Posted by: Alexandre Borovik | November 3, 2018

## Open Book Publishers

In my opinion, they deserve attention: Open Book Publishers ; it is likely that I will soon submit to them a book for publication. What follows is an excerpt from their recent email. Please notice a useful list of links at the end.

OBP is trialing a new platform to engage readers, encourage discussion and to keep our books alive and thriving. We have currently implemented hypothes.is on our  title: Hanging on to the Edges by Daniel Nettle – please take part! To annotate this book, all you have to do is click on the HTML version and look for the ‘Annotate this book’ button below the cover image.
https://www.openbookpublishers.com/
OBP recently weighed in on the dangers of participating in Knowledge Unlatched Open Funding and if you want to understand why OBP will not be participating, visit here.

Finally, for those interested, our new and updated Autumn 2018 catalogue is available to download here.

OA Week Blog Series: see an excerpt from An Academic’s Guide to Open Access, in which we explain why authors should choose to publish Open Access or, to read the whole series, visit here.

This blog series includes answers to the following questions:

Posted by: Alexandre Borovik | November 3, 2018

## Back to basics

My good colleague allowed me to distribute these extracts from his emails. They are quite interesting, in my opinion.

I have now come to the conclusion that if you want first year students to learn how to write mathematics properly, it is necessary and fully sufficient to spend two hours face-to-face with them, in front of a blackboard, and have them write any form of silly proof such as: a uniformly continuous function is continuous.

But they must hold the stick of chalk and write, and you must correct real-time every single f*****g comma.

I can mark homework week after week and am stupidly dedicated at that; but homework, even good-willed, will not force the lost ones to make any progress, as opposed to the above. Just a face-to-face two hour session, correcting every move.

What a gain of time it would be to simply teach them the trick! How delightfully readable would their papers be afterwards! And what a side-benefit for the whole society!

Why do we then not do this?

As you see I completely gave up on conveying intuitions to 1st year students. Not to mention my own research for the time being.

This term my teaching duties are in a small branch the University has in *******. Classes of 20; students are dedicated, which is not unprecedented. But we can afford being dedicated to them, which is.

So unfair to the crowd in *****! [However] it is not impossible to implement.

Of course the professor who lectures in front of 100+ (30+ is already quite too much) cannot afford it.

Our TA’s should be assigned such — pleasant — duties, could we rely on all of them.

For all I request is two (2, TWO) hours per student, not per student per year. The cost is reasonable.

And the savings are huge; homework and exams become less of a pain: you only have to grade the contents, not return every-single-mark-my-word-bloody-time to the difference between “if… then…” and “therefore…” (a linguistic ability quite useful in everyday life). Which I do in written form, every-single-mark-my-word-bloody-time.

This, for the student who will spend in the average two or three years studying or trying to study mathematics, takes overall more than two hours.

I can report on this more in the future. One of my duties (in *****) will be to mark and comment on exams for people applying to become secondary school teachers. We actually run a full course entitled “How to write mathematics”. It is a year 5 class!

I have not suggested my idea there yet.

Posted by: Alexandre Borovik | October 14, 2018

## Confident students do not cheat

This an abstract of a talk given by me at the Meeting “Mathematical Academic Malpractice in the Modern Age”, Manchester, Monday 21st May 2018.

Confident students do not cheat: how to build mathematical confidence in our students

I think it could be useful to address the question which, in my experience, is almost never asked: what pushes problem students to cheat by plagiarising work from their peers and, increasingly, from the Internet? Some answer can be found in Denizhan (2014):

“These students exhibit an inability to evaluate their own performances independent of external measurements.”

Plagiarism is one of the psychological defenses of a student who does not otherwise know whether his/her solution / answer is correct.

Mathematics provides a simple remedy: systematically teach students how they can check their solutions. This will boost their confidence in their answers – and in themselves.

I teach linear algebra; I have at least two dozen undergraduate linear algebra textbooks in my office — none of them provides systematic advice on these matters. The same applies, I think, to any other undergraduate subject.

In my view, the most efficient methods for checking answers in a particular class of problems usually provided by a more advanced point of view. For example,

• all these elementary problems about systems of linear equations can be effectively checked if the concepts of the rank of a matrix is used;
• the correctness of eigenvalues of a matrix can be checked by using the fact that the sum of eigenvalues is the trace of the matrix, and the product is its determinant, etc.

This retrospective reassessment of previous material can give students a chance to see how actually simple it is — and boost their mathematical confidence.

In my talk, I’ll discuss how to incorporate error-correcting aspects of mathematics into course design.

Posted by: Alexandre Borovik | October 7, 2018

## What are some common gaps in logic that students make in mathematical proofs that lead to inaccurate results?

My answer to a question in Quora:

What are some common gaps in logic that students make in mathematical proofs that lead to inaccurate results?

