## Posted tagged ‘Shormann Math’

### Letters of Euler to a German Princess, Vol. II, Letter XIII

July 17, 2018

This is the seventh of eleven Letters of Euler I will rewrite and post on the subject of infinitesimals (the infinitely small), an idea that is fundamental to a good understanding of calculus. Click here to read the previous letter.

Principle of the satisfying Reason, the strongest Support of the Monadists.

You must be perfectly sensible that one of the two systems, which have undergone such ample discussion, is necessarily true, and the other false, seeing they are contradictory. It is admitted on both sides, that bodies are divisible: the only question is, “Whether this divisibility is limited?”, or “Whether it may always be carried further, without the possibility of ever arriving at indivisible particles?”

The system of monads is established in the former case, since after having divided a body into indivisible particles, these very particles are monads, and there would be reason for saying that all bodies are composed of them, and each of a certain determinate number. Whoever denies the system of monads, must likewise, then, deny that the divisibility of bodies is limited. he is under the necessity of maintaining, that it is always possible to carry this divisibility further, without ever being obliged to stop; and this is the case of divisibility in infinitum, on which system we absolutely deny the existence of ultimate particles: consequently the difficulties resulting from their infinite number fall to the ground themselves. In denying monads, it is impossible to talk any longer of ultimate particles, and still less of the number of them which enters into the composition of each body.

You must have remarked, that what I have hitherto produced in support of the system of monads is destitute of solidity. I now proceed to inform you that its supporters rest their cause chiefly on the great principle of the sufficient reason, which they know how to employ so dexterously, that by means of it they are in a condition to demonstrate whatever suits their purpose, and to demolish whatever makes against them. The blessed discovery made, then, is this, “That nothing can be without a sufficient reason;” and to modern philosophers we stand indebted for it.

In order to give you  an idea of this principle, you have only to consider, that in every thing presented to you, it may always be asked, “Why it is such?” And the answer is what they call the sufficient reason, supporting it really to correspond with the question proposed. Wherever the “why” can take place, the possibility of a satisfactory answer is taken for granted, which shall, of course, contain the sufficient reason of the thing.

This is very far, however, from being a mystery of modern discovery. Men in every age have asked “why;” an incontestable proof of their conviction that every thing must have a satisfying reason of its existence. This principle, that nothing is without a cause,  was very well known to ancient philosophers; but unhappily this cause is for the most part concealed from us. To little purpose do we ask “why:” no one is qualified to assign the reason. It is not a matter of doubt, that every thing has its cause; but a progress thus far hardly deserves the name; and so long as it remains concealed, we have not advanced a single step in real knowledge.

You may perhaps imagine, that modern philosophers, who make such a boast of the principle of a satisfying reason, have actually discovered that of all things, and are in a condition to answer every why that can be proposed to them; which would undoubtedly be their very summit of human knowledge; but, in this respect, they are just as ignorant as their neighbors: their whole merit amounts to no more than pretension to have demonstrated, that wherever it is possible to ask the question “why,” there must be a satisfying answer to it, though concealed from us.

They readily admit, that the ancients had a knowledge of this principle, but a knowledge very obscure; whereas they pretend to have placed it in its clearest light, and to have demonstrated the truth of it: and therefore it is that they know how to turn it most to their account, and that this principle puts them in a condition to prove, that bodies are composed of monads.

Bodies, they say, must have their sufficient reason somewhere, but if they were divisible to infinity, such reason could not take place: and hence they conclude, with an air altogether philosophic, “that, as every thing must have its sufficient reason, it is absolutely necessary that  all bodies should be composed of monads:” which was to be demonstrated. This, I must admit, is a demonstration to be resisted.

It were greatly to be wished that a reasoning so slight could elucidate to us questions of this importance; but I frankly confess, I comprehend nothing of the matter. They talk of the sufficient reason of bodies, by which they mean to reply to a certain “wherefore,” which remains unexplained. But it would be proper, undoubtedly, clearly to understand, and carefully to examine a question, before a reply is attempted; in the present case, the answer is given before the question is formed.

