# 3 – 1 = 2

What follows is a translation of a fragment from Igor Arnold’s (1900—1948) paper of 1946 Principles of selection and composition of arithmetic problems (Известия АПН РСФСР, 1946, вып. 6, 8-28). I believe it is relevant to the current discussions around “modelling” and “real life mathematics”. For research mathematicians, it may be interesting that I.V. Arnold was V.I. Arnold’s father.

Existing attempts to classify arithmetic problems by their themes or by their algebraic structures (we mention relatively successful schemes by Aleksandrov (1887), Voronov (1939) and Polak (1944)} are not sufficient [...] We need to embrace the full scope of the question,  without restricting ourselves to the mere algebraic structure of the problem: that is, to characterise those operations which need to be carried out for a solution. The same operations can also be used in completely different concrete situations, and a student may draw a false conclusion as to why these particular operations are used.

Let us use as an example several problems which can be solved by the operation

$3 - 1 =2.$

1.  I was given 3 apples, and I have eaten one of them. How many apples are left?
2. A three meters long barge-pole reached the bottom of the river, with one meter of it remaining above the level of water. What is the depth of the river?
3. Tanya said: “I have three more brothers than sisters”. In Tanya’s family, how many more boys are there than girls?
4. A train was expected to arrive to a station an hour ago. But it is 3 hours late. When will it arrive?
5. How many cuts do you have to make to saw a log into 3 pieces?
6. I walked from the first milestone to the third one. The distance between milestones is 1 mile. For how many miles did I walk?
7. A brick and a spade weigh the same as 3 bricks. What is the weight of the spade?
8. The arithmetic mean of two numbers is 3, and half their difference is 1. What is the smaller number?
9. The distance from our house to the rail station is 3 km, and to Mihnukhin’s along the same road is 1 km. What is the distance from the station to Mihnukhin’s?
10. In a hundred years we shall celebrate the third centenary of our university. How many centuries ago it was founded?
11. In 3 hours I swim 3 km in still water, and a log can drift 1 km downstream. How many kilometers I will make upstream in the same time?
12. 2 December was Sunday. How many working days preceded the first Tuesday of that month? [This question  is historically specific: in 1946 in Russia, when these problems were composed, Saturday was a working day --AB]
13. I walk with speed of 3 km per hour; my friend ahead of me walks pushing his motobike with speed 1 km per hour. At what rate is the distance between us diminishing?
14. Three crews of ditch-giggers, of equal numbers and skill, dug a 3 km long trench in a week. How many such crews are needed to dig in the same time a trench that is 1 km shorter?
15. Moscow and Gorky are in adjacent time zones. What is the time in Moscow when it is 3 p.m in Gorky?
16. To shoot at a plane from a stationary anti-aircraft gun, one has to aim at the point three plane’s lengths ahead of the plane. But the gun is moving in the same direction as the plane with one third the speed. At what point should the gunner aim his gun?
17. My brother is three times as old as me. How many times my present age was he  in the year when I was born?
18. If you add 1 to a number, the result is divisible by 3. What is the reminder upon division of the original number by 3?
19. A train of 1 km length passes by a pole in minute, and passes right through through a tunnel at the same speed — in 3 minutes. What is the length of the tunnel?
20. Three trams operate on a two track route, with each track reserved to driving in one direction. When trams are on the same track, they keep 3 km intervals. At a particular moment of time  one of them is at crow flight distance of 1 km from a tram on the opposite track. What is the distance from the third tram to the the  nearest one?

These examples clearly show that teaching arithmetic involves, as a key component, the development of  an ability to negotiate situations whose concrete natures represent very different relations between magnitudes and quantities. The difference between the “arithmetic” approach to solving problems and the algebraic one is, primarily the need to make a  concrete and sensible interpretation of all the values which are used and/or which appear in the discourse.

[With thanks to Tony Gardiner]

# Gove reveals rethink on grades in new GCSE

From The Independent (not in Hansard yet):

[Mr Gove, speaking to Education Select Committee on 15 May) indicated he was]

planning to scrap the present grading system entirely and replace A* and A grade passes with a one, two, three or four pass. [...]

He said it could well be the case that the “band of achievement that is currently A* and A” was replaced by a new one, two, three or four pass. The new-style GCSEs will start to be taught in schools in September 2015.

Graham Stuart, the Conservative chairman of the committee, also argued that Mr Gove could be “deliberately” paving the way for “grade deflation” in the exam system through the changes.

He said that the pass rate could also go down in the first year of pupils sitting the new exam (2017) – “because schools don’t know how to work the system”.

