# The mathematical part of classical education

The mathematical part of classical education March 12, 2010

Those who are bringing back classical education as an alternative to the deadends evident in John Dewey’s progressive education are familiar with the trivium: grammar, logic, and rhetoric, the three liberal arts that lead to a mastery of language. The other four liberal arts, the quadrivium, though, gets short shrift: arithmetic, geometry, music, and astronomy.

Many people think that the liberal arts is just another word for the humanities, forgetting the quadrivium completely. Dorothy L. Sayers, whose essay “On the Lost Tools of Learning” was a major catalyst for the revival of classical education, thought that the quadrivium represented “subjects” that would be learned after the trivium provided the tools for doing so. In this she was just wrong. The quadrivium are “arts”; that is, powers of the human mind. They are essentially mathematical, even in the way music was approached. Thus, classical education embraces the two spheres that educators recognize are necessary for education: language and mathematics.

Anyway, my daughter, who has been studying Boethius, the great systematizer of the quadrivium, explained to me the connections between the arts of the quadrivium, in a way that also helped me see the way mathematics really does provide a unifying model for the order and design that underlies all existence.

arithmetic = numbers
geometry = numbers in space
music = numbers in time
astronomy = numbers in space and time

Do you see why music is numbers in time? And why astronomy is numbers in space and time?

Now what we need is to bring mathematical education back from the dead–it’s telling that progressive education, for all its claim of being scientific and all, is failing most dramatically precisely in teaching science and mathematics–by coming up with a classical way of teaching it. Does anyone have any ideas? (And by this I don’t mean just teaching it more effectively or traditionally, such as Saxon Math. That and similar methods still lift numbers out of any context, which is not the classical way.)

HT: Joanna

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• Philip Larson

Thank you for noticing this! For quite awhile I’ve told classical educators that their view couldn’t not account for mathematics and the natural sciences. I’m sure there is a way for a classical-minded person to be serious about them all.

Here’s a suggestion. Mathematics tends to be seen as distinct from the humanities so long as it is taught as pure mathematics (or even in a Saxon-like focus on symbol manipulation). Perhaps a bridge can be built by using a mathematics curriculum that is centered on solving realistic problems, problems that occur in science, engineering, business, human relations, literature, linguistics, etc. There are a few curricula resources out there that do this, or at least that would support a teacher who wanted to go this direction. I know by experience that this can be done at the junior high, senior high, and junior college levels. I assume it should be easy at the elementary school level.

• Philip Larson

Thank you for noticing this! For quite awhile I’ve told classical educators that their view couldn’t not account for mathematics and the natural sciences. I’m sure there is a way for a classical-minded person to be serious about them all.

Here’s a suggestion. Mathematics tends to be seen as distinct from the humanities so long as it is taught as pure mathematics (or even in a Saxon-like focus on symbol manipulation). Perhaps a bridge can be built by using a mathematics curriculum that is centered on solving realistic problems, problems that occur in science, engineering, business, human relations, literature, linguistics, etc. There are a few curricula resources out there that do this, or at least that would support a teacher who wanted to go this direction. I know by experience that this can be done at the junior high, senior high, and junior college levels. I assume it should be easy at the elementary school level.

• I would like to see a curriculum that incorporates math and the humanities. We homeschooled using a classical method, and despite my husband having a degree in math, our mathematical education was not classical, with the exception of my teaching it through the three stages of the trivium. My children are all excellent at math and all three are musicians, and I know they are related. The problem is that math is seen as a technical language, totally unrelated to verbal language.

• I would like to see a curriculum that incorporates math and the humanities. We homeschooled using a classical method, and despite my husband having a degree in math, our mathematical education was not classical, with the exception of my teaching it through the three stages of the trivium. My children are all excellent at math and all three are musicians, and I know they are related. The problem is that math is seen as a technical language, totally unrelated to verbal language.

• EricM

I wonder if the key to this issue is “context.” I see this in my work (I work in computer and network security). If I only talk about security controls, I fail. However, if I can frame the issue in a business context (risk to the business, impact on the business, assistance to the business), then it is much easier to make executives understand the issue and to get on board with a solution.

Math may be suffer from the same problem. Without a context, geometric proofs are boring. Without a context, why would someone want to work hard at trigonometry or calculus? As was noted by Philip, it is possible to show real world problems that are solved with mathematics when you get to high school and college. You must still teach the theory so that the student learns to apply the math to problems that are different than the examples, but putting the math into a real world context must be done.

BTW – I love the “math in space, math in time, and math in space and time” quote.

• EricM

I wonder if the key to this issue is “context.” I see this in my work (I work in computer and network security). If I only talk about security controls, I fail. However, if I can frame the issue in a business context (risk to the business, impact on the business, assistance to the business), then it is much easier to make executives understand the issue and to get on board with a solution.

Math may be suffer from the same problem. Without a context, geometric proofs are boring. Without a context, why would someone want to work hard at trigonometry or calculus? As was noted by Philip, it is possible to show real world problems that are solved with mathematics when you get to high school and college. You must still teach the theory so that the student learns to apply the math to problems that are different than the examples, but putting the math into a real world context must be done.

BTW – I love the “math in space, math in time, and math in space and time” quote.

• I would strongly agree with Mr. Philip Larson concerning applications. I happen to be a mathematical physicist (more applied even than applied mathematicians, and more theoretical than theoretical physicists) with a desire, eventually, to teach math, physics, and logic in a classical Christian school. I have examined the views of many, certainly not all, pure mathematicians: many of them tend to rejoice in doing work that is “unsullied” by application. That view is nonsense. Mind you, I’m not saying that pure mathematics has no value or that nobody should engage in it. I simply disagree with that higher-level philosophy; it seems to me to exhibit something of the universal desire of Babel: autonomy. I prefer to view mathematics more incarnationally. Mathematics, to me, is the discovery of numerical order and patterns in God’s creation. Pure mathematicians, such as I have described, should not teach mathematics at the grade level. The best math teachers for grade level, including in a classical Christian school, are, in my opinion, engineers. They are the ones who have really used this math in the workplace, and can furnish real real-world problems, as opposed to what usually passes for such.

