What is the most difficult concept to grasp in Calculus 1?Is there a more telling name for “Calculus...
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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?Good ways of explaining the idea of epsilon-delta limits to bio & chem majors?The purpose of mathematics in a liberal education when it is not a prerequisite to other subjectsNatural, rich, calculus questionsExample of function with *all* the features of differential calculus at first-year levelTutoring a recalcitrant/awkward/exasperating student---special needs?Difference between high school and college calculus coursesWould teaching nonstandard calculus in an introduction calculus course make it easier to learn?
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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.
Summary by 16Aug2019:
I did not anticipate such lack of consensus.
- @vonbrand: "the whole $epsilon/delta$ dance."
- @SueVanHattum: implicit derivatives and optimization
- @guest2: "multistep problem solving"
- @Maesumi: "the concept of a variable and that of a function"
- @JamesSCook: "the chain rule is most troublesome"
- @user1527: "the point of the mean value theorem"
undergraduate-education calculus limits
asked 2 days ago
Joseph O'RourkeJoseph O'Rourke
16.4k3 gold badges36 silver badges88 bronze badges
$endgroup$
|
show 2 more comments
$begingroup$
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.
Summary by 16Aug2019:
I did not anticipate such lack of consensus.
- @vonbrand: "the whole $epsilon/delta$ dance."
- @SueVanHattum: implicit derivatives and optimization
- @guest2: "multistep problem solving"
- @Maesumi: "the concept of a variable and that of a function"
- @JamesSCook: "the chain rule is most troublesome"
- @user1527: "the point of the mean value theorem"
undergraduate-education calculus limits
asked 2 days ago
Joseph O'RourkeJoseph O'Rourke
16.4k3 gold badges36 silver badges88 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
2 days 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
2 days 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
2 days 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
2 days ago
$begingroup$
This really depends on the level at which the course is being taught. If you're teaching the completeness axiom of the reals, with real proofs of things like the intermediate value theorem, then those are extremely deep logically, and not realistically something that any but the most talented 18-year-olds will be able to undrstand. If the course is being taught at a low level, then basically students are not learning any concepts at all. They're just learning how to calculate derivatives and integrals, without any understanding of what it means.
$endgroup$
– Ben Crowell
2 days ago
|
show 2 more comments
$begingroup$
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.
Summary by 16Aug2019:
I did not anticipate such lack of consensus.
- @vonbrand: "the whole $epsilon/delta$ dance."
- @SueVanHattum: implicit derivatives and optimization
- @guest2: "multistep problem solving"
- @Maesumi: "the concept of a variable and that of a function"
- @JamesSCook: "the chain rule is most troublesome"
- @user1527: "the point of the mean value theorem"
undergraduate-education calculus limits
asked 2 days ago
Joseph O'RourkeJoseph O'Rourke
16.4k3 gold badges36 silver badges88 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.
Summary by 16Aug2019:
I did not anticipate such lack of consensus.
- @vonbrand: "the whole $epsilon/delta$ dance."
- @SueVanHattum: implicit derivatives and optimization
- @guest2: "multistep problem solving"
- @Maesumi: "the concept of a variable and that of a function"
- @JamesSCook: "the chain rule is most troublesome"
- @user1527: "the point of the mean value theorem"
undergraduate-education calculus limits
undergraduate-education calculus limits
asked 2 days ago
Joseph O'RourkeJoseph O'Rourke
16.4k3 gold badges36 silver badges88 bronze badges
asked 2 days ago
Joseph O'RourkeJoseph O'Rourke
16.4k3 gold badges36 silver badges88 bronze badges
edited 6 hours ago
Joseph O'Rourke
asked 2 days ago
Joseph O'RourkeJoseph O'Rourke
16.4k3 gold badges36 silver badges88 bronze badges
asked 2 days ago
Joseph O'RourkeJoseph O'Rourke
16.4k3 gold badges36 silver badges88 bronze badges
asked 2 days ago
Joseph O'RourkeJoseph O'Rourke
16.4k3 gold badges36 silver badges88 bronze badges
16.4k3 gold badges36 silver badges88 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
2 days 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
2 days 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
2 days 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
2 days ago
$begingroup$
This really depends on the level at which the course is being taught. If you're teaching the completeness axiom of the reals, with real proofs of things like the intermediate value theorem, then those are extremely deep logically, and not realistically something that any but the most talented 18-year-olds will be able to undrstand. If the course is being taught at a low level, then basically students are not learning any concepts at all. They're just learning how to calculate derivatives and integrals, without any understanding of what it means.
