# Mathematics in the news this week

The week of 30 May 2016

# Why undegraduate students should not use online matrix calculators

Since 1 April 2011 I from time to time was trying to convince Wolfram Alpha to fix a bug in the way they computed eigenvectors, see my post of 28 April 2012. It survived until May 2016:

As you can see, Wolfram Alpha was thinking that the zero vector is eigenvector. On 5 May 2016 this bug was finally fixed:

But there is still one glitch which can send an undergraduate student on a wrong path. The use of round brackets as delimeters for both matrices and vectors suggests that the vector $$(1,0)$$ is treated as a $$1 \times 2$$ matrix, that is a row vector. This determines which way it can be multiplied by a $$2 \times 2$$ matrix: on the right, that way:
$(1,0) \left(\begin{array}{cc} 1 & 2 \\ 0 & 1\end{array}\right)$
and not that way
$\left(\begin{array}{cc} 1 & 2 \\ 0 & 1\end{array}\right)(1,0),$
the latter is simply not defined. Therefore the correct answer is not
$\mathbf{v}_1 = (1,0)$
but
$\mathbf{u} = (0,1) \quad\mbox{ or }\quad \mathbf{w} = (1,0)^T = \left(\begin{array}{c} 1 \\ 0\end{array}\right),$
depending on convention used for vectors: row vectors or column vectors. Indeed if
$A = \left(\begin{array}{cc} 1 & 2 \\ 0 & 1\end{array}\right),$
then
$\mathbf{v}_1A = (1,0)\left(\begin{array}{cc} 1 & 2 \\ 0 & 1\end{array}\right) = (1,2) \ne 1\cdot \mathbf{v}_1,$
while
$A\mathbf{w} = \left(\begin{array}{cc} 1 & 2 \\ 0 & 1\end{array}\right) \left(\begin{array}{c} 1 \\ 0\end{array}\right) = \left(\begin{array}{c} 1 \\ 0\end{array}\right) = 1\cdot \mathbf{w}$
and
$\mathbf{u} A = (0,1) \left(\begin{array}{cc} 1 & 2 \\ 0 & 1\end{array}\right) = (0,1) = 1\cdot \textbf{u}.$
The bug is likely to sit somewhere in the module which converts matrices and vectors from their internal representation within the computational engine into the format for graphics output. It should be very easy to fix. It is not an issue of computer programming, it is just lack of attention to basic principle of exposition of mathematics and didactics of mathematics education.

# New Report: The beginnings of school led teacher education

I have recently been involved in a five year project looking at the ascendance of school-led teacher training in England. We hope that you find our report interesting and we would be greatly appreciative if it were possible to make it available to your various networks of colleagues.

Yours sincerely

Tony Brown

——
The beginnings of school-led teacher training: New challenges for university teacher education.

The School Direct Research Project undertaken by a team of academics from Manchester Metropolitan University, UK, concludes five years of research into the effects of school–led training on the rationale and composition of university teacher education in England and considers the impacts of recent changes on the teaching profession.

In England, atypically perhaps among other countries, most teacher education has moved into schools with universities playing a more peripheral role. This is ostensibly a lower cost approach to teacher education that may appeal to other countries. The point of our report, however, is not to invite international readers to try this at home. The more general issue relates to how teacher education knowledge is conceptualised, how this shapes practice but also questioning how and why university contributions have been conceptualised in the way that they have been, and if they deliver on their promise. The report asks whether the choice between the benefits of school-based training and of university led teacher education is so obvious as it may first appear. By taking an atypical perspective on more familiar models the rationale for these models might be seen differently, whilst raising the more generic issue of how learning to teach happens differently across university and school locations.

Teacher education in England now comprises a vocational employment-based model of training located primarily in schools. This approach is in sharp contrast to models followed in the “European Teacher Education Area” where student teachers typically spend five years in university, followed by up to two years on school placement. “Almost all countries introduced reforms in initial primary teacher education after the initiation of the Bologna Process (1999)” (ENTEP), similarly for secondary subject teachers, and half of pre-primary sectors of education. These two approaches reveal radically different conceptions of how teacher quality might be improved in the name of international competitiveness. In the English model, teacher education has been wrested from its traditional home within the academy where universities play a support role to what has become “school-led” training where government funds for teacher education have been diverted to schools. Student teachers often spend as little as thirty days in university during a one-year postgraduate “training” course. Teacher professional identity has been referenced to skill development within this frame and the wider assessment culture. The wider European model, meanwhile, similarly claims to be concerned with “raising teacher quality … in a way which responds to the challenges of lifelong learning in a knowledge based society” (ENTEP). The model is characterised by reinvigorated faith in academic study and promotion of individual teachers, where a pedagogical dimension in included from the outset of undergraduate studies, but with relatively brief periods spent in school.