Is it asked, “Why do bodies exist?” It would be ridiculous, in my opinion, to reply, “Because they are composed of monads;” as if they contained the cause of that existence. Monads have not created bodies: and when I ask, “Why such a being exists?” I see no other reason that can be given but this, “Because the Creator has given it existence;” and as to the manner in which creation is performed, philosophers, I think, would do well honestly to acknowledge their ignorance.

But they maintain, that God could not have produced bodies, without having created monads, which were necessary to form the composition of them. This manifestly supposes, that bodies are composed of monads, the point which they meant to prove by this reasoning. And you are abundantly sensible, that it is not fair reasoning to take for granted the truth of a proposition which you are bound to prove by reasoning. It is a sophism known in logic by the name of a petitio principii, or, begging the question.

16th May, 1761.

### Letters of Euler to a German Princess, Vol. II, Letter XI

July 5, 2018

This is the fifth of eleven Letters of Euler I will rewrite and post on the subject of infinitesimals (the infinitely small), an idea that is fundamental to a good understanding of calculus. Click here to read the previous letter.

### Letter XI. Reflections on Divisibility in infinitum, and on Monads.

In speaking of the divisibility of body, we must carefully distinguish what is in our power, from what is possible in itself. In the first sense, it cannot be denied, that such a division of a body as we are capable of, must be very limited.

By pounding a stone we can easily reduce it to powder; and if it were possible to reckon all the little grains which form that powder, their number would undoubtedly be so great, that it would be matter of surprize, to have divided the stone into so many parts. But these very grains will be almost indivisible with respect to us, as no instrument we could employ would be able to lay hold of them. But it cannot with truth be affirmed that they are indivisible in themselves. You have only to view them with a good microscope, and each will appear itself a considerable stone, on which are distinguishable a great many points and inequalities; which demonstrates the possibility of a farther division, though we are not in a condition to execute it. For wherever we can distinguish several points in any object, it must be divisible into so many parts.

We speak not, therefore, of a division practicable by our strength and skill, but of that which is possible in itself, and which the Divine Omnipotence is able to accomplish.

It is in this sense, accordingly, that philosophers use the word ‘divisibility:’ so that if there were a stone so hard that no force could break it, it might be without hesitation affirmed as divisible, in its own nature, as the most brittle, of the same magnitude. And how many bodies are there on which we cannot lay any hold, and of whose divisibility we can entertain not the smallest doubt? No one doubts that the moon is a divisible body, though he is incapable of detaching the smallest particle from it: and the simple reason for its divisibility, is its being extended.

Wherever we remark extension, we are under the necessity of acknowledging divisibility, so that divisibility is an inseparable property of extension. But experience likewise demonstrates that the division of bodies extends very far. I shall not insist at great length on the instance usually produced of a ducat*: the artisan can beat it out into a leaf so fine, as to cover a very large surface, and the ducat may be divided into as many parts as that surface is capable of being divided. Our own body furnishes an example much more surprizing. Only consider the delicate veins and nerves with which it is filled, and the fluids which circulate through them. The subtility there discoverable far surpasses imagination.

##### *A ducat is a gold coin used in Euler’s day.

The smallest insects, such as are scarcely visible to the naked eye, have all their members, and legs on which they walk with amazing velocity. Hence we see that each limb has its muscles composed of a great number of fibres; that they have veins, and nerves, and a fluid still much more subtile which flows through their whole extent.

On viewing with a good microscope a single drop of water, it has the appearance of a sea; we see thousands of living creatures swimming in it, each of which is necessarily composed of an infinite number of muscular and nervous fibres, whose marvellous structure ought to excite our admiration. And though these creatures may perhaps be the smallest which we are capable of discovering by the help of the microsope, undoubtedly they are not the smallest which the Creator has produced. Animacules probably exist as small relatively to them, as they are relatively to us. And these after all are not yet the smallest, but may be followed by an infinity of new classes, each of which contains creatures incomparably smaller than those of the preceding class.

We ought in this to acknowledge the omnipotence and infinite wisdom of the Creator, as in objects of the greatest magnitude. it appears to me, that the consideration of these minute species, each of which is followed by another inconceivably more minute, ought to make the liveliest impression on our minds, and inspire us with the most sublime ideas of the works of the Almighty, whose power knows no bounds, whether as to great objects or small.

