OK, before everyone starts screaming about sexism, I used to go to my father for help with my maths homework, and this post is a reflection on the problem (grand)parents of my generation and/or younger are facing, or going to face, as the use of digital technology increases. Never mind about Gen-X or Gen-Y, the current cohort of students is definitely going to (have to) be e-Gen.
I have been ‘pushed’ towards this post because a number of different articles, talks and reflections have crystallized into a form which I feel satisfactorily explains or, at least, exemplifies what I think has always been a fundamental approach towards my own methodology in teaching mathematics, or which have helped me to clarify my own thoughts on it. It took another change in technology to bring it to the fore (or should that be four!). Be prepared for a bit of a ramble, but I feel I have to write this down because… I just do that’s why!
So, the articles…
First was an article in the BBC News magazine, September 2010, by Rob Eastaway (@robeastaway on Twitter, if you aren’t following him, and are interested in maths education, why not?) called : “Why parents can’t do maths today.” Basically, Rob explained that most of the problems you are having now, started with the National Numeracy Strategy (1999) which was aimed at primary education (in England). Of course, in my case, it was all about the new maths back in the ’70s…
Second was an article/broadcast by Keith Devlin (@nprmathguy now he could at least have used the ‘s’, being British and all…), last week (as of writing) on NPR in the US, called: “The Way You Learned Math Is So Old School.” Now, I’m not going to make any comments about the United States being a bit behind here – oops, just did! – but the content is essentially the same as Rob Eastaway was discussing in his item for the BBC.
Next, the talks…
Sir Ken Robinson (@SirKenRobinson) gave a talk to the RSA last year (2010), which I had a little rant about, and then I watched him again in a live presentation, streamed by Learning without Frontiers (edited highlights here), where he was again talking about creativity. I forgive him his harping on about maths, because he didn’t/doesn’t get it and admits as much, but in the interim I also watched John Cleese in a presentation he gave at the Creativity World Forum in Flanders in 2008. The main thing I took from John Cleese was that creativity can be taught, and I have long believed this myself. Interestingly, he also said “To know how good you are at something requires the same skills as to be good at that thing,” and “Most people who have no idea what they’re doing have absolutely no idea that they have no idea what they’re doing.” This takes some thinking about! Essentially, this means if you’re hopeless at maths, you lack exactly the skills you need in maths to know that you’re hopeless at it… However, he goes on to say “Teachers, who may not realize that they are not themselves creative, may not value creativity, even if they can recognize it.” (Yes! Yes! Yes!)
The Wolfram brothers. Yes, there are two. Let’s not confuse Stephen (who designed Mathematica, the Wolfram Alpha search engine, and wrote A New Kind of Science) with his (much) younger brother Conrad. [Oh, they’re both British by the way, just thought I’d mention that to the morons in the UK who are not investing in ICT research or education in computing technology… that would be the government, wouldn’t it?]. Now, Stephen clearly has delusions of grandeur, based on his idea that everything can be computed, but I’ll let him off, because he’s clearly a physicist – even though he mentioned Leibniz and not Newton! I still remain to be convinced that a computer/application/whatever can give me an answer to a somewhat vague question, though. “Single biggest idea of the Century,” hmm, I’m definitely reserving judgment on that one! By the way, I love what Mathematica and Wolfram Alpha provide…
I’m not sure that Stephen’s TED talk got blogged about, or retweeted, as much as his brother’s more recent one – Teaching kids real math with computers. Now, Conrad, we call it maths in the UK, and you should know that, particularly because you were giving your talk in Oxford… (Argh! It’s maths… maths… write it out 1,000 times… ) I don’t agree that mathematics has changed so much just or only because of computers, although I would agree that computers have speeded up the process. Learning Ancient Greek and Latin gave me excellent foundations in expanding my vocabulary in English, as well as helping with my other language acquisition, and I don’t remember ever being taught Ancient Greek in my maths class. Not using paper in mathematics, sorry, totally disagree… I am switching off now. Where is the connection between real world (origami) and the algebra/geometry to represent it on the computer? How are computers helping with visualization of 3D or understanding geometry and topology better? Yes, I agree with the programming, and almost always recommend the Project Euler site for that, but that is really only the algebraic part of mathematics. Teachers have to be able to understand (all or) most of the concepts before they can start throwing computers at students, and hoping they will ‘get it’. Personally, I got very little that was new from this talk because it’s something I’ve being doing (trying to do) already… Don Cohen (aka the Mathman) has also been doing things like this for many years, from which he drew up his calculus map.
