WHAT ARE SOME OF THE MYTHS SURROUNDING JOHN SAXON'S MATH BOOKS?
Whether I was attending a homeschool convention or browsing the online homeschool blogs, I kept hearing and seeing comments from homeschool parents that express the idea that: "You must use a separate geometry book to receive credit for geometry."
Myths that will be discussed in future news articles:
WHAT ARE SOME OF THE MYTHS SURROUNDING JOHN SAXON'S MATH BOOKS?
When you hear someone say that if you use John Saxon's Algebra 2 textbook, you will need a separate geometry book because "There are no two-column proofs in John Saxon's Algebra 2 textbook," they are telling you that either (1) they have never used that textbook or (2) if they did use it, they never finished the book – they stopped before reaching lesson 124, or (3) they used the new fourth edition which has no geometry content.
Whether they are using the second or third edition of John's Algebra 2 book, students will encounter more than forty informal and formal two-column proof problems in the last six lessons of the textbook. The first ten or so geometry proof problems students encounter in lesson 124 of the textbook are the more informal method of outlining a proof. John felt this introduction to the informal outline would get the students better prepared for the more formal two-column proofs that they will encounter later. Then, from lesson 125 through lesson 129, students will be asked to solve more than thirty formal two-column proofs that are as challenging as any the students will encounter using any separate geometry textbook.
If they proceed onto the Saxon Advanced Mathematics course the following school year, they will encounter two dozen informal proofs in the first ten or so lessons followed by more than forty-six formal two-column proofs in the next thirty or so lessons. They will encounter at least one formal two column proof problem in every lesson through lesson forty and then encounter them less frequently through the next twenty or so lessons of the book.
When I was teaching high school math in a rural public high school, I taught both Saxon Algebra 2 as well as John's Advanced Mathematics course. The students who took my Advanced Mathematics class came from my Algebra 2 class as well as another teacher's Algebra 2 class. I recall the students in my Advanced Mathematics class who had taken Saxon Algebra 2 from me would comment that the two-column proofs in the Advanced Mathematics book were easier than those they had encountered last year in our Algebra 2 book. "Perhaps you have learned how to do two-column proofs" was my reply.
However, the students who came from the other teacher's Algebra 2 class moaned and groaned about how tough these two-column proofs were in the Advanced Mathematics book. After discussing the situation with the other teacher, I found that she knew I would cover two-column proofs in the early part of the Advanced Mathematics textbook so she stopped at lesson 122 in the Algebra 2 course – never covering the introduction to two-column proofs.
The geometry concepts encountered in John Saxon's Algebra 2 textbook – whether the second or third edition – is the equivalent of the first semester material of a regular high school geometry course and that includes a rigorous amount of formal two-column proofs!
If you are using the new fourth edition of Algebra 2, you must also purchase a separate geometry textbook to acquire geometry credit as the new fourth editions of the revised HMHCO Algebra 1 and Algebra 2 textbooks have had the geometry content removed from them.
Myths that will be discussed in future news articles:
WHAT ARE SOME OF THE MYTHS SURROUNDING JOHN SAXON'S MATH BOOKS?
More than tthirty years ago, at a National Council of Teachers of Mathematics (NCTM) Convention, John and I encountered a couple of teachers manning their registration booth. When John introduced himself, they made a point to tell him that they did not use his math books because they felt the books were just "mindless repetition."
John laughed and then in a serious note told the two teachers that in his opinion it was the NCTM that had denigrated the idea of thoughtful, considered repetition. He quickly corrected them by reminding them that the correct use of daily practice results in what Dr. Benjamin Bloom of the University of Chicago had termed "Automaticity." Dr. Bloom was an American educational psychologist who had made significant contributions to the classification of educational objectives and to the theory of mastery-learning.
Years earlier, John Saxon had taken his Algebra 1 manuscript to Dr. Bloom to evaluate his manuscript's methodology. John wanted to find out if there was a term that described the way his math book was constructed. Dr. Bloom informed John that he had not created a new teaching method. He himself had named this same methodology in the early 1930's.
Dr. Bloom referred to this method of mastery — the same one contained in John's manuscript - as "Automaticity. He described it as the ability of the human mind to accomplish two things simultaneously so long as one of them had been overlearned (or mastered). He went on to explain to John that the two critical elements of this phenomenon were repetition and time. John had never heard this term used before, but while in military service, he had encountered military training techniques that used this concept of repetition over extended periods of time, and he had found them extremely successful.
