Saving mathematics

The Washington Post has sure been publishing some good articles about today’s education debacles, which tells us that even the liberal establishment is waking up to the necessity of actually educating children, as opposed to what contemporary educational theory is doing.  Today’s edition included a feature entitled Parents Rise Up Against A New Approach to Math.  It’s about a math textbook entitled “Investigations in Number, Data, and Space.” It tackles the problem of  multiplying six times three, by having students make six marks on a piece of paper in a little box and to do that with three boxes.  Then count how many marks.  And it does away with the traditional way of adding big numbers, in which you put them into columns, add each one, and carry as needed.  Instead, students are taught to make pyramids, in which they first add up the ones, then the tens, then the hundreds, then the thousands, then put them all together.      Defenders say this method, which scorns memorizing “math facts,” teaches the concepts better.  But it makes math harder, not easier, and it is doing nothing to improve test scores.    I admit that classical education may be lagging in the math department.  The new classical schools are doing little with the Quadrivium, the other four liberal arts (arithmetic, geometry, astronomy, and music).  The Trivium, which is being implemented to great effect (grammar, logic, and rhetoric), has to do with mastering language and what you can do with it.  The Quadrivium has to do with mathematics (yes, even in the way music was taught).    This, I think, is the new frontier for classical educators.  Yes, there is Saxon math, but it seems traditional (which is better than the contemporary), rather than classical, as such.  Does anyone have any suggestions about what a classical approach to mathematics  might look like?

About Gene Veith

Professor of Literature at Patrick Henry College, the Director of the Cranach Institute at Concordia Theological Seminary, a columnist for World Magazine and TableTalk, and the author of 18 books on different facets of Christianity & Culture.

  • http://www.pagantolutheran.blogspot.com Bruce

    I don’t know if this was classical math my wife was teaching, or just desperation. But she integrated math curricula from Singapore into her complex of Saxon, Math U See, and other Western curricula. With each child came a different recipe. It worked because it was tailored to the individual child.

    What all of the various curriculums had in common was mastery. Saxon does it to some extent, recycling through previous material on a regular basis to make sure it isn’t lost. What my wife insisted upon was that whatever level the child was on, it was understood and mastered–even if it meant drawing pyramids and counting boxes in order to grasp the concept. She had been hired to tutor a few kids from the local high school in algebra. The kids had passed the high school’s algebra course but the parents were uncomfortable with what they had learned (They were using Chicago Math). What my wife found out was that the kids had not grasped working with fractions. She just kept going further and further back in their education until she found mastery, and then started forward again. This is not rocket science.

  • http://www.pagantolutheran.blogspot.com Bruce

    I don’t know if this was classical math my wife was teaching, or just desperation. But she integrated math curricula from Singapore into her complex of Saxon, Math U See, and other Western curricula. With each child came a different recipe. It worked because it was tailored to the individual child.

    What all of the various curriculums had in common was mastery. Saxon does it to some extent, recycling through previous material on a regular basis to make sure it isn’t lost. What my wife insisted upon was that whatever level the child was on, it was understood and mastered–even if it meant drawing pyramids and counting boxes in order to grasp the concept. She had been hired to tutor a few kids from the local high school in algebra. The kids had passed the high school’s algebra course but the parents were uncomfortable with what they had learned (They were using Chicago Math). What my wife found out was that the kids had not grasped working with fractions. She just kept going further and further back in their education until she found mastery, and then started forward again. This is not rocket science.

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  • The Jones

    If the old way of teaching math educated the generation that sent men to the moon, created the computer, and invented the intercontinental balistic missile, then it seems that way was working fairly well for them. I don’t know about test scores back then, but I think it’s a stretch to say everyone was dumb. So why are we struggling to learn new ways to teach MATH? It seems like the subject that never changes. It’s logic and proven facts, pure and simple. Not much in the subject needs to be adapted and applied to new situations and cultures. Are we battling an “entertain and thrill me” classroom culture or bad instruction? I hope somebody who knows more about this than me can help me out on this.

  • The Jones

    If the old way of teaching math educated the generation that sent men to the moon, created the computer, and invented the intercontinental balistic missile, then it seems that way was working fairly well for them. I don’t know about test scores back then, but I think it’s a stretch to say everyone was dumb. So why are we struggling to learn new ways to teach MATH? It seems like the subject that never changes. It’s logic and proven facts, pure and simple. Not much in the subject needs to be adapted and applied to new situations and cultures. Are we battling an “entertain and thrill me” classroom culture or bad instruction? I hope somebody who knows more about this than me can help me out on this.

  • http://castingoutnines.wordpress.com Robert Talbert

    First of all, let me say that the Saxon method is horrible. I know that a lot of parents swear by it, but I’ve seen many students come into my college precalculus and calculus classes over the years and while they are very good at any problem that is highly similar to one found in the Saxon books, they have almost no ability in extrapolating their skills to new or even significantly different problems. Especially problematic are word problems that are not phrased EXACTLY LIKE those in the Saxon book. Almost all the Saxon kids I’ve had end up doing poorly in their college math courses because they have learned the Saxon problem sets rather than mathematics itself. (There have been happy exceptions, too.)

    I think a “classical” approach to teaching math would, going along with the spirit of the other classical education posts yesterday, teach the hypostatic union of content and process — the facts and the methods, yes (and without cutesy gimmicks), but also the processes of logical deduction, analytic problem-solving heuristics, and argumentation. Geometry is a very good place to start and an essential to include in any such approach. But I’d also throw in more esoteric topics as number theory and discrete math (counting and graph theory) — in whatever dosage and level is age-appropriate.

    At the university level, and maybe at the high school level for kids with a good basic arithmetic background, I’d love to be able to use the book “Essential College Mathematics” by Zwier and Nyhoff as a standard one-year course in mathematics (and in place of the usual year of calculus most such students take). It’s out of print, but the chapters are on sets; cardinal numbers; the integers; logic; axiomatic systems and the mathematical method; groups; rational numbers, real numbers, and fields; analytic geometry of the line and plane; and finally functions, derivatives, and applications. You have to see how the text is written to see why it does a good job with both content and process.

  • http://castingoutnines.wordpress.com Robert Talbert

    First of all, let me say that the Saxon method is horrible. I know that a lot of parents swear by it, but I’ve seen many students come into my college precalculus and calculus classes over the years and while they are very good at any problem that is highly similar to one found in the Saxon books, they have almost no ability in extrapolating their skills to new or even significantly different problems. Especially problematic are word problems that are not phrased EXACTLY LIKE those in the Saxon book. Almost all the Saxon kids I’ve had end up doing poorly in their college math courses because they have learned the Saxon problem sets rather than mathematics itself. (There have been happy exceptions, too.)

    I think a “classical” approach to teaching math would, going along with the spirit of the other classical education posts yesterday, teach the hypostatic union of content and process — the facts and the methods, yes (and without cutesy gimmicks), but also the processes of logical deduction, analytic problem-solving heuristics, and argumentation. Geometry is a very good place to start and an essential to include in any such approach. But I’d also throw in more esoteric topics as number theory and discrete math (counting and graph theory) — in whatever dosage and level is age-appropriate.

    At the university level, and maybe at the high school level for kids with a good basic arithmetic background, I’d love to be able to use the book “Essential College Mathematics” by Zwier and Nyhoff as a standard one-year course in mathematics (and in place of the usual year of calculus most such students take). It’s out of print, but the chapters are on sets; cardinal numbers; the integers; logic; axiomatic systems and the mathematical method; groups; rational numbers, real numbers, and fields; analytic geometry of the line and plane; and finally functions, derivatives, and applications. You have to see how the text is written to see why it does a good job with both content and process.

  • http://castingoutnines.wordpress.com Robert Talbert

    @the jones: The proponents of newer mathematics pedagogies would argue that while the subject hasn’t changed significantly, the students we are teaching *have* changed, and so we can’t expect teaching methods that worked in the 1960′s to work now without some adaptation.

    I sort of agree with those people, but the problem with it is that most of these people have no idea exactly *how* students have changed in their ways of conceiving mathematics. There is the popular “digital native” hypothesis that says that students learn best through digital media because they are saturated in user-controllable electronic content in their daily lives, for example, which is appealing but has no evidence to back it beyond the anecdotal.

    I would say, though, that certainly the youth culture out of which these kids are emerging today is very different than it was in the Space Race days, and it poses very difficult problems for math (or any other) educators. Kids are instructed by their culture, especially popular culture and entertainment media, that the intellect in general — and being good in math in particular — just isn’t important, and in fact bad. The cultural problem dwarfs any particulars about curriculum and pedagogy, and it’s sad to see people thinking that we can fight a cultural battle with smartboards and manipulatives.

    That’s why I think teaching mathematics is a deeply countercultural act, and one particularly well-suited for Christians to engage in. But that’s more than one comment field will allow!

  • http://castingoutnines.wordpress.com Robert Talbert

    @the jones: The proponents of newer mathematics pedagogies would argue that while the subject hasn’t changed significantly, the students we are teaching *have* changed, and so we can’t expect teaching methods that worked in the 1960′s to work now without some adaptation.

