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Newsletters
WHAT IS THE BASIC DIFFERENCE BETWEEN MATH 87 – AND – ALGEBRA ½?
There appear to be varying explanations regarding whether a student should use Math 87 or algebra ½ after completing the third or fourth edition of John Saxon's Math 76 textbook. Let me see if I can shed some light on the best way to determine which one, or when both, should be used.
First, whenever I discuss the Math 76 textbook in this article, I am talking about the third or fourth editions of that book. I am not talking about the old first or second editions of John's Math 76 books. These two older editions have been out of print for almost fifteen years now and their content, while acceptable at the time, would not now enable a math student to proceed successfully through either the second or third editions of Math 87 or the newer third edition of Algebra ½.
Second, whenever I discuss using the Math 87 textbook, I am talking about either the second or third editions of that book and not the older first edition which has also been out of print for more than a decade. Let me assure you that, except for the new soft cover, the addition of a solutions manual, and the varied numbering of the pages, there is absolutely no difference between the 120 lessons and the 12 Investigations of the hard cover second edition of Math 87 and the new soft cover third edition of Math 87. Oh yes, the new third edition has added a TOPIC A at the end of the book (after Investigation 12) dealing with Roman Numerals and Base 2. And even though the marketing folks have added the word "Prealgebra" under the cover title of the soft cover third edition textbook, these two additional topics, while interesting and nice to know, are not pre-algebra material and will not create any shortfall in either Algebra 1 or even later in Algebra 2.
Both the Math 87 and Algebra ½ courses prepare the student for the Algebra 1 course. The main difference between using the Math 87 and the Algebra ½ books depends upon the student's success in the Math 76 textbook. The Math 87 book starts out a little slower than the Algebra ½ book does because it assumes the student needs the additional review resulting from the student encountering difficulty in the latter half of the Math 76, textbook. Also, If you were to open to any lesson in the Algebra ½ textbook, you would immediately notice that the "Warm-Up" box common in Math 54 through Math 87, is noticeably absent in the Algebra ½ textbooks.
If there exists a math savvy student of John Saxon's Math 76 textbook, who received test grades of 80-85 or better on the last five tests in Math 76 (50 – 55 minute test period, no calculator, and no partial credit), then that student would be more challenged and, from my teaching experience, much better off in the Algebra ½ book. However, if his last five test scores are below 80, then from my experiences, that indicates that student should proceed to Math 87, and upon completion of that book, if his or her last five tests are now 80 or better (minimal mastery), then that student can easily skip Algebra ½ and go on to success in Algebra 1.
If however, a student encounters difficulty going through both Math 76 and Math 87, then proceeding through the Algebra ½ textbook before attempting Algebra 1 will allow the student to regain his confidence. Doing so will further ensure the student has mastered a solid conceptual base for success in any algebra one course.
Students will fail an Algebra 1 course if they have not mastered fractions, decimals and percents which are emphasized before the student reaches that course. I realize that not every child will fit neatly into a specific mold; however, John Saxon's Math 76, Math 87, and Algebra ½ series allows the Saxon Homeschool Educator sufficient flexibility to satisfy every student's needs, and to ensure the students' success in any algebra one course.
I realize that not every student is alike. Should you therefore encounter a situation not clearly described in this news article, please feel free to email or call (580) 234-0064 (CST) and I will do my best to assist you in resolving the issue facing your child.
Have a Blessed Christmas and a Happy New Year
WHAT SHOULD YOU DO WITH STUDENTS WHO CONTINUALLY MAKE SIMPLE MISTAKES ON THEIR DAILY WORK?
Often, I receive telephone calls or emails from homeschool educators who express concern that their sons or daughters continue to make simple mistakes in computations when doing their daily work.
