WHAT DETERMINES THE DIFFERENCE BETWEEN MASTERY AND MEMORY?
Think back to your days in high school and your math classes. Do you recall having your math teacher hand out a review sheet a few days before the big test? So what did you do with this review sheet? Right! You memorized it knowing that most of the questions would appear on the test in some form or other. We are the only industrialized nation in the world that I know of where parents proudly announce “Oh, I was never very good at math.” Not hard to explain considering you probably memorized the material for a passing test grade, and then after the test was over, quickly forgot the material.
I still see students in the local public school receiving a passing math grade using the “review” sheet technique, even though their test grades never get above a sixty. How can this happen? Easy! The student’s grades are based upon a grading system that ensures success even though the student cannot pass a single test (unless you consider a sixty a passing grade). Many students’ overall average grades are computed based upon fifty percent of their grade coming from the homework (easily copied by them) and another fifty percent determined from their test scores (following the review sheet). So the student who receives hundreds on the daily homework grades and fifties or sixties on the tests is cruising along with an overall grade average of a high “C” or a low “B.” Yet, that student cannot explain half of the material in the book.
I have often explained to parents of students who were struggling in my math classes that their struggle was akin to the honey bee struggling its way through the wax seal of the comb. It is that struggle that strengthens the bee’s wings and enables it to immediately fly upon its exit from the hive. Cut the wax away for the young bee and it will die because its wings are too weak to allow it to fly. Yes, there is a difference between struggling and frustration! The home educator as well as the classroom teacher must be ever vigilant to recognize the difference.
While we all would like the student to master the new concept on the day it is introduced, that does not always happen. Not every math student completely understands every math concept on the day it is introduced. It is because of this that John Saxon developed his incremental approach to mathematics. When John’s incremental development is coupled with a constant review of these concepts, “mastery” occurs.
Mastery occurs through a process referred to by Dr. Benjamin Bloom as “automaticity.” The term was coined by Dr. Bloom, of “Bloom’s Taxonomy,” while at the University of Chicago in the mid 1950’s. He described this phenomenon as the ability of the human mind to accomplish two things simultaneously so long as one of them was over-learned (or mastered). The two critical components for mastery are repetition over time.
Automaticity is another way to describe the placing of information or data into long term memory. The process requires that its two components—repetition over time—be used simultaneously. It is this process in John Saxon’s math books that creates the proper atmosphere for mastery of the math concepts. Violating either one of the two components negates the process. In other words, you cannot speed up the process by taking two lessons a day or doing just the odd or even numbered problems in each lesson.
Trying to take shortcuts with mathematics would be like trying to save meal preparation time every day. Why not just eat all the meals on weekends and save the valuable time spent preparing meals Monday through Friday. Just as your body will not permit this “short-cut,” your mind will not allow mastery of material squeezed into a short time frame for the sake of speeding up the process by reducing the amount of time spent on the individual math concepts.
In a single school year of nine months, the student using John Saxon’s math books will have taken more than twenty-five weekly tests. Since all the tests are cumulative in content, passing these tests with a minimum grade of “80” reflects “mastery” of the required concepts - not just memory!
While a student may periodically struggle with an individual test or two throughout the entire range of the tests, it is not their test “average” that tells how prepared they are for the next level math course, nor is it the individual test scores (good or bad) they received on the early tests that matter. What is important are the individual test scores the student receives on the last five tests in the course. It is these last five test scores that reflect whether or not the student is ready for the next level math course. Students who receive individual test scores of 80 or higher—first time tested—on their last five tests in any of John Saxon’s math books are well prepared for success in the next level math course.
“May you have a very Merry and Blessed Christmas -
and a Prosperous and Happy New Year!”
THIS YEAR AT THE 11th HOUR OF THE 11th DAY OF THE 11th MONTH THE WORLD WILL CELEBRATE THE 100th YEAR ANNIVERSARY OF THE SIGNING OF THE ARMISTICE OF WWI. THIS ARTICLE IS NOT ABOUT MATH – IT IS ABOUT A VETERAN OF THAT WAR – A DOUGHBOY – MY FATHER!
Each year on the 11th of November our country – along with France and England – celebrates Veteran’s Day. This is the day our nation has set aside to recognize military veterans of all branches of service for the sacrifices they have made throughout our country’s history, sacrifices that have ensured our continued freedom. This day of recognition in November of each year originated from the date of the signing of the armistice at the end of World War I. The armistice was signed in a railroad car in the forest near the French village of Compiegne. The document was signed exactly at the eleventh hour of the eleventh day in the eleventh month in the year 1918. There are no more living veterans of WWI, but if you know or meet a veteran of any armed conflict from WWII to the Gulf War, Iraq or Afghanistan, and you get the chance this coming Veteran’s Day, shake their hand and thank them for their service – and tell them “Welcome Home!”
Private John William Reed, Infantryman
Company F, 358th Infantry
90th Infantry Division
(Wounded at St. Mihiel, France on September 12, 1918)
My father was twenty-two years old when he received his induction notice from the local draft board in Minneapolis, Minnesota on April 22, 1918 (Order # 651, Serial # 356, Division #4). He was ordered to report to the draft board one week later on April 29, 1918 for immediate induction into the United States Army. Immediately after his induction, he was shipped to Camp Davis, Texas for training and deployment with the 358th Infantry Regiment of the 90th Infantry Division. In less than two months, he would be on a troop-ship headed overseas for the War in France. In less than five months from the day he was inducted, he would find himself in battle near the small French village of St. Mihiel.
The 90th Infantry Division was activated on August 25, 1917 at Camp Travis, Texas. It was nicknamed the “Alamo Division” and sometimes referred to by the enlisted men as the “Tough Ombres” (for Texas and Oklahoma). Initial members of the 90th Division came from Texas and Oklahoma; however, just before the division deployed to France in the summer of 1918, it received a large number of new recruits from other states like Minnesota. The division began its embarkation from Hoboken, New Jersey in early June of 1918, and by June 30th all of the units of the 90th Infantry Division had sailed from Hoboken. The division initially landed in England where, on July 4th, 1918, the 358th Infantry (including my father) paraded before the Lord Mayor of Liverpool. That evening, the entire 358th Infantry was hosted at a banquet given by the city of Liverpool, England.
The 358th Infantry arrived in France shortly thereafter and was stationed at Minot, France. In early September, the unit was moved about 192 km NE to a small village east of Paris in the northeast part of France. The name of the village was “Villers – en –Haye.” It had a population then of only 96 people. In 2007, the population of “Villers – en – Haye” was still only 167).
Their first engagement with the German army came on September 12, 1918, at a town called St. Mihiel. The town was much larger than “Villers – en – Haye” having a population in 1918 of slightly more than 2000 residents. It was located 42 km from “Villers – en- Haye” on the edge of the Meuse River. The town had grown around a Benedictine abbey founded in 709 A.D. At the time of the battle, there were still several Abbey buildings in the town constructed in the 17th and 18th century. The town church had a door that dated back to Roman times. Both the church and the Abby buildings are still there today, undamaged by the fierce fighting that occurred there more than 100 years ago. In 2008, the population of St. Mihiel had increased to 4,816.
The World War I battle that took place at St. Mihiel on September 12 - 14, 1918, was the first major American military offensive of the war. The campaign against the German fortifications at St. Mihiel involved more than 55,000 men of the U.S. First Army commanded by Gen. John J. “Black Jack” Pershing. The 90th Infantry Division (including the 358th Infantry Regiment) was part of that force. The three-day campaign - led by the U.S. First Army - was successful. The American Army forced the Germans to relinquish a military fortification held by the Germans since 1914.
In those first three days of battle in mid-September of 1918, the 90th Infantry Division suffered a total of 37 officers and 1,042 enlisted men killed in action and another 266 officers and 8,022 enlisted men wounded and mustard gassed during the battle with the German units. In just three days, the division had lost more than half of its men! Private John William Reed, Company F, 358th Infantry, was among those wounded and mustard gassed by the Germans that first day of battle, on September 12, 1918. Today, more than 4,150 of the American soldiers, killed in that September offensive, are buried in the American Military Cemetery at St. Mihiel.
Now - for the “Rest of the Story . . . !”
More than half a century later, while I was stationed with the U.S. Army in Heidelberg, Germany, my wife and I were visiting the nearby town of Schwetzingen, Germany - located several kilometers from Heidelberg. My wife wanted to visit the world famous historical doll maker Ilse Ludecke. While she visited with the doll maker, I practiced my German by conversing with Ilse’s older sister. After I mentioned that my father had fought in France during World War I, she smiled and commented that I was too young to have a father who was in the First World War. "Mein Vater diente im Ersten Weltkrieg" - “My father served in the First World War,” she said. "Sie sind gerade ein Baby. Sie sind zu jung, um einen Vater zu haben, der in diesem Krieg war." - “You are just a baby. You are too young to have a father who was in that war.”