If you think about it, professional sports players practice the basics of their sport until they can perform them flawlessly in a game without thinking about them. By "Automating" the basics, they allow their minds to concentrate on what is occurring as the game progresses. Basketball players do not concentrate on dribbling the basketball, they concentrate on how their opponents and fellow players are moving as each play develops and they move down the floor to the basket while automatically dribbling the basketball.
Baseball players perfect their batting stance and grip of the bat by practicing hitting a baseball for hours every day so that they do not waste time concentrating on their stance or their grip at the plate each time they come up to bat. Their full concentration is on the pitcher and the split second timing of each pitch coming at them at eighty or ninety miles an hour.
How then does applying the concept of "Automaticity" in a math book differentiate that math book from being just "mindless repetition?" John Saxon's math books apply daily practice over an extended period of time. They enable a student to master the basic skills of mathematics necessary for success in more advanced math and science courses. As I mentioned earlier, the two necessary and critical elements of "Automaticity" are repetition over time. If one attempts to take a short cut and eliminate either one of these components, mastery will not occur. You cannot review for a test the day before the test and call that process "Automaticity." Nor can you say that textbook provides mastery through review.
Just as you cannot eat all of your weekly meals on a Saturday or Sunday — to save time preparing meals and washing dishes daily — you cannot do twenty factoring problems one day and not do any of them again until the test without having to create a review of these concepts just before the test. When a math textbook uses this methodology, it does not promote mastery; it promotes memory of the concepts specifically for the test. That procedure would best be described as "Teaching the Test."
John Saxon's method of doing two problems of a newly introduced concept each day for fifteen to twenty days, then dropping that concept from the homework for a week or so, then returning to see it again, strengthens the process of mastery of the concept in the long term memory of the student. Saxon math books are using this process of thoughtful, considered repetition over time to create mastery!
Myths that will be discussed in future news articles:
WHAT ARE SOME OF THE MYTHS SURROUNDING JOHN SAXON'S MATH BOOKS?
This common myth is generated by public and private schools as well as homeschool educators who place a transfer student into the wrong level of Saxon math – usually a level above the student's ability. I recall a homeschool parent at one of the Homeschool Conventions a few years ago telling me she was going to switch to Saxon Math. She wanted to buy one of my Algebra 2 DVD tutorial series. I asked her what level book her son had just completed and did she know what company had printed the book. She thought for a moment – then she said it was an Algebra 1 textbook from (you fill in the name) company.
Since she lived in the area and was coming back to the convention the next day, I asked her if she would have her son take the Saxon Algebra 1 Placement test that night and come back the next day with the results so we could make sure he was being placed into the correct level Saxon math textbook. The next day, she came by the booth and informed me that her son had failed the Saxon Algebra 1 Placement Test. When I told her that test was the final exam in the Saxon pre-algebra course, she became quite concerned. I told her that the problem was not a reflection upon her son's intelligence.
The problem her son had encountered was that the previous textbook he had used taught the test. However, the cumulative nature of Saxon Math books requires mastery of the concepts, which is why there is a weekly test. Had her son used the Saxon Algebra 2, 3rd Ed book - by the time he reached lesson twenty - he would have become painfully aware of what he and his mother would believe to be the "Difficulty" of the book. They would have blamed the Saxon book as being "Too Difficult." They would never have realized that his difficulty in the Saxon Algebra 2 book was that the previous math book allowed him to receive good test grades through review for each test the night before, rather than requiring mastery of the concepts as Saxon books do through the weekly tests.
This parent is not alone. Every week I receive emails or telephone calls from homeschool educators who are trying to accomplish the same thing. And until they have their student take the Saxon Math Placement Test, homeschool educators do not realize that they could very well be placing the student in a Saxon math book at a level above the students' capabilities.
The Saxon Math Placement Tests were not designed to test the students' knowledge of mathematics; they were designed to seek out what necessary math concepts had been mastered by the student to ensure success in the next level Saxon math book. Low test results on a specific Placement Test tell us that the student has not mastered a sufficient number of necessary math concepts to be successful in that level Saxon Math book.
Saxon Placement Tests should not be used at the end of a Saxon math book to evaluate the student's progress. Classroom teachers as well as homeschool educators should use the student's last five test scores of the course to determine their ability to be successful in the next level course. If the last five test scores are clearly eighty or better, the student will be successful in the next level Saxon math course – or anyone else's math textbook should you elect to change curriculum.
