With your undivided attention

Both President Obama of the USA and President Putin of the Russian Federation have somewhat illogical positions. Obama repeats the ritual article 5 “An attack on one is an attack on all” but the Ukraine is not a fraction of NATO. So what is the USA going to do about the Ukraine ? Putin holds that Russia defends all Russians everywhere but claims that Russia is not involved with the combatants in the Ukraine. His proposed 7 point plan contains a buffer zone so that he creates fractions in a country on the other side of the border. Overall, we see the fractional division of the Ukraine starting, as already predicted in an earlier entry in this weblog.

What is it with fractions, that Presidents find so hard, and what they apparently didn’t master in elementary school, like so many other pupils ? There are two positions on this. The first position is that mathematics teachers are right and that kids must learn fractions, with candy or torture, whatever works best. The second position is that kids are right and that fractions may as well be abolished as both useless and an infringement of the Universal Declaration of Human Rights (article 1). Let us see who is right.

An abolition of fractions

Could we get rid of fractions ? We can replace 1 / a or one-per-a by using the exponent of -1, giving a‾¹ that can be pronounced as per-a.  In the earlier weblog entry on subtraction we found the Harremoës operator H = -1. The clearest notation is aH = 1 / a. Before we introduce the negative numbers we might consider to introduce the new notation for fractions. The trick is that we do not say that. We just introduce kids to the operator with the following algebraic properties:

0H =  undefined

a aH =  aH a =  1

( aH ) H = a

Getting rid of fractions in this manner is not my idea, but it was considered by Pierre van Hiele (1909-2010), a teacher of mathematics and a great analyst on didactics, in his book Begrip en Inzicht (1973:196-204), thus more than 40 years ago. His discussion may perhaps also be found in English in Structure and Insight (1986). Note that aH = 1 / a already had been considered before certainly in axiomatics, but the Van Hiele step was to consider it for didactics at elementary school.

From the above we can deduce some other properties.

Theorem 1:

(a b) H = aH bH

Proof. Take x = a b. From xH x =  1 we get  (a b)H (a b) =  1. Multiply both sides with aH bH, giving (a b)H (a b) aH bHaH bH, giving the desired. Q.E.D.

Theorem 2:

HH = H

Proof: From addition and subtraction we already know that H H = 1. Take a a H = 1, substitute a = H, get H HH = 1, multiply both sides with H, get H H HH = H, and thus HH = H. Q.E.D.

It remains to be tested empirically whether kids can follow such proofs. But they ought to be able to do the following.

Simplification

The expression 10 * 5H or ten per five can be simplified into 10 * 5H = 2 * 5 * 5H  = 2 or two each.

Equivalent fractions

Observing that 6 / 12 is actually 1 / 2 becomes 6 * 12H = 6 * (2 * 6)H = 6 * 2H * 6H = 2H. Alternatively all integers are factorised into the primes first. Note that equivalent fractions are part of the methods of simplification.

Multiplication

a bH * c d H  = (a c) (b d ) H

Comparing fractions

Determining whether a bH  > c dH or conversely: this reduces by multiplication by b d, giving the equivalent question whether a d > c b or conversely.

Rebasing

That (a / b = c)  ⇔  (a / c = b) may be shown in this manner:

a bH  = c

a bH  (b cH) = c  (b cH)

a cH  = b  

Addition

Van Hiele’s main worry was that we can calculate 2 / 7 + 3 / 5 = 31 / 35 but without much clarity what we have achieved. Okay, the sum remains smaller than 1, but what else ? Translating to percentages 2 / 7 ≈ 28.5714% and 3 / 5 = 60%, so the sum ≈ 88.5714%, is more informative, certainly for pupils at elementary school. This however requires a new convention that says that 0.6 is an exact number and not an approximate decimal, see Conquest of the Plane (2011). The argument would be that first calculating 31 / 35 and then transferring to decimals would give greater accuracy for the end result. On the other hand it is also informative to see the decimal constituants, e.g. observe where the greatest contribution comes from.

