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?
The two necessary elements of "automaticity" are "repetition over time." If one attempts to take a short cut and eliminate or shorten either one of these components, mastery will not occur. Just as you cannot eat all of your weekly meals only on Saturday or Sunday - to save time preparing meals and washing dishes every day - you cannot do twenty factoring problems one day and not do any of them again until the test in five weeks without having to review just before the test.
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.
WHICH SAXON HIGH SCHOOL MATH COURSES CAN BE TRANSCRIPTED AS HONORS COURSES?
Almost two decades ago – when I wrote the book on how to use John Saxon’s Math books – I did not include a chapter on honors courses because I mistakenly thought that by then everyone knew about them. My apologies, but the subject should have been included in the book. Since it was not, I thought I would publish the subject in the monthly news articles and make it available for homeschool educators to print a copy as an addendum to the book. (Click Here for a printable copy)
I would like to say that all of John Saxon’s math books are honors courses. The contents of John’s math books are no-nonsense, straightforward, rigorous, challenging, and conceptually sound. These outstanding math books enable mastery of the concepts, not just memorization; however – I would be stretching the accepted definition of honors courses. Generally, the title of honors course when applied to math courses is reserved for the higher-level math courses that cover more material and are therefore more rigorous and challenging than regular courses.
Yes, the term honors course can be applied to non-math courses as well; however, in this article, I will restrict use of the term to just mathematics – and more specifically – to John Saxon’s math courses.
Unless your State Board of Education has created its own standards regarding who can certify an honors curricula, the classroom mathematics teacher can authorize an honors course. There are no official rules or standards that list what defines an honors course. However; the term is generally applied to high school courses considered to be more rigorous and therefore more academically challenging. With some exceptions, a student must acquire the classroom teacher’s approval to enroll in an honors course along with an overall grade average of a B or higher in prerequisite math courses.
I am a qualified state certified secondary mathematics teacher with more than twelve years’ experience teaching high school mathematics while using John Saxon’s math books from algebra through calculus. There is no doubt in my mind that the courses in John Saxon’s high school math curriculum that qualify for honors courses are the Saxon Algebra 2 (only the 2nd or 3rd Ed), Saxon Advanced Mathematics (2nd Ed. – whether taught in a single year or in three or four semesters) and the Saxon Calculus textbook (1st or 2nd Ed). Let me briefly state why each of these qualify as honors courses.
Algebra 2, 2nd or 3rd Ed. Why not the new 4th Edition? In my opinion, the new fourth edition of this book will not allow the student to satisfactorily enter the Advanced Math textbook – nor would it qualify for the title of “Honors Course.” This new edition was not created by John Saxon. It was created by a publishing company that stripped all references to geometry from the fourth edition textbook. You can read more detail about the potential problems with using this non-Saxon edition in my Nov 2019 News Article. The challenges and rigorous nature of John Saxon’s Algebra 2, 2nd or 3rd Ed. textbook have been reduced to a standard high school algebra 2 textbook in this new non-Saxon 4th Ed. version.
Now, what is it that makes the 2nd or 3rd editions of John’s Algebra 2 textbook qualify as honors courses? When using the 2nd or 3rd Ed. of John’s Algebra 2 textbook, students have 30 problems to tackle every day through all 129 lessons as well as a weekly test to determine their mastery of the material. Unlike a regular algebra 2 course, students must not only master the daily menu of some very rigorous algebra 2 concepts, but they must also master the rigorous geometry concepts found in the first semester of a high school geometry course – plus the introduction of trigonometric functions midway in the book as well.
It is acceptable to use Algebra 2 w/Geom (1 credit) on the student’s HS transcript and in an appropriate place indicate honors credit for that course. Don’t forget when a student takes an honors course, the GPA is scored differently: an A is worth 5 pts, a B is worth 4 pts, a C is worth 3 pts, and a D is 2 pts – the grade of F is still 0.
I recall at a homeschool convention several years ago, a homeschool parent told me that she was told by a homeschool friend you could not award a semester of HS geometry because there were no two-column proofs in the Saxon Algebra 2 (2nd or 3rd Ed) textbooks. My reply was “Your friend did not finish the book, he probably stopped at lesson 122 (“Venn Diagrams”), because there are more than 15 rigorous two-column proofs in the six lessons between lesson 123 and 129 (the end of the book).
