What is the most difficult concept to grasp in Calculus 1?Is there a more telling name for “Calculus 2”?Evaluating the reception of (epsilon, delta) definitionsTeaching limits of sequences before limits of functions in Calculus?When should we get into limits in introductory calculus courses?The purpose of mathematics in a liberal education when it is not a prerequisite to other subjectsLinear algebra textbooks presenting an eclectic, geometric approach to the subjectNatural, rich, calculus questionsExample of function with *all* the features of differential calculus at first-year levelMost Commonly-Adopted Calculus TextbooksDifference between high school and college calculus coursesWould teaching nonstandard calculus in an introduction calculus course make it easier to learn?
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What is the most difficult concept to grasp in Calculus 1?
Is there a more telling name for “Calculus 2”?Evaluating the reception of (epsilon, delta) definitionsTeaching limits of sequences before limits of functions in Calculus?When should we get into limits in introductory calculus courses?The purpose of mathematics in a liberal education when it is not a prerequisite to other subjectsLinear algebra textbooks presenting an eclectic, geometric approach to the subjectNatural, rich, calculus questionsExample of function with *all* the features of differential calculus at first-year levelMost Commonly-Adopted Calculus TextbooksDifference between high school and college calculus coursesWould teaching nonstandard calculus in an introduction calculus course make it easier to learn?
.everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty margin-bottom:0;
$begingroup$
I would say it is not the Fundamental Theorem of Calculus,
but rather some notion connecting limits and continuity,
perhaps the $(epsilon,delta)$-definitions of limits and continuity.
But I would be interested to learn from experienced calculus
instructors, as it would help me in courses that have
Calculus as a prerequisite, to understand where to
anticipate weaknesses.
For the usual meaning of "Calculus 1" (or "Intro to Calculus"), see
Is there a more telling name for “Calculus 2”?
See also Wikipedia's List of calculus topics.
undergraduate-education calculus limits
asked 9 hours ago
Joseph O'RourkeJoseph O'Rourke
16.3k3 gold badges35 silver badges86 bronze badges
$endgroup$
add a comment |
$begingroup$
I would say it is not the Fundamental Theorem of Calculus,
but rather some notion connecting limits and continuity,
perhaps the $(epsilon,delta)$-definitions of limits and continuity.
But I would be interested to learn from experienced calculus
instructors, as it would help me in courses that have
Calculus as a prerequisite, to understand where to
anticipate weaknesses.
For the usual meaning of "Calculus 1" (or "Intro to Calculus"), see
Is there a more telling name for “Calculus 2”?
See also Wikipedia's List of calculus topics.
undergraduate-education calculus limits
asked 9 hours ago
Joseph O'RourkeJoseph O'Rourke
16.3k3 gold badges35 silver badges86 bronze badges
$endgroup$
2
$begingroup$
Do you want calculus-specific concepts, or would you accept the application of non-calculus topics, such as writing a function that performs a particular task (distance between two points, area of a basic shape, etc.)? For me, teaching calculus always implies teaching prerequisite material.
$endgroup$
– Nick C
8 hours ago
$begingroup$
@NickC: I was thinking of topics, but I would be interested as well to learn of skills that seem difficult to master.
$endgroup$
– Joseph O'Rourke
8 hours ago
1
$begingroup$
My guess is the point of the mean value theorem. Or based on impact, perhaps the Chain Rule. If the limit definition is not taught, it can hardly be difficult. If it not used in the course you're teaching, it's hardly important. Otherwise limits are probably one of the more difficult concepts, although it seems less difficult in the third or fourth semester of learning limits (calc 1 -> 2 -> 3 -> transition-to-proofs course)
$endgroup$
– user1527
7 hours ago
1
$begingroup$
Mechanically, I think the chain rule is most troublesome. However, logically, anything involving non-standard algebra and multi-step logic is an opportunity to fail; optimization, graphing with calculus, related rates, conceptual problems which illustrate theorems... anything not algorithmic.
$endgroup$
– James S. Cook
7 hours ago
add a comment |
$begingroup$
I would say it is not the Fundamental Theorem of Calculus,
but rather some notion connecting limits and continuity,
perhaps the $(epsilon,delta)$-definitions of limits and continuity.
But I would be interested to learn from experienced calculus
instructors, as it would help me in courses that have
Calculus as a prerequisite, to understand where to
anticipate weaknesses.