    I sort of agree with those people, but the problem with it is that most of these people have no idea exactly *how* students have changed in their ways of conceiving mathematics. There is the popular “digital native” hypothesis that says that students learn best through digital media because they are saturated in user-controllable electronic content in their daily lives, for example, which is appealing but has no evidence to back it beyond the anecdotal.

    I would say, though, that certainly the youth culture out of which these kids are emerging today is very different than it was in the Space Race days, and it poses very difficult problems for math (or any other) educators. Kids are instructed by their culture, especially popular culture and entertainment media, that the intellect in general — and being good in math in particular — just isn’t important, and in fact bad. The cultural problem dwarfs any particulars about curriculum and pedagogy, and it’s sad to see people thinking that we can fight a cultural battle with smartboards and manipulatives.

    That’s why I think teaching mathematics is a deeply countercultural act, and one particularly well-suited for Christians to engage in. But that’s more than one comment field will allow!

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  • http://www.bikebubba.blogspot.com Bike Bubba

    Having been a TA in college for intro math classes, my bete noire was the failure to teach young people basic arithmetic. Easily half the red marks I’d put on paper were basic arithmetic errors, not errors with the theory being presented.

    Didn’t have enough Saxon people in classes (or anybody not from the government schools for that matter) to judge Saxon–or really many of the non-government schools.

  • http://www.bikebubba.blogspot.com Bike Bubba

    Having been a TA in college for intro math classes, my bete noire was the failure to teach young people basic arithmetic. Easily half the red marks I’d put on paper were basic arithmetic errors, not errors with the theory being presented.

    Didn’t have enough Saxon people in classes (or anybody not from the government schools for that matter) to judge Saxon–or really many of the non-government schools.

  • Pinon Coffee

    I did Saxon math all through middle school and high school and hated every minute of it. I’d probably agree with Robert Talbert: I feel like I have a better grasp of Saxon problem sets than math itself. Physics was pretty rough.

    I don’t necessarily want to bash Saxon per se; I do know some “exceptions” as he put it, who came out positively brilliant. But I wonder if a different curriculum might have helped me learn to like math, rather than, well, turning out a dyed-in-the-wool literature major… I know when I discovered Aristotelian logic, I adored it; and logic isn’t really that different from algebra. But nobody introduced it to me that way.

    If classical educators could come up with a classical-friendly way to approach math and science, I want to hear about it. :-)

  • Pinon Coffee

    I did Saxon math all through middle school and high school and hated every minute of it. I’d probably agree with Robert Talbert: I feel like I have a better grasp of Saxon problem sets than math itself. Physics was pretty rough.

    I don’t necessarily want to bash Saxon per se; I do know some “exceptions” as he put it, who came out positively brilliant. But I wonder if a different curriculum might have helped me learn to like math, rather than, well, turning out a dyed-in-the-wool literature major… I know when I discovered Aristotelian logic, I adored it; and logic isn’t really that different from algebra. But nobody introduced it to me that way.

    If classical educators could come up with a classical-friendly way to approach math and science, I want to hear about it. :-)

  • Booklover

    Saxon math worked wonderfully for us when we were homeschooling, but we only used it through the “87″ book, I believe. I appreciated its incremental approach.

    About 10 years ago, a few of us Christian homeschooling moms put our thoughts together about classical Christian materials and methodology. Christine Miller put together a website with mostly her wisdom but some from us too. I wrote the articles on “Writing” and “Music” at the grammar level.

    Here’s the link to the suggestions for classical math materials at the grammar level. “Ray’s” won out among most of the group. Saxon was a little further down the list.

    http://www.classical-homeschooling.org/curriculum/math-grammar.html

  • Booklover

    Saxon math worked wonderfully for us when we were homeschooling, but we only used it through the “87″ book, I believe. I appreciated its incremental approach.

    About 10 years ago, a few of us Christian homeschooling moms put our thoughts together about classical Christian materials and methodology. Christine Miller put together a website with mostly her wisdom but some from us too. I wrote the articles on “Writing” and “Music” at the grammar level.

    Here’s the link to the suggestions for classical math materials at the grammar level. “Ray’s” won out among most of the group. Saxon was a little further down the list.

    http://www.classical-homeschooling.org/curriculum/math-grammar.html

  • http://www.cockahoop.com/ tODD

    Unfortunately, I can’t speak to the specifics of the approach mentioned in the article (beyond what the article mentions), and I definitely can’t speak to the specifics of any homeschooling programs, but … why let that stop me? :)

    I really don’t understand the antipathy I constantly hear against any new approaches to teaching. My wife is a math teacher, and, from the stories I hear, I sometimes think that parents rail against anything new more out of ignorance than constructive criticism. Some parents will hear that a new curriculum encourages group work and somehow turn that into the idea that their kids are just talking in math class, not working. That involves an ignorance of what is actually happening. Other times, I wonder if a new approach exposes a parent’s ignorance of the subject matter and makes them uncomfortable in trying to help their child.

    As an example, I’m having a hard time finding a problem with one of Veith’s examples: “It tackles the problem of multiplying six times three, by having students make six marks on a piece of paper in a little box and to do that with three boxes.” Obviously, that approach isn’t sustainable up into the higher levels of math (come time for exponentiation, that’s a lot of marks!), but as an introduction to what multiplication actually means, I think it’s a great example. And yet I have no trouble imagining some parent haranguing a teacher using this method (“Why not have them memorize that 6×3=18? It’s much quicker! This counting is ridiculous!”) without understanding that memorizing that fact doesn’t actually help one understand the problem, or solve multiplication in a word problem or, notably, the real world. That said, memorizing the multiplication table is helpful once one understands multiplication, since it aids in speed and consistency.

    I’m just as confused why anybody would find that, “When they multiply 23 times 5, they’ll do five 20s to get 100, and then add five 3s to get 15, and they put that all together and get 115.” (It is implied that Mr. Barlow had issues with a similar approach to adding large numbers.) That’s how I’d do 23×5 in my head — doing it the “old way” doesn’t lend itself easily to mental math, what with the carrying and the implied ones place and all. What’s more, doing math that way allows one to much more quickly arrive at a rough estimate, since the “old way” focuses first on the least significant digits.

    Of course, I’m just focusing on the examples I read about, since that’s all I’ve got.

  • http://www.cockahoop.com/ tODD

    Unfortunately, I can’t speak to the specifics of the approach mentioned in the article (beyond what the article mentions), and I definitely can’t speak to the specifics of any homeschooling programs, but … why let that stop me? :)

    I really don’t understand the antipathy I constantly hear against any new approaches to teaching. My wife is a math teacher, and, from the stories I hear, I sometimes think that parents rail against anything new more out of ignorance than constructive criticism. Some parents will hear that a new curriculum encourages group work and somehow turn that into the idea that their kids are just talking in math class, not working. That involves an ignorance of what is actually happening. Other times, I wonder if a new approach exposes a parent’s ignorance of the subject matter and makes them uncomfortable in trying to help their child.

    As an example, I’m having a hard time finding a problem with one of Veith’s examples: “It tackles the problem of multiplying six times three, by having students make six marks on a piece of paper in a little box and to do that with three boxes.” Obviously, that approach isn’t sustainable up into the higher levels of math (come time for exponentiation, that’s a lot of marks!), but as an introduction to what multiplication actually means, I think it’s a great example. And yet I have no trouble imagining some parent haranguing a teacher using this method (“Why not have them memorize that 6×3=18? It’s much quicker! This counting is ridiculous!”) without understanding that memorizing that fact doesn’t actually help one understand the problem, or solve multiplication in a word problem or, notably, the real world. That said, memorizing the multiplication table is helpful once one understands multiplication, since it aids in speed and consistency.

    I’m just as confused why anybody would find that, “When they multiply 23 times 5, they’ll do five 20s to get 100, and then add five 3s to get 15, and they put that all together and get 115.” (It is implied that Mr. Barlow had issues with a similar approach to adding large numbers.) That’s how I’d do 23×5 in my head — doing it the “old way” doesn’t lend itself easily to mental math, what with the carrying and the implied ones place and all. What’s more, doing math that way allows one to much more quickly arrive at a rough estimate, since the “old way” focuses first on the least significant digits.

    Of course, I’m just focusing on the examples I read about, since that’s all I’ve got.

  • http://www.hempelstudios.com Sarah in Maryland

    Let me tell you about my crazy math education. I was a very good student all throughout school. In first grade, after I had already memorized my addition and subtraction tables we had to learn a new method called “Touch Math.” In Touch Math we learned to make little dots on the numbers in a certain patter so that we could add them up by counting the dots (or subtracting by couting the dots backwards.) Let’s just say that I became the worst math student ever after this.

    Then my family moved to Japan. I was struggling in the 3rd grade with addition and subtration. Yet I could divide just fine because we didn’t learn that in Touch Math. I was reading at a 5th grade level, but couldn’t add up 5+8. So, my parents signed me up for soroban lessons. (Japanese abacus.) I soared to the head of the class in a matter of months and went on to attend a math and science center in high school. I am by no means a math genius, but this has made all the difference in the world. Want to know why the Japanese are leading the world in math? Soroban!