I then told her that my father had fought near Verdun at St. Mihiel, France in September of 1918 and that he was wounded and mustard gassed by the opposing German forces in that battle. She stared at me and momentarily looked somewhat confused, and then she excused herself and went upstairs, returning shortly clutching a scroll. She handed me the scroll and asked me to read it. As I unrolled the scroll and began reading it (mentally translating the German words into English), I could not believe what I was reading. It was a certificate addressed to Oberst (Colonel) Ludecke, Kommandant (Commander) of the 81st Chemical Brigade for a special mission against the American 90th Infantry Division in September of 1918. It was signed by Kaiser Wilhelm II, and dated in 1918.
Without thinking, I turned to her and said “Your father killed my father!” She turned pale and appeared weak-kneed. I quickly put my arm around her shoulders and, realizing the ramifications of what I had just blurted out, I said to her “But he knew enough to marry my mother who was German.” I then told her that my mother’s parents were born in the small town of Mohringen just on the outskirts of Stuttgart. She looked at me and laughed. "Sie sind nicht deutsch, Sie sind Swaibish" - “You are not German, you are Swaibish,” she said. It should be noted that the Swaibish are known as a hard headed (or bull headed) clan of Germans living in the Stuttgart area of Germany.
She said something to her sister Ilse and they laughed about the “Swaibish” revelation. Then the two of them invited my wife and me to accompany them upstairs to their home above the store. I learned later that day when speaking with one of the neighbors that Ilse Ludecke and her sister had never before invited Americans upstairs to their home. As we came up the stairway and entered the large living room, I noticed there were paintings of military officers lining the walls. Judging by the uniforms worn by each of the men in the paintings, most of them dated back before World War I. The older sister pointed to the painting of her father and grandfather as well as one of her great-grandfather telling me that all were once officers in the Prussian Army. She explained that when the American soldiers came through their town during WWII, she and her sister would take the military paintings down and hide them in the closet. When the American soldiers left, they would return the paintings to the wall.
Frau Ludecke walked over to a closet behind a beautiful ornate wood burning stove and returned with a small brown cardboard box. She opened the box and showed me a large piece of shrapnel from a WWI mustard gas shell. The shell fragment was about nine inches in length. She explained that her father did not want to be in the military, that he always wanted to be an artist. He had brought home this painting he had made depicting a battle scene near Verdun. Painted on the side of this large piece of shrapnel was a scene from one of the small French villages that her father’s unit had shelled. She explained that while the mustard gas had eventually killed my father from his wounds on the battlefield that day in France, her father also died of cancer just a few short years after returning from the war.
She believed her father’s cancer had developed from him mixing the chemicals and handling the mustard gas mortar rounds just as sure as she believed those mustard gas shells that her father had fired upon the American soldiers during the St. Mihiel campaign had caused them to later die of cancer as well. We talked for awhile longer and as we left, Ilse’s sister gave me a hug and whispered in my ear, "Ihr Vater machte eine kluge Wahl, welcher feiner Sohn, den er hat." – “Your father made a wise choice, what a fine son he has.”
Two weeks later, my family and I left Germany for stateside, and several months later the handmade dolls my wife had ordered arrived at our home. I thought one of the doll boxes was a bit heavy for just the doll and after opening the box and removing the doll, I noticed a second small brown cardboard box at the bottom. Upon opening the box, I noticed the note on top. It read "Besser haben Sie das als wir" – “Better you have this than us.” Inside was the piece of shrapnel she had showed me that day. It was the one her father had picked up on the battlefield and upon which he had painted a portrait of the French village he had shelled and where my father was wounded that September day in 1918.
Here is a photo of that mustard gas shell fragment painted by Frau Ludecke’s father:
I‹----------------------------- 8.5 inches ------------------------------›I
Below is a photo of my father taken just after he was released from a military hospital in December of 1918. He was medically discharged from the U.S. Army in January of 1919. He spent the next several decades going from one VA hospital to another, courageously fighting against the debilitating effects of the cancer caused by the mustard gas. Dad died in 1945 at the Hines VA Hospital, located just outside Chicago in Hines, Illinois – I was just nine years old!
Private John William Reed - January 1919
How Much Time Should Students Spend on Math Each Day?
One of several arguments advanced by home school educators regarding the efficiency of the Saxon math curriculum is that from Math 54 through Advanced Mathematics the courses require too much time to complete the daily assignment of thirty problems. Their solution to this often takes one of two approaches. Either they allow the student to take shortcuts to reduce the time spent on daily assignments, or they find another math curriculum that takes less time â you know â youâve heard them say, âWe found another math curriculum that is more fun, easier, and it does not require so much time.â
In several news articles published earlier this year, I addressed some of the ramifications of taking these shortcuts when using John Saxonâs math books. In these articles, I described in detail the effects upon students who used some or all of them, so I will not go over them again here. I would ask you to read those two newsletters if you have not already done so. What I want to discuss here is what may be causing the excessive amount of time taken by the students and also, what constitutes excessive time to an educator who taught in a public classroom using Saxon math books for over a decade.
While I was teaching high school mathematics in a rural Oklahoma high school, I would often go and watch my students who were on the the high school track, basketball, or football teams during their practice sessions after school. I was able to chat with the mothers and fathers who were also watching these practices. This one-on-one conversation often gave me an insight into their priorities regarding their childrenâs education.
While they sometimes complained about the rigors of my math classes, they never once complained about the length of time their sons and daughters were out on the field in the heat or cold - or on the basketball court â practicing â after just spending six academic hours in the classroom. In fact, when coaches were forced to cancel a practice for one reason or another, some of the parents would vocally complain that the practices should continue. They expressed concern that skipping practice would take the âedgeâ off their son or daughterâs playing ability and inhibit their athletic âsharpnessâ for the next game.
Strange then that some parents would complain the 45 â 60 minutes spent each day on mathematics âpracticeâ would be excessive - and more unusual - that they would seek an easier course of action. They never discussed the ramifications that doing so might take the âedgeâ off their childâs math âsharpnessâ for the next math course or the state mandated math test. I never heard the high school parents complain about watching the tough daily drills and practices run by the coaches. I never heard a parent complain about the hour spent each day by the students diligently practicing their piano lessons, or having to come in before school early each day to spend 45 minutes in the weight room.
At least several times each week I receive email from home school parents who express concern that their son or daughter was taking an inordinate amount of time on their daily math assignment in one of the books from Math 54 through Advanced Mathematics.
The solution to the excessive time spent by students using the Advanced Mathematics textbook is easy to resolve. The solution to that unique situation is explained in a short seven-minute video clip (Click Here to view that video).
I have interacted with several thousand parents and students in the twelve years that I taught mathematics at that rural high school. I have also advised thousands more home school educators and home school students in the succeeding decades after my retirement while serving as one of the Homeschool Curriculum Advisors (for Math 76 through Calculus and Physics) for Saxon Publishers and later for Harcourt-Achieve who bought the company from Johnâs children. And while every child and home school situation is different, my experiences have shown me that there exist several situations that contribute to excessive time spent on daily work by students, whether home schooled or attending a public or private classroom.
These situations are:
The Student is in The Wrong Level Math Course: If after lesson thirty in any Saxon math book, students continue to receive 80% on the weekly tests of twenty questions, within a maximum of fifty minutes with no partial credit (all right or all wrong) and no calculator (until Algebra 1), then they are in the correct level Saxon math book. If the test scores are constantly below that or if they fall below a 70-75 on their first five or so tests, then that is a good indication they are in the wrong level Saxon math book. This situation can result from any one or more of the following conditions:
â˘ They did not finish the previous Saxon math book.
â˘ They took shortcuts in the preceding math book.
â˘ Their previous math book was not a Saxon math book.
â˘ They did not take the weekly tests in the previous math book. Instead, they used the student's daily grades.
â˘ The studentâs last five tests in the preceding course were well below 80% (minimal mastery).
The Student is Required to Re-do Math Problems from Yesterdayâs Lesson: Why do we want students to get 100% on their daily practice for the weekly test? When we grade their daily work and have them go over the ones they missed on the previous dayâs assignment, nothing is accomplished except to âacademically harassâ the students. The daily work reflects nothing but the status of the studentsâ temporary learning curve. It is the weekly tests and not the daily work that reveal what the student has mastered from the previous weeks and months of work. Not every student masters every concept the day it is introduced, which is why there is a four to five day delay from when the concept is introduced to when it is tested. In the twelve years that I taught John Saxonâs math books in high school, I did not grade one homework paper â but I did grade the weekly tests which reflected what the students had mastered as opposed to their daily work which did not.
Remember, John Saxonâs math books are the only books I am aware of that use weekly tests to evaluate a studentâs progress. There are a minimum of twenty or more weekly tests in every one of Johnâs math books from Math 54 on.
Too Much Time is Spent on The Warm Up Box: From Math 54 through Math 87, there is what used to be called a âWarm Upâ box at the top of the first page of every lesson. I recall watching a sixth grade teacher waste almost thirty minutes of class time while three boys took turns giving different opinions as to how the âProblem of The Dayâ was to be solved â and arguing as to which had the better approach. After class, I reminded the teacher that the original purpose of the box was to get the students settled down and âfocusedâ on math right after the second bell rang. I said to her, âWhy not immediately review a couple of the problems from yesterdayâs lesson at the start of class for the few who perhaps did not grasp the concept yesterday? Then move immediately to the new lesson.â This process would take about 10 to 20 minutes and would leave students with about 40 minutes of remaining class time to work on their new homework assignment.
