## Archive for the ‘Teaching Mathematics’ category

### Defining Mathematics

July 11, 2014

Thinking of mathematics as the “ship,” and the definition as the “captain,” the way we define mathematics can greatly influence how we use it. (Wikipedia photo of the tall ship “Elissa.”)

#### The Challenge of Defining Mathematics

Throughout history, humans have never settled on one particular definition for mathematics. Part of the reason is the abstract nature of mathematics, and the way general mathematical truths can apply to an infinite number of situations. For example, think of numbers. Numbers are abstract ideas. The number 3 is an idea of “threeness,” and can be used to describe 3 bears, 3 cars, 3 words, etc.

Here are a few of the ways famous mathematicians and mathematics teachers have defined mathematics:

• The science which investigates the means of measuring quantity(L. Euler, Elements of Algebra, 1765).
• The foundation of exact thought as applied to natural phenomena(A.N. Whitehead, An Introduction to Mathematics, 1911).
•  Mathematics is the classification and study of all possible patterns(W. W. Sawyer, Prelude to Mathematics, 1955).
• A study of space and quantity (Kline, Mathematics and the Physical World, 1959).

#### If Math is the Ship, Then its Definition is the Captain

Do you think it matters how mathematics is defined in the math courses you or your children do? I’ve been thinking about this question for many years now, and I think the answer is most definitely “yes!” A good definition can set the foundation for the entire course. And for a mathematics curriculum writer like myself, it can set the foundation for not just one course, but the entire curriculum. Thinking of mathematics as the “ship” and the definition as its “captain,” a good captain can use the ship for what it’s designed for. A good captain knows who built the ship. A good captain can help others better understand what the ship is capable of.

#### How Shormann Mathematics Defines Math

At DIVE, we are getting close to launching our own standalone mathematics curriculum, Shormann Mathematics. Algebra 1 is the first course. For the first year, it will be available as a live, online class (click here to register). In Shormann Mathematics, we will use the following definition for mathematics:

mathematics: the language of science and a God-given tool for measuring and classifying pattern and shape.

This definition tells us that mathematics, with all of its unique symbols, is best thought of as a language. It is a language we can use to study creation. Next, this definition tells us mathematics is about measuring things. It also tells us mathematics helps us find truth, goodness and beauty as we classify pattern and shape.

But most importantly, this definition of mathematics tells us “who built the ship.” Mathematics is not man-made, it is God-given. Created in His image (Genesis 1:26), we are designed by God to use this tool to be creative, too! God designed us to be creative and to engage in fruitful, productive activities (Genesis 1:28).

#### What’s the “Common Core” of Your Math And Science Curriculum?

A lot is being discussed right now about “Common Core” curriculum promoted by the United States government. Unfortunately, man and his ever-changing ideas are at the core of this curriculum. At DIVE, we strive to place Jesus Christ at the core of all our products, and we pray that this will result in students learning math and science for His glory and the service of others. We would appreciate your prayers as we seek to put a new captain at the helm of the ship of mathematics, helping students use the ship for what God intended it for!

May 7, 2014

### A Brief History of University

Late in the 12th Century, a phenomenon unique to Europe appeared, the university. University is actually a combination of two words, unity and diversity. Originally, universities were schools that owned no real estate, but were instead an association of teachers or students. Although not always theologically or scholarly accurate, what under girded the university was the unification of all subjects by an all-encompassing worldview. Christianity provided the unity that connected the diversity of courses offered.*

In other words, Jesus Christ was at the core of the worldview of original universities! Unfortunately, in the 21st Century, Christianity is no longer at the core of most educational systems. In the United States, the government’s new Common Core program has a godless, purposeless, evolutionary worldview at its core.

### Who Interprets the Facts Matters

Cornelius Van Til (1895-1987), author of Essays on Christian Education, made the wise statement that, in this world, there exists a whole collection of facts. Would you rather have those facts interpreted to your child through a Christian or a non-Christian worldview?