    My suggestion: SOROBAN LESSONS

  • http://www.hempelstudios.com Sarah in Maryland

    Let me tell you about my crazy math education. I was a very good student all throughout school. In first grade, after I had already memorized my addition and subtraction tables we had to learn a new method called “Touch Math.” In Touch Math we learned to make little dots on the numbers in a certain patter so that we could add them up by counting the dots (or subtracting by couting the dots backwards.) Let’s just say that I became the worst math student ever after this.

    Then my family moved to Japan. I was struggling in the 3rd grade with addition and subtration. Yet I could divide just fine because we didn’t learn that in Touch Math. I was reading at a 5th grade level, but couldn’t add up 5+8. So, my parents signed me up for soroban lessons. (Japanese abacus.) I soared to the head of the class in a matter of months and went on to attend a math and science center in high school. I am by no means a math genius, but this has made all the difference in the world. Want to know why the Japanese are leading the world in math? Soroban!

    My suggestion: SOROBAN LESSONS

  • http://adsoofmelk.wordpress.com/ Adso of Melk

    We tried Saxon Math 1A with my child on the recommendation of Susan Wise Bauer’s _The Well-trained Mind_, dutifully working through the incredibly repetitious, soul-killing boredom of counting tally marks, memorizing addition facts, and learning almost nothing. We got most of the way — 75-80% through — before my child basically shut down on math altogether.

    This was not good.

    We gave her a break of a few weeks, consulted a math teacher for her advice, and went for Miquon Math. Miquon, with its focus on manipulatives, made concepts such as place value and regrouping completely understandable — it was no longer the mindless execution of a standard algorithm one memorized (as in Saxon) but the execution of a problem one understood from “hands-on” practice.

    I can’t say Miquon would necessarily work with everyone; I think math, like many subjects, is one which can be taught well using a variety of methods. However, that said, Saxon was a dreadful choice for us and for the way our child learned.

  • http://adsoofmelk.wordpress.com/ Adso of Melk

    We tried Saxon Math 1A with my child on the recommendation of Susan Wise Bauer’s _The Well-trained Mind_, dutifully working through the incredibly repetitious, soul-killing boredom of counting tally marks, memorizing addition facts, and learning almost nothing. We got most of the way — 75-80% through — before my child basically shut down on math altogether.

    This was not good.

    We gave her a break of a few weeks, consulted a math teacher for her advice, and went for Miquon Math. Miquon, with its focus on manipulatives, made concepts such as place value and regrouping completely understandable — it was no longer the mindless execution of a standard algorithm one memorized (as in Saxon) but the execution of a problem one understood from “hands-on” practice.

    I can’t say Miquon would necessarily work with everyone; I think math, like many subjects, is one which can be taught well using a variety of methods. However, that said, Saxon was a dreadful choice for us and for the way our child learned.

  • http://www.cockahoop.com/ tODD

    You know, the more I think about it, the more I conclude that the problem is that anyone thinks that there is any one perfect curriculum. It’s silly to think so — we all know how different people are. The main problem with children is that they don’t know how they’re different, or how a curriculum fails to meet their needs, or barring that, they lack the ability to communicate any of this.

    This, of course, is one advantage to home-schooling, or at least very small student-teacher ratios (or supplemental one-on-one teaching from a parent or tutor). The teacher can recognize that a curriculum isn’t working and perhaps even identify why. And hopefully adjust to meet the student’s needs and abilities. With many students per teacher, this simply isn’t feasible.

  • http://www.cockahoop.com/ tODD

    You know, the more I think about it, the more I conclude that the problem is that anyone thinks that there is any one perfect curriculum. It’s silly to think so — we all know how different people are. The main problem with children is that they don’t know how they’re different, or how a curriculum fails to meet their needs, or barring that, they lack the ability to communicate any of this.

    This, of course, is one advantage to home-schooling, or at least very small student-teacher ratios (or supplemental one-on-one teaching from a parent or tutor). The teacher can recognize that a curriculum isn’t working and perhaps even identify why. And hopefully adjust to meet the student’s needs and abilities. With many students per teacher, this simply isn’t feasible.

  • Joyce

    For us, it looks like memorizing your math facts and doing time tests for the grammer stage (addition, subtraction, multiplication and division) in grades 1-5. If you haven’t mastered your basic math facts, you are hindered at every new concept. To your basic math facts you add the basic principles of fractions, measuring, etc. A number of curriculums do this well. Around the sixth grade, or when the child is ready to think conceptually, we stepped away from Saxton for Algebra and Geometry and used Jacobs. For learning the concepts involved in Algebra and Geometry, Jacob’s teaching method was more geared to a step by step increasing understanding of each new concept introduced. Saxton has too much review and not enough problems in mastering a new concept. I considered this the logic of mathematics.

  • Joyce

    For us, it looks like memorizing your math facts and doing time tests for the grammer stage (addition, subtraction, multiplication and division) in grades 1-5. If you haven’t mastered your basic math facts, you are hindered at every new concept. To your basic math facts you add the basic principles of fractions, measuring, etc. A number of curriculums do this well. Around the sixth grade, or when the child is ready to think conceptually, we stepped away from Saxton for Algebra and Geometry and used Jacobs. For learning the concepts involved in Algebra and Geometry, Jacob’s teaching method was more geared to a step by step increasing understanding of each new concept introduced. Saxton has too much review and not enough problems in mastering a new concept. I considered this the logic of mathematics.

  • forty-two

    One word: proofs.

    From what I’ve read, the Quadrivium is focused on understanding and manipulating mathematical concepts, not just the ability to do calculations (a useful business skill, to be sure, but nothing that trained the mind). In other words, the ability to do proofs. (Certainly, Euclidean geometry is nothing but proofs, and in fact is the first example of doing math axiomatically.)

    Fast forward 2000 years, and it is still true that real math is primarily the business of proving theorems. Sadly, outside of a proof-based Euclidean geometry course (which should be redundant, but isn’t these days; in fact it seems to be a dying breed) in high school, there is practically no real math taught below the senior collegiate level. I took six university math courses (toward an engineering degree), and none went beyond cookbook calculus or applications of linear algebra – not a proof in sight. My sister, a math major, took nearly that many math-application-style courses before she took her first real math class – real analysis – which, being the first exposure to proofs after years of training in application-only math, has a reputation as a killer class.

    So the result is that pretty much everyone outside of mathematicians has never been exposed to real math (versus math appreciation or math applications) unless they were lucky enough to have taken a good Euclidean geometry class. Being a math nerd, I’ve been trying teach myself real math, as well as try to figure out how in the world I can teach my dd real, proofy math when the time comes. Thanks to the power of the internet, I’ve found a homeschooling math major who is blazing the teach-proofs-to-kids trail. (He is also an advocate of having Latin and math as the cornerstones of education.)

    I’ve bought many of the math books he recommends (he lists them at his website: http://www.oplink.net/~adrian/mathed.html), and am using them to slowly remediate myself. I am quite convinced that teaching math classically means teaching it as rigorously and axiomatically as possible.

  • forty-two

    One word: proofs.

    From what I’ve read, the Quadrivium is focused on understanding and manipulating mathematical concepts, not just the ability to do calculations (a useful business skill, to be sure, but nothing that trained the mind). In other words, the ability to do proofs. (Certainly, Euclidean geometry is nothing but proofs, and in fact is the first example of doing math axiomatically.)

    Fast forward 2000 years, and it is still true that real math is primarily the business of proving theorems. Sadly, outside of a proof-based Euclidean geometry course (which should be redundant, but isn’t these days; in fact it seems to be a dying breed) in high school, there is practically no real math taught below the senior collegiate level. I took six university math courses (toward an engineering degree), and none went beyond cookbook calculus or applications of linear algebra – not a proof in sight. My sister, a math major, took nearly that many math-application-style courses before she took her first real math class – real analysis – which, being the first exposure to proofs after years of training in application-only math, has a reputation as a killer class.

    So the result is that pretty much everyone outside of mathematicians has never been exposed to real math (versus math appreciation or math applications) unless they were lucky enough to have taken a good Euclidean geometry class. Being a math nerd, I’ve been trying teach myself real math, as well as try to figure out how in the world I can teach my dd real, proofy math when the time comes. Thanks to the power of the internet, I’ve found a homeschooling math major who is blazing the teach-proofs-to-kids trail. (He is also an advocate of having Latin and math as the cornerstones of education.)

    I’ve bought many of the math books he recommends (he lists them at his website: http://www.oplink.net/~adrian/mathed.html), and am using them to slowly remediate myself. I am quite convinced that teaching math classically means teaching it as rigorously and axiomatically as possible.

  • http://www.oldsolar.com/currentblog.php Rick Ritchie

    I think sometimes the very concrete methods are helpful in the early stages. I remember being in first grade, sitting in our school learning center with the little counters working out a math problem with place value until I was convinced that the math rules were not arbitrary but represented something real. Carrying really was something that could be represented with counters as well as on paper. After that, it was easier to trust that other rules would also reflect reality rather than being an arbitrary process cooked up by teachers.