Another argument is that 2 / 7 + 3 / 5 = 31 / 35 would provide practice for algebra. But why practice a particular format if it is unhandy ? The weighted sum can also be written in terms of multiplication. Compare these formats, and check what is less cluttered:

a / b + c / d = (a / b + c / d) (b d) / (b d) = (a d + c b) / (b d)

a bH + c d H  = (a bH + c d H) (b d) (b d) H = (a d + c b) (b d) H

Subtraction

In this case kids would have to see that H can occur at two levels, like any other symbol.

a bH + H c d H  = (a bH + H c d H) (b d) (b d) H = (a d + H c b) (b d) H

Mixed numbers

A number like two-and-a-half should not be written as two-times-a-half or 2 1/2. Elegance with Substance (2009) already considers to leave it at 2 + 1/2. Now we get 2 + 2H .

Division

Part of division we already saw in simplification. The major stumbling block is division by another fraction. Compare:

a / b / {c / d} = (a / b)(d / c) / { (c / d) (d / c) } = (a / b)(d / c)  / { 1 } = (a d) / (b c)

a bH * (c d H)H =   = a bH  * c H d = (a d) (b c ) H

Supposedly, kids get to understand this by e.g. dividing 1/2 by 1/10 so that they can observe that there are 5 pieces of 10H = 1/10 that go into 2H = 1/2. Once the inversion has been established as a rule, it becomes a mere algorithm that can also be applied to arbitrary numbers like 34H (127H)H = 1/34 / (1/127). The statement “divide per-two by per-10” becomes more general: divide by per-a is multiply by a.

Dynamic division

A crucial contribution of Elegance with Substance (2009:27) and Conquest of the Plane (2011:57) is the notion of dynamic division, that allows an algebraic redefinition of calculus.

With y xH = y / x as normal static division then dynamic division y xD = y // x becomes::

y xD ≡ { y xH, unless x is a variable and then: assume x ≠ 0, simplify the expression y xH, declare the result valid also for the domain extension x = 0 }.

A trick might be to redefine y / x as dynamic division. It would be somewhat inconsistent however to train on xH and then switch back to the y / x format that has not been trained upon. On the other hand, some training on the division slash and bar is useful since it are formats that occur.

Van Hiele 1973

Van Hiele in 1973 includes a discussion of an axiomatic development of addition and subtraction and an axiomatic development of multiplication and division. This means that kids would be introduced to group theory. This axiomatic development for arithmetic is much easier to do than for geometry. Since mathematics is targeted at “definition, theorem, proof” it makes sense to have kids grow aware of the logical structure. He suggested this for junior highschool rather than elementary school, however. It is indeed likely that many kids at that age are already open to such an insight in the structure of arithmetic. This does not mean a training in axiomatics but merely a discussion to kindle the awareness, which would already be a great step forwards.

His 1973 conclusions are:

Advantages
  1. In the abolition of fractions 1/a a part of mathematics is abolished that contains a technique that stands on its own.
  2. One will express theorems more often in the form of multiplication rather than in the form of division, which will increase exactness. (See the problem of division by zero.)
  3. Group theory becomes a more central notion.
  4. In determining derivatives and integrals, it no longer becomes necessary to transform fractions by means of powers with negative exponents. (They are already there.)
Disadvantages
  1. Teachers will have to break with a tradition.
  2. It will take a while before people in practice write 3 4H instead of 3 / 4.
  3. Proponents will have to face up to people who don’t like change.
  4. We haven’t studied yet the consequences for the whole of mathematics (education).

His closing statement: “We do not need to adopt the new notation overnight. It seems to me very useful however to consider the abolition of the algorithms involving fractions.”

Conclusion

Given the widespread use of 1 / a, we cannot avoid explaining that aH = 1 / a. The fraction bar is obviously a good tool for simplication too, check 6 * (2 * 6)H .

Similarly, issues of continuity and limits x →1 for expressions like (1 – )  (1 + x) H would benefit from a bar format too. This would also hold alternatively for (1 – ) (1 + x) D.

But, awareness of this, and the ability to transform, is something else than training in the same format. If training is done in algorithms in terms of aH then this becomes the engine, and the fraction slash and bar merely become input and output formats that are of no significance for the actual algebraic competence.

Hence it indeed seems that fractions as we know them can be abolished without the loss of mathematical insight and competence.

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