Referring back to Dr. Veith’s original post, I am aware of the (hopefully) friendly debate between those who defend the Sayers insight, and the approach more along the lines of Littlejohn and Evans in their book Wisdom and Eloquence. While I think the differences are real, and while I side with Sayers, I think that the two approaches tend to end up in more or less the same place. Littlejohn and Evans advocate using a technique very like the Sayers-type Trivium in practice, as a method of teaching all 7 liberal arts. They are simply against the idea of calling the Trivium a pedagogical method. I recognize that the Sayers Insight, as Douglas Wilson has called it, is NOT the medieval way of teaching the liberal arts. The Sayers Insight is new with Sayers.

So you’ll forgive me, I hope, for using Sayers-type language here. You can substitute the Littlejohn and Evans language instead if you like; I don’t think the results would be all that different.

There are several goals that I believe a classical Christian math education should have. 1. It should be Christ-centered. 2. It should develop the imaginations of the students so as to improve their problem-solving skills. 3. It should give them a standard toolbox of strategies for solving certain kinds of problems.

Anyone in cC education should have no problem agreeing with #1, although they might not, like me, have a very good idea how to do that. I think the Association of Christians in the Mathematical Sciences (you can google them) has some great papers on their website. I’ve read a couple, and I hope to work my way through all of them. James Nickel’s Mathematics: Is God Silent, is good, as is Vern Poythress’s Redeeming Science.

#2 might puzzle many people who view math as a collection of formulas. I am a professional mathematician, and I can say with a great deal of confidence that the most important aspect of the working mathematician is imagination. Imagine a theorem. (Ok, you needed imagination to do that, even!) How are you going to prove it? Or imagine a problem. How to solve it? (The preceding sentence, incidentally, is the title of a fantastic book by George Polya that I believe should be incorporated into the curriculum.) No one tells you how to do it. You have to imagine a solution. Perhaps you’ve had some experience close to the problem at hand. Fine, you can use your memory. But there will be plenty of times when you don’t have that experience to reference.

#3 is simply what many people see as wrong with the public school systems. Students aren’t taught their multiplication tables, or loads of other facts that are definitely worth knowing off the top of your head.

With these goals in mind, I propose the following system.

1. Determine the list of sub-topics in mathematics which you desire the students to know before they graduate from high school. I would advocate the following: arithmetic, algebra, geometry, trigonometry, calculus, differential equations. The reason I include differential equations is that there is no more applied field than that. It’s a fun topic, because then you can actually model how an electrical circuit behaves, or how a mass-spring system behaves.

2. Teach the grammar of each sub-topic during the poll-parrot years (grades 3-6). Teach the logic or dialectic of each sub-topic during the pert years (grades 7-9), and teach the rhetoric (I would say here is where you really hit imaginative problem-solving the hardest – including the presentation of your solution, an incredibly important aspect of problem-solving. The Challenger disaster was the indirect result of poor presentation!) of these sub-topics during the poetic years (grades 10-12).

The students coming out of a program like this may or may not score well on the SAT. While I certainly want students to have the opportunities that suit them well, and thus would like them to do decently well on the SAT, I don’t think good test scores are the best indicator. Does the student understand what mathematics is? Does the student see how mathematics fits in with theology and the other liberal arts? Does the student love mathematics? Does the student have any idea how to tackle a completely unfamiliar problem? Can the student present a problem solution in a meaningful way to people who might even be hostile to the solution (as was the case with the Challenger disaster)? I believe the proposed system could, if done well, do all these things.

In Christ.

• I would strongly agree with Mr. Philip Larson concerning applications. I happen to be a mathematical physicist (more applied even than applied mathematicians, and more theoretical than theoretical physicists) with a desire, eventually, to teach math, physics, and logic in a classical Christian school. I have examined the views of many, certainly not all, pure mathematicians: many of them tend to rejoice in doing work that is “unsullied” by application. That view is nonsense. Mind you, I’m not saying that pure mathematics has no value or that nobody should engage in it. I simply disagree with that higher-level philosophy; it seems to me to exhibit something of the universal desire of Babel: autonomy. I prefer to view mathematics more incarnationally. Mathematics, to me, is the discovery of numerical order and patterns in God’s creation. Pure mathematicians, such as I have described, should not teach mathematics at the grade level. The best math teachers for grade level, including in a classical Christian school, are, in my opinion, engineers. They are the ones who have really used this math in the workplace, and can furnish real real-world problems, as opposed to what usually passes for such.

Referring back to Dr. Veith’s original post, I am aware of the (hopefully) friendly debate between those who defend the Sayers insight, and the approach more along the lines of Littlejohn and Evans in their book Wisdom and Eloquence. While I think the differences are real, and while I side with Sayers, I think that the two approaches tend to end up in more or less the same place. Littlejohn and Evans advocate using a technique very like the Sayers-type Trivium in practice, as a method of teaching all 7 liberal arts. They are simply against the idea of calling the Trivium a pedagogical method. I recognize that the Sayers Insight, as Douglas Wilson has called it, is NOT the medieval way of teaching the liberal arts. The Sayers Insight is new with Sayers.

So you’ll forgive me, I hope, for using Sayers-type language here. You can substitute the Littlejohn and Evans language instead if you like; I don’t think the results would be all that different.

There are several goals that I believe a classical Christian math education should have. 1. It should be Christ-centered. 2. It should develop the imaginations of the students so as to improve their problem-solving skills. 3. It should give them a standard toolbox of strategies for solving certain kinds of problems.

Anyone in cC education should have no problem agreeing with #1, although they might not, like me, have a very good idea how to do that. I think the Association of Christians in the Mathematical Sciences (you can google them) has some great papers on their website. I’ve read a couple, and I hope to work my way through all of them. James Nickel’s Mathematics: Is God Silent, is good, as is Vern Poythress’s Redeeming Science.

#2 might puzzle many people who view math as a collection of formulas. I am a professional mathematician, and I can say with a great deal of confidence that the most important aspect of the working mathematician is imagination. Imagine a theorem. (Ok, you needed imagination to do that, even!) How are you going to prove it? Or imagine a problem. How to solve it? (The preceding sentence, incidentally, is the title of a fantastic book by George Polya that I believe should be incorporated into the curriculum.) No one tells you how to do it. You have to imagine a solution. Perhaps you’ve had some experience close to the problem at hand. Fine, you can use your memory. But there will be plenty of times when you don’t have that experience to reference.

