Posted tagged ‘Leonhard Euler’

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

May 22, 2017

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

May 18, 2017

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.


It Only Takes an Instant

December 1, 2014
Two images of a wave in Waimea Bay, Hawaii, taken 0.1 s apart, reveal how quickly things can change from beauty to chaos (or vice versa).

Two images of a wave in Waimea Bay, Hawaii, taken 0.1 s apart, reveal how quickly things can change from beauty to chaos (or vice versa). © 2014 David E. Shormann

I never cease to be amazed by the technology of today’s digital cameras. By pushing a few buttons, anyone crazy enough to bob around in a 6-8′ high shore break can capture some amazing beauty in God’s creation! But as the photos above reveal, the beauty only lasts for an instant.

It seems impossible that things can change from beauty to chaos, or vice-versa, in so short a time. But big changes can and do occur in even less time than the tenth of a second that elapsed between these two photos.

Strangely enough, instantaneous change, something us humans really can’t fully comprehend, is behind almost every major technological achievement of the past 300+ years!  How can that be? How can something we will never fully understand help us make all sorts of useful devices? Well, some things we just have to take on faith. Faith is at the heart of the branch of mathematics known as calculus. And calculus is all about the study of instantaneous change.

Subjects like calculus are easier to grasp when we consider the Author of every instant of time, and Creator of the biggest  and best instantaneous changes of all. Paul writes in 1 Corinthians 15:52 how we will be changed “in a moment, in the twinkling of an eye, at the last trumpet. For the trumpet will sound, and the dead will be raised imperishable, and we shall be changed.”

Leonhard Euler (1707-1783), a devout Christian man considered the best mathematician ever, wrote that “it is God, therefore, who places men, every instant, in circumstances the most favourable, and from which, they may derive motives the most powerful, to produce their conversion; so that men are always indebted to God, for the means which promote their salvation.”

Euler understood God’s relationship with man and creation very well. He also understood mathematics really well, too! Much of the way we teach mathematics today comes from Euler’s textbooks on the subject.

In our new Shormann Mathematics curriculum, we believe that all 10 major topics covered, including and especially calculus, are best understood by connecting the study of mathematics to Jesus Christ, the founder of all knowledge, and the founder and perfecter of our faith (Hebrews 12:2).

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

February 8, 2012

Add to FacebookAdd to DiggAdd to Del.icio.usAdd to StumbleuponAdd to RedditAdd to BlinklistAdd to TwitterAdd to TechnoratiAdd to Yahoo BuzzAdd to Newsvine

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

Physical Differences

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

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

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

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

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

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

Content differences

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

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

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

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

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

Differences in methodology

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Time to DIVE

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

June 2015 update:

Our new curriculum, Shormann Math, is here! Click here to learn more.

Understanding vs. Memorization

April 29, 2011

Add to FacebookAdd to DiggAdd to Del.icio.usAdd to StumbleuponAdd to RedditAdd to BlinklistAdd to TwitterAdd to TechnoratiAdd to Yahoo BuzzAdd to Newsvine

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

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

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

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

understand-perceive the meaning of

memorize-commit to memory; learn by heart

analogy– a comparison between two things

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

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

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

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

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

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