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

Posted May 29, 2017 by gensci
Categories: Uncategorized

This is the third 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 IX. Whether this Divisibility in Infinitum Takes Place in Existing Bodies?

It is, then, a completely established truth, that extension* is divisible to infinity, and that it is impossible to conceive parts so small as to be unsusceptible of farther division. Philosophers accordingly do not impugn this truth itself, but deny that it takes place in existing bodies. They allege, that extension, the divisibility of which to infinity has been demonstrated, is merely a chimerical object, formed by abstraction; and that simple extension, as considered in geometry, can have no real existence.

*if a body has extension, that means you can measure it (length, mass, etc.).

Here they are in the right; and extension is undoubtedly a general idea, formed in the same manner as that of man, or of tree in general, by abstraction; and as man or tree in general exists not, no more does extension in general exist. You are perfectly sensible, that individual beings alone exist, and that general notions are to be found only in the mind; but it cannot therefore be maintained that these general notions are chimerical; they contain, on the contrary, the foundation of all our knowledge.

Whatever applies to general notion, and all the properties attached to it, of necessity take place in all the individuals comprehended under that general notion. When it is affirmed that the general notion of man contains an understanding and a will, it is undoubtedly meant, that every individual man is endowed with those faculties. And how many properties do these very philosophers boast of having demonstrated as belonging to substance in general, which is surely an idea as abstract as that of extension; and yet they maintain, that all these properties apply to all individual substances, which are all extended. If, in effect, such a substance had not these properties, it would be false that they belonged to substance in general.

If then bodies, which infallibly are extended beings, or endowed with extension, were not divisible to infinity, it would be likewise false, that divisibility in infinitum is a property of extension. Now those philosophers readily admit that this property belongs to extension, but they insist that it cannot take place in extended beings. This is the same thing with affirming, that the understanding and will are indeed attributes of the notion of man in general; but that they can have no place in individual men actually existing.

Hence you will readily draw this conclusion: if divisibility in infinitum is a property of extension in general, it must of necessity likewise belong to all individual extended beings; or if real extended beings are not divisible to infinity, it is false that divisibility in infinitum can be a property of extension in general.

It is impossible to deny that the one or the other of these consequences without subverting the most solid principles of all knowledge; and that philosophers who refuse to admit divisibility in infinitum in real extended beings, ought as little to admit it with respect to extension in general; but as they gran this last, they fall into a glaring contradiction.

You need not to be surprised at this; it is a failing from which the greatest men are not exempt. But what is rather surprising, these philosophers, in order to get rid of their embarrassment, have thought proper to deny that body is extended. They say, that it is only an appearance of extension which is perceived in bodies, but that real extension by no means belongs to them.

You see clearly that this is merely a wretched cavil, by which the principal, and the most evident property of body is denied. It is an extravagance similar to that formerly imputed to the Epicurean philosophers, who maintained that every thing which exists in the universe is material, without even excepting the gods whose existence they admitted. But as they saw that these corporeal gods would be subjected to the greatest difficulties, they invented a subterfuge similar to that of our modern philosophers, alleging, That the gods had not bodies, but as it were bodies, (quasi corpora) and that they had not senses, but senses as it were; and so of all the members. The other philosophical sects of antiquity made themselves abundantly merry with these quasi-corpora and quasi-sensus; and they would have equal reason, in modern times, to laugh at the quasi-extension which our philosophers ascribe to body; this term quasi-extension seems perfectly well to express that appearance of extension, without being so in reality.

Geometricians, if they meant to confound them, have only to say, that the objects whose divisibility in infinitum they have demonstrated, were likewise only as it were extended, and that accordingly all bodies extended as it were, were necessarily divisible in infinitum. But nothing is to be gained with them; they are resolute to maintain the greatest absurdities rather than acknowledge a mistake. You must have remarked, that this is the character of almost all scholars.*

3rd May, 1761

*NOTE: In 2017, beware of the scholar who claims that, in nature, what we see is merely quasi-design, the appearance of design. Making such an absurd statement is evidence of a stubborn refusal to acknowledge the Designer.