As one parent stated:
"My son is taking Algebra 1 and constantly makes silly mistakes, like forgetting to put a negative sign in front of his answer when his work clearly shows the answer should reflect a negative number. He understands the concepts well, but because of these simple, careless, errors he gets a fourth or more of the problems wrong on his daily work."Mistakes like those described above are normal with most students working on the daily assignment preparing for the upcoming weekly test. Have you noticed that they make fewer, if any, of these same mistakes when they take a test? I like to use the phrase that "students put on their Test Hat" when taking a test, and they will not accept the same mistakes they do on their daily practice work. However, if you reward them for making these mistakes on a test by giving them partial credit, they will continue making them on the tests as well. No matter how much we try to eliminate these mistakes, some students will never stop making them, no matter how good they become at mathematics.
That is why experienced engineers always check each other's work before releasing a new project for testing or production. Several years ago I read in the daily newspaper that Spanish engineers working on a new submarine for the Spanish Navy did not do this verification check. After building a new submarine, it was found that the engineers had overlooked the erroneous placement of a decimal point in their computations. The embarrassing – and costly – result was that the Spanish Navy ended up with a new submarine so heavy that it would not surface if it were ever submerged.
Most students make fewer mistakes in performing simple mental arithmetic calculations on paper than they do when pressing the wrong button on a calculator, which still constitutes a human error, although the student will try to blame the calculator!
Even students looking to achieve perfection can be found guilty of "rushing" through their daily work for one reason or another. It might help to ensure students develop the habit of checking the work of the problem they just finished before moving on to the next. This process of review would enable them to find many, if not all, of these types of simple mistakes and while it may add a few minutes to the time spent on the daily assignment, it might get them to slow down a bit to avoid making them in the first place.
So long as you do not reward the student for making these simple calculation errors on the weekly tests–like giving them partial credit for using the right concept but getting the wrong answer–they will eventually overcome that shortcoming.
And if they do not, but their weekly test scores remain constantly at an 80 or better, I would not worry about it. Remember, the cumulative and repetitive nature of John Saxon's math books and tests is what creates the mastery as opposed to other math curriculums reviewing for–and teaching–the test.
So making a few computational errors, while maintaining a minimum score of 80 on the thirty-some cumulative weekly tests, is truly outstanding. While I fully understand that everyone considers an acceptable target grade for tests at 95 – 100, receiving an 80 on one of John Saxon's weekly cumulative math tests is equivalent to the 95 one would receive on the periodic test using some other math curriculum that teaches the test.
WHAT ARE SOME OF THE MYTHS SURROUNDING JOHN SAXON'S MATH BOOKS?
(Myth 6)
You Do Not Have to Finish the Last Twenty or So Lessons of a Saxon Math Book.
Over the past several decades, I have heard hundreds of homeschool educators as well as parents of my high school classroom students tell me that there was no need to finish a Saxon math book because the last twenty or so lessons of any Saxon math book are repeated in the review of the first thirty or so lessons of the next level Saxon math book.
There is a bit of truth to that observation. A few of the concepts encountered in the later lessons of a book are repeated in the early lessons of the next level book because that important concept came late in the book and did not allow sufficient time for the student to master it before reaching the end of the course. But when repeated, the re-introduction of these concepts assumes the student had encountered the concepts in a simpler format in the previous level textbook.
But anyone who would attempt to skip the last twenty or so lessons of any Saxon math book under the misguided impression that all of that material is repeated in the first thirty lessons of the next math book is in for a shocking surprise. Someone may tell you their son or daughter did just that while using the Saxon Algebra 1 textbook and their child did quite well in the Saxon Algebra 2 book the following school year.
While there are always exceptions that justify the rule, what most of these home educators will not tell you is that – because of this shortcut – their child struggled through the Saxon Algebra 2 course and the student either repeated the course a second year, or failed to master the required concepts – having to enroll in a no credit algebra course as a freshman in college the following year.
The concept of automaticity requires the application of repetition over time and violating either one of these conditions greatly reduces the student's chances of mastering the necessary math concepts to be successful in the next level math course. There is a third factor involved in the process of automaticity. When the student encounters a concept, works with it over several weeks and then does not encounter it again until as much as a month later, that delay in repeating – coupled with a slight change in the level of difficulty of that concept – challenges the student's level of mastery and some students who have not quite mastered the entire concept have to review it from previous lessons before continuing. However, once mastered a second time – following the delay – the concept is more strongly imbedded in their long term memory.
