Mathematical Scales

November 6, 2010

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.

16 Comments | Mathematics | Tagged: , , , , , , | Permalink
Posted by gelada

Education and Research

October 15, 2010

A twitter discussion this morning with Ian Hopkinson (@smallcasserole) Rory Duncan (@HardyDuncan) and Mary Wombat (@little_mavis), led me to try to formulate what I think about the relationship between education and research. There is a very good case that these should not be required to be together. Any academic knows cases of researchers who did well and did teach even though they really should not have been unleashed on the students. Equally talented teachers, perhaps because they spend more than the minimum time working on their teaching, can often struggle with jobs or promotion. The government have also talked about some of this as a cost cutting measure. But…

[Edit 25/1/11: Another, interesting opinion from David Colquhoun (@david_colquhoun) on the need to separate the two, though I feel that this runs into the eliteism issues raised by Stephen Curry (@Stephen_Curry)]

Education subsidises Research

Universities in general receive more money to educate than do research. Yet in many departments the staff do more research than education. Of course things are muddied by the fact that the research will take over all remaining time! Education does not pay for time spent on research over the weekend. The research money given out, therefore, goes further as it does not have to pay the full cost of the researchers. Even if the teaching is bought out this will normally pay for someone to come in, or a teaching fellow position to cover the teacher at much less than the researchers salary.

[Edit 15/10/10: Apparently at Ediburgh income from research is now >40% I would be interested in other statistics that people have on this. Both the percentage of income and the percentage of staff time on research]

[Edit 25/1/11: A more informed version of this argument, and how higher fees might just go to research can from John Holmwood]

On the other hand…

Research subsidises Education

A big attraction for me of the academic career was the freedom and time available. I am lucky that I am a mathematician and so not having a research grant does not make research impossible. Most people completing a PhD have a wide variety of jobs available. Many of which pay a lot more than academic ones. The thrill of research, the freedom and the ability to be part of a vibrant international community all make up for the lack of cash. Take this away and it would be harder to attract the best into academic teaching. Is having a PhD too much? Firstly I think that the comment (of other jobs) would apply to anyone who you would want teaching at this high level. Secondly, although it is not necessary to be research active in order to teach a particular subject, being research capable (~having a PhD) could be. You need teachers who have seen beyond the limits of the material they are putting forward if they are to truly inspire with the ideas.

In conclusion, even if they are not logically combined, the historical situation that has linked teaching and research should be unpicked carefully!

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Posted by gelada

The Laws of Gelada (How to be a grad student)

March 3, 2010

Irving Herman‘s Laws for graduates can make good little scientists. How can we make misbehaved big scientists?1 The original Herman rules have Hx numbers my versions are down under Gx.

H1. Your vacation begins after you defend your thesis.

G1. (Force yourself if necessary) to take some time off.2

H2. In research, what matters is what is right, and not who is right.

G2. In research, what matters is good and useful answers, and not who gives them.3

H3. In research and other matters, your adviser is always right, most of the time.

H4. Act as if your adviser is always right, almost all the time.

H5. If you think you are right and you are able to convince your adviser, your adviser will be very happy.

G3. In research, your adviser is probably right more often than you.

G4. Assume your adviser is wrong if you do not agree with him.

G5. If you are right and are able to convince your adviser, everyone gains. 4

H6. Your productivity varies as (effective productive time spentper day)1,000.

H7. Your productivity also varies as 1/(your delay in analysing acquired data)1,000.

G6. Your productivity varies and is not necessarily tied to effort.

G7. Keep on top of routine tasks, but do not be ruled by them.5

GH8. Take data today as if you know that your equipment will break tomorrow.

GH9. If you would be unhappy to lose your data, make a permanent back-up copy of them within five minutes of acquiring them.6

H10. Your adviser expects your productivity to be low initially and then to be above threshold after a year or so.

G10. Realise your productivity will not be high initially. Aim to be more productive, but always allow for variation.7

H11. You must become a bigger expert in your thesis area than your adviser.

G11. You must be more passionate about your thesis area than your adviser.8

H12. When you cooperate, your adviser’s blood pressure will go down a bit.

H13. When you don’t cooperate, your adviser’s blood pressure either goes up a bit or it goes down to zero.

G12. Do not care about your advisors blood pressure.

G13. Cooperate with your advisor. You will get more out of them. They should know a lot that you care about. Thats why you picked them isn’t it?9

H14. Usually, only when you can publish your results are they good enough to be part of your thesis.

H15. The higher the quality, first, and quantity, second, of your publishable work, the better your thesis.

G14. Ideas are only good enough for your thesis when you are proud of them, you can do things with them and you can communicate them to others.

