Newsletters
December 2016
MASTERY - vs - MEMORY
More than two decades ago, at one of the annual mathematics conventions of the National Council of Teachers of Mathematics (NCTM), John Saxon and I were walking the floor looking at the various book publisher’s exhibits, when we encountered a couple of teachers manning the registration booth of the NCTM. When I introduced John to them, they instantly recognized him as the creator of the Saxon Math books and, after gleefully mentioning that they did not use his math books, they proceeded to tell him that they felt his math books were nothing more than 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 over time results in what Dr. Benjamin Bloom of the University of Chicago had described as “Automaticity.” Dr. Bloom was an American educational psychologist who made 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. Benjamin Bloom (known for Bloom’s Taxonomy) at the University of Chicago. John wanted to find out if there was a term that described the way his math book was constructed. Dr. Bloom examined the book’s content and then told John that the technique used in his book was called “automaticity,” which describes the ability of the human mind to do two things simultaneously - so long as one of them was overlearned.
If you think about it, every professional sports player practices the basics of his sport until he can perform them flawlessly in a game without thinking about them. By “automating” the basics, players allow their thoughts to concentrate on what is occurring as the game progresses. Basketball players do not concentrate on their dribbling the basketball as they move down the floor towards the basket. They have overlearned the basics of dribbling a basketball and they concentrate on how their opponents and fellow players are moving on the floor as the play develops.
The great baseball players practice hitting a baseball for hours every day so that they do not spend any time concentrating on their stance or their grip on the bat at the plate each time they come up to bat. Their full concentration is on the movements of the pitcher and the split second timing of each pitch coming at them at eighty or ninety miles an hour. How then does the term “automaticity” change John’s math books from being called “mindless repetition” to math books that - through daily practice over time - enable a student to master the basic skills of mathematics necessary for success?
Both John and I taught mathematics at the university level. And we both encountered freshman students who could not handle the freshman algebra course. These students had failed the entrance math exam and were forced to take a “no-credit” algebra course before they were allowed to enroll in the freshman algebra course for credit. In my book, I refer to them as “at risk adults.” I tell about asking for and receiving permission from the university to use John’s high school Algebra 2 textbook for this “no-credit” course and adjusting the instruction to enable covering the entire book in a college semester.
The results were astounding. More than 90% of those who received a “C” or higher passed their freshman algebra course the following semester.
They had all taken an Algebra 2 course in high school and they had all passed the course. They could not understand why they had failed the math entrance requirement. The day John and I had encountered the NCTM teachers at the registration booth, I would have given anything to have had some of these “at risk adults” tell those teachers just what they thought of their teaching the test, rather than requiring them to master the concepts. They would also have given them a piece of their mind about their teachers using “fuzzy” grading practices that allowed them to pass a high school Algebra 2 course while failing the university’s basic entrance exam several weeks later. They would have also given these NCTM representatives an earful about the difference between being taught the test and receiving a warm fuzzy passing grade and mastering the necessary math concepts to be successful in math at the collegiate level.
There are some new math curriculums out there today using the word “mastery” in their advertisements-attempting to show that their “fun” curriculum is as good if not better than John’s-but to date, I know of none of them that use a cumulative review of the math concepts coupled with weekly tests to reflect mastery by the student rather than re-packaging what my “at risk adults” encountered more than a quarter of a century ago.
"May you have a very Merry and Blessed Christmas,
And a Prosperous and Happy New Year"!
November 2016
DO MATH SUPPLEMENTS REALLY HELP STRUGGLING STUDENTS?
- or -
CAN YOU TEACH A DROWNING CHILD HOW TO SWIM WHILE HE IS DROWNING?
Before addressing that question directly, let me first relate a story about a man walking across a bridge spanning a river. As he looked down at the water, he noticed a boy who had fallen into the swift current. It was apparent from the boy's struggle that he could not swim. The man realized he had only two alternatives. He could shout instructions to the boy on how to overcome the swift current and perhaps enable him to dog paddle to safety on the shore, or he could dive into the water and rescue him. Without hesitating, the man dived into the water and immediately swam to the side of the struggling boy. Now the man had to face another dilemma. Should he pull the struggling boy to safety or should he immediately try to teach him how to swim?
Everyone would agree that when people are drowning, that is not the time to try to teach them how to swim. All one can do at that time is try to get them to a place of safety where they can overcome the swift current of the river. So it is with mathematics. In any of John Saxon's math textbooks from Math 54 through Calculus, if student's begin struggling before reaching lesson thirty or sooner, it is a sign that they will drown in the later lessons of the book unless they are taken to a place of safety where they can better manage and learn the concepts that they are now unfamiliar with. Concepts that are dragging them into deep water! It should become apparent that they are not prepared for the book they are in, and no amount of supplemental material or expensive tutors will overcome those shortcomings.