For Christians, the answer should be simple. Jesus Christ is, or should be, the common core of all the subjects we teach our children. To the extent that we are able to, we should try to use curriculum taught from a Christian worldview. But that also doesn’t mean we need to fear books and courses that are taught from a non-Christian, or even anti-Christian worldview. We just need to train our children how to think critically about these things. If we teach them the Truth, they will be more capable of detecting false “philosophy and empty deceit.” (Colossians 2:8)

Pray that more Christian families will realize the problems of sending their children day after day to be taught in schools where Jesus Christ is not the common core. Pray that we can find new and creative ways to help those who, for various reasons, it would be extremely difficult or impossible to home school or attend a private Christian school. Pray for Christian families in countries where homeschooling and Christian schooling is illegal, that they will be able to help their children test everything they are learning, holding onto the good (I Thessalonians 5:21).

If you are interested in learning more about DIVE Math and Science courses, where Jesus Christ is the common core, click here.

*Paraphrased from Mathematics, is God Silent?, by James Nickel.

### Build a Better Engine

April 17, 2014

Over the last few years, I have talked to a lot of people who spew propaganda claiming Bible-believing Christians are “anti-science.” Because people like me are skeptical of the history claims of evolutionists and futurology claims of global warming alarmists, we are labeled “anti-science.” Fortunately, discerning between what is and is not a scientific claim is as easy as understanding a chocolate chip cookie recipe. Unfortunately, some refuse to acknowledge the differences, using the “Christians are anti-science” fallacy to create political division. For others,  it’s just another excuse to hate their neighbor.

One pattern I’ve noticed among the “Christians are anti-science” crowd is that the most outspoken individuals tend to have little or no background in science or engineering. When God gives me an opportunity to talk to unbelievers that promote this agenda, I have learned to 1) let them know Christians like me are most assuredly pro-science, 2) present the Gospel, and 3) encourage them to stop doing what they are doing and get into a science and engineering field.

Something I have encouraged more than one unbeliever to do is “build me a better engine.” Promoting the idea that fossil fuels are causing catastrophic global warming is foolish. In spite of increased atmospheric CO2 levels, there has been no warming for 17 years and 8 months now. If, instead of promoting unscientific future climate ideas and labeling those who disagree as “anti-science,” why not do something meaningful?  Why not be actively involved in designing less expensive, more fuel efficient engines, ones that could reduce air pollution and provide better lives for the poor? Wouldn’t a pro-science, love-your-neighbor mindset be better than an unscientific, hate-your-neighbor one? Well, of course it would, but the former is a difficult concept for those who don’t believe the foolishness of God is wiser than men (I Corinthians 1:25).

Unless God changes their hearts and they repent and turn to Christ, foolish actions are to be expected from unbelievers (Psalm 14:1). Fortunately, there are young Christian men and women out there who love God and His creation, and want to “build a better engine” for His glory. Listen to this testimony from David K., a homeschooling senior that is currently using our DIVE Calculus course (bold emphasis mine):

“Thank you also for all the work you have put into the DIVE CDs. Your teaching is clear, easy to understand, and you explain everything really well. Your lectures have helped me immensely, and I don’t know where I would be in math with out them. I definitely agree with you, in that God has allowed us to understand math so that we can get to know Him better. I love looking in Creation and seeing God Himself! I am a senior in high-school, and I plan to go to college to study Engineering Physics, with mechanical emphasis. I want to eventually perform engine research to produce a more financially feasible engine. I would do this by creating a new energy conversion process that does more work per unit of fuel than engines today. I have always had a love for science and math, and I really look up to people like you who know so much and use it for the glory of God. Thank you for being a great example for me to follow.”

While David K’s words are incredibly kind and humbling to me, I hope they are an encouragement to you! A lot of people are surrounded by hopelessness and despair, but there’s also a lot of hope out there, too!

Are you a young person like David K who loves the Lord and wants to take what God has made and use it to design things that will serve others? Are you currently an unbeliever? Whoever you are, it is important to be intellectually honest and spread the word that Christians are pro-science. History proclaims this truth, as do present actions of humans all over the world.  So, enough of this blog post, get out there and build a better engine!

### Sowing DIVE Seeds

October 11, 2012

“……But others fell on good ground, sprang up, and yielded a crop a hundredfold.”  Luke 8:4-8

In the parable of the sower, Jesus used the analogy of a seed to help His disciples understand how the Word of God works in the lives of people. When God’s word falls on the “good ground” of a noble and good heart, the Word is kept and bears fruit with patience (Luke 8:15). It is because Christ is author of both the spiritual and material things that He is able to make such amazing comparisons between His word’s effect on a human, and what happens to a real seed when planted in the right soil.

