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January 2022

Perhaps this is part of my having now circled the sun more than 85 times – but in last month’s news article I mentioned that In the first two news articles of 2022, I would discuss what I refer to as the essential Do’s and Don’ts together with my comments and recommendations on how to correct them and have your child enjoy mathematics – using the best mathematics curriculum on the market today – John Saxon’s Math books!  Well – my apologies, but – I had already done that in the November 2020 (the Do’s) and in the December 2020 (the Don’ts) news articles.

So let’s talk this month about another of our favorite subjects – Geometry!


Have you ever seen an automobile mechanic’s tool chest?  Unless things have changed, auto mechanics do not have three or four separate tool chests.  They have one tool chest that contains numerous file drawers separating the tools necessary to accomplish their daily repair work.  But the key is that all of these tools are in a single tool chest.

What if the auto mechanic purchased several tool chests thinking to simplify things by neatly separating the specific types of tools from each other into separate tool chests rather than in separate drawers in the one tool chest?  Each separate tool chest would then contain a series of complete but distinctly different tools.  If mechanics did this, there would now exist the possibility that they would find themselves trying to remember which tool chest contained which tools – and – the extra tool chests would cost them more!   It is somewhat like that in mathematics.  Each division of mathematics has its strengths and weaknesses and like the auto mechanic who selects the best tool for a specific job, so the physicist, engineer, or mathematician selects the best math procedure to meet the needs of what they are doing.

But how do we address the argument that geometry provides a distinct and essential thought process unlike that used in algebra?  The advent of computers has provided educators with an alternative course titled Computer Programming.  A computer programming course teaches students the same methodology or thought process that the two-column proofs of geometry do.  Basically, it teaches the student that he cannot go through a door until he has opened it – meaning – the student must use valid statements that are logically and correctly placed to reach a valid conclusion and to prove that conclusion valid by having the computer program work correctly.

Before computers, educators in the United States felt that providing the separate geometry course would benefit those students interested in literature and the arts, who enjoyed the challenge of geometry without the burden of algebra, while still allowing students entering the fields of science and engineering, who had to take more math, to take the course also.  When I was in high school, most geometry teachers taught only geometry, they never taught an algebra course – as I soon learned!

I recall encountering that little known fact when I took high school geometry from one such teacher.  I was sharply rebuked early in the school year when I kept using the term “equal’” to describe two triangles that had identical measures of sides and angles.  The first time I said the two triangles that contained identical angles and sides were “equal,” she told me I was wrong. She then proceeded to tell me and the class that the only correct term to describe two identical triangles was the term “congruent.”  She did not say my answer was technically correct, but that in the geometry class, we used the term “congruent” rather than “equal”  –  she specifically pointed out that I was “wrong.”

The next day in geometry class, I really got in trouble when I stood and read Webster’s definition of the word congruent.

Con-gru-ent – adj. 1) In agreement or harmony: . .  2) Geometry (of figures): identical in form: Coinciding exactly when superimposed.  . .

Just before I was told to go to the office and tell the principal that I was being rude, I asked her why the two triangles could not also be said to be equal since they had identical angles and sides and were equal in size.  Then I drove the final nail in my coffin when I proceeded to read Webster’s definition of “equal.”

Equal – adj. 1) Being the same in quantity or size . . . 

Half a century later, when I taught both the algebra and geometry concepts simultaneously while using John Saxon’s math books at a rural high school, I made it clear to the students that while they should become familiar with the terminology of the subject, they were free to interchange terms as long as they were correctly applied.  I also made it clear to them that the object of learning geometry and algebra was to challenge and expand their thought processes and for them to understand the strengths and weaknesses of each and apply whichever math tool best served the problem being considered.

While many tout the separate geometry textbook as necessary to enable a child to concentrate on a single subject rather than attempting to process both geometry and algebra simultaneously, I would ask them how a young geometry student can solve for an unknown side in a particular triangle without some basic knowledge of algebraic equations.   In other words, if you are going to use a separate geometry textbook, it cannot be used by a student who has not yet learned how to manipulate algebraic equations.  This means that a separate geometry book is best introduced after the student has successfully completed an algebra 1 course.

For most students this means placing the separate geometry course between the Algebra 1 and Algebra 2 courses, creating a gap of some fifteen months between them. (Two summers off, plus the nine month geometry course).  As was true in my high school days, this situation creates a problem for the vast majority of high school students who enter the Algebra 2 course having forgotten much of what they had learned in the Algebra 1 course fifteen months earlier.

So what am I getting at?  Must we have a separate geometry textbook for students who cannot handle the geometry and algebra concurrently?  Well, let me ask you, if students can successfully study a foreign language while also taking an English course or successfully study a computer programming course while also taking an algebra course, why can’t they study algebra and geometry at the same time, as John Saxon designed it?

Successful completion of John Saxon’s Algebra 2, (2nd or 3rd editions) not only gives the student a full years’ credit for the Algebra 2 course, but it also incorporates the equivalent of the first semester of a regular high school geometry course.  I said “Successful Completion” for several reasons.  FIRST:  The student has to pass the course and SECOND: The student has to complete all 129 lessons. 

Whenever I hear home school educators make the comment that “John Saxon’s Algebra 2 book does not have any two-column proofs,” I immediately know that they stopped before reaching lesson 124 of the book which is where two-column proofs are introduced.  The last six lessons of the Algebra 2 textbook (2nd or 3rd editions) contain thirty-one problems dealing with two-column proofs. The following year, in the first half of the Advanced Mathematics textbook, they not only encounter some heavy duty algebra concepts, but they also complete the equivalent of the second semester of a regular high school geometry course.  The first thirty of these sixty lessons contain more than forty problems dealing with two-column proofs.

So why then did John Saxon not want to publish a separate geometry textbook?  As I mentioned in one of my newsletters several years ago, a group of professors who taught mathematics and science at the University of Chicago bemoaned the fact that educators continued to place a geometry course between basic algebra (Algebra 1) one and the advanced algebra course (Algebra 2) to the detriment of the student. AND THIS WAS MORE THAN 110 YEARS AGO!   In the preface to their book titled “Geometric Exercises for Algebraic Solution,” the professors explained that it is this lengthy “void” that prevents students from retaining the necessary basic algebra concepts learned in basic algebra to be successful when encountering the rigors of Advanced Algebra.

We remain one of the only – if not the only – industrialized nations that have separate math textbooks for each individual math subject.  When foreign exchange students arrive at our high schools, they come with a single mathematics book that contains geometry, algebra, trigonometry, and when appropriate, calculus as well.  Is it any wonder why we are falling towards the bottom of the list in math and science?

When students take a separate geometry course without having gradually been introduced to its unique terminology and concepts, they encounter more difficulty than do students using John Saxon’s math books The beauty of using John’s math books, from Math 76 through Algebra 1 is that students receive a gradual introduction to the geometry terminology and concepts.

If you are going to use John Saxon’s math books through Advanced Mathematics or Calculus you do not need a separate geometry book.  This means you must use the third editions of John’s Algebra 1 and Algebra 2 books, because HMHCO has stripped all of the geometry from the new fourth editions of their versions of Algebra 1 and Algebra 2.  And you do not want a student to go from the fourth edition of Algebra 2 to the Saxon Advanced Mathematics textbook.

NOTE:  Please Click Here to watch a short video on how to receive credits for Geometry, Trigonometry and Pre-calculus using John Saxon’s Advanced Mathematics textbook – and how to record them on the student’s transcript.


           Have a Blessed, Safe, and Happy New Year!


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