Knowing God – And How to Think About Him

I’ve enjoyed discovering this blog in the past year. It’s called Beauty for Truth’s Sake and includes frequent reflection on the surprising delights of thinking truthfully instead of being driven by the appetites for power or pleasure.

When you read something like this, you know the writer thinks into things more than we normally do:

Mathematics, in its own way (and you won’t hear this said too often!), is a picture of love.

This was in an article on analogy and how we can, since the incarnation, think about God. Take a look – add it to your blog list.

On Teaching Math

One trouble with learning math is that children who are accustomed to thinking of learning as retention of information (no they wouldn’t put it that way) have a hard time adjusting to an art.

In the art of mathematics, you’re goal is not to remember facts. Your goal is to be able to do things with those facts – to develop skill and understanding.

It seems important to me, therefore, that in the early grades the teacher deliberately combine the three different aspects of learning:

  1. Drill the facts into their heads (five minutes two or three times a day for 12 years is probably enough)
  2. Give students problems that demand imagination. These problems need gaps that the mind has to leap over, even when there is no given process. The older and more confident students get with math, the bigger these gaps can and need to become.
  3. Make sure they know the facts well enough to remember them before you start drilling them.

1 and 3 create a dynamic process in which the teacher needs to be aware of her students state. If he does not know a math fact, he should not be drilled on it. If he does know it, he should be drilled.

Drilling is necessary for speed of recall and, even more, for adaptability and use.

So you need all three steps.

1. Tell them the facts as isolated facts. Over and over again. 2+3=5, 2+3=5, 2+3=5.
2. As they gain recall (and this can take a long time – these are the hardest lessons a person ever has to learn in math), begin to drill them by mixing up the facts they have to recall.
3. Present them with challenges that demand mathematical imagination and both challenge and expand their understanding/perception of how numbers behave.

What Abraham Lincoln studied

Abraham Lincoln, it is widely known, carried around two books: the Bible and Shakespeare. Did you know that, according to this blog, he carried a third? That’s right, Euclid’s Elements. More evidence of the decline of the western mind and its devaluing of both knowledge and the human faculties of perception is found in its neglect.

Incarnational Teaching in Kindergarten

I am increasingly amazed at the power of classical modes of instruction to enable students and even teachers to better understand ideas.

During yesterday’s apprenticeship phone call, Buck Holler, an apprentice from Geneva School of Manhatten, described how a kindergarten teacher applied the mimetic mode to guide her kindergarten students to understand what a polygon is.

He said that when the lesson was over, the kids understood it so well they didn’t need to do a worksheet.

That launched my thoughts into a comparison of modern math programs with the classical approach. The differences are too vast to explore deeply here, but one in particular stood out to me.

As always, the difference is rooted in the priority given to ideas and therefore to thinking.

One popular math teacher, for example, stated very clearly that  he developed his math program to improve student scores on standardized tests – not, by implication, so they would be able to think better mathematically. As a result, low scoring schools have consistently found that if they switch to this program, their test scores improve.

But unless they have mathematically and pedagogically sound teachers, schools using this program have not produced a vast quantity of students who can think mathematically.

I believe the reason for this is in the developer’s approach to teaching math. His pedagogy is rooted firmly in the behavioral sciences, so he sees learning as a stimulus-response activity.

If you stimulate the mind to perform an operation and then reward it when it does it the correct way, then eventually it will perform that operation whenever confronted with a similar context. Of course, it becomes very elaborate, being the human mind and all, but that’s the fundamental idea behind this program’s techniques and its why it uses a cyclical approach. More on that in a moment.

In the classical tradition, by contrast, mathematics was treated as a contemplative activity. In other words, the students were not treated to a series of intellectual stimuli when they were taught. Instead, they were presented with types of the idea to be learned and they learned how to think by attending to those types. That probably sounds scary to an unfamiliar modern teacher, but in fact it is gloriously simple.

If the idea is polygons, then the teacher presents multiple examples of polygons to the students. The students describe them in as much detail as they can to aid their attentive perception. Then they compare them with each other. In a very short time, they will have learned what a polygon is.

If the student is being taught an algebraic principle, they are shown that principle at work in various contexts. They attend perceptively to each individual type. Then they compare the types with each other. Pretty soon, through the teacher’s guidance, they come to see for themselves, to perceive, the idea that has been embodied in the types.