The parable also applies to other things, including our DIVE math and science instructional materials. When they are planted in the “good ground” of a child and family that are willing to take the instruction and “bear fruit with patience”, the results are sometimes truly remarkable. The following are just two of many examples we hear almost every week.

The Brooks family wrote us on Oct. 8, 2012,

“As a child with autism, my son had an IEP (Individualized Education Program) in the public school system, but somehow, he was left behind. After changing schools in sixth grade, I was informed my son was three years behind his peers and his transcripts didn’t reflect his performance level. Their outlook was bleak, and less than aggressive in working with me to help catch him up. So I removed him from the public school system and started homeschooling.

Making a long story short, I’m proud to share with you, my son is [now] well ahead of his public school peers. He’s currently in his senior year taking AP Calculus, AP Physics, AP Government, AP Macro, AP US History.

While my son did all this work himself, I really owe you a huge piece of recognition for your quality CD’s. My son is proof, even a once diagnosed “low functioning” autistic can gain success. Thank you so very much for your contribution to his success. My son plans to work in the computer animation and engineering field. He too appreciates and recognizes your CD’s to help him achieve success.”

And here is an excerpt from Tiffini, who made a comment recently on my Producer Math post:

“I couldn’t seem to wrap my brain around algebra. Instead of help, I got a transfer to “consumer math”. I was so embarrassed, I couldn’t tell my friends about the class I was in. I had been labeled “dumb”, so therefore I thought that was the case. Unfortunately I didn’t make it through college, but I believe strongly in education, so I worked very hard to put my husband through college, med school, and residency. In 6th grade our son was having trouble in school. I didn’t want the same experience to happen to him. After much prayer and fasting, we decided to homeschool. It is definitely not the easy way out. I’m thrilled that as a freshman this year he’s doing very well in algebra 2 with geometry. I’m so thankful for the dive cd’s.”

Some things to notice from those two stories include:

• Both are examples of individuals and families who have the “good ground” that is so essential for taking the “DIVE seed” and bearing fruit.
• Both are examples of government schools failing to properly educate, and that should be a huge warning to you if you currently have a child, especially one who is struggling, in a government school. And for Christians, government schools are really no place to put your children anyways, because the goal is not to help your child become a creative Christian, rich in good works and ready to give, willing to share (I Timothy 6:18).

So what about your family? Is it “good ground” for sowing not just “DIVE seeds”, but any seeds? Is it filled with the thorns (Luke 8:7) of whining, complaining, hopelessness, anger, and laziness? If so, it is never too late to turn your thorny, rocky soil into rich, fertile ground. It may take lots of prayer, patience, and daily repentance and reforming, but with God, all things are possible (Matthew 19:26). Think about it, if the two families highlighted above can turn impossible things into possible ones, then by God’s grace you can, too!

### Producer Math

June 27, 2012

When you eat the labor of your hands, you shall be happy, and it shall be well with you. Psalm 128:2

### Failing at producing

You may have heard the phrase that America has turned into a “nation of consumers”. If you think that is an incorrect assessment, take a look at this graph:

United States Imports Minus Exports, 1960-2011

This year, 2012, will likely be the 37th consecutive year that the United States of America has imported more than it has exported. In other words, we are consuming hundreds of billions of dollars more than we produce. As a whole, Americans are less productive, which means we are also less creative, than we were back in the 1960s.

For Christians, a trend like this should be unsettling, because it goes against the most basic of Christian principles. One of God’s very first commands to humans was to “be fruitful and multiply” (Genesis 1:28). Created in His image, we are designed to create, too. To be productive. To bear fruit and “eat the labor of your hands”, as Psalm 128:2 teaches. “Bearing fruit” is not just about having children, or raising corn and cattle, building houses, bearing spiritual fruits, etc. Bearing fruit is ultimately about fulfilling the Great Commission by spreading the Gospel to the ends of the Earth (Matthew 28:18-20). And God didn’t make us all clones, and give us all the same exact plan for fulfilling the Great Commission. He designed us to be creative in this task.

Unfortunately in America, we answered the trend towards excessive consumption by developing “consumer math” classes for high school and college students. Such classes usually contain basic arithmetic and very little algebra, and are designed to help students understand common-sense ideas such as not spending more money than you earn. Less obvious topics like interest rate are also covered. However, most topics are a review of what students already learned in elementary and middle-grade math courses. Also called “business math”, Wikipedia describes these courses as “subjects taught to students who are not planning a university education.” In other words, the classes are for people who are not planning to be producers, just consumers.