NOTE: In any of John Saxonâs math books from Math 54 through Algebra 2, the âAâ and âBâ students will get
their 30 problems done in less than 40-50 minutes. The âCâ students will require more than an hour.
The Student is Required to Do All of The Daily Practice Problems: The daily practice problems were created for teachers to use on the blackboard when teaching the lessonâs concept so they did not have to create their own or use the homework problems for demonstrating that concept. Many of the lessons from Math 54 through Math 87 have as many as six or more such problems and if the student understands the concept, they are not necessary. If the student has not yet grasped the concept, having the student do six or more additional practice problems of the same concept will only further frustrate him. Remember, not every student grasps every concept on the day it is introduced. The five minutes spent on review each day is essential to many students.
The Student is a Dawdler or a Dreamer: There is nothing wrong with being a âDreamer,â but some students just look for something to keep them from doing what they should be doing. I call these students âDawdlers.â I recall the first year I taught. I had to constantly tell some students in every class to stop gazing out the window at the cattle grazing in the field outside our classroom â and get on their homework. That summer, I replaced the clear glass window and frame with a frosted glass block window - and in the following eleven years I had absolutely no problem with my âDawdlers.â
The Student is Slowed by Distractions: Is the student working on the daily assignment in a room filled with activity and younger siblings who are creating all sorts of distractions? Even the strongest math student will be distracted by excessive noise or by constantly being interrupted by younger siblings seeking attention. Did you leave the student alone in his room only to find he was on his cell phone talking or texting with friends or listening to the radio? Or worse, does he have a television or computer in his room and does he use the computer to search the internet for a solution to his math problems - or engage in something equally distracting by watching the television?
Please do not misinterpret what I have discussed here. If you desire to do all of the above and the student takes two hours to complete a daily assignmentâand both you and the student are satisfiedâthen that is acceptable. But if you are using this excessive time as an excuse for your childâs frustrationâand as an argument against John Saxonâs textbooksâI would remind you of what John once told a school district that did everything John had asked them not to do. They were now blaming Johnâs math books for their districtâs low math test scores.
John told them âIf you want to continue your current practices, get rid of my books and buy someone elseâs book to blame.â
Just What is the Difference Between Math 87 and Algebra Â˝?
There appear to be varying explanations regarding whether a student should use Math 87 or Algebra Â˝ after completing the third or fourth edition of John Saxonâs Math 76 textbook. Let me see if I can shed some light on the best way to determine which one, or when both, should be used.
First, whenever I discuss the Math 76 textbook in this article, I am talking about the third or fourth editions of that book. I am not talking about the old first or second editions of Johnâs Math 76 books. These two older editions have been out of print for almost twenty years now and their content, while acceptable at the time, would not now enable a math student to proceed successfully through either the second or third editions of Math 87 or the newer third edition of Algebra Â˝.
Second, whenever I discuss using the Math 87 textbook, I am talking about either the second or third editions of that book and not the older first edition which has also been out of print for more than a decade. Let me assure you that, except for the new soft cover, the addition of a solutions manual, and the varied numbering of the pages, there is absolutely no difference between the 120 lessons and the 12 Investigations of the hard cover second edition of Math 87 and the new soft cover third edition of Math 87. Oh yes, the new third edition has added a TOPIC A at the end of the book (after Investigation 12) dealing with Roman Numerals and Base 2. And even though the marketing folks have added the word âPrealgebraâ under the cover title of the soft cover third edition textbook, these two additional topics, while interesting and nice to know, are not pre-algebra material and will not create any shortfall in either Algebra 1 or even later in Algebra 2.
Both the Math 87 and Algebra Â˝ courses prepare the student for the Algebra 1 course. The main difference between using the Math 87 and the Algebra Â˝ books depends upon the studentâs success in the Math 76 textbook. The Math 87 book starts out a little slower than the Algebra Â˝ book does because it assumes the student needs the additional review resulting from the student encountering difficulty in the latter half of the Math 76, textbook. Also, If you were to open to any lesson in the Algebra Â˝ textbook, you would immediately notice that the âWarm-Upâ box common in Math 54 through Math 87, is noticeably absent in the Algebra Â˝ textbooks.
If there exists a math savvy student of John Saxonâs Math 76 textbook, who received test grades of 80-85 or better on the last five tests in Math 76 3rd or 4th Ed (50 â 55 minute test period, no calculator, and no partial credit), then that student would be more challenged and, from my teaching experience, much better off in the Algebra Â˝ book. However, if his last five test scores are below 80, then from my experiences, that indicates that student should proceed to Math 87, and upon completion of that book, if his or her last five tests are now 80 or better (minimal mastery), then that student can easily skip Algebra Â˝ and go on to success in Algebra 1.
If however, a student encounters difficulty going through both Math 76 and Math 87, then proceeding through the Algebra Â˝ textbook (The Saxon Pre-Algebra textbook) before attempting Algebra 1, will allow the student to regain his confidence. Doing so will further ensure the student has mastered a solid conceptual base for success in any algebra one course.
Students will fail an Algebra 1 course if they have not mastered fractions, decimals and percents which are emphasized before the student reaches that course. I realize that not every child fits neatly into a specific mold, but John Saxonâs Math 76, Math 87, and Algebra Â˝ series allow the Saxon home school educator sufficient flexibility to satisfy every studentâs needs and to ensure the studentsâ success in any algebra one course.
I realize that not every student is alike. Should you therefore encounter a situation not clearly described in this news article, please feel free to email or call and I will do my best to assist you in resolving the issue facing your child. If you have not yet read the July 2018 news article, now would be a good time to read (or re-read) that article also.
THAT OLD âGEOMETRY BEARâ KEEPS RAISING HIS UGLY HEAD
Home School Educators frequently ask me about students taking a non-Saxon geometry course between algebra 1 and algebra 2, as most public schools do. They also ask if they should buy the new geometry textbook recently released to homeschool educators by HMHCO (the new owners of Saxon). As I mentioned in a previous newsletter earlier this year, a group of professors who taught mathematics and science at the University of Chicago bemoaned the fact that educators continued to place a geometry course between basic algebra (Algebra 1) and the advanced algebra course (Algebra 2) to the detriment of the student - AND THIS WAS MORE THAN 110 YEARS AGO!
The danger of using a separate geometry textbook as described by these professors more than a hundred years ago - still exists today! Placing a nine month geometry course between the Algebra 1 and Algebra 2 courses creates a void of some fifteen months between the two algebra courses. How did I arrive at fifteen months? In addition to the nine month geometry course, you must also add the additional six months of summer between the two algebra courses when no math is taken. The professors went on to explain in their book that it was this âlengthy voidâ that prevented most students from retaining the necessary basic algebra concepts from the basic algebra (Algebra 1) to be successful when encountering the rigors of the advanced algebra (Algebra 2) concepts.
Home school educators also asked about using the new fourth editions of Saxon Algebra 1 and Algebra 2 recently released by HMHCO (the new Saxon owners) together with their new separate geometry textbook now offered for homeschool use. To create the new fourth editions of both the Algebra 1 and Algebra 2 textbooks, all the geometry was gutted from the previous third editions of both Algebra 1 and Algebra 2. Using the new fourth editions of their revised Saxon Algebra 1 and Algebra 2 now requires also purchasing their new Saxon Geometry book to receive any credit for geometry. That makes sense, if you consider that publishers make more money from selling three books than they do from selling just two. Regardless of which editions you finally choose to use, I would add a word of caution. If you intend to use Johnâs Advanced Mathematics, 2nd Ed textbook, do not use the new fourth editions of Algebra 1 or Algebra 2.
So what Saxon math books should you use? The editions of John Saxonâs math books from fourth through twelfth grades that should be used today appear on page 15 of my book â or you can take a look at the end of my June 2018 news article on this website. These editions remain the best math books on the market today, and they will remain so for decades to come.
If you desire more information about the pros and cons of using a separate Geometry textbook, please read my May 2018 news article. Should you still have questions or reservations, please feel free to either email me at email@example.com or call my office any week-day at 580-234-0064 (CST).
WHICH BOOK SHOULD BE USED AFTER MATH 76 â MATH 87 or ALGEBRA Â˝?
HOW CAN STUDENTS OVERCOME THEIR DIFFICULTY WITH ALGEBRA?
When John Saxon published his original series of math textbooks, they were designed to be taken in order from Math 54 to Math 65, followed by Math 76, then Math 87, then Algebra Â˝, on to Algebra 1, then Algebra 2, followed by Advanced Mathematics (which, coupled with Algebra 2, gave the high school geometry and trigonometry credits) culminating with the calculus textbook for some students.
