Sharp readers will have observed that Vladimir Putin of Russia closely follows the suggestions in this weblog. After the last weblog discussion of “To invade or not to invade ?” we now see the “Alea iacta est” with Russian tanks crossing the Ukrainean border.
Putin’s dilemma reminded of Shakespeare and the Danish prince Hamlet: “To be or not to be ?” We shouldn’t be surprised that we got a response from Peter Harremoës from Denmark as well.
On the issue of taking a loss, be it the Crimea or now larger parts of the Ukraine, or children losing their fingers in Iraq-Syria or Israel-Gaza, but rather mathematically more general in the form of the subtraction of numbers in arithmetic, and thus the creation of negative numbers, Harremoës has developed a creative new approach that might stop the combatants in amazement. His 2000 article might stop you too, since it still is in Danish, and Google Translate still isn’t perfect. Harremoës mentions that he considers an extension in English at some time, so let us keep our fingers crossed till then – while we still have those.
In the mean time I would like to take advantage of some minor points on subtraction, partly relying on Peter’s article and thanking him for some additional explanation too.
Namely, in the last weblog discussion on confusing math in elementary school I stated that it is important to distinguish the operator minus from the sign min. Peter referred to a – (-b) and commented that problems of subtraction better be transformed into addition, and that subtraction can be seen on an abstract level as much more complex (or mathematically simple) than commonly thought.
One of his proposals is to create a separate symbol for -1 without the explicit showing of the min-sign. He took an example from history in which 1-with-a-dot-on-top already stood for -1. I have wondered about this, and would suggest to take a symbol that is available on the keyboard without much ado, where we e.g. already have i = Sqrt[-1].
A-ha ! Doesn’t the reader hear the penny drop ? Let us take i = quarter turn, H = half turn = i² = -1, then i³ = H i = – i = 3 / 4 turn = three per four turn, and H H = full turn = 1. It would appear that H best be pronounced as ‘eta’, both for international exchange, and in sympathy for German teachers who would otherwise have to pronounce H H as ‘haha’, which would form a challenge for the German sense of humour. I considered suggesting small η or h but the nice thing about H is that it has a shade of -1 in it. In elementary school we can use just the Harremoës-operator H = -1 without the complex numbers. Later in highschool when complex numbers would arise we can usefully refer to H as something that would already be known (or forgotten).
Kids can understand that a debt is an opposite from a credit, or that losing the Ukraine is opposite to winning it. Thus if a is an asset then H a is a liability of the same absolute size. Calculation of gains and losses could be done with a + H b for counting down, or H b + a for counting up. If you lose a debt, then you gain. Losing a debt H b then would be introduced as a + H (H b) = a + b. Actually, I suppose that it would be even better to start with the absolute difference between two numbers, Δ[a, b]. A sum would be to determine that Δ[a, H b] = a + b, presuming that a and b are nonnegative integers.
Thus H would be used in the creation of the negative numbers and the introduction of subtraction, and for later remedial teaching for who didn’t get it or lost it. Peter Harremoës seems to be of the opinion that there would be no need, in principle, to introduce minus and min, but agrees that people would currently want to stick to common notions. Once the basics of H are grasped, it is no use to grind them in, since it is better to switch to minus and min that must be ground in because of that commonality.
First the min sign and the negative integers are introduced by extending the number line: -1 = H 1 , -2 = H 2, … -100 = H 100 and so on. The teacher can show that applying H means making half turns, or moving from the right to the left, or back.
Subsequently the minus operator is introduced as a – b = a + H b.
Hence there arises the exercise a – (-b) = a – H b = a + H (H b) = a + 1 b = a + b.
Or the relation between minus and min: –b = 0 – b = 0 + H b = H b.
A pupil who has mastered arithmetic will do a – (-b) = a + b directly. Otherwise return to remedial teaching and practice with H again.
Arithmetic seems simplest in a positional system. Earlier, we already discussed that English better is regarded as a dialect of mathematics. A number like 15 is better pronounced as ten-five than as fifteen. A sum 15 + 36 then fluently (yes!) translates into “ten-five plus three-ten-six equals (one plus three)-ten-(five plus six), equals four-ten plus ten-one, equals five-ten-one” which is 51. Let me introduce the suggestion that children can use balloons in handwriting or brackets in typing to indicate not only the digits but also the values in the positional system.
In the same manner, the positional system allows us to state –1234 = [-1][-2][-3][-4], where we might rely on H if needed.
For subtraction, the algorithm for a – b is to keep that order if a ≥ b, or otherwise reverse and calculate -(b – a). But, it is useful to show pupils the following method if they forget about reversing the order. For example, 16 – 34 = 16 + [-3][-4] and the rest follows by itself.
One might compare the above with other expositions on subtraction. An obvious source is the wikipedia article on subtraction, while google gives some pages e.g. from the UK or the USA. Some texts seem somewhat overly complex.
Originally I thought that the subtraction a – b for a ≥ b would be harmless, but on close consideration there is a snake in the grass. A point is that corrections are made above the subtraction line, so that the original question is altered. In the Wikipedia example of the Austrian method the final sum doesn’t add up any more. The Wikipedia example of the American method is okay, provided that indeed 7 is replaced by 6, and 5 is replaced by 15. But this is not a proper positional notation anymore. The method also assumes that you use pen and paper, which is infeasible in a keyboard world. Below on the right there are two examples that keep the original sum intact, and that only use the working area below that original sum. One approach is to rewrite 753 =  and the other approach is to do the borrowing a bit later, which is faster. These methods rely on the trick of using balloons or brackets to put values and sub-calculations into a positional place. If we allow for adaptation above the minus line, then the use of H = -1 and T = 10 would work as well, without the need to dash out digits. The second column combines the American & Austrian methods with the Harremoës operator H but now treated as a digit, and using [H][T] = HT0 = 0.
Evaluating these methods, my preference is for the last column. It follows the work flow, in which the negative value is discovered by doing the steps. The method accepts negative numbers instead of creating some fear for them. A practiced pupil would not need the 2[10-4]2 line and directly jump to the answer, so that the number of lines is the same as in the first and second column. The American / Dutch method with HT0 = 0 inserted as a help line creates the suggestion as if borrowing is required before one can do the subtraction, which goes against the earlier training to be able to do such a subtraction that results into a negative value. The borrowing is only required to finalize into a final number in standard notation.
Overall, my conclusion is that the emphasis in teaching should be on the positional system. The understanding of this makes arithmetic much easier. Secondly, the Harremoës operator H indeed is useful to first understand the handling of credit and debt, before introducing the number line and the notation a – b. Thirdly, in a combination of the two earlier points, this operator also appears useful into decomposing –1234 = [-1][-2][-3][-4]. I want to thank Peter again for starting all this (apart from the more advanced ideas in his article). For completeness, let me refer to the 2012 paper A child wants nice and not mean numbers, with a discussion of the pronunciation of the numbers and some more exercise on the positional system.
But these mathematical operations don’t explain that Ukraineans first lose the Ukraine but subsequently gain it once they have turned into Russians.