We observe that some natural mathematical definitions are lifting properties relative to simplest counterexamples, namely the definitions of surjectivity and injectivity of maps, as well as of being connected, separation axioms \(T_0\) and \(T_1\) in topology, having dense image, induced (pullback) topology, and every real-valued function being bounded (on a connected domain).

We also offer a couple of brief speculations on cognitive and AI aspects of this observation, particularly that in point-set topology some arguments read as diagram chasing computations with finite preorders.

An experimental MOOC (Massive Open Online Course) Citizen Maths is launched and the first phase of the course is open for registration. It is free and open for everyone; its motto is Powerful Ideas in Action.

The readers of this blog may like to register for the course, because, as the organisers say,

The success of this first phase of Citizen Maths will depend crucially on the feedback that we obtain. We are particularly keen to get feedback from:

learners who do the course;

those with an interest in the learning and teaching of maths, and in the design of online courses.

There is a link to a feedback form on every page of the Citizen Maths web site, and there will be a similar link on every page of the course when it goes live on or around 12 September.

The first pilot stage will run for four weeks and cover the first “powerful idea”: proportion. An admirable choice (a detailed discussion of the role of proportions in elementary mathematics can be found in this paper by Tony Gardiner).

[Republished from Terry Tao's Blog -- because of the importance of the issue.-- AB]

[This guest post is authored by Matilde Lalin, an Associate Professor in the Département de mathématiques et de statistique at the Université de Montréal. I have lightly edited the text, mostly by adding some HTML formatting. -T.]

Mathematicians (and likely other academics!) with small children face some unique challenges when traveling to conferences and workshops. The goal of this post is to reflect on these, and to start a constructive discussion what institutions and event organizers could do to improve the experiences of such participants.

The first necessary step is to recognize that different families have different needs. While it is hard to completely address everybody’s needs, there are some general measures that have a good chance to help most of the people traveling with young children. In this post, I will mostly focus on nursing mothers with infants ( months old) because that is my personal experience. Many of the suggestions will apply to other cases such as non-nursing babies, children of single parents, children of couples of mathematicians who are interested in attending the same conference, etc..

The mother of a nursing infant that wishes to attend a conference has three options:

My older brother was the person who got me interested in science in general. He used to tell me what he learned in school. My first memory of mathematics is probably the time that he told me about the problem of adding numbers from 1 to 100. I think he had read in a popular science journal how Gauss solved this problem. The solution was quite fascinating for me. That was the first time I enjoyed a beautiful solution, though I couldn’t find it myself. [...]

I was very lucky in many ways. The war ended when I finished elementary school; I couldn’t have had the great opportunities that I had if I had been born 10 years earlier. I went to a great high school in Tehran – Farzanegan – and had very good teachers.[...]

Our school was close to a street full of bookstores in Tehran. I remember how walking along this crowded street, and going to the bookstores, was so exciting for us. We couldn’t skim through the books like people usually do here in a bookstore, so we would end up buying a lot of random books. Also, our school principal was a strong-willed woman who was willing to go a long way to provide us with the same opportunities as the boys’ school.

Later, I got involved in Math Olympiads that made me think about harder problems. As a teenager, I enjoyed the challenge. But most importantly, I met many inspiring mathematicians and friends at Sharif University. The more I spent time on mathematics, the more excited I became.

[Originally posted on Edmund Harriss' blog Maxwell's Demon, this is a transcript of a talk at the Twitter Math Camp 2014, a truly energising event, teacher organised peer professional development. Anyone interested in education, whether parent, academic, teacher or administrator should check it out.]

Zero…

Start by clearing your mind.

One…

Now imagine one dot pop into view.

Two…

A second dot joins it. Let the two dots flow around each other, rotating and getting closer and further apart.

Three…

Now a third dot, creating a line or a triangle…

Four…

Keep on adding, with each addition try to see all the dots, find a shape you like…

Five…

Six…

Seven…

Eight…

Nine…

Ten…

Eleven…

Twelve…

Thirteen…

Fourteen…

Fifteen…

Sixteen…

Seventeen…

This is about having fun and playing with math, which often sounds a little:

This is not the holy grail, it is not even a challenge to bring into the classroom. Teachers have too many challenges, sometimes the challenge is just to get through the day without messing up too badly. It is an encouragement to relax and have fun, yet remember that this fun is part of your teaching prep!

Playing with maths can often start with going back, returning to something you know well, and trying something new, testing an idea. If it fails try something a little different, or go back to work out how it went wrong. If it works, can you try everything? Mathematicians can say everything and really mean it! Even then do not settle, go back with your new knowledge and try something new. You might notice once you have started you cannot escape! You can always just stop. This is play not work. Though it might not be relaxing, just as playing a sport is exciting, fun and cathartic but you put effort in.

