Category: Mathematics

Last week, Simon Job – the creator of MathsLinks and its attendant sites – wrote a post called The transcience of sharing.

Simon is a sharer par excellence, not to mention a generally thoughtful and down-to-earth guy. So when he talks (types), I listen (read). Essentially, in his post he is posing this very valid question:

Why is sharing happening on social media (where it is transient) rather than on platforms that are clearly built for it and superior to it in almost every way (e.g. MathsLinks)?

This is a question I’ve thought about too – and it’s bugged me. Over the last few months, these have been the thoughts percolating around my head.

1. (a) It’s where the community is active, which motivates the poster. In the right space, at the right time, it will gain a responsive audience and that response is a very powerful motivator.
   (b). It’s where people visit, every day and for no particular reason, which is how the viewer sees it in the first place. People come to dedicated sites like MathsLinks when they (i) are after something, (ii) have the presence of mind to look for what someone else has made/found first, and (iii) have the time to commit to browsing for a little while. That happens far less often than people pulling up their social media feed of choice (which seems to happen reflexively once people get to a bus stop or train station these days).
2. Precisely because it does not aim to preserve, only the trendy and really engaging things bubble up to the top (either through Facebook’s black magic sorting algorithm or Twitter’s more organic system of retweets).
3. I alluded to this above, but MathsLinks (and other similar repositories like TES Australia and Scootle) has become its own worst enemy by being so good. There are hundreds of objects there – which is awesome, but also means that a new user doesn’t even know what’s there or where to begin. There’s awesome stuff there but (coming back to the time issue that has already been identified) someone needs to commit to searching thoughtfully through it to find what will be useful to them in the present moment. This is an issue with faculty resource files just like it is for MathsLinks.

So what can be done to improve the situation? I have a handful of thoughts, corresponding to the points above.

1. Clearly, MathsLinks is awesome as it is. We just need to connect it with the community more effectively. I feel like this is a market problem – it’s a great product, in a quiet spot. Stick it in the middle of George Street and it’ll go nuts because people will be exposed to it more frequently and the conversation about how good it actually is will spread from there. How practically to do that in our context is another question entirely, though.
2. Maybe there needs to be a dedicated team (and by team, I mean more than just Simon) of people dedicated to capturing those cool posts when they come up on social media and then preserving them. We don’t want to discourage the spontaneous sharing and ensuing discussion; we want to leverage it and keep it somewhere that it can be found for future reference.
3. Perhaps we need to do something like a “weekly featured resource”? I have considered doing something like that in my department with “my best lesson this week” as a regular feature of faculty meetings. It would just help people become aware of the riches that are hidden away there, rather than letting them gather digital dust in the cellar of the internet.

Just some food for thought.

What strategies can we use to find out the area of the top playground? What information will we need?

Last time I wrote some thoughts I had after completing my interview for the TER Podcast about maths education. You can go back and read that if you’re interested in thinking through some of the big-picture issues surrounding the problematic state of maths education in Australia. Following on from that post, I want to share some more practical pointers that I’ve observed to be helpful in a variety of different classes and contexts. Each one is its own idea, so I’m going to devote a few posts to unpacking them in a bit of detail.

What are some of the effective approaches you’ve seen people use? Answer number one: adopting a growth mindset.

First things first. It’s vital that teachers regard their students with a real growth mindset. This is a phrase familiar to anyone who has read the work of psychology professor Carol Dweck, who gives the best summary of what the idea is about:

In a fixed mindset students believe their basic abilities, their intelligence, their talents, are just fixed traits. They have a certain amount and that’s that, and then their goal becomes to look smart all the time and never look dumb. In a growth mindset students understand that their talents and abilities can be developed through effort, good teaching and persistence. They don’t necessarily think everyone’s the same or anyone can be Einstein, but they believe everyone can get smarter if they work at it.

You can see why this is such a big deal to maths education. Maths, perhaps more than any other subject in school, is dominated by a fixed mindset. There are people who are good at maths and then there are the rest of us. In fact, the phrase, “I’m no good at maths” has entered into our cultural vernacular and sadly become an acceptable response to anything encountered in everyday life that involves numbers or numerical thought.

The problem here is that this kind of thinking becomes a sort of self-fulfilling prophecy. When we think of ourselves as unable to do mathematics, we don’t bother trying – and hence deprive ourselves of the very experience that will allow us to develop mathematical skill (namely, struggling to grasp numerical concepts and master the tools necessary to solve problems that require their application).

It doesn’t take too much imagination to realise that our self-concept when it comes to our mathematical ability isn’t just self-generated. It is formed, in large part, by those we trust to nurture and develop us as mathematicians – our maths teachers. What kind of an effect do we as maths teachers expect to have if we consistently communicate that “This is too hard for you, and you will never be able to succeed at it no matter how hard you try”?

And sadly, whether it’s through the bevy of tests that end in failure, or the advice to children to take the mathematics course that will maximise their ATAR rather than challenge and enrich them, or even just the little interactions with students every lesson that erode their self-confidence – this is the message that students often pick up from us, their maths teachers. Some students survive this process, but many don’t. They aren’t just disempowered – they’re paralysed. No wonder “maths anxiety” is a thing (who ever heard of any other subject that has its own psychological malady associated with it?). What a tragedy.

