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

Posted May 22, 2017 by gensci
Categories: Shormann Math, Teaching Mathematics

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This is the second 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 VIII. Divisibility of Extension in Infinitum

The controversy between modern philosophies and geometricians to which I have alluded, turns on the divisibility of body. This property is undoubtedly founded on extension, and it is only in so far as bodies are extended that they are divisible, and capable of being reduced to parts.

You will recollect that in geometry it is always possible to divide a line, however small, into two equal parts. We are likewise, by that science, instructed in the method of dividing a small line, ai, into any number of equal parts at pleasure, and the construction of this division is there demonstrated beyond the possibility of doubting its accuracy.

You have only to draw a line AI (plate II. fig. 23) parallel to ai of any length, and at any distance you please, and to divide it into as many equal parts AB, BC, CD, DE, etc. as the small line given is to have divisions, say eight. Draw afterwards, through the extremities A, a, and I, i the straight lines AaO, IiO, till they meet in the point O: and from O draw toward the points of division B, C, D, E, etc. the straight lines OB, OC, OD, OE, etc., which shall likewise cut the small line ai into eight equal parts.

Plate II, Fig 23, line AI parallel to line ai, each divided into 8 equal segments.

This operation may be performed, however small the given line ai, and however great the number of parts into which you propose to divide it. True it is, that in execution we are not permitted to go too far; the lines which we draw always have some breadth, whereby they are at length confounded, as may be seen in the figure near point O; but the question is not what may be possible for us to execute, but what is possible in itself. Now in geometry lines have no breadth*, and consequently can never be confounded. hence it follows that such division is illimitable.

*In Shormann Math, a line is defined as a widthless length, which is the same thing Euler is describing. In fact, all normal geometry courses define a line this way. The idea is that we are not concerned with how thick, or wide the line is. When you draw a line though, it has to have some thickness to it in order to be able to see it.

If it is once admitted that a line may be divided into a thousand parts, by dividing each part into two it will be divisible into two thousand parts, and for the same reason into four thousand, and into eight thousand, without ever arriving at parts indivisible. However small a line may be supposed, it is still divisible into halves, and each half again into two, and each of these again in like manner, and so on to infinity.

What I have said of a line is easily applicable to a surface, and, with greater strength of reasoning, to a solid endowed with three dimensions, length, breadth, and thickness. Hence is is affirmed that all extension is divisible to infinity, and this property is denominated divisibility in infinitum.

Whoever is disposed to deny this property of extension, is under the necessity of maintaining, that it is possible to arrive at last at parts so minute as to be unsusceptible of any farther division, because they ceased to have any extension. Nevertheless all these particles taken together must reproduce the whole, by the division of which you acquired them; and as the quantity of each would be a nothing, or cypher (0), a combination of cyphers would produce quantity, which is manifestly absurd. For you know perfectly well, that in arithmetic, two or more cyphers joined never produce any thing.

This opinion that in division of extension, or of any quantity whatever, we may come at last to particles so minute as to be no longer divisible, because they are so small, or because quantity no longer exists, is, therefore, a position absolutely untenable.

In order to render the absurdity of it more sensible, let us suppose a line of an inch long, divided into a thousand parts, and that these parts are so small as to admit of no farther division; each part, then, would no longer have any length, for if it had any, it would be still divisible. Each particle, then, would of consequence be a nothing. But if these thousand particles together constituted the length of an inch, the thousandth part of an inch would, of consequence, be a nothing; which is equally absurd with maintaining, that the half of any quantity whatever is nothing. And if it be absurd to affirm, that the half of any quantity is nothing, it is equally so to affirm, that the half of a half, or that the fourth part of the same quantity, is nothing; and what must be granted as to the fourth, must likewise be granted with respect to the thousandth, and the millionth part. Finally, however far you may have already carried, in imagination, the division of an inch, it is always possible to carry it still farther; and never will you be able to carry on your subdivision so far, as that the last parts shall be absolutely indivisible. These parts will undoubtedly always become smaller, and their magnitude will approach nearer and nearer to 0, but can never reach it.

The geometrician, therefore, is warranted in affirming, that every magnitude is divisible to infinity; and that you cannot proceed so far in your division, as that all farther division shall be impossible. But it is always necessary to distinguish between what is possible in itself, and what we are in a condition to perform. Our execution is indeed extremely limited. After having, for example, divided an inch into a thousand parts, these parts are so small as to escape our senses, and a farther division would to us, no doubt, be impossible.

But you have only to look at this thousandth part of an inch through a good microscope, which magnifies, for example, a thousand times, and each particle will appear as large as an inch to the naked eye; and you will be convinced of the possibility of dividing each of these particles again into a thousand parts: the same reasoning may always be carried forward, without limit and without end.

It is therefore an indubitable truth, that all magnitude is divisible in infinitum, and that this takes place not only with respect to extension, which is the object of geometry, but likewise with respect to every other species of quantity, such as time and number.

28th April, 1761.

