A long awaited** Report of Professor Sir Adrian Smith’s review of post-16**

**mathematics** is published.

# Author Archives: Alexandre Borovik

# Theorem proving components for educational software

# Call for Extended Abstracts & Demonstrations (2nd)

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ThEdu’17

Theorem proving components for Educational software

6 August 2017

http://www.uc.pt/en/congressos/thedu/thedu17

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at CADE26

International Conference on Automated Deduction

6-11 August 2017

Gothenburg, Sweden

http://www.cade-26.info/

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## THedu’17 Scope:

Computer Theorem Proving is becoming a paradigm as well as a technological base for a new generation of educational software in science, technology, engineering and mathematics. The workshop brings together experts in automated deduction with experts in education in order to further clarify the shape of the new software generation and

to discuss existing systems.

## Invited Talk

Francisco Botana, University of Vigo, Spain, “Theorem Proving Components in GeoGebra”

## Important Dates

- Extended Abstracts: 18 June 2017
- Author Notification: 2 July 2017
- Final Version: 16 July 2017
- Workshop Day: 6 August 2017

## Topics of interest include:

- methods of automated deduction applied to checking students’ input;
- methods of automated deduction applied to prove post-conditions for particular problem solutions;
- combinations of deduction and computation enabling systems to propose next steps;
- automated provers specific for dynamic geometry systems;
- proof and proving in mathematics education.

## Submission

We welcome submission of extended abstracts and demonstration proposals presenting original unpublished work which is not been submitted for publication elsewhere.

All accepted extended abstracts and demonstrations will be presented at the workshop. The extended abstracts will be made available online.

Extended abstracts and demonstration proposals should be submitted via easychair,

https://easychair.org/conferences/?conf=thedu17

formatted according to

http://www.easychair.org/publications/easychair.zip

Extended abstracts and demonstration proposals should be approximately 5 pages in length and are to be submitted in PDF format.

At least one author of each accepted extended abstract/demonstration proposal is expected to attend THedu’17 and presents his/her extended abstract/demonstration.

## Program Committee

Francisco Botana, University of Vigo at Pontevedra, Spain

Roman Hašek, University of South Bohemia, Czech Republic

Filip Maric, University of Belgrade, Serbia

Walther Neuper, Graz University of Technology, Austria (co-chair)

Pavel Pech, University of South Bohemia, Czech Republic

Pedro Quaresma, University of Coimbra, Portugal (co-chair)

Vanda Santos, CISUC, Portugal

Wolfgang Schreiner, Johannes Kepler University, Austria

Burkhart Wolff, University Paris-Sud, France

## Proceedings

The extended abstracts and system descriptions will be published as a CISUC Technical Report series (ISSN 0874-338X). After presentation at the conference, selected authors will be invited to submit a substantially revised version, extended to 14-20 pages, for

publication by the Electronic Proceedings in Theoretical Computer Science (EPTCS).

# Teacher Preparation: 25 May, De Morgan House, Russell Square

**LMS Education Day 2017**

**Thursday 25th May, 11am – 3pm
**

**De Morgan House, London**

Teacher shortages in mathematics: how can HE mathematics departments help reverse the trend?

University mathematics departments depend on teachers to prepare their

own students, and they have an important role in training future generations of mathematics teachers. To do this effectively at a national level, it is critical that colleagues from across the sector understand the current state of Initial Teacher Training and the challenges that face teacher recruitment.

The day will be split into two parts. During the morning, participants will have the opportunity to learn about the challenges of teacher recruitment and find out how a number of maths departments have attempted to encourage students to think of mathematics teaching as a career. We are delighted that Simon Singh has agreed to introduce this session. After lunch, discussion, led by Tony Gardiner, will be focused around a document being developed by the LMS education committee on this subject for which input and feedback is sought. A detailed programme, including information about invited contributors, will follow in the coming weeks.

Whilst the theme for the day may seem somewhat removed from everyday teaching and learning activity within mathematics departments in HE, we do hope to get participants from a large number of mathematics departments to participate in the event and share their experiences and ideas.

The event is free to attend and a light lunch and other refreshments will be provided.

Please register interest in attending the event by emailing

Katherine.Wright@lms.ac.uk

# Ronnie Brown: from Esquisse d’un Programme by A Grothendieck

those said to be “advanced”) with a modest (and frequently less than mod-

est) mathematical baggage, led me to a Draconian renewal of the themes

of reﬂection I proposed to my students, and gradually to myself as well.

