Algebra vs. Calculus
While interacting with some students of 12th grade (possibly some undergraduates also) in Orkut Math Community, I found something about which I feel very sad. This has something to do with their approach of problem solving. What I found out was that their brains have been so trained to a formula based or rule based approach. They don't understand the concepts and their meaning. The whole procedure of solving problems has been mechanized. After a certain amount of thinking I found the root cause of this behavior.
Well, it has something to do with the difference between Algebra and Calculus. A more respectable name for Calculus is Analysis but this term is normally not known to students in 12th grade, so I will stick to Calculus only. Right from the school level most of the mathematics is concerned with algebra. And it involves manipulating expressions consisting of some variables and numbers with operations like +, -, *, / and =. Using these manipulations a lot of standard formulae like (a + b)^2 = a^2 + 2ab + b^2 are derived and then these are memorized.
What the student understands is that for a given problem he has to use some such formula and the answer will be arrived at easily. So the learning acquired is mostly mechanical. You can program a computer to do such stuff by giving all standard rules and formulae to it. Some bright students do understand the meaning of these algebraic operations also. But somehow they are kept away from the notions of inequalities. Thus we don't have much of symbols like <> in algebra. The order relations are simply ignored. Whatever inequalities the student learns (like AM > GM) is more like an exercise in algebraical manipulations, rather than understanding what happens in those order relations.
Another thing entirely missing out entirely from the school curriculum is the concept of an irrational number. Now this is somewhat difficult to grasp at high school level, but definitely possible in 12th grade. This also involves an appreciation of the order relations <, >. The student having trained algebraically has hell lot of problem understanding the concept of irrationals. This is primarily because irrationals cannot be derived out of rationals by means of algebraical operations of +, -, * and /. So at the end of 10th grade the students has the knowledge of few irrationals consisting of radicals like sqrt(2), 3 + sqrt(5) and a few isolated numbers like pi.
The concepts of irrationals numbers as a whole class is entirely alien to them. Some smart students do understand the class of irrationals as non terminating and non repeating decimals. But they have no idea about what such decimals mean. Again this is because of a lack of emphasis on order relations like <>.
And that brings us to calculus. Calculus primarily deals with the properties of real numbers (rationals and irrationals together) and most of these properties have to do with order relations. And the student is left wondering about these messy concepts of zero and infinity. Somehow he reaches upto derivatives and integrals without having any notions of limits and continuity and he becomes happy. There are some standard cookbook rules of differentiation and integration which resemble algebraical formulae and the students again gets into this mechanical learning mode. Ask him a problem related to mean value theorems and he is lost. Ask him the meaning of an integral he is lost.
The sole problem in calculus is the lack of understanding of concepts like real numbers, and order relations. Rather than teaching the students these concepts in 11th grade, they are taught infinite series like binomial, logarithmic and exponential series. These series and the functions involved in them cannot be appreciated without developing calculus first. Students are again kept wondering about these. Add to that the plethora of books by S L Loney, Hall & Knight who don't teach a grain of concepts and load the student with a pile of mechanical problems.
By the time he is ready to learn calculus at 12th grade, it is ensured that he will not understand a single line except those mechanical rules of differentiation and integration. The reason for not providing conceptual framework for calculus is that these concepts are difficult to grasp at that level. This is not the actual reason though, it is the reason provided by book authors and curriculum designers while the real reason is that they don't know how to present these concepts.
As an example to illustrate my point, I just need to tell that these concepts are not presented even at the undergraduate level or post graduate level in a standard textbook. Thus Walter Rudin, supposedly the God of Analysis Books, says in preface to Principles of Mathematical Analysis, "Experience has convinced me that it is pedagogically unsound to start off with the construction of real numbers from the rational ones". How the hell does Rudin think it is unsound? It is because of such a pedagogical attitude from school level that the student will never appreciate these concepts. On the contrary I think that any student, who can understand the tough concepts of metric spaces, point set topology and to top it all the Lebesgue measure, can definitely and much more easily understand the concept of a real number. Its a child's play for someone with the ability to grasp Lebesgue Measure. Rudin explains all these tough concepts with lot of proofs, examples and problems while he dedicates only 3-4 pages (that also in appendix) to the theory of reals with no proper examples.
It is teachers/authors like Rudin who have made that whole thing pedagogically unsound. It is not that the student cannot understand these concepts, but in reality it is Rudin who does not know how to teach these concepts to a student. Given proper training to students at 11th or 12th grade about the importance of order relation and introducing them to theory of real numbers will definitely get rid of this pedagogically unsound situation.
