Mathematics

Matrices Quiz – worked solutions

Posted on

Hey class – as promised, you can view the worked solutions to your Matrices Quiz. Complete your corrected solutions and then start working on the AT5!

Advertisements

Index Noughts & Crosses

Posted on

index_noughts_crosses

I’m endlessly searching for new ways to present and engage with old ideas. Some of the most wonderful experiences come from seeing or working with familiar things in a new way. So I was delighted when I discovered the game Nuxo on the iOS App Store recently.

It’s a simple game and if you’ve got an iOS device you can go ahead and play it yourself. But as I played it I realised there was so much scope for using this in the classroom. I was teaching index laws recently so I modified Nuxo into a game I call Index Noughts & Crosses. You can see how it works below:

Download the spreadsheet and generate your own boards!

Riding the wave: why I started a new Youtube channel

Posted on Updated on

I really admire intentional people. I mean the kinds of people who think about what they want to achieve, set goals to get them on the right path, and then go out to strategically accomplish their objectives. Those people inspire me: they show me what’s possible when you set your mind on something and then pursue it with all your strength and discipline.

But I’m not one of them. At least, I definitely don’t feel like one. Many of the things I would consider my greatest achievements have arisen from things that took place accidentally or out of my control. I feel like a surfer who just happened to be at the right spot at the right time to ride a wave into the shore.

For instance, my Youtube channel – which is what I’m most well known for at the moment – was never intended for wide public consumption. I started shooting and uploading videos for just one boy I taught two years ago. He was very unwell and missed huge amounts of school due to hospital visits, so I put things up on the channel so he could keep up with us when he was away. This is a little embarrassing now, but I initially made the videos public simply because I didn’t want to deal with the headache of adjusting privacy settings on every single thing I uploaded!

Since the videos were freely available, others started asking if they were allowed to watch too. It started with the other students in my class – no one else knew I was doing this, and they only knew because they saw me setting up my recording equipment each lesson. From there, it spread to students in other classes, mainly Extension 2 students who had glossed over the 2-Unit topics in class and wanted an actual explanation of things like Series and Sequences. And before I knew it, students from other schools – and even other countries! – were watching along.

All of this blew me away. As I said, none of it had been intentional. I never set out to have a popular Youtube channel; it just kind of happened by itself. But at the same time, it wasn’t all that surprising. Mathematics is, at its essence, the same everywhere – and in fact for most of school across the world, it’s compulsory – so it wasn’t that unusual that I found a wider audience of students. What surprised me most, though, was when the people who commented and emailed me weren’t students. They were teachers.

Why would teachers be watching my videos? The videos are aimed at students, not educators. But then I realised why it made sense. In mathematics – not to mention a hundred other areas of human endeavour – one of the key ways to learn is by example. Yes, you can explain the concepts or principles as much as you like – but what does it look like when you _use_ those principles? What do those concepts look like in action? This is why observations form such a helpful part of initial (and continued!) teacher education: there’s nothing quite like seeing someone else actually do something to help you wrap your head around how you might do it.

And that’s what my videos were helping to provide. Just one example of how to go about explaining things. Not the only way, nor the best way – just one way. And apparently there’s a need out there for this kind of thing. Then I thought to myself – are there things I would say to or upload for teachers if I had a channel just for them? And I realised the answer was yes.

So, in light of the opportunity that’s standing right there – to offer practical and thought-provoking content to hundreds of pre-service, early career and even experienced mathematics teachers out there, for free – I’m starting Wootube²: videos for anyone and everyone interested in mathematics education. I’d be lying if I said I have a huge amount of time to invest in the channel – this is most definitely a side project – but I already have tons of ideas for material to post that I hope will be helpful in developing teachers and cultivating constructive discussion around teaching and learning mathematics. If you are a maths teacher and you have ideas or requests for things you’d like me to do on the channel, please contact me and let me know. Otherwise, subscribe and stay tuned!

Number Systems

Posted on

We have spent so many years doing maths with the same set of numbers that we often forget that our way of writing numbers is just one way among many. We use the Arabic numerals in Base 10, but there are many alternatives each with their own story. For two other number systems, research the following:

  1. What is the purpose of this number system, and what are its distinctive features?
  2. Explain where and how this number system is used today.
  3. What are the advantages and disadvantages of this number system?

Put these into your own post and submit them by the end of the lesson!

A response to: the transcience of sharing

Posted on Updated on

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.

Working out irregular areas

Posted on Updated on

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

CTHS_aerial_map

Practical tips for maths teachers: the growth mindset (TER Podcast follow-up #2)

Posted on Updated on

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.