So after taking a break for the summer, is it not wise to start the next level Saxon math book with a small amount of review material to ensure the student retained the necessary skills to succeed in the next course?
But wait, would that apply to homeschool students who do not take a summer break? The argument is that if they finish the entire Algebra 1 book, and then go straight into Algebra 2, they can easily skip the first twenty or so lessons in the Algebra 2 text. That is also a dangerous procedure to follow for at least two reasons.
FIRST: Remember I said that some of the concepts introduced late in the previous textbook are repeated to allow mastery – I did not say all of them. The student will go down in flames around lesson forty or so, never having been introduced to a dozen or more concepts involving both algebra and geometry.
Additionally, the Algebra 2 book assumes the students mastered their basic introduction to these new concepts in the earlier lessons (the ones the student skipped) and it now combines them with other concepts. Now students start struggling as test scores begin to fall. This is where the parent or teacher blames the book as being too difficult to use and leaves Saxon math for an easier math course.
SECOND: While collegiate and professional athletes practice almost year round, they do take several months off sometime between their seasons to rest the mind as well as the body. In mathematics, it is good to take a month or so off between levels of math to allow students to refresh their thought processes. As I mentioned earlier, this break also allows them to better evaluate what concepts they have truly mastered. Once mastered a second time – following the delay – the concept is more strongly imbedded in their long term memory.
I believe these are two valid reasons not to skip lessons under any circumstances.
WHAT ARE SOME OF THE MYTHS SURROUNDING JOHN SAXON'S MATH BOOKS?
(Myth 5)
Advanced Mathematics Can Easily be Taken in a Single School Year.
Several decades ago, while teaching John's Advanced Mathematics textbook my second year at the high school, I encountered a problem with my Saxon Advanced Mathematics students. The students who had received an A or B in the Saxon Algebra 2 course the previous year were now struggling with low B and C grades – and we were only in our first nine weeks of the course.
I called John and explained the situation to him. He asked me if I was following the same procedure I had used in the Algebra 2 course last year (e.g. a lesson a day – all thirty problems assigned every day – and a test every Friday). I told him that we did all thirty problems every day and took a test every Friday just as we had in the Algebra 1 course as well. I went on to tell John that the students were frustrated. In Algebra 2, they had easily completed their daily work done in forty-five to fifty minutes, but now they were spending several hours each night to complete their daily assignments – and most of them were not even getting all of the assigned homework finished in that period of time.
John's response was quick and to the point. He asked me if I had read the preface to his book, and when I told him I had not, he told me to read the preface of the book and then he hung up. This was not an unusual trait of John's. I had known him for several decades and, like many other experienced fighter pilots I had encountered in my military service, he seldom went into any lengthy explanation when someone was not following instructions.
In the preface of the Advanced Mathematics textbook, I found that John had written in detail about the textbook's in-depth coverage of trigonometry, logarithms, analytic geometry, and upper-level algebraic concepts. He explained that the textbook could easily be broken into two 5-semester hour courses at the college level. But he cautioned – that at the high school level – teachers should break the course into three or four semesters.
I immediately chose the four semester option, calculating that this would allow two days for each lesson. The students could do the odd numbered problems one day and the even numbered problems the second day. By doing it this way, the students would encounter all of the concepts covered in the thirty problems both days since the concepts taught in each lesson were arranged in pairs. Also, they would not have to spend more than an hour each night on their daily assignment.
Is it possible for high school students to successfully complete the entire Advanced Mathematics textbook in a single school year? Yes, but both John and I were in agreement that those students are the exception rather than the rule. In all the years that I taught using John's math books, I have encountered only one student who completed the entire 125 lessons of the Advanced Mathematics textbook in a single year – with a test average of over 90 percent! She was a National Merit Scholar and her father taught mathematics with me at the local university.