The same principle applies in a literature or history class, though the ideas will be less precisely defined. For example, if a school wants a student to understand and appreciate justice, then it will ensure that students spend many years contemplating types of justice – i.e. just people, just actions, and just events: stories.

Aesop’s Fables provide priceless instances of justice embodied, which is why Martin Luther, for one instance, regarded them as priceless. “Needless to say”, the perfection of justice is embodied in Christ Himself, so the school that hopes to bear the spiritual fruit of just students will spend a great deal of time contemplating the words and actions of our Lord.

The main reason this approach to teaching has been dropped seems to be that, since Dewey, education is rooted in a behavioral psychology (even before Skinner developed the dogmas of behaviorism) in which experience is the dominant mode of learning and ideas are at best words and at worst meaningless. Combine that with the need to appear to teach large classes of students and there seems to be no motivation for contemplating ideas. Thus the cyclical approach, in which the stimulus-response sequence is stretched over time, but the students are never deliberately guided to contemplate the idea for its own sake.

This is why I often argue that, while the stages of a subject and of a child’s development are powerful concepts, the real glory of the trivium as three stages is in the individual lesson: grammar – present types; logic – compare types; rhetoric – express and apply the idea.

But when we stop contemplating ideas, we may be doing a lot of things, but one thing we are not doing is providing a classical education. Nor are we wisely leading children on the path to wisdom. So thanks, Buck, for reminding us how much children love ideas and how easily they can absorb them when we teach them the way God teaches us: incarnating what we want them to understand.

My Great Mathematical/Historical Discovery

While I was in Austin, at Veritas Academy, we went through a discussion speculating about the signs used in addition (+ adds a vertical line to -, – subtracts the vertical line from +, = are two equal lines). It was pretty exciting for me, because it indicated that the signs actually do mean something.

You can imagine the thrill I felt when I opened my History of Mathematics by Carl Boyer, opened it rather randomly, and read these marvelous confirming words by Robert Recorde (who, as we all know, made the two equal parallel lines the “equals” sign):

I will sette as I doe often in woorke use, a paire of paralleles, or Gemowe [twin] lines of one lengthe, thus: =, bicause noe 2. thynges, can be moare equalle.

Not much for spelling and grammar ; ), but what a great historical discovery this is, at least for me who loves these cheap thrills that connections and detections give me.

it’s really just simple math

Thanks to Buck Holler for this fabulous video on simple math that explains the bailout:

This serves to illustrate how a theory can direct us away from reality, doesn’t it!

The Simplified Curriculum

When we think of curricula, we tend to think of classes or subjects and materials to read or study in those subjects. That’s a very fine thing to do and we should keep doing it. I want to suggest that there might be more to think about and it’s one of those “mores” that make things over all “less” – that is, less confusing, less work, less anxious.

The more that I’m referring to is logic.

But wait! Don’t go so fast. Let me explain myself.


Look at it this way. A lot of subjects, especially in the sciences, end with “logy.” Why is that? Because “logy” comes from logos, which means word, or reason, or idea (or quite a few other things). If the Greek word logos has a core meaning (and I’m not sure it does), it would be something like “a unifying principle or reason.” Thus, the unifying principle of “biology” is “bios” or life. The unifying principle of physiology is “physio” but I’m not sure what that is. It’s obviously got something to do with the physical body.

In other words, each subject has a unifying principle that makes it the subject that it is. Strictly speaking, classical educators used to call these subjects “sciences,” which meant, to them, a domain of knowledge or inquiry. To us, science usually refers only to what they called the natural sciences or even natural philosophy. So, with your permission, I am going to use the old language, and refer to subjects that are ordered around a unifying principle “sciences” instead of subjects. You’ll see why in just a moment.

The main point I’ve made so far is that each science has a unifying principle or idea that makes it what it is. (life for biology, God for theology, etc.)

I should point out that this is true even for those sciences that don’t end in logy. For example, some subjects end with “nomy,” such as “economy,” “astronomy,” etc. In this case, the ending comes from “nomos” which means laws or customs, the main point being that something happens regularly. “Economy” comes from a funny Greek word: oikonomos, and I would argue that it literally means “household customs.”

You can see how words can lose their attachment to their heritage! Astronomy is already a Greek word. It means the laws of the stars. Some sciences are very precise, like astronomy, which strictly follows the laws of physics. Others are much less precise, like economics, which strictly follow only one law: “If momma ain’t happy, ain’t nobody happy!”