    When I hear such methods are used, I am happy, so long as I don’t have the impression that this is all that there is to the math teaching. This kind of thing is good for solving basic conceptual problems at the foundation.

  • http://www.oldsolar.com/currentblog.php Rick Ritchie

    I think sometimes the very concrete methods are helpful in the early stages. I remember being in first grade, sitting in our school learning center with the little counters working out a math problem with place value until I was convinced that the math rules were not arbitrary but represented something real. Carrying really was something that could be represented with counters as well as on paper. After that, it was easier to trust that other rules would also reflect reality rather than being an arbitrary process cooked up by teachers.

    When I hear such methods are used, I am happy, so long as I don’t have the impression that this is all that there is to the math teaching. This kind of thing is good for solving basic conceptual problems at the foundation.

  • http://www.geneveith.com Veith

    Thanks, forty-two! The use of “proofs” and this website may well be the key to a classical approach to math.

  • http://www.geneveith.com Veith

    Thanks, forty-two! The use of “proofs” and this website may well be the key to a classical approach to math.

  • http://www.hempelstudios.com Sarah in Maryland

    We did timed math tests in grades 1-3 and all they did for me was give me anxiety attacks. Can you imagine an otherwise very confident little girl literally sweating profusely and shaking over a timed math test? It was terrible and I wasn’t able to shake the anxiety until college. :-(

    THUMBS DOWN on timed math. THUMBS DOWN!

  • http://www.hempelstudios.com Sarah in Maryland

    We did timed math tests in grades 1-3 and all they did for me was give me anxiety attacks. Can you imagine an otherwise very confident little girl literally sweating profusely and shaking over a timed math test? It was terrible and I wasn’t able to shake the anxiety until college. :-(

    THUMBS DOWN on timed math. THUMBS DOWN!

  • Kathrine

    Robert Talbert,
    I appreciate your comments. We used Saxon math until halfway through the third grade book. I began to see what you were writing about and decided to switch to Singapore. Singapore has placement tests so you can start your child at the right level. We took the placement test for their 1B book (second half of first grade), and I was amazed at how many problems my son SHOULD’VE known how to do, but couldn’t even take an educated guess because there was literally one word’s difference in the way they asked the problem from the way Saxon asked the problem. He was not learning math, he was learning Saxon math. I’m glad to have a college prof. corroborate my non-expert opinion. I wish more of you would get together and publish some critiques of the math curricula available so we home schoolers can make a more educated choice.

  • Kathrine

    Robert Talbert,
    I appreciate your comments. We used Saxon math until halfway through the third grade book. I began to see what you were writing about and decided to switch to Singapore. Singapore has placement tests so you can start your child at the right level. We took the placement test for their 1B book (second half of first grade), and I was amazed at how many problems my son SHOULD’VE known how to do, but couldn’t even take an educated guess because there was literally one word’s difference in the way they asked the problem from the way Saxon asked the problem. He was not learning math, he was learning Saxon math. I’m glad to have a college prof. corroborate my non-expert opinion. I wish more of you would get together and publish some critiques of the math curricula available so we home schoolers can make a more educated choice.

  • Pinon Coffee

    Sarah in Maryland: yes! Time-tests were HORRIBLE for me too. I really think they convinced me I was dumb at math, and I’ve hated it ever since.

    If it works for some people, lovely. But for me–ew!!

  • Pinon Coffee

    Sarah in Maryland: yes! Time-tests were HORRIBLE for me too. I really think they convinced me I was dumb at math, and I’ve hated it ever since.

    If it works for some people, lovely. But for me–ew!!

  • http://www.pagantolutheran.blogspot.com Bruce

    Forty Two: I pasted your website into an email to my wife, who has homeschooled for nineteen years and is the resident math curriculum guru. I asked her, “Seen this?”

    Her reply:

    “Um, yeah…this is the program both J and R [our two sons, now aged 18 & 25] used. The primary math I
    mean. That goes through pre-algebra and then becomes an integrated
    math that doesn’t fit into the American system unless you are a math
    whiz and can make the transitions between systems (not our kids).

    Robin [child #3, now 13] hated this program so much she bailed after the 4th level (there
    are 6). She uses something else now. Not as thorough, but she’s never
    going to like math anyway.”

    Our kids are really, really good at Latin. #2 son became proficient enough at math to do well in a high school honors physics course. Otherwise, we’re the typical language-emphasis homeschooling family. Don’t look to homeschoolers to solve the huge math/science gap between us and the Asians any time soon. It remains our Achilles Heel (to use a non-math metaphor).

  • http://www.pagantolutheran.blogspot.com Bruce

    Forty Two: I pasted your website into an email to my wife, who has homeschooled for nineteen years and is the resident math curriculum guru. I asked her, “Seen this?”

    Her reply:

    “Um, yeah…this is the program both J and R [our two sons, now aged 18 & 25] used. The primary math I
    mean. That goes through pre-algebra and then becomes an integrated
    math that doesn’t fit into the American system unless you are a math
    whiz and can make the transitions between systems (not our kids).

    Robin [child #3, now 13] hated this program so much she bailed after the 4th level (there
    are 6). She uses something else now. Not as thorough, but she’s never
    going to like math anyway.”

    Our kids are really, really good at Latin. #2 son became proficient enough at math to do well in a high school honors physics course. Otherwise, we’re the typical language-emphasis homeschooling family. Don’t look to homeschoolers to solve the huge math/science gap between us and the Asians any time soon. It remains our Achilles Heel (to use a non-math metaphor).

  • http://www.cumberlandisland.blogspot.com Adrian Keister

    Reply to # 14.

    My name is Adrian Keister; ever since I was a sophomore in college, my goal has been to develop a K-12 math curriculum for classical education. Since my bachelor’s degree (in physics, computers, and math), I’ve gone on to get a master’s and doctorate in mathematical physics.

    I have taken a look at the webpage you mentioned. I’ve certainly heard of some of those books mentioned. I have to say, though, that I wouldn’t recommend “baby Rudin”. Rudin is an incredibly smart mathematician; as a result, his proofs are so ridiculously polished, that students get the idea that “mathematicians do math” that way. If they don’t come up with the perfect, most elegant proof immediately, they’ve failed. Math is much rougher than that, usually.

    My discipline of mathematical physics gives me an interesting outlook on math and math teaching (I have taught college calculus twice). There are a number of factors that are either missing from current curriculums or it is the case that no curriculum has more than about one or two of them.

    Continued…

  • http://www.cumberlandisland.blogspot.com Adrian Keister

    Reply to # 14.

    My name is Adrian Keister; ever since I was a sophomore in college, my goal has been to develop a K-12 math curriculum for classical education. Since my bachelor’s degree (in physics, computers, and math), I’ve gone on to get a master’s and doctorate in mathematical physics.

    I have taken a look at the webpage you mentioned. I’ve certainly heard of some of those books mentioned. I have to say, though, that I wouldn’t recommend “baby Rudin”. Rudin is an incredibly smart mathematician; as a result, his proofs are so ridiculously polished, that students get the idea that “mathematicians do math” that way. If they don’t come up with the perfect, most elegant proof immediately, they’ve failed. Math is much rougher than that, usually.

    My discipline of mathematical physics gives me an interesting outlook on math and math teaching (I have taught college calculus twice). There are a number of factors that are either missing from current curriculums or it is the case that no curriculum has more than about one or two of them.

    Continued…

  • http://www.cumberlandisland.blogspot.com Adrian Keister

    …Continued

    1. Hardly any curriculum is Christ-centered. According to James Nickol, author of Mathematics: Is God Silent?, it does matter, in math, whether Jesus Christ is Lord over all or not. Yet math is almost always taught as the archetypal “neutral” subject. I would put forth that any approximation to an “ideal” curriculum (I put ideal in quotes because it’s probably not attainable) must be Christ-centered.

    2. There is not enough review. I went through Saxon myself, and while it is seriously flawed, one point in its favor is the long-term retention of anything it does teach. You can be sure, in Saxon, that in Lesson 130, you’ll be doing problems you learned how to do in Lesson 1. That is useful for remembering things over the long haul.

    3. It is not oriented to the Trivium. No curriculum that I know of teaches math according to the Trivium. The Trivium works so well for so many things: why not arithmetic, algebra, geometry, trigonometry, calculus, and differential equations?

    4. It is not applications-oriented. I’ve seen maybe one or two math books that are really applications oriented. Most “applications” consist of showing how we can more easily prove… yet another theorem.

    Continued…

  • http://www.cumberlandisland.blogspot.com Adrian Keister

    …Continued

    1. Hardly any curriculum is Christ-centered. According to James Nickol, author of Mathematics: Is God Silent?, it does matter, in math, whether Jesus Christ is Lord over all or not. Yet math is almost always taught as the archetypal “neutral” subject. I would put forth that any approximation to an “ideal” curriculum (I put ideal in quotes because it’s probably not attainable) must be Christ-centered.