To imagine that after having divided a body into a great number of parts, we arrive, at length, at particles so small as to defy all farther division, is therefore the indication of a very contracted mind. But supposing it is possible to descend to particles so minute as to be, in their own nature, no longer divisible, as in the case of the supposed monads; before coming to this point, we shall have a particle composed of only two monads, and this particle will be of a certain magnitude or extension, otherwise it could not have been divisible into these two monads. Let us farther suppose, that this particle, as it has some extension, may be the thousandth part of an inch, or still smaller if you will; for it is of no importance, what I say of the thousandth part of an inch may be said with equal truth of every smaller part. This thousandth part of an inch, then, is composed of two monads, and consequently two monads together would be the thousandth part of an inch, and two thousand times nothing, a whole inch; the absurdity strikes at first light.

The partisans of the system of monads accordingly shrink from the force of this argument, and are reduced to a terrible nonplus when asked how many monads are requisite to constitute an extension. Two, they apprehend, would appear insufficient, they therefore allow that more must be necessary. But, if two monads cannot constitute extension, as each of the two has none; neither three, nor four, nor any number whatever will produce it; and this complexity subverts the system of monads.

9th May, 1761.

### Building Good Study Habits with Shormann Math

May 16, 2016

We just completed the beta-test of Shormann Algebra 2, our second course in the Shormann Math series. We learned a lot about what does and doesn’t work last year in the Shormann Algebra 1 course, so in building Shormann Algebra 2, we applied the good and cast the bad into the lake of fire.

A key part of Shormann Math is TruePractice™, the result of our efforts to design the most efficient system for building fluency in mathematics. If you want to be good at something, whether it’s baseball, piano, math, etc., there is simply no substitute for the need to practice. A lot.  If, however, you think you can be good at something by receiving magical superhero powers while sitting on your couch, then you either watch way too many movies, or you’re weird. Or both! But there are more and less efficient systems for practice, and we are finding that our TruePractice™ system that includes 100 lessons with 20 problems per lesson is achieving good results, compared to John Saxon-authored math courses which average 120 lessons and 30 problems per lesson.

With Shormann Math, students build fluency through 1) Practice Sets that are designed with the understanding that “practice time” is different than “game time,” 2) Weekly Quizzes that are like a “practice game,” and 3) Quarterly Exams that equate with “game day,” “piano recital,” etc.

Regarding Quarterly Exams, take a look at the graph of average student score vs. study effort. On the week of a quarterly exam, we provide detailed instructions on what we believe are the best methods for studying for an exam. The key, as you probably know, is to practice a lot. Because our eLearning campus provides data on some, but not all aspects of student study effort, we can group students into those who followed our study guidelines (blue line) and those who did not (red line).

The results are not surprising at all and show that we have a good system in place for helping students build fluency in math. Follow the system and make an A. Don’t follow the system and make a B or worse. Our study guidelines are based on years of teaching experience, combined with years more of learning from good college math, science and engineering professors at top universities.

Are you a parent who wants a good and God-glorifying math curriculum for your child? Or, even better, are you a student who wants to know God better by using math as a tool for studying His creation, and you’ve been looking for a curriculum that will help you do this? If yes, take a look at Shormann Math today.

December 3, 2015

### Common Core Cancer

A brave Texas 7th-grader alleges that during an assignment about Common Core’s fake version of critical thinking, the teacher directed students to label God as a myth. This type of “anchor chart” assignment forces students to wrongfully classify all statements as either fact, opinion, or commonplace assertion. Here’s how these categories are defined in a typical Common Core classroom:

• fact: Something that is true about a subject and can be tested or proven.
• opinion: What someone thinks, feels or believes
• commonplace assertion: Stating something is true without supporting it with facts or proof.

Notice how only facts are considered “true”, while opinions and commonplaces assertions are categorized as things that are either false or just “true for me but not necessarily true for you.”

This type of assignment is at the heart of Common Core educational standards (standards that supposedly aren’t taught in Texas. Surprise!). In his excellent March 2015 New York Times article about this fundamental problem with Common Core , philosopher Justin McBrayer described how students are required to fit things into one, and only one of these categories. In other words, you can’t believe in a fact, and only facts can be true. So, God can’t be believed in AND also be a fact! But neither can you believe that 2+2 = 4, the sky is blue, or grass is green. Those are just facts, not things you also believe, you silly non-Common Core indoctrinated person!