It’s obvious that people can take the growth mindset too far. One of the most enduring characteristics of truth in all spheres is that it will always be abused by someone with the wrong idea about how it should be interpreted, and this is no exception. If you’re curious about this and want to know how to avoid that particular trap, you can watch a vlog I recorded about it a few months ago.

But that isn’t most of us. For most of us, the growth mindset is a breath of fresh air. Yes, anyone can master maths! Sure, it takes some more time than others – but who is surprised by that given that every human being is unique and brings a new perspective and set of skills to the table? Rather than view those differences as a cage locking us into a certain level of achievement, let’s embrace them and see how they can be brought to bear on the pursuit of mathematical understanding that we should all be a part of.

2014 has been a year full of firsts for me. First year teaching in a comprehensive school (student teacher placements notwithstanding). First year as a head teacher (which has produced a whole lot of firsts of its own). First time recognised in public because of the videos I make. And as I type this, I’m at my first MANSW Annual Conference (arguably the biggest gathering of maths teachers in the state all year), my head spinning from considering new ideas and meeting new people (or in some cases, seeing people face-to-face who I’ve been interacting with online for a long time now).

Another first happened last month, when I participated in a phone interview with Corinne Campbell (@corisel) for the TER Podcast. The topic was Is Maths Education Broken?, and I was there to provide a sort of foil to an interview that Corinne had with the luminary Conrad Wolfram (of Wolfram Alpha fame). You can listen to the entire episode (and if you’re in education and haven’t subscribed to the podcast, you really ought to). It was an interesting experience, not least because it was so unusual to actually interact with a voice that I was so used to just listening to passively through a podcast.

I’m the kind of person who thinks of the perfect witty comeback or joke ten minutes after the conversation is over. So even though I had prepared my own thoughts and notes before the interview and tried my best to cover everything that would be important, I found myself in the shower that evening thinking, “Oh, _this_ would have been the perfect answer to that question!” and “How on earth did I forget to say _that_?” So here are a few of the things that I should have said, but forgot to. In this post I’m going to talk about the large-scale issues that are related to the “STEM crisis” Australia is experiencing, and in a follow-up post I’ll talk about some of the smaller practical strategies that can be employed in the classroom to help our students from day to day.

How do we improve STEM skills in Australian schools?
There’s no simple solution to this one – so you can know with a fair degree of certainty that if someone tells you they have a straightforward way to fix this problem, they’re probably just oversimplifying the situation. The so-called STEM crisis is a perfect storm of different factors and so there won’t be a single actionable item to fix things.

But there are definitely many identifiable aspects of the challenge. For instance, the syllabus is by-and-large divorced from real mathematical practice (both in everyday life and in vocational contexts). Here’s a great little quote from Optimising the Future with Mathematics (via The Conversation):

Current mathematics education, in schools and universities, is satisfied with programming students to carry out certain mathematical processes, and assessment rewards students who can calculate everything even if they understand nothing.

So what can we work on at the ground level? Firstly, it’s vital to recognise the enormous continuity of learning in maths. All key learning areas exhibit a degree of intra-dependence within their skills and knowledge, but it seems to be especially noticeable in maths where a single “weak link” in the chain can be disastrous! Once student confidence is lost, it is hard (not impossible, but significantly challenging) to rebuild it.

Secondly, top-down (syllabus level) change is required, but we can’t wait for that to happen. Policy is always hard and slow to change (it must be in a democratic and bureaucratic environment), but we can push the envelope of our daily practices right now and see what works. We can undertake action research projects into what is effective and helpful. Another insightful quote:

We need mathematics “to be taught more like it is done” by those engaged in it, in both the innovations economy and research. This is a cultural change that involves the discipline itself, one that must be mainstreamed into school and university systems.

These cultural changes almost never come as mandates from above – they are typically born out of grassroots movements from below that are then recognised and ratified by authorities.

I’m a pretty slow thinker. In an argument, I’m that guy who comes up with the perfectly witty comeback… about two hours after the conversation is over. This doesn’t just happen to me in social settings – it also happens to me in the classroom! Often I’ll teach through a skill or concept only to realise, after the lesson is over, that there was something else I should have said, or some other analogy that would have been immensely helpful, which would have added valuable insight or made things clearer.

Something like that happened when I taught this lesson on congruence transformations:

In geometry, a congruence transformation is – roughly speaking – a way that we can change (“transform”) a figure in such a way that it is still the same shape and size (“congruent”). There are three main kinds of transformation that we cover in year 7: translation (sliding the shape to a different position), reflection (flipping a shape over) and rotation (spinning the shape around).

After introducing the idea of transformations and how they work, we start to think about “composite transformations” – what happens when you combine more than one transformation and consider the total effect from original to final image. An interesting fact emerges: that not all transformations are created equal. In fact, one of the transformations is more simple and basic than the others. One transformation can be used to create each of the other two. Which do you think it is?

It turns out that the most basic transformation is reflection. You can make translation and rotation out of a series of reflections, but not the other way around; though it’s counter-intuitive, reflection is the most basic kind of geometric transformation. Perhaps that’s why reflectional symmetry is so deeply ingrained into our natural sense of beauty.