|
7
|
|
edited Jan 30 2011 at 6:32 |
Maybe I have misunderstood, and please forgive me if so, but this question, as asked, seems like asking why we should teach understanding as opposed to rote memorization. There are options that include understanding other than limits, such as finding the linear approximation of a function. This is computable for polynomials by re - expansion, i.e. setting x = (x-a) +a, then picking off the coefficient of the linear term in (x-a). But just teaching rules denies the student the ability to use the concept in any situation other than the ones met before. It is worth noting that there is a lot of historical precedent for teaching it as a limit, which occurs already in Euclid. I.e. Euclid characterizes the tangent to a circle as the unique line such that between it and any other line through the same point, one can interpose a secant (Prop. 16, Book III). (Strictly, he says equivalently that one cannot interpose another line between the tangent and the circle itself, i.e. every other line through the point is a secant.) Thus the tangent is the limit of those secants. Thus I believe one can easily say that the limiting point of view is the original one of Euclid. From this point of view, the idea of limit is the one used so fruitfully by the Greeks, and the contribution of the mathematicians of later times is to make that notion more precise.
On the other hand, if you want to avoid the conceptual difficulty students have with limits, you can follow Descartes instead, at least for derivatives of polynomials, and characterize the tangent line as the unique line such that subtracting its equation from the original function gives a polynomial with a double root at the given point. This leads to motivating the Zariski cotangent space, as M/M^2.
Both points of view also have a nice dynamic interpretation as realizing the tangent as the unique line intersecting the curve doubly at the point, understood as the limit of the two secant intersections,and measured by the presence of a double root.
But if you want a defense of limits, I suggest Euclid Prop. 16, Book III as ample precedent.
If you want a defense of making students practice using the limit definition, I propose that as noted above, this is the only way to get them to appreciate the fundamental theorem of calculus. That theorem cannot be appreciated by memorizing rules for derivatives, One must understand the definition and apply it to an abstractly defined area function. I suggest that one reason many students do not understand why the fundamental theorem of calculus is true, is that (again as noted above) they have not grasped either what an abstractly defined function is, nor what a derivative truly means.
So if you want them to understand the relation between the derivative and the integral, then I agree with others that they need to know what a function is and derivative is. The reasoning here is that once someone understands something, he can use it in more settings than could possibly be covered by any set of rules.
Another practical benefit of testing the use of the h-->0 definition to obtain derivatives of simple functions, is that it forces practice in algebra, trig identities, and exponentials, skills which most of my students are almost completely lacking.
However, I recommend you teach it any way that makes sense to you. after all you understand it, so whatever you say based on that understanding will be useful. Make up your mind what seems important to you, and go for it!
|
|
|
|
|
6
|
|
edited Jan 30 2011 at 6:23 |
Maybe I am afraid have misunderstood, and please forgive me if so, but this question, as asked, is rather seems like asking why we should teach understanding as opposed to rote memorization. There are options that include understanding other than limits, such as finding the linear approximation of a function. This is computable for polynomials by re - expansion, i.e. setting x = (x-a) +a, then picking off the coefficient of the linear term in (x-a). But just teaching rules denies the student the ability to use the concept in any situation other than the ones met before.
It is worth noting that there is a lot of historical precedent for teaching it as a limit, which occurs already in Euclid. I.e. Euclid characterizes the tangent to a circle as the unique line such that between it and any other line through the same point, one can interpose a secant (Prop. 16, Book III). (Strictly, he says equivalently that one cannot interpose another line between the tangent and the circle itself, i.e. every other line through the point is a secant.) Thus the tangent is the limit of those secants. Thus I believe one can easily say that the limiting point of view is the original one of Euclid. From this point of view, the idea of limit is the one used so fruitfully by the Greeks, and the contribution of the mathematicians of later times is to make that notion more precise.
On the other hand, if you want to avoid the conceptual difficulty students have with limits, you can follow Descartes instead, at least for derivatives of polynomials, and characterize the tangent line as the unique line such that subtracting its equation from the original function gives a polynomial with a double root at the given point. This leads to motivating the Zariski cotangent space, as M/M^2.
Both points of view also have a nice dynamic interpretation as realizing the tangent as the unique line intersecting the curve doubly at the point, understood as the limit of the two secant intersections,and measured by the presence of a double root.
But if you want a defense of limits, I suggest Euclid Prop. 16, Book III as ample precedent.