Key points, as I see or expand on them, from the Wolfram brothers’ talks:
- Stephen had to “create a new kind of science” as a result of the use of technology.
- The ability to ask questions does not mean that the question asked was valid, nor that it has been critically explored by the person asking it.
- There are huge links between language, technology, science and mathematics, and… where were the arts here?
- Programming probably only addresses the algebraic aspects of mathematics, at least to begin with.
- Computation speeds up calculations and allows mathematically trained minds to move faster towards a solution.
Finally, the reflections…
I posted a little about my programming background when I introduced my first online manipulative. This post is essentially a follow-up to that one. I don’t think that anyone would disagree that the introduction of new technologies, applications and, in particular, computational power are increasing exponentially. It is ridiculous to ask human beings to perform tasks which can be carried out more efficiently and effectively by machines. We are currently entering a transitional phase between what I see as a tech-available and a tech-reliant society. Just considering what has happened during my own lifetime in mathematics, the impact is going to be very much greater than being able to use a calculator, and then being required to use one, in examinations.
When I was taught mathematics, algorithmic methods were taught and used because they helped to make the calculations more efficient. There were books of tables of sines, cosines, logarithms and so on. Part of my O-level required the use of a slide rule. By the time I came to take my A-levels, calculator use was allowed, but only certain types – none of your programmable calculators if you please! Move on to 1995, when I did my PGCE, and there’s a debate brewing about whether or not graphing calculators should be allowed, let alone required. The introduction of this type of technology allows the user to go much farther in the areas of mathematics which they can explore: wherein lies the problem…
Allowing or encouraging the use of new technology is great, and I am all in favour of it. However, the curricula, examinations, qualifications, systems of training and so on which are in place have had a good deal of time and money invested in them, and they are not going to be changed overnight. I think it is reasonably accurate to say that the effects of a new approach to content or methodology are going to take at least five, probably more like ten, years to start to make themselves known.
So that is why, when you were at school ten or more years ago, the mathematics (and let’s be fair many other subjects) you were taught are not and will not be the same as your children are being or going to be taught. The introduction of powerful calculating machines means that mathematics can now be taught or explored at deeper levels at earlier ages, and in ways that children understand. This also means there needs to be a paradigmatic shift in thinking to accommodate the freedom from the slavery of tedious calculation. This is not a new idea, Wilhelm Schickard wrote to Johannes Kepler in 1623 about his progress on developing a machine to completely automate the tedious calculations required to do astronomy at that time, despite the fact that John Napier had only recently discovered logarithms, which speeded up the process of multiplying and dividing a lot!
The e-Gen, as I called them earlier, are going to be bombarded with huge amounts of information which is easily accessible. More time will need to be invested on improvement of critical thinking skills, assessing and filtering the information, making connections and models in the virtual world which help us to understand the real world better… For mathematics, this means being able to understand number systems, thinking in a more algebraic way for programming, thinking more geometrically for design, thinking more probabilistically to handle data, and so on. That is why you probably won’t be able to help your child or grandchild or nieces and nephews with their maths, and it’s probably just as well you can’t! However, you can help by asking them to try and explain what they’re doing and why it works – if you have the patience to do so – since the latest changes to the curriculum are asking students to communicate more and show their understanding of the subject. Now, if you don’t mind, I’m going back to read my book of tables and play with my slide-rule!