**It seemed important to me to start from an intuitive baggage common to**

everyone, independent of any technical language used to express it, and

anterior to any such language– it turned out that the geometric and topo-

everyone, independent of any technical language used to express it, and

anterior to any such language

logical intuition of shapes, particularly two-dimensional shapes, formed such

a common ground.

# Chinese maths textbooks to be translated for UK schools

The Guardian, 20 March 2017. Some quotes:

British students may soon study mathematics with Chinese textbooks after a “historic” deal between HarperCollins and a Shanghai publishing house in which books will be translated for use in UK schools.

HarperCollins signs ‘historic’ deal with Shanghai publishers amid hopes it will boost British students’ performance.

The textbook deal is part of wider cooperation between the UK and China, and the government hopes to boost British students’ performance in maths, Hughes added.

Most likely, an attempt to introduce Chinese maths textbooks in English schools will lay bare the basic fact still not accepted by policymakers. Quoting the article,

Primary school maths teachers in Shanghai are specialists, who will have spent five years at university studying primary maths teaching. They teach only maths, for perhaps two hours a day, and the rest of the day is spent debriefing, refining and improving lessons. English primary teachers, in contrast, are generalists, teaching all subjects, all of the time.

See the whole article here.

# 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.

# David Pierce: The geometry of numbers in Euclid

**Reposted from David Pierce’s blog Polytropy**

This is about how the *Elements* of Euclid shed light, even on the most basic mathematical activity, which is counting. I have tried to assume no more in the reader than elementary-school knowledge of how whole numbers are added and multiplied.

How come 7 ⋅ 13 = 13 ⋅ 7? We can understand the product 7 ⋅ 13 as the number of objects that can be arranged into seven rows of thirteen each.

If we turn the rows into columns, then we end up with thirteen rows of seven each; now the number of objects is 13 ⋅ 7.

In the end, it doesn’t matter whether we have arranged the objects into rows or columns of thirteen. Either way, when we gather up the objects and count them, we must always get the same result.

Must we really? We believe from childhood that we must. As children, we learn to say certain words in a certain order: one, two, three, four,

and so on. We learn to say these words as we move objects, one by one, from one pile to another. As we move the last object, the last word we say is supposed to be the number of objects in the original pile. We have now counted that pile. In the process, we have removed the pile; but if we count the new pile, we get the same number.

At least we think we do. Does anybody ever question this? If we do question it, and if we are familiar with some mathematical terminology, we may decide that, in technical terms, what we are asking is whether all *linear orderings* of the same finite set are *isomorphic,* or whether all *one-to-one functions* from the set to itself are also *onto* the set. We can prove that they are, in either case, by the method of *mathematical induction.* However, I suppose it takes some mathematical sophistication, not only to understand the terminology, but to believe that anything is accomplished by its use. If one is being asked to learn the method of mathematical induction for the first time, I doubt one will be impressed by its usefulness in establishing that, no matter how many times you count a bag of bottle-caps, you will always get the same number.

Meanwhile, there is a more fundamental question: what is a number in the first place? As it happens, for me, the best theoretical answer is that a number, a *counting* number, is a nonempty *ordinal* that neither contains a *limit* nor is itself a limit. An ordinal is a set with two properties: (1) it contains every member of each of its members, and (2) among the members of each of its nonempty subsets, there is one that has no other as a member. The empty set is an ordinal, and if a set called alpha is an ordinal, then so is the set that contains every member of alpha, along with alpha itself. This new set is the *successor* of alpha, and every ordinal that is neither the empty set nor a successor is a limit. Now, using the method given by von Neumann in 1923, I have defined counting numbers in simple terms, but in a complicated way that cannot be made sense of without some work. I am not going to do that work here, but I shall instead suggest that Euclid’s *Elements* offers an understanding of numbers that is unmatched, as far as I know, until the work of Dedekind in 1888. It some ways it may remain unmatched in the twenty-first century.

For Euclid, a number is a *magnitude.* A pile of bottle-caps might be called a magnitude; at least it has a weight, to which may be assigned a number. No matter how the bottle-caps are piled into the pan of a scale, we expect the same weight to be found; but it is hard to see how this observation can be made into a mathematical principle.

Euclid’s typical magnitudes—the ones seen in his diagrams—are bounded straight lines, or what we call line segments. What makes one of these a *number* is that some specified segment *measures* it, or goes into it evenly. This is the fundamental notion. The measuring segment is a *unit,* as is any other segment that is equal to it—equal in the sense of being congruent.

