What Is Elementary Mathematics?

On might ask whether the first five years of the curriculum even include such a thing as mathematical instruction. The subject taught in those years is what used justly to be designated not math, but "arithmetic."

H. M. Enzensberger
Drawbridge Up: Mathematics? A Cultural Anathema,
A K Peters, 1999, p. 35


Elementary mathematics is mathematics that is elementary. It is not necessarily the subject taught in elementary school; more critically, calling what is taught in elementary school mathematics is mostly a misnomer, in the first place. But what is mathematics and what makes some of it elementary?

The answer to this question stems from a very distinct feature of mathematics, a feature that holds mathematics apart from other sciences. What is it? I believe that both professional mathematicians and those who had to struggle with mathematics throughout the school would agree that at the heart of mathematics stands proof, i.e., a logical derivation of a fact from other, often simpler (but not necessarily so) facts. The difficulty of the derivation may be judged by two factors: its length and the level of the auxiliary facts that were used in the derivation. The elementary mathematics may then be defined as that part of human activities that employs short proofs based on the simplest facts possible.

How short may be the proofs? To draw an analogy with learning to play chess, there are non-trivial problems that fall under the rubric of "mate in one", or "mate in two" moves. And how simple should be those simplest facts that derivations of elementary mathematics may be based upon? If any are needed at all, they ought to be intuitive in some sense. For example, this is well known that human brain is hard wired for simple arithmetic and number comparison. So claims, like those that follow from the Pigeonhole Principle, should be quite intuitive to the majority of the human race. (The Pigeonhole Principle asserts that if the number of pigeons exceeds the number of nests, then there is a nest with at least two pigeons.) The necessary intuition may also come from experience. For example, hardly any one would object to the fact that the combined length of two sticks aligned end to end does not depend on the order in which the sticks are placed one after the other.

I'll give an examples: Why 1/3 + 1/2 = 5/6?

The fraction 5/6 arises with the need to divide 5 items into 6 parts, say, 5 apples between 6 boys. Upon the completion of the task when all boys have received an equal share of 5 apples, each got 5/6 of an apple. But how can one accomplish the task? One way is to divide each apple into 6 parts, getting 30 equal pieces and hand each boy 5 (= 30/6) of these. However, there is another way of achieving the same goal. Divide 2 apples into 3 pieces each. There will be 6 pieces each equal to 1/3 of an apple. Pass them to the boys. Divide the remaining 3 apples into 2 equal pieces, getting 6 pieces each equal to 1/2 of an apple. Pass these to the boys. At this point, each of the boys has 1/3 + 1/2 of an apple and, as we have seen, each got an equal portion of the five apples, i.e., 5/6. The conclusion is inescapable: 1/3 + 1/2 = 5/6.

As another example, can you draw two identical curves one on, say, a sphere and the other on a cube? Stop for a second and give it a thought. Do not read what follows right away. Well, imagine a ghost of a cube and a ghost of a sphere. As is common among the ghosts, the two may pass with ease through each other. At any time they do, their exteriors intersect in a curve that can be said to be drawn on both of them. Obviously, you can substitute other shapes, even much less regular, for either sphere or cube or both.

When one recollects the early experiences with mathematics, the first thing that springs to mind is numbers. But numbers on their own form only a tiny part of mathematics. The numerical tricks we learn in arithmetic are only the tip of an iceberg [Stewart, p. 31]. Many examples of elementary mathematics - both numeric and not - are strewn all over the site. Here are some:

  1. Breaking chocolate bars
  2. Heads and Tails
  3. A Candy Game
  4. Counting Chips On a Circle
  5. Counting Triangles
  6. Counting Triangles II
  7. Pigeon Checkers
  8. Shuttle Puzzle
  9. Halving a square
  10. Shortest Fence in a Quarter-Circle Pasture
  11. Farmer and Wife to Catch Rooster and Hen
  12. One Dimensional Ants
  13. A Formal Framework

References

  1. Ian Stewart, Nature's Numbers, BasicBooks, 1995

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