For the usual meaning of "Calculus 1" (or "Intro to Calculus"), see
Is there a more telling name for “Calculus 2”?
See also Wikipedia's List of calculus topics.
undergraduate-education calculus limits
asked 9 hours ago
Joseph O'RourkeJoseph O'Rourke
16.3k3 gold badges35 silver badges86 bronze badges
$endgroup$
I would say it is not the Fundamental Theorem of Calculus,
but rather some notion connecting limits and continuity,
perhaps the $(epsilon,delta)$-definitions of limits and continuity.
But I would be interested to learn from experienced calculus
instructors, as it would help me in courses that have
Calculus as a prerequisite, to understand where to
anticipate weaknesses.
For the usual meaning of "Calculus 1" (or "Intro to Calculus"), see
Is there a more telling name for “Calculus 2”?
See also Wikipedia's List of calculus topics.
undergraduate-education calculus limits
undergraduate-education calculus limits
asked 9 hours ago
Joseph O'RourkeJoseph O'Rourke
16.3k3 gold badges35 silver badges86 bronze badges
asked 9 hours ago
Joseph O'RourkeJoseph O'Rourke
16.3k3 gold badges35 silver badges86 bronze badges
asked 9 hours ago
Joseph O'RourkeJoseph O'Rourke
16.3k3 gold badges35 silver badges86 bronze badges
asked 9 hours ago
Joseph O'RourkeJoseph O'Rourke
16.3k3 gold badges35 silver badges86 bronze badges
asked 9 hours ago
Joseph O'RourkeJoseph O'Rourke
16.3k3 gold badges35 silver badges86 bronze badges
16.3k3 gold badges35 silver badges86 bronze badges
2
$begingroup$
Do you want calculus-specific concepts, or would you accept the application of non-calculus topics, such as writing a function that performs a particular task (distance between two points, area of a basic shape, etc.)? For me, teaching calculus always implies teaching prerequisite material.
$endgroup$
– Nick C
8 hours ago
$begingroup$
@NickC: I was thinking of topics, but I would be interested as well to learn of skills that seem difficult to master.
$endgroup$
– Joseph O'Rourke
8 hours ago
1
$begingroup$
My guess is the point of the mean value theorem. Or based on impact, perhaps the Chain Rule. If the limit definition is not taught, it can hardly be difficult. If it not used in the course you're teaching, it's hardly important. Otherwise limits are probably one of the more difficult concepts, although it seems less difficult in the third or fourth semester of learning limits (calc 1 -> 2 -> 3 -> transition-to-proofs course)
$endgroup$
– user1527
7 hours ago
1
$begingroup$
Mechanically, I think the chain rule is most troublesome. However, logically, anything involving non-standard algebra and multi-step logic is an opportunity to fail; optimization, graphing with calculus, related rates, conceptual problems which illustrate theorems... anything not algorithmic.
$endgroup$
– James S. Cook
7 hours ago
add a comment |
2
$begingroup$
Do you want calculus-specific concepts, or would you accept the application of non-calculus topics, such as writing a function that performs a particular task (distance between two points, area of a basic shape, etc.)? For me, teaching calculus always implies teaching prerequisite material.
$endgroup$
– Nick C
8 hours ago
$begingroup$
@NickC: I was thinking of topics, but I would be interested as well to learn of skills that seem difficult to master.
$endgroup$
– Joseph O'Rourke
8 hours ago
1
$begingroup$
My guess is the point of the mean value theorem. Or based on impact, perhaps the Chain Rule. If the limit definition is not taught, it can hardly be difficult. If it not used in the course you're teaching, it's hardly important. Otherwise limits are probably one of the more difficult concepts, although it seems less difficult in the third or fourth semester of learning limits (calc 1 -> 2 -> 3 -> transition-to-proofs course)
$endgroup$
– user1527
7 hours ago
1
$begingroup$
Mechanically, I think the chain rule is most troublesome. However, logically, anything involving non-standard algebra and multi-step logic is an opportunity to fail; optimization, graphing with calculus, related rates, conceptual problems which illustrate theorems... anything not algorithmic.
$endgroup$
– James S. Cook
7 hours ago
2
2
$begingroup$
Do you want calculus-specific concepts, or would you accept the application of non-calculus topics, such as writing a function that performs a particular task (distance between two points, area of a basic shape, etc.)? For me, teaching calculus always implies teaching prerequisite material.