The books were not originally intended to be âgradeâ oriented textbooks, but were intended to be taken in sequential order based upon a studentâs knowledge and capabilities without regard to the studentâs grade level. But schools and homeschool educators quickly assigned Math 54 to the fourth grade level, Math 65 to the fifth grade level, Math 76 to the sixth grade, and Math 87 to the seventh grade level to be followed by the pre-algebra course titled Algebra 1/2. When the new third edition of Math 76 came out in the summer of 1997, it was much stronger academically than its predecessor, the older second edition textbook. It did not take long for confusion to develop around which textbooks were now the correct editions to be used and what the correct sequencing would be.
In the thousands of telephone calls I received over the years I served as Saxon Publishersâ Curriculum Director for Math 76 through calculus, the question that arose most often among classroom teachers as well as homeschool educators was whether the student should go from the new stronger Math 76 book to Math 87 or to Algebra Â˝ as both the Math 87 and the Algebra Â˝ textbooks appeared to contain basically the same material. Adding to the confusion, after John Saxonâs death, was the fact that the new soft cover third edition of Math 87 had the title changed to read Math 8/7 âwith pre-algebra.â
So what Saxon math book does a student who has completed Math 76 use? Well, that depends upon how well the student did in the Math 76 book. The key word is âsuccessfully completed,â not just âcompletedâ Math 76. If a student completed the entirety of the Math 76 textbook and his last five tests in that book were eighty or better, he would have âsuccessfully completedâ Math 76 and he could move on to the Algebra Â˝ book. However, if the studentâs last five test grades were all less than seventy-five, that student has indicated that he will in all likelihood experience difficulty in the Algebra Â˝ materials and should therefore proceed first through the Math 87 textbook.
While both the Math 87 and the Algebra Â˝ textbooks prepare the student for any Algebra 1 course, the Math 87 book starts off a bit slower with more review, allowing the student to âcatch up.â The student who then moves successfully through the Math 87 textbook, receiving eighties or better on the last five tests, can then skip the Algebra Â˝ book and move directly to an Algebra 1 textbook.
However, if the student finishes the Math 87 book and the last five test grades reflect difficulty with the material, that student should then be moved into the Algebra Â˝ book to receive another â but different â look at âpre-algebraâ before attempting the Algebra 1 course. Students fail algebra because they do not understand fractions, decimals and percents; they fail calculus because they do not understand the basics of algebra. Attempting to âfast trackâ a student who had weak Math 76 test scores - into Algebra Â˝ - then on to Algebra 1, will most certainly result in frustration if not failure in either Algebra Â˝, or Algebra 1 â or cause the homeschool educator to seek an âeasierâ math curriculum..
So what have we been talking about? If students have to take all three courses (Math 76, Math 87 and Algebra Â˝), how will they ever get through algebra? When I taught Saxon math in a public high school, we established three math tracks for the students. Fast, Average, and Slower math tracks to accommodate the difference in learning styles and backgrounds of the students. Listed below are the recommended three math tracks. Please note there are no grade levels associated with these courses, but Math 76 was generally taught in the 6th grade at the middle school.
The course titled âIntroduction to Algebra 2â was the studentâs first attempt at the Algebra 2 course which resulted in low test scores, so the course was titled as an âIntroduction to Algebra 2â and the student repeated the entirety of the same book the following year.
Over ninety-five percent of all these students received an âAâ or âBâ their second year through the Algebra 2 course. In the ten years we used the system, I only had one student who received a âDâ in the course and he did so because he did little or no studying the second year and still passed the course with a 65 percent test average.
I will make you the same promise I made to the parents of my former students. If students can accomplish no more than âmasteringâ John Saxonâs Algebra 2 course by the time they are seniors in high school, they will pass any collegiate freshman algebra course from MIT to Stanford (provided they go to class). Remember, they can still take calculus at the university if they have changed their mind and need the course in their new major field of study. And because they now have a strong algebra background, they will be successful!
FAST MATH TRACK: Math 76 â Algebra Â˝ â Algebra 1 â Algebra 2 â Geometry with Advanced Algebra â Trigonometry and Pre-Calculus â Calculus. NOTE: The Saxon Advanced Mathematics textbook was used over a two year period allowing the above underlined two full math credits after completing Saxon algebra 2. (TOTAL Possible High School Math Credits: 4)
AVERAGE MATH TRACK: Math 76 â Math 87 â Algebra Â˝ â Algebra 1 â Algebra 2 â Geometry with Advanced Algebra â Trigonometry and Pre-Calculus. (TOTAL Possible High School Math Credits: 4)
SLOWER MATH TRACK: Math 76 â Math 87 â Algebra Â˝ â Algebra 1 â Introduction to Algebra 2 â Algebra 2 â Geometry with Advanced Algebra. (TOTAL Possible High School Math Credits: 4)
NOTE 1: YOU SHOULD USE THE FOLLOWING EDITIONS AS THEY ARE ACADEMICALLY STRONGER THAN THE EARLIER EDITIONS ARE. USING OLDER EDITIONS WILL RESULT IN FRUSTRATION OR FAILURE FOR THE STUDENT.
Math 76: Either the hardback 3rd Ed or the new soft cover 4th Ed. (Math content of both editions is the same)
Math 87: Either the hardback 2nd Ed or the new soft cover 3rd Ed. (Math content of both editions is the same)
Algebra Â˝: Use only the 3rd Edition. (Book has the concept lesson reference numbers added also)
Algebra 1: Use only the 3rd Edition. (Book has the concept lesson reference numbers added also)
Algebra 2: Use the 2nd or 3rd Editions. (Content is same- Concept lesson reference numbers added to 3rd Ed)
Advanced Mathematics: Use only the 2nd Edition: (Concept lesson reference numbers will be found in the
Calculus: Either the 1st or 2nd Edition will work. However, if the student needs DVD tutorial assistance, then they
will need the 2nd Edition textbook.
NOTE 2: WHEN RECORDING COURSE TITLES ON THE TRANSCRIPT, USE THE FOLLOWING TITLES:
Math 76: Use âSixth Grade Math.â
Math 87: Use âPre-Algebra.â(If student must also take Algebra Â˝, then use âSeventh Grade Mathâ)
Algebra Â˝: Use âPre-Algebra.â
Advanced Mathematics: Use âGeometry with Advanced Algebra (1 credit) if they only complete the First 60 â 70 lessons of that textbook. Add âTrigonometry and Pre-calculusâ (1 credit) if they successfully complete the entirety of the Advanced mathematics textbook. Under no circumstances should you record âAdvanced Mathematicsâ on the studentâs high school transcript as colleges and universities will not know what math this course contains. They will ask you for a syllabus for the course.
Each child is unique and what works for one will not always work for another. Whatever track you use, you must decide early to allow students sufficient time to overcome any hurdles they might encounter in their math journey. If you have any questions, please feel free to email me at firstname.lastname@example.org or call me at my office any weekday (580) 234-0064 (CST).
JOHN WAS RIGHT! â SOME THINGS HAVENâT CHANGED - EVEN AFTER MORE THAN A HUNDRED YEARS!
Homeschool educators are constantly faced with the dilemma of deciding whether or not their son or daughter needs to take a separate high school geometry course because some academic institution wants to see geometry on the high school transcript. Or, because the publishers offer it as a separate math textbook in their curriculum â implying it is to be taken as a separate course. Remembering, of course, that selling three different math textbooks books brings in thirty-three percent more revenue than selling just two.
John Saxonâs unique methodology of combining algebra in the geometric plane and geometry in the algebraic plane all in the same math textbook had solved that dilemma facing home school educators for these past twenty-five years. However, unknown to John, this same problem had been addressed over a hundred years earlier at the University of Chicago.
Knowledge of this information came to me by way of a gift from my wife and her sisters. Since 2003, after their mom and dad had passed away, my wife and her sisters have been going through some fifty years of papers and books accumulated by their parents and stored in the attic and basement of the house they all grew up in. When asked by friends why it was taking them so long, one of the daughters replied âMom and Dad took more than a half century to fill the house with their memories. It won't hurt to take a couple more years to go through them.â
Among some of the treasures they found in the basement were letters to their great-grandfather written by a fellow soldier while both were on active duty in the Union Army. One of these letters was written to their great-grandfather while his friend was assigned to âPicket Dutyâ on the âPicket Line.â His fellow Union Soldier and friend was describing to his friend (their great-grandfather) the dreary rainy day he was experiencing. He wrote that he thought it was much more dangerous being on âPicket Dutyâ than being on the front lines, as the âRebelsâ were always sneaking up and shooting at them from out of nowhere.
The treasure they found for me was an old math book that their father had used while a sophomore in high school in 1917. The book is titled âGeometric Exercises for Algebraic Solution â Second Year Mathematics for Secondary Schools.â It was published by the University of Chicago Press in October of 1907.
The authors of the book were professors of mathematics and astronomy at the University of Chicago, and they addressed the problem facing high school students in their era. Students who had just barely grasped the concepts of the algebra 1 text, only to be thrown into a non-algebraic geometry textbook and then, a year or more later being asked to grasp the more complicated concepts of an algebra 2 textbook. The book they had written contained algebraic concepts combined with geometry. It was designed as a supplement to a geometry textbook so the students would continue to use algebraic concepts and not forget them.