I hope you agree with me that it is absurd to force students to categorize all statements into only one of three “anchor chart” categories, and then call it a “critical thinking” assignment. It is sad that so many millions of students are being taught “how to think” using such irrational methods. And it doesn’t just start in 7th grade; McBrayer spotted the same type of anchor chart assignment in his child’s 2nd grade classroom!

The Katy ISD 7th-grade teacher directed the class to categorize the statement “There is a God” as opinion by labeling God as a myth. This is a fine tactic for someone who hates God to employ, because when most people, not just Common Core indoctrinated schoolchildren, hear the word myth, they think “legend,” or “fake story about the past.”

### C.S. Lewis to the Rescue

But, could a great story about the past also be true? Why does myth have to always make us think “fake Greek sky gods?” Here’s where C.S. Lewis rescues us from oversimplifying our world in a way that gives us a false view of reality:

Now the story of Christ is simply a true myth: a myth working on us the same way as the others, but with this tremendous difference that it really happened: and one must be content to accept it in the same way, remembering that it is God’s myth where the others are men’s myths: i.e., the Pagan stories are God expressing Himself through the minds of poets, using such images as He found there, while Christianity is God expressing Himself through what we call ‘real things’.

In one long, beautiful, eloquent, God-glorifying sentence, C.S. Lewis destroys the Common Core’s ridiculous “anchor chart.” Lewis words reassure us that legends can also be true! Or in Common Core language, opinions can also be facts, facts can be assertions, etc.

### Tools to Use in Your Thinking

You see, school-aged children don’t need to be trained “what to think,” nor do they need to be trained “how to think.” As math-teaching legend John Saxon once said,

God gives students the ability to think. Society does not give children that ability.

God designed us with the ability to think critically. The 7th grade Katy ISD student is a perfect example of that, as she was able to spot the flaw in her teachers’ fake “critical thinking” assignment, an assignment that will no longer be taught in Katy ISD thanks to her efforts.

What students need are tools to use in their thinking. And one of the best tools is mathematics. Some math curriculum to consider include any John Saxon-authored courses, as well as my company’s new curriculum, Shormann Math, a curriculum built on a solid foundation of mathematics’ legends, with Jesus Christ as the common core. Logic is another course worth considering. At a minimum, study this logical fallacy poster. Another resource is Introductory Logic by Roman Roads Media.  Books by Nancy Pearcey are also excellent resources for understanding the negative impact of oversimplifying the ‘real things’ C.S. Lewis was describing. Total Truth, Saving Leonard0, and Finding Truth are all excellent. And of course, any books or essays by C.S. Lewis! And last but not least, the Bible, without which we would not know that we are supposed to reason together (Isaiah 1:18).

### Shormann Math Builds Effective Study Habits

October 19, 2015

With Shormann Math, using 21st Century technology to create a math course allows us to obtain valuable information revealing that, regardless of skill level, students who want to learn math, can, and Shormann Math has the tools for them to do so.

For example, during quarterly exam week, students are provided with two full-length practice exams. Practice exams allow students to prove to themselves that they really do (or don’t) know the material covered that quarter. Besides the practice exams, they are given other guidelines on how to prepare for the exams. The guidelines are based on years of teaching experience, as well as observing university professors. Between my bachelor’s in aerospace engineering, and a PhD in aquatic science, I had a lot of professors and exams! And the best professors, the ones who really wanted you to learn the material, did two things: 1) they kept a file of previous exams in the library that students could check out and study, and 2) they had office hours so students could ask questions. Shormann Math provides both, with 1) practice exams that reward students for a good study effort and 2) free email Q&A any time.

But are the practice exams helpful? Well, see for yourself. The following graph displays the recent results of Quarterly Exam 1 scores for Shormann Algebra 1 and 2(beta) students.* The bottom line is that students with “Good” study habits made A’s on the exam. The graph is a display of the obvious fact that good study habits build fluency, resulting in good scores on the actual exam. Being fluent in math means you know how to use the rules to solve new problems. And the purpose of the Practice Exams in Shormann Math is to provide new problems so the student can prove to themselves whether they are fluent, and if not, what they need to review.