#3 is simply what many people see as wrong with the public school systems. Students aren’t taught their multiplication tables, or loads of other facts that are definitely worth knowing off the top of your head.

With these goals in mind, I propose the following system.

1. Determine the list of sub-topics in mathematics which you desire the students to know before they graduate from high school. I would advocate the following: arithmetic, algebra, geometry, trigonometry, calculus, differential equations. The reason I include differential equations is that there is no more applied field than that. It’s a fun topic, because then you can actually model how an electrical circuit behaves, or how a mass-spring system behaves.

2. Teach the grammar of each sub-topic during the poll-parrot years (grades 3-6). Teach the logic or dialectic of each sub-topic during the pert years (grades 7-9), and teach the rhetoric (I would say here is where you really hit imaginative problem-solving the hardest – including the presentation of your solution, an incredibly important aspect of problem-solving. The Challenger disaster was the indirect result of poor presentation!) of these sub-topics during the poetic years (grades 10-12).

The students coming out of a program like this may or may not score well on the SAT. While I certainly want students to have the opportunities that suit them well, and thus would like them to do decently well on the SAT, I don’t think good test scores are the best indicator. Does the student understand what mathematics is? Does the student see how mathematics fits in with theology and the other liberal arts? Does the student love mathematics? Does the student have any idea how to tackle a completely unfamiliar problem? Can the student present a problem solution in a meaningful way to people who might even be hostile to the solution (as was the case with the Challenger disaster)? I believe the proposed system could, if done well, do all these things.

In Christ.

• EricM

Adrian – that is a wonderful post! Thank you very much for writing it. I am an engineer (although these days I have moved into the IT world) and I agree with your comment completely.

In our home school, we are working toward calculus as a requirement for a HS diploma. I had not thought about Diff Eq. My older son is studying engineering now and is headed toward Diff Eq in his sophmore year. I remember my time in math-beyond-calculus very fondly as it was fun! Seeing how math relates to physics, electronics, mechanical systems, etc. is exciting. Things begin to make much more sense when the math works out instead of just having the teacher stand up and tell you this is how it is without explaining the details.

• EricM

Adrian – that is a wonderful post! Thank you very much for writing it. I am an engineer (although these days I have moved into the IT world) and I agree with your comment completely.

In our home school, we are working toward calculus as a requirement for a HS diploma. I had not thought about Diff Eq. My older son is studying engineering now and is headed toward Diff Eq in his sophmore year. I remember my time in math-beyond-calculus very fondly as it was fun! Seeing how math relates to physics, electronics, mechanical systems, etc. is exciting. Things begin to make much more sense when the math works out instead of just having the teacher stand up and tell you this is how it is without explaining the details.

• WebMonk

No word of biology, chemistry, geology, medicine, etc, though astronomy is mentioned. The trivium/quadrivium is good stuff that has been sadly neglected, but let’s remember there are worlds beyond them that need to be taught as well.

• WebMonk

No word of biology, chemistry, geology, medicine, etc, though astronomy is mentioned. The trivium/quadrivium is good stuff that has been sadly neglected, but let’s remember there are worlds beyond them that need to be taught as well.

• Rev. Alexander Ring

As a math teacher at a classical school, I really enjoyed Adrian’s post, and would love to pursue it in a conversation. I think it strikes the balance that I look for in math education. But more than curriculum, what is needed is competant and energized teachers of math. Many of the people teaching math (especially at the middle and jr high level) didn’t like the subject themselves, and their lack of enthusiasm is communicated. The effect is like having the local unitarian minister teach your confirmation class.

The fact is, the beauty of mathematics really isn’t hard to communicate to kids, but it takes a teacher who is excited about the subject themselves and can communicate that excitement to students.

Alex

• Rev. Alexander Ring

As a math teacher at a classical school, I really enjoyed Adrian’s post, and would love to pursue it in a conversation. I think it strikes the balance that I look for in math education. But more than curriculum, what is needed is competant and energized teachers of math. Many of the people teaching math (especially at the middle and jr high level) didn’t like the subject themselves, and their lack of enthusiasm is communicated. The effect is like having the local unitarian minister teach your confirmation class.

The fact is, the beauty of mathematics really isn’t hard to communicate to kids, but it takes a teacher who is excited about the subject themselves and can communicate that excitement to students.

Alex

• I would definitely agree with you, Rev. Ring, about the need for enthusiastic teachers. We need John Milton Gregory-like teachers.

I would also claim we need parents not to undermine the work teachers do. Mind you, I believe the parents are in charge of their children’s education, period. However, they are, in a cC school, delegating some of that authority to the teachers. If a parent tells his kid that he doesn’t like math, and thinks it’s boring and too hard, what is that going to do to the kid? Math education has fallen on hard enough times that we don’t need that additional burden!

In Christ.

• I would definitely agree with you, Rev. Ring, about the need for enthusiastic teachers. We need John Milton Gregory-like teachers.

I would also claim we need parents not to undermine the work teachers do. Mind you, I believe the parents are in charge of their children’s education, period. However, they are, in a cC school, delegating some of that authority to the teachers. If a parent tells his kid that he doesn’t like math, and thinks it’s boring and too hard, what is that going to do to the kid? Math education has fallen on hard enough times that we don’t need that additional burden!

In Christ.

• Carl Vehse

Just make sure the updated quadrivium system excludes courses in numerology, astrology, and geomancy.

• Carl Vehse

Just make sure the updated quadrivium system excludes courses in numerology, astrology, and geomancy.

• Rev. Alexander Ring

Adrian,

Amen again. In touching base with parents on a struggling student I often hear, “Yeah, I didn’t like math either.” I’ll say, “I think if you ask your child you’ll find they *like* the class; they just need you to help them [get homework finished, get it to me on time, call me for help, etc].”

It also needs to be communicated to students that learning math takes some work. In Algebra I’m constantly saying, “Yes, I know this is hard. I sympathize with you. It is something you’ve never done before, and it will take us a while to learn it. But you will learn it; you will even understand it and enjoy it.” As that happens with more concepts, their confidence grows, and now not only does their love for mathematics grow but they’ve learned an important life-lesson: hard work = achievement.