### Training up backward-thinking consumers

Now, don’t get me wrong. I certainly believe non-university bound students can also be productive members of society! However, by taking “consumer math” instead of an advanced math or calculus class in high school, you are essentially falling in line with secular and non-Christian education standards. For example, the National Council of Teachers of Mathematics claims “For those whose formal education will end with high school, the needs of citizens and consumers for increasing mathematical sophistication dictate a collection of courses based on consumer and career needs”. See, there it is again! Non-university-bound students are just consumers. And citizens. But I didn’t say that, “they” did! The average government school is training students to be consumers and citizens who are told of their supposed not-so-special origin from a monkey-man. Shouldn’t they, shouldn’t we, instead be training students to be forward-thinking producers? Of course! If you are in a government school, you should fight against this kind of demoralizing miseducation. If you homeschool or private school, don’t use the government schools as your guide! Instead make sure your child gets a good dose of Christ-centered science, and it’s language, mathematics.

### You are more than a consumer

University bound or not, current Christian or not, I hope you can see the problem with consumer math. Of course, some consumer math is a good idea, but “producer math” should be the priority, especially in high school and college. Because human beings are designed by God to be creative, creativity comes naturally for us. But creativity always requires tools, and in the 21st Century, good mathematics skills are definitely one tool that will help spark creativity, and in turn, productivity. All humans are consumers, but life is about so much more than that. Being a producer as well means that you and/or the company you work for need to 1) Create something to sell and 2) have the ability to sell it AND make a profit. And it is the profit (fruit) that you can use to grow your family, grow your church, and be a wise ruler of God’s creation as you fulfill the Great Commission.

Three of the many math skills that are important for 21st Century producers, two of which you won’t see much of until Algebra 2 or later, include 1) Unit multipliers (conversion factors), 2) Analytical Geometry, and 3) Calculus. And in all three of these, an understanding of fractions is key.

Good skills with unit multipliers are helpful when you are designing a new cancer-fighting nanotechnology, and you need to convert micrograms per liter per hour to ppm per day. Or, maybe you are setting up a spreadsheet to help you determine cost per unit of an invention that you patented, and now want to sell. Analytical geometry is helpful in computer graphics and other applications, where knowledge of not only shape, but exact spatial positioning is important. And calculus is where rates of change are studied, which has applications in more areas than you will ever imagine in a lifetime.

### Math for producers

John Saxon (1923-1996) wrote some of the best “producer math” books available. While newer editions are moving away from his tried and true methods, the pre-2009 Saxon textbook editions are the best I’ve seen at helping students learn producer math. Avoid the newer, blue-covered hardback Saxon texts, published by Houghton Mifflin Harcourt, and NOT written by John Saxon. In texts written and approved by John, unit multipliers are taught beginning in the elementary grades, and continue through Calculus. Consumer math topics are also included. For example, sales tax, a topic that would be taught in a high school “consumer math” course, is introduced in the elementary-level Saxon Math 5/4. Students continue building their consumer math skills from this point on through Saxon Calculus.

As I get closer to creating my own mathematics curriculum, I hope to take the best of John Saxon’s principles, and build on those. As I develop this curriculum, I am taking note of the fact that John Saxon  never wrote a “consumer math” textbook. Indeed he frowned upon the very idea of placing students in these classes.  Regarding consumer, or “basic” math, John Saxon said “We cannot take kids and relegate them to the trash heap in this technological society. We label them as failures when we put them in basic math”(from John Saxon’s Story, by Niki Hayes, p. 276). And Saxon wasn’t the only successful teacher opposed to these courses. The book Standing and Delivering by Henry Gradillas highlights the story of how he and teacher Jaime Escalante eliminated “dumb dumb” math classes from Garfield High School in Los Angeles, and by doing so, turned around math education, with many students passing the AP Calculus exams.

So is “producer math” harder than “consumer math”? Well, is buying a blueberry bush, planting it, watering it, nurturing it, harvesting the fruit and then taking it to market to sell, harder than consuming a bowl of blueberries? Yes! But what does Scripture say about doing hard things? Does it say to run from them? Certainly not! It says to count our trials as joy (James 1:2-3). Parents and teachers who seek to help students be producers will get more heartache, more complaints, and more trials to deal with. But 10 years later, those parents will probably get more “thank you’s” from their children than from the ones who failed to challenge.