John never mentioned these authors â or the book â so I can only assume he never knew it existed. For if he had, I feel certain that it would have been one more shining light for him to shine in the faces of the high-minded academicians that he â as did these authors â thought were wreaking havoc with mathematics in the secondary schools.
In the preface of their textbook, the professors wrote:
âThe reasons against the plan in common vogue in secondary schools of breaking the continuity of algebra by dropping it for a whole year after barely starting it, are numerous and strong . . . With no other subject of the curriculum does a loss of continuity and connectiveness work so great a havoc as with mathematics . . . To attain high educational results from any body of mathematical truths, once grasped, it is profoundly important that subsequent work be so planned and executed as to lead the learner to see their value and to feel their power through manifold uses.â
So, should you blame the publishers for publishing a separate geometry textbook? Or is it the fault of misguided high-minded academicians who â after more than a hundred years â 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 Saxonâs math books for the original Homeschool third editions of John Saxonâs algebra one and algebra two textbooks still contain geometry as well as algebra â as does the advanced mathematics textbook.
Any home school student using John Saxonâs Homeschool math textbooks who successfully completes Algebra one, (2nd or 3rd editions), Algebra 2, (2nd or 3rd editions), and at least the first half of the Advanced Mathematics (2nd edition) textbook, has covered the same material found in any high school algebra one, algebra two and geometry math textbook â including two-column formal proofs. Their high school transcripts â as I point out in my book â can accurately reflect completion of an algebra one, algebra two, and a separate geometry course.
When home school educators tell me they are confused because the school website offers different materials than what is offered to them on the Homeschool website, I remind them that - unless they want to purchase a hardback version of their soft back textbook - they do not need anything being offered on the Saxon school website. In fact, they are getting a better curriculum by staying on the Homeschool website. You can still purchase the original versions of John Saxonâs math textbooks that he intended be used to develop âmasteryâ as recommended by the University of Chicago mathematics professors over a hundred years ago.
Because many of you do not have a copy of my book, I have reproduced that list from page 15 of the book so you can see what editions of John Saxonâs original math books are still good whether acquired used or new. These editions will easily remain excellent math textbooks for several more decades.
Math 54 - hard cover second edition or the new soft cover third edition.
Math 65 - hard cover second edition or the new soft cover third edition.
Math 76 - hard cover third edition or the new soft cover fourth edition.
Math 87 - hard cover second edition or the new soft cover third edition.
Algebra Â˝ - hard cover third edition.
Algebra 1 - hard cover third edition.
Algebra 2 - hard cover second or third editions (content is identical).
Advanced Mathematics - hard cover second edition.
Calculus - hard cover first or second edition.
Physics - hard cover first edition (there is no second edition of this book).
IS THERE ANY VALUE TO USING A SEPARATE GEOMETRY TEXTBOOK?
Have you ever seen an automobile mechanicâs tool chest? Unless things have changed, auto mechanics do not have three or four separate tool chests. They have one tool chest that contains numerous file drawers separating the tools necessary to accomplish their daily repair work. But the key is that all of these tools are in a single tool chest.
Con-gru-ent â adj. 1) In agreement or harmony: . . 2) Geometry (of figures): identical in form: Coinciding exactly when superimposed. . .
What if the auto mechanic purchased several tool chests thinking to simplify things by neatly separating the specific types of tools from each other into separate tool chests rather than in separate drawers in the one tool chest? Each separate tool chest would then contain a series of complete but distinctly different tools. If mechanics did this, there would now exist the possibility that they would find themselves trying to remember which tool chest contained which tools â and â the extra tool chests would cost them more! It is somewhat like that in mathematics. Each division of mathematics has its strengths and weaknesses and like the auto mechanic who selects the best tool for a specific job, so the physicist, engineer, or mathematician selects the best math procedure to meet the needs of what they are doing.
But how do we address the argument that geometry provides a distinct and essential thought process unlike that used in algebra? The advent of computers has provided educators with an alternative course titled Computer Programming. A computer programming course teaches students the same methodology or thought process that the two-column proofs of geometry do. Basically, it teaches the student that he cannot go through a door until he has opened it â meaning â the student must use valid statements that are logically and correctly placed to reach a valid conclusion and to prove that conclusion valid by having the computer program work correctly.
Before computers, educators in the United States felt that providing the separate geometry course would benefit those students interested in literature and the arts, who enjoyed the challenge of geometry without the burden of algebra, while still allowing students entering the fields of science and engineering, who had to take more math, to take the course also. When I was in high school, most geometry teachers taught only geometry, they never taught an algebra course â as I soon learned!
I recall encountering that little known fact when I took high school geometry from one such teacher. I was sharply rebuked early in the school year when I kept using the term âequalââ to describe two triangles that had identical measures of sides and angles. The first time I said the two triangles that contained identical angles and sides were âequal,â she told me I was wrong. She then proceeded to tell me and the class that the only correct term to describe two identical triangles was the term âcongruent.â She did not say my answer was technically correct, but that in the geometry class, we used the term âcongruentâ rather than âequalâ â she specifically pointed out that I was âwrong.â
The next day in geometry class, I really got in trouble when I stood and read Websterâs definition of the word congruent.
Just before I was told to go to the office and tell the principal that I was being rude, I asked her why the two triangles could not also be said to be equal since they had identical angles and sides and were equal in size. Then I drove the final nail in my coffin when I proceeded to read Websterâs definition of âequal.â
Equal â adj. 1) Being the same in quantity or size . . .
Half a century later, when I taught both the algebra and geometry concepts simultaneously while using John Saxonâs math books at a rural high school, I made it clear to the students that while they should become familiar with the terminology of the subject, they were free to interchange terms as long as they were correctly applied. I also made it clear to them that the object of learning geometry and algebra was to challenge and expand their thought processes and for them to understand the strengths and weaknesses of each and apply whichever math tool best served the problem being considered.
While many tout the separate geometry textbook as necessary to enable a child to concentrate on a single subject rather than attempting to process both geometry and algebra simultaneously, I would ask them how a young geometry student can solve for an unknown side in a particular triangle without some basic knowledge of algebraic equations. In other words, if you are going to use a separate geometry textbook, it cannot be used by a student who has not yet learned how to manipulate algebraic equations. This means that a separate geometry book is best introduced after the student has successfully completed an algebra 1 course.
For most students this means placing the separate geometry course between the Algebra 1 and Algebra 2 courses, creating a gap of some fifteen months between them. (Two summers off, plus the nine month geometry course). As was true in my high school days, this situation creates a problem for the vast majority of high school students who enter the Algebra 2 course having forgotten much of what they had learned in the Algebra 1 course fifteen months earlier.
So what am I getting at? Must we have a separate geometry textbook for students who cannot handle the geometry and algebra concurrently? Well, let me ask you, if students can successfully study a foreign language while also taking an English course or successfully study a computer programming course while also taking an algebra course, why canât they study algebra and geometry at the same time, as John Saxon designed it?
Successful completion of John Saxonâs Algebra 2, (2nd or 3rd editions) not only gives the student a full yearsâ credit for the Algebra 2 course, but it also incorporates the equivalent of the first semester of a regular high school geometry course. I said âSuccessful Completionâ for several reasons. FIRST:
The student has to pass the course and SECOND:
The student has to complete all 129 lessons.
Whenever I hear home school educators make the comment that âJohn Saxonâs Algebra 2 book does not have any two-column proofs,â I immediately know that they stopped before reaching lesson 124 of the book which is where two-column proofs are introduced. The last six lessons of the Algebra 2 textbook (2nd or 3rd editions) contain thirty-one problems dealing with two-column proofs. The following year, in the first half of the Advanced Mathematics textbook, they not only encounter some heavy duty algebra concepts, but they also complete the equivalent of the second semester of a regular high school geometry course. The first thirty of these sixty lessons contain more than forty problems dealing with two-column proofs.
So why then did John Saxon not want to publish a separate geometry textbook? As I mentioned in one of my newsletters several years ago, a group of professors who taught mathematics and science at the University of Chicago bemoaned the fact that educators continued to place a geometry course between basic algebra (Algebra 1) one and the advanced algebra course (Algebra 2) to the detriment of the student. AND THIS WAS MORE THAN 110 YEARS AGO!
In the preface to their book titled âGeometric Exercises for Algebraic Solution,â the professors explained that it is this lengthy âvoidâ that prevents students from retaining the necessary basic algebra concepts learned in basic algebra to be successful when encountering the rigors of Advanced Algebra.
We remain one of the only â if not the only â industrialized nations that have separate math textbooks for each individual math subject. When foreign exchange students arrive at our high schools, they come with a single mathematics book that contains geometry, algebra, trigonometry, and when appropriate, calculus as well. Is it any wonder why we are falling towards the bottom of the list in math and science?
When students take a separate geometry course without having gradually been introduced to its unique terminology and concepts, they encounter more difficulty than do students using John Saxonâs math books The beauty of using Johnâs math books, from Math 76 through Algebra 1 is that students receive a gradual introduction to the geometry terminology and concepts.