If you want a defense of making students practice using the limit definition, I propose that as noted above, this is the only way to get them to appreciate the fundamental theorem of calculus. That theorem cannot be appreciated by memorizing rules for derivatives, One must understand the definition and apply it to an abstractly defined area function. I suggest that one reason many students do not understand why the fundamental theorem of calculus is true, is that (again as noted above) they have not grasped either what an abstractly defined function is, nor what a derivative truly means.
So if you want them to understand the relation between the derivative and the integral, then I agree with others that they need to know what a function is and derivative is. The reasoning here is that once someone understands something, he can use it in more settings than could possibly be covered by any set of rules.
Another practical benefit of testing the use of the h-->0 definition to obtain derivatives of simple functions, is that it forces practice in algebra, trig identities, and exponentials, skills which most of my students are almost completely lacking.
However, I recommend you teach it any way that makes sense to you. after all you understand it, so whatever you say based on that understanding will be useful. Make up your mind what seems important to you, and go for it!
|
|
|
|
|
5
|
|
edited Jan 30 2011 at 6:06 |
I am afraid this question, as asked, is rather like asking why we should teach understanding as opposed to rote memorization. There are options that include understanding other than limits, such as finding the linear approximation of a function. This is computable for polynomials by re - expansion, i.e. setting x = (x-a) +a, then picking off the coefficient of the linear term in (x-a). But just teaching rules denies the student the ability to use the concept in any situation other than the ones met before.
It is worth noting that there is a lot of historical precedent for teaching it as a limit, which occurs already in Euclid. I.e. Euclid characterizes the tangent to a circle as the unique line such that between it and any other line through the same point, one can interpose a secant (Prop. 16, Book III). (Strictly, he says equivalently that one cannot interpose another line between the tangent and the circle itself, i.e. every other line through the point is a secant.) Thus the tangent is the limit of those secants. Thus I believe one can easily say that the limiting point of view is the original one of Euclid. From this point of view, the idea of limit is the one used so fruitfully by the Greeks, and the contribution of the mathematicians of later times is to make that notion more precise.
On the other hand, if you want to avoid the conceptual difficulty students have with limits, you can follow Descartes instead, at least for derivatives of polynomials, and characterize the tangent line as the unique line such that subtracting its equation from the original function gives a polynomial with a double root at the given point. This leads to motivating the Zariski cotangent space, as M/M^2.
Both points of view also have a nice dynamic interpretation as realizing the tangent as the unique line intersecting the curve doubly at the point, understood as the limit of the two secant intersections,and measured by the presence of a double root.
But if you want a defense of limits, I suggest Euclid Prop. 16, Book III as ample precedent.
If you want a defense of making students practice using the limit definition, I propose that as noted above, this is the only way to get them to appreciate the fundamental theorem of calculus. That theorem cannot be appreciated by memorizing rules for derivatives, One must understand the definition and apply it to an abstractly defined area function. I suggest that one reason many students do not understand why the fundamental theorem of calculus is true, is that (again as noted above) they have not grasped either what an abstractly defined function is, nor what a derivative truly means.
So if you want them to understand the relation between the derivative and the integral, then I agree with others that they need to know what a function is and derivative is. The reasoning here is that once someone understands something, he can use it in more settings than could possibly be covered by any set of rules.
Another practical benefit of testing the use of the h-->0 definition to obtain derivatives of simple functions, is that it forces practice in algebra, trig identities, and exponentials, skills which most of my students are almost completely lacking.
However, I recommend you teach it any way that makes sense to you. after all you understand it, so whatever you say based on that understanding will be useful. Make up your mind what seems important to you, and go for it!
|
|
|
|
4
|
|
edited Nov 6 2010 at 15:40 |
It is worth noting that there is a lot of historical precedent both for teaching it algebraically, as is done in algebraic geometry, i.e. via the concept of orders of vanishing, but also as a limit, which occurs already in Euclid. I.e. Euclid characterizes the tangent to a circle as the unique line such that between it and any other line through the same point, one can interpose a secant (Prop. 16, Book III). (Strictly, he says equivalently that one cannot interpose another line between the tangent and the circle itself, i.e. every other line through the point is a secant.) Thus the tangent is the limit of those secants. Thus I believe one could can easily say that the limiting point of view is the original one of Euclid. From this point of view, the idea of limit is the one used so fruitfully by the Greeks, and the contribution of the mathematicians of later times is to make that notion more precise. If you want a defense of making students practice using the limit definition, I propose that as noted above, this is the only way to get them to appreciate the fundamental theorem of calculus. That theorem cannot be appreciated by memorizing rules for derivatives, One must understand the definition and apply it to an abstractly defined area function. I suggest that one reason many students do not understand why the fundamental theorem of calculus is true, is that (again as noted above) they have not grasped either what an abstractly defined function is, nor what a derivative truly means. So if you want them to understand the relation between the derivative and the integral, then I agree with others that they need to know what a function is and derivative is. The reasoning here is that once someone understands something, he can use it in more settings than could possibly be covered by any set of rules. Another practical benefit of testing the use of the h-->0 definition to obtain derivatives of simple functions, is that it forces practice in algebra, trig identities, and exponentials, skills which most of my students are almost completely lacking.