$endgroup$
– Ben Crowell
2 days ago
|
show 2 more comments
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
2 days 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
2 days 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
2 days 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
2 days ago
$begingroup$
This really depends on the level at which the course is being taught. If you're teaching the completeness axiom of the reals, with real proofs of things like the intermediate value theorem, then those are extremely deep logically, and not realistically something that any but the most talented 18-year-olds will be able to undrstand. If the course is being taught at a low level, then basically students are not learning any concepts at all. They're just learning how to calculate derivatives and integrals, without any understanding of what it means.
$endgroup$
– Ben Crowell
2 days 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
2 days 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
2 days 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
2 days 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
2 days 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
2 days 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
2 days 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
2 days 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
2 days ago
$begingroup$
This really depends on the level at which the course is being taught. If you're teaching the completeness axiom of the reals, with real proofs of things like the intermediate value theorem, then those are extremely deep logically, and not realistically something that any but the most talented 18-year-olds will be able to undrstand. If the course is being taught at a low level, then basically students are not learning any concepts at all. They're just learning how to calculate derivatives and integrals, without any understanding of what it means.
$endgroup$
– Ben Crowell
2 days ago
$begingroup$
This really depends on the level at which the course is being taught. If you're teaching the completeness axiom of the reals, with real proofs of things like the intermediate value theorem, then those are extremely deep logically, and not realistically something that any but the most talented 18-year-olds will be able to undrstand. If the course is being taught at a low level, then basically students are not learning any concepts at all. They're just learning how to calculate derivatives and integrals, without any understanding of what it means.
$endgroup$
– Ben Crowell
2 days ago
|
show 2 more comments
4 Answers
4
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 2 days ago
Sue VanHattum♦Sue VanHattum
10.1k1 gold badge22 silver badges64 bronze badges
$endgroup$
1
$begingroup$
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
2 days ago
3
$begingroup$
@SueVanHattum: "terrain" and "grounded"---kinda clashing metaphors :-)
$endgroup$
– Joseph O'Rourke
2 days ago
2
$begingroup$
(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♦
2 days ago
3
$begingroup$
What is calculus if not analysis? In my language the name for the calculus translates literally to "mathematical analysis"...
$endgroup$
– Džuris
2 days ago
2
$begingroup$
In the U.S., the beginning calculus courses are not so rigorous. Students learn how to use calculus to solve problems. In analysis, everything is carefully proved. (Does someone else have a more precise way of describing the difference? My first calculus course, honors at UMich, was actually an analysis course.)
$endgroup$
– Sue VanHattum♦
2 days ago
|
show 2 more comments
$begingroup$
Related rates problems show the most issues on AP grading. (No reference, just my own observation) 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.
edited 2 days ago
JoeTaxpayer
5,66120 silver badges47 bronze badges
answered 2 days ago
guest2guest2
862 bronze badges
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add a comment |
$begingroup$
I'd say the whole $epsilon/delta$ dance. Just because it involves inequalities, while students just have trained equalities all their life. To just work with (sometimes very rough) bounds, when you were drilled to get exact answers goes against the grain.
answered yesterday
vonbrandvonbrand
9,3431 gold badge17 silver badges51 bronze badges
$endgroup$
$begingroup$
In many universities, $epsilon/delta$ is not used in introductory Calc I - III. It may appear in Advanced Calculus (for Honor students with AP experience in calc.).
$endgroup$
– Namaste
10 hours ago
$begingroup$
@Namaste, the above observation is from my experience (now long ago).
$endgroup$
– vonbrand
8 hours ago
add a comment |
$begingroup$
In terms of issues affecting most students I believe the concept of a variable and that of a function are still the most difficult concepts for calculus 1 students, even though the concepts are introduced in precalculus. Writing a full and correct mathematical sentence is a topic most students struggle with.
answered yesterday
MaesumiMaesumi
1786 bronze badges
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add a comment |
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4 Answers
4
active
oldest
votes
4 Answers
4
active
oldest
votes
active
oldest
votes
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 2 days ago
Sue VanHattum♦Sue VanHattum
10.1k1 gold badge22 silver badges64 bronze badges
$endgroup$
1
$begingroup$
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
2 days ago
3
$begingroup$
@SueVanHattum: "terrain" and "grounded"---kinda clashing metaphors :-)
$endgroup$
– Joseph O'Rourke
2 days ago
2
$begingroup$
(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♦
2 days ago
3
$begingroup$
What is calculus if not analysis? In my language the name for the calculus translates literally to "mathematical analysis"...