$endgroup$
– Nick C
8 hours ago
$begingroup$
Do you want calculus-specific concepts, or would you accept the application of non-calculus topics, such as writing a function that performs a particular task (distance between two points, area of a basic shape, etc.)? For me, teaching calculus always implies teaching prerequisite material.
$endgroup$
– Nick C
8 hours ago
$begingroup$
@NickC: I was thinking of topics, but I would be interested as well to learn of skills that seem difficult to master.
$endgroup$
– Joseph O'Rourke
8 hours ago
$begingroup$
@NickC: I was thinking of topics, but I would be interested as well to learn of skills that seem difficult to master.
$endgroup$
– Joseph O'Rourke
8 hours ago
1
1
$begingroup$
My guess is the point of the mean value theorem. Or based on impact, perhaps the Chain Rule. If the limit definition is not taught, it can hardly be difficult. If it not used in the course you're teaching, it's hardly important. Otherwise limits are probably one of the more difficult concepts, although it seems less difficult in the third or fourth semester of learning limits (calc 1 -> 2 -> 3 -> transition-to-proofs course)
$endgroup$
– user1527
7 hours ago
$begingroup$
My guess is the point of the mean value theorem. Or based on impact, perhaps the Chain Rule. If the limit definition is not taught, it can hardly be difficult. If it not used in the course you're teaching, it's hardly important. Otherwise limits are probably one of the more difficult concepts, although it seems less difficult in the third or fourth semester of learning limits (calc 1 -> 2 -> 3 -> transition-to-proofs course)
$endgroup$
– user1527
7 hours ago
1
1
$begingroup$
Mechanically, I think the chain rule is most troublesome. However, logically, anything involving non-standard algebra and multi-step logic is an opportunity to fail; optimization, graphing with calculus, related rates, conceptual problems which illustrate theorems... anything not algorithmic.
$endgroup$
– James S. Cook
7 hours ago
$begingroup$
Mechanically, I think the chain rule is most troublesome. However, logically, anything involving non-standard algebra and multi-step logic is an opportunity to fail; optimization, graphing with calculus, related rates, conceptual problems which illustrate theorems... anything not algorithmic.
$endgroup$
– James S. Cook
7 hours ago
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
At my (community) college, I believe many teachers skip the (𝜖,𝛿)-definitions of limits. I give the main definition, and teach it with an example I call the cookie crispness index. But I don't assign problems in that section, and I don't test on it. I figure actually working with it is for an analysis course.
Of the topics that I do test on, implicit derivatives and related rates seem to be the hardest for students to understand. They also have trouble with optimization, because too many of them can't actually do anything with math other than follow the steps. (Hmm, I wonder if I can somehow have them optimize throughout the course. It seems like the most important topic that they do badly on.)
answered 7 hours ago
Sue VanHattum♦Sue VanHattum
9,9721 gold badge22 silver badges63 bronze badges
$endgroup$
$begingroup$
Optimization requires a lot of algebraic skill and it involves words. I also find graphing with calculus gets a lot of students, I think the logic and nuance flies in the face of the cookie-cutter problem solving mode so many of them are in before calculus. I used to test on $epsilon-delta$... it is even worse, which makes sense as it is even more removed from canned-problem solving.
$endgroup$
– James S. Cook
7 hours ago
$begingroup$
Once you learn how to do those (epsilon delta) problems, they don't seem so removed from canned procedures to me. What I think makes them hard is how symbolic and distant from practical concerns they are. They feel, to me, like the terrain of mathematicians. I like that terrain, but my students usually need to be more grounded.
$endgroup$
– Sue VanHattum♦
6 hours ago
$begingroup$
@SueVanHattum: "terrain" and "grounded"---kinda clashing metaphors :-)
$endgroup$
– Joseph O'Rourke
6 hours ago
$begingroup$
(I was thinking of the mathematical terrain as mountains in the sky, sort of, and grounded referring to down on the flatter ground.)
$endgroup$
– Sue VanHattum♦
4 hours ago
add a comment |
$begingroup$
Related rates problems show the most issues on AP grading. Source PIOOMA. Topic involves word problems, geometry and some multistep problem solving.
I don't think the concept of epsilon delta is so difficult as the tediousness of the algebra along with students openly or unconsciously questioning the value add for science and engineering. Even for the calculus class itself there is an issue that you do it st the beginning and set it aside. Not really a tool you are building with. These create an engagement issue.