At some point in your life, you will be tested on a large amount of information. Whether it’s for a job you really want, a driver’s license, an SAT, ACT, MCAT, etc., sooner or later, test day is coming. And if you really want that license, or that job, etc., you are going to put the personal effort into it to study. Shormann Math is designed to help students build effective study habits in a less important setting where the stakes aren’t as high. But, as the results above reveal, the best curriculum in the world won’t make a bit of difference if the student doesn’t put that personal effort into following directions and studying effectively.

*Graph details: Scores are from Quarterly Exam 1 taken by students in Dr. Shormann’s live online Algebra 1 and 2 classes, October 2015. The three categories are based on student performance on the 2 practice exams take prior to the actual exam. The students are allowed to take the practice exam, review mistakes using the solutions manual provided, and then take it again. Students who put the effort into retaking each practice exam were rewarded for their effort with a higher grade. Students are also encouraged to show work on their paper, solving each problem by hand. For the actual exam, they are required to submit handwritten work on each problem. The practice exams were counted as one of their homework grades, providing further encouragement to complete them. The three categories were broken down as follows: “Good” students averaged 95% or better on the practice exams, all of which took at least one of the exams more than once in order to get a higher score, which means they took the time to correct their mistakes and study the problems they missed. “Mediocre” students took each exam once, but averaged below 95%, and showed little to no effort to try the exam again, missing a valuable opportunity to review and build fluency. “Poor” students did not attempt either practice exam. Of special note is the fact that the trend was consistent, regardless of which course students were doing (Algebra 1 or 2).  Also, because the students had the opportunity to retake each practice exam until they received a 100, study effort, and not skill level, was the main factor influencing performance on the actual exam. Not all students are equally gifted in math (or any subject), but students who are less-skilled at math can do better by studying harder. These results provide good evidence that, with Shormann Math, students who want to learn math, can, regardless of skill level!

August 6, 2015

### A Great Question

We recently received a great question about our new Shormann Algebra 1 course:

Are your courses best for mathy children, or can average students also complete them?

While “mathy” really isn’t a word, anyone with any teaching experience knows what this parent was talking about. Some students just “get” math quicker than others. They’re able to go farther and faster in math than most children their age. So, is Shormann Math mainly for these students, or is it more for students who are gifted in other, “non-mathy” areas?

### An Illustration

The best answer is that Shormann Math is for everyone! To help me explain how, first take a look at this photo I shot a few months ago of a Hawaiian green sea turtle. The photo appears at the top of Shormann Algebra 2, Lesson 25. You’ll see what this has to do with answering the parent’s question shortly:

Everyone loves sea turtles, right? I mean, do you know anyone who hates sea turtles? I don’t. There are some things in this photo that everyone can relate to, like beauty, design, color, and function, to name a few. There are also things that individuals gifted in certain areas would appreciate that others won’t. Photographers, for example, may be curious about what type of camera was used, resolution, lighting, etc. Everyone might notice how the magnified view of the eye is blurred, and composed of rows and columns of tiny squares. But only someone with a good knowledge of computers and/or digital photography could explain the “why” behind the tiny squares (called pixels).

### Connecting Students to Their World and Their Creator

But what if your child is a future computer scientist, engineer, etc., and they just don’t know it yet? What if they, or you, haven’t already drawn the line between “mathy” and “non-mathy?” Well, Shormann Math is for you, too! Because everyone is created in God’s image (Genesis 1:26-28), everyone is designed to be creative like Him, too. But while God can just create by speaking (John 1:1-5), we humans need tools. And mathematics is like a giant treasure chest of tools, waiting to be discovered and put to use.

But the primary focus of Shormann Math is not about math. It’s about relationship. It’s about using math to help a child discover more about God’s Word and His creation, and build their relationship with Christ.

If you study the greatest mathematicians in history, like we do in Shormann Math, you find that all their new mathematical discoveries were connected to their study of Creation. While not all of them acknowledged God, a lot of them did, and in doing so it allowed them to see farther and discover more than any of their predecessors. The rich Christian heritage of modern mathematics is not something to hide in the back of a dark closet, but, like a favorite painting, it should be placed in the right frame, with the right lighting, and set in a prominent place.