This is also why what Adrian said about having math taught by engineers rings true. Most pure mathematicians are really, really gifted in the subject, and so have a hard time comprehending that anyone doesn’t understand it. You’re usually better of with someone who had to struggle a bit and so has some sympathy with others who struggle, too.

• Rev. Alexander Ring

Adrian,

Amen again. In touching base with parents on a struggling student I often hear, “Yeah, I didn’t like math either.” I’ll say, “I think if you ask your child you’ll find they *like* the class; they just need you to help them [get homework finished, get it to me on time, call me for help, etc].”

It also needs to be communicated to students that learning math takes some work. In Algebra I’m constantly saying, “Yes, I know this is hard. I sympathize with you. It is something you’ve never done before, and it will take us a while to learn it. But you will learn it; you will even understand it and enjoy it.” As that happens with more concepts, their confidence grows, and now not only does their love for mathematics grow but they’ve learned an important life-lesson: hard work = achievement.

This is also why what Adrian said about having math taught by engineers rings true. Most pure mathematicians are really, really gifted in the subject, and so have a hard time comprehending that anyone doesn’t understand it. You’re usually better of with someone who had to struggle a bit and so has some sympathy with others who struggle, too.

• Steven

As a geographer I would also say that there is a definite mathematical component, at least in many parts of the discipline. Much of what I work with could be considered applied graph theory (networks) and geometry underlies much of the technology of geographic information systems such as projections, and datums; spatial statistics relies upon geometry in determining point patterns, spatial dependence, as well as important concepts such as directionality and distance. An excellent example of the practical application of geometry, mathematics and geography is the book “Circumference” by Nicholas Nicastro. It also provides a perfect real-world mathematical experiment for middle or high schoolers.

• Steven

As a geographer I would also say that there is a definite mathematical component, at least in many parts of the discipline. Much of what I work with could be considered applied graph theory (networks) and geometry underlies much of the technology of geographic information systems such as projections, and datums; spatial statistics relies upon geometry in determining point patterns, spatial dependence, as well as important concepts such as directionality and distance. An excellent example of the practical application of geometry, mathematics and geography is the book “Circumference” by Nicholas Nicastro. It also provides a perfect real-world mathematical experiment for middle or high schoolers.

• Nanette

I like the Math-It program by Elmer Brooks for the grammar stage. As an adult I was amazed by and enjoyed learning the ‘tricks’! Here is some information about the program:

“A fresh and creative approach to the basic skill of computation
* Reduction of voluminous memory facts to simple rules
* Chance to be far more skillful in numbers than most adults
* Ability to easily add 3-digit numbers in one breath
* Simple and ingenious techniques to computing math problems
* Ability to add large columns of numbers quickly
* A fun way to remove the blocks to learning basic math
* Shortcut skills for various math functions as in the following example:

Did you know that the 11s times table is easy to figure in your head? Here’s the trick: When you multiply 63 by 11 put a space between the 6 and the 3, add the 6 and 3 together, and place their sum (9) in the space. The answer is 693. Now try multiplying 43 by 11 this way. Did you get 473?

Math-It is a program designed by Elmer Brooks to teach basic facts of addition, subtraction, and multiplication. Full of “tricks” to help students become whizzes in the basics. Children who can count back from 20 as they can tie their shoes with their eyes closed are ready for Math-It. “

• Nanette

I like the Math-It program by Elmer Brooks for the grammar stage. As an adult I was amazed by and enjoyed learning the ‘tricks’! Here is some information about the program:

“A fresh and creative approach to the basic skill of computation
* Reduction of voluminous memory facts to simple rules
* Chance to be far more skillful in numbers than most adults
* Ability to easily add 3-digit numbers in one breath
* Simple and ingenious techniques to computing math problems
* Ability to add large columns of numbers quickly
* A fun way to remove the blocks to learning basic math
* Shortcut skills for various math functions as in the following example:

Did you know that the 11s times table is easy to figure in your head? Here’s the trick: When you multiply 63 by 11 put a space between the 6 and the 3, add the 6 and 3 together, and place their sum (9) in the space. The answer is 693. Now try multiplying 43 by 11 this way. Did you get 473?

Math-It is a program designed by Elmer Brooks to teach basic facts of addition, subtraction, and multiplication. Full of “tricks” to help students become whizzes in the basics. Children who can count back from 20 as they can tie their shoes with their eyes closed are ready for Math-It. “

• Peter Leavitt

WebMonk’s right that the quadrivium should reflect modern subject matter. In math a good high school should take students through some math analysis and calculus. Statistics including some elementary regression analysis wouldn’t hurt. All of these subjects in the hands of an able math teacher can be related to the real world.

Interesting that Plato viewed math as an essential part of serious
education. He regarded math as the vital first step in learning to turn from the sensible realm to transcendent forms of reality. In his day arithmetic and geometry were important. In our day Algebra, trig, and calculus are essential to understanding forms of reality. However, in the final analysis the highest goal in all of education, Plato believed, is knowledge of the Good, the culmination of the best in learning.

• Peter Leavitt

WebMonk’s right that the quadrivium should reflect modern subject matter. In math a good high school should take students through some math analysis and calculus. Statistics including some elementary regression analysis wouldn’t hurt. All of these subjects in the hands of an able math teacher can be related to the real world.

Interesting that Plato viewed math as an essential part of serious
education. He regarded math as the vital first step in learning to turn from the sensible realm to transcendent forms of reality. In his day arithmetic and geometry were important. In our day Algebra, trig, and calculus are essential to understanding forms of reality. However, in the final analysis the highest goal in all of education, Plato believed, is knowledge of the Good, the culmination of the best in learning.

• As one who has struggled through a touch of Newton’s Principia, I agree wholeheartedly with the idea that it’s got to be real world. Teach basic arithmetic in the context of everyday tasks–cooking (double that recipe!), cleaning, and so on–and you’re quickly going to find that it makes a whole lot more sense.

And you get kids who can drive people nuts by applying those skills to everyday situations. :^)

Right now, I’m familiar with Sayers, Wilson’s book about Sayers, Wise & Bauer, and such. If anyone has some books to follow on, that would be a blessing.

