The View From Mathematics

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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.

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(As inspired by Serfaty and Ben Orlin)

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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.

Teach for Australia Stories

Last week Teach for Australia published an article I wrote on their Stories blog. It is called “The School That We Need: A lesson in change“. Enjoy.

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.

Learning Mathematics: The struggle, the tools and the way forward from Michaela Epstein on Vimeo.

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

 

A sales pitch to the disinterested

My job is to sell a product that the client doesn’t want but has to have – so Dan Meyer has aptly described the job of a maths teacher.

Don’t get me wrong, there are some students who do love maths. They see the beauty of it and are curious about the language that underlies mathematics. They know that maths provides them with a break from the everyday and opens up a new world in which they can creatively explore.

For others, however, maths is a seemingly unattainable subject. The language of maths (and, quite often, the language of English) provides a brick wall that halts their ability to reason with any problems or puzzles they face. “I’m dumb at maths” becomes an easy catch cry and excuse for giving up. Many of these students have spent years not understanding this supposedly important subject and now, in high school, are one, two or sometimes five years behind.

These students’ behaviours unsurprisingly manifests in ways that show a disinterest in and lack of responsibility towards their learning.

At the passive end of the spectrum are students who ask for textbook exercises. Instead of being given open-ended or discursive problems to solve, they would rather follow procedural tasks that have a single answer to be ticked off by flipping to the end of the book. These students are simply satisfied with knowing they’re right and then moving on. The more interesting question of understanding ‘why’  is not of concern.

Somewhere behind these students are those who make an active choice to hand over responsibility for all aspects of their learning to their teacher. I mean all. These students, despite having had around a decade of education thus far, have over this period lost the ability to bring a book or pen to class. Education is of such little significance that these basic items are not perceived as an essential part of their day. Once the work gets started in class, these students will then often lack impetus or ability to get started, often needing help step-by-step to understand exactly what they need to think about and write down.

Finally, at the aggressive end of the spectrum, are those students who care so little about the learning of their peers and of themselves that they actively disrupt classes. These students take attention away from constructive discussions and productive stages of a lesson, shifting it onto their own behaviour. They display an unwillingness to concede that a classroom is made up of more than one individual and that those other individuals are there in the room trying to become smarter, not learn about their best friend’s Facebook status.

While these four personalities will exist in most classes, the difficulty multiplies when the balance of students shifts towards the disengaged end of the spectrum. In these classrooms, the teaching and learning becomes more about improving behavioural and attitudinal skills rather than specific mathematical concepts. To turn this around, takes not only much work from a teacher, but also from the school, support services and parents.

 

Friday (week 7, term 1)

CHRISTOPHER PYNE: Well, I think the measure of a very good teacher is a teacher who has been given every opportunity to give their students the opportunity to reach their full potential. So I think a great teacher should have time in their class to actually teach… I want to give you the opportunity, if we get elected, to have the time to teach your students the English, the history, the geography, the maths that you want to teach them, the basic skills.

Although I don’t fully follow this comment by Mr Pyne, I sense that according to his definition I would not be “a very good teacher”.

On Friday I gave a class a test. One student handed back a blank test. Others gave it a good shot. One or two showed excellent understanding of the content.

There are 26 students on my roll. On any given day, however, about 20 will show up. Have I been given every opportunity to give my students the opportunity to reach their full potential? (That apparently being the indicator of good teaching!)

My goal for this class is to help them to progress 1.5 years in their learning – two if I can really succeed – in the space of a year. Currently, they are on average four years behind where they should be.

Does this mean that every maths teacher that has preceded me has failed to “have time in their class to actually teach”? Clearly these students do not know the basics. To outrightly say that these predecessors are not great teachers, would be a simple statement. It would be unfair to not give them the benefit of the doubt that they have tried and that, in fact, the needs of these students are deeply multi-faceted and not at all straightforward.

I hope that Mr Pyne, as with others in the public domain, hold their tongue before blindly pouncing on this notion of teacher quality. Certainly, it is fundamental that teachers have particular capabilities in order to carry out their work to some degree of success. But let us not assume that these capabilities are fixed within teachers (and from the moment they begin their university degree). Instead, we should assume that teachers, as with others, hold the potential for change and development.

It would also be misguided, of course, to assume that all student success or failure rests in the hands of the teacher. By recognising the teacher as a facilitator, it is possible to see that a student’s trajectory of learning and achievement is necessarily beyond the means of that one adult. Even if it is just the possibility of providing students with rich learning opportunities “to reach their full potential”, sometimes this is only something we can hope for.