That is not to say that others could not have accomplished the same feat, but these exceptions only tend to justify the rule. The beauty of John's Advanced Mathematics book is its flexibility that allows students to use the book at a pace comfortable to them whether that pace takes two, three or four semesters. There is no academic dishonor in a bright home school math student taking three or four semesters to complete John Saxon's Advanced Mathematics textbook if that student needs the extra study time to take care of other tough academic subjects being taken at the same time. There is no need to bunch everything up and rush through the math just to get to calculus before the student graduates from high school.
Students fail calculus in college not because of the difficulty of the calculus concepts, but because their background in algebra and trigonometry is weak. It is the student with the weak mastery of algebra and trigonometry in high school who fails the calculus course – or – perhaps the student who has mastered the algebra and trigonometry in high school, but because of this knowledge, elects not to attend the daily calculus lectures.
Please Click Here to watch a short video that describes how the Advanced Mathematics course is taught and credited.
The sixth and last myth to be discussed in next month's news article is:
- Oct - Myth 6 - You Do Not Have to Finish the Last Twenty or So Lessons in any Saxon Math Book.
WHAT ARE SOME OF THE MYTHS SURROUNDING JOHN SAXON'S MATH BOOKS?
(Myth 4)
You Must Use the New Saxon Geometry Textbook to Receive Geometry Credit.
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."
More than two decades ago, when HMHCO started to publish their separate Geometry textbook, I received an Examination Copy of the new fourth edition Algebra 1 book prepared by the new owners of John Saxon's Publishing Company. They had gutted the book of all references to geometry. The index had one reference to several pages late in the textbook titled "Geometric Sequences," but that term refers to an algebraic formula dealing with common ratios – it is not a geometry formula found in any geometry textbook.
Unlike the old second or third editions of Algebra 2, their new fourth edition of Algebra 2 has also had all references to geometry removed from it. Why did the new owners do this? Well I can come up with several reasons:
FIRST: the marketing people would tell you that you make more money from three books than you do from two. Several decades ago, I learned from the Corporate Executives at the company that first bought Saxon Publishers from John's children in 2004 that they truly believed that "A math book is a math book is a math book." In my dealings with them as they transitioned John's Publishing Company into theirs, it was apparent that they failed to realize or accept the uniqueness of John's math books. To them one math book was just like another. If a particular state did not buy their math book this year another state was switching from someone else's math book to theirs. So as long as this phenomenon went on why waste profit margin selling a unique math book and explaining or defending its content.
Why? Because the perception was that failure in the math program of any particular public school was never the fault of the teacher; it was always the poor quality of the math book which required switching to more "improved" math books every four to five or so years as math test scores either did not improve or fell. And the publishers would be more than happy to tout their new and improved math textbooks which they said would result in higher test scores. One book publisher even went so far as to openly advertise that since they also published the annual student state math tests and that their books were geared to ensure student success with these mandated state tests.
I recall telling a high school principal several decades ago that it never ceased to amaze me that after a decade or two of schools switching math books every few years – because of low math test scores – that sooner or later school administrators would realize it might be the teachers or the poor quality of the math books responsible for the low test scores. So why not do as everyone else does and create three separate and distinct math books for the algebra one, algebra two, and geometry courses? That not only makes it easier to sell the books, but it increases the quarterly profit margins because of the requirement for the additional geometry book.
SECOND: Some math teachers would tell you that students cannot learn geometry while they are trying to master the algebra. They therefore demand a separate geometry textbook. The second and third editions of John Saxon's Algebra 2 textbooks contain the equivalent of the first semester of a regular high school geometry textbook – to include rigorous two-column proofs. But wait! Isn't it true that students cannot handle the geometry while they are also trying to master the algebra? Not so! European students have been combining algebra geometry and trigonometry in a single math book as long as I can remember. And they consistently come out ahead of us in comparative math comprehension tests.
This myth makes about as much sense as telling a high school student that they cannot take a mandatory sophomore English course while also taking a separate journalism course. 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 master 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? Must the content be in two separate textbooks taken at two different times in order for the student to master their content?