Edit. Just noticed your question on prereqs. I think for calc 2, 3, and diffy Qs, the most important issue is just having solid drilled ability to work problems. And having solidified algebra. Calculus tends to do that. For real analysis, the course itself tends to cover the dive into theory versus assuming it. So even here main benefit of calculus is the muscle building. Although of course having seen definitions and theorems before once has a benefit in terms of repetition when doing a theoretical calc course.
answered 6 hours ago
guest2guest2
662 bronze badges
$endgroup$
1
$begingroup$
"Source PIOOMA." Could you change that please? I just spend some time to look it up, assuming it was something relevant only to find it's a casual expression. To be clear my concern is more that it is hard to understand (not that the expression is inappropriate).
$endgroup$
– quid♦
17 mins ago
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
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active
oldest
votes
$begingroup$
At my (community) college, I believe many teachers skip the (𝜖,𝛿)-definitions of limits. I give the main definition, and teach it with an example I call the cookie crispness index. But I don't assign problems in that section, and I don't test on it. I figure actually working with it is for an analysis course.
Of the topics that I do test on, implicit derivatives and related rates seem to be the hardest for students to understand. They also have trouble with optimization, because too many of them can't actually do anything with math other than follow the steps. (Hmm, I wonder if I can somehow have them optimize throughout the course. It seems like the most important topic that they do badly on.)
answered 7 hours ago
Sue VanHattum♦Sue VanHattum
9,9721 gold badge22 silver badges63 bronze badges
$endgroup$
$begingroup$
Optimization requires a lot of algebraic skill and it involves words. I also find graphing with calculus gets a lot of students, I think the logic and nuance flies in the face of the cookie-cutter problem solving mode so many of them are in before calculus. I used to test on $epsilon-delta$... it is even worse, which makes sense as it is even more removed from canned-problem solving.
$endgroup$
– James S. Cook
7 hours ago
$begingroup$
Once you learn how to do those (epsilon delta) problems, they don't seem so removed from canned procedures to me. What I think makes them hard is how symbolic and distant from practical concerns they are. They feel, to me, like the terrain of mathematicians. I like that terrain, but my students usually need to be more grounded.
$endgroup$
– Sue VanHattum♦
6 hours ago
$begingroup$
@SueVanHattum: "terrain" and "grounded"---kinda clashing metaphors :-)
$endgroup$
– Joseph O'Rourke
6 hours ago
$begingroup$
(I was thinking of the mathematical terrain as mountains in the sky, sort of, and grounded referring to down on the flatter ground.)
$endgroup$
– Sue VanHattum♦
4 hours ago
add a comment |
$begingroup$
At my (community) college, I believe many teachers skip the (𝜖,𝛿)-definitions of limits. I give the main definition, and teach it with an example I call the cookie crispness index. But I don't assign problems in that section, and I don't test on it. I figure actually working with it is for an analysis course.
Of the topics that I do test on, implicit derivatives and related rates seem to be the hardest for students to understand. They also have trouble with optimization, because too many of them can't actually do anything with math other than follow the steps. (Hmm, I wonder if I can somehow have them optimize throughout the course. It seems like the most important topic that they do badly on.)
answered 7 hours ago
Sue VanHattum♦Sue VanHattum
9,9721 gold badge22 silver badges63 bronze badges
$endgroup$
$begingroup$
Optimization requires a lot of algebraic skill and it involves words. I also find graphing with calculus gets a lot of students, I think the logic and nuance flies in the face of the cookie-cutter problem solving mode so many of them are in before calculus. I used to test on $epsilon-delta$... it is even worse, which makes sense as it is even more removed from canned-problem solving.
$endgroup$
– James S. Cook
7 hours ago
$begingroup$
Once you learn how to do those (epsilon delta) problems, they don't seem so removed from canned procedures to me. What I think makes them hard is how symbolic and distant from practical concerns they are. They feel, to me, like the terrain of mathematicians. I like that terrain, but my students usually need to be more grounded.
$endgroup$
– Sue VanHattum♦
6 hours ago
$begingroup$
@SueVanHattum: "terrain" and "grounded"---kinda clashing metaphors :-)
$endgroup$
– Joseph O'Rourke
6 hours ago
$begingroup$
(I was thinking of the mathematical terrain as mountains in the sky, sort of, and grounded referring to down on the flatter ground.)
