Sunday, December 4, 2016

Then versus Now: Geometry

The question I’ve been wondering about is when did we lard up secondary mathematics courses to the point where there was too much content to teach in a year? Or has it always been this way?

Quite honestly, I don’t remember my high school math courses: too long ago.

Thus my mission to acquire math textbooks from the time when I sat in the desk watching a teacher explain math.

Recently, I was alerted to a series of textbooks written and co-written by Mary Dolciani. My correspondents said she was the best. I then turned to Amazon to find used books and scored, that is, I was able to find the books. I had been searching for a year.

Here is the comparison for Geometry between a 1969 Dolciani text and the current Pearson text via the table of contents:

Basic elements of Geometry: definitions and terms
Tools of Geometry: definitions, terms, and initial postulates and theorems
Principles of Logic
Reasoning and proof: logic (plus vertical angles
Initial Postulates and Theorems
Parallel and perpendicular lines
Perpendicular lines, basic angle relationships (vertical, supplementary, complementary)
Congruent Triangles
Parallel lines and planes
Points of concurrency and other relationships in triangles
Congruent Triangles
Quadrilaterals; triangle inequalities
Right triangles and trigonometry
Right triangles and trigonometry
Volume and Surface Area
Coordinate Geometry: methods
Coordinate Geometry: proofs


Volume and Surface Area

Of all the secondary math courses, Geometry is the least subject to cramming down the curriculum because you can move radical functions from Algebra 2 to Algebra 1, but there is nowhere to push down a topic like triangle congruency, not at least as long as we continue to follow the century-old practice of teaching the branches of mathematics in isolation.

(Yes, Algebra 2 students complain loudly about taking a year off from algebra to study geometry and then we expect them to recall perfectly everything they learned in Algebra 1.)

(No, the rest of the world does not do this. They create integrated courses that move the students through their years of study in which all branches of mathematics are present.)

Two big takeaways:

                One: I was wrong that trigonometry being introduced in geometry was a recent development. It has been there all along. What is added to what had been was additional study of triangles. The points of concurrency do not appear in the 1969 textbook.

The problem with the pedagogy that is pushed upon teachers and students by non-experts is that the non-experts believe that if something is nice to know, it has to be known and therefore added to the curriculum. Or as someone said 10 years ago, Florida has never found an additional benchmark it didn’t like. We are always adding and never subtracting until there is simply too much to learn in a year.

Dopes. Children are not computers; you cannot upgrade the CPU and memory chips and get more capacity.

                Two: The 1969 textbook takes longer to establish the basics before moving onto more advanced ideas. The current textbook throws ideas out quickly and assumes children will make sense of them with little work.

In practice, that is the deadly sin that dooms district curriculums written to satisfy state demands. Math, and Geometry in particular, is sequential. What you did not understand yesterday will hurt you today. If that goes on long enough, despair and failure are the only ends at which students will arrive.

Next Up: A comparison of Algebra 2 textbooks.