The geometry concepts encountered in John Saxon's Algebra 2 textbook – whether the second or third edition – are 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! However, if you choose to use the new fourth edition of Algebra 2, you must also purchase a separate geometry textbook to acquire geometry credit. As I previously mentioned, the new fourth editions of the revised HMHCO Algebra 1 and Algebra 2 textbooks do not contain any geometry concepts.
THIRD: The new fourth editions of Algebra 1 and Algebra 2 – as well as the new first edition of Geometry – do not have a responsible author, and therefore the new owners of John Saxon's company do not have to pay any royalties! If you look at the inside cover of the new fourth editions of Algebra 1 and Algebra 2 as well as the new first edition of the new Geometry textbooks, you will not be able to find the name(s) of an author or authors of these books. Why? Because they were created by a committee hired by marketing people and the committee that constructed that edition of the algebra textbooks may or may not have had any extensive math or teaching experience. The publishers paid a one-time fee to a "committee" to create the new editions releasing them from paying future royalties to an author.
So, do we blame the profit minded publishers for publishing a separate geometry textbook, or is it the fault of misguided high-minded academicians who – after more than a hundred years – still demand a separate geometry text from the publishers? I am not sure, but thankfully, this decision need not yet face the homeschool educators using John Saxon's math books. The original homeschool third editions of John Saxon's Algebra 1 and Algebra 2 textbooks still contain geometry as well as algebra – as does the Advanced Mathematics textbook which follows the Algebra 2 textbook..
Any homeschool student using John Saxon's homeschool math textbooks who successfully completes Algebra 1, (2nd or 3rd editions), Algebra 2, (2nd or 3rd editions), and at least the first sixty lessons of the Advanced Mathematics (2nd edition) textbook, has covered the same material found in any high school Algebra 1, Algebra 2, and Geometry textbook – including two-column formal proofs. Their high school transcripts – as I point out in my book – can accurately reflect a full credit for completion of an Algebra 1, Algebra 2, and a separate Geometry course.
NOTE: Just as you do not record "Smith's Biology" on the student's transcript when awarding credit for a year of biology, you should not use Saxon Algebra 1, or Saxon Algebra 2, etc., when recording Saxon math on the student's transcript either. Just record Algebra 1, Algebra 2, Geometry, etc.
Myths that will be discussed in future news articles:
- Sept - Myth 5 - Advanced Mathematics Can be Taken Easily in a Single School Year!
- Oct - Myth 6 - You Do Not Have to Finish the Last Twenty or So Lessons in any Saxon Math Book.
WHAT ARE SOME OF THE MYTHS SURROUNDING JOHN SAXON'S MATH BOOKS?
(Myth 3)
Saxon Algebra 2, 2nd or 3rd Ed Does Not Contain Formal Two-Column Proofs.
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:
- Aug - Myth 4 - You Must Use the New Saxon Geometry Textbook to Receive Geometry Credit.
- Sept - Myth 5 - Advanced Mathematics Can be Taken Easily in a Single School Year!
- Oct - Myth 6 - You Do Not Have to Finish the Last Twenty or So Lessons in any Saxon Math Book.
WHAT ARE SOME OF THE MYTHS SURROUNDING JOHN SAXON'S MATH BOOKS?
(Myth 2)
Saxon Math is Just Mindless Repetition
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:
- July - Myth 3 - Saxon Algebra 2 does not Contain Two-Column Proofs.
- Aug - Myth 4 - You Must Use the New Saxon Geometry Textbook to Receive Geometry Credit.
- Sept - Myth 5 - Advanced Mathematics Can be Taken Easily in a Single School Year!
- Oct - Myth 6 - You Do Not Have to Finish the Last Twenty or So Lessons in any Saxon Math Book.
WHAT ARE SOME OF THE MYTHS SURROUNDING JOHN SAXON'S MATH BOOKS?
(Myth 1)
Saxon Math is too Difficult
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.
NOTE: Students should be given no more than 60 minutes to complete each test of any individual Saxon math course. Each test question is awarded five points if correct. Test questions should be graded as either right or wrong with no partial credit awarded for partially correct answers.
