Maths education: again

I have written a few posts about education.  But I’d not seen this presentation by Conrad Wolfram – brother of someone who may be one of the intellectual giants of our time – Stephen. (Since Stephen is a good deal older – born in 1959 with Conrad born in 1970 – perhaps one might call Conrad “Wolfram Beta”, but I digress).

Anyway, Conrad’s TED talk is very well worth watching. His case is simple and compelling. I couldn’t agree more.  It’s kind of tantalising, frustrating to have something so obvious within our intellectual grasp and yet to be so far off in terms of realisation, so far off because the workplace is a mass of routines. Even small routines can be difficult to break but usually they come in numbers which form a thicket which somehow kills off its enemies which die the death of a thousand cuts.

For those of you who don’t want to watch the video – I sympathise – after all you could read the words, jumping in and out at points of greater and less interest in a fraction of the time. If that’s you, you can read the words here.  Even better, I’ll summarise the basic message which is pretty straightforward.

Maths, Wolfram argues consists of four steps.

Steps to doing math

And as Wolfram says:

Here’s the funny thing. We insist that the entire population learns how to do step 3 by hand. Perhaps 80% of doing math education at school is step 3 by hand and largely not doing steps 1, 2, and 4. And yet step 3 is the step that computers can do vastly better than any human at this point, so it’s kind of bizarre that that’s the way around we’re doing things. Instead, I think we should be using computers to do step 3 and we should be using students to do steps 1, 2, and 4 to a much greater extent than we are.

Remarkably like the teaching of economics too – though it focuses on both calculation and model building, but only in passing on 1 and 2 and just a bit on 4.

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Patrick
Patrick
10 years ago

That is in fact a compelling summary. I am going to rethink how I teach my kids math now :(

Rohan
Rohan
10 years ago

Data mining competitions (kaggle.com) also tend to be all about step 3 too. A well designed competition includes as much of steps 2 and 4 as is possible in a competition framework.

conrad
conrad
10 years ago

I think he’s wrong for quite a number of reasons. He reminds of the anti-phonics people who offered easy solutions that sounded really good, but in the end didn’t work very well in the long term. The main difference is that very few people have any idea about what you learn from maths that isn’t directly related to solving maths problems, unlike reading, which became a political issue. Here’s a few complaints and a few observations:

1) The reason you teach maths like you do in early years at least (and possibly later years too) is that it leads to the development of lots of things you need for later in life, like the ability to understand inequalities and do visual-spatial tasks. On this note,there’s an interview with Stephen Hawkings somewhere, and he suggests that one of the reasons he became so good at physics was because he had to try and visualize and remember everything due to his illness, and he got really good at this which helped him with his physics problems a lot.

I’m sure I’ve pointed this out before, but if you want a good example of this that isn’t just a simple observation of one person, then you can look at people that learn calculus with and without graphical calculators. The ones that use the calculators never end up learning how different graphs looks, and importantly, this seems to affect their ability to visualize how different shapes look. So, for example, if they were an engineer, and you asked them a question like “what do you think is the best type of bridge for this” I assume that not being able to visualize the solution would affect them. Now perhaps there a computer packages where you can just look at ever common type of bridge, but one wonders how works of true genius (or indeed anything a bit different) would ever get created if everyone relied on these (I’m glad Gaudi didn’t have a computer when he drew up the plans for the Sagrada Familia).

2) Whether teaching theory first or pratical stuff first leads to better outcomes is an empirical question. Last year a paper came out in Nature showing that at least university students in introductory maths courses learn better if you teach the theory first. This is why you need to test things rather than just assume that what sounds good is good.

3) Doing the calculations really does help you understand things. If you want an example of this, then look at people taught press-the-button-and-hope-magic-happens statistics, and look at people who had to do the calculations for the early simple stuff. You’ll find the first group really has no idea what very simple things are, such as what variance is in, say, a between groups vs within groups design (or what variance is at all). Alternatively, the group that had to do the calculations will have a vastly better understanding.

4) Programming isn’t mathematics. There are many programmers than don’t know the first things about mathematics and arn’t able to apply the type of logic and reasoning you get from learning it — perhaps most that have done business computing type courses where you never learn any real mathematics. So the analogy he makes is poor. I’m better at programming than maths, for example, and so if I have to program hard(-ish) maths problems, it really gives my brain a strain and takes me comparatively ages. Alternatively, writing big programs for complex procedures that don’t involve hard maths I could do blind-folded.