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

Posted May 18, 2017 by gensci
Categories: Shormann Math, Teaching Mathematics

Tags: , , , , , ,

This is the first of eleven Letters of Euler I will rewrite and post on the subject of infinitesmals (the infinitely small), an idea that is fundamental to a good understanding of calculus. I am rewriting them from a 1795 English translation, and will edit some of the awkward character usage (among other things, the first “s” used in any word actually looks like an “f”), but otherwise, for the most part, I will leave it unchanged. Additions and edits will be marked by braces, […].

Considered by scholars as the best mathematician in history, Euler’s influence is everywhere present in modern mathematics. Yet as smart as he was, he still took time to bring difficult concepts down to a level where a non-mathematician might learn some things. And, as you will see, defend the Christianity at the same time.

Although these posts from Letters of Euler are for students in my Shormann Calculus course(available Summer 2018), any curious prince or princess is welcome to read them, too! The idea Euler (and myself) is trying to convey is that any real object can be divided, and divided again. And again, until it is in such small parts (infinitesimals) we can’t see them. Nevertheless, they exist. But how? To understand that, let’s begin with Euler’s description of the properties of any real object, which he refers to as a body.  Enjoy!

Letter VII. The True Notion of Extension

I have already demonstrated, that the general notion of body necessarily comprehends these three qualities, extension, impenetrability, and inertia*, without which no being can be ranked in the class of bodies. Even the most scrupulous must allow the necessity of these three qualities, in order to constitute a body; but the doubt with some is, Are these three characters sufficient? Perhaps, say they, there may be several other characters, which are equally necessary to the essence of body.

*if a body has extension, that means you can measure it (length, mass, etc.); if it has impenetrability, that means you can feel it, which is possible with any solid, liquid or gas; if it has inertia that means it has the physical property of resisting a change in motion.

But I ask: were God to create a being divested of these other unknown characters, and that it possessed only the three above mentioned, would they hesitate to give the name of body to such a being? No, assuredly; for if they had the least doubt on the subject, they could not say with certainty, that the stones in the street are bodies, because they are not sure whether the pretended unknown characters are to be found in them or not.

Some imagine, that gravity is an essential property of all bodies, as all those which we know are heavy; but were God to divest them of gravity, would they therefore cease to be bodies? Let them consider the heavenly bodies, which do not fall downward; as must be the case, if they were heavy as the bodies which we touch, yet they give them the same name. And even on the supposition that all bodies were heavy, it would not follow that gravity is a property essential to them, for a body would still remain a body, though its gravity were to be destroyed by a miracle.

But this reasoning does not apply to the three essential properties mentioned. Were God to annihilate the extension of a body, it would certainly be no longer a body; and a body divested of impenetrability would no longer be a body; it would be a spectre, a phantom: the same holds as to inertia.

You know that extension is the proper object of geometry, which considers bodies only in so far as they are measurable. [Geometry does not consider impenetrability and inertia.] The object of geometry, therefore, is a notion much more general than that of body, as it comprehends not only bodies, but all beings simply extended without impenetrability, if any such there be. Hence it follows, that all the properties deduced in geometry from the notion of extension must likewise take place in bodies, in as much as they are extended; for whatever is applicable to a more general notion, to that of a tree, for example, must likewise be applicable to the notion of an oak, an ash, an elm, etc. And this principle is even the foundation of all the reasonings in virtue of which we always affirm and deny of the species,  and of individuals, every thing that we affirm and deny of the genus.

There are however, philosophers, particularly among our contemporaries, who boldly deny, that the properties applicable to extension, in general, that is, according as we consider them in geometry, take place in bodies really existing. They allege that geometrical extension is an abstract being, from the properties of which it is impossible to draw any conclusion, with respect to real objects: thus, when I have demonstrated that the three angles of a triangle are together equal to two right angles, this is a property belonging only to an abstract triangle, and not at all to one really existing.

But these philosophers are not aware of the perplexing consequences which naturally result from the difference which they establish between objects formed by abstraction, and real objects; and if it were not permitted to conclude from the first to the last, no conclusion, and no reasoning whatever could subsist, as we always conclude from general notions to particular.

Now all general notions are as much abstract beings as geometrical extension; and a tree, in general, or the general notion of trees, is formed only by abstraction, and no more exists out of our mind than geometrical extension does. The notion of man in general is of the same kind, and man in general no where exists: all men who exist are individual beings, and correspond to individual notions. The general idea which comprehends all, is formed only by abstraction.

The fault which these philosophers are ever finding with geometricians, for employing themselves about abstractions merely, is therefor groundless, as all other sciences principally turn on general notions, which are no more real than the objects of geometry. The patient, in general, who the physician has in view, and the idea of whom contains all patients really existing, is only an abstract idea; nay the very merit of each science is so much the greater, as it extends to notions more general, that is to say, more abstract.

I shall endeavor, by next post, to point out the tendency of the censures pronounced by these philosophers upon geometricians; and the reasons why they are unwilling that we should ascribe to real, [measurable] beings, that is, to existing bodies, the properties applicable to [measurement] in general, or to abstracted [measurement]. They are afraid lest their metaphysical principles should suffer in the cause.

25th April, 1761.