Some other common Myths that will be discussed in future news articles:
Myth 2: Saxon Math is Just Mindless Repetition.
Myth 3: Saxon Algebra 2 does not Contain Two-Column Proofs.
Myth 4: You Must Use the New Saxon Geometry Textbook to Receive Geometry Credit.
Myth 5: Advanced Mathematics Can be Easily Taken in a Single School Year!
Myth 6: You Don't Have to Finish the Last Twenty or So Lessons of any Saxon Math Book.
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!
With the current school year more than half through and looking towards purchasing textbooks for the next school year – undoubtedly, some questions will arise about which editions of John Saxon's Math Books should be used. So I thought it would be advantageous to go over with you the correct editions of his math books that I recommend you use! While there is more detail in my book, I believe the following information will help you select the correct level and edition of John Saxon's math books from the 4th through the 12th grades.
All of the textbooks listed below also include an appropriate level of introduction to basic geometry as well as a review of the geometric terms associated with geometry at the appropriate introductory grade level. As the student moves from Math 54 to Algebra 1, the repetition of these terms and concepts allows for a gradual increase in their level of difficulty.
However, this geometry still remains at the introductory level and there is no formal credit for any geometry until successful completion of the Algebra 2 textbook (3rd Ed). After completing all 129 lessons of the textbook, the student has also earned a full semester credit for the first semester of a regular high school geometry course. Acquiring credit for the second semester of geometry is addressed by the link at the end of next-to-last paragraph at the end of this article.
If after reading the information provided below, you feel your particular student's situation has not been addressed, please feel free to email me at art.reed@usingsaxon.com or call me at 580-234-0064 (CST) before you purchase any Saxon math textbooks.
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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.
Math 76 (3rd or 4th Ed): The kingpin book in the Saxon series. This book follows successful completion of the Math 65 textbook. Again, successful completion of Math 65 means completing the entire book as well as all of the tests. To be successful in Math 76, students should have received scores no lower than an eighty on the last four or five tests in the Math 65 course. Either the hard cover 3rd edition or the newer soft cover 4th edition can be used.
As with the previous two math courses, there is no difference between the math content of the hard cover 3rd edition and the softcover 4th edition textbooks. I recommend acquiring a copy of the solutions manual as this is a challenging textbook. Students who score eighty or better on the last five tests in this level book indicate they are ready to move to Algebra ½, 3rd edition. Student's who encounter difficulty in the last part of Math 76, reflected by lower test scores, can easily make up their shortcomings by proceeding to Math 87 rather than Algebra ½. Calculators should not be used at this level.
Math 87 (2nd or 3rd Ed): Again, there is little if any difference between the hardcover 2nd edition and the softcover 3rd edition textbooks. Even though the older second edition does not have "with pre-algebra" printed on its cover as the 3rd edition softcover book does, the two editions are identical in math content. Students who successfully complete the entire textbook and score eighty or better on their last five or six tests can skip the Algebra ½ textbook and proceed directly to the Algebra 1, 3rd edition textbook. Both the Math 87 and the Algebra ½ textbooks get the student ready for Algebra 1; however, the Math 87 textbooks start off a bit slower with a bit more review of earlier concepts than does the Algebra ½ book.
This enables students who encountered difficulty in Math 76 to review earlier concepts they had difficulty with and to be successful later in the textbook. Students who encounter difficulty in the last part of this book will find that going into Algebra ½ before they move to the Algebra 1 course will strengthen their knowledge and ability of the basics necessary to be successful in the Algebra 1 course. Their frustrations will disappear and they will return to liking mathematics when they do encounter the Algebra 1 course. Calculators should not be used at this level.
Algebra ½ (3rd Ed): This is John's version of what other publishers title a "Pre-algebra" book. Depending upon the students earlier endeavors, this book follows successful completion of either Math 76 or Math 87 as discussed above. Use the 3rd edition textbook rather than the older 2nd edition as the 3rd edition contains the lesson concept reference numbers which refer the student back to the lesson that introduced the concept of the numbered problem they're having trouble with. These concept lesson reference numbers save students hours of time searching through the book for a concept they need to review – but they do not know the name of what they are looking for. From this course through calculus, all of the textbooks have hard covers, and tests occur every week, preferably on a Friday.