5) You can check things out cross-culturally to see what works. In Shanghai, everyone learns maths via the traditional Chinese style beat’n learn learning, as they do in HK too. Both these countries always win the we’re good at maths comparisons, excluding Singapore, where they also learn via that style. This type of learning involves liberals doses of repetitive problems, learning your 15 times tables etc. . Does it lead to better outcomes apart from the cross-country comparisons ? Well, anybody that’s taught in a university in Hong Kong will realize that in terms of mathematical understanding, there’s a grand canyon of difference between the students there and Australia. I was shocked when I came back to Aus and found that just getting many people to read a graph correctly was hard.

desipis
10 years ago

I’m not convinced. Particularly about being able to do steps 2 and 4 well without a strong understanding of how things work in step 3. There are certainly artefacts in the way mathematics is taught that a left over from the pre-computer age that are now largely unnecessary. However I see that a shift from pen and paper to computer provides an opportunity for a evolutionary step in how mathematics is learnt, not a revolutionary new way to teach it. The main advantage I see for teaching mathematics is being able to demonstrate the concepts using interactive visualisation tools, and perhaps have students do mathematical calculations (step 3 stuff) without requiring them to be intimately familiar with the language of mathematics as it is currently written.

I think more appropriate way to use computer based tools that perform calculations is in teaching other subjects that rely on mathematics. Science, geography or economics could focus learning how to apply the mathematics tools while leaving the detailed internals of the calculations to the mathematics subject.

conrad
conrad
10 years ago

“The main advantage I see for teaching mathematics is being able to demonstrate the concepts using interactive visualisation tools”

This is really a poison challice. Like I noted, if you want an example of where this gives you the opposite result you want, just look at the effect of graphical calculators. I’m not exactly sure what the underlying reason is for this, but it might simply be something to do with human memory — people remember things much better that are slow and take a lot effort and thinking to do (like when you graph functions by hand, work out minima and maxima and so on) compared to when they don’t (like when you type in the equation and press the button).

I might say, that, in my books, the real burning question (which probably only burns for a few weird people :) is not so much whether everyone learns maths well, it’s why the right tail of the distribution has been shrinking in the last decade or two — i.e., why we arn’t getting as many kids that are really good at maths as before. Personally, I think it doesn’t make piles of difference if the average person is good, bad, or indifferent at maths, but it does make a big difference if you don’t have enough people really good at it (i.e., you don’t have a pool of people that can become engineers, scientists etc.).

desipis
10 years ago

conrad, I wasn’t suggesting that the tools replace learning to do things like plot graphs manually, but useful as an introductory tool. For example, rather than have static graphs in a textbook showing what quadratic, cubic, and so on graphs look like, you could have a tool where you can manipulate the variables and have the graph adjust itself. Then once the basic concept is established, the manual process of plotting and sketching is taught to facilitate the understanding. Of course this idea relies on students (and teachers) having the discipline to follow through with the actual work rather than relying on the tool (which I suspect is the problem with graphics calculators).

Steve X
Steve X
10 years ago

On what basis do you assert that we are not getting as many kids who are really good at math as before?

Why does the removal of calculation, i.e. the somewhat mechanical part of things, always apply to the teaching of math? Are there advocates who suggest less spelling should be learnt as computers do it now? Why not remove grammar as well?

Also, given that it’s a serious possibility that we’ll have computers that can put together essays given a few bulletpoints would it then be wise to dispense with essay writing as well?

conrad
conrad
10 years ago

“On what basis do you assert that we are not getting as many kids who are really good at math as before?”

Andrew Leigh had some data on this on his blog some years ago. It may still be there since he became a politician. I’m not sure what he did with the data in the end (I presume he must have published in somewhere). If you want to look at the government figures, then what you’ll find is a slow and steady decline of kids doing advanced maths in year 12 also (and an increase in those doing veggie/business maths). You’ll also find that advanced maths became easier than what it was 30 years ago if you care to dig up the old textbooks.