In a nutshell, here’s what Shormann Math is about:

Shormann Math is designed to connect students to their world and their Creator by using an incremental approach with continual review to teach 10 major math concepts from a Christian foundation.

### But Does it Work?

But does this “incremental approach with continual review” work? Well, the results of our Shormann Algebra 1 beta-test say “yes!” Pioneered by the late John Saxon (1923-1996), his “incremental approach with continual review”  has achieved astounding results. The results of Saxon Math in a traditionally low-performing Dallas public school were highlighted in this 1990 interview on 60 Minutes.

If the 60 Minutes interview doesn’t convince you of the merits of John Saxon’s approach, then maybe this historic quote by President Ronald Reagan will:

I’m sure you’ve probably heard about that new math textbook. It’s by a fellow named John Saxon, that has average I.Q. students scoring above high I.Q. students and has Algebra I students who use this textbook doing better on tests than Algebra II students who use the traditional text…

(Remarks at a White House Reception for the National Association of Elementary School Principals and the National Association of Secondary School Principals, July 29, 1983)

Even a former U.S. President saw the merits of a teaching method that could help the average student go farther in mathematics than they ever dreamed.

Scholars describe mathematics as “the language of science.” And what is a good way to learn a new language (or a sport, or an instrument)? Well, you learn some of the basics, practice for a while, and then learn some more. You use an “incremental approach with continual review!” And like a language, sport, or instrument, mathematics is not a passive, textbook-only activity. It’s an active, pencil and paper pursuit. The method is instrumental in making Shormann Math for everyone!

July 31, 2015

### Why do results matter?

Shormann Math builds on a solid foundation of time-tested teaching methods, including the incremental development + continual review format pioneered by John Saxon(1923-1996). And not just Saxon’s teaching methods, but his teaching thoughts as well, including his thought that

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

One of the primary reasons John Saxon developed his math curriculum in the 1980s was because new ways of teaching math were not working. Math “educrats” at the time were promoting their untested “visions” of math teaching. But with 3 engineering degrees, John was a math user before he became a math teacher. Not only that, he was a test pilot. If anyone knew the extreme value and importance of testing a new product, it was John!

Results matter because they reveal whether or not a new product really works. And while statistics certainly don’t reveal everything about a new product, they can certainly reveal many things. Most math curricula don’t provide this level of detail on student performance. But with Shormann Math, each new course is beta-tested in a live, online setting first before releasing it to the general public. The following are statistics from the beta-test of Shormann Algebra 1. The results show that the majority of students made an A! The following statistics, plus other detailed information about the course, can also be found in our Shormann Algebra 1 teacher’s guide. To purchase Shormann Math, click here.

### Overall Performance

Discussion: The average student in our beta test made an A in the class! Because each new Shormann Math course is beta-tested in a live online class setting, Dr. Shormann gets to know the students on more than just a “numbers only” basis. And we all know that God doesn’t make clones, so the fact that not every student performed the same should not be a surprise. Natural talent definitely matters, but so do things like attitude and maturity.  Dr. Shormann spends time during the video lectures encouraging students to develop fruits like patience and self-control (Galatians 5:22-23), as well as persevering with joy (James 1:2-3), and gratefulness (I Thessalonians 5:18).

### Practice Sets

Discussion: You’ve probably never seen statistics on student performance in a math class before, which is why it is important to discuss the data! The decreasing trend over time is exactly what we expected. Two big factors are responsible for the trend: 1) There’s more review of previously-learned concepts at the beginning, so it’s easier and 2) student effort tends to decrease the closer you get to the end of the year!

What we had hoped for was a Practice Set average above 85%, and that was achieved in all 4 quarters! 85% is a good cutoff for determining whether students are understanding, and retaining most of the concepts learned.

Note also the high first quarter average. Because Shormann Math is built on John Saxon’s method of integrating geometry and algebra, students using Saxon Math 8/7 or Saxon Algebra ½ will be most comfortable starting Shormann Math. However, not all beta-test students used Saxon previously, so the high first quarter average is a good indication that students who successfully completed any pre-algebra course should do just fine in Shormann Math.

### Weekly Quizzes

Discussion: Weekly Quizzes show a similar trend to the Practice Sets, challenging the students more as the year progressed. A score of 8 out of 10 or higher is a good indication of whether students understood the lessons covered that week. We are pleased that scores were well above this in all four quarters!