• As one who has struggled through a touch of Newton’s Principia, I agree wholeheartedly with the idea that it’s got to be real world. Teach basic arithmetic in the context of everyday tasks–cooking (double that recipe!), cleaning, and so on–and you’re quickly going to find that it makes a whole lot more sense.

And you get kids who can drive people nuts by applying those skills to everyday situations. :^)

Right now, I’m familiar with Sayers, Wilson’s book about Sayers, Wise & Bauer, and such. If anyone has some books to follow on, that would be a blessing.

• I’ve recently come across the Life of Fred Series written by Stanley Schmidt. These books take a little character named Fred using] real life application all the way through a series of college prep math.

• I’ve recently come across the Life of Fred Series written by Stanley Schmidt. These books take a little character named Fred using] real life application all the way through a series of college prep math.

• I attended high school in Australia in the 197os, where Dewey’s ideas also prevailed [you Americans have a lot to answer for ;0)], and I flunked out on maths in a major way. It just made no sense to me – (I’m a humanities oriented person). My maths teacher used to say to me, “I’ve talked to the other subject teachers, you’re a smart guy, why don’t you get it?” Now I know I didn’t “get it” because it was presented as an abstract system unrelated to the other sublects I was learning about. If my maths teacher had given me the “big picture” first, showing how maths is integrated with all knowledge, I’m sure I would at least have enjoyed the subject, even if I didn’t pass the tests!

• I attended high school in Australia in the 197os, where Dewey’s ideas also prevailed [you Americans have a lot to answer for ;0)], and I flunked out on maths in a major way. It just made no sense to me – (I’m a humanities oriented person). My maths teacher used to say to me, “I’ve talked to the other subject teachers, you’re a smart guy, why don’t you get it?” Now I know I didn’t “get it” because it was presented as an abstract system unrelated to the other sublects I was learning about. If my maths teacher had given me the “big picture” first, showing how maths is integrated with all knowledge, I’m sure I would at least have enjoyed the subject, even if I didn’t pass the tests!

• David Carver

Teach Euclid! And if children can learn Greek early enough, teach him in the original, which is much simpler and cleaner than any translation.

(I realize I’m probably overly biased, but I am a little over halfway through him right now and am enjoying every proof of it.)

• David Carver

Teach Euclid! And if children can learn Greek early enough, teach him in the original, which is much simpler and cleaner than any translation.

(I realize I’m probably overly biased, but I am a little over halfway through him right now and am enjoying every proof of it.)

• Great, helpful, and important comments! Classical educators–here are some splendid ideas. Run with them.

Webmonk and Peter, chemistry, physics, biology, etc., WERE taught in classical schools and universities (such as Oxford and Cambridge). They are not liberal arts–which deal with processes and skills–but they come under the liberal sciences (the word literally meaning “knowledge), specifically, Natural Science. (The others are Moral Science [knowledge of Man, including history, law, the humanities, etc.], and Theological Science [the knowledge of God, theology being the “Queen of the Sciences,” in the sense of comprehending the source of all of the others.)

Astronomy as an “art” teaches empirical observation to which is applied mathematical analysis. As such, it teaches the conceptual “art” necessary in those other sciences.

I guess I should next post something asking how to teach the natural sciences classically!

• Great, helpful, and important comments! Classical educators–here are some splendid ideas. Run with them.

Webmonk and Peter, chemistry, physics, biology, etc., WERE taught in classical schools and universities (such as Oxford and Cambridge). They are not liberal arts–which deal with processes and skills–but they come under the liberal sciences (the word literally meaning “knowledge), specifically, Natural Science. (The others are Moral Science [knowledge of Man, including history, law, the humanities, etc.], and Theological Science [the knowledge of God, theology being the “Queen of the Sciences,” in the sense of comprehending the source of all of the others.)

Astronomy as an “art” teaches empirical observation to which is applied mathematical analysis. As such, it teaches the conceptual “art” necessary in those other sciences.

I guess I should next post something asking how to teach the natural sciences classically!

• Josie

Just posting to say we’ve begun supplementing this year with the Life of Fred books and they really do add a bit of real world and fun at the same time to our math studies.

• Josie

Just posting to say we’ve begun supplementing this year with the Life of Fred books and they really do add a bit of real world and fun at the same time to our math studies.

• Thank you, Dr. Veith, for pointing out that there can be more to classical education than just the trivium.

I taught high school science for a year at a successful classical Christian school in the Midwest. Before the school year started, the headmaster made it clear that science would take a back seat to the trivium. I was to give only a minimum amount of homework so that students would be able to dedicate several hours every night to more important matters.

While teaching at this school, one of the leaders of the Christian classical education movement came to the area to give a workshop. I had the chance to talk to him about the place of science in the curriculum, and I got the impression that he really had not given it a whole lot of thought. Science, like math, was useful when it could be used to teach logic or some other topic related to the trivium, but it had minimum value beyond that.

A related problem I have seen in classical Christian curricula (as well as in Christian education in general) is an attitude that science is useful in education primarily as an apologetic tool. This often results in a twisting of science in order to bolster students’ faith. This is not wise.

• Thank you, Dr. Veith, for pointing out that there can be more to classical education than just the trivium.

I taught high school science for a year at a successful classical Christian school in the Midwest. Before the school year started, the headmaster made it clear that science would take a back seat to the trivium. I was to give only a minimum amount of homework so that students would be able to dedicate several hours every night to more important matters.

While teaching at this school, one of the leaders of the Christian classical education movement came to the area to give a workshop. I had the chance to talk to him about the place of science in the curriculum, and I got the impression that he really had not given it a whole lot of thought. Science, like math, was useful when it could be used to teach logic or some other topic related to the trivium, but it had minimum value beyond that.

A related problem I have seen in classical Christian curricula (as well as in Christian education in general) is an attitude that science is useful in education primarily as an apologetic tool. This often results in a twisting of science in order to bolster students’ faith. This is not wise.

• Randy E

Very thought provoking. It has been helpful for me to see how the contemporary study of astronomy misses the enchantment of the Medieval study of the same. If astronomy (numbers in space and time) is both the combination of geometry (numbers in space) and music (numbers in time), then astronomy is truly the the study of ‘the music of the spheres’.