$endgroup$
– Sue VanHattum♦
4 hours ago
add a comment |
$begingroup$
At my (community) college, I believe many teachers skip the (𝜖,𝛿)-definitions of limits. I give the main definition, and teach it with an example I call the cookie crispness index. But I don't assign problems in that section, and I don't test on it. I figure actually working with it is for an analysis course.
Of the topics that I do test on, implicit derivatives and related rates seem to be the hardest for students to understand. They also have trouble with optimization, because too many of them can't actually do anything with math other than follow the steps. (Hmm, I wonder if I can somehow have them optimize throughout the course. It seems like the most important topic that they do badly on.)
answered 7 hours ago
Sue VanHattum♦Sue VanHattum
9,9721 gold badge22 silver badges63 bronze badges
$endgroup$
At my (community) college, I believe many teachers skip the (𝜖,𝛿)-definitions of limits. I give the main definition, and teach it with an example I call the cookie crispness index. But I don't assign problems in that section, and I don't test on it. I figure actually working with it is for an analysis course.
Of the topics that I do test on, implicit derivatives and related rates seem to be the hardest for students to understand. They also have trouble with optimization, because too many of them can't actually do anything with math other than follow the steps. (Hmm, I wonder if I can somehow have them optimize throughout the course. It seems like the most important topic that they do badly on.)
answered 7 hours ago
Sue VanHattum♦Sue VanHattum
9,9721 gold badge22 silver badges63 bronze badges
answered 7 hours ago
Sue VanHattum♦Sue VanHattum
9,9721 gold badge22 silver badges63 bronze badges
answered 7 hours ago
Sue VanHattum♦Sue VanHattum
9,9721 gold badge22 silver badges63 bronze badges
answered 7 hours ago
Sue VanHattum♦Sue VanHattum
9,9721 gold badge22 silver badges63 bronze badges
9,9721 gold badge22 silver badges63 bronze badges
$begingroup$
Optimization requires a lot of algebraic skill and it involves words. I also find graphing with calculus gets a lot of students, I think the logic and nuance flies in the face of the cookie-cutter problem solving mode so many of them are in before calculus. I used to test on $epsilon-delta$... it is even worse, which makes sense as it is even more removed from canned-problem solving.
$endgroup$
– James S. Cook
7 hours ago
$begingroup$
Once you learn how to do those (epsilon delta) problems, they don't seem so removed from canned procedures to me. What I think makes them hard is how symbolic and distant from practical concerns they are. They feel, to me, like the terrain of mathematicians. I like that terrain, but my students usually need to be more grounded.
$endgroup$
– Sue VanHattum♦
6 hours ago
$begingroup$
@SueVanHattum: "terrain" and "grounded"---kinda clashing metaphors :-)
$endgroup$
– Joseph O'Rourke
6 hours ago
$begingroup$
(I was thinking of the mathematical terrain as mountains in the sky, sort of, and grounded referring to down on the flatter ground.)
$endgroup$
– Sue VanHattum♦
4 hours ago
add a comment |
$begingroup$
Optimization requires a lot of algebraic skill and it involves words. I also find graphing with calculus gets a lot of students, I think the logic and nuance flies in the face of the cookie-cutter problem solving mode so many of them are in before calculus. I used to test on $epsilon-delta$... it is even worse, which makes sense as it is even more removed from canned-problem solving.
$endgroup$
– James S. Cook
7 hours ago
$begingroup$
Once you learn how to do those (epsilon delta) problems, they don't seem so removed from canned procedures to me. What I think makes them hard is how symbolic and distant from practical concerns they are. They feel, to me, like the terrain of mathematicians. I like that terrain, but my students usually need to be more grounded.
$endgroup$
– Sue VanHattum♦
6 hours ago
$begingroup$
@SueVanHattum: "terrain" and "grounded"---kinda clashing metaphors :-)
$endgroup$
– Joseph O'Rourke
6 hours ago
$begingroup$
(I was thinking of the mathematical terrain as mountains in the sky, sort of, and grounded referring to down on the flatter ground.)
$endgroup$
– Sue VanHattum♦
4 hours ago
$begingroup$
Optimization requires a lot of algebraic skill and it involves words. I also find graphing with calculus gets a lot of students, I think the logic and nuance flies in the face of the cookie-cutter problem solving mode so many of them are in before calculus. I used to test on $epsilon-delta$... it is even worse, which makes sense as it is even more removed from canned-problem solving.