### Quarterly Exams

Discussion: Notice the Quarterly Exams do not follow the same trend as Practice Sets or Weekly Quizzes, with Quarter 1 having the lowest average. And this is where beta-testing a new product is so valuable. We realized that we were asking a lot for 9th-grade level students, most of which had never taken a cumulative exam like this. The solution? Practice exams! Just like when learning a sport, a musical instrument, etc., good practice results in good performance. The beta-test students clearly performed best on first quarter Practice Sets and Quizzes. Most likely, if they were given practice exams prior to their quarterly exam 1, this would have been their highest exam average. Now, all quarterly exams have two practice exams that students use to study for their actual exam.

85%+ is an indicator of good retention and understanding of concepts covered in a quarter. For all 4 quarters, student averages were at, or well above 85%. Because of Shormann Math’s format of continual review, we are basically asking students to be responsible for “all their math, all the time.” These results show that on average, students are responding very well!

### The New Shormann Math vs. Saxon Math and Common Core

April 21, 2015

Summary: If mathematics is the language of science, then fluency should be the goal, which means the traditional “layer cake” approach to 3 years of high school math (Algebra 1, Geometry, Algebra 2) is probably not the best approach. The shapers of Common Core’s math standards realized this, and now states that adopt their standards can choose between two high school “pathways”, either the layer cake or integrated approach, where students learn algebra and geometry together. John Saxon* actually pioneered the integrated approach in the United States back in the 1980’s, but his integrated approach was only one small part of his textbooks’ successes. His method of “incremental development with continual review,” combined with a constant encouragement for students to learn by doing, were the keys. Shormann Math builds on John Saxon’s efforts to really teach math like the language of science that it is, by not just connecting students to their world, but, more importantly, to their Creator. In doing so, students learn to wisely mingle concepts like science and Scripture, faith and reason. Doing so makes it easier to learn subjects like calculus, which really does require a faith commitment in order to make sense of it. Because of its obvious connections to God’s attributes, secular calculus courses steer clear of this, and in so doing make it much more difficult to learn. Shormann Math will change that.

*John Saxon passed away in 1996, and the company he founded, Saxon Publishers, is now owned by Houghton-Mifflin/Harcourt. They have since created some Saxon-in-name-only Algebra 1, 2, and Geometry textbooks. Click here to read our review of the new books and to learn why we don’t recommend them.

### The 10 Major Topics of Shormann Math

The 10 major topics of Shormann Math, compared to John Saxon’s books and Common Core standards.

Measurement is a topic that is a natural part of any math course seeking to teach math as the language of science. That it’s missing from three years of Common Core high school math is a huge problem. As a science class and lab teaching assistant during graduate school, one of the biggest math-related struggles I remember was students’ inability to convert from one unit to another. And it’s not just Common Core, most government school standards are weak in teaching measurement-related topics.

All Shormann Math high school courses will keep students fresh with working with measurements. Computers are also a very real part of every students’ world, so knowing about some of the mathematics behind them should be a priority. And, as mentioned in the Summary above, calculus becomes a normal part of high school math when one of the priorities is to connect students to their Creator.

### Foundations and Pedagogy

As Euclid famously said centuries ago, there is no royal road to learning. However, some methods are definitely better than others, and the Common Core’s integrated pathway (click here to read an Education Week review) is definitely a step in the right direction. However, the integrated approach has it’s own challenges. To really teach math like the active, hands-on language of science that it is, you have to teach it like languages are taught, or sports, or instruments, etc. You teach students a little bit about something, give them time to practice it, and then build on it. John Saxon called this “incremental development with continual review,” which is missing from Common Core.

Also missing from Common Core is the importance of math history. Understanding why they are learning the different math topics makes math more relevant to students. Learning some things about the people behind the math concepts they are learning, as well as some of the great, and not-so-great things they did, makes math more meaningful. And the connection to history also shines a bright light on the rich Christian heritage of mathematics, especially regarding algebra and calculus. Showing students how God’s attributes are clearly revealed in mathematics can make a huge difference in their comprehension and success in the course.