$endgroup$
– Džuris
2 days ago
2
$begingroup$
In the U.S., the beginning calculus courses are not so rigorous. Students learn how to use calculus to solve problems. In analysis, everything is carefully proved. (Does someone else have a more precise way of describing the difference? My first calculus course, honors at UMich, was actually an analysis course.)
$endgroup$
– Sue VanHattum♦
2 days ago
|
show 2 more comments
$begingroup$
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 2 days ago
Sue VanHattum♦Sue VanHattum
10.1k1 gold badge22 silver badges64 bronze badges
$endgroup$
1
$begingroup$
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
2 days ago
3
$begingroup$
@SueVanHattum: "terrain" and "grounded"---kinda clashing metaphors :-)
$endgroup$
– Joseph O'Rourke
2 days ago
2
$begingroup$
(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♦
2 days ago
3
$begingroup$
What is calculus if not analysis? In my language the name for the calculus translates literally to "mathematical analysis"...
$endgroup$
– Džuris
2 days ago
2
$begingroup$
In the U.S., the beginning calculus courses are not so rigorous. Students learn how to use calculus to solve problems. In analysis, everything is carefully proved. (Does someone else have a more precise way of describing the difference? My first calculus course, honors at UMich, was actually an analysis course.)
$endgroup$
– Sue VanHattum♦
2 days ago
|
show 2 more comments
$begingroup$
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 2 days ago
Sue VanHattum♦Sue VanHattum
10.1k1 gold badge22 silver badges64 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 2 days ago
Sue VanHattum♦Sue VanHattum
10.1k1 gold badge22 silver badges64 bronze badges
answered 2 days ago
Sue VanHattum♦Sue VanHattum
10.1k1 gold badge22 silver badges64 bronze badges
answered 2 days ago
Sue VanHattum♦Sue VanHattum
10.1k1 gold badge22 silver badges64 bronze badges
answered 2 days ago
Sue VanHattum♦Sue VanHattum
10.1k1 gold badge22 silver badges64 bronze badges
10.1k1 gold badge22 silver badges64 bronze badges
1
$begingroup$
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.
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– James S. Cook
2 days ago
3
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@SueVanHattum: "terrain" and "grounded"---kinda clashing metaphors :-)
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– Joseph O'Rourke
2 days ago
2
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(I was thinking of the mathematical terrain as mountains in the sky, sort of, and grounded referring to down on the flatter ground.)
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– Sue VanHattum♦
2 days ago
3
$begingroup$
What is calculus if not analysis? In my language the name for the calculus translates literally to "mathematical analysis"...
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– Džuris
2 days ago
2
$begingroup$
In the U.S., the beginning calculus courses are not so rigorous. Students learn how to use calculus to solve problems. In analysis, everything is carefully proved. (Does someone else have a more precise way of describing the difference? My first calculus course, honors at UMich, was actually an analysis course.)
$endgroup$
– Sue VanHattum♦
2 days ago
|
show 2 more comments
1
$begingroup$
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
2 days ago
3
$begingroup$
@SueVanHattum: "terrain" and "grounded"---kinda clashing metaphors :-)
$endgroup$
– Joseph O'Rourke
2 days ago
2
$begingroup$
(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♦
2 days ago
3
$begingroup$
What is calculus if not analysis? In my language the name for the calculus translates literally to "mathematical analysis"...
$endgroup$
– Džuris
2 days ago
2
$begingroup$
In the U.S., the beginning calculus courses are not so rigorous. Students learn how to use calculus to solve problems. In analysis, everything is carefully proved. (Does someone else have a more precise way of describing the difference? My first calculus course, honors at UMich, was actually an analysis course.)
$endgroup$
– Sue VanHattum♦
2 days ago
1
1
$begingroup$
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
2 days 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
2 days ago
3
3
$begingroup$
@SueVanHattum: "terrain" and "grounded"---kinda clashing metaphors :-)
$endgroup$
– Joseph O'Rourke
2 days ago
$begingroup$
@SueVanHattum: "terrain" and "grounded"---kinda clashing metaphors :-)
$endgroup$
– Joseph O'Rourke
2 days ago
2
2
$begingroup$
(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♦
2 days 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♦
2 days ago
3
3
$begingroup$
What is calculus if not analysis? In my language the name for the calculus translates literally to "mathematical analysis"...