• Randy E

Very thought provoking. It has been helpful for me to see how the contemporary study of astronomy misses the enchantment of the Medieval study of the same. If astronomy (numbers in space and time) is both the combination of geometry (numbers in space) and music (numbers in time), then astronomy is truly the the study of ‘the music of the spheres’.

• For those interested, the Life of Fred materials are available here:
http://www.stanleyschmidt.com/FredGauss/index2.html
I did not see at Amazon.

• For those interested, the Life of Fred materials are available here:
http://www.stanleyschmidt.com/FredGauss/index2.html
I did not see at Amazon.

• Peter Leavitt

Sharaya, thanks for that link to Steve Scmidt’s Life of Fred materials. Stan Schmidt, a devout Christian and former high school and college professor of math, has put together a very clever and attractive set of math books from elementary school through beginning college. While apparently a serious mathematician who loves the subject, he presents the rigorous material with a sense of humor and lightness of touch.

Many years ago at the Groton School I was fortunate enough to have two teachers who knew their math and poor students well enough to tease some of us with wit humor into an appreciation of math. Frankly, the books were the usually dull math affairs, though these teachers had a way of overcoming this. Both these teachers could be seen with all the others at early morning chapel, which was one of our school’s ways of suggesting what was really important.

Dr. Veith, I quite agree with you that a good classical curriculum should combine the Trivium and a modern version of the Quadrivium. Dorothy Sayers was right that grammar, dialectic, and rhetoric are in the long rum more important, though there is no reason that math, music, and science can’t be taught concurrently with the view of deepening the Trivium . Personally, I think that both Latin and Greek should be part of the mix in both high school and college. This was mostly the case in the better New England schools and colleges until about fifty-years ago when the corrosive influence of Dewey et al had its effect.

• Peter Leavitt

Sharaya, thanks for that link to Steve Scmidt’s Life of Fred materials. Stan Schmidt, a devout Christian and former high school and college professor of math, has put together a very clever and attractive set of math books from elementary school through beginning college. While apparently a serious mathematician who loves the subject, he presents the rigorous material with a sense of humor and lightness of touch.

Many years ago at the Groton School I was fortunate enough to have two teachers who knew their math and poor students well enough to tease some of us with wit humor into an appreciation of math. Frankly, the books were the usually dull math affairs, though these teachers had a way of overcoming this. Both these teachers could be seen with all the others at early morning chapel, which was one of our school’s ways of suggesting what was really important.

Dr. Veith, I quite agree with you that a good classical curriculum should combine the Trivium and a modern version of the Quadrivium. Dorothy Sayers was right that grammar, dialectic, and rhetoric are in the long rum more important, though there is no reason that math, music, and science can’t be taught concurrently with the view of deepening the Trivium . Personally, I think that both Latin and Greek should be part of the mix in both high school and college. This was mostly the case in the better New England schools and colleges until about fifty-years ago when the corrosive influence of Dewey et al had its effect.

• Rev. Alexander Ring

I am all a-quiver over this topic, and want to comment on all the posts, but I will try and focus. I will thank everyone for their insightful comments.

The Astronomy thing, however, is going to set me off again. I am doing Astronomy right now with my 5th & 6th grade science class. It is a great science course for middle school students because it helps teach what science is: We observe the natural world and based on our observations we come up with an explaination. We observe the sun rise and set and the planets move, and explain that the sun revolves around the earth. We find problems with that explaination and make better observations and explain that planets revolve around the sun. We find problems with that, and our observations tell us the orbits are ellipses. Astronomy also easily integrates the other subjects, and so students see how moments in history come together. That, and kids just find it fascinating.

• Rev. Alexander Ring

I am all a-quiver over this topic, and want to comment on all the posts, but I will try and focus. I will thank everyone for their insightful comments.

The Astronomy thing, however, is going to set me off again. I am doing Astronomy right now with my 5th & 6th grade science class. It is a great science course for middle school students because it helps teach what science is: We observe the natural world and based on our observations we come up with an explaination. We observe the sun rise and set and the planets move, and explain that the sun revolves around the earth. We find problems with that explaination and make better observations and explain that planets revolve around the sun. We find problems with that, and our observations tell us the orbits are ellipses. Astronomy also easily integrates the other subjects, and so students see how moments in history come together. That, and kids just find it fascinating.

• Peter Leavitt

Thinking about this, it is wondrous that serious Christians from home -school parents, Dorothy Sayers, Stan Schmidt, Rev. Ring and Dr. Veith have arrived at this Christian/classical view of what a serious education is about. This is hardly some random accident of history.

• Peter Leavitt

Thinking about this, it is wondrous that serious Christians from home -school parents, Dorothy Sayers, Stan Schmidt, Rev. Ring and Dr. Veith have arrived at this Christian/classical view of what a serious education is about. This is hardly some random accident of history.

• Josie

Another place one can find the Life of Fred books is Rainbow Resource. Try this link: http://www.rainbowresource.com/index.php

Stan Schmidt’s site is great to have though as it gives a good overview of the books.

Rev. Ring…would you mind sharing what curriculum you use with your students for Science? Do you think its approached best through the secular textbook cur. or does Apologia or BJU fit the bill better? Thank you!

• Josie

Another place one can find the Life of Fred books is Rainbow Resource. Try this link: http://www.rainbowresource.com/index.php

Stan Schmidt’s site is great to have though as it gives a good overview of the books.

Rev. Ring…would you mind sharing what curriculum you use with your students for Science? Do you think its approached best through the secular textbook cur. or does Apologia or BJU fit the bill better? Thank you!

• The Jungle Cat

I didn’t have time to read all of the comments, so I am sure that someone has probably already said it or something like it, but part of the problem I think is that classical educators have not yet shed the tendency to understand mathematics as a science rather than an art.

• The Jungle Cat

I didn’t have time to read all of the comments, so I am sure that someone has probably already said it or something like it, but part of the problem I think is that classical educators have not yet shed the tendency to understand mathematics as a science rather than an art.

• WebMonk

Might I suggest here that no one here so far talking about astronomy has the foggiest idea of what they’re talking about if they really think modern astronomy is about “numbers in space and time”.

It’s not numbers in space, it is numbers in nuclear physics. Sure, 200-500 years ago it could be thought of as numbers in space and time, but no longer. It is diffraction, brightness levels, gravitational effects, relativity, nuclear physics, pressure/temperature/brightness/gravitational-pull calculations.