    2. There is not enough review. I went through Saxon myself, and while it is seriously flawed, one point in its favor is the long-term retention of anything it does teach. You can be sure, in Saxon, that in Lesson 130, you’ll be doing problems you learned how to do in Lesson 1. That is useful for remembering things over the long haul.

    3. It is not oriented to the Trivium. No curriculum that I know of teaches math according to the Trivium. The Trivium works so well for so many things: why not arithmetic, algebra, geometry, trigonometry, calculus, and differential equations?

    4. It is not applications-oriented. I’ve seen maybe one or two math books that are really applications oriented. Most “applications” consist of showing how we can more easily prove… yet another theorem.

    Continued…

  • http://www.cumberlandisland.blogspot.com Adrian Keister

    …Continued

    4. Continued. Why my emphasis on applications? because, prior to about the 1700′s or possibly 1800′s, the concept of Pure Mathematics was practically unknown. What you had, though they weren’t called this, were mathematical physicists. People studied math in order to be able to solve real-world problems – either to make people’s lives better (technology) or simply to find out more about how God’s creation worked (science). Both activities fall under the Dominion Mandate in Genesis. Pure mathematicians tend to rejoice in doing work that is “unsullied” by applications. And for that attitude, I really don’t have much respect. Christianity is incarnational. We are not gnostics. In my hopes of being a teacher at a classical Christian school, I am currently working as an engineer so as to be able to tell my students, “You know, this stuff really matters. You can use it to do THIS!!!”

    5. It does not teach the students how to think mathematically. The Trivium is all about this, so perhaps this should fall under my previous category 3.

    6. As I mentioned in my comment about Rudin’s book, most math students are given the impression that math is originally developed in definition, theorem, proof format. That is NOT TRUE!!! Math is hammered out painfully, slowly, more like refining diamonds. Logic, so touted as the method geometry is so important to teach, is really quite limited. Note I did not say useless – far from it. Logic is absolutely essential. We all want, philosophically, to get from truth A to truth B. Logic can’t get us there, actually. You have to use your imagination to do that. Logic won’t even tell us what A and B are! Why should we want to prove this theorem? Gosh, I don’t know! You see? You need a worldview for that. Logic, in the final analysis, is most useful in telling whether, GIVEN a particular method of getting from A to B, whether it’s valid or not. So you can never go against logic, but it is extremely limited.

    Continued…

  • http://www.cumberlandisland.blogspot.com Adrian Keister

    …Continued

    4. Continued. Why my emphasis on applications? because, prior to about the 1700′s or possibly 1800′s, the concept of Pure Mathematics was practically unknown. What you had, though they weren’t called this, were mathematical physicists. People studied math in order to be able to solve real-world problems – either to make people’s lives better (technology) or simply to find out more about how God’s creation worked (science). Both activities fall under the Dominion Mandate in Genesis. Pure mathematicians tend to rejoice in doing work that is “unsullied” by applications. And for that attitude, I really don’t have much respect. Christianity is incarnational. We are not gnostics. In my hopes of being a teacher at a classical Christian school, I am currently working as an engineer so as to be able to tell my students, “You know, this stuff really matters. You can use it to do THIS!!!”

    5. It does not teach the students how to think mathematically. The Trivium is all about this, so perhaps this should fall under my previous category 3.

    6. As I mentioned in my comment about Rudin’s book, most math students are given the impression that math is originally developed in definition, theorem, proof format. That is NOT TRUE!!! Math is hammered out painfully, slowly, more like refining diamonds. Logic, so touted as the method geometry is so important to teach, is really quite limited. Note I did not say useless – far from it. Logic is absolutely essential. We all want, philosophically, to get from truth A to truth B. Logic can’t get us there, actually. You have to use your imagination to do that. Logic won’t even tell us what A and B are! Why should we want to prove this theorem? Gosh, I don’t know! You see? You need a worldview for that. Logic, in the final analysis, is most useful in telling whether, GIVEN a particular method of getting from A to B, whether it’s valid or not. So you can never go against logic, but it is extremely limited.

    Continued…

  • http://www.cumberlandisland.blogspot.com Adrian Keister

    …Continued

    If you were to ask me what is the most important quality in a person to really do mathematics, I would say it is this: imagination. And by the way, what is mathematics? It’s certainly not a bunch of formulas and rules. It’s not definition, theorem, proof; though if you were to get an advanced degree in math you might come to think that. I would define it, though others might disagree, as the recognition of numeric patterns in God’s creation. Any math program that fails to go into the philosophical basis for it is useless.

    Continued…

  • http://www.cumberlandisland.blogspot.com Adrian Keister

    …Continued

    If you were to ask me what is the most important quality in a person to really do mathematics, I would say it is this: imagination. And by the way, what is mathematics? It’s certainly not a bunch of formulas and rules. It’s not definition, theorem, proof; though if you were to get an advanced degree in math you might come to think that. I would define it, though others might disagree, as the recognition of numeric patterns in God’s creation. Any math program that fails to go into the philosophical basis for it is useless.

    Continued…

  • http://www.cumberlandisland.blogspot.com Adrian Keister

    …Continued

    7. Rather tied to 4: it is not integrated with physics. Physics and math are inextricably entwined – physics without math would be nothing. I would also argue from history that math without physics would be nothing – just look at the Greeks. If you can teach an idea in the math, and then use it in the physics a week later, that provides a wonderful way to hammer things in.

    And so what, pray, would I put forth as a curriculum? I can stand here all day and tear down other curriculums, but if I can’t put forth a substitute, why should I waste your time?

    Here it is.

    First of all, it is Christ-centered. If Jesus Christ is not Lord over all, then 2 + 2 equals whatever we want.

    Second of all, it uses the Trivium as the method.

    Third of all, it has as its goal differential equations before graduation from high school. Why diff eq? Because it is diff eq where the real-world applications start to arise in gigantic proportions. Most real-world engineering problems and scientific problems are posed as differential equations, and the solution of them gives us amazing predictive power.

    I propose this: in the grammar years, teach the grammar of the following subjects: arithmetic, algebra, geometry, trigonometry, calculus, and differential equations. In the dialectic years, teach the logic of the subjects I just mentioned. In the rhetoric years, teach the applications and real problem-solving skills. In other words, what do you do when you are faced with a problem you know nothing about and have no idea how to solve? How to you hack away at it? Someone mentioned heuristics earlier: that is the idea here. We look at possible methods of solution and eliminate infeasible ones; there are all kinds of ways to do this.

    Last and certainly not least: no program is successful without parents and teachers who are behind it. Teachers in particular: you need to know and be a master of everything that is to be taught. I would put forth applied mathematicians or physicists as the best math teachers: they know where they can use it, since they have used it.

    These are my ideas. I’ve posted them at the risk of someone else stealing them and writing their curriculum first. So be it. I’d be ok with that, and be content just to teach the stuff.

    In Christ.

  • http://www.cumberlandisland.blogspot.com Adrian Keister

    …Continued

    7. Rather tied to 4: it is not integrated with physics. Physics and math are inextricably entwined – physics without math would be nothing. I would also argue from history that math without physics would be nothing – just look at the Greeks. If you can teach an idea in the math, and then use it in the physics a week later, that provides a wonderful way to hammer things in.

    And so what, pray, would I put forth as a curriculum? I can stand here all day and tear down other curriculums, but if I can’t put forth a substitute, why should I waste your time?

    Here it is.

    First of all, it is Christ-centered. If Jesus Christ is not Lord over all, then 2 + 2 equals whatever we want.

    Second of all, it uses the Trivium as the method.

    Third of all, it has as its goal differential equations before graduation from high school. Why diff eq? Because it is diff eq where the real-world applications start to arise in gigantic proportions. Most real-world engineering problems and scientific problems are posed as differential equations, and the solution of them gives us amazing predictive power.

    I propose this: in the grammar years, teach the grammar of the following subjects: arithmetic, algebra, geometry, trigonometry, calculus, and differential equations. In the dialectic years, teach the logic of the subjects I just mentioned. In the rhetoric years, teach the applications and real problem-solving skills. In other words, what do you do when you are faced with a problem you know nothing about and have no idea how to solve? How to you hack away at it? Someone mentioned heuristics earlier: that is the idea here. We look at possible methods of solution and eliminate infeasible ones; there are all kinds of ways to do this.

    Last and certainly not least: no program is successful without parents and teachers who are behind it. Teachers in particular: you need to know and be a master of everything that is to be taught. I would put forth applied mathematicians or physicists as the best math teachers: they know where they can use it, since they have used it.

    These are my ideas. I’ve posted them at the risk of someone else stealing them and writing their curriculum first. So be it. I’d be ok with that, and be content just to teach the stuff.

    In Christ.

  • http://www.cumberlandisland.blogspot.com Adrian Keister

    Errata: my entire post was a reply to #15, not #14. Sorry for the confusion.

    In Christ.

  • http://www.cumberlandisland.blogspot.com Adrian Keister

    Errata: my entire post was a reply to #15, not #14. Sorry for the confusion.