$endgroup$
– Džuris
2 days ago
$begingroup$
What is calculus if not analysis? In my language the name for the calculus translates literally to "mathematical analysis"...
$endgroup$
– Džuris
2 days ago
2
2
$begingroup$
In the U.S., the beginning calculus courses are not so rigorous. Students learn how to use calculus to solve problems. In analysis, everything is carefully proved. (Does someone else have a more precise way of describing the difference? My first calculus course, honors at UMich, was actually an analysis course.)
$endgroup$
– Sue VanHattum♦
2 days ago
$begingroup$
In the U.S., the beginning calculus courses are not so rigorous. Students learn how to use calculus to solve problems. In analysis, everything is carefully proved. (Does someone else have a more precise way of describing the difference? My first calculus course, honors at UMich, was actually an analysis course.)
$endgroup$
– Sue VanHattum♦
2 days ago
|
show 2 more comments
$begingroup$
Related rates problems show the most issues on AP grading. (No reference, just my own observation) 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.
edited 2 days ago
JoeTaxpayer
5,66120 silver badges47 bronze badges
answered 2 days ago
guest2guest2
862 bronze badges
$endgroup$
add a comment |
$begingroup$
Related rates problems show the most issues on AP grading. (No reference, just my own observation) 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.
edited 2 days ago
JoeTaxpayer
5,66120 silver badges47 bronze badges
answered 2 days ago
guest2guest2
862 bronze badges
$endgroup$
add a comment |
$begingroup$
Related rates problems show the most issues on AP grading. (No reference, just my own observation) 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.
edited 2 days ago
JoeTaxpayer
5,66120 silver badges47 bronze badges
answered 2 days ago
guest2guest2
862 bronze badges
$endgroup$
Related rates problems show the most issues on AP grading. (No reference, just my own observation) 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.
edited 2 days ago
JoeTaxpayer
5,66120 silver badges47 bronze badges
answered 2 days ago
guest2guest2
862 bronze badges
edited 2 days ago
JoeTaxpayer
5,66120 silver badges47 bronze badges
edited 2 days ago
JoeTaxpayer
5,66120 silver badges47 bronze badges
edited 2 days ago
JoeTaxpayer
5,66120 silver badges47 bronze badges
5,66120 silver badges47 bronze badges
answered 2 days ago
guest2guest2
862 bronze badges
answered 2 days ago
guest2guest2
862 bronze badges
answered 2 days ago
guest2guest2
862 bronze badges
862 bronze badges
add a comment |
add a comment |
$begingroup$
I'd say the whole $epsilon/delta$ dance. Just because it involves inequalities, while students just have trained equalities all their life. To just work with (sometimes very rough) bounds, when you were drilled to get exact answers goes against the grain.
answered yesterday
vonbrandvonbrand
9,3431 gold badge17 silver badges51 bronze badges
$endgroup$
$begingroup$
In many universities, $epsilon/delta$ is not used in introductory Calc I - III. It may appear in Advanced Calculus (for Honor students with AP experience in calc.).
$endgroup$
– Namaste
10 hours ago
$begingroup$
@Namaste, the above observation is from my experience (now long ago).
$endgroup$
– vonbrand
8 hours ago
add a comment |
$begingroup$
I'd say the whole $epsilon/delta$ dance. Just because it involves inequalities, while students just have trained equalities all their life. To just work with (sometimes very rough) bounds, when you were drilled to get exact answers goes against the grain.
answered yesterday
vonbrandvonbrand
9,3431 gold badge17 silver badges51 bronze badges
$endgroup$
$begingroup$
In many universities, $epsilon/delta$ is not used in introductory Calc I - III. It may appear in Advanced Calculus (for Honor students with AP experience in calc.).
$endgroup$
– Namaste
10 hours ago
$begingroup$
@Namaste, the above observation is from my experience (now long ago).
