[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:
[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.]
Start by clearing your mind.
Now imagine one dot pop into view.
A second dot joins it. Let the two dots flow around each other, rotating and getting closer and further apart.
Now a third dot, creating a line or a triangle…
Keep on adding, with each addition try to see all the dots, find a shape you like…
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…
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!
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.
When we really need to send the message that international STEM students
will get a warm welcome in the UK, they’re getting the cold shoulder and
they are heading elsewhere. We’ve seen over the last few years how
international student numbers have fallen dramatically. As a result
we’re missing out on the talent and the economic and cultural
contribution that international students bring when they come here to
study, and our competitors are reaping the rewards.
Please sign the petition and ask your friends to sign it. The more
support we can get behind this campaign, the better chance we have of
MY REASON FOR STARTING THIS PETITION
The LMS June newsletter drew my attention to the House of Lords Science
and Technology report (April 2014) http://tinyurl.com/nbqbvca that
recommends a change in immigration practices relating to foreign
This is a campaign of national and international importance, and not just
about a few individuals, but I decided to act because this week some
friends of ours from Chennai met with the unpleasant face of UK
immigration practices, and academic visitors coming to work with us have
had similar experiences. Mandira, age 17, who has permanent residency in
the UK, and was on her way to the UK to do a 2 week course for High
School students in Cambridge that starts this week, was turned back and
put on a plane back to India because she has not been in the UK for 2
years. She was with her mother and younger sister, and they also planned
to visit some other universities because Mandira wants to apply to UK
universities to study medicine starting in September 2015. This was
probably a mistake on the part of some officious individual but
never-the-less it is typical.
D. Edwards, The Math Myth, The De Morgan Gazette 5 no. 3 (2014), 19-21.
I’ve been concerned with what skills those who are working as scientists and engineers actually use. I find that the vast majority of scientists, engineers and actuaries only use Excel and eighth grade level mathematics. This suggests that most jobs that currently require advanced technical degrees are using that requirement simply as a filter.
[A version of this text appeared in the August, 2010 issue of The Notices of The American Mathematical Society.]
Some current developments in UK Higher Education Institutions raise serious concerns for mathematicians. The issues involve complex changes in the relationships between career development, the impact agenda, and external funding. While many of these changes affect academics in other fields, I will concentrate here on their particular effects on those working in the mathematical sciences. These effects are, broadly speaking, of two sorts: changes in our working conditions as individual mathematical scientists, and changes in the overall structure of academic mathematical science in the UK. Here are some examples of the sort of thing I have in mind: the first 6 predominantly concern individuals, at least initially, while the remainder are more structural.
award of sabbatical leave only to those winning Research Council (RC) grants;
allocation of PhD students only to those winning RC grants;
supervision of research student(s) a necessary condition for promotion;
substantial external research income a necessary condition for promotion;
move to “tenured” status dependent on winning external income and/or PhD supervision;
non-submission of an individual’s outputs to the REF, despite availability of a full set of internationally-published outputs;
departmental decisions on number of outputs submitted to the REF influenced by the number of sufficiently strong Impact Statements;
decisions on research fields to support or appointments to make dependent on likelihood of future Impact Statements being generated;
loss of service teaching leading to reduced student FTE numbers and reductions in staffing.
The purpose here is not to provide a detailed analysis of each of the above issues—rather, I want to open a dialogue, letting others develop topics which they feel are of particular concern, whether from the above list or not. Instead, I’ll simply comment briefly below on a couple of the points.
Of course, not everyone will think that each of these developments is by definition “a bad thing”. Regarding point 8, for instance, areas of research focus and consequently of appointments must change over time if our subject is to remain vital. The increased focus on Impact in the UK is part of a world-wide trend which we as mathematical scientists cannot and should not try to oppose—rather we must continue with and redouble our efforts to make funders’ definitions of and ways of measuring impact more in tune with the full range of our activities. We must also continue to emphasise the huge long-term impact of the mathematical sciences, as catalogued for example in the Deloitte report; and we should develop a portfolio of examples of the profound influence of the mathematical sciences on all aspects of our lives—one excellent example is the USA’s National Research Council report “The Mathematical Sciences in 2025”. (This is available as a free download at http://www.nap.edu/catalog.php?record_id=15269.)
On point 4, we all know that RC grant income in the mathematical sciences is very low compared to many other STEM subjects. This is in part because the main costs of much of most of the research in the mathematical sciences has been for people and for time, costs which, though very significant, have in the past been adequately covered for many of us in the UK by the dual support system of funding. Perhaps also it is the case that what we do has historically been undervalued, thanks to long lag times for impact, but also—let’s be honest—thanks to our sometimes relaxed attitude in the past to the need to make the case for more funding. The LMS, both on its own and in conjunction with the Council for the Mathematical Sciences, has been working hard to make these cases and to assemble relevant data, for grant income and more broadly: for example, the Deloitte report, produced with CMS backing last year, has generated a lot of publicity, and the LMS is producing data documents on UK HEI staffing in the mathematical sciences (November 2013 http://www.lms.ac.uk/policy/statistics-mathematics), and on UK research funding in the mathematical sciences (to be published July 2014). A CMS report on the “people pipeline” in the mathematical sciences will come out later this year.
