Archive for November, 2011

ICT in the Classroom

November 18, 2011

This week’s MEd session was all about ICT in the classroom. We were asked last week to think about anything we could present on the topic for the other students, and I thought it made sense for me to talk about the way NRICH promote ICT in the mathematics classroom so I put together a few ideas for a five minute talk. It makes sense to write those ideas up here so they are not lost.

Firstly I made the point that as a web-based project NRICH is dependent on ICT to get its resources to its audience, but once they have visited the website, for a lot of the problems the teacher needs nothing more complicated than a board and pen/chalk to introduce the tasks. For others, there will be printable resources that can be handed out. But it is the problems with interactive elements that seem to raise the most interesting questions in terms of ICT use in the classroom.

One of my favourite interactive resources is Dozens. The interactivity allows students to keep trying, getting feedback if they get it wrong, and to generate unlimited examples at each stage. There are levels of difficulty corresponding to divisibility by two, three, four and six. Once students are confident with the mathematics, there is a final challenge to work on with pencil and paper. I like this resource because the feedback allows a whole class of students to work at their own rate without relying on the teacher to tell them how they are doing.

Charlie’s Delightful Machine also offers students the chance to work on unlimited examples that are different each time, but that’s not why I like it. The coloured lights in the problem are an enticing hook to draw students in, and the mathematics needed to completely solve the problem of when it’s possible to turn all four lights on is quite sophisticated. Again, there is the opportunity for students to work individually on the problem before the teacher brings the class together and draws out discussion points.

GOT IT is an old NRICH favourite. Again, there is the opportunity for students to work on their own and get feedback from the computer, perhaps studying the computer’s strategy and trying to work out why it works. One nice technique that we suggest in the teachers’ notes to several problems is to set a challenge, in this case something along the lines of “In a while, I will stop you and set up a game with a different target and maybe the numbers from 1-6 or 1-7, I haven’t decided yet. You need to have a strategy so that you can quickly work out the best way of winning my game, whatever it is.” This forces students to generalise, as they are being asked to come up with a way of solving an as yet unknown case.

Regular visitors to NRICH may have noticed a few videos appearing lately. This problem is a new version of something published on the site some time ago. There is a lot of power in showing children something in silence and inviting them to make sense of it. Of course, the teacher could demonstrate what’s in the videos on the board, but by showing the videos it removes the teacher’s position as the expert who knows what’s going on – “Shall we watch this video together and see if we can make sense of it?” rather than “I know what’s going on and you have to figure it out”.

Finally I mentioned that we publish problems where technology makes a solution more accessible, for example problems where graphing software, or dynamic geometry, or using a spreadsheet makes routine calculation or graph drawing or example creation much easier, and offers routes into the problem that pencil and paper methods wouldn’t allow. Whenever we do this, we try to signpost it in the problem and the teachers’ notes. I think part of a maths education should be learning to use such tools, which is why when I published Which List Is Which? last month, I included the data in a spreadsheet for students to download and manipulate.

There are other ways of using ICT in the mathematics classroom, and other reasons for doing so, that I haven’t mentioned. In fact, I’m sure there are ways that NRICH support and promote ICT that I haven’t thought of, so I guess the comment space below would be a good place to talk about anything obvious I have missed!

Conventional Wisdom

November 10, 2011

There are some truths in mathematics that are true because they are true because they are true. For example, if I have a right-angled triangle in the plane, the square on the hypotenuse has to be equal to the sum of the squares on the other two sides. Other truths are true in a different way. It’s true to say that this:

is a square, but it’s not true in the same way that the statement of Pythagoras’s Theorem above was true. And of course, saying that the statement is Pythagoras’s Theorem is true, but not in the same way that the Theorem itself is true.

Confused yet? Great!
The notion of different types of truth has been around for a very long time, and has been recast using many different descriptions. I’m rather fond of Dave Hewitt’s designations “arbitrary (socially agreed names and conventions) and necessary (properties and relationships)”, so calling that red shape a square is arbitrary, because I could call it anything I liked, I just choose to follow the social convention to call it a square, but Pythagoras’s theorem is necessarily true, it is a property of right-angled triangles, it is a relationship I can derive for myself.

These necessary truths are vital to mathematics, in fact, perhaps in some sense they are mathematics. And in teaching mathematics, the NRICH philosophy draws on lots of great thinkers in maths education and comes to the conclusion that these truths are something that children should have the opportunity to explore and discover for themselves. They will never be able to discover that a square is called a square without some external influence (an adult, older child or dictionary telling them that a regular quadrilateral has the name “square”).

But despite my strong feeling that it is the necessary truths that are core to mathematics, I also think that educating children into the conventions of mathematics is important. Part of being a mathematician is being able to speak a common language with other mathematicians. This means knowing the definitions, being fluent in the notation, understanding the conventions.

I ranted a bit on Facebook earlier about the questions like “20 + 20 x 0 + 1” that have been popping up, with a popularity contest where people vote on what the answer should be. An idea that came out of that discussion was that one of the reasons that people don’t remember BIDMAS, BODMAS, PEMDAS or whatever it’s called in their local language is that they don’t see a need for such a convention. For me, the link between arithmetic and algebra means that order of operations is firmly embedded – if I was evaluating 2 + 3n, of course I would do 2 + (3 x n), so if I’m doing 2 + 3 x 4 I think of it in the same way – in my mind, the “three times four” is grouped together. Given the widespread lack of awareness that mathematicians have a convention for order of operations though, I think if I have need to write a calculation down for others I will use extra brackets just to be on the safe side!