A mother humpback whale and calf. That the calf is a scale model of it’s mother is no accident or mere “appearance of design.” It is clear evidence of design, and their Designer.

Comparing Khan Academy’s Mastery Approach to Shormann Math’s Fluency Approach for Teaching High School Math

Posted May 25, 2017 by gensci
Categories: Shormann Math, Teaching Mathematics

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This is the first in a series of posts comparing Khan Academy’shormann khan comparison memes online math courses to our new Shormann Math curriculum. Shormann Math is part of DIVE Math and Science.

When Sal Khan started tutoring his cousins in 2003 using digital whiteboard video lectures, my family’s small business was already in its third year of producing similar products. Since then, both Khan Academy and DIVE have continued to offer digital video lectures for learning mathematics and other subjects, leveraging the ever-improving array of digital content delivery methods. Today in 2017, both Khan Academy and DIVE offer self-paced eLearning math courses. So, while there are obvious similarities between the two, there are also some pretty big differences.

Khan’s Store-Bought Layer Cake vs. Shormann Math’s Homemade Pizza

To begin, it is important to understand that Khan Academy teaches a Common Core-based “layer cake” approach to high school math. Shormann Math, on the other hand, teaches an integrated approach pioneered in Europe and Asia (you know, the countries that always beat us on international math tests), and introduced to the United States by the late John Saxon. John Saxon believed in the common-sense idea that results, not methodology are what mattered most(we’ll explore Khan Academy vs. Shormann Math results in a later post).

With the layer-cake approach, high school math is taught in layers, beginning with Algebra 1. Geometry is next, then Algebra 2. Each course is typically covered in one school year.

With the integrated approach, students learn algebra and geometry together. For example, Shormann Math does not have a separate geometry course, because geometry is integrated into Algebra 1 and 2. A geometry credit is included in Shormann Algebra 1 and 2, so students can take 3 years to complete the two courses and be “on track.”

Using a food analogy, both a store-bought layer cake and a homemade pizza can be delicious. With a store-bought layer cake, everybody gets the same thing, which is not always a bad thing, but can be. With a layer cake, if you just like vanilla, you can focus on eating the vanilla and not the chocolate. Or vice-versa.

With a homemade pizza, at least the one my family has made for years, each bite is an integrated medley of cheeses, meats, and vegetables. It is unique, diverse, and anchored in an amazing story that originated on float trip adventures deep in the Alaskan wilderness. Our homemade pizzas are about more than pizza, just like our Shormann Math courses are about more than math.

homemade pizza

Shormann Math, like our family’s legendary pizza, is a savory recipe developed over time with proven ingredients.

From my experience, a layer-cake approach is good for reviewing a particular subject, or specific concepts within a subject. Our CLEP and AP test prep courses employ this method. In other words, the layer-cake approach is good for review, which means Khan Academy math is good for testing your math skills in a particular subject. Khan Academy can help you determine if you have mastered a particular concept for the short-term.

Shormann Math, on the other hand, can help you build fluency in mathematics, which means you develop mastery and you retain it for the long term because you keep using it. In the beginning of every Shormann Math course, we define mathematics as the language of science and a God-given tool for measuring and classifying pattern and shape. I am not sure how Khan Academy defines mathematics, as I couldn’t find a definition anywhere. When you learn a new language, you don’t just learn nouns for a year, then verbs for a year, etc. You learn a little of the basics of everything, then you start putting them into sentences, which you practice. And practice some more. And then you review some more. And practice some more.

Building on fundamentals is how you learn a language, or a sport, or an instrument, or just about anything that you, eventually, hope to excel at. Building fluency takes time. Up next, we’ll explore Khan Academy’s “math procedures” to Shormann Math’s “math story,” followed by real data from real Shormann Math students on how the fluency approach can turn a bad math student into a good one. Stay tuned and thanks for reading!




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).


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.


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

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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).