MAKE SURE YOU BUY AND USE THE CORRECT EDITIONS OF JOHN SAXON'S MATH BOOKS
NOTE: Three years ago, I wrote this article and published it in my 2019 news articles. When January 2022 came around, we dropped the 2019 articles and since then I have had numerous homeschool educators asking questions dealing with which edition should they use – or – some just asking what happened to the 2019 article about which editions of John's books should be used. Some asked me to email them a copy of the article. Before reading this article, make sure you first read the March 2021 news article about which of John's books are classed as Honors Courses. I believe the best plan then would be to reprint that 2019 news article – so – here it is!
Math 54 (2nd or 3rd Ed): You can use either the hard cover 2nd edition textbook or the newer soft cover 3rd edition as they have identical math content. In fact, they are almost word for word and problem for problem the same textbooks. The page numbers differ because of different graphics and changed page margins, and the newer soft cover 3rd edition homeschool packet has an added solutions manual. However, my experience with that level of mathematics is that most home school educators will not need a solutions manual until they encounter Math 76. If you can acquire a less expensive homeschool kit without the solutions manual, I would recommend acquiring that less expensive set. Calculators should not be used at this level.
Math 65 (2nd or 3rd Ed): This book is used following successful completion of the Math 54 textbook. Successful completion is defined as completing the entire Math 54 textbook, doing every problem and every lesson on a daily basis, and taking all of the required tests. To be successful in this textbook, students must have scored eighty or better on the last four or five tests in the Math 54 textbook. As with the Math 54 textbooks, the 2nd edition hard cover book and the newer soft cover 3rd edition have identical math content. The newer 3rd edition series also has a solutions manual, but if you're on a tight budget, I do not believe that it is necessary at this level of mathematics either. Calculators should not be used at this level.
FIRST: The fourth edition has had all references to geometry removed from it also requiring the purchase of an additional geometry book.
Advanced Mathematics (2nd Ed): Do not use the older first edition, use the 2nd Edition. Make sure you also purchase the separate solutions manual as the lesson concept reference numbers are found in the solutions manual – not in the textbook! Students who attempt this book must have successfully completed all of Saxon Algebra 2 using either the 2nd or 3rd edition textbooks. Upon successful completion of just the first sixty lessons of this textbook, the student will have completed the equivalent of the second semester of a regular high school geometry course. An inexpensive scientific calculator is all that is needed for this course. For more information on how to transcript the course to receive credit for a full year of geometry as well as a semester of trigonometry and a second semester of pre-calculus, please Click Here.
HOW MUCH TIME SHOULD STUDENTS SPEND ON MATH EACH DAY?
One of several arguments advanced by home school educators regarding the efficiency of the Saxon math curriculum is that from Math 54 through Advanced Mathematics the courses require too much time to complete the daily assignment of thirty problems. Their solution to this often takes one of two approaches. Either they allow the student to take shortcuts to reduce the time spent on daily assignments, or they find another math curriculum that takes less time – you know – you’ve heard them say, “We found another math curriculum that is more fun, easier, and it does not require so much time.”
The Student is in The Wrong Level Math Course: If after lesson thirty in any Saxon math book, students continue to receive 80% on the weekly tests, within a maximum of fifty minutes - with no partial credit (all right or all wrong) and no calculator (until Algebra 1), then they are in the correct level Saxon math book. If the test scores are constantly below that or if they fall below a 70-75 on their first five or so tests, then that is a good indication they are in the wrong level Saxon math book. This situation can result from any one or more of the following conditions:
The Student is Required to Re-do Math Problems from Yesterday’s Lesson: Why do we want students to get 100% on their daily practice for the weekly test? When we grade their daily work and have them go over the ones they missed on the previous day’s assignment, nothing is accomplished except to “academically harass” the students. The daily work reflects nothing but the status of the students’ temporary learning curve. It is the weekly tests and not the daily work that reveal what the student has mastered from the previous weeks and months of work. Not every student masters every concept the day it is introduced, which is why there is a four to five day delay from when the concept is introduced to when it is tested. In the twelve years that I taught John Saxon’s math books in high school, I did not grade one homework paper – but I did grade the weekly tests which reflected what the students had mastered as opposed to their daily work which did not.
NOTE: In any of John Saxon’s math books from Math 54 through Algebra 2, the “A” and “B” students will get their 30 problems done in less than 40-50 minutes. The “C” students will require more than an hour.
The Student is Required to Do All of The Daily Practice Problems: The daily practice problems were created for teachers to use on the blackboard when teaching the lesson’s concept so they did not have to create their own or use the homework problems for demonstrating that concept. Many of the lessons from Math 54 through Math 87 have as many as six or more such problems and if the student understands the concept, they are not necessary. If the student has not yet grasped the concept, having the student do six or more additional practice problems of the same concept will only further frustrate him. Remember, not every student grasps every concept on the day it is introduced. The five minutes spent on review each day is essential to many students.