There are indeed very few books of Calculus which present the theory of reals and most of them present the theory in a set theoretic notation without any examples and problems. The way student is taught about rational numbers in class 6 or 7 with lot of examples of +, -, *, / of rationals, that way he is never taught the theory of reals. I believe that learning a theory of reals at 12th grade takes at least as much time and material as it takes to learn rationals in 6th or 7th grade. That means roughly 30 pages of textbook and 3-4 weeks of learning time for solved examples and problems.
The teachers/authors want to keep the algebraical attitude as long as possible and never let the student have an appreciation of the wonderful world of calculus with its deepest concepts. And when they do teach the stuff, it is made boring by a ton of set theoretic concepts like metric spaces, topological spaces, and measure theory. All that is presented in the fashion of abstract algebra. But there is a difference. In algebra the students know the basic algebra of integers, rationals and polynomials before learning the Abstract Algebra concepts of groups, rings and fields. But in calculus the students don't know the real number system and thus limit/continuity/derivative integral concepts in reals and are taught Abstract Analysis consisting of metric spaces, point set topology, topological spaces and measure theory. Looks like a step fatherly approach towards calculus.
It is because of these reasons that students never appreciate calculus and I sometime feel like laughing when I hear about these notions from a student of 12th grade. Most of them live in some dreamland alien from the real world of calculus concepts.
Those of you who had the patience to read this much of the post, for them I have a good news. Its never late in learning and there is a book (and it is the only one) which teaches calculus the right way beginning with a chapter on real numbers with almost 30 pages. That is the masterpiece created by G. H. Hardy and is titled "A Course of Pure Mathematics". This book is suitable for students of 11th grade but it needs good language skills to understand and appreciate the book. Go and get it from your college library or get it from Amazon if you have $50 to spend. The value of the book is more than the value of all books written by Rudin/Royden/Apostol and other Analysis gurus.
Well, it has something to do with the difference between Algebra and Calculus. A more respectable name for Calculus is Analysis but this term is normally not known to students in 12th grade, so I will stick to Calculus only. Right from the school level most of the mathematics is concerned with algebra. And it involves manipulating expressions consisting of some variables and numbers with operations like +, -, *, / and =. Using these manipulations a lot of standard formulae like (a + b)^2 = a^2 + 2ab + b^2 are derived and then these are memorized.
What the student understands is that for a given problem he has to use some such formula and the answer will be arrived at easily. So the learning acquired is mostly mechanical. You can program a computer to do such stuff by giving all standard rules and formulae to it. Some bright students do understand the meaning of these algebraic operations also. But somehow they are kept away from the notions of inequalities. Thus we don't have much of symbols like <> in algebra. The order relations are simply ignored. Whatever inequalities the student learns (like AM > GM) is more like an exercise in algebraical manipulations, rather than understanding what happens in those order relations.
Another thing entirely missing out entirely from the school curriculum is the concept of an irrational number. Now this is somewhat difficult to grasp at high school level, but definitely possible in 12th grade. This also involves an appreciation of the order relations <, >. The student having trained algebraically has hell lot of problem understanding the concept of irrationals. This is primarily because irrationals cannot be derived out of rationals by means of algebraical operations of +, -, * and /. So at the end of 10th grade the students has the knowledge of few irrationals consisting of radicals like sqrt(2), 3 + sqrt(5) and a few isolated numbers like pi.
The concepts of irrationals numbers as a whole class is entirely alien to them. Some smart students do understand the class of irrationals as non terminating and non repeating decimals. But they have no idea about what such decimals mean. Again this is because of a lack of emphasis on order relations like <>.
And that brings us to calculus. Calculus primarily deals with the properties of real numbers (rationals and irrationals together) and most of these properties have to do with order relations. And the student is left wondering about these messy concepts of zero and infinity. Somehow he reaches upto derivatives and integrals without having any notions of limits and continuity and he becomes happy. There are some standard cookbook rules of differentiation and integration which resemble algebraical formulae and the students again gets into this mechanical learning mode. Ask him a problem related to mean value theorems and he is lost. Ask him the meaning of an integral he is lost.
The sole problem in calculus is the lack of understanding of concepts like real numbers, and order relations. Rather than teaching the students these concepts in 11th grade, they are taught infinite series like binomial, logarithmic and exponential series. These series and the functions involved in them cannot be appreciated without developing calculus first. Students are again kept wondering about these. Add to that the plethora of books by S L Loney, Hall & Knight who don't teach a grain of concepts and load the student with a pile of mechanical problems.
