Proof should be taught in geometry — I highly recommend it. But today, unfortunately, there is too much of a trend — pushed by the NCTM and others — to take proof out of geometry and to use geometry as time to review algebra 1. Students miss a great deal of cognitive training that way, training (in how to reason and construct arguments) important for law, philosophy, writing, mathematics, and science. The ancient Greek philosopher Plato is purported to have had a sign above his school, the Academy, saying “Let no one who is not a geometer enter here ” — meaning, he wanted only people in his school who knew geometry or were capable of learning it, since the ancient Greeks knew that geometry taught reasoning, and thus could be used to prepare people for the higher rigors of philosophy.

I’d recommend you find a geometry text that has proofs in it. Students should have to learn a number of proofs, memorize some proofs, and do a plethora of proofs.

It would be a good idea to spread geometry over several years, too. Students — from even the elementary years — need a good introduction to shapes and their interrelationships before the students even get to the proofs, i.e., before they get to systematically organizing and structuring geometry. You can get some ideas from Montessori methods; here are some videos:

Circles (39 sec)

Triangles, polygons and other shapes (57 sec)

Prisms and cylinders (1 min 20 sec)

Triangles and polygons (2 min 28 sec)

Cylinders (1 min 6 sec)

Arithmetic (prep for algebra), mostly (3 min, 40 sec)

Child with cylinder blocks (1 min, 24 sec)

Excerpt from “Nurturing the Love of Learning” produced by the American Montessori Society (10 min, 5 sec)

Montessori School in Phuket (13 min, 21 sec)

In view of the value we get out of geometry, it would be highly beneficial to study formal geometry (that is, study geometry as a science using proofs) over two years or three instead of cramming it in one year and then running from it.

You can read how I generally run my geometry classes and tutoring sessions in my prior post “A Typical Geometry Class at MGTutoring“, where I wrote up some highlights and commentary from a recent tutoring session.

My own memories of geometry class about 40 years ago now was that it was almost exclusively devoted to developing proofs. Unfortunately this process of putting together a proof was never integrated to my other schooling including mathematics in most cases. Had it been I would have learned so much more and even today would be much more able to properly reason out solutions and more importantly properly communicate the reasoning behind the solutions I have arrived at.

Michael, in providing this integration is giving all of his students the ability to use their minds to logically and efficiently solve problems of all kinds, not just those in geometry and mathematics. In the end, isn’t this exactly what an education should do for you?!

Thanks!

And I am glad you grasp what I am doing — thanks for that, too.

Yes, in mathematics, students need to be educated to reason about things in the real world — which is what my students get. Math education is too influenced and corrupted by Platonic methods and ideas and Deweyian methods and ideas. Math is sometimes taught as some kind of isolated deductive system; sometimes it’s taught as a game with no real relation to reality; sometimes it’s taught as a set of pragmatic rules that “just work.” None of these three things are good for a child’s proper cognitive development.

If we want to see what math is, we need to look to its greatest practitioners: Archimedes, Newton, Gauss — rational, practical and focused on the real world, all; they did not do math isolated from facts and the everyday world, nor did they hyper-focus on facts, like an ostrich with its head in the ground, without regard for abstractions and principles.

What’s more, most students are not going to be mathematicians, so they should not be treated as if they were. (And even the future mathematicians need to have a proper conception of mathematics, and what it’s about. Theory divorced from reality is irrational and non-objective.)

Math should be used to train students to reason and be objective. Which is what my students get.