G15. The more interesting you find your results the better your thesis.10

H16. Remember, it’s your thesis. You (!) need to do it.

G16. Remember, it’s your thesis, your research.11

H17. Your adviser wants you to become famous, so that he/she can finally become famous.

G17. Care about your work and find it important. Do not chase fame.

H18. Your adviser wants to write the best letter of recommendation for you that is possible.

G18. Be aware of politics, sell what you do well.12

H19. Whatever is best for you is best for your adviser.

H20. Whatever is best for your adviser is best for you.

G19. Think hard and decide what is best for you.

G20. Listen to authorities (like your advisor), but do not be ruled by them.13

Footnotes

1 BACK TO POST
A few years ago Irving Herman, a physicist at Columbia published a set of laws for graduate students in Nature. To be fair he does say that some of his comments are slightly exaggerated and should not be taken completely seriously. However he also claims these as laws. Which is a very strong rhetoric. On my side my laws can also be slightly exaggerated and are usually highly idealistic, but if you are not idealistic about science you are probably better in a different career anyway.

I came across these recently in Eric Weinstein’s twitter and his comments were the spark and much of the inspiration for this post. I have included his comments on specific laws below. Here is his overall opinion:

New Topic: Thoughts on Prof. Irving Herman’s “20 Laws All Grad Students Should Follow” or “On Being Science ‘Help’ ” as published in Nature. Tweet

I am delighted that colleagues in academe are starting to write down the ‘meta-rules’ of new science that select against strong scientists. Tweet

My goal as taxpayer & scientist is to get you, the young scientist, out of Irving Herman’s dystopia before he can help you become ‘better’. Tweet

and he concludes:

Yet being a scientist isn’t about any of this idiocy. This is about survival in universities & why basic research must reform or leave them. Tweet

2 BACK TO POST
A PhD is hard work, but…Practically taking time off can renew focus, give perspective and thus generate more ideas. More importantly, you are not a robot or slave. Take time off to remember why you are doing this crazy thing.

Herman’s Law 1:”Your vacation begins after you defend your thesis.”
Weinstein’s Excercise:Translate into German without use of a dictionary.

Eric Weinstein Tweet

3 BACK TO POST
It is often better to be productively wrong than unproductively right. Liebnitz/Newton’s Calculus was wrong (and many, most noticably Berkerley spotted this) but those who ignored or were ignorant of this did better maths for 100 years.

4 BACK TO POST
Of course you should treat your advisor with respect, especially for the work that he or she has done. They do have more experience and know more, so they are probably right. However they are also better at arguing than you. Give your intuitions confidence and be persuaded out of them by reasoning not authority.

(“in other matters” your advisor is just another human being, saying that they are mostly right there is crazy!)

3. In research and other matters, your adviser is always right, most of the time.
Just who is this guy? Nature? Physics? Columbia? Anyone?

Eric Weinstein Tweet

Herman’s Law #4. “Act as if your adviser is always right, almost all the time.”
That would be ‘Science … with Benefits’…wouldn’t it?

Eric Weinstein Tweet

5 BACK TO POST
Productive time is essential, but what is it? It is certainly more than time spent in the lab/office. Learn what helps make you productive. Maybe a weekend of surfing leads regularly to great results on a Monday. Routine tasks do need to be done. Keep on top of them so you can relax and think, do not hope you can get PhD students on day to do them, or become a lab assistant for your supervisor!

6 BACK TO POST
Agreement for both of these, it is good to put a little time into insurance against disasters.

7 BACK TO POST
Coming back to the “productivity is complicated” idea. Sometimes you have to get lost, following blind alleys for weeks or months to chase up the great result. It is easy to be productive by finding more routine tasks to do, is that your ambition?

8 BACK TO POST
If you are passionate about your area you will of course think about it and study it more. You should be doing all this for passion and not because your supervisor says that it is interesting.

9 BACK TO POST
Cooperation is a good thing. We do need to work together, to help find the right or useful ideas and communicate them. It is useful to achieve other goals however, not as a goal in its own right.

10 BACK TO POST
Find out what is important to you, what you feel are the big questions. Chase them. Take into account the opinions of others (such as journals) but remember they can be wrong.

11 BACK TO POST
You are paying for this and working very hard on it. Take pride in it, make it your own and do the best job you can. For yourself not your supervisor.

12 BACK TO POST
It is good to show your work achievements and ideas in the best light. Do not however do something for no other reason than it looks good.

13 BACK TO POST
The summary of this post. To be a good scientist is to respect authority while questioning it.

19. Whatever is best for you is best for your adviser.
20. Whatever is best for your adviser is best for you.
So,we may catch Pyonyang yet?

