A SPECIAL ANNOUNCEMENT
When I undertook this challenge more than 15 years ago, I felt my mission was not to make a lot of money – but to assist homeschool educators and their children master John's math books as I had seen my high school students do. SOoooo – I want to announce – that effective the first of June – I am reducing the cost of the newly created Online Math Series to $49.95 for a single subscription. They are about ten dollars less in expenses because there are no packaging or mailing costs in the Online Math Series – and I am passing that savings on to you.
MY CONGRATULATIONS TO ALL THE 2021 GRADUATES
HAVE A GREAT SUMMER!
Before you start with this news article, I recommend you first read the April 2021 news article. John did not think much of the policies of the National Council of Teachers of Mathematics (the NCTM). On numerous occasions he would accuse them of revising the mathematics program in the United States – to the detriment of the students.
When I first heard that the state of Virginia was going to implement a new program of mathematics titled "Equity-Based Mathematics," I immediately went to the NCTM web-site and saw they had something to do with that program. Why did I go there? Because the headquarters of the NCTM is located in Reston, Virginia, and in my opinion it is no coincidence that it was the State of Virginia who caved first. A few days ago, I see where the State of California has followed suit. If you reside in either state, be thankful you homeschool.
From the NCTM website I read the following:
"Equity-based mathematics requires more than implementing new curriculum or using specific practices because it involves taking a stand for what is right. It requires mathematics teachers to reflect on their own identity, positions, and beliefs in regard to racist and sorting-based mechanisms."
Remember the Pythagorean Theorem from your algebra days? It was named after the 5th Century B.C. Greek mathematician and philosopher Pythagoras. Until the NCTM came up with this "Equity-Based Gobbly-Gook," I had no idea that Pythagoras could be called a racist because – to him – mathematics was "pure" with no cultural or political strings attached, as in the Pythagorean Equation of a2 + b2 = c2.
I had always thought that regardless of one's color, cultural background or status in life that 2+2 = 4 and that the square root of 169 is – and always will be – a positive 13. And regardless of the "Gobbly-Gook" printed above from the NCTM website some students, regardless of their skin color or cultural background, will excel in mathematics and some students, regardless of their skin color or cultural background, will struggle.
What the states of Virginia and California now want to implement is a version of the NCTM's "Gobbly-Gook" guidance that requires public school math teachers at the high school level to teach mathematics to all high school students at the struggling student's level, and let the knowledgeable math students fend for themselves. When math teachers slow the course down and teach the math to the struggling students, that will bring the rest of the class down and restrict learning. Nor – as the NCTM would have it – should we restructure our entire math program for those few unfortunate struggling math students – struggling because of any number of reasons – none of which are associated with racism.
When I started teaching high school mathematics almost a half century ago, I ran into what I would soon learn were normal circumstances in many public schools. I encountered a middle school math teacher who made the students use scientific calculators rather than teaching them how to deal with fractions, decimals, and percents. Remember, one of the reasons students fail Algebra 1 – is not because of the word algebra – but because they never learned how to deal with fractions, decimals and percents in their Middle School math classes.
So when these students arrived at their first algebra encounter in my Algebra 1 class at the high school – and found that some fractions had letters making the calculator useless – they started to fail. Every student from that teacher's 8th grade pre-algebra class – who had left that class with solid "A's" and "B's" – was now receiving "C's" and "D's on their weekly tests. Some of the parents told me they thought the Saxon Algebra 1 textbooks were too difficult to use. A few accused me of being an incompetent math teacher.
So one evening I got all of the Algebra 1 parents together. Among the parents of the students were several black families, a Marshallese mother, several Mexican families, and a Vietnamese doctor who was the father of two Vietnamese students of mine. I made them all a promise that night. I promised them that no child would fail the course unless that student gave up and stopped trying. I asked them a simple question. "Would you rather your children attempted to climb the tallest mountain and only got half way up – or would you rather they ran around in the lowlands and were always happy?"
I received their approval of the fact that it would be unfair for me to slow the course down at the expense of the students who were doing great in the class and had not been cheated by this middle grade teacher. However, I had an idea that would allow every student to experience their fullest capability without fear of failing. Recall that the middle school subjects or grades do not have anything to do with the HS transcript – so a student who received a final grade of a low "C"- or lower - in my Algebra 1 class, would receive a grade of "B" in a lesser inclusive math course titled either "Pre-algebra" or "Intro to Algebra 1" – and I would have them back in class next year for a very successful Algebra 1 course.
I assured the parents that night that this same concept would allow me to enable a struggling student to take two years to complete the Saxon Algebra 2 material as well. If necessary, year one would be transcripted as "Intro to Algebra 2" – and year two as "Algebra 2." I explained to the parents that it was the cumulative nature of John Saxon's math books that allow this to occur without harming the student.