I wish to comment on two specific flaws exhibited by students who encounter proofs first time in their lives.

The first one is

inability to accept the Identity Principle: “$$A$$ is $$A$$”, and arguments related to it, as a valid ingredient of proofs.

For many students, a basic observation

For all sets $$A$$, $$A \subseteq A$$ ($$A$$ is a subset of $$A$$) because every element of $$A$$ is an element of $$A$$

is very hard to grasp because of the appearance of the same words about  the same set $$A$$ twice in the sentence: “element of  $$A$$  is an element of  $$A$$”.  I have observed that many times and I think that students cannot overcome a mental block created by their

expectation that a proof should yield some new information about objects involved

– and this is the second fundamental flaw.

And, of course, reduction, removal of unnecessary information, is seen by many students as something deeply unnatural.

Every year, I hear from my Year 1 students the same objection:

How can we claim that 2 is less or equal than 3, that is, $$2\leqslant 3$$, if we already know that 2 is less than 3, $$2 < 3$$?

I think we encounter here a serious methodological (and perhaps philosophical) issue which I have never seen explicitly formulated in the literature on mathematics education:

a proof of a mathematical statement can illuminate and explain this statement, it may contain new knowledge about mathematics which goes far beyond the statement proved; but

• elementary steps in proofs frequently do not produce any new information, moreover, sometimes they remove unnecessary information from consideration.

A proof can be compared with a living organism built from molecules which can hardly be seen as living entities — and even worse, from atoms which are definitely not living objects.

This is closely related to another issue which many students find difficult to grasp: statements of propositional logic have no meaning, they have only logical values (or truth values, as they are frequently called) TRUE or FALSE. Any two true statements are logically equivalent to each other because they are both true; moreover, the statement

if London is a capital of England then tea is ready

makes perfect sense, and can be true or false, even if constitution of the country has no relation to the physical state of my teapot.

When my students express their unhappiness about logic which ignores meaning (and I provoke them to express their emotions), I provide an eye-opening analogy: numbers also have no meaning. The statement

The Jupiter has more moons than I have children

compares two numbers, and this arithmetic statement makes perfect sense (and is true) even if Jupiter has no, and cannot have any, connections whatsoever with my family life. Numbers have no meaning; they have only numerical values. Arithmetic, the most ordinary, junior school, sort of arithmetic is already a huge and deep abstraction. We did not notice that because we are conditioned that way.

Learning proofs also involves some degree of cultural conditioning. As a side remark, I suspect (but have no firm evidence) that the role of family — presence of clear rational argumentation in everyday conversations within family — could be important.

Posted by: Alexandre Borovik | September 7, 2018

## Mathematics for teachers of mathematics

My new paper at The De Morgan Gazette:

A. Borovik, Mathematics for teachers of mathematics, The De Morgan Gazette 10 no. 2 (2018), 11-25. bit.ly/2NWECtn

Abstract:

The paper contains a sketch of a BSc Hons degree programme Mathematics (for Mathematics Education). It can be seen as a comment on Gardiner (2018) where he suggests that the current dire state of mathematics education in England cannot be improved without an improved structure for the preparation and training of mathematics teachers:

Effective preparation and training requires a limited number of national institutional units, linked as part of a national effort, and subject to central guidance. For recruitment and provision to be efficient and effective, each unit should deal with a significant number of students in each area of specialism (say 20–100). In most systems the initial period of preparation tends to be either

•  a “degree programme” of 4–5 years (e.g. for primary teachers), with substantial subject-specific elements, or
• an initial specialist, subject-based degree (of 3+ years), followed by (usually 2 years) of pedagogical and didactical training, with some school experience.

This paper suggests possible content, and didactic principles, of

a new kind of “initial specialist, subject-based degree” designed for intending teachers.

This text is only a proof of concept; most details are omitted; those that are given demonstrate, I hope, that a new degree would provide a fresh and vibrant approach to education of future teachers of mathematics.

Posted by: Alexandre Borovik | September 5, 2018

## UKRI: Accelerating the transition to full and immediate Open Access to scientific publications

Yesterday, 4 September 2018, UKRI announced their

Since the LMS critcally depends on income from publishing, it has serious implications for out Society.

The key principle of the Plan is as follows:

“After 1 January 2020 scientific publications on the results from research funded by public grants provided by national and European research councils and funding bodies, must be published in compliant Open Access Journals or on compliant Open Access Platforms.”

