When, as a child, I was told by my teacher that I had to be careful with
“named” numbers and not to add apples and people, I remember asking her why in that case we can divide apples by people:
Even worse: when we distribute 10 apples giving 2 apples to a
person, we have
Where do “people” on the right hand side of the equation come
from? Why does “people” appear and not, say, “kids”? There
were no “people” on the left hand side of the operation! How do
numbers on the left hand side know the name of the number on the
right hand side?
I did not get a satisfactory answer from my teacher and only
much later did I realize that the correct naming of the numbers should be
It is a commonplace wisdom that the development of mathematical skills in a student goes alongside the gradual expansion of the realm of
numbers with which he or she works, from natural numbers to integers, then to rational, real, complex numbers:
What is missing from this natural hierarchy is that already at the
level of elementary school arithmetic children are working in a much more sophisticated structure, a graded ring
of Laurent polynomials in variables over , where symbols stand for the names of objects involved in the calculation: apples, persons, etc. This explains why educational psychologists confidently claim that the two operations above have little in common (P. Bryant and S. Squire, The influence of sharing on children’s initial concept of division, J. Experimental Child Psychology 81 no. 1 (January 2002) 1–43.). Indeed, the second operation involves operands of much more complex nature.
Usually, only Laurent monomials are interpreted as having physical (or real life) meaning. But the addition of heterogeneous quantities still makes sense and is done componentwise: if you have a lunch bag with , and another bag, with , together they make
Notice that this gives a very intuitive and straightforward approach to vectors.
(By the way, this “lunch bag” approach to vectors allows a natural
introduction of duality and tensors: the total cost of a purchase of amounts of some goods at prices is a “scalar product”-type expression . We see that the quantities and $p_i$ could be of completely different nature. The standard treatment of scalar (dot) product in undergraduate linear algebra usually conceals the fact that dot product is a manifestation of duality of vector spaces, creating immense difficulties in the subsequent study of tensor algebra.)
Of course, there is no need to teach Laurent polynomials to kids; but it would not harm to teach them to teachers. I have an ally in Francois Viete who in 1591 wrote in his Introduction to the Analytic Art that
If one magnitude is divided by another, [the quotient] is heterogeneous to the former … Much of the fogginess and obscurity of the old analysts is due to their not paying attention to these [rules].
It pays to be attentive to the dimensions of quantities involved in a physical formula: the balance of names of units (dimensions) on the left and right hand sides may suggest the shape of the formula. Such dimensional analysis quickly leads to immensely deep results, like, for example, Kolmogorov’s celebrated “ Law” for the energy spectrum of turbulence.
My book is full of examples which all lead to the same conclusions:
- We should not underestimate the immense richness of basic elementary mathematics.
- Glossing over difficulties presented by hidden structures may seriously imperil students’ progress.
- The teacher has to be aware about the hidden structures and be able to guide pupils around dangerous spots—perhaps without needlessly alerting them every time.