The Student is a “Dawdler” or a “Dreamer”: There is nothing wrong with being a “Dreamer,” but some students just look for something to keep them from doing what they should be doing. I call these students “Dawdlers.” I recall the first year I taught. I had to constantly tell some students in every class to stop gazing out the window at the cattle grazing in the field outside our classroom – and get on their homework. That summer, I replaced the clear glass window and frame with a frosted glass block window - and in the following eleven years I had absolutely no problem with my “Dawdlers.”
The Student is Slowed by Distractions: Is the student working on the daily assignment in a room filled with activity and younger siblings who are creating all sorts of distractions? Even the strongest math student will be distracted by excessive noise or by constantly being interrupted by younger siblings seeking attention. Did you leave the student alone in his room only to find he was on his cell phone talking or texting with friends or listening to the radio? Or worse, does he have a television or computer in his room and does he use the computer to search the internet for a solution to his math problems or engage in something equally less distracting by watching the television?
Please do not misinterpret what I have discussed here. If you desire to do all of the above and the student takes two hours to complete a daily assignment—and both you and the student are satisfied—then that is acceptable. But if you are using this excessive time as an excuse for your child’s frustration—and as an argument against John Saxon’s textbooks—I would remind you of what John once told a school district that did everything John had asked them not to do. They were now blaming John’s math books for their district’s low math test scores.
WHAT ARE FORMULA CARDS? WHAT ARE THEY USED FOR? AND WHERE CAN I GET THEM?
Having been repeatedly threatened by my high school math teachers that I would be doomed to fail their tests if I did not memorize all those math formulas, I was somewhat surprised later in a college calculus course when the professor handed out formula cards containing over ninety geometry, trigonometry identities, and calculus formulas. He explained that they could be used on his tests. He did not bat an eye as he handed them out and reminded us that selecting the correct formulas and knowing how and when to use them was far more important than trying to memorize them or write them on the desk top.
So, when I started teaching at the high school, I announced to the students that they could make formula cards by using 5 x 8 inch cards, lined on one side and plain on the other. It never failed. Immediately, one of the students would ask why I did not have them printed off and handed out, saving them a considerable amount of time and money creating their own.
I told my students that whenever they encountered a formula in their textbooks, writing it down would strengthen the connection more than if they just read it and tried to recall it later while working a problem. Reading the formula in the textbook was their first encounter and there would not yet be a strong connection between what they were reading and what they tried to remember. However, when they took the time to create a formula card for that particular formula, they would be strengthening that connection. As they used the card when doing their daily assignments, they would continue the process and eventually place the formula in their long term memory.
So, how can you get formula cards? Simple! Each student makes his own. I allowed my students to use them starting with Math 87 or Algebra 1/2. One young lady in my Algebra 2 class used blue cards for geometry formulas and white cards for the algebra formulas to save her time looking through the cards. The cards should be destroyed after completion of the course, requiring the next student to make his own. Then how do you make formula cards?
Have the students use 5 x 8 cards and tell them to write or print clearly and big. On the plain side of the card they print the title of the formula such as the formula for the area of a sector found on page 16 of the third edition of the Algebra 2 textbook. Then, on the front of the card (the plain side) in the center of the card the student would print:
AREA OF A SECTOR
When you turn the card over, in the upper right hand corner is the page number of the formula to enable the student to immediately go to that page should he need more information (in this case page 18 in the 2nd Ed & page 16 in the 3rd Ed). Recording the page number saves flipping through the book looking for the information and wasting time, especially when the student encounters a difficult problem some twenty lessons later.
After writing down the appropriate page number, they neatly record the formula: (double checking to make sure they have recorded it correctly.)
Area of Sector = Pc/360 times pi(r)2 - Where Pc equals the part of the sector given.
NOTE: If diameter is given remember to divide by two before squaring the value.
Remember, students may also use the formula cards on tests, and if you watch them, the dog eared cards seldom get looked at after awhile.
For those of you concerned about students taking the ACT or SAT, unless they have changed their policy, students are given a sheet of formulas for the math portion of the test. Again, this requires the student to know which formula to select and what to do with it, rather than remembering all those formulas!
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!
IS THERE ANY VALUE TO USING A SEPARATE GEOMETRY TEXTBOOK?
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.
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.”
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!