We were sent a link to a blog post by Katie Steckles for the Math/Maths Podcast a couple of weeks ago. I’m preparing for the recording of episode 52 in a few hours and I thought I would share my thoughts on the topic here.
The blog post quotes another, ‘The Mathematics Generation Gap‘. This starts with “Profs do not know how their students were taught mathematics, what their students know, what their students don’t know – and have no idea how to help their students bridge those gaps.” This makes me think of the document written by MEI and published by my employer with others, “Understanding the UK Mathematics Curriculum Pre-Higher Education – a guide for Academic Members of Staff“. The problem this looks to address is that “it is not always clear what mathematics content, methods and processes students will have studied (or indeed can be expected to know and understand) as they commence their university-level programmes”.
However, the main thrust of the article is on what is called “The arithmetic gap”: “profs over a certain age (and some immigrant profs) were drilled in mental math;… students under a certain age haven’t been. Some implications of the arithmetic gap are familiar: profs who can’t understand why students insist on using calculators; students who can’t understand why their profs are so unreasonable. …” The article goes on to talk about analogue clocks and even Google Maps as forming a difference in understanding and approach between students and their professors.
The blog post Katie sent a link to, titled ‘“The Mathematics Generation Gap”‘, talks about “mental arithmetic tricks”. I don’t want to quote the whole thing here and stop you going to read the other post so I’ll take out a lot of the detail (…), but it gives an example: “to multiply any single digit number by nine, just add a zero to the end and subtract the number… Then, it’s easy to generalize, 9 times any two digit number is the number with a zero attached minus the number… Then extend further … This can be generalized further… This also leads directly to the proof…” Then we come to the main argument:
How do you discover this rule, and learn how to take it to a proof, without rote exercises that force you to search for shortcuts? I understand that the response to all of the above is to use a calculator instead, these tricks aren’t needed if you have a calculator at hand, but that isn’t the point. The point is that these exercises lead to additional insights, proofs, etc. and those insights are critical for more advanced insights and more complex proofs.
The inductive type reasoning that emerges from these exercises is valuable in many settings — I’d guess learning to find patterns is a skill that is useful beyond pure mathematics — and I worry that an over reliance on calculators will erode the development of these skills. I am absolutely convinced, for example, that forcing people to do econometric and statistical exercises by hand develops intuition that you cannot get any other way, and this is a key to moving on to doing proofs.
A related area is whether to allow use of computers for solving advanced mathematics. At work in January we ran the HE Mathematics Curriculum Summit, the report of which is now available. This included a debate on, basically, whether students should be expected to use memory, acquire subject knowledge and demonstrate technical fluency, or whether the computation part of mathematics could be left to computers, leaving the students to worry about when and why a particular calculation is used. However, the compelling arguments for me of students performing mathematics by hand there lay in understanding what a computer would be doing and what its limitations would be, whereas the arguments in the blog post seem to be that performing mental arithmetic develops other skills that a mathematical thinker ought to have.
What is my view? Certainly the point isn’t finding the numbers; if it were a calculator or computer can be used for certainty. Having said that, there are other areas of mathematics that are well suited to developing this mathematical thinking. I appreciate the desire to encourage pattern searching, logical reasoning, abstraction and extension, but I’m not sure forcing students who haven’t been brought up on mental arithmetic to do such tricks is a productive way of doing so. If everybody has a phone or calculator in their pocket that can solve the question in a millisecond, then forcing them to not use that device and do it by some mental trick instead is just going to put people off, I would say. Beyond this, a lot of people have a genuine anxiety, or some even a disability that can produce a panicked reaction when faced with numbers. Doing something in a non-numerical area might be much more effective. Tilings seem to be a good option, and at work we are running a workshop at Greenwich in a couple of weeks, led by Noel-Ann Bradshaw and at which Katie is a presenter, on using problems, puzzles and games to develop mathematical thinking. Areas such as these can be used to develop the same skills but don’t have the hangups of mental arithmetic. In fact, I have a group of people coming round this afternoon to plan our stall at a local science fair in a couple of weeks. I intend our stall to be themed around using puzzles for developing problem solving skills. Beyond this, mental arithmetic forms part of a number of magic tricks for which a calculator would give the game away, so perhaps encouraging students to play around with this sort of thing may give a motivation to learn some mental arithmetic tricks. (Of course, this all depends what topic you are trying to teach.)
Overall, I think the battle is lost – the distinction between profs and students is not as clear as this article would have it because plenty of (and increasingly many) lecturers will have been brought up on calculators as well. I agree there are differences between how lecturers and students approach mathematics, some of which will be generational due to the increasing availability of technology; some will be due to the lecturers being unusual (perhaps more capable and motivated than average) students in their day. Still, if the aim is to develop a mathematical topic, using modern tools to make this more efficient is a good thing; if the aim is develop mathematical thinking I think there are more interesting approaches for developing the kinds of skills the blog post author would like to develop.
The blog post ends: “But what is your view on all of this?” Katie has sent me her view for the podcast and she may choose to repeat it in the comments but because I have been sent it for one purpose I don’t feel I should copy it out here. Perhaps you will share your views in the comments.