Confusing math in elementary school

The problems in Russia-Ukraine, Irak-Syria and Israel-Gaza are so large since the combatants are hardly aware of the concept of fair division and sharing. Something must have gone wrong in elementary school with division and fractions. Let us see whether we can improve education, not only for future dictators but for kids in general.

English as a dialect

In 2012 I suggested that English can best be seen as a dialect of mathematics. The case back then was the pronunciation of the integers, e.g. 14 as “fourteen” (English) instead of “ten-four” (math & Chinese). The decimal positional system isn’t merely a system of recording but it contains switches in the unit of account. In this system the step from 9 to 10 means that ten becomes a new unit of account, and the step from 99 to 100 means that hundred (ten-ten) becomes a new unit of account. This relies on the ability to grasp a whole and the notion of cardinality. Having a new unit of account means that it is valid to introduce the new words “ten” and “hundred”, so that 1456 as a number differs from a pin-code with merely mentioning of the digits. When the numbers are pronounced properly then pupils will show greater awareness of these elements and become better in arithmetic – and arithmetic is crucial for division and fractions.

When education is seen as trying to plug mathematics into the mold of English as a natural language, then this is an invitation to trouble. It is better to free mathematics from this mold and teach it in its own structural language. It is a task for the teaching of English to show that it is a somewhat curious dialect.

Rank numbers

After the recent discussion of ordinal or cardinal 0, it can be mentioned that the ordinals are curiously abused in the naming of fractions. Check the pronunciation of 1/2, 1/3, 1/4, 1/5, … With number 4 = four and the rank 4th = fourth, the fraction 3/4 is pronounced as “three-fourths”. What is rank “fourth” doing in the pronunciation of 3/4 ? School kids are excused to grow confused.

Supposedly, when cutting up a cake in four parts, one can rank the pieces into the first, second, third and fourth piece. Assuming equal pieces, or fair division, then one might borrow the name of the last rank number “fourth” to say that all pieces are “a fourth”. This is inverse cardinality. Presumably, this is how natural language developed in tandem with budding mathematics. Such borrowing of terms is conceivable but not so smart to do. It is confusing.

The creation of “a fourth”, as a separate concept in the mind, also takes up attention and energy, but it doesn’t produce anything particularly useful. Malcolm Gladwell alerted us to that the Chinese language pronounces 3/4 as “out of four parts, take three”. Shorter would be “3 out of 4”. This directly mentions the parts, and there is no distracting step in-between.

For a reason discussed below we better avoid the “of” in “out of”. Thus it might be even shorter to use “3 from 4”, but a critical reader alerted me to that his might be seen as subtraction. Thus “3 out 4” seems shortest. However, there is also the issue of ratio versus rate. In a ratio the numerator and the denominator have the same dimension (say apples) while in a rate they are different (say meter per second). Thus the overall best shortest pronunciation would be “3 per 4″, which is neutral on dimensions, and actually can be used in most European languages that are used to “percent”.

This pronunciation thus facilitates direct calculation, like “one per four plus three per four gives four per four, which gives one”.

Dividing and sharing

The Dutch word for “divide” also means “share” (Google Translate). Sharing a cake tends to generate a new unit of account, namely the part. In fair division each participant gets a part of the same size, which becomes: the same part. This process focuses on the denominator and generates a larger number and not a smaller number. It actually relies on multiplication: the denominator times the new unit of account (the part) gives the original cake again. The process of sharing is rather opposite to the notion of division that gives a fraction, that maintains the old unit of account and generates a smaller number on the number line.

A fraction 3 / 4 or “three per four”, when three cakes must be shared by four future dictators, requires the pupil to establish the proportional ratio with “three cakes per four cakes” (virtually giving each a cake even though there are no four cakes but only three), and then rescale from the four hypothetical cakes down to one cake. PM. The pupil must have a good control of active versus passive voice. The relation is that “4 kids share 3 cakes” (active) and “3 cakes are being shared by 4 kids” (passive). Thus “3 per 4” or “3 out of 4” is shorthand for “3 units taken out of 4 units” (or “4 (kids) take out of 3 (cakes)”) but not for “3 (kids) take out of 4 (cakes)” (which would give 1 + 1/3 per kid, and would require a discussion of mixed numbers).