As I promised my students and their parents – and I will promise you – if students get no further than successful mastery of the Saxon Algebra 2 textbook (2nd or 3rd Ed) when they graduate from high school, they will be able to pass any freshman college algebra course from MIT to Stanford – provided they attend class every day, pay attention, complete assignments, and do not sleep in class. Oh, and – one more minor requirement – show up on test days!
Advanced Mathematics, 2nd Ed: John designed this course to be taken in three, or four semesters. I taught the textbook as a four semester (2 year) course. If you would go to this link on my website, you can watch a short seven minute video on why and how you transcript the course: https://usingsaxon.com/flvplayer.html
Unless textbooks have drastically changed in the field of collegiate freshman mathematics, this textbook is tougher than any collegiate freshman algebra textbook I have seen or previewed. Students who complete the entirety of the textbook and successfully master the material presented will score in the 90th – or higher – percentile on either the ACT or SAT math score. As described in the referenced video, both of the course titles described in the four semester use of the book qualify as honors courses.
Calculus (1st or 2nd Ed.): Both calculus textbooks qualify as honors courses in a high school environment. And, while successful completion of all 117 lessons of the older 1st edition textbook prepares students for the AB portion of the College Board’s Advanced Placement (AP) program for calculus, I recommend you use the newer 2nd edition. That edition prepares students for both the AB (through lesson 102) – and the BC (all 148 lessons) portions of the College Board’s Advanced Placement (AP) program for calculus. The 2nd Ed. of John’s Calculus textbook contains 31 more lessons than the older 1st Ed.
Lastly, the new 2nd Ed. of John’s Calculus textbook has the added feature of the lesson reference numbers which appear in parenthesis under each problem number as used earlier in the third editions of Saxon’s Algebra 1 and Algebra 2 textbooks. They direct students to the lesson that introduced the concept of that problem they may need to revisit. It saves the student wasted hours of valuable time trying to find the lesson that introduced the concept without knowing the correct terminology of what it is they are looking for.
SHOULD STUDENTS TAKE CALCULUS AT HOME?
Calculus is not difficult! Students fail calculus not because the calculus is difficult – it is not – but because they never mastered the required algebraic concepts necessary for success in a calculus course. However, not everyone who is good at algebra needs to take a calculus course.
A number of the students I taught in high school never got to calculus their senior year because they could not complete the advanced mathematics textbook by the end of their junior year. They ended up finishing their senior year with the second course from the advanced math book titled “Trigonometry and Pre-calculus” and then taking calculus at the university level. This worked out just fine for them as they were more than adequately prepared and had an opportunity to share the challenge with likeminded contemporaries on campus.
Some of my students advanced no further than completing Saxon Algebra 2 by the end of their senior year in high school. They were able to take a less challenging math course their first year of college by taking the basic college freshman algebra course required for most non-engineering or non-mathematics students. These students would never have to take another math course again – unless of course they switched majors requiring a higher level of mathematics. And, if they did, they would be adequately prepared for the challenge.
I believe the answer for homeschool students in these same situations is what we in Oklahoma call “concurrent enrollment.” In other words, don’t take a calculus course at home by yourself. Under the guidelines of “concurrent” or “dual”’ enrollment – or whatever your state calls it – take the course at a local college or university and share the experience with likeminded contemporaries. If your state has such a program a high school student can also receive both high school and college credit for the course. I would not recommend taking calculus under “concurrent” or “dual” enrollment at a local community college unless you first verified that the college or university your child was going to attend will accept that level credit for the course. Many of them will accept those credits but only as electives and not as required courses in the student’s major field of studies. Check with the head of the mathematics department or the registrar’s office before you enroll in the local community college.
The concept of “concurrent” or “dual” enrollment was just beginning to take hold in the field of education when I was teaching and there were not many high school students taking these college courses enabling them to receive both a high school and college math credit for their efforts. As we gained experience with the new program, we learned that our high school juniors and seniors who had truly mastered John Saxon’s Algebra 2 course could easily enroll at the local university in the freshman college algebra course and could – provided they went to class – easily pass the course. And, if they were English or Art majors, they would never have to take another math course if they so desired.
Students who were eligible and wanted to take a calculus course their senior year looked forward to taking it at the local university and receiving “concurrent” or “dual” credit for the course. Many of these same students went on to become research technicians in the field of bio-chemistry and physics. However, several of them never took another math course in their college careers because they were English or Art History majors. They took the college freshman calculus course because they wanted to prove they could pass the course. They wanted to be able to say “I took college calculus my senior year of high school.”