When astronomy was about “numbers in space and time” they were calculating orbits of planets and moons. Saying that modern astronomy is like that is similar to saying building modern cars is about good welding – sure, there are parts that still rely on welding, but those parts are rare, a shrinking portion, and completely inadequate to the larger goal of making a car.

• WebMonk

Might I suggest here that no one here so far talking about astronomy has the foggiest idea of what they’re talking about if they really think modern astronomy is about “numbers in space and time”.

It’s not numbers in space, it is numbers in nuclear physics. Sure, 200-500 years ago it could be thought of as numbers in space and time, but no longer. It is diffraction, brightness levels, gravitational effects, relativity, nuclear physics, pressure/temperature/brightness/gravitational-pull calculations.

When astronomy was about “numbers in space and time” they were calculating orbits of planets and moons. Saying that modern astronomy is like that is similar to saying building modern cars is about good welding – sure, there are parts that still rely on welding, but those parts are rare, a shrinking portion, and completely inadequate to the larger goal of making a car.

• ptl

to WebMonk above……yes modern physics is too advanced for most people to comprehend (at least in my opinion) although things like black holes, string theory, super string theory (or super duper string theory!), parallel universes, etc. all usually take at least a good understanding of graduate level Mathematics, and even then most of those folks don’t really understand it (am thinking of Richard Feynman’s comments on that topic, if you want references) but they pretend they do, so they sound intelligent and hip (if it’s possible for a super nerd to be hip!). In any case, it makes good fun and great cocktail conversation, and justifies alot of grant money to explore the outer edges of the universe and theories of the universe. Well, enough of that…my main point would be to say that from an education point of view, and in particular the introductory level of science education, it would be perfectly natural and useful to talk about astronomy as numbers in time and space. afterall, those are the observations that began the whole journey and in my opinion, a good way to kick it off for the young and others. then as their math ability gets better you can move on to other topics that require more math. elliptical orbits of the planets, gravitational attraction between masses, even Einstein’s Theory (Special and General) at the introductory level.

So lighten up WebMonk and think about how to interest young people and those without advanced mathematics, you’ll be in good company and the sky’s the limit 🙂

• ptl

to WebMonk above……yes modern physics is too advanced for most people to comprehend (at least in my opinion) although things like black holes, string theory, super string theory (or super duper string theory!), parallel universes, etc. all usually take at least a good understanding of graduate level Mathematics, and even then most of those folks don’t really understand it (am thinking of Richard Feynman’s comments on that topic, if you want references) but they pretend they do, so they sound intelligent and hip (if it’s possible for a super nerd to be hip!). In any case, it makes good fun and great cocktail conversation, and justifies alot of grant money to explore the outer edges of the universe and theories of the universe. Well, enough of that…my main point would be to say that from an education point of view, and in particular the introductory level of science education, it would be perfectly natural and useful to talk about astronomy as numbers in time and space. afterall, those are the observations that began the whole journey and in my opinion, a good way to kick it off for the young and others. then as their math ability gets better you can move on to other topics that require more math. elliptical orbits of the planets, gravitational attraction between masses, even Einstein’s Theory (Special and General) at the introductory level.

So lighten up WebMonk and think about how to interest young people and those without advanced mathematics, you’ll be in good company and the sky’s the limit 🙂

• WebMonk

Depends on the level you’re teaching at, and what you’re trying to teach. Non-math astronomy and history of astronomy, but unless you’re teaching calculus, you aren’t going to be teaching math while teaching “astronomy”. I guess I would call teaching the classical astronomy today a history course rather than actually learning astronomy.

I realize I might be a bit of a purist on this issue! Try telling a baseball fan that all you have to do to follow baseball is know what balls and strikes are, and you’ll probably get a similar reaction. 😀

• WebMonk

Depends on the level you’re teaching at, and what you’re trying to teach. Non-math astronomy and history of astronomy, but unless you’re teaching calculus, you aren’t going to be teaching math while teaching “astronomy”. I guess I would call teaching the classical astronomy today a history course rather than actually learning astronomy.

I realize I might be a bit of a purist on this issue! Try telling a baseball fan that all you have to do to follow baseball is know what balls and strikes are, and you’ll probably get a similar reaction. 😀

• ptl

to WebMonk above….try this link and watch video 1. have not seen it in a while, but it seems this wonderful professor does a great job and am pretty sure used no (or very, very little…perhaps once could say an infinitesimal amount) of calculus 🙂

if that doesn’t work, try the following:

It may require you download some free software, so do it and be patient, it will be worth it!

Enjoy and may the force be with you 🙂

• ptl

to WebMonk above….try this link and watch video 1. have not seen it in a while, but it seems this wonderful professor does a great job and am pretty sure used no (or very, very little…perhaps once could say an infinitesimal amount) of calculus 🙂

if that doesn’t work, try the following:

It may require you download some free software, so do it and be patient, it will be worth it!

Enjoy and may the force be with you 🙂

• WebMonk

*grump, grump, grump* I’m feeling like an old fart yelling “Get offa my grass!”

I love Feynman – he was a great scientist and a dynamic teacher to boot. But, that is teaching ABOUT astronomy for the most part (and physics, quantum mechanics, etc.), not actually teaching astronomy. There’s a difference between teaching about astronomy and the actual teaching of astronomy.

What is in those videos is a fantastic intro to physics to let people know the wonderful and interesting aspects of it. It’s a great way to give laymen the 50,000-ft overview of what astronomy can cover. It presents the terms and VERY general concepts to people in an understandable fashion. He covers general principles of physics, and concepts of quantum physics even more than he covers astronomy.

It’s not teaching any of those topics though. (at least not in the two that I watched – 1 and 2) He is teaching about the various topics, and that’s his intention. If I were a teacher, I would happily make my students watch those videos to get them excited about physics, astronomy, and quantum physics. But after that, I would proceed to actually teach astronomy or quantum physics.

Like I said, “math in time and space” isn’t modern astronomy except in the vaguest possible sense.

Until now I had only seen the seventh video in that series, which is something of a classic among Feynman fan boys. I am ashamed to admit I didn’t realize it was part of an integrated series. I thought it was just another one-off lecture.