Christians have been called to handle the hard stuff with grace and thanksgiving. Parents, you know your child best. Are they capable of doing more producer math? The majority of them are, so push them with much love, patience, and perseverance. And if they fail the first time, give them a second chance the next year. And the next. But if you are certain your child is not capable of things like calculus, then do what it takes to teach them as much producer math as you can. Being a producer is not just the American way, it’s the Christian way, and us parents need to make sure we are training our children up to be more than consumers. Much more!

### 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.

### Understanding vs. Memorization

April 29, 2011

A Google search of the phrase “understanding versus memorization” yielded some of the following comments:

• “The old style of teaching used to stress memorization.”
• “More and more top-level research on how we learn backs up the benefits of ‘teaching for understanding’ versus memorization.”
• “Productive thinking is defined as thinking based on an understanding of the nature of problems rather than on memorization of facts and rules.”

Another popular homeschool math curriculum states that only 5% of mathematics should be learned by rote and 95% should be understood.

So, it seems that memorization is “out”, and understanding is “in!” Or is it? Before we continue, I think it is important to define some terms:

understand-perceive the meaning of

memorize-commit to memory; learn by heart

analogy– a comparison between two things

When looking for a curriculum, whether it’s math or science, I think it would be a bad idea to pick the curriculum based on its emphasis on understanding versus memorization. In a good curriculum, there is no competition going on between the two. Both understanding AND memorization are important. What is even more important though, is the use of an analogy, a comparison between two things. Think about it, what was one of the first assignments God gave to a human? To name the animals. When we name things, we are making a comparison of two things, 1) the object and 2) its name. Jesus used parables all the time, and parables are a form of an analogy. And think about what Jesus did, he would use an example of something people were already familiar with, or had committed to memory, to perceive the meaning of something else. He would use a common item like a seed to help people understand things like faith (Matthew 17:20) and God’s word (Luke 8:11). God designed our brains to learn new things using analogies.

A good curriculum will use the analogy of something familiar to learn about something new. I have a copy of “Elements of Algebra”, originally written in 1765 by Leonhard Euler. This book is important because 1) it was written by the man that most consider to be the greatest mathematician EVER, and 2) the basic layout of most modern algebra textbooks is based on this book. One thing you will not find in this book is a discussion of “understanding versus memorization.” What you will find though is the extensive use of analogies. For example, to help a student understand adding fractions, Euler begins by explaining how to add fractions with common denominators. Euler familiarizes the student with this simple example. Next he uses it as an analogy to teach the more complex subject of addition of fractions with different denominators.

Any good math curriculum would teach addition of fractions in this way. Saxon math, for example, begins teaching addition and subtraction of fractions in their 3rd grade text, Math 3. This is continued in Math 5/4, as they build on understanding and memorizing how to simplify fractions. Adding fractions with different denominators is not covered in detail until Math 6/5, when the student has had ample time gaining experience adding and simplifying fractions, and committing the techniques involved to memory.

One reason I like the Saxon method so much is that it is not an “understanding versus memorization” approach, but instead relies heavily on the use of analogies. A student is given time to understand a concept and commit it to memory. Over time, something that may have been difficult for a student to do becomes familiar and easy to do. Once it is familiar, then Saxon presents a new concept that builds on the old one. The familiar concept is used as an analogy to help the student understand a new concept.

Understanding and memorization are both important. A student who has memorized basic math facts and rules will have a strong foundation on which to build. Memorization provides a foundation on which to advance learning. It is not an “old style of teaching” or “5% important”.  A good math curriculum will challenge a student’s understanding of the facts and rules by giving them opportunities to apply the facts and rules in new situations to solve new problems. A good math curriculum is one where memorization and understanding work together, and new concepts are gently introduced, mainly through the use of analogies. This is how Leonhard Euler taught math, it is how traditional Saxon textbooks teach math, and it is how I teach math, and it works!

Understanding and memorization are important in all learning, including advancing your learning of God and His perfect plan for you. If you spend time on Scripture memory, and you go to a good church that helps you understand God’s word, you will advance your learning of Him. Hebrews 5:12-13 is an excellent example of the importance of having a strong foundation for our Christian walk. Advance your learning through understanding and memorization, not just one or the other.