The flexibility of the Saxon Math system also allows for several corrections to occur before the student reaches the high school level. However, failing in high school math started long before they got to the high school. It started in the fifth and sixth grades where well meaning, but unqualified teachers started watering down the students learning abilities in mathematics by trying to get everyone a passing grade – trying to keep the parents happy. This non-racist phenomenon passes on into the high school classroom as well.
The gifted math student is not slowed down by an unqualified math instructor at this point. But it definitely affects the average math student who really needs a knowledgeable math teacher to improve their level of understanding whenever possible. There are a multitude of young men and women of all colors, creeds, and social backgrounds who enter high school with the capability to successfully pass an Algebra 2 course as a high school freshman – and it is not always done just so they can take a high school calculus course as the NCTM would like us to believe.
I trust that those who read this article fully understand that the flexibility within John Saxon's math series from 6th grade math (Math 76) through the upper level high school math courses – affords their child the best ability to succeed to any level they desire. It also affords them the ability to meet the state requirements for graduation from high school, and to go and pursue the field of learning they desire at any university from MIT to Stanford – regardless of their color, creed, or social background.
As a final note, when I first arrived at the high school, the average ACT math scores were pitifully below the national average – and less than half of the students were enrolled in upper level math classes. A couple of years after implementing John Saxon's math books, our ACT math scores exceeded the national average. This would hold true for many more years as well – and – we reached and maintained a minimum of 90 percent of our freshmen, sophomores, juniors and seniors enrolled in the more advanced Saxon math classes as well.
THE COLLEGE LEVEL EXAMINATION PROGRAM (CLEP)
Over the years, I have had parents ask about the advantages of having their child take a test under the College Level Examination Program (CLEP) - or as some of my students would say "CLEP out of a Course." For those not familiar with the program, the 90-minute CLEP tests are administered by The College Board at any of their more than 1800 test centers or at one of the 2900 colleges or universities that accept them.
The College Board states it is a not-for-profit membership association whose mission is to connect students to college success and opportunity. It was founded in 1900, and the association has a membership of more than 5,600 schools, colleges, universities and other educational organizations. Each year, the College Board serves more than seven million students through major programs and services in college readiness, college admissions, guidance, assessment, financial aid, enrollment, and teaching and learning.
While most homeschool educators are more familiar with the College Board's SAT® and AP® programs, their CLEP Program can also save students considerable course fees if they can pass the appropriate tests. For a fee of $80.00 per course, students can take CLEP tests in any of the more than 33 subjects in the areas of Literature, Foreign Languages, History and Social Studies, Science and Mathematics, and Business.
One word of caution - the College Board advises students that:
"Before you take a CLEP exam, learn about your college's CLEP policy. Most colleges and universities grant credit for CLEP exams, but not all. There are 2,900 institutions that grant credit for CLEP and each of them sets its own CLEP policy; in other words, each institution determines for which exams credit is awarded, the scores required and how much credit will be granted. Therefore, before you take a CLEP exam, check directly with the college or university you plan to attend to make sure that it grants credit for CLEP and review the specifics of its policy."
Not every university or college may accept every College Board CLEP test score, and not all have the same scoring levels for credit. For example, while one university may award three credit hours for a score of 55 on the college algebra CLEP test, another may require a higher score, while still a third university may not accept the College Board CLEP results for that particular test at all. It may require that students take their individual university CLEP test for a particular subject.
In the area of mathematics, parents also need to know what levels of high school math courses correspond to what level CLEP test. For example, the student who takes the college algebra CLEP test before mastering John Saxon's Algebra 2 course will, in all likelihood receive a failing grade. Each of the CLEP math tests indicate the subject matter included in the tests. Following the math book's index will give you a pretty good idea of whether or not the student can handle that particular test.
I will say this about John Saxon's second edition of Advanced Mathematics. All students who have mastered the first ninety lessons in that book should easily pass the College Board's CLEP test for College Algebra and College Mathematics. If they have mastered the entire Advanced Mathematics book and also finished the first 25 lessons of calculus, they can easily pass not only those same two course tests, but the College Board pre-calculus CLEP test as well.
I recall that when I was teaching in the high school, one of my calculus students went down to the OU campus and took the calculus CLEP test and passed it. While in my senior calculus class, he was happy with just a "C" because he was going to study "Communications" at OU and openly admitted that he did not really need the math. He never took another math course in his life. When I asked him why he did not just take the college algebra CLEP test, he smiled and said, "I just wanted to be able to tell people that I had passed college calculus at OU."
The College Board tests are a great way to get a few basic courses out of the way and save mounting college tuition costs, but if the students are going into engineering or research science, I would recommend they not use the CLEP tests to replace core courses in their field. They need to revisit these courses at the collegiate level.
NEXT MONTH'S ARTICLE WILL DISCUSS THE NEW "EQUITY-BASED MATHEMATICS" STARTING TO CREEP INTO OUR PUBLIC SCHOOL EDUCATIONAL SYSTEM.
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!