## Posted tagged ‘Algebra’

### The 10 Major Concepts of Shormann Mathematics

August 26, 2014

The following is the fifth in a series of posts covering Shormann Mathematics, Algebra 1, the newest product from DIVE Math and Science! Click here to read the complete document that covers Shormann Math core ideas, course description, and Algebra 1 table of contents.

After years of teaching mathematics, researching math curricula and math history, and applying mathematics as a scientist and engineer, I concluded mathematics can be taught by covering 10 major concepts. The 10 major concepts are: number, ratio, algebra, geometry, analytical geometry, measurement, trigonometry, calculus, statistics, and computer math. While all 10 concepts can be taught in any K-12 course, specific concepts will be emphasized more or less at appropriate times. For example, number and ratio will be emphasized in younger grades, algebra in Algebra 1 and 2, etc.

I know what you are thinking right now, and that is “But CALCULUS is one of the 10 major concepts! How can you possibly teach calculus to an Algebra 1 student?!” Well, if you have even an 8th grade level of math proficiency, you know that if it took you exactly one hour to drive 60 miles, your average speed would be 60 mph. If you understand that, you already understand something about calculus, because calculus is really nothing more than studying rates of change. And yes, it gets more complicated than that example, but it also gets less complicated, too, so much so that there are things about calculus you could teach a kindergartner!

Most state mathematics standards do not include calculus, and none that I know of require calculus in high school. And the federal Common Core math standards include no calculus, and almost no precalculus either! However, the discovery of calculus is one of the greatest mathematical achievements ever! All the great technological achievements of the last 300+ years are in some way or another related to calculus! And proficiency in calculus opens the door for a student to choose any college major, while an inability to pass calculus limits a student to about 20% of college majors.

For high school mathematics, most home schools and private schools simply parrot whatever their state standards are, which means they complete Algebra 1, 2, and Geometry, and check off math on their transcript, not really knowing why they did math this way. With Shormann Math though, we want you to know why you are doing math differently. We are going to paint a broader brush than most math curricula, teaching math like a language, while at the same time helping you become proficient in standard Algebra 1, 2 and Geometry concepts. Along the way, rather than avoiding calculus because you heard it was scary, you are gently introduced to it. And, before you know it, you will be understanding more calculus than all your peers, and probably even your parents, ever did! Rather than an afterthought or a scary thought, Shormann Math makes calculus a normal, natural part of the curriculum, and culminates with a formal (and yes, it’s optional!) calculus course that will prepare students to receive college credit via CLEP or AP Calculus.

Done in a thoughtful and age-appropriate way, all 10 major concepts listed above can most definitely be represented in one way or another in a K-12 mathematics curriculum.

### Weighing the Differences in 3rd and 4th Edition Saxon Algebra 1

February 8, 2012

Over the past few months, many parents have contacted us, asking if we plan to make a DIVE CD to teach the new Saxon Algebra 1, 4th edition textbook.  The short answer is “no”, and the short reason is that we believe the newer Saxon textbooks have strayed too far from John Saxon’s (1923-1996) original, tried and tested vision for teaching mathematics.  This new textbook was not published by John Saxon, but by Houghton Mifflin Harcourt (HMH). If you would like to know more about our reasons, please read on.

Physical Differences

Saxon Algebra 1 4th ed. (left), Saxon Algebra 1 3rd ed. (center), and Leonhard Euler’s Elements of Algebra (right), a text that most modern algebra books are based upon.

The 4th edition cover is noticeably different from earlier Saxon editions. For comparison, I have included a copy of Leonhard Euler’s Elements of Algebra, a textbook whose subject matter is the foundation of most modern algebra courses. Euler lived from 1707-1783, and is considered by most scholars to be one of the best, if not the best, mathematician ever. While I am in awe of his ability to write original research at the rate of 800 pages per year for most of his adult life, I am more impressed by his understanding of God. One of my favorite quotes is from his book, Letters to a German Princess:

“It is God, therefore, who places men, every instant, in circumstances the most favourable, and from which, they may derive motives the most powerful, to produce their conversion.”

Euler was a genius, but he was also a humble, Christian family man, and I think his biblical understanding of the world helped him excel at mathematics. Helping students understand the importance of a biblical foundation to their education is one way our DIVE Math lectures differ from instruction found in either new or traditional Saxon textbooks.