    In Christ.

  • forty-two

    A few thoughts re: Adrian’s comments, in no particular order:

    I disagree that pure mathematics is somehow an unworthy pursuit – that math is only valuable inasmuch it has utility. Classical education is all about focusing on things that are true, beautiful, and good, simply because they ARE true, beautiful, and good. Those qualities are their own justification – nothing further is required. Math is beautiful, and it describes things that are true. If a part of math should have applications, that certainly doesn’t “sully” the math, but neither does the existence of applications somehow confer worthiness on otherwise useless math.

    As well, I disagree with the idea that the elegant proofs of Rudin shouldn’t be studied simply because of the fact that most proofs – especially those attempted by students – are not nearly as beautiful. Should we not study the Great Books simply because most writing – especially that of students – is not nearly as polished? Of course not! Classical education is all about studying the masterpieces of the world, the very best that mankind has accomplished, in order to learn from them. Limiting one’s study to mediocrity ensures that one will never rise above it.

    con’t

  • forty-two

    A few thoughts re: Adrian’s comments, in no particular order:

    I disagree that pure mathematics is somehow an unworthy pursuit – that math is only valuable inasmuch it has utility. Classical education is all about focusing on things that are true, beautiful, and good, simply because they ARE true, beautiful, and good. Those qualities are their own justification – nothing further is required. Math is beautiful, and it describes things that are true. If a part of math should have applications, that certainly doesn’t “sully” the math, but neither does the existence of applications somehow confer worthiness on otherwise useless math.

    As well, I disagree with the idea that the elegant proofs of Rudin shouldn’t be studied simply because of the fact that most proofs – especially those attempted by students – are not nearly as beautiful. Should we not study the Great Books simply because most writing – especially that of students – is not nearly as polished? Of course not! Classical education is all about studying the masterpieces of the world, the very best that mankind has accomplished, in order to learn from them. Limiting one’s study to mediocrity ensures that one will never rise above it.

    con’t

  • forty-two

    Con’t…

    And, while your plan to teach math via the Trivium is intriguing, I must admit I haven’t the faintest idea what the grammar of any of the higher math topics may be, though I assume the logic stage would be where the proofs come in? That’s my best guess, anyway. It sounds very ambitious, but without more details, impossible to evaluate. Have you worked it out any further? I’m very interested in how it would break down.

  • forty-two

    Con’t…

    And, while your plan to teach math via the Trivium is intriguing, I must admit I haven’t the faintest idea what the grammar of any of the higher math topics may be, though I assume the logic stage would be where the proofs come in? That’s my best guess, anyway. It sounds very ambitious, but without more details, impossible to evaluate. Have you worked it out any further? I’m very interested in how it would break down.

  • http://www.cumberlandisland.blogspot.com Adrian Keister

    Reply to forty-two.

    I didn’t actually say that pure mathematics is not valuable to study. I said I didn’t have much respect for the attitudes of many pure mathematicians. As for beauty in mathematics, of course there’s loads of it, and I love that about math. But I, personally, see the beauty IN the applications, not in some man-made treadmill where we make up our own tiny rules and follow our own tiny logic. Ironically, most pure mathematics these days has real-world applications.

    I would disagree with your statement, “neither does the existence of applications somehow confer worthiness on otherwise useless math.” That depends on how you define “worthiness.”

    I would propose the Moore method, though difficult, as a better model than Rudin.

    I have to run now. Possibly more later.

    In Christ.

  • http://www.cumberlandisland.blogspot.com Adrian Keister

    Reply to forty-two.

    I didn’t actually say that pure mathematics is not valuable to study. I said I didn’t have much respect for the attitudes of many pure mathematicians. As for beauty in mathematics, of course there’s loads of it, and I love that about math. But I, personally, see the beauty IN the applications, not in some man-made treadmill where we make up our own tiny rules and follow our own tiny logic. Ironically, most pure mathematics these days has real-world applications.

    I would disagree with your statement, “neither does the existence of applications somehow confer worthiness on otherwise useless math.” That depends on how you define “worthiness.”

    I would propose the Moore method, though difficult, as a better model than Rudin.

    I have to run now. Possibly more later.

    In Christ.

  • http://www.oplink.net/~adrian Adrian Durham

    Alright Keister, you have provoked the troll from under the bridge.

    It’s not definition, theorem, proof; though if you were to get an advanced degree in math you might come to think that. I would define it, though others might disagree, as the recognition of numeric patterns in God’s creation. Any math program that fails to go into the philosophical basis for it is useless.

    But I, personally, see the beauty IN the applications, not in some man-made treadmill where we make up our own tiny rules and follow our own tiny logic.

    This is a common attitude. You’re view is actually far more prolific than those of pure mathematicians — perhaps even those of most applied mathematicians. The problem I have with it is more a philosophical one than a mathematical or scientific one. First of all, math is not man made. This kind of a question, in fact, has a lot more to do with the thrust of 20th century development in axiomatics than some sort of call for arbitrary rigor. And, as it turns out, it doesn’t really work out to think of mathematics that way as sort of a convention or language constrained by logic. It definitely is not just a sophisticated expression of logic. It is its own subject. Recognizing numeric patterns in God’s creation is so clearly science and not math. I don’t know what more can be said about that. You cannot possibly define math that way. To do so is to define it out of existence. So, you want to do science and not math — fine — but it does not and cannot answer the question of how to teach math. Math is a well defined subject — better defined even than physics or science and with a longer history — it doesn’t just go back to the 1800s.

    Math is not an empirical science. It is a formal a priori philosophy. Now, if one’s world view does not admit of the a priori (particularly anything like a “synthetic a priori” proposition, to borrow some terminology), then they will have a lot of trouble dealing with this issue. But, it is unavoidable. Not even the most radical empiricists will deny the a priori nature of mathematics these days. And, the truth is that the idea that math is something like “logic on steroids” or really just our way of describing the physical universe has been so sufficiently trounced by now. It is its own subject like any other subject with a philosophically vague foundation, its own unique objects of study that make it distinct from other subjects, and so on.

    And I’ll say two more things:

    1) It is factually incorrect to talk about math like it was always just a subarea of physics. That may be true about the current math department at your local university, but originally math as it has been recognized for millenia, now, was actually more than anything else an outgrowth of Platonist philosophers doing geometry without even the aid of algebra — essentially a philosophical matter. In fact, it is the influence of that philosophy that makes math math, and that influence is, in fact, where rigor comes from. The only reason we have ever gotten away from that is because of centuries of British empiricism.

    2) You may find the execution to be lacking, but what K-14 education consists of is almost entirely in the service of your view of math. Math is done empirically and heuristically which also means that it is being done fallaciously. The reason they don’t do more applications in K-14 is because they are too busy enduring the consequences of settling for a vague heuristic idea of the mathematics rather than a true and rigorous understanding of it. Instead of settling on the material covered, we have settled on how we cover the material. And, that manner has been almost entirely influenced by the “math is a science” view of math.

    Teaching something like differential equations to school aged students is so antithetical to any version of classical education, I cannot imagine a greater digression from it. You want a theoretical science oriented curriculum — that is the very opposite of classical. It is through the influence of science that we have departed from the classical and started doing what we do now. If we wanted a classical education we would do just what Richard says on his blog — teach number theory and geometry. We can throw in the modern subject of combinatorics and then build eventually up to topology and abstract algebra in college and only after that would we be permitted to tackle real analysis. We would not try to do it heuristically through science the way we do in the standard freshman calculus.

    And incidentally, Baby Rudin, is mathematics at its best. The only thing wrong with it is that it is so elementary. Baby Rudin is to classical real analysis as the Elements is to geometry. You may think it too polished or something, but, again, that just means you don’t like math — not that math bends around your interests to become theoretical physics just because that’s what you, personally, want to do. The Moore method and, for that matter, Moore’s subject (point-set topology) is entirely consistent with a Baby Rudin approach to analysis because you don’t read Baby Rudin. You do the problems and that is why the book is almost legendary now. It is the text equivalent of the Moore method — a little key stimulation and lots of truly difficult problems and the rest is up to the student. In fact, I really cannot think of a text more consistent with the idea of the Moore method than one like Baby Rudin in which so much material is built into the problems with so little in the way of coaching and example given in the text. (Perhaps it is no accident — Mary Rudin was Moore’s student, after all.)

    I truly cannot believe my eyes to see a mathematician say such things. You truly have turned into an engineer.

    But I, personally, see the beauty IN the applications, not in some man-made treadmill where we make up our own tiny rules and follow our own tiny logic.

    This is a common attitude. You’re view is actually far more prolific than those of pure mathematicians — perhaps even those of most applied mathematicians. The problem I have with it is more a philosophical one than a mathematical or scientific one. First of all, math is not man made. This kind of a question, in fact, has a lot more to do with the thrust of 20th century development in axiomatics than some sort of call for arbitrary rigor. And, as it turns out, it doesn’t really work out to think of mathematics that way as sort of a convention or language constrained by logic. It definitely is not just a sophisticated expression of logic. It is its own subject. Recognizing numeric patterns in God’s creation is so clearly science and not math. I don’t know what more can be said about that. You cannot possibly define math that way. To do so is to define it out of existence. So, you want to do science and not math — fine — but it does not and cannot answer the question of how to teach math. Math is a well defined subject — better defined even than physics or science and with a longer history — it doesn’t just go back to the 1800s.