A number then is a magnitude that can be divided into units. Unless it is *prime,* it can be divided into numbers as well. Thus a number consisting of fifteen units can be divided into those fifteen units, or into five numbers of three units each, or three numbers of five units each. In the last case, we might refer to each of those three numbers as five; but then, strictly speaking, we are using the adjective five

as a noun meaning five units. The units may vary. All fives are *equal*—equal in number— but they are not all the *same.*

Dividing is not the same as measuring, but complementary. Dividing fifteen apples among five children is a different activity from measuring how many five-apple collections can be formed from fifteen apples. In the first case, each child gets three apples, in the second, three collections of apples are formed. A number of three things arises in each case, because multiplying three by five has the same result as multiplying five by three.

This is only a special case of Euclid’s general result, which is Proposition 16 of Book VII of the *Elements*:

If two numbers multiply one another,

their products will be equal to one another.

Actually the Greek is a bit wordier: If two numbers, multiplying one another, make some things, the products of them [that is, the original numbers] will be equal to one another.

This multiplication is defined as follows:

Ἀριθμὸς ἀριθμὸν πολλαπλασιάζειν λέγεται,

ὅταν,

ὅσαι εἰσὶν ἐν αὐτῷ μονάδες,

τοσαυτάκις συντεθῇ ὁ πολλαπλασιαζόμενος,

καὶ γένηταί τις.A number is said to multiply a number

whenever,

however many units are in it,

so many times is the the number being multiplied laid down,

and something is produced.

Jonathan Crabtree argues strenuously that multiplying *A* by *B* does not mean adding *A* *to itself* *B* times, since this would result in a sum of *B* + 1 copies of *A*.

A product is the multiple of a *multiplicand* by a *multiplier.* Euclid proves that the roles of multiplicand and multiplier are interchangeable: in modern terms, multiplication is *commutative.* The proof uses a theory of *proportion.* In this theory, there are several ways to say the same thing:

- the numbers
*A*,*B*,*C*, and*D*are in proportion; - the ratio of
*A*to*B*is the same as the ratio of*C*to*D*; *A*is to*B*as*C*is to*D*.

I shall abbreviate these by writing

*A*:

*B*::

*C*:

*D*.

The meaning of this for Euclid may not be crystal clear to modern readers; but I think it can only mean that when the so-called *Euclidean algorithm* is applied to *C* and *D*, the algorithm has the same steps as when applied to *A* and *B*.

Applied to any two magnitudes, each step of the Euclidean algorithm has the following two parts:

- Judge whether one of the magnitudes is greater than the other.
- If it is, then subtract from it a piece that is equal to the less magnitude.

Repeat as long as you can. There are three possibilities. In the simplest case, you will keep subtracting pieces equal to the same magnitude, until what is left of the other magnitude is equal to it. In this case, the one magnitude measures the other: alternatively, it is a *part* of the other. In case the algorithm never ends, then the original magnitudes must not have been numbers, but they were *incommensurable.* In the third case, you end up with a *greatest common measure* of the original two numbers, and each of these numbers is said to be *parts* of the other.

That is Euclid’s terminology. By the definition at the head of Book VII of the *Elements,* *A* : *B* :: *C* : *D* means *A* is the same part, or parts, or multiple of *B* that *C* is of *D*. Again, if we assume Euclid knows what he’s doing, this can only mean that, at each step of the Euclidean algorithm, the same magnitude (first or second, left or right) is greater, whether we start with *A* and *B* or *C* and *D*. Thus 8 : 6 :: 12 : 9, because

2 < 6, 6 – 2 = 4;

2 < 4, 4 – 2 = 2;

while in the same way

3 < 9, 9 – 3 = 6;

3 < 6, 6 – 3 = 3;

the pattern >, <, < is the same in each case. We discover incidentally that the greatest common measure of 8 and 6 is 2; and of 12 and 9, 3. In fact

The repetition of the multipliers 4 and 3 here also ensures the proportion 8 : 6 :: 12 : 9, but only because 3 and 4 are *prime to one another*: they have no common measure, other than a unit. If we did not impose this condition on the multipliers, then the definition of proportion alone would not ensure the *transitivity* of sameness of ratios: the definition alone would not guarantee that ratios that were the same as the same ratio were the same as one another. But every kind of sameness has this property. Therefore, although Euclid does not quite spell it out, I contend that his definition of proportion of numbers has the meaning that I have given.

