The purpose of mathematics in classical education

[Monica_in_Switzerland]

I very much enjoyed this article. (It is from a conservative website, but I think the subject and article are very neutral.)

[Down_the_Rabbit_Hole]

Thank you. Interesting. I need to process it more, sometime later when I am more awake.

[ladydusk]

I very much enjoyed this article. (It is from a conservative website, but I think the subject and article are very neutral.)

 

I can't wait until they post about early math education! I loved this article :)

[mathwonk]

i agreed with basically every word of that essay. I was not surprised at the end to see that the author is a trained mathematician. The word conservative has many strange uses. E.g I do not believe multi-millionaires should pay income taxes at less than half the rate paid by the average middle class family, which view some people think is liberal, but I agree that research in mathematics is an exploration of an especially beautiful realm of God's creation. Apparently that latter view is considered conservative.

 

how interesting. no offense meant to anyone. thank you for the very eloquent and inspiring link.

Edited August 9, 2012 by mathwonk

[serendipitous journey]

This is close to what I think of the role of math in classical education, but has been a Hot Topic on the boards in the past ... this is not the role of maths in the medieval university system, to which the Trivium traces its routes, and so it is different to many folks' conception of "classical" -- at least that is my understanding. And if one goes back to the ancient Greeks, there seems to be a great deal of ambivalence about mapping math theory onto the physical world: the relationship was seen to be imperfect (as opposed to maths revealing or underlying natural relationships, the natural world could be seen as an imperfect reflection of purer, more mathematical realities) and it seems to me that a major goal of the maths education was in training logical thinking and also for its relationship to music, not for a greater comprehension of the natural order. So the "classical" label there is also an interesting fit.

 

This is not to be pedantic -- it's just that this is very close to my own conception of the role of maths in classical ed., b/c I take classical education to encompass mainly the goals of the Classical Greek & Roman cultures and not to be particularly tied to either their methods or to the medieval/Trivium model of education; however, when people talk about a Classical Education they are usually referring to this European tradition that stems from the medieval educational model and includes, for example, Latin. I'm curious what other types of classical educators think of the importance of math education for its relationship to the natural order.

 

FWIW, I wish SWBauer would update her high school math goals with this sort of focus, and not the current enough-maths-to-qualify-for-university goals :).

 

Also, disagree with the authors RE the calculus, esp. for homeschooled children. Calculus is one of the great cultural achievements of humanity, and it's super helpful for understanding the real world -- rates of change, and changing rates of change, underlie so many systems and a feel for them helps with understanding models of the economy and of complex, dynamic systems generally. Classically homeschooled children esp. should be able to study calculus easily -- if our standard were simply to start teaching math in K or 1st, and to teach any good program right through summer breaks (take, say, 4 or so weeks off math per year) calculus would easily be reached by high school. Any child capable of a WTM reading load is capable of calculus. But I understand this last idea of mine is probably Not Standard and I acknowledge that it may be wrong-headed;).

 

thanks for the link, and the thread!

 

ETA: the article contains, in the reader comment section, a reference to Lockheart's soon-to-be-released Measurement, which suggests itself as a guide to early maths educators (or self-educators). I don't know from Lockheart, but there's a strong recommendation from Strogatz, whose work on chaos and dynamical systems I've really enjoyed (at a non-expert level).

Edited August 9, 2012 by serendipitous journey

[a27mom]

What an excellent article. One of my goals for my children's education is that they will understand math as an expression of the order and beauty of creation. The idea of a transition from concrete to abstract is fascinating. I wish I had been taught that way. I may have enjoyed calculus more.

[kemilie]

good read, thanks for sharing!

[mathwonk]

I suspect the author's bias against calculus early, which I share for most students, stems from our experience trying to teach calculus to college kids who have studied it in high school before coming to college. To just make up an analogy, imagine trying to teach the king's English to someone who had learned to speak in the street, oh yes, that example has already been dramatized, in my fair lady! That's exactly what college professors face in teaching calculus. But not all of us are henry higgins, and not all our students are as motivated as eliza doolittle, (nor born speaking as well as audrey hepburn.)

 

Of course this means nothing to those who can manage it at home, it is just a reality check coming from decades of data as to the failure of the experiment of teaching it early in high school, mainly by the marginally qualified teacher. (At my sons' excellent private school, a year or so of college calculus was considered ample training for a high school calculus teacher. The English classes however were taught by life long specialists in English literature, one a former PhD college English professor at Agnes Scott. And the math classes there were way above average for high school.)

 

The best background for understanding calculus is the kind of geometry taught in Euclid and Archimedes. High schools, to make room for the more prestigious calculus course have cut out precisely the courses that would have been adequate preparation. Of course it can be done. I tried to introduce calculus last summer on the last day of epsilon camp, after spending 11 days on Euclidean and Archimedean geometry, and with a very rarified group of kids. I don't know how well the kids liked it but one of the fathers, an engineer, said something nice afterwards about the way it was explained.