So, what does all this mean? Home school students whose major will require calculus at the college level should adjust their math sequence to complete John Saxon’s advanced mathematics textbook (2nd Ed) by the end of their junior year of high school, and then take calculus the first semester of their senior year at a local college or university. Not only will this enable them to receive “concurrent” or “dual” – unless their state prohibits it – but they will enjoy the camaraderie of other likeminded college students taking the course with them.
There is a final serendipity to all of this. When enrolling at most universities, honors freshman and freshman with college credits enroll before the “masses” of other freshman students. This would virtually guarantee the student with college credits the courses and schedule they desire – not to mention the potential for scholarship offers with high ACT or SAT scores and earned college credits in a course titled “Calculus I”” recorded on their high school transcript.
Next month’s article will be about Saxon Math Honors Courses.
DOES THE STUDENT'S GRADE IN THE COURSE REFLECT THE STUDENT'S UNDERSTANDING OF THE CONCEPTS?
Some years ago I read a math teacher’s syllabus that stated how their seventh grade Saxon math class would be graded. The syllabus stated that the grading scale would be the standard 90-100 A; 80-89 B; 70-79 C; 60-69 D; 59 and below was failing. The syllabus then explained that 10% of the student’s grade would be awarded for class participation and timely submission of the daily work. Accuracy of the daily work comprised another 40% of the student’s grade, and test grades comprised the remaining 50% of the student’s overall grade.
What this means is that a student who does not understand the material, reflected by weekly test grades in the 50’s, but who has enough initiative to copy his friend’s homework paper via the telephone, email, or other means – and who then receives a daily homework grade of 100 – will receive an overall math grade of a 75 (a good solid ‘C) reflecting he understands the work – which he clearly does not! How did I arrive at that passing grade? Easy. Fifty percent of a homework grade of one hundred is 50. Fifty percent of a test grade of only fifty is 25. Adding them together, you can easily see how the student quickly calculates the critical value of the daily assignment grades.
The greatest mistake a classroom teacher or a home school educator can make in establishing a grading system for a mathematics course is to put too much weight upon the daily grade as this does not reflect mastery of the material. Teachers have little or no idea how students acquired the answers to the daily work unless they stand over the students as they do their work – which is not a recommended course of action.
The beauty of the Saxon math curriculum is the weekly tests which tell the parent or teacher how the student is progressing. The daily work is nothing more than practice for that weekly test as the 20 test questions come from the 150 questions the student encounters in the previous five days of daily work. However, unlike students using some textbooks which provide a “test review” section, the Saxon students have no idea which of the 150 problems will be on the upcoming test. The Saxon students cannot memorize the concepts they encounter. They must understand them.
Oh yes, I almost forgot. The syllabus went on to explain to the parents and students that “after every test, students will be given the opportunity to retake a similar test, after more practice, and be given full credit.” A sure way to ensure students will pass the course - whether they understood the concepts or not. Have you ever known any student to receive a lower grade on a re-take of the same test? I say re-take because the Saxon classroom test booklet has an A and a B version of each test. Both versions are identical in content except the numbers are changed resulting in different numerical answers. The two versions were designed – not for re-takes – but for make-up tests to ensure the student taking the make-up test on Monday, did not receive the answers from another student who took the test on Friday.
John Saxon’s math books are the only math books on the market today (that I am aware of) that require a weekly test to determine how well the student is progressing. That means that in a school year of about nine months, the student takes about 30 tests. My youngest grandson was in his sophomore high school math class for over eight weeks before he took his first test. He passed it with a 94, but what if he had received a 60? How do you review material covered in over two months of instruction? In a Saxon math curriculum, if the teacher or parent never looked at the student’s homework - and the student never asked for help - the teacher or parent would know on a weekly basis how the student is progressing, allowing sufficient time for review and remediation if necessary.
The two scenarios I have discussed above are what I would define as the difference between “Memorizing” and “Mastering.” Both reflect “knowledge”, but the mastery reflects what the student has placed in long term memory as opposed to what the student has memorized for the short term benefit of a good test grade. In a Saxon curriculum, the mastery enables the student to effortlessly move from middle school math (the foundation for upper level math) to the challenges of upper level algebra, trigonometry and geometry, pre-calculus and calculus should they so desire.
Grades in the Saxon curriculum (after K – 3) are based upon test scores. It is the test scores that determine mastery or acquisition of knowledge – not the daily assignment grades.
May You Have a Blessed and Happy New Year!