    Math is not an empirical science. It is a formal a priori philosophy. Now, if one’s world view does not admit of the a priori (particularly anything like a “synthetic a priori” proposition, to borrow some terminology), then they will have a lot of trouble dealing with this issue. But, it is unavoidable. Not even the most radical empiricists will deny the a priori nature of mathematics these days. And, the truth is that the idea that math is something like “logic on steroids” or really just our way of describing the physical universe has been so sufficiently trounced by now. It is its own subject like any other subject with a philosophically vague foundation, its own unique objects of study that make it distinct from other subjects, and so on.

    And I’ll say two more things:

    1) It is factually incorrect to talk about math like it was always just a subarea of physics. That may be true about the current math department at your local university, but originally math as it has been recognized for millenia, now, was actually more than anything else an outgrowth of Platonist philosophers doing geometry without even the aid of algebra — essentially a philosophical matter. In fact, it is the influence of that philosophy that makes math math, and that influence is, in fact, where rigor comes from. The only reason we have ever gotten away from that is because of centuries of British empiricism.

    2) You may find the execution to be lacking, but what K-14 education consists of is almost entirely in the service of your view of math. Math is done empirically and heuristically which also means that it is being done fallaciously. The reason they don’t do more applications in K-14 is because they are too busy enduring the consequences of settling for a vague heuristic idea of the mathematics rather than a true and rigorous understanding of it. Instead of settling on the material covered, we have settled on how we cover the material. And, that manner has been almost entirely influenced by the “math is a science” view of math.

    Teaching something like differential equations to school aged students is so antithetical to any version of classical education, I cannot imagine a greater digression from it. You want a theoretical science oriented curriculum — that is the very opposite of classical. It is through the influence of science that we have departed from the classical and started doing what we do now. If we wanted a classical education we would do just what Richard says on his blog — teach number theory and geometry. We can throw in the modern subject of combinatorics and then build eventually up to topology and abstract algebra in college and only after that would we be permitted to tackle real analysis. We would not try to do it heuristically through science the way we do in the standard freshman calculus.

    And incidentally, Baby Rudin, is mathematics at its best. The only thing wrong with it is that it is so elementary. Baby Rudin is to classical real analysis as the Elements is to geometry. You may think it too polished or something, but, again, that just means you don’t like math — not that math bends around your interests to become theoretical physics just because that’s what you, personally, want to do. The Moore method and, for that matter, Moore’s subject (point-set topology) is entirely consistent with a Baby Rudin approach to analysis because you don’t read Baby Rudin. You do the problems and that is why the book is almost legendary now. It is the text equivalent of the Moore method — a little key stimulation and lots of truly difficult problems and the rest is up to the student. In fact, I really cannot think of a text more consistent with the idea of the Moore method than one like Baby Rudin in which so much material is built into the problems with so little in the way of coaching and example given in the text. (Perhaps it is no accident — Mary Rudin was Moore’s student, after all.)

    I truly cannot believe my eyes to see a mathematician say such things. You truly have turned into an engineer.

  • http://www.oplink.net/~adrian Adrian Durham

    Alright Keister, you have provoked the troll from under the bridge.

    It’s not definition, theorem, proof; though if you were to get an advanced degree in math you might come to think that. I would define it, though others might disagree, as the recognition of numeric patterns in God’s creation. Any math program that fails to go into the philosophical basis for it is useless.

    But I, personally, see the beauty IN the applications, not in some man-made treadmill where we make up our own tiny rules and follow our own tiny logic.

    This is a common attitude. You’re view is actually far more prolific than those of pure mathematicians — perhaps even those of most applied mathematicians. The problem I have with it is more a philosophical one than a mathematical or scientific one. First of all, math is not man made. This kind of a question, in fact, has a lot more to do with the thrust of 20th century development in axiomatics than some sort of call for arbitrary rigor. And, as it turns out, it doesn’t really work out to think of mathematics that way as sort of a convention or language constrained by logic. It definitely is not just a sophisticated expression of logic. It is its own subject. Recognizing numeric patterns in God’s creation is so clearly science and not math. I don’t know what more can be said about that. You cannot possibly define math that way. To do so is to define it out of existence. So, you want to do science and not math — fine — but it does not and cannot answer the question of how to teach math. Math is a well defined subject — better defined even than physics or science and with a longer history — it doesn’t just go back to the 1800s.

    Math is not an empirical science. It is a formal a priori philosophy. Now, if one’s world view does not admit of the a priori (particularly anything like a “synthetic a priori” proposition, to borrow some terminology), then they will have a lot of trouble dealing with this issue. But, it is unavoidable. Not even the most radical empiricists will deny the a priori nature of mathematics these days. And, the truth is that the idea that math is something like “logic on steroids” or really just our way of describing the physical universe has been so sufficiently trounced by now. It is its own subject like any other subject with a philosophically vague foundation, its own unique objects of study that make it distinct from other subjects, and so on.

    And I’ll say two more things:

    1) It is factually incorrect to talk about math like it was always just a subarea of physics. That may be true about the current math department at your local university, but originally math as it has been recognized for millenia, now, was actually more than anything else an outgrowth of Platonist philosophers doing geometry without even the aid of algebra — essentially a philosophical matter. In fact, it is the influence of that philosophy that makes math math, and that influence is, in fact, where rigor comes from. The only reason we have ever gotten away from that is because of centuries of British empiricism.

    2) You may find the execution to be lacking, but what K-14 education consists of is almost entirely in the service of your view of math. Math is done empirically and heuristically which also means that it is being done fallaciously. The reason they don’t do more applications in K-14 is because they are too busy enduring the consequences of settling for a vague heuristic idea of the mathematics rather than a true and rigorous understanding of it. Instead of settling on the material covered, we have settled on how we cover the material. And, that manner has been almost entirely influenced by the “math is a science” view of math.

    Teaching something like differential equations to school aged students is so antithetical to any version of classical education, I cannot imagine a greater digression from it. You want a theoretical science oriented curriculum — that is the very opposite of classical. It is through the influence of science that we have departed from the classical and started doing what we do now. If we wanted a classical education we would do just what Richard says on his blog — teach number theory and geometry. We can throw in the modern subject of combinatorics and then build eventually up to topology and abstract algebra in college and only after that would we be permitted to tackle real analysis. We would not try to do it heuristically through science the way we do in the standard freshman calculus.

    And incidentally, Baby Rudin, is mathematics at its best. The only thing wrong with it is that it is so elementary. Baby Rudin is to classical real analysis as the Elements is to geometry. You may think it too polished or something, but, again, that just means you don’t like math — not that math bends around your interests to become theoretical physics just because that’s what you, personally, want to do. The Moore method and, for that matter, Moore’s subject (point-set topology) is entirely consistent with a Baby Rudin approach to analysis because you don’t read Baby Rudin. You do the problems and that is why the book is almost legendary now. It is the text equivalent of the Moore method — a little key stimulation and lots of truly difficult problems and the rest is up to the student. In fact, I really cannot think of a text more consistent with the idea of the Moore method than one like Baby Rudin in which so much material is built into the problems with so little in the way of coaching and example given in the text. (Perhaps it is no accident — Mary Rudin was Moore’s student, after all.)

    I truly cannot believe my eyes to see a mathematician say such things. You truly have turned into an engineer.

    But I, personally, see the beauty IN the applications, not in some man-made treadmill where we make up our own tiny rules and follow our own tiny logic.

    This is a common attitude. You’re view is actually far more prolific than those of pure mathematicians — perhaps even those of most applied mathematicians. The problem I have with it is more a philosophical one than a mathematical or scientific one. First of all, math is not man made. This kind of a question, in fact, has a lot more to do with the thrust of 20th century development in axiomatics than some sort of call for arbitrary rigor. And, as it turns out, it doesn’t really work out to think of mathematics that way as sort of a convention or language constrained by logic. It definitely is not just a sophisticated expression of logic. It is its own subject. Recognizing numeric patterns in God’s creation is so clearly science and not math. I don’t know what more can be said about that. You cannot possibly define math that way. To do so is to define it out of existence. So, you want to do science and not math — fine — but it does not and cannot answer the question of how to teach math. Math is a well defined subject — better defined even than physics or science and with a longer history — it doesn’t just go back to the 1800s.

    Math is not an empirical science. It is a formal a priori philosophy. Now, if one’s world view does not admit of the a priori (particularly anything like a “synthetic a priori” proposition, to borrow some terminology), then they will have a lot of trouble dealing with this issue. But, it is unavoidable. Not even the most radical empiricists will deny the a priori nature of mathematics these days. And, the truth is that the idea that math is something like “logic on steroids” or really just our way of describing the physical universe has been so sufficiently trounced by now. It is its own subject like any other subject with a philosophically vague foundation, its own unique objects of study that make it distinct from other subjects, and so on.