Hence it is unfortunate that the Dutch language uses the same word for both sharing and dividing. Fraction 3/4 reads in Dutch as “3 shared by 4 gives three-fourths” (“3 gedeeld door 4 geeft drie-vierde”), which thus combines the two major stumbling blocks: (a) the sharing/dividing switch in the unit of account, (b) the curious use of rank words. When 3/4 = “three per four” would be used, then the stumbling blocks disappear, and teaching could focus on the difference between the process of dividing and the result of the fractional number on the number line.

David Tall (2013) points to a related issue in the language on sharing and dividing: “The notion of a fraction is often introduced as an object, say ‘half an apple’. This works well with addition. (…) What does ‘half an apple multiplied by half an apple’ mean? (…) However, if a fraction is seen flexibly as a process, then we can speak of the process ‘half [halve] an apple’ and then take ‘a third of half an apple’ (…) the idea is often simply introduced as a rule, ‘of means multiply’, which can be totally opaque to a learner meeting the idea for the first time.” (p97) Note that Tall’s book is rather confused so that you better wait for a revised edition. He indeed does not mention above issues (a) and (b). But this latter observation on the process and result of division is correct.

The rank words thus are abused not only as nouns but also as verbs (“take a third of half of an apple”). We better translate into “(one per three) of (one per two)”, which gives “(one times one) per (three times two)”. The mathematical procedure quickly generates the result. The didactic challenge becomes to help kids understand what is involved rather than to master confused language.

Multiplication

Speaking about Tall and multiplication: Apparently the English pronunciation of the tables of multiplication can be wrong too. E.g. ‘two fours are eight’ refers to two groups of four, and thus implies an order, while merely ‘two times four is eight’ gives the symmetric relation in arithmetic. Said book p94 compares a table with 3 rows and 4 columns, and Tall argues: “the idea of three cats with four legs is clearly different from that of four cats with three legs. The consequence is that some educators make a distinction between 4 x 3 and 3 x 4. (…) I question whether it is a good policy to teach the difference. (…) [ reference to Piaget ] (…) So a child who has the concept of number should be able to see that 3 x 4 is the same as 4 x 3.”

An exercise in marbles

An exercise in marbles

Tall doesn’t explain this: Pierre van Hiele focuses on the distinction “concrete versus abstract”, would focus on the table, so that children would master the insight that the order does not matter for arithmetic. Once they have mastered arithmetic, they might consider “reality versus model” cases like on the cats and their legs without becoming confused by arithmetical issues hidden in those cases. Instead, Hans Freudenthal with his “realistic mathematics education” (RME) would present kids with the “reality versus model” cases (e.g. also five cups with saucers and five cups without saucers, a 3D table), and argue that this would inspire kids to re-invent arithmetic, though with some guidance (“guided re-invention”). Earlier, I wondered why Freudenthal blocked empirical research in what method works best (and my bet is on Van Hiele).

Conclusion

Overall, the scope for improvement is huge. It is advisable that the Parliaments of the world investigate failing math education and its research. When kids have improved skills in arithmetic and language, they would have more time and interest to participate in and understand issues of fair division. Hurray for World Peace !

PM 1. Conquest of the Plane pages 77-79 & 207-210 discuss proportions and fractions.

PM 2. See also COTP for the distinction between dynamic division y // x and standard static division y / x.

PM 3. Some say “3 over 4” for 3/4, hinting at the notation with a horizontal bar.¬† I wonder about that. The “3 per 4” is actually shorter for “3 taken from 4”, and this puts emphasis on what is happening rather than on the shape of the notation. An alternative is “3 out of 4” but my inclination was to avoid the “of” as this is already used for multiplication. Also, my original training has been to reserve “n over k” for the binomial coefficient (that can be taught in elementary school too). However, a reader alerted me to Knuth’s suggestion to use “n choose k” for the binomial coefficient, and that is better indeed. In that case I would tend to avoid the “over”. It was also commented that “3 from 4” sounded like subtraction: but my proposal is to adhere to “3 minus 4” for “3 – 4” as opposed to “3 plus 4” for the addition. It is just a matter to introduce plus and minus into general usage, so that it is always clear what they are. Note that we are speaking about mathematics as a language and not about English as a natural language. Also -1 would be “negative-1” or “min-1”, with the sign “min” differing from the operator “minus”.

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