To be successful in John Saxon's Algebra 1 course, the student must complete the entire Algebra ½ textbook, scoring eighty or better on the last five tests of the course. Students who encounter difficulty by time they reach lesson 30 indicate problems related to something that occurred earlier in either Math 76 or Math 87. Parents should seek advice and assistance before proceeding as continuing through the book will generally result in frustration and lower test scores since the material in the book becomes more and more challenging very quickly. Calculators should not be used at this level.
Algebra 1 (3rd Ed): I strongly recommend you use the academically stronger 3rd edition textbook. The new owners of the Saxon Publishers (HMHCO) have produced a new fourth edition that does not meet the Saxon methodology. The new fourth edition of Algebra 1 has had all references to geometry removed from it and using it will require also buying a separate geometry book. While the associated solutions manual is an additional expense, I strongly recommend parents acquire it at this level to assist the student when necessary. Depending upon the students earlier successes, this book follows completion of either Math 87 or Algebra ½ as discussed above. Calculators are recommended for use at this level after lesson 20.
While lesson 114 of the book contains information about using a graphing calculator, one is not necessary at this level. That lesson was inserted because some state textbook adoption committees wanted math books to reflect the most advanced technology. The only calculator students need from algebra through calculus is an inexpensive scientific calculator that costs about ten dollars at one of the local discount stores. I use a Casio fx260 solar which costs about $9.95 at any Target, K-Mart, Wal-Mart, Radio Shack, etc. If the 3rd edition of Saxon Algebra 1 is used, a separate geometry textbook should not be used between Saxon Algebra 1 and Algebra 2 because the required two semesters of high school geometry concepts will be covered in Saxon Algebra 2 (1st semester) and in the first sixty lessons of the Advanced Mathematics book (2nd semester).
Algebra 2 (2nd or 3rd Ed): Either the 2nd or 3rd editions of the Saxon Algebra 2 textbooks are okay to use. Except for the addition of the lesson concept reference numbers in the newer 3rd edition, the two editions are identical. These lesson concept reference numbers save students hours of time searching through the book for a concept they need to review – but they do not know the name of what they are looking for. If you already have the older 2nd edition textbook, and need a solutions manual, you can use a copy of the 3rd edition solution manual which also has solutions to the daily practice problems not in the older 2nd edition solutions manual.
Also, the 3rd edition test booklet has the lesson concept reference numbers as well as solutions to each test question – something the 2nd edition test booklet does not have. An inexpensive scientific calculator is all that is needed for this course. Upon successful completion of the entire book, students have also completed the equivalent of the first semester of a regular high school geometry course in addition to the credit for Algebra 2. I strongly recommend you not use the new fourth edition of Algebra 2 for several reasons.
FIRST: The fourth edition has had all references to geometry removed from it also requiring the purchase of an additional geometry book.
SECOND: The Advanced Mathematics textbook assumes the student has just successfully completed the 2nd or 3rd edition of the Saxon Algebra 2 textbook with their inclusive geometry. If the student took a separate geometry course between the fourth editions of Algebra 1 and Algebra 2, they have not had any exposure to geometry for as much as fifteen months (nine months of school plus two summer breaks). This gap will result in the student encountering extreme difficulty in the Advanced Math textbook.
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.
Calculus: The original 1st edition is still an excellent textbook to master the basics of calculus, but the newer 2nd edition affords students the option to select whether they want to prepare for the AB or BC version of the College Boards Advanced Placement (AP) Program. To prepare for the AB version, students must only complete through lesson 100. To prepare for the BC version, they must complete all 148 lessons of the book. While the 2nd edition reflects use of a graphing calculator, students can easily complete the course using an inexpensive scientific calculator. I recommend that students who use a graphing calculator first attend a course on how to use one at the local tech school or at a local community college – over the summer – before attempting upper level math as they need to concentrate on the math and not on how their fancy calculator works. It is not by accident that the book accompanying the graphing calculator is over a half inch thick.