This is why this can build into your teaching, once you have fun you have a chance to help your students have fun. If they have fun they will put far more effort in than if you have to push them. Also I do not feel that mathematics has a huge number of facts, but isolated they are not that useful, going back and playing with ideas helps build the dense web of connections that really drives understanding.

General strategies are great, but it can be hard to know where to start, I will describe two tools:

Analogy and the concept of same/different (mathematics is the world’s greatest metaphor!)

Breaking rules! (yes mathematics is often about creating them, but also about changing them and seeing what happens).

To get further, we need an example, and not one that will lose half the audience just with its title so…

Counting.

Three dots, are they the same or different? They are in different positions, but are the same shape. We have to be clear what we mean.

Now we take pairs of dots, we can spin them around and pull them apart. We could say they were the same if they can be moved on top of each other. Yet to define that precisely we have to use most of plane geometry. We have not even counted past two and we already need that!

Getting to three the line and the triangle, different in ways that the pair of dots can never be.

Lets change tack, we have been looking at how the same number of dots can be different, what about how different numbers of dots can be the same?

These patterns for four, six and eight have some similar features. How might we describe those precisely so we can identify other ones? Saying that the numbers are all even is an obvious way to do it, but maybe they also share something with this:

Like the earlier examples nine dots drawn like this form a rectangle (specifically a square). Following this definition we can define prime numbers (technically composite numbers!).

Here are another collection of dot patterns that share features, one dimension, two dimension and three dimension, and at this point reality gives up on us. Yet we really went past our page after two, we can use the notions of analogy to push further. We know the next pattern will have sixteen dots. For example we can make this image, with lines to show the structure. Can you find the eight cubes?

With a little work from here we can work out that an nn-dimensional cube has 2n (n-1)-dimensional faces. So we know very little about 172 dimensional space, but we do know that a hypercube in that space has 344 faces! Playing with some of these tricks we can get this:

There is a lot more to discover in this image. If you are interested in getting a version send me an email, I am looking into options.

Lets move to the other trick, breaking the rules. Mathematics is made of rules, yet there is not one rule that is not broken somewhere else in mathematics. For example this might make you uncomfortable:

7 + 7 = 2

2 + 1 = 2

If I say instead that seven months after July (the seventh month) is February then the first makes perfect sense. In this case 7+7 is still 14 but 14 is the same as 2, we have modular arithmetic.

That trick will not work for 2 + 1 = 2. Yet in Chemistry two hydrogen molecules combine with an oxygen molecule to create two water molecules. There is an even greater rule, though one that has been enshrined in legend. Yet this image shows what happens when we divide by zero (at the centre)!

(the mathematical trick is to use what is called the Riemann sphere).

In conclusion playing with math can happen with the simplest structures and lead to a variety of thoughts and adventures. No one should be shy of having a go!

Here is a neat animation from my play:

Notes

I have a list of some other materials to inspire your mathematical play, and there is a whole world of examples in Sue van Hattan’s book Playing with mathematics. That should be available for pre-order soon!

A. E. Kyprianou: The UK financial mathematics M.Sc. arXiv:1405.6739v2 [math.HO]

Abstract:

Postgraduate taught degrees in financial mathematics have been booming in popularity in the UK for the last 20 years. The fees for these courses are considerably higher than other comparable masters-level courses. Why? Vendors stipulate that they offer high-demand, high-level vocational training for future employees of the financial services industry, delivered by academics with an internationally recognised research reputation at world-class universities.

We argue here that, as the UK higher education system moves towards a more commercial environment, the widespread availability of the M.Sc. in financial mathematics exemplifies a practice of following market demand for the sake of income, without due consideration for the broader consequences. Indeed, we claim that, as excellent as such courses can be in intellectual content and delivery, they are mismatching needs and expectations for such education and confusing the true value of what is taught.

The story of the Mathematical Finance MSc serves as a serious case study, highlighting some of the incongruities and future dangers of free-market education.

An article by Elizabeth Green in the NYT; it is about America, but is very timely in the context of the National Curriculum reform in England. A quote:

It wasn’t the first time that Americans had dreamed up a better way to teach math and then failed to implement it. The same pattern played out in the 1960s, when schools gripped by a post-Sputnik inferiority complex unveiled an ambitious “new math,” only to find, a few years later, that nothing actually changed. In fact, efforts to introduce a better way of teaching math stretch back to the 1800s. The story is the same every time: a big, excited push, followed by mass confusion and then a return to conventional practices.

The trouble always starts when teachers are told to put innovative ideas into practice without much guidance on how to do it. In the hands of unprepared teachers, the reforms turn to nonsense, perplexing students more than helping them. [Emphasis is mine -- AB.]