$endgroup$
– James S. Cook
7 hours ago
$begingroup$
Optimization requires a lot of algebraic skill and it involves words. I also find graphing with calculus gets a lot of students, I think the logic and nuance flies in the face of the cookie-cutter problem solving mode so many of them are in before calculus. I used to test on $epsilon-delta$... it is even worse, which makes sense as it is even more removed from canned-problem solving.
$endgroup$
– James S. Cook
7 hours ago
$begingroup$
Once you learn how to do those (epsilon delta) problems, they don't seem so removed from canned procedures to me. What I think makes them hard is how symbolic and distant from practical concerns they are. They feel, to me, like the terrain of mathematicians. I like that terrain, but my students usually need to be more grounded.
$endgroup$
– Sue VanHattum♦
6 hours ago
$begingroup$
Once you learn how to do those (epsilon delta) problems, they don't seem so removed from canned procedures to me. What I think makes them hard is how symbolic and distant from practical concerns they are. They feel, to me, like the terrain of mathematicians. I like that terrain, but my students usually need to be more grounded.
$endgroup$
– Sue VanHattum♦
6 hours ago
$begingroup$
@SueVanHattum: "terrain" and "grounded"---kinda clashing metaphors :-)
$endgroup$
– Joseph O'Rourke
6 hours ago
$begingroup$
@SueVanHattum: "terrain" and "grounded"---kinda clashing metaphors :-)
$endgroup$
– Joseph O'Rourke
6 hours ago
$begingroup$
(I was thinking of the mathematical terrain as mountains in the sky, sort of, and grounded referring to down on the flatter ground.)
$endgroup$
– Sue VanHattum♦
4 hours ago
$begingroup$
(I was thinking of the mathematical terrain as mountains in the sky, sort of, and grounded referring to down on the flatter ground.)
$endgroup$
– Sue VanHattum♦
4 hours ago
add a comment |
$begingroup$
Related rates problems show the most issues on AP grading. Source PIOOMA. Topic involves word problems, geometry and some multistep problem solving.
I don't think the concept of epsilon delta is so difficult as the tediousness of the algebra along with students openly or unconsciously questioning the value add for science and engineering. Even for the calculus class itself there is an issue that you do it st the beginning and set it aside. Not really a tool you are building with. These create an engagement issue.
Edit. Just noticed your question on prereqs. I think for calc 2, 3, and diffy Qs, the most important issue is just having solid drilled ability to work problems. And having solidified algebra. Calculus tends to do that. For real analysis, the course itself tends to cover the dive into theory versus assuming it. So even here main benefit of calculus is the muscle building. Although of course having seen definitions and theorems before once has a benefit in terms of repetition when doing a theoretical calc course.
answered 6 hours ago
guest2guest2
662 bronze badges
$endgroup$
1
$begingroup$
"Source PIOOMA." Could you change that please? I just spend some time to look it up, assuming it was something relevant only to find it's a casual expression. To be clear my concern is more that it is hard to understand (not that the expression is inappropriate).
$endgroup$
– quid♦
17 mins ago
add a comment |
$begingroup$
Related rates problems show the most issues on AP grading. Source PIOOMA. Topic involves word problems, geometry and some multistep problem solving.
I don't think the concept of epsilon delta is so difficult as the tediousness of the algebra along with students openly or unconsciously questioning the value add for science and engineering. Even for the calculus class itself there is an issue that you do it st the beginning and set it aside. Not really a tool you are building with. These create an engagement issue.
Edit. Just noticed your question on prereqs. I think for calc 2, 3, and diffy Qs, the most important issue is just having solid drilled ability to work problems. And having solidified algebra. Calculus tends to do that. For real analysis, the course itself tends to cover the dive into theory versus assuming it. So even here main benefit of calculus is the muscle building. Although of course having seen definitions and theorems before once has a benefit in terms of repetition when doing a theoretical calc course.
answered 6 hours ago
guest2guest2
662 bronze badges
$endgroup$
1
$begingroup$
"Source PIOOMA." Could you change that please? I just spend some time to look it up, assuming it was something relevant only to find it's a casual expression. To be clear my concern is more that it is hard to understand (not that the expression is inappropriate).