$endgroup$
– vonbrand
8 hours ago
add a comment |
$begingroup$
I'd say the whole $epsilon/delta$ dance. Just because it involves inequalities, while students just have trained equalities all their life. To just work with (sometimes very rough) bounds, when you were drilled to get exact answers goes against the grain.
answered yesterday
vonbrandvonbrand
9,3431 gold badge17 silver badges51 bronze badges
$endgroup$
I'd say the whole $epsilon/delta$ dance. Just because it involves inequalities, while students just have trained equalities all their life. To just work with (sometimes very rough) bounds, when you were drilled to get exact answers goes against the grain.
answered yesterday
vonbrandvonbrand
9,3431 gold badge17 silver badges51 bronze badges
edited 8 hours ago
answered yesterday
vonbrandvonbrand
9,3431 gold badge17 silver badges51 bronze badges
answered yesterday
vonbrandvonbrand
9,3431 gold badge17 silver badges51 bronze badges
answered yesterday
vonbrandvonbrand
9,3431 gold badge17 silver badges51 bronze badges
9,3431 gold badge17 silver badges51 bronze badges
$begingroup$
In many universities, $epsilon/delta$ is not used in introductory Calc I - III. It may appear in Advanced Calculus (for Honor students with AP experience in calc.).
$endgroup$
– Namaste
10 hours ago
$begingroup$
@Namaste, the above observation is from my experience (now long ago).
$endgroup$
– vonbrand
8 hours ago
add a comment |
$begingroup$
In many universities, $epsilon/delta$ is not used in introductory Calc I - III. It may appear in Advanced Calculus (for Honor students with AP experience in calc.).
$endgroup$
– Namaste
10 hours ago
$begingroup$
@Namaste, the above observation is from my experience (now long ago).
$endgroup$
– vonbrand
8 hours ago
$begingroup$
In many universities, $epsilon/delta$ is not used in introductory Calc I - III. It may appear in Advanced Calculus (for Honor students with AP experience in calc.).
$endgroup$
– Namaste
10 hours ago
$begingroup$
In many universities, $epsilon/delta$ is not used in introductory Calc I - III. It may appear in Advanced Calculus (for Honor students with AP experience in calc.).
$endgroup$
– Namaste
10 hours ago
$begingroup$
@Namaste, the above observation is from my experience (now long ago).
$endgroup$
– vonbrand
8 hours ago
$begingroup$
@Namaste, the above observation is from my experience (now long ago).
$endgroup$
– vonbrand
8 hours ago
add a comment |
$begingroup$
In terms of issues affecting most students I believe the concept of a variable and that of a function are still the most difficult concepts for calculus 1 students, even though the concepts are introduced in precalculus. Writing a full and correct mathematical sentence is a topic most students struggle with.
answered yesterday
MaesumiMaesumi
1786 bronze badges
$endgroup$
add a comment |
$begingroup$
In terms of issues affecting most students I believe the concept of a variable and that of a function are still the most difficult concepts for calculus 1 students, even though the concepts are introduced in precalculus. Writing a full and correct mathematical sentence is a topic most students struggle with.
answered yesterday
MaesumiMaesumi
1786 bronze badges
$endgroup$
add a comment |
$begingroup$
In terms of issues affecting most students I believe the concept of a variable and that of a function are still the most difficult concepts for calculus 1 students, even though the concepts are introduced in precalculus. Writing a full and correct mathematical sentence is a topic most students struggle with.
answered yesterday
MaesumiMaesumi
1786 bronze badges
$endgroup$
In terms of issues affecting most students I believe the concept of a variable and that of a function are still the most difficult concepts for calculus 1 students, even though the concepts are introduced in precalculus. Writing a full and correct mathematical sentence is a topic most students struggle with.
answered yesterday
MaesumiMaesumi
1786 bronze badges
answered yesterday
MaesumiMaesumi
1786 bronze badges
answered yesterday
MaesumiMaesumi
1786 bronze badges
answered yesterday
MaesumiMaesumi
1786 bronze badges
1786 bronze badges
add a comment |
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.
$endgroup$
– Nick C
2 days 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
2 days 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
2 days 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
2 days ago
$begingroup$
This really depends on the level at which the course is being taught. If you're teaching the completeness axiom of the reals, with real proofs of things like the intermediate value theorem, then those are extremely deep logically, and not realistically something that any but the most talented 18-year-olds will be able to undrstand. If the course is being taught at a low level, then basically students are not learning any concepts at all. They're just learning how to calculate derivatives and integrals, without any understanding of what it means.
$endgroup$
– Ben Crowell
2 days ago