 

I am sure many of you also have your own ideas about to do it at home appropriately. But I can say there are no books leaping to mind that I can recommend for this, certainly nothing as user friendly and insightful as Harold Jacobs. Maybe the book in the New Math Library series? I don't recall being much impressed in the 1970's by it but I have not looked again lately.

 

Maybe the best explanation I know of in a textbook of the calculus is in a book by Michael Comenetz, called Calculus: the Elements.

 

It was written for his course at St Johns Univ., the "great books" school. (They also read Newton.)

 

http://www.amazon.com/Calculus-The-Elements-Michael-Comenetz/dp/9810249047

Edited August 9, 2012 by mathwonk

[Kathryn]

Great article! Thank you!

[Monica_in_Switzerland]

Mathwonk- I enjoyed your perspective. Thank you!

 

I happened to have an excellent high school Calculus teacher, and felt really prepared for the next level in college (ultimately I got my degree in physics), but I know that is not always the case.

 

I suspect the author's bias against calculus early, which I share for most students, stems from our experience trying to teach calculus to college kids who have studied it in high school before coming to college. To just make up an analogy, imagine trying to teach the king's English to someone who had learned to speak in the street, oh yes, that example has already been dramatized, in my fair lady! That's exactly what college professors face in teaching calculus. But not all of us are henry higgins, and not all our students are as motivated as eliza doolittle, (nor born speaking as well as audrey hepburn.)

 

Of course this means nothing to those who can manage it at home, it is just a reality check coming from decades of data as to the failure of the experiment of teaching it early in high school, mainly by the marginally qualified teacher. (At my sons' excellent private school, a year or so of college calculus was considered ample training for a high school calculus teacher. The English classes however were taught by life long specialists in English literature, one a former PhD college English professor at Agnes Scott. And the math classes there were way above average for high school.)

 

The best background for understanding calculus is the kind of geometry taught in Euclid and Archimedes. High schools, to make room for the more prestigious calculus course have cut out precisely the courses that would have been adequate preparation. Of course it can be done. I tried to introduce calculus last summer on the last day of epsilon camp, after spending 11 days on Euclidean and Archimedean geometry, and with a very rarified group of kids. I don't know how well the kids liked it but one of the fathers, an engineer, said something nice afterwards about the way it was explained.

 

I am sure many of you also have your own ideas about to do it at home appropriately. But I can say there are no books leaping to mind that I can recommend for this, certainly nothing as user friendly and insightful as Harold Jacobs. Maybe the book in the New Math Library series? I don't recall being much impressed in the 1970's by it but I have not looked again lately.

 

Maybe the best explanation I know of in a textbook of the calculus is in a book by Michael Comenetz, called Calculus: the Elements.

 

It was written for his course at St Johns Univ., the "great books" school. (They also read Newton.)

 

http://www.amazon.com/Calculus-The-Elements-Michael-Comenetz/dp/9810249047

[Roadrunner]

I suspect the author's bias against calculus early, which I share for most students, stems from our experience trying to teach calculus to college kids who have studied it in high school before coming to college. To just make up an analogy, imagine trying to teach the king's English to someone who had learned to speak in the street, oh yes, that example has already been dramatized, in my fair lady! That's exactly what college professors face in teaching calculus. But not all of us are henry higgins, and not all our students are as motivated as eliza doolittle, (nor born speaking as well as audrey hepburn.)

 

Of course this means nothing to those who can manage it at home, it is just a reality check coming from decades of data as to the failure of the experiment of teaching it early in high school, mainly by the marginally qualified teacher. (At my sons' excellent private school, a year or so of college calculus was considered ample training for a high school calculus teacher. The English classes however were taught by life long specialists in English literature, one a former PhD college English professor at Agnes Scott. And the math classes there were way above average for high school.)

 

The best background for understanding calculus is the kind of geometry taught in Euclid and Archimedes. High schools, to make room for the more prestigious calculus course have cut out precisely the courses that would have been adequate preparation. Of course it can be done. I tried to introduce calculus last summer on the last day of epsilon camp, after spending 11 days on Euclidean and Archimedean geometry, and with a very rarified group of kids. I don't know how well the kids liked it but one of the fathers, an engineer, said something nice afterwards about the way it was explained.

 

I am sure many of you also have your own ideas about to do it at home appropriately. But I can say there are no books leaping to mind that I can recommend for this, certainly nothing as user friendly and insightful as Harold Jacobs. Maybe the book in the New Math Library series? I don't recall being much impressed in the 1970's by it but I have not looked again lately.

 

Maybe the best explanation I know of in a textbook of the calculus is in a book by Michael Comenetz, called Calculus: the Elements.

 

It was written for his course at St Johns Univ., the "great books" school. (They also read Newton.)

 

http://www.amazon.com/Calculus-The-Elements-Michael-Comenetz/dp/9810249047

 

Thank you for your perspective. What are your thoughts on Kisilev geometry books as a foundation?