• Authors retain copyright of their publication with no restrictions. All publications must be published under an open license, preferably the Creative Commons Attribution Licence CC BY. In all cases, the license applied should fulfil the requirements defined by the Berlin Declaration;
• The Funders will ensure jointly the establishment of robust criteria and requirements for the services that compliant high quality Open Access journals and Open Access platforms must provide;
• In case such high quality Open Access journals or platforms do not yet exist, the Funders will, in a coordinated way, provide incentives to establish and support them when appropriate; support will also be provided for Open Access infrastructures where necessary;
• Where applicable, Open Access publication fees are covered by the Funders or universities, not by individual researchers; it is acknowledged that all scientists should be able to publish their work Open Access even if their institutions have limited means;
• When Open Access publication fees are applied, their funding is standardised and capped (across Europe);
• The Funders will ask universities, research organisations, and libraries to align their policies and strategies, notably to ensure transparency;
• The above principles shall apply to all types of scholarly publications, but it is understood that the timeline to achieve Open Access for monographs and books may be longer than 1 January 2020;
• The importance of open archives and repositories for hosting research outputs is acknowledged because of their long-term archiving function and their potential for editorial innovation;
• The `hybrid’ model of publishing is not compliant with the above principles;
• The Funders will monitor compliance and sanction non-compliance.
Posted by: Alexandre Borovik | September 1, 2018

## Tony Gardiner: Towards an effective national structure for teacher preparation and support in mathematics

A new paper at The De Morgan Gazette:

A. D. Gardiner, Towards an effective national structure for teacher preparation and support in mathematics, The De Morgan Gazette 10 no. 1 (2018), 1-10. bit.ly/2N9NU7W

Posted by: Alexandre Borovik | August 30, 2018

## Trigger reflex

When I see a statement like that, I just cannot stop myself from pulling the trigger:

Why does Wittgenstein want surveyability? He seems to think that to be capable of the specific use of a theorem which a new proof makes possible we must be able  to reproduce its proof. This is just false, indeed perversely so — without understanding anything about Wiles’ proof of Fermat’s Last Theorem you can use it to rule out the truth of $$a^17 + b^17 = c^17$$ where $$a$$, $$b$$ and $$c$$ are any three integers, even hundreds of digits long — for example I know that $$123456789^17 + 12233445566778899^17$$ can’t be equal to $$12345678901234567890^17$$ without needing to calculate any of the three powers. [Edwin Coleman, The surveyability of long proofs, Foundations of Science, 14, Issue 1–2pp 27–43.]
Indeed, I believe most mathematicians will make an instant observation that $$17$$ is a odd natural number, and therefore the last digit of $$9^17$$ is $$9$$, and therefore the last digit of  $$123456789^17 + 12233445566778899^17$$ is $$8$$ and does not equal to the last digit of  $$12345678901234567890^17$$ , which is, of course, $$0$$. One does not need Fermat’s Last Theorem for that (and, for the sake of historical integrity of the narrative, the case $$n = 17$$ had been settled by Kummer in 1847).
Posted by: Alexandre Borovik | August 30, 2018

## Trigger reflex

When I see a statement like that, I just cannot stop myself from pulling the trigger:

Why does Wittgenstein want surveyability? He seems to think that to be capable of the specific use of a theorem which a new proof makes possible we must be able  to reproduce its proof. This is just false, indeed perversely so — without understanding anything about Wiles’ proof of Fermat’s Last Theorem you can use it to rule out the truth of $$a^17 + b^17 = c^17$$ where $$a$$, $$b$$ and $$c$$ are any three integers, even hundreds of digits long — for example I know that $$123456789^17 + 12233445566778899^17$$ can’t be equal to $$12345678901234567890^17$$ without needing to calculate any of the three powers. [Edwin Coleman, The surveyability of long proofs, Foundations of Science, 14, Issue 1–2pp 27–43.]
Indeed, I believe most mathematicians will make an instant observation that $$17$$ is a odd natural number, and therefore the last digit of $$9^17$$ is $$9$$, and therefore the last digit of  $$123456789^17 + 12233445566778899^17$$ is $$8$$ and does not equal to the last digit of  $$12345678901234567890^17$$ , which is, of course, $$0$$. One does not need Fermat’s Last Theorem for that (and, for the sake of historical integrity of the narrative, the case $$n = 17$$ had been settled by Kummer in 1847).
Posted by: Alexandre Borovik | August 22, 2018

## Do mathematicians need funding?

My answer to a question on Quora: Do mathematicians need funding?

There was a number of answers in this thread regurgitating the old joke about mathematicians and philosophers – apparently the latter require even less funding. What really matters is that mathematicians really need (very modest, admittedly) funding for meeting regularly other mathematicians – either at conferences, or, which is much more important, in direct visits to their universities, or in various programs like “Research in Pairs” run by the famous Mathematisches Forschungsinstitut Oberwolfach. There are serious psychological and neurophysiological reasons why doing serious mathematics in isolation from the community of fellow mathematicians is very difficult, next to impossible. I served, for a number of years, on a committee of a learned society which was awarding quick, easy to apply, small grants for international collaborative research in mathematics (basically, short visits, for about a week) . Quality of papers arising from this research is astonishingly high, on the value for money count it beats any state-funded bureaucratic schemes.