When I titled this post “Weighing the Differences”, I meant it, literally! I put the books on a scale, and the 4th edition is quite a monster at 4.75 lbs, a 58.3% increase over the 3rd edition.

The 4th edition weighs 58.3% more than the 3rd edition.

Content differences

You might be scratching your head right now, wondering “what does book weight have to do with anything?” Well, it matters to students! A bigger book means more weight to lug around in the backpack, but even more dreadful, it means more content! Sure enough, the 4th edition sports a whopping 66.3% increase in the number of pages.

The 4th edition has 374 more pages than the 3rd edition.

Some of that increase is because the 4th edition often has more practice problems for each lesson, which may be helpful to some students, but most of the increase is from new content. Both textbooks have 120 “Lessons”, but in the 4th edition, there are an additional 59 lessons the student must learn.

The 4th edition has an almost 50% increase in the number of individual lessons a student must learn.

To understand why there is such an increase in content, understand that in order to sell textbooks to all government schools, publishers must include content that satisfies the educational standards of every state in the nation. The increase in the 4th edition’s content is partly because states don’t all agree on what should and shouldn’t be taught in algebra class. Like any business, textbook publishers must be profitable. If their main goal is to sell to government schools, they will make more money if they can satisfy every state’s requirements. Selling to government schools is clearly the priority for HMH, which results in really large books! Something else to keep in mind is that the goal of publishers is to satisfy state standards; whether or not their books produce good results is often overlooked. Surprised? John Saxon wasn’t.

Differences in methodology

One cool math teacher. John Saxon was a test pilot for the U.S. Air Force in the 1950s. Photo courtesy of Niki Hayes, author of John Saxon’s Story, A Genius of Common Sense in Math Education.

John Saxon was known for his “Saxonisms”, one of which was

“Results, not methodology, should be the basis of curriculum decisions. Results matter.”

A methodology, or way of doing something, does make a difference, but what John Saxon meant is that when it comes to educating a child, the methodology should never trump the results. An Air Force test pilot with three engineering degrees, after retirement Saxon started teaching algebra at the local junior college. Appalled at the results he was seeing, Saxon wrote and published his first algebra book in 1981. His methodology produced good results, so he stuck with it, and when he died in 1996, Saxon Publishers annual sales were at \$27 million. You can read more about John Saxon in Niki Hayes’ book, John Saxon’s Story, A Genius of Common Sense in Math Education.

You will hear many people say mathematics is the “language of science”, but to my knowledge, math books published by John Saxon and the original Saxon Publishers are about the only books that actually teach math this way. Just like learning a language, the original Saxon methodology begins with the fundamentals and provides students ample time to practice these before gently introducing more advanced material.

Original Saxon textbooks are also the best I’ve seen at teaching mathematics as one subject. Traditional American government math courses teach algebra and geometry separately. Many home educators follow the lead of government schools, without realizing that most European and Asian countries teach algebra and geometry together. You know, the same countries that consistently outperform the United States on international math exams (click here, see p. 7).  It makes sense that a student who is learning algebra and geometry together will probably understand all math better and be more ready to apply it in science and engineering fields. High school students will probably be able to outperform other students on college admissions tests, because these tests present algebra and geometry together.

Another distinguishing feature of John Saxon’s methodology was his desire for high school students to learn calculus. Again, Saxon shows its uniqueness in that, to my knowledge, it is one of the only curricula with a high school calculus course.

And finally, John Saxon was proud of his work. He had created something that produced good results, and he wanted to share it with others. Putting his name on the front of every book and naming the company after his family were ways of claiming ownership and responsibility for what he had done.

How true to the Saxon methodology is the new Algebra 1, 4th edition text?

While the 4th edition retains some of the pattern of incremental development with review, there is an obvious lack of understanding of what John Saxon was trying to accomplish. One thing John Saxon was fairly insistent on was teacher training for using his books. To an outside observer, the Saxon format looks fairly random, and if the teacher or student didn’t understand this was part of the incremental process, the format could be confusing. This is one reason our DIVE CDs have been helpful to so many learners, because we help students make the proper connections between lessons. However, many students are able to make the connections on their own, because when a new lesson builds on previous ones, John would normally mention this. For example, Lesson 25 in the 3rd edition begins with “In Lesson 23 and 24, we were introduced to the …” This gives the student a great reminder that what they are learning now is not all new, but is building on previous material, and they know where to find it if they need to review. By contrast, Lesson 23 in the 4th edition builds on Lesson 19 and 21, but no mention of this is made in Lesson 23, making it less likely the student will know where to go to review that concept. The lack of connection to previously learned material should be of particular concern to homeschoolers, as it makes self-directed learning more difficult.

