Gosh, I’ve not done very well in keeping up with this series of blog posts, have I? For the past week I’ve been caught up in the Edinburgh International Science Festival, helping to chair some talks and run an exhibition at the National Museum of Scotland about game theory. More on those in another post perhaps.

So, Friday. Although it’s been over two weeks since this particular Friday, I remember it very well. It was on this day that we were lucky enough to have a visit by the distinguished professor Eric Mazur.

Although Mazur is distinguished in his field of physics (lasers, semi-conductors, optical fibres), it wasn’t a physics lecture that everyone turned out to hear. Surprisingly (even to himself) he has become most well known for his radical teaching method, known as **peer instruction**. The talk that he gives about this is really fantastic, and I recommend that everyone watches it on YouTube.

It is ironic that Mazur should be touring universities around the world, giving lectures about peer instruction, when a fundamental tenet of the theory is that we shouldn’t be lecturing to students! The idea is that people don’t take in information when they are forced to sit and listen to something; they have to be **doing** and **discussing** the subject matter in order to really engage with it. This became especially clear when Harvard University physics students were given a simple exam which tested their basic understanding of Newtonian physics.

For example, suppose that a heavy truck and a small car crash into each other. At the moment of impact, is the force of the truck on the car (a) larger than, (b) smaller than or (c) equal to the force exerted by the car on the truck? Think about this for a moment…….. Your intuition is probably telling you that the truck exerts a larger force on the car than the car on the truck. Yet anyone who can remember their high-school mechanics should know Newton’s 3rd law: that forces are always equal and opposite.

What was interesting was that Harvard physics students got this question wrong almost as often as the general public did. It’s an extremely strong indication that students are only superficially learning information, memorising things in order to pass an exam but not really internalising the concepts. We see this all the time with our maths undergraduates too. They can compute all manner of difficult integrals and solve complex matrix equations, but in the end very few know what the answers mean or why they are important. What actually *is* the determinant of a matrix, or what does it mean to have an infinite decimal expansion?

But if lecturing doesn’t work as a means of education, then what else can we do? Eric Mazur’s answer is **peer instruction**, which works something like this. The students read the material in a textbook before the lecture, submit a list of things they don’t understand to the lecturer, then during class the students work through questions designed to address and correct their misconceptions. Questions are often presented in multiple-choice format and students have the chance to vote on an initial answer before discussing with their peers and then voting again on a new answer. From research into this method, it seems that students teach each other a lot more effectively than a lecturer can. They understand each other’s problems and can more easily get to the heart of the explanation. And nobody can just sit and sleep through the lecture, because there is constant discussion of the material in the class.

At Edinburgh, we would love to try and implement this method with our first year undergraduates in September. The main difficulty in starting out is getting those multiple choice questions which can really change people’s opinions about a subject. What are the common misconceptions in maths? When does our natural intuition override the definitions we are given in lectures, like in the physics example before?

The only university which has really implemented teaching like this in mathematics is Cornell University, and you can take a look online at their list of ‘Good Questions‘. Here’s one to get you talking:

**Was there a time in your life when you were exactly pi feet tall?**

A good question, to me, is one where you have an instinctive immediate answer, but then when you think more carefully you really get to very deep questions about the subject.

For example, someone might immediately think that the answer to the above question is “no”, because pi is an infinite non-repeating decimal and nothing can be ‘exactly’ pi feet long. But then they might think, is it really because pi is irrational that the answer is “no”? Was there ever a time when they were exactly 3 feet tall? Or they might think, there was a time when they were less than pi feet tall and now they are bigger, so surely there must have been a moment (however brief) when they **were** exactly pi feet tall. Discussions of this will get to the heart of the real number system and questions of approximation, which are essential for anyone studying analysis to master .

I would be really interested to hear from anyone with opinions about this. Do you remember which concepts in maths you struggled with the most? When do you think you learnt your specialist subject: through listening to lectures, or at home with a textbook, or chatting to friends? Do you believe that peer instruction can work or do you think the system is fine as it is?

Send me your comments!

Posted by Nathan on April 20, 2011 at 12:18 pm

I think that peer instruction, and getting people to do things and discuss them, is at the heart of how people really get the most from learning. It’s definitely at the core of the kind of things that I like to get involved with: telling people about something, getting them to have a go at it, then getting them to reflect on what it is that they’ve done and how it worked – I think this is just as important in maths, science, english literature etc as it is in skills training.

Was there ever a time I was pi feet tall?I think that there was (I’m six feet four inches now, so it was a while ago!); I’m kind of throwing ideas around in my head for ways to show that a person is pi feet tall, and I’m thinking at the minute that it could be done – in principle. But it’s a fun problem so am also happy to be proven wrong! đź™‚

Posted by Cynthia on April 21, 2011 at 2:55 am

I am a dedicated Peer Instructor, but aside from that I wanted to answer your other question about where I had troubles in maths. I am a physicist who routinely did horrible in math classes, mainly because I didn’t see the point (in physics, the math describes a physical system, so I could tolerate it more).

Three examples stand out, though.

One was an essay in high school that I was forced to write in my math class about why you cannot divide by zero. It started off being the worst assignment ever, but by the end i really enjoyed the abstract thinking about it.

The second occasion was when I realized the graphical connection between a function and its derivative (and second derivative). This was through my first year physics teacher who always made us graph everything. To this day, if i need to know the derivative of a -sin curve, I make a sketch.

The third time was when I finally realized the big picture behind riemann’s squares and integration. This was also my first year at university and again the math teacher wouldn’t rest until you answered your homework in words. It took some practice to get used to, but it was my favorite math class (and that coming from a math-hater).

I’m not sure if you’ll find this information useful, but I was prompted to share it after reading your blog post. Note that I didn’t really address Peer Instruction, but both of the two courses mentioned above (in college) were peer learning environments, and are to this day the best courses I ever took (and feel I learned the most from the actual course as opposed to teaching myself out of a text book).

Posted by Weekly Picks « Mathblogging.org — the Blog on April 28, 2011 at 4:04 am

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Posted by Peter Krautzberger on April 28, 2011 at 4:50 am

Have you thought about asking this on mathoverflow? Seems like a great big-list question to me.

Posted by Paul Caira on May 5, 2011 at 2:53 pm

Superficially, I’ve no doubt that there was a day when I was 3.14 feet tall (to 3sf) and that there was a (presumably subsequent) day when I was 3.15 feet tall (to 3sf). If height increases continuously, then there must have been an intervening time when I was exactly pi feet tall, but we might not be able to identify it the instant.

However, the ‘granularity of reality’ must be a reason to suspect that this is not a meaningful thing to believe. Since many phenomena are quantum, and happen only in jumps, at a low enough scale of distance, perhaps height cannot increase smoothly.

The simplest idea here is to say that I can’t increase my height by fewer than one atom, but even that’s not simple. What about the distance between the atoms changing? Is that continuous?

Harder is the question of whether it is meaningful to talk about the location of an atom, and therefore the ‘limit of extent’ of my height. Heisenberg’s Uncertainty Principle suggests otherwise. So even though we might know that height has increased from one number to the next, we can’t guarantee, it seems to me, that it has been all the heights in between, even in principle.

Well, the mock exam I’m invigilating is nearly over, so time to stop.