    And I’ll say two more things:

    1) It is factually incorrect to talk about math like it was always just a subarea of physics. That may be true about the current math department at your local university, but originally math as it has been recognized for millenia, now, was actually more than anything else an outgrowth of Platonist philosophers doing geometry without even the aid of algebra — essentially a philosophical matter. In fact, it is the influence of that philosophy that makes math math, and that influence is, in fact, where rigor comes from. The only reason we have ever gotten away from that is because of centuries of British empiricism.

    2) You may find the execution to be lacking, but what K-14 education consists of is almost entirely in the service of your view of math. Math is done empirically and heuristically which also means that it is being done fallaciously. The reason they don’t do more applications in K-14 is because they are too busy enduring the consequences of settling for a vague heuristic idea of the mathematics rather than a true and rigorous understanding of it. Instead of settling on the material covered, we have settled on how we cover the material. And, that manner has been almost entirely influenced by the “math is a science” view of math.

    Teaching something like differential equations to school aged students is so antithetical to any version of classical education, I cannot imagine a greater digression from it. You want a theoretical science oriented curriculum — that is the very opposite of classical. It is through the influence of science that we have departed from the classical and started doing what we do now. If we wanted a classical education we would do just what Richard says on his blog — teach number theory and geometry. We can throw in the modern subject of combinatorics and then build eventually up to topology and abstract algebra in college and only after that would we be permitted to tackle real analysis. We would not try to do it heuristically through science the way we do in the standard freshman calculus.

    And incidentally, Baby Rudin, is mathematics at its best. The only thing wrong with it is that it is so elementary. Baby Rudin is to classical real analysis as the Elements is to geometry. You may think it too polished or something, but, again, that just means you don’t like math — not that math bends around your interests to become theoretical physics just because that’s what you, personally, want to do. The Moore method and, for that matter, Moore’s subject (point-set topology) is entirely consistent with a Baby Rudin approach to analysis because you don’t read Baby Rudin. You do the problems and that is why the book is almost legendary now. It is the text equivalent of the Moore method — a little key stimulation and lots of truly difficult problems and the rest is up to the student. In fact, I really cannot think of a text more consistent with the idea of the Moore method than one like Baby Rudin in which so much material is built into the problems with so little in the way of coaching and example given in the text. (Perhaps it is no accident — Mary Rudin was Moore’s student, after all.)

    I truly cannot believe my eyes to see a mathematician say such things. You truly have turned into an engineer.

  • Pingback: Math Resources Blog » Saving mathematics

  • Pingback: Math Resources Blog » Saving mathematics

  • Kathrine

    Adrian wrote:
    “2. There is not enough review. I went through Saxon myself, and while it is seriously flawed, one point in its favor is the long-term retention of anything it does teach. You can be sure, in Saxon, that in Lesson 130, you’ll be doing problems you learned how to do in Lesson 1. That is useful for remembering things over the long haul.”

    Adrian,
    I believe this is actually Saxon’s biggest flaw. If the students learned the lesson thoroughly the first time there would be no need for review. But the students do not learn it thoroughly the first time. Each day, a new and completely unrelated lesson is tossed into the mix of math problems, without substantial practice on the new concept all by itself.
    This is why on international tests, the USA seems to be keeping up with the rest of the world–until the 4th grade, that is. After the 4th grade the rest of the world moves on and leaves review behind, while American students are forced to review, review again, and review some more. The difference is that the students in the rest of the world concentrated on one concept, practiced it, learned it well, and THEN moved on to the next concept. Only after every 6 weeks or so are they to do reviews where all the problems they’ve learned that year are worked in all together, unlike those tedious DAILY reviews that Saxon forces upon our students. If they had learned it well enough the first time they wouldnt’ be slowed down with all the reviewing.

  • Kathrine

    Adrian wrote:
    “2. There is not enough review. I went through Saxon myself, and while it is seriously flawed, one point in its favor is the long-term retention of anything it does teach. You can be sure, in Saxon, that in Lesson 130, you’ll be doing problems you learned how to do in Lesson 1. That is useful for remembering things over the long haul.”

    Adrian,
    I believe this is actually Saxon’s biggest flaw. If the students learned the lesson thoroughly the first time there would be no need for review. But the students do not learn it thoroughly the first time. Each day, a new and completely unrelated lesson is tossed into the mix of math problems, without substantial practice on the new concept all by itself.
    This is why on international tests, the USA seems to be keeping up with the rest of the world–until the 4th grade, that is. After the 4th grade the rest of the world moves on and leaves review behind, while American students are forced to review, review again, and review some more. The difference is that the students in the rest of the world concentrated on one concept, practiced it, learned it well, and THEN moved on to the next concept. Only after every 6 weeks or so are they to do reviews where all the problems they’ve learned that year are worked in all together, unlike those tedious DAILY reviews that Saxon forces upon our students. If they had learned it well enough the first time they wouldnt’ be slowed down with all the reviewing.

  • Kathrine

    Todd writes:
    “I really don’t understand the antipathy I constantly hear against any new approaches to teaching. ”

    Todd, speaking as a parent, I can give several reasons for this.
    1. The REAL problems in teaching are not being addressed. That is, teachers spend too much time in methods classes and not enough time earning a liberal arts education.
    2. Bad teachers are not fired–or it takes 12 years to fire on bad teacher.
    3. New “methods” are almost always introduced with NO proof that the new method works. Every other year we attempt a new method, test scores do not go up, and so we switch to the next fad, with no empirical proof that students will learn better.
    The problem is not the the method is necessarily bad, the problem is that teachers (generally speaking) do NOT know math. They are incapable of having a math “conversation” with their students, or of utilizing the socratic method to help them think mathematically. They are literal slaves to the teacher’s guide and the latest method they have been taught. Whenever I observe a grade school teacher, I am shocked at how often they stand up there and read verbatim from the teaching script. This is shameful.

    So with each new method introduced, we are promised that things will get better. But the real problems are not being fixed. It is just one empty promise after another.

  • Kathrine

    Todd writes:
    “I really don’t understand the antipathy I constantly hear against any new approaches to teaching. ”

    Todd, speaking as a parent, I can give several reasons for this.
    1. The REAL problems in teaching are not being addressed. That is, teachers spend too much time in methods classes and not enough time earning a liberal arts education.
    2. Bad teachers are not fired–or it takes 12 years to fire on bad teacher.
    3. New “methods” are almost always introduced with NO proof that the new method works. Every other year we attempt a new method, test scores do not go up, and so we switch to the next fad, with no empirical proof that students will learn better.
    The problem is not the the method is necessarily bad, the problem is that teachers (generally speaking) do NOT know math. They are incapable of having a math “conversation” with their students, or of utilizing the socratic method to help them think mathematically. They are literal slaves to the teacher’s guide and the latest method they have been taught. Whenever I observe a grade school teacher, I am shocked at how often they stand up there and read verbatim from the teaching script. This is shameful.

    So with each new method introduced, we are promised that things will get better. But the real problems are not being fixed. It is just one empty promise after another.

  • Kyralessa

    I wonder if the fundamental problem in a lot of these new methods of mathematics is that they’re overly perfectionistic.

    Learning is a messy business. You learn a lot of facts by rote, and it may be only much later that you learn the theory that binds them all together and allows you to start seeing the patterns. Lately I find this is especially true with languages (in Romanian, I learned a lot of individual verbs, adjectives, etc. before discovering the patterns they shared), but I suspect it’s true of math as well. Once you have a good command of a lot of basic facts, you can discover the theory behind them.

    And that discovery may be part of the problem as well. Perhaps a lot of these new methods are so excited to teach the theory behind various parts of math that they don’t leave anything to be discovered…thus robbing learning of much of its joy.

    But I’m just speculating here. My kid’s only three; I may have more opinions in a few years. :)

  • Kyralessa

    I wonder if the fundamental problem in a lot of these new methods of mathematics is that they’re overly perfectionistic.

    Learning is a messy business. You learn a lot of facts by rote, and it may be only much later that you learn the theory that binds them all together and allows you to start seeing the patterns. Lately I find this is especially true with languages (in Romanian, I learned a lot of individual verbs, adjectives, etc. before discovering the patterns they shared), but I suspect it’s true of math as well. Once you have a good command of a lot of basic facts, you can discover the theory behind them.

    And that discovery may be part of the problem as well. Perhaps a lot of these new methods are so excited to teach the theory behind various parts of math that they don’t leave anything to be discovered…thus robbing learning of much of its joy.

    But I’m just speculating here. My kid’s only three; I may have more opinions in a few years. :)

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    Does anyone else have any experience with this?

  • http://www.bestrussiantour.com Travel Russia

    Does anyone else have any experience with this?

  • Karen Scheidhauer

    I am looking for any ideas for a high school classical christian math curriculum. Any suggestions?

  • Karen Scheidhauer

    I am looking for any ideas for a high school classical christian math curriculum. Any suggestions?


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