$endgroup$
– quid♦
17 mins ago
add a comment |
$begingroup$
Related rates problems show the most issues on AP grading. Source PIOOMA. Topic involves word problems, geometry and some multistep problem solving.
I don't think the concept of epsilon delta is so difficult as the tediousness of the algebra along with students openly or unconsciously questioning the value add for science and engineering. Even for the calculus class itself there is an issue that you do it st the beginning and set it aside. Not really a tool you are building with. These create an engagement issue.
Edit. Just noticed your question on prereqs. I think for calc 2, 3, and diffy Qs, the most important issue is just having solid drilled ability to work problems. And having solidified algebra. Calculus tends to do that. For real analysis, the course itself tends to cover the dive into theory versus assuming it. So even here main benefit of calculus is the muscle building. Although of course having seen definitions and theorems before once has a benefit in terms of repetition when doing a theoretical calc course.
answered 6 hours ago
guest2guest2
662 bronze badges
$endgroup$
Related rates problems show the most issues on AP grading. Source PIOOMA. Topic involves word problems, geometry and some multistep problem solving.
I don't think the concept of epsilon delta is so difficult as the tediousness of the algebra along with students openly or unconsciously questioning the value add for science and engineering. Even for the calculus class itself there is an issue that you do it st the beginning and set it aside. Not really a tool you are building with. These create an engagement issue.
Edit. Just noticed your question on prereqs. I think for calc 2, 3, and diffy Qs, the most important issue is just having solid drilled ability to work problems. And having solidified algebra. Calculus tends to do that. For real analysis, the course itself tends to cover the dive into theory versus assuming it. So even here main benefit of calculus is the muscle building. Although of course having seen definitions and theorems before once has a benefit in terms of repetition when doing a theoretical calc course.
answered 6 hours ago
guest2guest2
662 bronze badges
edited 6 hours ago
answered 6 hours ago
guest2guest2
662 bronze badges
answered 6 hours ago
guest2guest2
662 bronze badges
answered 6 hours ago
guest2guest2
662 bronze badges
662 bronze badges
1
$begingroup$
"Source PIOOMA." Could you change that please? I just spend some time to look it up, assuming it was something relevant only to find it's a casual expression. To be clear my concern is more that it is hard to understand (not that the expression is inappropriate).
$endgroup$
– quid♦
17 mins ago
add a comment |
1
$begingroup$
"Source PIOOMA." Could you change that please? I just spend some time to look it up, assuming it was something relevant only to find it's a casual expression. To be clear my concern is more that it is hard to understand (not that the expression is inappropriate).
$endgroup$
– quid♦
17 mins ago
1
1
$begingroup$
"Source PIOOMA." Could you change that please? I just spend some time to look it up, assuming it was something relevant only to find it's a casual expression. To be clear my concern is more that it is hard to understand (not that the expression is inappropriate).
$endgroup$
– quid♦
17 mins ago
$begingroup$
"Source PIOOMA." Could you change that please? I just spend some time to look it up, assuming it was something relevant only to find it's a casual expression. To be clear my concern is more that it is hard to understand (not that the expression is inappropriate).
$endgroup$
– quid♦
17 mins ago
add a comment |
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2
$begingroup$
Do you want calculus-specific concepts, or would you accept the application of non-calculus topics, such as writing a function that performs a particular task (distance between two points, area of a basic shape, etc.)? For me, teaching calculus always implies teaching prerequisite material.
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– Nick C
8 hours ago
$begingroup$
@NickC: I was thinking of topics, but I would be interested as well to learn of skills that seem difficult to master.
$endgroup$
– Joseph O'Rourke
8 hours ago
1
$begingroup$
My guess is the point of the mean value theorem. Or based on impact, perhaps the Chain Rule. If the limit definition is not taught, it can hardly be difficult. If it not used in the course you're teaching, it's hardly important. Otherwise limits are probably one of the more difficult concepts, although it seems less difficult in the third or fourth semester of learning limits (calc 1 -> 2 -> 3 -> transition-to-proofs course)
$endgroup$
– user1527
7 hours ago
1
$begingroup$
Mechanically, I think the chain rule is most troublesome. However, logically, anything involving non-standard algebra and multi-step logic is an opportunity to fail; optimization, graphing with calculus, related rates, conceptual problems which illustrate theorems... anything not algorithmic.
$endgroup$
– James S. Cook
7 hours ago