage in an interplay of offering to hand out sheets, clean the board, becoming argumentative about starting their work and endlessly wandering around the room.

On catching up with the Maths Anxiety Kid at a subsequent lunch or recess, he/she will have little to no recollection of the content of the lesson. Every persuasive technique picked up in English class will be tried on me to avoid revisiting the learning that was supposed to happen and putting pen to paper as we talk through a problem.

The Maths Anxiety Kid is a serial avoider, who lacks persistence and drive. In short, they hold a fixed mindset, perceiving their abilities in maths to be innate.

As teachers or parents or friends, our job is to help them turn this mindset around. Any display of effort and understanding should be celebrated and opportunities to use maths in a fun way, for example with games and puzzles, should be seized.

* Possible exception is ‘science anxiety’. I’m sure I had a mild form of it in high school.

Reliability and perfection

I

reliability, n. the extent to which an experiment, test, or measuring procedure yields the same results on repeated trials.

perfection, n. the action or process of improving something until it is faultless.

II

“There’s a paradox here. Ask most research physicians how a profession can advance, and they will talk about the model of ‘evidence-based medicine’. …[I]n a 1978 ranking of medical specialities according to their use of hard evidence from randomised clinical trials, obstetrics came in last. Yet almost nothing else in medicine has saved lives on the scale that obstetrics has.”

III

I have found myself in a state of slowly trying to procure the most perfect lesson. To fix things up so that it runs minute-by-minute according to a plan, in which I am asking just the right questions, have tasks that are carefully scaffolded at the precise level for each of my students, with every stage of the lesson considered and noted down. The problem with this is that the complexity of the details that I’ve built up in my mind across the 75 minutes of a lesson don’t play out as planned. This, of course, is because I’m dealing with actual people – and young, inquisitive, social teenagers at that.

Instead of needing perfectly tailored lessons they need reliability.

IV

The Apgar Score is a measurement tool that is used in obstetrics. It is a very, very simple tool to administer, consisting of five questions worth two points each. As a result of having this standardised measure used for diagnostic purposes and for comparing the health of newborns, this tool has saved the lives of many babies and their mothers who would otherwise have been at risk of death. And part of the beauty of this tool is that it concentrates on the notion of overall reliability as opposed to the perfect measurement of many characteristics for each and every child.

For an individual doctor, it may seem more useful to work with a detailed tool that goes into the particulars of the baby they are delivering. But consider this one child in the context of all of the babies that are born and how the doctor can learn from the vast numbers of babies and apply that learning to the one individual baby.

If you can gather data on a general population, then as a consequence you will be able to have greater insight into what goes on within that population, including broader patterns or trends, in contrast to the perspective that would be gained by looking at a smaller case study in a greater amount of detail.

V

Given the simplicity of the Apgar score, it can be delivered by anyone, no matter the level of expertise and no matter the location. The same sort of thinking needs to be applied in schools.

In the complex environment of a school, we are consistently guilty of trying to find the most perfect way of measuring where a student is at or planning a lesson or analysing the performance and progress of a teacher year by year. Now, don’t get me wrong, all of this is important and I don’t deny the utility in understanding each of these insights. However, if we are to better consider the progress of a student and the progress of a number of students overall, then we need to be looking at them as a conglomerate. We need to be thinking, what is a more reliable way in which we can assess improvement and be able to do this in an efficient way?

Equanimity

So I’ve been teaching for a bit over three years now. During that time I’ve begun a journey of building my craft. As a profession, teaching is something where you are constantly accumulating new knowledge and skills. There is no end-point or definitive stage in which you become The Good Teacher.

Some days I am on top of the world. The most difficult student in my class voluntarily spends half of his lunchtime with me doing revision for an upcoming assessment. A former student emails me with the exciting news that she has become School Captain. Another student writes me a note thanking me for teaching her that term. Are those the signs of success? There’s no NAPLAN score or developmental continuum against which these events can be marked, yet they are met with the feeling that some good has been done.

At other times I am low. I think perhaps the job isn’t for me; that I work behind this façade that will soon be removed to reveal that the competence that people have come to see in me is not the reality. This is the case when that difficult student destabilises yet another lesson. Or when I fumble my way through a meeting I am leading. Or when students again let out the disparaging remarks of “I hate maths” – to which, in an attempt to inject humour into the situation, I respond “You’ve just killed a Maths Fairy”. Although, in reality I feel it is a small part of my confidence and my professional image that’s just been killed.

To all this, I’m learning to remind myself that I am doing a good job. There are enough signs around me that tell me this is the case. And it’s not surprising that after a bit over three years of teaching a difficult age-group a subject that is frequently disliked, and within schools where large numbers of students come from families where there is little history of educational attainment, that there are tough days. The idealised image of how lessons will run or the progress that all students will make is just that: idealised.

I recently came across this Humans of New York post that is a nice reminder of the need to put such highs and lows in perspective:

“When is the time you felt most broken?”
“I first ran for Congress in 1999, and I got beat. I just got whooped. I had been in the state legislature for a long time, I was in the minority party, I wasn’t getting a lot done, and I was away from my family and putting a lot of strain on Michelle. Then for me to run and lose that bad, I was thinking maybe this isn’t what I was cut out to do. I was forty years old, and I’d invested a lot of time and effort into something that didn’t seem to be working. But the thing that got me through that moment, and any other time that I’ve felt stuck, is to remind myself that it’s about the work. Because if you’re worrying about yourself—if you’re thinking: ‘Am I succeeding? Am I in the right position? Am I being appreciated?’ — then you’re going to end up feeling frustrated and stuck. But if you can keep it about the work, you’ll always have a path. There’s always something to be done.”

In a job in which you care so deeply about what you are trying to achieve, you need to make it about the work and do so with a tone of equanimity.

The Sixth Sense for Multiplication

The crux of problems faced in middle school maths – now in visual form.

Or, Seeing Arrays (Less Cinematic Than Seeing Dead People, But More Useful)

This year, I’m teaching younger students than I’ve ever taught before. These guys are 11 and 12. They’re newer than iPods. They watched YouTube before they learned to read.

And so, instead of derivatives and arctangents, I find myself pondering more elemental ideas. Stuff I haven’t thought about in ages. Decimals. Perimeters. Rounding.

And most of all: Multiplication.

It’s dawning on me what a rich, complex idea multiplication is. It’s basic, but it isn’t easy. So many of the troubles that rattle and unsettle older students (factorization, square roots, compound fractions, etc.) can be traced back to a shaky foundation in this humble operation.

What’s so subtle about multiplication? Well, rather than just tell you, I’ll try to show you, by using a simple visualization of what it means to multiply.

Multiplication is making an array.

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Learning Mathematics: The struggle, the tools and the way forward

On Friday 4 July I presented the Pecha Kucha below at the 2014 TransformEd Conference, Melbourne.

This is Maths: Pythagoras’ theorem

1. Concrete model: Pythagorean water demonstration

2. Representational model:

The square on the left is dissected into two blue triangles, two red triangles, a green square and a yellow square. The square on the right also has two blue triangles and two red triangles, as well as a light blue square whose sides are the hypotenuse of the triangles. “By subtracting equals from equals, it now follows that the square on the hypotenuse is equal to the sum of the squares on the legs [of the triangles]”.

3. Abstract model:

a2+b2 = c2