At this point, you may have noticed something very interesting. Given all these “logies” and “nomies” and other endingies (the accent goes on the second syllable in that very subtle AGreek word), and given that some are known more precisely than others, it seems that each of them needs to be studied differently! I will not study the customs of the household the same way I study the movement of the stars. I will not study literature and history the same way I study chemistry and physics. Each science asks a different set of questions. Each gets answers to its questions in a different way. Each has its own logic.

Bang! That’s it! I told you it would be worth it if you stayed.

At the beginning I pointed out that when we think about the curriculum, we need to think about the subjects we study and the materials we use to study them. Now I hope we can see that we also need to think about the logic of the subject (or science) that we are studying. Until we get the logic of a science we don’t get the science, no matter how well we know the content of the science. That’s why when you teach, no matter what you are teaching, you always want to teach your students how to think in the given science you are teaching.

Now here’s where it gets especially exciting. I said above that this “more” would make things “less” confusing, less work, and less anxiety. But at this point you might be thinking, “What!? Now I have to teach logic too!? Augghh!!!!”

Be still, oh restless heart!

Above I pointed out that the “logy” ending comes from logos, and that logos, at least in that context, means a unifying principle or what we can now call “the logic of a science.”

Think about this: what would you have if you dropped all the particular sciences and started studying the “logies” themselves? That’s right, you’d have logic. As every particular science has its own logic, so all the logics combined make up Logic itself. In the same way, as every particular science has its own unifying principle, so every science combined has one common unifying principle or Logos. And that is Christ. He truly is the one in whom all things are held together, not only physically but in their very essence.

There are a number of ways we can apply these facts to our thinking, teaching, and curriculum development, most of which I haven’t thought of. If you have any ideas, I’d love to hear them, either here or in the CiRCE forum.

Here are some thoughts on relating this to our curriculum, and this is where I’ll keep my promise up above, when I assured you that you would see why I call subjects “sciences” in this essay. Subjects has become a bit of a lazy word. We throw everything we study into that category. So art is a subject, music is a subject, gymn is a subject, as are math, science, literature, history, etc. By calling everything a subject, we are enabled to not think about important distinctions between types of subjects.

That doesn’t mean we should never call anything a subject, it just means we also need to learn ways to distinguish types of subjects. And the way we distinguish them is by, one, their purpose, and two, the logic each employs.

To bring this into the classroom, the classical theorists distinguished first of all between arts and sciences. An art is a way of doing something. A science is a domain of knowing. This is an amazingly important distinction if only for this somewhat obvious reason: to know something you have to do something: you have to study it. May I add that you have to study it correctly, according to its nature, using the sort of logic that is native to that subject.

So if I’m going to learn physics, I first need to learn math. If I’m going to learn biology, I need to become skilled at inductive logic. If I’m going to learn literature, I need to learn grammar, logic, and rhetoric.

In this sense, math and the trivium are not merely subjects; they are arts. They are the arts that enable us to gain the knowledge that makes up the science. Historically, they have been recognized as the arts that are universally necessary for study of every science, for leadership, and for human excellence. They have been called the humanities (Cicero’s term) and the liberal arts. They are the necessary foundation for civilized society and culture.

The fulness of this idea can be seen when we look at it at its highest level of abstraction and its most immediate level of concrete application. At the highest level of abstraction, we see that every object of study (every science) is united by a common unifying principle that we call the logos and that the scriptures and experience identify as Christ. At the most immediate level, we can see that the foundation of all learning is mastery of the tools of thinking. These can be divided into the seven liberal arts. I would suggest that those seven are then bound together by the central art of the trivium and the art that links the trivium with the quadrivium, then binds the seven into a unified package, and, in turn, bridges learning with experience and with the development of the soul. That unifying art is, just as the unifying science is, logos, which in its concrete unifying activity, we call logic.

Perhaps you can see that the only way to really integrate a curriculum is through the trivium, understood not only as psychological stages, but also as the tools of learning that Dorothy Sayers wanted us to restore. The arts of the trivium: grammar, logic, and rhetoric.

Every subject has its object, the thing you are thinking about when you study it. Every subject also has its logic, the way you think about the subject. And the object of every study and the logic of every study is contained in the unifying study of logic itself.

If I’m able to, I will describe how logic needs to be taught if this potential is to be fulfilled. Let me add that it is an awesome, transforming potential.