|
|
|
|
|
3
|
|
edited Nov 2 2010 at 15:31 |
I recommend you teach it any way that makes sense to you. after all you understand it, so whatever you say will be valid. go for it!
However it It is worth noting that there is a lot of historical precedent both for teaching it algebraically, as is done in algebraic geometry, i.e. via the concept of orders of vanishing, but also as a limit, which occurs already in Euclid. I.e. Euclid characterizes the tangent to a circle as the unique line such that between it and any other line through the same point, one can interpose a secant (Prop. 16, Book III). Thus the tangent is the limit of those secants. Thus one could say that the limiting point of view is the original one of Euclid.
On the other hand, if you want to avoid the conceptual difficulty students have with limits, you can follow Descartes instead, at least for derivatives of polynomials, and characterize the tangent line as the unique line such that subtracting its equation from the original function gives a polynomial with a double root at the given point. This leads to motivating the Zariski cotangent space, as M/M^2.
Both points of view also have a nice dynamic interpretation as realizing the tangent as the unique line intersecting the curve doubly at the point, understood as the limit of the two secant intersections,and measured by the presence of a double root.
But if you want a defense of limits, I suggest Euclid Prop. 16, Book III as ample precedent.
If you want a defense of making students practice using the limit definition, I propose that this is the only way to get them to appreciate the fundamental theorem of calculus. That theorem cannot be appreciated by memorizing rules for derivatives, One must understand the definition and apply it to an abstractly defined area function. I suggest that one reason many students do not understand why the fundamental theorem of calculus is true, is that they have not grasped either what an abstractly defined function is, nor what a derivative truly means. So if you want them to understand the relation between the derivative and the integral, then they need to know what a derivative is.
However, I recommend you teach it any way that makes sense to you. after all you understand it, so whatever you say based on that understanding will be useful. Make up your mind what seems important to you, and go for it!
|
|
|
|
|
2
|
|
edited Nov 2 2010 at 15:24 |
not having read the previous answers, i I recommend you teach it any way that makes sense to you. after all you understand it, so whatever you say will be valid. go for it! I myself teach However it is worth noting that there is a lot of historical precedent both for teaching it algebraically, as an intersection problemis done in algebraic geometry, and use only i.e. via the idea concept of double roots orders of polynomials vanishing, but also as a limit, which occurs already in Euclid. I.e. Euclid characterizes the tangent to cary a circle as the unique line such that between it outand any other line through the same point, one can interpose a secant (Prop. 16, Book III). Thus the tangent is the limit of those secants. Thus one could say that the limiting point of view is the original one of Euclid. On the other hand, if you want to avoid the conceptual difficulty students have with limits, you can follow Descartes instead, at least for those functionsderivatives of polynomials, and characterize the tangent line as the unique line such that subtracting its equation from the original function gives a polynomial with a double root at the given point. This leads to motivating the Zariski cotangent space, as M/M^2. Both points of view also have a nice dynamic interpretation as realizing the tangent as the unique line intersecting the curve doubly at the point, understood as the limit of the two secant intersections,and measured by the presence of a double root. But if you want a defense of limits, I suggest Euclid Prop. 16, Book III as ample precedent. If you want a defense of making students practice using the limit definition, I propose that this is the only way to get them to appreciate the fundamental theorem of calculus. That theorem cannot be appreciated by memorizing rules for derivatives, One must understand the definition and apply it to an abstractly defined area function. I suggest that one reason many students do not understand why the fundamental theorem of calculus is true, is that they have not grasped either what an abstractly defined function is, nor what a derivative truly means. So if you want them to understand the relation between the derivative and the integral, then they need to know what a derivative is.
|
|
|
|
|
1
|
|
answered Sep 28 2010 at 4:04 |
not having read the previous answers, i recommend you teach it any way that makes sense to you. after all you understand it, so whatever you say will be valid. go for it!
I myself teach it as an intersection problem, and use only the idea of double roots of polynomials to cary it out, at least for those functions.
|
|
|