Mathematics is like the swift current that challenged the drowning boy. Like the river, upper level mathematics is challenging and can easily become unforgiving. Looking for a slower moving or shallower river may create a temporary solution, but eventually that water will again become swifter and deeper and unless one is prepared, all the advice and assistance given at the time of the struggle will come too late.
While it is a noble goal for students to strive towards taking a calculus course in their senior year of high school, it is critical that they first master the algebra. The calculus is easy! It is the challenge of the algebra and to a lesser degree the trigonometry that causes students to fail calculus. Any student with a solid algebra background, entering any college or university, will pass that school's math entrance exam and will be successful in a calculus course should they choose to do so.
When classroom teachers or home school educators take shortcuts with one of John Saxon's math books, they are not adequately preparing the student for the deeper water ahead. More than a quarter of a century of experience with Saxon Math textbooks has shown me that classroom teachers and parents who take shortcuts with his curriculum (instead of going slowly and deliberately through as John intended) cause students to "flounder" as they encounter the "deeper" water. At this point, they find it easier to blame the book - and they look for an easier math course!
The classroom instructions contained within my DVD "video" tutorial series are not math supplements. They contain actual classroom instruction on each concept in the book. Like the book, the classroom instruction is designed for the homeschool student who is in the appropriate level math book. The instruction enhances the written word they have already read from the textbook. Many of the lessons present a different explanation by an experienced Saxon math teacher that helps the student through the difficult reading of the lesson.
However, regardless of who creates them, neither the CD white-board presentations nor my DVD classroom tutorials will help students who are taking a course they are ill prepared for - and they find themselves frustrated and floundering in "deep" water.
October 2016
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 not putting the negative sign in front of his answer when his
work reflects it is a negative number. He understands the concepts
well, but he gets a fourth or more of the problems wrong on his daily
work because of these simple, careless, computational errors."
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. I recently 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.
September 2016
MAKE SURE YOU BUY AND USE THE CORRECT EDITIONS OF JOHN SAXON‘S MATH BOOKS
As we approach textbook purchasing time for homeschool educators the questions about which editions are to be used increase, so I thought it would be advantageous to go over with you the correct editions of John Saxon‘s math books to use. I would also like to provide you with some recommendations on how to use the textbooks correctly and reduce students‘ frustration with mathematics. While there is more detail in my book, I believe the following information will help you select the correct level and edition of one of John Saxon‘s math books.
All of the textbooks listed below also include an introduction to basic geometry as well as a review of the geometric terms associated with geometry at the introductory 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 remains at the introductory level and there is no formal credit for any geometry until successful completion of the Algebra 2 textbook (3rd Ed) where the student also earns a full credit the first semester of a regular high school geometry course.
If after reading this newsletter, you feel your particular 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 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-five 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 30. 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. 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 go 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 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.
August 2016
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. However, as I said earlier, 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.
July 2016
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.
Please Click Here to watch a short video that describes how the Advanced Mathematics course is taught and credited.
Next month we will discuss the sixth and final Myth:
Myth 6: You Do Not Have to Finish the Last Twenty or So Lessons of a Saxon Math Book.
June 2016
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 am at a homeschool convention or browsing the online homeschool blogs, I keep hearing and seeing comments from homeschool parents that express the idea that: “You must use a separate geometry book to receive credit for geometry.”
A little over four years ago, 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. Almost a decade 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 (see last month‘s news article). 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” when recording Saxon math on the student‘s transcript either. Just record Algebra 1 or Algebra 2.
Myths that will be discussed in future News Articles:
Advanced Mathematics Can Easily be Taken in a Single School Year!
You Do Not Have to Finish the Last Twenty or So Lessons of a Book.
May 2016
WHAT ARE SOME OF THE MYTHS SURROUNDING JOHN SAXON‘S MATH BOOKS?"
MYTH 3: Saxon Algebra 2 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.
Future myths that will be discussed:
- You Must Use the New Saxon Geometry Textbook to Receive Geometry Credit.
- Advanced Mathematics Can Easily be Taken in a Single School Year!
- You Do Not Have to Finish the Last Twenty or So Lessons of a Book.
April 2016
WHAT ARE SOME OF THE MYTHS SURROUNDING JOHN SAXON‘S MATH BOOKS?"
MYTH 2: Saxon Math is Just Mindless Repetition
More than a decade 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 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!
Future myths that will be discussed:
- Saxon Algebra 2 does not Contain Two-Column Proofs.
- You Must Use the New Saxon Geometry Textbook to Receive Geometry Credit.
- Advanced Mathematics Can Easily be Taken in a Single School Year!
- You Do Not Have to Finish the Last Twenty or So Lessons of a Book.
