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
\[3 - 1 =2. \]
- I was given 3 apples, and I have eaten one of them. How many apples are left?
- 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?
- Tanya said: “I have three more brothers than sisters”. In Tanya’s family, how many more boys are there than girls?
- A train was expected to arrive to a station an hour ago. But it is 3 hours late. When will it arrive?
- How many cuts do you have to make to saw a log into 3 pieces?
- I walked from the first milestone to the third one. The distance between milestones is 1 mile. For how many miles did I walk?
- A brick and a spade weigh the same as 3 bricks. What is the weight of the spade?
- The arithmetic mean of two numbers is 3, and half their difference is 1. What is the smaller number?
- 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?
- In a hundred years we shall celebrate the third centenary of our university. How many centuries ago it was founded?
- 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?
- 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]
- 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?
- 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?
- Moscow and Gorky are in adjacent time zones. What is the time in Moscow when it is 3 p.m in Gorky?
- 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?
- My brother is three times as old as me. How many times my present age was he in the year when I was born?
- 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?
- 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?
- 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]
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”.
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
- University of Cambridge
- Massachusetts Institute of Technology (MIT)
- Harvard University
- University of California, Berkeley (UCB)
- University of Oxford
- Princeton University
- University of California, Los Angeles (UCLA)
- Stanford University
- (=10) ETH Zurich (Swiss Federal Institute of Technology)
- (=9) National University of Singapore (NUS)
British universities in the top 50:
46-49: Manchester (shared with Nanyang Technological, Auckland and Queensland)
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.
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. [...]
Foundations of the Formal Sciences VIII: History & Philosophy of Infinity
organized by the network INFTY in the series FotFS
20-23 September 2013
Corpus Christi College
Proposals deadline: 31 May 2013
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.
Read the whole article.
…. the poignant and hugely entertaining theatre production of “The Universal Machine” at the New Diorama in central London. On April 23 there was a special performance with various various prominent ATY supporters in the audience. It was a great treat to see the nieces of Alan Turing there, familiar to many from their engaging TV interviews, with fascinating memories of their uncle Alan.
The uniformly wonderful company, and Diorama staff, must have been really relieved to hear all the positive comments. The music and cleverly crafted lyrics gave a special lightness to the essentially sad story, and both intensified, and lifted the impact to a new level. Turing’s niece Janet was especially happy to see her grandmother (Turing’s mother Sara) played so brilliantly by Judith Paris. Judith also attracted high praise from The Guardian.
There were lots of reviews in the national press. There was a thoughtful piece by Daisy Bowie-Sell in the Telegraph: http://bit.ly/ZLaiWH with our favourite review by the ever perceptive Libby Purves in The Times: http://www.thetimes.co.uk/tto/arts/stage/theatre/article3751815.ece
If you live within reach of London, don’t miss it! Some nights are already sold out, but it’s on at New Diorama (just 15 minutes walking from Kings Cross) until May 11: http://newdiorama.com/whats-on/the-universal-machine