If you are going to use John Saxonâs math books through Advanced Mathematics or Calculus you do not need a separate geometry book. This means you must use the third editions of Johnâs Algebra 1 and Algebra 2 books, because HMHCO has stripped all of the geometry from the new fourth editions of their versions of Algebra 1 and Algebra 2. And you do not want a student to go from the fourth edition of Algebra 2 to the Saxon Advanced Mathematics textbook.
NOTE: Please Click Here to watch a short video on how to receive credits for Geometry, Trigonometry and Pre-calculus using John Saxonâs Advanced Mathematics textbook â and how to record them on the studentâs transcript.
HOW TO SUCCESSFULLY USE JOHN SAXON'S MATH BOOKS FROM MATH 54 THROUGH CALCULUS
AND PHYSICS (PART 3)
Here is the final series describing situations I have encountered for more than three decades while teaching Saxon in a rural high school as well as providing curriculum advice to homeschool educators for John Saxon. As with the previous two parts of the series, I have added my thoughts about why you want to avoid them:
1) ATTEMPTING THE ADVANCED MATHEMATICS TEXTBOOK IN A SINGLE YEAR: Since there are only 125 lessons in the textbook, it seems reasonable to assume this is possible.
RATIONALE: âMy son had absolutely no trouble in the Algebra 2 book and I believe he will have no trouble in this book either. The book has fewer lessons than the Algebra 2 book has. Besides, he is a junior this year and we want him to be in calculus before he graduates from high school.â
FACTS: The second edition of John Saxonâs advanced mathematics textbook is tougher than any college algebra textbook I have ever encountered. The daily assignments in this book are not impossible, but they are time consuming and can take most math students more than several hours each evening to complete the thirty problems. This generally results in students doing just doing the odd or even numbered problems to get through the lessons. I must have said this a thousand times âCalculus is easy; students fail calculus because they do not understand the algebra.â Speeding through the advanced mathematics textbook by taking shortcuts does not allow the student the ability to master the advanced concepts of algebra and trigonometry to be successful in calculus. And if the only argument is that the student will not take calculus in high school, then what is the rush?
The DVD tutorial series for the second edition of Johnâs Advanced Mathematics book that I have prepared allows students three different choices based upon their needs and capabilities.
A) They can follow my advice and take the course in two years (doing a lesson every two days). Thereby gaining credit for the first academic year of âGeometry w/Advanced Algebra,â with a first semester credit for Trigonometry and a second semester credit for Pre-calculus in their second academic year.
B) They can take the course in three semesters. Their first semester credit would be titled Geometry, followed by a second semester credit for Trigonometry w/Advanced Algebra; ending with a third semester credit for Pre-calculus.
C) While not recommended â they can take the entire 125 lessons in the Advanced Mathematics book in a single school year gaining credit for a full year of Geometry along with a semester credit for Trigonometry w/Advanced Algebra. In all the years that I taught the subject, I only had one student complete the entire Advanced Math course of 125 lessons in a single school year â with a test average above ninety percent - and she was a National Merit Scholar whose father taught mathematics with me at the local university.
The specific details of how the transcript is recorded are covered in my book, but if you have any questions regarding your son or daughterâs high school transcript, please feel free to send me an email.
2) IS IT CRITICAL FOR STUDENTS TO TAKE CALCULUS IN HIGH SCHOOL?
Students lacking a solid base in algebra and a basic knowledge of trigonometry will find taking calculus at any level difficult, if not impossible.
RATIONALE: âI want our son to take calculus his senior year in high school. The only way we can accomplish that is to have him speed through the Saxon Algebra 2 and Advanced Mathematics book to finish them by the end of his junior year. He may even have to use the summer months for math as well.â
Even if students successfully complete a calculus course their senior year in high school, whether at home or at a local community college, I would strongly recommend that they enroll in âCalc Iâ as a freshman at the university or college they choose to attend for several reasons.
If they truly understand enough of their Calculus I course (usually encompassing derivatives) they can enjoy a solid five hours of âAâ on their transcript for their first five hours of math as a freshman. They can also make some nice extra money tutoring their less fortunate classmates
While they think they understand everything there is about calculus, they will see much more as they sit back and âunderstandâ what the professor is talking about. They might even learn something they never fathomed in the high school textbook they went through.
Their solid âAâ the first semester in a calculus I class lets the professors know what kind of student they are. That perception by the professor makes a big difference should they encounter difficulties later in their second semester of calculus II (usually through integrals). Finishing John Saxonâs second edition of Advanced Mathematics at a pace that allows the student to grasp all of the material in that textbook, without being frustrated or discouraged, is paramount to their success in calculus at the university or college level.
3) DO HIGH SCHOOL STUDENTS NEED A SEPARATE GEOMETRY TEXTBOOK? To reflect that a student has received a well rounded math background, states that require three or more math courses require that geometry be recorded on a studentâs high school transcript, along with algebra 1, algebra 2, trigonometry, etc.
RATIONALE: “It is too difficult for high school students to learn both algebra and geometry at the same time. My son did just fine in the Saxon Algebra 1 textbook. However, he is only on lesson 35 in the Saxon Algebra 2 book, and he is already struggling.”– or their rationale is - "I have been told by other home school parents that there are no two-column proofs in John Saxon’s Algebra 2 textbook.”
FACTS: Many of my top studentsâ worst test in the Saxon Algebra 2 course was their very first test. This happened because they did not realize the book covered so much geometry review from the algebra 1 text, as well as several key new concepts taught early in the Algebra 2 text. They quickly recovered and went on to master both the algebra and the geometry concepts. From my experiences, most students who encountered difficulty early in John Saxonâs Algebra 2 textbook did so - not because they did not understand the geometry being introduced - but because their previous experiences with the Saxon Algebra 1 course did not result in mastery of the math concepts necessary to handle the more complicated algebra concepts introduced early in the Algebra 2 textbook. I would not recommend students attempt John Saxonâs Algebra 2 math book if they have done any one or more of the following:
1) Never finished all of the lessons in the Saxon Algebra 1 textbook.
2) Hurried through the Saxon Algebra 1 textbook doing two lessons a day and then only did the odd or even numbered problems from each lesson.
3) Received multiple test scores of less than seventy-five on their last four or five tests in the Algebra 1 textbook (not counting partial credit).
What about the students who never took the tests, because parents used the studentsâ daily homework grades to determine their grade average? What does that reveal about the studentsâ ability? Establishing a studentsâ grade average based upon their daily work reflects what they have âmemorized.â The weekly tests determine what they have âmastered.â
The successful completion of John Saxonâs Algebra 2 textbook (2nd or 3rd Editions) gives students an additional equivalent of the first semester of a high school geometry course (including two-column proofs). Successful completion of the first sixty lessons of the Saxon Advanced Mathematics textbook (2nd Ed) ensures they receive the equivalent of the second semester of high school geometry, in addition to the advanced algebra and trigonometry concepts they also receive in the latter half of the book.
But what about the lack of two-column proofs in the Saxon Algebra 2 book (2nd or 3rd Ed)? Whenever I hear Homeschool Educators make the comment that "John Saxon's Algebra 2 book does not have any two-column proofs,â I immediately know they stopped before reaching lesson 124 of the book which is where two-column proofs are introduced. The last six lessons of the Saxon Algebra 2 textbook (2nd or 3rd editions) contain thirty-one different problems dealing with two-column proofs. The following year, in the first half of the Advanced Mathematics textbook, they not only encounter some heavy duty algebra concepts, but they will also complete the equivalent of the second semester of a regular high school geometry course. The first thirty of these sixty lessons contain more than forty different problems dealing with two-column proofs.
So why then did John Saxon not want to publish a separate geometry textbook? As I mentioned in a previous newsletter, a group of professors who taught mathematics and science at the University of Chicago bemoaned the fact that educators continued to place a geometry course between basic algebra (Algebra 1) and the advanced algebra course (Algebra 2) to the detriment of the student. AND THEY PUBLISHED THEIR BOOK MORE THAN 110 YEARS AGO!
In the preface to their book titled "Geometric Exercises for Algebraic Solution," published in 1907, the professors explained that it is this lengthy "void" between the two algebra courses that prevents students from retaining the necessary basic algebra concepts learned in basic algebra (algebra 1) to be successful when encountering the rigors of advanced algebra (algebra 2).
Then apparently aware of this situation, and knowing John Saxonâs position on the subject, why did HMHCO (the current owners of Johnâs books) go ahead and create and publish the new fourth editions of Saxon Algebra 1, Algebra 2, and a separate first edition Saxon Geometry textbook? I do not know why they did, but I do know that three textbooks will make more money for a publisher than two textbooks will. I also know that the new books â while initially sold only to the schools on the companyâs school website, are now offered to Homeschool Educators as well. Having to decide between the two different editions of algebra makes the selection process more confusing; however, I would not recommend any student go from the fourth edition of Saxon Algebra 2 to John Saxonâs Advanced Mathematics textbook.
If you stick with the editions of John Saxonâs math books that I listed in my Sep 2016 Newsletter, you will have the best math books on the market today good for several more decades to come.
