Students are their own best teachers, including those who need extra help. This is true even though they will not take a very efficient approach to learning on their own and they will probably miss some essentials. This does say that teachers and parents have an important function to perform. But that function is far too often seen as sharing an insight or explaining something.

We can explain to our students that if an altitude is drawn to the hypotenuse of a right triangle, then either leg of the given right triangle is the mean proportional between the hypotenuse of the given right triangle and the segment of the hypotenuse adjacent to that leg, or of course the projection of that leg on the hypotenuse.  The amazing thing is not that we get some blank stares for our efforts, it is that we get as many students as we do looking like they got something. We then hope the homework will solidify it for them and make it clear to a reasonable number of the others who gave us the blank stare.

And we can explain to them the FOIL approach to working with quadratic equations and some of them will solve many of the problems correctly. But most of them will not know what they’re doing or why it works. They are not fishing; they are eating the fish we gave them.

In between the extremes of unaided student discovery and teacher explaining and modeling is guided student discovery. See the post on “Guided Student Discovery” for a little more on this topic.