## Abstract and concrete simultaneously

In most education literature I have seen going from concrete to abstract concepts as a ladder.

The metaphor has always started with concrete objects before “ascending the ladder” to abstract ideas.

I was researching something for a different (as yet unrevealed) project when I came across the following.

[Source.]

That is, someone used the same metaphor but reversed the ladder.

This is from a paper on the Davydov curriculum, used in parts of Russia for the first three grades of school. It has the exotic position of teaching measuring before counting. Students compare objects with quantity but not number — strips of paper, balloons, weights:

Children then learn to use an uppercase letter to represent a quantitative property of an object, and to represent equality and inequality relationships with the signs =, ≠, >, and B, or A<B. There is no reference to numbers during this work: “A” represents the unmeasured length of the board.

[Source.]

A later exercise takes a board A and a board B which combine in length to make a board C, then have the students make statements like “A + B = C” and “C – B = A”.

Number is eventually developed as a matter of comparing quantities. A small strip D might need to be used six times to make the length of a large strip E, giving the equation 6D = E and the idea that number results from repetition of a unit. This later presents a natural segue into the number line and fractional quantities.

The entire source is worth a read, because by the end of the third year students are doing complicated algebra problems. (The results are startling enough it has been called a “scam”.)

I found curious the assertion that somehow students were starting with abstract objects working their way to concrete ones. (The typical ladder metaphor is so ingrained in my head I originally typed “building their way down” in the previous sentence.) The students are, after all, handling boards; they may be simply comparing them and not attaching numbers. They give them letters like A and B, sure, but in a way that’s no less abstract than naming other things in the world.

After enough study I realized the curriculum was doing something clever without the creators being aware of it: they were presenting situations that (for the mind) were concrete and abstract at the same time.

For a mathematician’s perspective, this is impossible to do, but the world of mental models works differently. By handling a multitude of boards without numbers and sorting them as larger and smaller, an exact parallel is set up with the comparison of variables that are unknown numbers. Indeterminate lengths work functionally identical to indeterminate number.

This sort of thing doesn’t seem universally possible; it’s in this unique instance the abstract piggybacks off the concrete so nicely. Still it may be possible to hack it in: for my Q*Bert Teaches the Binomial Theorem video I used a split-screen trick of presenting concrete and abstract simultaneously.

Although the sequence in the video gave the concrete example first, one could easily imagine the concrete being conjoined with an abstract example cold, without prior notice.

(For a more thorough treatment of the Davydov curriculum itself, try this article by Keith Devlin.)