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.
This to a certain degree defines the difference of problems where it is natural to request an arithmetic solution from problems which are essentially algebraic. For the latter, an arithmetic solution could be seen as a higher level exercise that goes beyond the mandatory minimal requirements of education. In many problems relations between the data and the unknowns are such that an unsophisticated normal approach naturally leads to corresponding algebraic equations. Meanwhile an arithmetic solution would require difficult, hard to retain in memory, algebraic by their nature operations over unknown quantities.
This happens, for example, in solution of the the following problem.
If 20 cows were sold, then hay stored for cow’s feed would last for 10 days longer; if, on the contrary, 30 cows were bought than hay would be eaten 10 days earlier. What is the number of cows and for how many days hay will last?
Some basic understanding of relations between the quantities appearing in the problem suffices for its conversion in an algebraic form. But to demand from pupils that they independently came to the formula
\[(200+300) \div 10\]
means pursuing a level of sophistication in operation with unknown quantities that is unnecessary in practice and unachievable in large scale education.
[With thanks to Tony Gardiner]