“Are there advocates who suggest less spelling should be learnt as computers do it now”

Yes. The best example of this happened in Japan, where many people started using computers to generate Hiragana (I could confusing this — I mean the Chinese-like writing form). The consequence of this is now they have many people that can’t write well and are now entirely dependent on these devices. A lot of this is implicit, and it’s not clear to me how you could stop it (who doesn’t use a spell checker these days?) unless you stop kids writing stuff on computers.

“Why not remove grammar as well?”

Are they teaching grammar in schools again:) ?

Cathy
Cathy
10 years ago

You can consult the Barrington Brown Report on Year 12 students in Mathematics. It is getting a bit old now (2006), but I doubt that things have improved much since then.

As a maths teacher, I must say that I completely agree with Conrad. At the school were I work (in NSW) we occasionally get students from other states such as WA, where graphics calculators are used extensively. Those students are generally several years behind in their mathematical development. They can’t perform or follow basic algebraic manipulations and they haven’t developed any higher order conceptual understanding, either.

Btw Conrad, the Chinese type script is called “kanji”. Hiragana and katakana are the two phonetic Japanese scripts.

wizofaus
wizofaus
10 years ago

There’s certain nothing wrong with accepting certain skills are no longer required due to better technology – presumably at one point learning how to fill ink bottles/prepare quills was an important part of basic education, and we gave that up happy to make ourselves dependent on better pen technology.
But I do wonder how far you can take it. It does instinctively bother me that people utterly depend on computing technology for skills as basic as arithmetic and spelling. The ability to manually draw your own graphs seems less obviously essential, and if there is strong evidence that kids that don’t learn this suffer with other more advanced skills, then it’s worth asking whether graph-drawing is the most effective way to prepare for those later skills.

Mark Heydon
Mark Heydon
10 years ago

I haven’t watched the talk – I seem to be lacking a plug-in in my (newly installed) browser. However, based on the description and dicussion, I reckon the computational step 3 is pretty vital to be drilled into students as a foundation to learning maths.
I work in a very numbers intensive field and get presented with work regularly where, within seconds, it is obvious that there are mistakes. As a result of many many hours of computational practice when I was young, I am fortunate to have the ability to make very quick estimates of ballpark answers and when the results presented aren’t in that ballpark, there is almost always an error. The problem seems to be that younger people today (god, am I that old already!), even when they are quite bright, have not done the same level of foundation work and if the computer says X, then X must be the answer. However it is very easy to make either a data entry error or a small logic error and get to the wrong answer.

murph the surf.
murph the surf.
10 years ago

Hiragana evolved from the kanji- each symbol a simplified phonetic script based on the sound of a kanji character.
Katakana is reserved for foreign words- and the difference in script is to show that the word is foreign while being based on the same phonetic sound.
How is that for discrimination?
The point Conrad highlights shows up in memory skills too – learning the JoYo kanji character list of approx 2100 chararcters by the age of 14 is quite a feat and trains the brain in skills that others just can’t access.It would possibly be more so with the chinese system where you need 2500 characters to read newspaper.

aidan
aidan
10 years ago

Count me in as a skeptic of Wolfram Beta’s thesis.

Doing a reasonable amount of boring repetitive mathematics is how we program our brains, which are sort of mush computers after all, to do this stuff.

Once our brains are capable of doing ‘3’ it crucially informs how we do the other steps.

Patrick
Patrick
10 years ago

I think some of us might be taking the thesis too far. I couldn’t imagine not teaching ‘3’ quite comprehensively. But in my experience of schooling, both my own and that of my kids, we could do much more on learning how to frame a problem.

For example I try and teach my kids (well for now the eldest) how to break down apparently complex problems into sth more manageable, but this doesn’t appear to get a lot of airplay at school. I do think that in many more areas than just mathematics we don’t spend enough time asking what is the question rather than answering the question.

Steve Perlman said basically the same thing at his latest lecture at (UC perhaps?) in that innovation is mainly asking the right question.

Amanda
Amanda
10 years ago

RE: Mark Heydon’s comments on estimates and mistakes, I remember seeing research some time ago showing that aircraft pilots often override their own knowledge and instinct when presented with conflicting information from a computer. This frequently had (potentially) disastrous consequences. We need to have the skills and the scepticism to question computers.