MANY THANKS for the link!

• WebMonk

*grump, grump, grump* I’m feeling like an old fart yelling “Get offa my grass!”

I love Feynman – he was a great scientist and a dynamic teacher to boot. But, that is teaching ABOUT astronomy for the most part (and physics, quantum mechanics, etc.), not actually teaching astronomy. There’s a difference between teaching about astronomy and the actual teaching of astronomy.

What is in those videos is a fantastic intro to physics to let people know the wonderful and interesting aspects of it. It’s a great way to give laymen the 50,000-ft overview of what astronomy can cover. It presents the terms and VERY general concepts to people in an understandable fashion. He covers general principles of physics, and concepts of quantum physics even more than he covers astronomy.

It’s not teaching any of those topics though. (at least not in the two that I watched – 1 and 2) He is teaching about the various topics, and that’s his intention. If I were a teacher, I would happily make my students watch those videos to get them excited about physics, astronomy, and quantum physics. But after that, I would proceed to actually teach astronomy or quantum physics.

Like I said, “math in time and space” isn’t modern astronomy except in the vaguest possible sense.

Until now I had only seen the seventh video in that series, which is something of a classic among Feynman fan boys. I am ashamed to admit I didn’t realize it was part of an integrated series. I thought it was just another one-off lecture.

MANY THANKS for the link!

• There is a Mathematics course which restores classical deductive Math , starting with Arithmetic and progressing through Geometry, Music and Astronomy. Mr. William Michael is the author and director of the Classical Liberal Arts Academy http://classicalliberalarts.com , where the true Christian classical liberal arts are being thought to hundreds of children online.

To see the overview of Mathematics look here: http://www.classicalliberalarts.com/Courses/QUADRIVIUM/index.htm

Philosophy of Mathematics is explained here: http://www.classicalliberalarts.com/Courses/QUADRIVIUM/philosophy_math.htm

• There is a Mathematics course which restores classical deductive Math , starting with Arithmetic and progressing through Geometry, Music and Astronomy. Mr. William Michael is the author and director of the Classical Liberal Arts Academy http://classicalliberalarts.com , where the true Christian classical liberal arts are being thought to hundreds of children online.

To see the overview of Mathematics look here: http://www.classicalliberalarts.com/Courses/QUADRIVIUM/index.htm

Philosophy of Mathematics is explained here: http://www.classicalliberalarts.com/Courses/QUADRIVIUM/philosophy_math.htm

• Reply to iwka.

Interesting website. However, I don’t think much of his philosophy of mathematics. He doesn’t appear to believe in the resurrection of the body after Christ comes back!?! That’s a doctrine found in the Apostles’ Creed (“the resurrection of the body”) and the Nicene Creed (same wording). For Bible verses, you need look no further than Job 19:26 and 1 Cor. 15:42ff, not to mention the entire gospel of John. In my opinion, John’s Gospel and Plato are incompatible.

Unfortunately, Platonism appears to be the driving force behind his philosophy, instead of the Scriptures. The result is a non-incarnational philosophy, which will always have a tendency toward “pure” mathematics that eschews applications. The tendency will be towards thinking, “Why should we apply mathematics to the real world (which is ‘temporary’, and therefore ‘not real’), when the ‘more interesting’ thing is the theorem, which exists in the world of ‘forms’ (to use Platonic language)?”

Mind you, I’m not denying the importance of abstraction, propositions, and the like. The Bible has plenty of that as well. I’m just saying that the pattern of quite a few biblical books (Romans and Ephesians come to mind) is propositional truth followed by application to this present world and how we are to live in it. It seems to me that mathematics might profitably duplicate that pattern.

In Christ.

• Reply to iwka.

Interesting website. However, I don’t think much of his philosophy of mathematics. He doesn’t appear to believe in the resurrection of the body after Christ comes back!?! That’s a doctrine found in the Apostles’ Creed (“the resurrection of the body”) and the Nicene Creed (same wording). For Bible verses, you need look no further than Job 19:26 and 1 Cor. 15:42ff, not to mention the entire gospel of John. In my opinion, John’s Gospel and Plato are incompatible.

Unfortunately, Platonism appears to be the driving force behind his philosophy, instead of the Scriptures. The result is a non-incarnational philosophy, which will always have a tendency toward “pure” mathematics that eschews applications. The tendency will be towards thinking, “Why should we apply mathematics to the real world (which is ‘temporary’, and therefore ‘not real’), when the ‘more interesting’ thing is the theorem, which exists in the world of ‘forms’ (to use Platonic language)?”

Mind you, I’m not denying the importance of abstraction, propositions, and the like. The Bible has plenty of that as well. I’m just saying that the pattern of quite a few biblical books (Romans and Ephesians come to mind) is propositional truth followed by application to this present world and how we are to live in it. It seems to me that mathematics might profitably duplicate that pattern.

In Christ.

• @ Adrian Keister

I don’t think you have looked enough around. 🙂

It’s a Catholic Academy, therefore in agreement with Catholic Catechism http://www.usccb.org/catechism/text/, therefore resurrection of Christ and the saints is a CENTRAL truth. The teachings of the Academy are based on the Scriptures, are teaching Scriptures (daily Liturgy of the Hours, prayers, daily Bible readings), every course has annotations to the Scriptures and there will be a five year Bible course offered starting this fall.

The core curriculum is based on Catechism, Grammar and Arithmetic.

As for Platonism. It would be enough to have a look at the Curriculum’s overview to realize what it is based on:
http://www.classicalliberalarts.com/curriculum/index.htm

Blessings.

• @ Adrian Keister

I don’t think you have looked enough around. 🙂

It’s a Catholic Academy, therefore in agreement with Catholic Catechism http://www.usccb.org/catechism/text/, therefore resurrection of Christ and the saints is a CENTRAL truth. The teachings of the Academy are based on the Scriptures, are teaching Scriptures (daily Liturgy of the Hours, prayers, daily Bible readings), every course has annotations to the Scriptures and there will be a five year Bible course offered starting this fall.

The core curriculum is based on Catechism, Grammar and Arithmetic.

As for Platonism. It would be enough to have a look at the Curriculum’s overview to realize what it is based on:
http://www.classicalliberalarts.com/curriculum/index.htm

Blessings.