I should also briefly explain what I have in mind with the point 5. At least two Russell Group universities have recently introduced contracts for newly-appointed lecturers, which lead the appointee through a career path set up to complete probation in 2-3 years, with an expectation of promotion to Senior Lecturer or Reader (possibly called something different), within 5-7 years of initial appointment. All to the good, you might think—except that milestones expected to be passed en route to promotion include winning substantial RC grant income, and supervising a PhD student to completion. The consequences of failing to achieve these targets in the specified time frame are left unclear.
So, why am I writing this article? The first and very important reason is to gather information. At the moment our community has no way of knowing how widespread are these and similar changes. Those of us directly affected can feel isolated, powerless and undervalued. I’m therefore inviting two sorts of response. First, I will very much welcome information about particular cases along the lines of those listed above. It will be equally valuable to learn of examples of good practice with regard to these issues. Naturally, I’ll treat all such communications in the utmost confidence, but will hope to share what global data I can gather, in due course. More generally, it will be good to hear other views on the issues raised here: perhaps, for example, some of these changes should be welcomed? Most importantly, we need to consider what we as a community should be doing in the face of these developments. What should the LMS be doing?
In late February I was a member of a delegation representing HM Government that visited the three Chinese provinces of Shanghai, Beijing and Hubei with a specific focus on mathematics education.
I have waited until now to reflect on my visit to China because I wanted to go back into some English schools to test out the thinking I developed while there. The differences in maths outcomes for our young people between the two countries are stark and worrying for us, unless we act now to catch up – and I do not mean just in terms of PISA test scores. I am coming at this not only from an inspector’s point of view, but also from my background of being a physics teacher and so frequent user of maths, reliant on pupils being able to handle and manipulate numbers confidently. In this respect, Chinese children are streets ahead of ours, so the benefits of their high standards in mathematics go way beyond just this core subject.
As everyone knows, Her Majesty’s Inspectors are not concerned about the ‘how’ but ‘how effective’ with teaching. This approach requires a clear focus on the outcomes for the pupils and their response to the teaching, including crucially the evidence of learning and progress over time in their work books and folders. These were impressive in the classes we observed in China, and told a story of a consistency of approach and expectations that has led to the pupils being confident mathematicians, willing to have a go and able to tackle problems in different contexts.
For example, given this problem…:
X = 2√ (7/14 x 28/7 x 3/9 x 24/8 x 18/9)
… none of the 12-year-old pupils reached for the calculator; they couldn’t because they have been banned from their classrooms. They calmly looked for the potential to cancel and reduce the fractions, and spotted that this expression is really just the square root of 4. Not a job for the calculator; not for them at least. This was clearly not about them learning ‘tricks’ either. This problem was one of just 4 or 5 set by the teacher in a 5 minute burst of practice, to help the pupils master the concepts covered by her in the latest part of the lesson before they moved on confidently together to the next stage of increasingly challenging maths. The key was not the teacher’s ‘performance’ in this lesson, but the demonstration of the depth of the pupils’ mathematical learning over time and the impressive armoury of knowledge and skills they had built up to deploy as and when needed. Evidence of solidly knowing their times tables was absolutely apparent across the pupils, as was the ability to use efficient methods of calculation without having to really think. Their mathematical toolkit was there to be used as surely as a mechanic’s spanners, or a surgeon’s scalpel
What if we figured out ways for young kids to play with ideas from calculus, algebra, and other mathematical subjects beyond arithmetic? Last week, The Atlantic published Dr. Maria Droujkova’s interview, “5-Year-Olds Can Do Calculus” by Luba Vangelova. It started a broad international discussion, with follow-up interviews by Canada’s “Globe and Mail” and UK’s “The Times,” and translations into Japanese and Russian by news agencies. Droujkova and her colleagues at Natural Math are aggregating major themes from the comments:
How can we create and sustain environments where kids are free to learn, and adults are free to help them?
Can young children understand abstractions? Can they deal with the formal language of mathematics? If they can, will it hurt their development in some way?
Many grown-ups believe that young math will finally give them a second chance at making sense of algebra and calculus.
But what about calculating and memorizing? We need more research on balancing concepts and technical skills.
What can young kids actually do with algebra or calculus? How can they play with these ideas, or apply them to their daily lives?
Many people recognized our activities as similar to what they are doing with their kids – or what their parents did with them. What difference does this casual, everyday early math make for kids whose parents understand and love mathematics?
Alexandre Borovik invited me to join the writers here at The DeMorgan Forum. (Thank you, Alexandre. I am honored.) He got me started by reposting my Top Ten Issues in Math Education post, which I have now edited. (There were links to dead blogs and disappeared posts, mentions of ‘last week’ four years ago, and opinions I’ve changed.)
I blog most often at Math Mama Writes, and will bring some of my favorite posts from there over here. If you want to hear more from me, please follow me there.