The 4th edition also seems less adept at providing continual review. A good example is unit multipliers (a.k.a unit conversions, conversion factors, etc.). Saxon is the only curriculum I have seen that makes a real effort to teach students how to convert from one unit to another, a necessary skill when analyzing data collected in a science experiment, when building almost anything, in financial transactions, etc. Unfortunately, the 4th edition only has one lesson on unit multipliers (Lesson 8). One thing I liked was that they included an example of converting from one foreign currency to another, a useful skill in our increasingly global economy. What I didn’t like though was that after Lesson 30, I could not find any more homework problems on unit multipliers. Contrast that with the 3rd edition, where unit multipliers are taught in Lesson 4, 10, and 53, and students continue to have homework problems through Lesson 90. For assuring that a student learn a fundamental concept with such important implications, the choice is clear.

Besides the 66.3% increase in pages, the most glaring difference between the editions is that in the 4th edition, geometry was shoved to the “Skills Bank” section in the back of the book. Students are never taught these concepts beforehand, and I couldn’t really find where they ever practice many of the skills banks concepts. There is 1 geometry problem in every homework set, compared to 2 or more geometry problems in the 3rd edition. The geometry/algebra integration is essentially gone, and with their new, 887-page Geometry course, the newer Saxon editions look more like all the other government school textbooks that teach algebra and geometry separately.

A final, perplexing difference between the 3rd and 4th edition texts is that the authors’ names are not listed on the 4th edition (or on the new Geometry for that matter). I have quite a few math textbooks in my office, and all of them display the author’s name on the front or the spine. So who authored the new Saxon books? Is HMH being a little deceptive by not putting the real authors’ names on the book so that people will think John Saxon (dead since 1996) wrote them? Is HMH embarrassed by the new editions? I don’t know what the answer is.

So how different are the 3rd and 4th editions of Saxon Algebra 1? The differences remind me of a scene from a favorite childhood movie, Chitty Chitty Bang Bang. Caractacus Potts has built a “fantasmagorical motor car”, named Chitty Chitty Bang Bang, that can drive, fly, and swim, and the evil King of Vulgaria wants one, too! The king tries to kidnap Caractactus, but mistakenly kidnaps his father instead, who has no knowledge of how to build a Chitty Bang Bang. The king then uses the father to direct a crackpot team of engineers to build his own Chitty Bang Bang, which fails miserably. Like the Vulgarian king’s engineers, Houghton Mifflin Harcourt seems to have built Saxon Algebra 1, 4th edition with a limited understanding of the original designer’s plan and purpose. When it comes to understanding “incremental development with continual review”, a hallmark of the Saxon methodology, HMH doesn’t seem to get it.

When you try to build something without fully understanding what you’re doing, disaster usually results.

To conclude, I believe Saxon Algebra “peaked” with its 3rd editions of Algebra 1 and 2, so I won’t be making a DIVE CD for the 4th editions.  Fortunately, HMH is still selling the 3rd editions, and they show no signs of discontinuing them.

Time to DIVE

Since Saxon Publishers was first sold in 2004, I’ve feared that any new editions might lose their original Saxon methodology that strives to teach mathematics like the language of science that it is. The new 4th edition confirms this. And finally, since the sale, I have thought that, Lord willing, if the Saxon curriculum took a turn for the worst, then I would be ready to stand on the shoulders of giants like John Saxon, Leonhard Euler, Isaac Newton, Euclid and others, learn from them, and build a better curriculum. My goal has never been to just be a “Saxon Math Teacher”, but to teach students math and science so they can know their Savior and better serve Him and their fellow man. As providence would have it, John Saxon created a curriculum that I thought was the best, and so that’s what I’ve been teaching with since 1997. But since Saxon Publisher’s sale, I’ve had 8 years to think and pray about what to do, and I believe the time to act has come. Coming soon from DIVE,  look for a new, and I trust better, way to learn math.