### 2011 Catalog

April 20, 2011

My favorite part of making this catalog was taking the pictures! Click the link below to view it:

2011 Catalog

### Student finds error in Saxon Calculus, 2nd ed

January 20, 2011

Last week, a student who uses our DIVE CDs emailed me to verify an error in problem number 24 of Saxon Calculus, 2nd edition, Problem Set 63. Although this error did not actually affect the final answer, it was an error nonetheless, and was missed by the editors of the Saxon Solutions Manual. EVERY curriculum has a few errors, including the products my company, DIVE LLC, produces, so the point of this article is not to ridicule Saxon Publishers for their errors. The point is to talk about educating children. If you visit the DIVE website or read a DIVE catalog, you will quickly understand that I think it is time to “raise the standard” in K-12 education, and this includes completing calculus in high school. I realize not every student can accomplish this goal, but there is nothing wrong with at least making that the goal with the possibility of falling short.

So what is the big deal about this student who found an error? Well, he’s 12 years old. And he is home educated. By his parents. Obviously, this boy has an aptitude for math that exceeds that of most 12-year olds and even many adults, but the fact remains that he is almost halfway-through a course that is the equivalent of college Calculus I. And if there are 12-year olds out there who can do calculus, there are even more 13, 14, 15+ year olds who can do calculus.

Error in Saxon Calculus 2nd edition solutions manual. Okay, so the handwriting is not super-neat, but give the kid a break, he's 12!

Most people today realize the potentials of homeschooling, but strangely enough, there are still those who want to eliminate it. According to an article in the November/December 2010 issue of The Home School Court Report, American homeschooling is entering a “Third Wave” of persecution. The first wave had to do with education. Could parents really educate children at home? While there are some parents who do a really bad job at educating, the same can be said of many government and private schools. And in today’s world, 12-year olds doing calculus would be a real surprise in a government school and most private schools, but not in a home school.

The second wave of persecution was about the issue of socialization, but the evidence now weighs heavily against that idea, too. While the first two waves of persecution were based on false premises, the third wave of persecution is, according to HSLDA Chairman Mike Farris, essentially correct. According to Farris “Christian homeschooling parents are effectively transmitting values to their children that the elitists believe are dangerous to the well-being of both these very children and society as a whole.”

I believe God gave us the ability to do mathematics so that we could better understand His creation. I believe we should study math and science so we can know Him better and as Christians, act on the faith He gave us and get out there and be excellent at doing the work He has already prepared for us (Ephesians 2:8-10). I also believe we can study math and science without acknowledging its Author, but to do so, we miss half the story. There are those in society who actually think it would be better for us all if Christian 12-year old boys with above-average math aptitude were taken from their loving parents, placed in a large group, and taught a below-aptitude, godless curriculum. The boys would not learn that they were created in the image of the greatest Creator, and are therefore creative, too. Instead, their creativity will be stifled, supposedly so they fit in better with the group, and they will instead learn they were not created by a loving Father, but rather evolved from a meaningless pile of goo in a mysterious land that existed before history. And the people who promote such ideas are referred to as the “elites”? Really?

The third wave of homeschool persecution is nothing more than misguided “elitists” wanting to replace Christian religion, where we are taught to love the sinner but hate the sin, with a secular fundamentalist religion that teaches to hate some sinners and love many sins. The first two waves taught us home schoolers to surf. Skills learned in the past will help us surf this bigger, badder, third wave, too, and ensure the current freedoms American parents have to teach calculus to their math-gifted 12-year olds. But we won’t keep these freedoms if we sit around and do nothing, so get out there and surf!

### Why Use DIVE Math

August 15, 2009

DIVE Math is simply CD-ROM Lectures that teach Saxon Mathematics. Designed with homeschooling families in mind, DIVE CDs eliminate teacher preparation and lecture time, and allow students to go at their own pace. Dr. Shormann teaches using a learn-by-doing approach, with each lecture starting with a brief introduction, followed by several practice problems that the student works on their own paper. Taught from a Christian worldview, Dr. Shormann teaches that mathematics is the language of science and a tool for studying God’s creation. Watch the video to learn more on how DIVE compares to other curriculum, and how your child can get a great mathematics education using DIVE and Saxon!