 

Google’s 20% policy

At some point in the short history of Google, employment practices have included what is known as a 20% time policy. The policy specifies that staff at Google spend 20% of their working week putting their feet up and being creative. During this time, staff do not work on specified Google projects, but instead focus on innovation, thinking about whatever it is they want to think about…

Earlier this year I became aware of this policy and began thinking about the implications that it could have when applied to education. My ideas have since taken two different paths.
20% creative time for teachers

An obvious alignment of Google’s policy to education, is to provide teachers with time off for 20% of their working week. This does not mean a day off each week to sleep in or go to the beach. Rather, the 20% would be set aside for teachers to develop projects at school, construct innovative curricula, do professional readings, design learning spaces and so forth. If a teacher with particular expertise wanted to spend some of this time working with groups of special needs students in order to achieve certain goals or explore learning pathways, then perhaps this could also be made possible.

Over the course of this year, I have been fortunate to have my own “20% time” of sorts. Specifically set aside for study that I am concurrently doing, I have used much of this spare time to consolidate my learning inside the classroom whilst being in my own space, free from distraction. Other graduate teachers at my school, working a full load for the first time, have looked on enviably at this apparent freedom. As they have told me, the extra classes that they are now teaching limit time for reflection and enhancement of teaching practices. For these young professionals, who are keen to perfect their craft, they are finding themselves managing to just teach and nothing more.  Could allocated time to think, reflect, plan, discuss and be creative make all the difference?

20% creative time for students

The idea of providing students with ‘creative time’ in which to explore their own ideas and interests under the guidance of a teacher sounds ideal in the current pedagogical climate of differentiation. Since a key role of education is to foster the development of young minds, why not try something similar at school? Why not push aside the time constraints set by the curriculum and allow students to experiment a little?

I initially had no idea of how this might play out. So many factors seem to limit this being feasible: disruptive classrooms arising from problematic behaviours, a crowded curriculum needing to be taught, and limited face-to-face time each week (not to mention excursions, camps, assemblies, strike days, professional development days, etc.).

For one of my classes in particular I have been wanting to make this happen though. In this class are students who have been a year ahead of the expected level since the start of the year, and who I feel I have not adequately catered for.

Over the weekend I saw a number of students from this class participating in the school production: on stage singing and dancing, in the band and backstage managing props and scene changes. The diversity of these students’ talents and interests jumped out at me, in a way that made me realise that the class work and whatever differentiation I had been providing up until then only pushed them back. Sure the odd bit of group work, competition or interesting, yet challenging mathematical problem spiced things up, yet the bulk of the learning itself has been traditional, expected and uncreative.

To implement this “20% time policy”, next term one session a week will be spent on an inquiry-based learning project. Whether it is science, English, music or mathematics itself that they are passionate about, students will research that area, devising a key question for exploration that is linked in some respect to their learning in maths. For example, if a student is a keen musician, they might look into the music of pi and potentially compose a song of their own. At the end of this process, students will then teach their peers and myself about their findings.

 

The ideas spelled out above are only preliminary thoughts that are still to develop. These thoughts are propelled by the sense that thinking outside the square will facilitate new understandings of concepts and ideas. Potentially, the allocation of time for creativity will also enable paradigm shifts in learning amongst students and teachers alike .

In defense of Algebra

Recently I have started reading the book Unknown Quantity: A real and imagined history of algebra by John Derbyshire. For any history (or mathematics) students, the book is a fascinating read, detailing the development of the number system, algebraic notation and use from over four thousand years ago to as we know it today.

From time to time I have shared with my students snippets of what I have read. For example, the ancient Babylonians didn’t have a conceptual understanding of the numbers 1 to 10. Instead they worked with a base 60 notation, which explains why this same notation appears in various instances of our numerate world today. For my students, who were studying trigonometry and the breakdown of degrees into minutes and seconds, this explanation proved helpful.

Derbyshire’s book has highlighted for me how complex the development of algebraic thinking, language and writing has been. Mathematicians struggled over the course of generations trying to figure out how to represent the unknown number (i.e. a pronumeral or variable, often written as ‘x’). Each step of the way, they didn’t know how their findings would be of use to secondary students today, nor did they often know what had immediately preceded them or was being simultaneously discovered somewhere else in the world.

The study of Algebra today gives many students grief. They struggle with its abstractness and the ideas of simplifying an expression or working with a balanced equation.

Alongside these students, however, there are others who enjoy it and who get a sense of achievement when they find that they do understand this new language they are learning.

In response to recent debate over the worthiness of algebra in the classroom, why limit what our students are learning because it is difficult for some? Learning is not intended to be easy. It is intended to set students up to navigate and make sense of our world. And who knows, maybe there is another Diophantus or al-Khwarizmi growing up today, who will in the years to come make some amazing leap forward in mathematical thinking.

A great contribution to this debate can be found here in an article by Judy Bolton-Fasman.