## Mathematical Scales

Thanks to the move to the US, my son has a new piano teacher. He is playing at an advanced level, beyond grade 8 (for the UK audience), with pieces by Bach, Mozart and Chopin often ringing out. Yet for the last couple of months he has been taken right back to the basics. Looking again at simple techniques on how fingers hit the keys and going over scales.

I am in love with this idea of training, taking someone who has proved incredibly able in an area and taking them back to the most basic ideas. I started to wonder what the equivalent might be for mathematics. What exercises should we be giving to starting PhD students?What exercises could we ourselves try in order to gain intuition and insight into the basic workings of our subject. I have a first proposal, but am sure there are others? What do you think? Of the idea itself, or of suggestions of possible exercises?

**Multiplication Exercise**

Multiply all possible pairs of numbers from 1 to 99, that is 4950 different calculations. At a conservative estimate of 120 per hour (most will be a lot quicker than 30s, some will be longer!) that is just over 40 hours work. That could spread quite nicely over a month, maybe two along with other activities. It would be 40 hours of meditation on the most fundamental of mathematical operations, what might come from that?

**Other suggestions**

A couple of excellent suggestions from commentors in a lively debate on reddit:

1) Teaching, which of course is already a significant part of graduate training in the US, unfortunately less so in the UK (those being the two systems I have worked in).

2) Deep study of proofs, with mention of this beautiful paper of Dykstra.

Sometimes mechanical repetition of the same operation brings fruit, but what’s the point of multiplying numbers? You want to produce researchers, not computers?

@noseynetty The point is to develop intuition of the basic ideas of mathematics and so build the firm foundations required for the creativity of research. In other fields, such as music, this is accomplished in part by detailed study of the mechanics. Perhaps the same will help in mathematics. What exercises would you suggest?

That’s a great idea, like having some kind of “exercises” to “solve” while you learn a new theory! I wonder why nobody has thought of this before!

@naem I would hope that for most people starting a PhD in maths multiplication would not be a new theory😉

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@gelada: For most people starting a PhD, I suspect that the foundations of numbers, counting, and the various operations that can be created would be a good refresher.

I like the idea!

At a first thought, it seems like this is done already in some sense at least as a lead-in to many higher math courses. For example, revisiting what distance and measurement really mean in a real analysis or measure theory course, starting with the simple (the length of the interval [0,1] is 1) and moving to more complicated with boundary-pushing counterexamples like the Cantor set. Or revisiting the most basic of operations, addition, in an abstract algebra or number theory course: understanding what makes addition addition, which characteristics can generalize to other groups than the integers, figuring out how addition works in modular arithmetic, etc.

Is this similar to what you had in mind? I think the multiplication exercise you suggest has the added benefit of slowing down to think more deeply about a simple topic, without pushing too quickly on to pushing the boundaries. But boundary-pushing is essential for growth!

You are right, in many courses we introduce ideas and then lead them into abstraction. Pushing the obvious boundaries of the methods and concepts. The motivation for this is different, to push at the boundaries we take for granted.

We can assume that we understand things that we can do. As you say the multiplication slows one down, helping to find the footing so you can push more firmly on the boundaries.

I don’t think an exercise such as this would compel any arithmetic insight as opposed to just playing with some basic number theory. You can do some impressive calculations when you understand something as simple as prime factorization.

For example, take every positive divisor of 20! and multiply them together. The answer, which you can get in less than a minute, is (20!)^20520. That’s an amusingly large number and a nice reward for understanding some basic number theory.

@Eric By learning number theory you gain other people’s insights. That is not the same as having insights of your own. Even if someone else has had them before you.

I see your point if the intent is to find your own fundamental insights as opposed to creatively discovering new insights upon given fundamentals.

On the other hand, once you see integers in terms of their factors, most products sound the same as any other, but for the occasional timbre of a new prime. For example: 80 * 80 dissolves into 2^8 * 5^2 and from here I see that 80 * 80 was a simple variation on a theme of two primes. In this factorized form it’s easy to see relationships between 80, 80^2 and other integers and I could spend a lot of time exploring, though my new insights wouldn’t be as fundamental as discovering the utility of factorization itself.

I suppose without some specific ideas, it’s hard for me to see what I could gain from the straight products themselves, which to me seem to hide both the common simplicity and rich connections between integers. I think factorization is a simple tuning system within the cacophony of possibilities. Perhaps your idea is for people to find their own tuning system, which isn’t a bad idea, but you can be a jazz musician without discovering equal temperament yourself.

Ok, I get it now. Probably you’ve heard about R.L.Moore and his method of mathematical education?

http://en.wikipedia.org/wiki/Moore_method

In the original version of Moore’s method students are supposed to discover theory on their own, without textbooks or help from the instructor. Experience shows that many of them hate it, but probably PhD students would be more motivated. The only problem is it would take a lot longer to learn something using this method, and so modern 3-4 years on a PhD probably won’t be enough.

@noseynetty Yep I know the Moore method, have even taught using it. I agree that it is slower, but that is because it is also teaching other things. Doing everything that way would be a bad idea. One should not have to reinvent every wheel!

The idea here is related, but not quite the same. Take students back to the basics that they think they know already, rather than try to get them to create their next material.

I think this is a beautiful idea. Going back to the bare basics of a subject and exploring them with the experienced mind is a humble exercise and can be applied to most subjects. The actual approach and timeframe is probably a material for debate but the core of the idea is fascinating and can enhance one´s understanding of any subject. However, I fear that this could create frustration in some people, perhaps those who are arrogant enough to think they are somehow above the simplicity of the basic ideas of the subject they have acquired such advanced understanding in. Take the great successors in most fields for example. They sometimes tend to appear as arrogant bastards but when it comes to face the essence of their field they are humble.

This idea is also well known in teaching. In fact, this is what most teachers are doing everyday. I, like most teachers, have experienced this myself; after teaching something very simple and basic for few times, realised that my own understanding had changed, therefore my horizon broadened.

As per my comment on Twitter, the process of constantly returning to the basics when you study martial arts at an advanced level is a great parallel here. While it’s necessary anyway to keep reviewing the basics, appreciation of them, and of their application, changes over time.

I really like the example of teaching, too. When I teach the very basic stuff (experimental design, statistics, fundamentals of visual perception), going through it is a little different each time, because I am simultaneously thinking about more complex examples from recent experience, and how the basics apply to those. Sometimes these wonderings make it as far as the students; sometimes I just shut up and do the basics. Both are useful🙂

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