We can now describe Euclid’s proof of the commutativity of multiplication as follows. We accept that addition is commutative:

*A*+

*B*=

*B*+

*A*.

This means, if you pick up a rod, turn it end to end, and put it back down, it will still occupy the same distance. One might try to imagine a geometry in which this is not true; but we assume it is true. It follows then that, for any multiplier *x*,

*A*

*x*+

*Bx*= (

*A*+

*B*)

*x*,

that is,

*A*+ … +

*A*+

*B*+ … +

*B*=

*A*+

*B*+ … +

*A*+

*B*,

where the ellipses represent the same number of missing terms in each case.

Given *A* : *B* :: *C* : *D*, we now show *A* : *B* :: (*A* + *C*) : (*B* + *D*). Assuming, as we may, that *A* is less than *B*, we have two possibilities:

*A*is part of*B*, and so, for some*x*, we have*B*=*Ax*,*D*=*Cx*.*A*is parts of*B*, and so, for some*x*and*y*that are prime to one another, for some*E*and*F*, we have*B*=*Ex*,*A*=*Ey*,*D*=*Fx*,*C*=*Fy*.

In the first case, *B + D* = *Ax + Cx* = *(A + C)x*. Similarly, in the second case, *B + D* = *(E + F)x*, while *A + C* = (*E* + *F*)*y*. In either case, we have the desired conclusion, *A* : *B* :: (*A* + *C*) : (*B* + *D*). As special cases, we have

*A*:

*B*::

*A*+

*A*:

*B*+

*B*::

*A*+

*A*+

*A*:

*B*+

*B*+

*B*

and so on; in general, *A* : *B* :: *Ax* : *Bx*.

Given again *A* : *B* :: *C* : *D*, we now show *A* : *C* :: *B* : *D*. We consider the same two cases as before. In case (1), we have *A* : *C* :: *Ax* : *Cx* :: *B* : *D*. In the same way, in case (2), we have *A* : *C* :: *E* : *F* and *B* : *D* :: *E* : *F*, so again *A* : *C* :: *B* : *D*.

Finally, denoting a unit by 1, since by definition we have 1 : *A* :: *B* : *BA* and 1 : *B* :: *A* : *AB*, and the latter gives us now 1 : *A* :: *B* : *AB*, we conclude *BA* = *AB*. This is Proposition 16 of Book VII of Euclid’s *Elements.*

Thus, if we lay out seven sticks end to end, each thirteen units long, we reach the same length as if we lay out thirteen sticks, each seven units long. This is not obvious, even though, if you follow the rules of computation learned in school, you will find that 7 ⋅ 13 and 13 ⋅ 7 are equal. Euclid proves that this will be so, without any need for computation—which anyway will apply only to the particular example in question.

# V. Solomonov: Short Rules for Russians Teaching Calculus and Lower-Level Classes in USA

A new paper in The De Morgan Gazette:

V. Solomonov, Short Rules for Russians Teaching Calculus and Lower-Level Classes in USA, The De Morgan Gazette 8 no. 5 (2016), 79–84 ISSN 2053–1451. bit.ly/2izEyR2

# PhD studentships at Sheffield Hallam University

PhD studentships in mathematics education at Sheffield Hallam University. Further information and contact details are below – please contact us to discuss your proposal.

Full details and application forms are available here: http://www.jobs.ac.uk/job/AWG226/phd-studentships-4-posts/ Closing date: 17:00 on 1st February 2017

# Thales and the Nine-point Conic

New paper in the De Morgan Gazette:

David Pierce, Thales and the Nine-point Conic, The De Morgan Gazette 8 no. 4 (2016) 27-78. bit.ly/2hlyHzZ. ISSN 2053-1451

Abstract:The nine-point circle is established by Euclidean means; the nine-point conic, Cartesian.Cartesian geometry is developed from Euclidean by means of Thales’ s Theorem. A theory of proportion is given, and Thales’s Theorem proved, on the basis of Book I of Euclid’sElements, without the Archimedean assumption of Book V. Euclid’s theory of areas is used, although this is obviated by Hilbert’s theory of lengths. It is observed how Apollonius relies on Euclid’s theory of areas. The historical foundations of the name of Thales’s Theorem are considered. Thales is thought to have identified water as a universal substrate; his recognition of mathematical theorems as such represents a similar unification of things.