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.”
In news articles published several years ago in November and December of 2020, I addressed some of the ramifications of taking these shortcuts when using John Saxon’s math books. In these articles, I described in detail the effects upon students who used some or all of them, so I will not go over them again here. I would ask you to read those two newsletters if you have not already done so. What I want to discuss here is what may be causing the excessive amount of time taken by the students and also, what constitutes excessive time to an educator who taught in a public classroom using Saxon math books for more than a decade.
While I was teaching high school mathematics in a rural Oklahoma high school, I would often go and watch my students who were on the the high school track, basketball, or football teams during their practice sessions after school. I was able to chat with the mothers and fathers who were also watching these practices. This one-on-one conversation often gave me an insight into their priorities regarding their children’s education.
While they sometimes complained about the rigors of my math classes, they never once complained about the length of time their sons and daughters were out on the field in the heat or cold - or on the basketball court – practicing – after just spending six academic hours in the classroom. In fact, when coaches were forced to cancel a practice for one reason or another, some of the parents would vocally complain that the practices should continue. They expressed concern that skipping practice would take the “edge” off their son or daughter’s playing ability and inhibit their athletic “sharpness” for the next game.
Strange then that some parents would complain the 45 – 60 minutes spent each day on mathematics “practice” would be excessive - and more unusual - that they would seek an easier course of action. They never discussed the ramifications that doing so might take the “edge” off their child’s math “sharpness” for the next math course or the state mandated math test. I never heard the high school parents complain about watching the tough daily drills and practices run by the coaches. I never heard a parent complain about the hour spent each day by the students diligently practicing their piano lessons, or having to come in before school early each day to spend 45 minutes in the weight room.
At least several times each week I receive email from home school parents who express concern that their son or daughter was taking an inordinate amount of time on their daily math assignment in one of the books from Math 54 through Advanced Mathematics.
The solution to the excessive time spent by students using the Advanced Mathematics textbook is easy to resolve. The solution to that unique situation is explained in a short seven-minute video clip (Click Here to view that video).
I have interacted with several thousand parents and students in the twelve years that I taught mathematics at that rural high school. I have also advised thousands more home school educators and home school students in the succeeding decades after my retirement while serving as one of the Homeschool Curriculum Advisors (for Math 76 through Calculus and Physics) for Saxon Publishers and later for Harcourt-Achieve who bought the company from John’s children. And while every child and home school situation is different, my experiences have shown me that there exist several situations that contribute to excessive time spent on daily work by students, whether home schooled or attending a public or private classroom.
These situations are:
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:
- They did not finish the previous Saxon math book.
- They took shortcuts in the preceding math book.
- Their previous math book was not a Saxon math book.
- They did not take the weekly tests in the previous math book. Instead, they are using the daily grade to determine the student’s level of proficiency.
- The student’s last five tests in the preceding course were well below 80% (minimal mastery).
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.
Remember, John Saxon’s math books are the only books I am aware of that use weekly tests to evaluate a student’s progress. There are a minimum of twenty or more weekly tests in every one of John’s math books from Math 54 on.
Too Much Time is Spent on The Warm Up Box: From Math 54 through Math 87, there is what used to be called a “Warm Up” box at the top of the first page of every lesson. I recall watching a sixth grade teacher waste almost thirty minutes of class time while three boys took turns giving different opinions as to how the “Problem of The Day” was to be solved – and arguing as to which had the better approach. After class, I reminded the teacher that the original purpose of the box was to get the students settled down and “focused” on math right after the second bell rang. I said to her, “Why not immediately review a couple of the problems from yesterday’s lesson at the start of class for the few who perhaps did not grasp the concept yesterday? Then move immediately to the new lesson.” This process would take about 10 to 20 minutes and would leave students with about 40 minutes of remaining class time to work on their new homework assignment.
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.
John told them “If you want to continue your current practices, get rid of my books and buy someone else’s textbooks to blame.”
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.
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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