Shormann Math’s emphasis on math history means that, in developing the course, I dove deep into the classic works of Euclid, Newton, Euler, etc. Rather than reinventing the wheel, this study of the classics allowed me to develop a curriculum that stands on the shoulders of giants (a phrase often attributed to Isaac Newton). It should be a huge confidence-booster to parent and student alike to know your course is built on time-tested and proven methods for learning math.

### What 3 Years of Math Covers

By focusing on what matters most, Shormann Math does more in 3 years than either Common Core or Saxon. In the first two years, Shormann Math covers all the concepts presented on the SAT (the new 2016 version), the ACT, and both the CLEP College Algebra and CLEP College Math exam. A full credit of geometry is integrated into the first two years as well. This is different from John Saxon-authored texts, which include the geometry credit in years 2 and 3 (Algebra 2 and the first 1/3 of Advanced Math). And before you think Shormann Math couldn’t possibly have enough geometry, consider that we will cover all the standards, like perimeter/area/volume, similarity and congruence, circle and triangle theorems, and proofs. In addition, we will show students how the proof technique is not some isolated subject you only learn in geometry class, which is what most students, and parents, think it is. We’ll introduce students to proofs by studying the master, Euclid, covering several of his propositions. We’ll do the standard triangle proofs and circle proofs, but will also apply proof technique in other topics like algebra. And students will learn how proof is used in the real world. They’ll even learn how geometry is used in art and architecture. And on top of all that, we’ll introduce non-Euclidean geometry in Algebra 2, diving deeper in Precalculus. We’ll also use CAD programs like Geometer’s Sketchpad to complete proofs and more.

Finally, Shormann Math will introduce calculus fundamentals. By year 3 (precalculus), Shormann Math students will be very comfortable finding limits, and will have a solid grasp of derivatives and integrals. We hope all students will continue on to Shormann Calculus, but if not, they will be more than ready for college-level calculus. Of all the courses in college, calculus is the subject that opens the door to virtually every college major, or if the student cannot pass the class, closes the door on about 80% of majors. The first three years of Shormann Math will give students the confidence they need to take college calculus, and be at a level to help their peers learn it, which can also open up opportunities to build relationships and share the gospel. And completing 4 years of Shormann Math will allow students to possibly prepare for and pass either the CLEP or AP Calculus exam, receiving college credit for their efforts.

### But Saxon + DIVE Lectures do a lot of this already. Why make a new curriculum?

There are many reasons, here are a few:

• We can build the curriculum on a Christian and historical foundation, rather than bringing these fundamentals in from the side, like we do with the DIVE Lectures that teach Saxon Math.
• The one topic John Saxon didn’t integrate was calculus. We think it just might be the most important topic to integrate, and our current Shormann Math students are proving Algebra 1-level students can learn some calculus fundamentals!
• We don’t know how long Houghton Mifflin/Harcourt will continue to sell John Saxon-authored textbooks.
• We can take advantage of 21st Century technology and e-learning to provide more efficient and effective learning. Our self-paced e-learning format includes many powerful learning tools, including video lectures and video solutions to homework, all for about the same price as the Saxon home study kits. The following table lists some detailed differences between Shormann Math and Saxon Math.

### What is the prerequisite for Shormann Math Algebra 1?

Students who have successfully completed a standard pre-algebra course, including either Saxon 8/7 or Saxon Algebra Half, are ready for Shormann Math Algebra 1.

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Click here for a detailed description of Shormann Math, including sample video lectures and pricing information.

### 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.

### Shormann Math Core Ideas: Jesus Christ is the “Common Core.”

August 5, 2014

The following is the third 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.

Jesus Christ is the “Common Core” of Shormann Math

Perhaps you have heard of the United States government’s “Common Core” curriculum. Perhaps you have also heard that a lot of people are concerned about it. Leading experts believe the Common Core’s mathematics standards will not prepare students to study science, technology, engineering, and math (STEM) in a selective four-year college. And a white paper by the Pioneer Institute concludes by saying

“At this time we can conclude only that a gigantic fraud has been perpetrated on this country, in particular on parents in this country, by those developing, promoting, or endorsing Common Core’s standards.”

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. So, even though our primary goal is NOT to prepare students for STEM, we believe by putting Christ at the foundation, just like the world’s original universities did, students will naturally learn to use mathematical tools that will connect them to their world and their Creator.