 

4 Key Math Concepts You Won’t Find on Standardized Tests

Posted November 28, 2016 by gensci
Categories: Uncategorized

For the most part, standardized tests like the SAT and ACT are good indicators of math aptitude and college readiness. Since the tests are timed, you have about a minute to answer each problem. Success on these tests means having your fundamental math rules memorized and being fluent with their use.

However, there are some problems types that are really good for students to learn, and that take more than a minute to solve, even for the most fluent student. And these are not found on the SAT, PSAT or ACT. So, if a teacher and/or a math course is designed to “teach to the test,” it may be lacking some key concepts that are fantastic at building good problem-solving skills. More importantly, these concepts are (or should be) vital for teaching math as the “language of science,” which is what makes math real and useful and connects students to their world and their Creator.

The following is a list of 4 key concepts, all of which are present in Shormann Mathematics, but are normally missing from the SAT and ACT. Shormann Math teaches these concepts in more basic forms starting in Algebra 1, progressing to more complex forms later. Much of the text below was pulled directly from our Shormann Math lessons.

Measurement/Unit analysis

“To measure is to know” is a quote by William Thompson, Lord Kelvin(1824–1907), a Christian and scientist. What Kelvin meant was that if we can measure something, we then know something about it. As Christians, we must be careful about faulty reasoning that says by building our knowledge of nature, we gain enough evidence to conclude God’s existence. We should never think we need to “conclude” God from the evidence. On the contrary, God designed us to know He exists (Romans 1:20), so we start with God, who is the beginning of knowledge (Proverbs 1:7).

When we measure things, we often have to convert the measurement from one unit to another. Measuring and converting units are essential skills in everything from cooking to engineering. Measuring accurately, and honestly, is also important to God (Proverbs 20:10 and elsewhere).

Proofs

To understand any subject well, not just math, one must start with rules and definitions. As the famous math teacher John Saxon said, fundamentals like these form the “basis of creativity,” and this is true. Likewise, to understand God, you have to start with some foundational rules. And while Scripture is much more than a “rulebook”, it contains Truth that helps us know who He is, how to build a relationship with Him, and how to do the things He has called us to do. It is self-evident that to learn anything, we must do so using the deductive process of applying rules.

While postulates are statements assumed to be true without proof, theorems (propositions) are true statements requiring proof. One mark of a maturing Christian is that they are able to use Scripture to “give a reason” for the hope that is in them (I Peter 3:15). In the same way, a mature math student should be able to give a reason for the steps they use to complete a problem. In mathematics, proof and the techniques used to write proofs require us to be prepared to have an answer we can back up. It forces us to slow down and think things through a little more before we answer.

Infinite Series

Leonhard Euler (1707-1783) said that infinite series are a subject that should be studied with “the greatest attention.” Unfortunately, in most modern math courses, infinite series are studied little, if any, until calculus, where they tend to create a lot of confusion because students have a poor foundation. But Euler put them in his algebra book, Elements of Algebra, a book that most modern Algebra 1 and 2 courses are based off. If you start Shormann Math in Algebra 1, you will learn a lot about Euler and other famous mathematicians, and you will probably know more about series and infinite series than the average student your age.

But why did Euler think infinite series were so important, especially in regards to fractions? Well, what is calculus? It’s the study of speed, right? Or even more generally, it’s the study of rates of change. It’s a study of how this changes as that changes, and when we compare this to that, we are studying fractions! Not only that, when we break a fraction into an infinite series of discrete pieces, we are doing computations that computer programs must do. Building fluency with infinite series can really go far in connecting students to fundamental aspects of computers.

Vectors

If you understand that traveling North at 60 mph is different than traveling South at 60 mph, then you have a basic understanding of vectors. Vectors allow us to consider two things at the same time, such as an object’s speed and it direction of travel. And something called the Parallelogram Law provides a simple way for understanding how to add vectors. In fact, the famous mathematician Alfred North Whitehead (1861-1947) believed that the Parallelogram Law “is the chief bridge over which the results of pure mathematics pass in order to obtain application to the facts of nature.” In other words, vectors are a really important tool for studying God’s creation!

Concepts like these are not usually learned overnight. Like learning a language or a new instrument, sport, etc., it take patient practice over several years. That’s why Shormann Math introduces these concepts in more basic forms starting in Algebra 1, giving students time to gradually build skills through practice and repetition. Click here to learn more about Shormann Math, and how Shormann Algebra 1 and 2 also help prepare students for the SAT, ACT and CLEP exams. Thanks for reading this post!

Building Good Study Habits with Shormann Math

Posted May 16, 2016 by gensci
Categories: Teaching Mathematics

Tags: , , , ,

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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.

 

C.S. Lewis Destroys Common Core in One Sentence

Posted December 3, 2015 by gensci
Categories: Teaching Mathematics

Tags: , , , , , , , , , ,

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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-diseased 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 absolutely 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

Posted October 19, 2015 by gensci
Categories: Teaching Mathematics

Tags: , , , , , , ,

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.

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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!

Shormann Math is for Everyone

Posted August 6, 2015 by gensci
Categories: Teaching Mathematics

Tags: , , , , , , ,

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:

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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!

Click here if you want to learn more about Shormann Math, including pricing, sample lectures and homework, a detailed teacher’s guide, and more.