By the time he is ready to learn calculus at 12th grade, it is ensured that he will not understand a single line except those mechanical rules of differentiation and integration. The reason for not providing conceptual framework for calculus is that these concepts are difficult to grasp at that level. This is not the actual reason though, it is the reason provided by book authors and curriculum designers while the real reason is that they don't know how to present these concepts.
As an example to illustrate my point, I just need to tell that these concepts are not presented even at the undergraduate level or post graduate level in a standard textbook. Thus Walter Rudin, supposedly the God of Analysis Books, says in preface to Principles of Mathematical Analysis, "Experience has convinced me that it is pedagogically unsound to start off with the construction of real numbers from the rational ones". How the hell does Rudin think it is unsound? It is because of such a pedagogical attitude from school level that the student will never appreciate these concepts. On the contrary I think that any student, who can understand the tough concepts of metric spaces, point set topology and to top it all the Lebesgue measure, can definitely and much more easily understand the concept of a real number. Its a child's play for someone with the ability to grasp Lebesgue Measure. Rudin explains all these tough concepts with lot of proofs, examples and problems while he dedicates only 3-4 pages (that also in appendix) to the theory of reals with no proper examples.
It is teachers/authors like Rudin who have made that whole thing pedagogically unsound. It is not that the student cannot understand these concepts, but in reality it is Rudin who does not know how to teach these concepts to a student. Given proper training to students at 11th or 12th grade about the importance of order relation and introducing them to theory of real numbers will definitely get rid of this pedagogically unsound situation.
There are indeed very few books of Calculus which present the theory of reals and most of them present the theory in a set theoretic notation without any examples and problems. The way student is taught about rational numbers in class 6 or 7 with lot of examples of +, -, *, / of rationals, that way he is never taught the theory of reals. I believe that learning a theory of reals at 12th grade takes at least as much time and material as it takes to learn rationals in 6th or 7th grade. That means roughly 30 pages of textbook and 3-4 weeks of learning time for solved examples and problems.
The teachers/authors want to keep the algebraical attitude as long as possible and never let the student have an appreciation of the wonderful world of calculus with its deepest concepts. And when they do teach the stuff, it is made boring by a ton of set theoretic concepts like metric spaces, topological spaces, and measure theory. All that is presented in the fashion of abstract algebra. But there is a difference. In algebra the students know the basic algebra of integers, rationals and polynomials before learning the Abstract Algebra concepts of groups, rings and fields. But in calculus the students don't know the real number system and thus limit/continuity/derivative integral concepts in reals and are taught Abstract Analysis consisting of metric spaces, point set topology, topological spaces and measure theory. Looks like a step fatherly approach towards calculus.
It is because of these reasons that students never appreciate calculus and I sometime feel like laughing when I hear about these notions from a student of 12th grade. Most of them live in some dreamland alien from the real world of calculus concepts.
Those of you who had the patience to read this much of the post, for them I have a good news. Its never late in learning and there is a book (and it is the only one) which teaches calculus the right way beginning with a chapter on real numbers with almost 30 pages. That is the masterpiece created by G. H. Hardy and is titled "A Course of Pure Mathematics". This book is suitable for students of 11th grade but it needs good language skills to understand and appreciate the book. Go and get it from your college library or get it from Amazon if you have $50 to spend. The value of the book is more than the value of all books written by Rudin/Royden/Apostol and other Analysis gurus.
posted by Paramanand at 10:05 AM
5 Comments:
wow, thats a nice piece of info.
Thats a common ill - not understanding the problem. And unfortunately something that extends to many fields and education sytems (I am not just talking India here).Unfortunately, we are now turning into math historians rather than mathematicians - equipped with a "knowledge" of math, with scarce abilities to use them. Martin Gardner, the well known problemist has a lot to say about this in his book 'a gardeners workout' (not available on amazon for some reason). The book 'how to solve it by g.polya is s good book providing insight to the technique of problem solving. There are a few other good books on math too, that introduce one to problem solving, than solving problems given that they belong to a particular chapter - that alone makes a lot of difference. Nice post!
I feel like you've probably described the core of what was wrong with my math education, several decades ago. I'm going to buy that book. Thanks!
Yes but, your preaching the the masses... not an individual which is harder to teach concepts on a mass level.. thats the true problem...
Really like this explanation. Thanks!
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