Why should a new idea in mathematics start with a problem the child cannot yet finish?
An idea makes sense as the answer to a question. When a problem defeats the child’s present tools, the child feels the need, and the idea arrives to meet it.
Begin with a problem the child can almost do. Will the new sofa fit through the door?
Becausethe child already has a method, putting two things side by side, and trusts it.
Let that method fail. The door cannot be carried to the sofa.
Becausethe Soviet psychologist Vasily Davydov, working with Daniil Elkonin, defined a learning task as a situation where the old way of acting falls short. The child must then look for a general way to solve a whole class of problems.
Let the child find a go-between. A string cut to the door’s width can travel to the sofa.
Becausea third thing equal to one length can be laid against the other. In an American version of Davydov’s first-grade lessons, the lengths are drawn on the wall, where they cannot be moved. So the children invent the go-between.
Let the string fail too. Now the sofa is in a shop across town, and the question comes by phone.
Becausea string cannot be read aloud, but a count can. The door is nine hand-spans wide, for example. Davydov’s curriculum introduces number through measurement, as how many times a unit fits into a quantity.
Only now name the ideas of unit, measure and number.
Becauseeach name then lands on something the child has done and needed. Whose hand to measure with is the next learning task.
Another way
Start from the ruler. Teach inches, then measure the door.
Becauseit is faster and works on the problems in the book. Its weakness shows later. Researchers in Davydov’s school observe that solving one practical problem correctly does not protect a child from mistakes on the next of the same kind.
Check
Before teaching anything new, finish the sentence “Without this idea, the child cannot yet …”. If it has no ending, there is no learning task yet. Later, ask the child why the idea exists. A child who met the problem first answers with it, “So we can tell how wide the door is without carrying it.”
KeepThe need first. Then the idea.