As I mentioned previously, there will always be exceptions that justify the rule. However, just because one parent tells you their child did any one or all of the above - and had no trouble with the Advanced Math course - does not mean you should also attempt it with your child. Those parents might not have told you that one or more of the following occurred.
1) Their children encountered extreme difficulty when they reached Saxon Algebra 2, and even more difficulty and frustration or even failure with the Saxon Advanced Mathematics or Calculus courses.
2) They had switched curriculum after experiencing difficulty in Saxon Algebra 1.
3) Their son or daughter had to take remedial (no credit) college algebra because that received low scores on the universities mathematics entrance exam.
For those readers who do not have a copy of my book, please donât forget to take a minute and read that Sep 2016 news article for information that will help you select the correct level and edition of John Saxon's math books. As I said earlier, these editions will remain excellent math textbooks for many more decades.
If your child is already experiencing difficulty in one of the Saxon series math books, and you need to find a workable solution, please email me at: email@example.com. Or feel free to call me at (580) 234-0064 (CST).
HOW TO SUCCESSFULLY USE JOHN SAXON'S MATH BOOKS FROM MATH 54 THROUGH CALCULUS
AND PHYSICS (PART 2)
As I promised last month here are several more of the common misuses I have encountered during the past three decades of teaching and providing curriculum advice to homeschool educators. I have added my thoughts about why you want to avoid them:
1) THE EFFECTS OF DOING JUST THE ODD OR EVEN PROBLEMS: Allowing the student to do just the odd or even problems in each daily lesson may appear to save time, but it creates a false sense of mastery of the concepts.
RATIONALE: âEach lesson shows two of each of the different problems, and it saves us valuable time by doing just one of the pair. Besides, since they both cover the same concept, why take the extra time doing both of them?â
FACT: The reason there are pairs of each of the fifteen or so concepts found in the daily assignments is because each of the problems in each pair is different from the other. While both problems in each pair address the same concept, they are different in their approach to presenting that concept. One goes about presenting the concept one way while the second one approaches the concept from a totally different perspective. Doing both of them gives the student a broader basis for understanding the concept and prevents the student from memorizing a particular procedure rather than mastering the concept based upon solving the two different formats or procedures.
Whenever I receive an email from a homeschool educator or student, and they need help with solving a particular problem on one of the tests remarking that they never saw this test question in any of their daily work, I can tell that they have been doing either the âoddsâ or the âevensâ in their daily work because this test question resembled an approach to the concept that was contained in the set they never did. Additionally, doing only half of the daily assignment restricts the studentâs ability to more quickly and easily master the concepts. Doing two a day for fourteen days increases the studentâs ability to more quickly master those concepts than doing just one a day for that same period of time.
The âAâ or âBâ student who has mastered the material should take no more than fifty minutes to complete the daily assignment of thirty problems if their grade is based upon their weekly test scores and not upon their daily homework. The âCâ student should complete the daily assignment of thirty problems in about ninety minutes. The additional time above the normal fifty minutes is usually the result of the âCâ student having to look up formulas or concepts that might not have yet been mastered. This is why I recommend using âformula cards.â
Use of the formula cards saves students many hours of time flipping through the book looking for a formula to make sure they have it correctly recorded. The details on how to implement using these cards is explained in detail on page 94 of my book. If you have not yet acquired that book, you can find information on how to make and use them in my January 2018 Newsletter.
2) THE EFFECTS OF DOING MORE THAN ONE LESSON A DAY:
Permitting the students to do two or three lessons a day believing this will allow them to complete the course faster.
RATIONALE: âMy son wants to finish the Saxon Calculus course by the end of his junior year. The only way he can do that is to finish the Algebra 2 book in six rather than nine months. Besides, he told me that he already knows how to do most of the material from the previous Algebra 1 book.â
To those who feel it necessary to âspeedâ through a Saxon math book, I would use the analogy of eating oneâs daily meals. Why not just eat once or twice a week to save time preparing and eating three meals each day? Not to mention the time saved doing all those dishes. The best way I know to answer both of these questions is to remind the reader that our bodies will not allow us to implement such a time saving methodology any more than our brains will allow us to absorb the new math concepts by doing multiple lessons at one sitting.
I have heard just about every reason to support doing multiple lessons, skipping tests to allow another lesson to be taken, or taking a lesson on a test day. All of these processes were attempted solely to speed up completing the textbook. Students who failed calculus did so, not because they did not understand the language and concepts of calculus, but because they did not sufficiently master the algebra.
Why must students always be doing something they do not know? What is wrong with students doing something they are familiar with to allow mastery as well as confidence to take over? Why should they become frustrated with their current material because they ârushedâ through the previous prerequisite math course?
The two components of âautomaticityâ are time and repetition and violating either one of them in an attempt to speed through the textbook (any math book) results in frustration or failure as the student progresses through the higher levels of mathematics. I recall my college calculus professor filling the blackboard with a calculus problem and at the end, he struck the board with the chalk, turned and said âAnd the rest is just algebra.â To the dismay of the vast majority of students in the classroom - that was the part they did not understand and could not perform. When I took calculus in college, more than half of my class dropped out of their first semester of calculus within weeks of starting the course, because their algebra backgrounds were weak.
3) ENTERING THE SAXON MATH CURRICULUM AFTER MATH 76:
Switching to Saxon Algebra 1 or Algebra 2 because you have found the curriculum you were previously using was not preparing your child for the ACT or SAT and you wanted them to be more challenged.
âWe were having trouble with math because the curriculum we were using, while excellent in the lower grades, did not adequately prepare our son and daughter for the more advanced math concepts. We needed a stronger more challenging math curriculum, so we switched to Saxon algebra 1.â
Switching math curriculums is always a dangerous process because each math curriculum attempts to bring different math concepts into their curriculum at different levels. Constantly moving from one math curriculum to another - looking for the perfect math book - creates âmathematical holesâ in the studentsâ math background. It also creates a higher level of frustration for these students because, rather than concentrating on learning the mathematics, they must concentrate on what the new textbookâs system of presentation is and spend valuable time trying to analyze the new format, method of presentation, test schedule, etc.
If you intend to use Saxon in the middle and upper level math courses because of its excellence at these levels of mathematics, I would strongly recommend that you start with the Math 76, 3rd or 4th Ed textbook. The cumulative nature of the Saxon Math textbooks requires a solid background in the basics of fractions, decimals and percentages. All of these basics, together with the necessary prerequisites for success in pre-algebra or algebra 1 are covered in Saxonâs Math 76, 3rd or 4th Edition textbook. This math textbook is what I refer to as the âHINGE TEXTBOOKâ in the Saxon math curriculum. Successful completion of this book will take care of any âMath Holesâ that might have developed from the math curriculum you were using in grades K â 5.
Successful completion of this book can allow the student to move successfully to the Saxon algebra Â˝ textbook (a pre-algebra course). Should students encounter difficulty in the latter part of the Math 76 text, they can move to the Saxon Math 87, 2nd or 3rd Ed and, upon successful completion of that book, move either to the Algebra Â˝ or the Algebra 1 course depending on how strong their last 4 or 5 test scores were. Yes, some students have been successful entering the Saxon curriculum at either the Algebra 1 or the Algebra 2 levels, but the number of failures because of weak math backgrounds from using other curriculums, roughly exceeds the number of successes by hundreds!
As I mentioned last month, there will always be exceptions that justify the rule. However, just because one parent tells you their child did any one or all of the above, and had no trouble with their advanced math course, does not mean you should also attempt it with your child. That parent might not have told you that:
(1) Their child encountered extreme difficulty when they reached Saxon Algebra 2, and even more difficulty and frustration or failure with the Saxon Advanced Mathematics course, or â
(2) They had switched curriculum after experiencing difficulty in Saxon Algebra 1, or â
(3) Their child had to take remedial college algebra when they enrolled at a university, because they had received a low score on the universityâs math entrance exam.
For those readers who do not have a copy of my book, please read my September 2016 news article for information that will help you select the correct level and edition of John Saxon's math books. These editions will remain excellent math textbooks for several more decades.
If your child is already experiencing difficulty in one of the Saxon series math books, and you need to find a workable solution, please email me at: firstname.lastname@example.org. Or feel free to call me at (580) 234-0064 (CST).
In next month's issue, I will cover:
1) ATTEMPTING THE ADVANCED MATH TEXTBOOK IN A SINGLE YEAR:
2) IS IT CRITICAL FOR STUDENTS TO TAKE CALCULUS IN HIGH SCHOOL?
3) DO HIGH SCHOOL STUDENTS NEED A SEPARATE GEOMETRY TEXTBOOK?
HOW TO SUCCESSFULLY USE JOHN SAXONâS MATH BOOKS FROM MATH 54 THROUGH CALCULUS
AND PHYSICS (Part 1)
Both homeschool educators as well as public and private school administrators often ask me âWhy do John Saxonâs math books require special handling? Another question I am also frequently asked is âIf John Saxonâs math books require special instructions to use them successfully, why would we want to use themâ? Before the end of this newsletter, I hope to be able to answer both of these questions to your satisfaction.