Students who previously were awarded an A grade pass could be awarded a four  under the new system (a one or two would be roughly equivalent to an A* while three or four would equate to an A grade). Academics argue a four would not be seen by employers and universities as a top grade pass.  Numbers are likely to replace grades throughout the system so instead of A* to G grade passes students would be awarded one to 10 passes.

However, Mr Gove replied that that the current exam system meant teachers were spending “too much time on exam technique and not enough on content”.

# Rutherford Schools Physics Project

A new five-year project aimed at developing the skills of sixth-form physicists has been awarded a £7 million grant by the Department for Education.

The Rutherford Schools Physics Project, led by Cambridge University Professor of Theoretical Physics Mark Warner, and Cavendish Laboratory Outreach Officer Dr Lisa Jardine-Wright, will work collaboratively with teachers, schools and other partner universities to deliver extension materials, on-line learning, workshops for students and support for physics teachers.[...]

The project will also work closely with its two sister initiatives, the Cambridge Mathematics Education Project, led by Professor Martin Hyland and also supported by the DfE, and “i-want-to-study-engineering.org”, led by Professor Richard Prager and supported by the Underwood Trust.

Since Archimedes, mathematics and physics have been inseparable, and the interdependence continues into the 21st century — Professor Mark Warner

# Rating of mathematics in universities of the world

1. University of Cambridge
2. Massachusetts Institute of Technology (MIT)
3. Harvard University
4. University of California, Berkeley (UCB)
5. University of Oxford
6. Princeton University
7. University of California, Los Angeles (UCLA)
8. Stanford University
9. (=10) ETH Zurich (Swiss Federal Institute of Technology)
10. (=9) National University of Singapore (NUS)

British universities in the top 50:

1. Cambridge
5. Oxford
12. Imperial
23. Warwick
38. Bristol
46-49: Manchester (shared with Nanyang Technological, Auckland and Queensland)

# Lewis Carroll – Man of Science

Conference jointly organised by The British Society for the History of Mathematics
and The Lewis Carroll Society in association with The Birmingham and Midland Institute

The Birmingham and Midland Institute, Margaret Street, Birmingham, B3 3BS

Saturday, 18th May 2013.  10:30 – 17:00.

# Report from Demos: Detoxifying school accountability

A report from Demos, published today. From Executive Summary:

This report strongly argues that the current model of accountability is profoundly toxic and is failing to achieve its stated goal of improving education. It sets out an alternative
regime, which would allow all children to achieve their potential, while ensuring the quality of education in schools is of a high standard. [...]

# Labour would reverse Gove’s A-level plan

From BBC:

Labour will reverse many of the coalition’s changes to A-levels if it wins the next election, shadow schools minister Kevin Brennan has told England’s exam regulator.

In a letter to Ofqual, Mr Brennan said Labour could not support “a policy that undermines both rigour and equity”[...]

Mr Brennan, writing to the chief exams regulator, Glenys Stacey, said “the weight of opposition” to decoupling the two sets of qualifications [A and AS levels --AB] was “overwhelming

He said the move would narrow students’ A-level choices, remove a key indicator for assessing university applicants and undermine progress in widening access to higher education. [...]

I understand that the secretary of state’s position on this constitutes a policy direction to you, but in undertaking your work we think that it is important to signal clearly what our position will be following the next general election.

It is on this basis that I write to you to inform you that a future Labour government in 2015 would not proceed with the decoupling of AS and A-levels.

The letter says that under Labour AS-levels would continue to be building blocks towards A-levels and students would continue to choose which AS-level subjects they take as full A-levels.

Mr Brennan also raises concerns about other aspects of the government’s plan, including “linear assessment for all subjects at the end of two years of study, the rushed timetable for implementation, and the limited evidence base on which the proposals have been made“.

A Labour spokesman added that further consultation with subject experts was needed before deciding the exact form of assessment for each A-level.

# Children learn to tie shoelaces later than ever before

One of many cultural shifts undermining the traditional model of mathematics education: loss of dexterity in children. From The Telegraph:

Today’s children may be whiz kids at hi-tech gadgets, but they now learn to tie their shoelaces at a later age than ever before, a new report has found.

Few master the art before the age of six, and some still have difficulty tying their own laces when they are nine or ten years old, it is claimed.

The findings represent a major shift in social habits – just thirty years ago, being able to tie shoelaces was regarded as a skill to be learnt by the age of four, but changes in shoe design and footwear fashions means the skill is no longer essential until much older.

Gary Kibble, retail director for Littlewoods.com who carried out the study, said: “Today’s children now learn how to operate complex technology long before they know how to tie shoe laces. They understand navigation paths and algorithms – yet still don’t know how to make a knot.