The report, written by Tony Brown, Harriet Rowley and Kim Smith, shows how the reconfiguration of how training in the English model is distributed between university and school sites consequential to School Direct altering how the content and composition of that training is decided. Most notably, local market conditions rather than educational principles can determine the design of training models and how the composition of teacher preparation is shared across sites. This contingency means that the content and structure of School Direct courses varies greatly between different partnership arrangements across the country, leading to greater fragmentation within the system as a whole. Thus, there is not only increased diversification in terms of type of training route but also diversification of experience within each route. School Direct has also altered the balance of power between universities and schools, and in turn, their relationship with one another. The ascendance of school-led training has changed how the responsibilities of each party are decided and impacted on how the categories ‘teacher educator’, ‘teacher’ and ‘trainee’ are defined. In particular, the function of ‘teacher educator’ has been split across the university and school sites, displacing traditional notions of what it means to be a ‘teacher’ and ‘teacher educator’. The flux is leading to uncertainty across role boundaries and, in turn, changes in practice. Furthermore, as those in different locations negotiate territorial boundaries, this can activate anxiety and tension within the workforce. The particular impact on different school subjects as a result of these contrasting approaches relates to the way in which conceptions of the subjects derive from where understandings of them are developed, whether in schools or in universities.

For those training in schools little more may be done than enable teachers to work through commercial schemes as implementers of curriculum, as opposed our European neighbours following university intensive courses where relatively low attention is given to the practical school aspects during the university element. Lower cost school-based teacher education may yet appeal to other countries in building and influencing the practice of their teaching forces. But four questions immediately present themselves: Does School Direct provide a viable alternative to university based teacher education? Does it alter the composition of the pedagogical subject knowledge it seeks to support? Is it low cost, or at least good value for money (National Audit Office, 2016)? How will it eventually impact on England’s reputation in international comparative testing?

The report can be found at:

http://www.esri.mmu.ac.uk/resgroups/schooldirect.pdf

Other project publications:

Brown, T, Rowley, H & Smith, K (2015) Sliding subject positions: knowledge and teacher educators. British Educational Research Journal
Brown, T., Rowley, H. & Smith, K. (2014). Rethinking research in teacher education. British Journal of Educational Studies. 62 (3), 281-296.

# From Russia with Math

Some readers may find interesting this issue, From Russia with Math,  of online magazine Higher Education in Russia and Beyond.

Many authors of articles in “From Russia with Math” are prominent
research mathematicians; Stanislav Smirnov, for example, is a Fields
Medallist. Not surprisingly, they tend to focus on education for people like them.

In my assessment, this is the key message of the publication:

in Russia, high quality academically selective mathematics education remains possible even after the collapse of the system of mass equal-for-all education.

# Adapted from StackExchange,  Getting Students to Not Fear Confusion

QUESTION:  I’m a fifth year grad student, and I’ve taught several classes for freshmen and sophomores. This summer, as an “advanced” (whatever that means) grad student I got to teach an upper level class: Intro to Real Analysis.

Since this was essentially these student’s first “real” math class, they haven’t really learned how to study for or learn this type of thing. I’ve continually emphasized throughout the summer that they need to put in more work than just doing a few homework problems a week.

Getting a feel for the definitions and concepts involved takes time and effort of going through proofs of theorems and figuring out why things were needed. You need to build up an arsenal of examples so some general picture of the ideas are in your head.

Most importantly, in my opinion, is that you wallow in your confusion for a bit when struggling with problems. Spending time with your confusion and trying to pull yourself out of it (even if it doesn’t work!) is a huge part of the learning process. Of course asking for help after a point is important too.

