# What Students Like

A new paper  in The De Morgan Gazette:

A. Borovik, What Students Like, The De Morgan Gazette 9 no.~1 (2017), 1–6. bit.ly/2ie2WLz

Abstract: I analyse students’ assessment of tutorial classes supplementing my lecture course and share some observations on what students like in mathematics tutorials. I hope my observations couldbe useful to my university colleagues around the world. However, this is not a proper sociologicalstudy (in particular, no statistics is used), just expression of my personal opinion.

# Call for Nominations for the 2017 ICMI Felix Klein and Hans Freudenthal Awards

[All nominations must be sent by e-mail to the Chair of the Committee (annasd >>at<< edu.haifa.ac.il, sfard >>at<< netvision.net.il) no later than 15 April 2017.]

Since 2003, the International Commission on Mathematical Instruction (ICMI) awards biannually two awards to recognise outstanding accomplishments in mathematics education research: the Felix Klein Medal and the Hans Freudenthal Medal.

The Felix Klein medal is awarded for life-time achievement in mathematics education research. This award is aimed at acknowledging excellent senior scholars who have made a field-defining contribution over their professional life. Past candidates have been influential and have had an impact both at the national level within their own countries and at the international level. We have valued in the past those candidates who not only have made substantial research contributions, but also have introduced new issues, ideas, perspectives, and critical reflections. Additional considerations have included leadership roles, mentoring, and peer recognition, as well as the actual or potential relationship between the research done and improvement of mathematics education at large, through connections between research and practice.

The Hans Freudenthal medal is aimed at acknowledging the outstanding contributions of an individual’s theoretically robust and highly coherent research programme. It honours a scholar who has initiated a new research programme and has brought it to maturation over the past 10 years. The research programme is one that has had an impact on our community. Freudenthal awardees should also be researchers whose work is ongoing and who can be expected to continue contributing to the field. In brief, the criteria for this award are depth, novelty, sustainability, and impact of the research programme.
For further information about the awards and for the names of past awardees (seven Freudenthal Medals and seven Klein Medals, to date), see http://www.mathunion.org/icmi/activities/awards/the-klein-and-freudenthal-medals/

# Misha Gavrilovich: Expressing the statement of the Feit-Thompson theorem with diagrams in the category of finite groups

Misha Gavrilovich’s paper Expressing the statement of the Feit-Thompson theorem with diagrams in the category of finite groups, available from

is a follow-up to his paper in The De Morgan Gazette,

M. Gavrilovich, Point-set topology as diagram chasing computations, The De Morgan Gazette, 5 no. 4 (2014), 23-32

The paper raises important questions about optimal approaches to exposition of elementary group theory: quite a number of group-theoretic concepts (for example, solvable, nilpotent group, p-group and prime-to-p group, abelian, perfect, subnormal subgroup, injective and surjective homomorphism) can be expressed as diagram chasing in the category theoretic language.

# 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.