March 2016
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 this past summer 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 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 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.
Future myths that will be discussed:
- Saxon Math is Just Mindless Repetition.
- Saxon Algebra 2 does not Contain Two-Column Proofs.
- You Must Use the New Saxon Geometry Textbook to Receive Geometry Credit.
- Advanced Mathematics Can Easily be Taken in a Single School Year!
- You Do Not Have to Finish the Last Twenty or So Lessons of a Book.
February 2016
DO YOU REALLY HAVE TO DO THE DAILY "WARM-UP" BOX AND "PRACTICE PROBLEMS?"
Not a week goes by that I do not receive a telephone call or an email about the excessive amount of time some home school student is spending on his math assignments each day. In almost every case, the student had spent between thirty minutes and an hour on the “Warm-Up” box and the six to eight “Practice Problems” before he even get started on the thirty problems of the Daily Assignment.
It has been more than a decade since I have been in a public classroom, and I am not sure if public school middle school math teachers still lean on what they used to call a math “Warm-Up“ at the start of each class. The purpose of the “Warm-Up” was to settle their students down and get them ready for the math regimen of the day.
Using the “Warm-Up” box at the beginning of each lesson in the Saxon Math 54 through Math 87 textbooks can become quite frustrating to students who do not have the advantage of a seasoned classroom math teacher gently guiding them in the direction of the correct solution for the problem of the day – knowing that problem came from a concept not yet introduced to the students.
But what about the “Daily Math Facts Practice” and the “Mental Math”; how will students receive training in those areas? While these two areas are essential to the student becoming well-grounded in the old pen and pencil format of adding, subtracting, multiplying and dividing, graded by the teacher, that format has been improved with a computer model. Using the computer format allows the students to instantly know whether their answers are right or wrong. Additionally, while the home educators can easily spot the results tallied on the computer as the student moves along, it saves them the time spent manually grading the documents. I have placed a link to a wonderful Math Facts site on my website. Readers can find it by going to my home page, and from the list on the left side of the home page, click on “Useful Links.” When the new window appears, select the second link from the top labeled “On-Line Math Facts Practice.”
That link takes you to a math facts practice site that allows the student to select from seven different levels of difficulty in adding, subtracting, multiplying and dividing. If the answer is correct, a smiling gold star appears and if the answer is wrong, the correct solution appears along with an unhappy red stop sign. Five to ten minutes on this site every day at the appropriate level for the student to be challenged without being frustrated is just as good as the mental math or facts practice found in the “Warm-Up” box.
While the Math 87 book still reflects the same “Warm-Up” box that the previous three math textbooks do, a student should have mastered the facts practice by this time. If this is the case, skipping the entire box is acceptable – unless – the student particularly enjoys the challenge of the “Problem Solving” exercise.
Now let‘s see if I can explain why I am recommending you stop having the student take time to do the six to eight practice problems at the front of each of the mixed practices (the daily assignments). The original purpose of these practice problems was for the classroom teacher to use all or some of them in explaining the concept on the board so that the teacher did not have to make up their own or use the homework problems. Sometimes teachers would use some of them to have students come to the board to show their understanding of the new concept.
My experience in teaching John‘s method of mastering math has shown me that there are only two possibilities that can exist after the student has read and/or had the concept of the daily lesson explained to them.
Possibility 1: The student understands the concept and after doing the two homework problems dealing with that new concept, completely understands what to do and has no trouble doing them. Mastery of this concept will occur over the next five to six days as the student does two more each of these days. If this is a critical concept linked to other steps in the math sequence, they will keep seeing this concept periodically throughout the rest of the book.
Possibility 2: When students encounter the two homework problems that deal with the new concept, they have difficulty doing them. So, on their own, should they go back to these practice problems and get another six to eight more problems wrong? If they did the practice problems before they started their daily work, would anything have changed? If they cannot do the two homework problems because they do not understand the new concept, why give them another six to eight problems dealing with the new concept to also get wrong? This approach ultimately leads to more frustration on the part of the student.
Students will have spent thirty minutes or more on these additional six to eight practice problems and still not understand the new concept. Not every student completely grasps a new concept on the day it is introduced which is why John‘s books do not test a new concept until the student has had five to ten days to practice that concept.
Those practice problems were not placed there to give the student more problems to do in addition to the thirty they are assigned every day. They were placed there for the classroom teacher to use on the blackboard to teach the new concept so they did not have to develop their own or use the student‘s homework problems. There is nothing wrong with a home school educator asking a student to do one or two of them to show them the student understands the new concept; however, doing more than that could be a waste of time and effort in either possibility.