Question: What is a good way to convince students that spending time lost and confused is a reasonable thing and how do you actually motivate them to do it?

Anecdote: Despite trying all quarter to explain this in various ways, I would consistently have people come in to office hours who had barely touched the homework because “they were confused”. But they hadn’t tried anything. Then when I talk around an answer to try to get them to do certain key parts on their own or get them to understand the concept involved, they would get frustrated and ask “so does it converge or not?!”

It is incredibly hard to shake their firm belief that the answer is the important thing. Those that do get out of this belief seem to get stuck at writing down a correct proof is the important thing. None seem to make it to wanting to understand it as the important thing. (Probably a good community wiki question? Also, real-analysis might be an inappropriate tag, do what you will)

ANSWER from Ronnie Brown: Has anyone tried as an additional technique the “fill-in” method?

This is based on the tried and tested method of teaching called “reverse chaining”. To illustrate it, if you are teaching a child to put on a vest, you do not throw it the vest and say put it on. Instead, you put it almost on, and ask the child to do the last bit, and so succeed. You gradually put the vest less and less on, the child always succeeds, and finally can put it on without help. This is called “error-less learning” and is a tried and tested method, particularly in animal training (almost the only method! ask any psychologist, as I learned it from one).

So we have tried writing out a proof that, say, the limit of the product is the product of the limits, (not possible for a student to do from scratch), then blanking out various bits, which the students have to fill in, using the clues from the other bits not blanked out. This is quite realistic, where a professional writes out a proof and then looks for the mistakes and gaps! The important point is that you are giving students the structure of the proof, so that is also teaching something.

This kind of exercise is also nice and easy to mark!

Finally re failure: the secret of success is the successful management of failure! That can be taught by moving slowly from small failures to extended ones. This is a standard teaching method.

Additional points: My psychologist friend and colleague assured me that the accepted principle is that people (and animals) learn from success. Another way of getting this success is to add so many props to a situation that success is assured, and then gradually to remove the props. There are of course severe problems in doing all this in large classes. This will require lots of ingenuity from all you talented young people! You can find some more discussion of issues in the article discussing the notion of context versus content.

My own bafflement in teenage education was not of course in mathematics, but was in art: I had no idea of the basics of drawing and sketching. What was I supposed to be doing? So I am a believer in the interest and importance of the notion of methodology in whatever one is doing, or trying to do, and here is link to a discussion of the methodology of mathematics.

Dec 10, 2014 I’d make another point, which is one needs observation, which should be compared to a piano tutor listening to the tutees performance. I have tried teaching groups of say 5 or 6, where I would write nothing on the board, but I would ask a student to go to the board, and do one of the set exercises. “I don’t know how to do it!” “Well, why not write the question on the board as a start.” Then we would proceed, giving hints as to strategy, which observation had just shown was not there, but with the student doing all the writing.

In an analysis course, when we have at one stage to prove AB, I would ask the class: “What is the first line of the proof?” Then: “What is the last line of the proof?” and after help and a few repetitions they would get the idea. I’m afraid grammar has gone out of the school syllabus, as “old fashioned”!

Seeing maths worked out in real time, with failures, and how a professional deals with failure, is essential for learning, and at the research level. I remember thinking after an all day session with Michael Barratt in 1959: “Well, if Michael Barratt can try one damn fool thing after another, then so can I!”, and I have followed this method ever since. (Mind you his tries were not all that “damn fool”, but I am sure you get the idea.) The secret of success is the successful management of failure, and this is perhaps best learned from observation of a professional.

# Basic Ratio Capacity May Serve as Building Block for Math Knowledge

Contact: Anna Mikulak
Association for Psychological Science
amikulak@psychologicalscience.org

Understanding fractions is a critical mathematical ability, and yet it’s one that continues to confound a lot of people well into adulthood. New research finds evidence for an innate ratio processing ability that may play a role in determining our aptitude for understanding fractions and other formal mathematical concepts. Continue reading

# Masterclass: Alexander Shen, “Geometry in Problems”

Classes given by Alexander Shen at Summer School “Vanechki” in August 2014 in Portugal, based on his book Geometry in Problems. It seems that the audience are children of Russian diaspora, classes are conducted in mixture of English and Russian. However, an English speaking teacher of mathematics may make a lot of interesting observations.