Eric Weinstein Tweet

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Posted by gelada

How to write machines

November 18, 2009

(If you are coming from Zeilberger’s opinions, the appropriate article is here)

Maths fun was had by all

Last weekend I was in Gothenburg at the incredibly inspiring Free Society conference FSCONS. Of course I was talking about mathematics, specifically how to get people learning it through fun, rather than “because it is useful”. My talk was called “Street Maths” (click for slides).

In discussions with many including Smári McCarthy and Marcin Jacubowski the idea developed further and one result is this (highly opinionated ;) manifesto for literacy.

In 1964 Paulo Freire was arrested and exiled from Brazil for teaching peasants to read. Both sides recognised the power of literacy, as a threat to oppression and a path towards a better life for individuals.

Today in the developed world we take it as an essential. Those who cannot read are not merely marginalised but kept out of society. Yet new skills are becoming necessary. Our formal interactions are now almost more likely to be through a computer than a pen. This change is sweeping through so fast that it can be hard to keep up. We have all joked that the kids teach the adults how to use the latest device.

Lets give the education system its due. The schools curriculum in the UK recognises that for Information and Computer technology (ICT):

…creative and productive use of ICT an essential skill for life.

National Curriculum (ICT) Key Stage 3

How do they suggest we try to achieve this?

The study of ICT should include:

  1. use of a range of information, with different characteristics, structures and purposes, and evaluation of how it matches requirements and its fitness for purpose
  2. use of a variety of information sources, including large data sets, in a range of contexts
  3. use and review of the effectiveness of different ICT tools, including a range of software applications, in terms of meeting user needs and solving problems
  4. developing an understanding of the need to:
    * employ safe working practices in order to minimise physical stress
    * keep information secure
    * manage information organisation, storage and access to secure content and enable efficient retrieval
  5. the impact of ICT on individuals, communities and society, including the social, economic, legal and ethical implications of access to, and use of, ICT.

National Curriculum (ICT) Key Stage 3

Think about these for a second as we consider the skill of literacy. It has two parts. Reading is of course important, but teaching people to read only allows one way communication. We also teach to write. We are taught to use written content, but also to create it. Think about this as you again read the list above. It only talks about learning to “use” ICT.

We need the skills to write and create as well as simply use.  Firstly, for some a bright idea will result in a new use for computers. Just as for some the ability to write leads to a published book. For others some simple creations will help their lives or those close by them, just as some write diaries. Finally there are many who do not write much at all. Yet learning to write writing still helps us develop our reading. The same is true for technology, but it is even more essential. Reading is a fixed skill. A language develops too slowly for reading skills to need much change. This is not the case with computers. The skills to use a particular piece of software can change with a single upgrade, even when we are not forced to change to a more advanced competitor. The usage skills therefore can easily go out of date. The more fundamental skills teach not just the skills to create but the ability to learn; to adapt to rapid changes.

So what skills are needed to create technology? Programming is obviously first. There is, however, a lot more to technology than computers. There are a vast number of ways that gadgets can be used, and will be used. Should we leave people waiting for someone else to make something close enough to what they need? What about adding the basic skills to make things?

Unlike literacy and use of computers these are not new skills. They are in fact ancient. Not a very long time ago if you wanted something you either had to make it yourself, or go to someone who could make it for you. Then we had the industrial revolution. The economy of scale. We came to rely on factories. This now goes so deep we hardly think of making something ourselves. For truly mass items like a hammer or a car, we are probably right.  What about a more specialised device though, like say a tractor? Or a 3d printing machine? Here plans are freely available that require some skill, but not expertise, to build. Including money for building time the product can be made for a fraction of the cost (in many cases 1/10 or less).  Even better, with such open design comes a powerful new option. Take the generic solution and adapt it to your own situation.  With time the design improves as individuals using it make refinements and add options. To do this takes a certain mindset and some basic skills.  A literacy of making.

The natural response to this is that, on top of the skills, tools are required and those tools are themselves prohibitively expensive. Though this is true right now, it is changing. Movements such as FabLabs and Hacker spaces have the tools and make them available for free, or at a small cost.  Even better, the machines can be part of the change.  One of the machines above is a 3d printer, this is not just cheap to produce, it is capable of making itself. The development of other machines has begun, with the ambitious goal of creating a RepLab a multipurpose factory that can create itself at a cost of less than $10000. Even commercially the machines only cost about $100000. Things are changing. Fast. The question is can we get the people in place with the creativity and skills to take full advantage of them?

4 Comments | General | Tagged: , , , , , , , , | Permalink
Posted by gelada

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