Not every child is the same and I realize that because of a particular child‘s temperament, there may be some instances where the parent has to go over more than one or two of the practice problems with the child - and this is okay - but for most students this is not necessary. If the student really enjoys the challenge of the daily “Problem Solving:” that is okay - except parents should make sure that the student does not spend an excessive amount of time on that individual challenge and allow the real goal of completing the thirty problems of the Daily Assignment to become a secondary goal – and later a bother to the student.
January 2016
WHAT ARE THE DIFFERENCES AMONG THE VARIOUS SAXON MATH TUTORIALS ON THE MARKET TODAY?
While at Home School Conventions, I was repeatedly asked by homeschool educators to explain to them the difference between the "DIVE" CD's, the "Saxon Teacher" CD's, the "Teaching Tapes Technology" DVD series, and the DVD series "MASTERING ALGEBRA, John Saxon's Way." That is an excellent question because some companies confuse the situation when they advertise their CD's as being "video" products when in fact they are not DVD's, but only CD's containing a graphic presentation with audio. The abbreviation DVD stands for "Digital Video Disc" and these DVD products will work on either a computer or a television DVD player, while CD products will only work on a computer with a CD player.
Basically, here are the differences.
DIVE CD's: The product covers John Saxon's math books from Math 54 through Calculus. Each level textbook has a single CD containing instruction corresponding to each individual lesson in that textbook. The presentation is a whiteboard presentation which means there is no teacher to watch at the board. The student hears the voice in the background and watches writing appear on the screen. The CD will not work in a television DVD player because it is not a true "Digital Video Disc," but rather a graphic presentation with audio. As a CD, it is restricted to being played only on a computer. Each individual CD costs just under $50.00. They are available for the current editions of Saxon Math from Math 54 through Calculus-and for some of the older Saxon editions as well.
THE SAXON TEACHER CD's: The product is sold by HMHCO and supports John Saxon's math books from Math 54 through Advanced Mathematics. Similar to the DIVE CD, these CDs are a graphic whiteboard presentation which means there is no one to watch presenting the material. The student hears the voice in the background as the writing appears on the board. The individual in each of the individual series of CD's goes over every problem in the textbook and the individual problems on the tests as well, which is why there are four or more CD's to this product as opposed to the single CD sold by DIVE. HMHCO sells these CD graphic "audio" solutions for $99.00. There is a printed version of the solution for each of the daily problems. The printed version is called the "solutions manual" (which contains the same "step-by-step" information as the more expensive CD). The printed solutions manual sells for between $27.00 for the Math 54 course to $45.00 for the Advanced Mathematics course. If you have purchased the new soft cover editions of Math 54, 65, 76 or 87, the solutions manuals are included in the price of the Homeschool Kit for these four courses. These CD's are not "videos" and they can only be used on a computer. They cannot be viewed on a television set using a DVD player.
TEACHING TAPES TECHNOLOGY: The product is a DVD "video" set of lessons which means they can be used on either a television or computer DVD player. The entire series covers Math 54 through Calculus. As advertised by the company, the individual lessons are taught by a state certified math teacher. The individual series for a particular math book in the upper level math series sell for anywhere from $175 for the Math 54 series to $200 for the Algebra ½ series to $245 for the Calculus series. The Calculus series requires the first edition of calculus. Each DVD series for a specific textbook contains from twelve to fifteen individual discs. Like the Saxon Teacher CD series, the teacher on these videos goes over every assigned and practice problem in the book, which explains why there are so many DVD's in each individual series. Unlike the DIVE or HMHCO CD's, these are DVD "Digital Video Disc" presentations and as true DVDs they will work on either a television or a PC running on Linux or Windows operating system-or MAC computer-as long as they have a DVD player (internal or external).
MASTERING ALGEBRA "John Saxon's Way" - by Art Reed: This product is also a DVD video presentation which means the DVD's will work on both a computer as well as a television DVD player. This capability would enable a group of homeschool students (or a Co-Op) to watch together, on a single television set, as they would in a regular math classroom. The concepts of every lesson are taught by an experienced Saxon math teacher with over twelve years teaching experience using Saxon Math books in a rural public classroom. The students see an experienced Saxon math teacher at the board teaching the concepts contained in that lesson. There are ten to twelve DVD's in each of the series which run from Math 76 through Calculus. Each individual series from Math 76 through Calculus sells for $59.95 (which includes postage within the U.S.A. and its Protectorates-including APO and FPO addresses).
Before you buy any of these products, sit down with your student and look at each of the samples provided by the companies on their websites. Make sure the student is comfortable with the instructor and the material—as presented by that instructor.
Here are the four websites:
diveintomath.com ; saxonhomeschool.com ; teachingtape.com ; usingsaxon.com
For a more information on my DVD series, you can also read my June 2014 Newsletter.
HAVE A HAPPY AND BLESSED NEW YEAR