There is nothing âmagicâ about John Saxonâs math books. They were published as a series of math textbooks to be taken sequentially. Math 54 followed by Math 65, and then Math 76, followed by either Math 87 or Algebra Â˝, and then algebra 1, etc. While other publishers were âdumbing-downâ the content of their new math books, John Saxon was publishing his new editions with stronger, more challenging content. Homeschool families, attempting to save money by buying older used Saxon Math books and inter-mingling them with the newer editions were unaware that the older out-of-print editions were often incompatible with these newer, more challenging editions. The same problem developed in the public and private school sector adding to the confusion about the difficulty of Johnâs math books.
For example, a student using the old first or second edition of Math 76 would experience a great deal of difficulty entering the newer second or third editions of Math 87 because the content in the outdated first or second editions of Math 76 was about the same as that of the material covered in the newer editions of Math 65 (the book following Math 54 and preceding Math 76). Jumping from the outdated older edition of Math 76 to the newer editions of either Math 87 or Algebra Â˝ would ultimately result in frustration or even failure for most, if not all, of the students who attempted this.
Many homeschool educators and administrators were also unaware that â when finishing a Saxon math book, they were not to use the Saxon placement test to determine the studentâs next book in the Saxon series. The Saxon placement test was designed to assist in initially placing non-Saxon math students into the correct entry level Saxon math book. The test was not designed to show parents what the student already knew, it was designed to find out what the student did not know. Students taking the placement test, who are already using a Saxon math book, receive unusually high âfalseâ placement test scores. These test results may erroneously recommend a book one or even two levels higher than the level book being used by the student (e.g. from their current Math 65 textbook to the Math 87 textbook â skipping the Math 76 textbook).
By far, the problems homeschool educators as well as classroom teachers encounter using â or shall I say misusing â Johnâs math books are not all that difficult to correct. However, when these âshort-cutsâ are taken, the resulting repercussions are not at first easily noticed. Later in the course, when the student begins to encounter difficulty with their daily assignments â in any level of Saxon math books, the parent or teacher assumes the student is unable to handle the work and determines that the student is not learning because the book is too difficult for the student.
Here are some of the most common misuses I have encountered literally hundreds of times during these past twenty years of teaching and providing curriculum advice to home school educators:
1) NOT FINISHING THE ENTIRETY OF THE TEXTBOOK: Not requiring the student to finish the entirety of one book before moving on to the next book in the sequence.
RATIONALE:âBut the beginning of the new book covers the same material as that in the last lessons of the book we just finished, so why repeat itâ?
FACT: The student encounters some review of this material in the next book, but this review assumes the student has already encountered the simpler version in the previous text. The review concepts in the new book are more challenging than the introductory oneâs they skipped in the previous book. This does not initially appear to create a problem until the student gets to about lesson thirty or so in the book, and by then both the parent and the student have gotten so far into the new book that they do not attribute the studentâs problem to be the result of not finishing the previous textbook. They start to think the material is too difficult to process correctly and do not see the error of their having skipped the last twenty to thirty or so lessons in the previous book. They now fault the excessive difficulty of the current textbook as the reason the student is failing.
Always finish the entirety of every Saxon math textbook! Because all students are not alike, if as youâre reading this article you have already encountered this particular phenomenon with your child, there are several steps you can take to satisfactorily solve the problem without harming the childâs progress or self-esteem. So that we can find the correct solution, please email me and include your telephone number and I will call you that same day â on my dime!
2) MISUSE OF THE SAXON PLACEMENT TEST: Skipping one of the books in the sequence (e.g. going from Math 54 to Math 76) because the âSaxon Placement Testâ results clearly showed the student could easily handle the Math 76 material.
RATIONALE: âHe even got some of the Math 87 level questions correct. Besides, we had him look at the material in the Math 65 book and he said that he already knew that material, so why bother doing the same concepts again.â
FACT: First, as I wrote earlier, the Saxon Placement Test was designed to place non-Saxon math students into the correct level math book. It was designed to see what the child had not encountered or mastered, not what he already knew. Saxon students who take the Saxon placement test receive unusually high âfalseâ test scores. The only way to determine if the student is ready for the next math book is to evaluate their last four or five tests in their current Saxon math book to determine whether or not they have mastered the required concepts to be successful in the next level book. The brain of young students cannot decipher the difference between recognizing something and being able to provide solutions to the problems dealing with those concepts. So when they thumb through a book and say âI know how to do thisâ what they really mean is âI recognize this.â Recognition of a concept or process does not reflect mastery.
3) USING DAILY HOMEWORK TO DETERMINE A STUDENTâS GRADE: Skipping the weekly tests and using the studentâs daily assignments to determine their grade for the course reflects memory rather than mastery of the material.
RATIONALE: I cannot count the number of times I have been told by a parent âHe does not test well, so I use the daily assignment grades to determine his course grade. He knows what he is doing because he gets ninetyâs or hundreds on his daily work.â
FACT: Just like practicing the piano, violin, or soccer, the student is not under the same pressure as when they have to perform in a restricted time frame for a musical solo or a big game. The weekly tests determine what a student has mastered through daily practice. The daily homework only reflects what they have temporarily memorized as they have access to information in the book not available on tests. Answers are provided for the odd numbered problems and some students quickly learn to âback-peddle.â This phenomenon occurs when the student looks at a problem and does not have the foggiest idea of how to work the problem. So they go to the answers and after seeing the answer to that particular problem, suddenly recall how to solve the problem. However, later, when they take the test, there are no answers to look up preventing them from âback-peddlingâ through to the correct solution.
As with anything, there are always exceptions that justify the rule. However, just because one parent says their child did any one or all of the above, and had no trouble with their math, does not mean you should also attempt it with your child. That parent might not have told you that (1) their child encountered extreme difficulty when they reached Saxon Algebra 2, and even more difficulty with the Saxon Advanced Mathematics textbook, or (2) they had switched curriculum after experiencing difficulty in Saxon Algebra 1, or (3) their child had to take remedial college algebra when they enrolled at the university because they had received a low score on the universityâs math entrance exam.
If your child is already experiencing trouble in one of the Saxon series math books, and you need to find a workable solution, please email me at: email@example.com.
1) THE EFFECTS OF DOING JUST THE ODD OR EVEN PROBLEMS:
2) THE EFFECTS OF DOING MORE THAN ONE LESSON A DAY:
3) ENTERING THE SAXON MATH CURRICULUM AFTER MATH 76:
WHAT ARE FORMULA CARDS? WHAT ARE THEY USED FOR? AND WHERE CAN I GET THEM?
AREA OF A SECTOR
Having been repeatedly threatened by my high school math teachers that I would be doomed to fail their tests if I did not memorize all those math formulas, I was somewhat surprised later in a college calculus course when the professor handed out âformula cardsâ containing over ninety geometry, trigonometry identities, and calculus formulas. He explained that they could be used on his tests. He did not bat an eye as he handed them out and reminded us that selecting the correct formulas and knowing how and when to use them was far more important than trying to memorize them or write them on the desk top.
So, when I started teaching at the high school, I announced to the students that they could make âformula cardsâ by using 5 x 8 inch cards, lined on one side and plain on the other. It never failed. Immediately, one of the students would ask why I did not have them printed off and handed out, saving them a considerable amount of time and money creating their own.
I told my students that when they encountered a formula in their textbooks, writing it down would strengthen the connection more than if they just read it and tried to recall it later while working a problem. Reading the formula in the textbook was their first encounter and there would not yet be a strong connection between what they were reading and what they tried to remember. However, when they took the time to create a formula card for that particular formula, they would then be strengthening that connection. As they used the card when doing their daily assignments, they would continue the process and eventually place the formula in their long term memory.
So, how can you get formula cards? Simple! Each student makes his own. I allowed my students to use them starting with Math 87 or Algebra 1/2. One young lady in my Algebra 2 class used blue cards for geometry formulas and white cards for the algebra formulas to save her time looking through the cards. The cards should be destroyed at the end of the course, requiring the next student to make his own. Then how do you make formula cards?
Have the students use 5 x 8 cards â and write or print clearly and big. On the plain side of the card they print the title of the formula such as the formula for the area of a sector found on page 16 of the third edition of the Algebra 2 textbook. So, on the front of the card (the plain side) in the center of the card the student would print:
When you turn the card over, in the upper right hand corner is the page number of the formula to enable the student to immediately go to that page should he need more information (in this case p 16). Recording the page number saves flipping through the book looking for the information - wasting time - especially when the student encounters a difficult problem some twenty lessons later. After writing down the appropriate page number, they neatly record the formula: double checking to make sure they have recorded it correctly.)
Area of Sector = Pc/360 times [(pi)(r)]^2 (where the piece (Pc) equals the part of the sector given.)
NOTE: If diameter is given remember to divide by two before squaring the value.
Remember, students may also use the formula cards on tests - and if you watch them - the dog eared cards seldom get looked at after awhile.
For those of you concerned about students taking the ACT or SAT, unless they have changed their policy, students are given a sheet of formulas for the math portion of the test. Again, this requires the student to know which formula to select and what to do with it â rather than remembering all those formulas!