All Questions
Tagged with mathematical-pedagogy calculus
38
questions
2
votes
3
answers
303
views
Is this a viable Calculus 1 question?
A person is standing next to a hot air balloon. At the same time, the person starts moving away from the balloon at 5 ft/sec and the balloon rises straight into the air at a rate of 12 ft/sec. Is the ...
15
votes
15
answers
7k
views
Students can't seem to grasp the intent of tangent lines and getting general trends of derivatives from graphs
Background
I'm informally helping a few students with college Calc 1. This isn't the first time I've aided people with calculus, and so they've sought me for help, though I don't consider myself to ...
3
votes
6
answers
1k
views
Is this motivation for the concept of a limit a good one?
tldr: There is a simple intuitive definition of a limit for monotone sequences, and I suggest that it can be used to motivate the (more complicated) standard definition. I am asking for feedback on my ...
4
votes
0
answers
792
views
What are your experiences with Buck’s Advanced Calculus?
I stumbled across the book when searching for rigorous alternatives to Rudin with some solutions. It’s an “old school” (1965) calculus text but, I think, covers similar material to Rudin in a more ...
8
votes
4
answers
756
views
Exponential & logarithm in a high school calculus class
So recently I was teaching high school calculus to a high school class and I was wondering about the pedagogically best way to make students actually understand why the derivatives of the exponential &...
5
votes
1
answer
275
views
Making the leap from Pre-Calculus to Calculus
This question is targeted at teachers who taught both low and high level mathematics. I have a group of students that I'm currently teaching precalculus and they seem to be doing really well in all ...
2
votes
1
answer
460
views
Are there any university programs that "supersize" calculus courses?
Most differential calculus courses begin with the theory (and analysis) of differentiation, followed by computations, and likewise integral calculus courses. That's a lot for a three credit course, ...
21
votes
6
answers
6k
views
How rigorous should high school calculus be?
In the UK, calculus taught in secondary school focuses mainly on computation of derivatives and integrals and solving simple differential equations. There is a small amount of discussion about limits ...
2
votes
3
answers
147
views
How to teach integrals motivated by the work done in moving an object?
I am now teaching Calculus of several variables this semester. In apllications of integrals, the problem of finding the work done in moving an object under a force $F$ is one of the most common ...
7
votes
5
answers
2k
views
Teaching asymptotic notations at the beginning of calculus [duplicate]
I'm thinking about teaching calculus by firstly introducing the asymptotic notations (big-Oh, little-oh, and $\sim$), secondly explaining their "arithmetic" (things like how to sum little-oh's and ...
6
votes
3
answers
210
views
(Riemann integrability) How do you explain this to a high school student?
The following question was in a high school teacher's guide:
Let $f\colon\mathbb{R}\rightarrow\mathbb{R}$ defined by
$$f(x)=\begin{cases} x & x\in\mathbb{R}\setminus\mathbb{Q}\\
2x & x\...
1
vote
2
answers
108
views
Retain problems and combat regression in learning
Regressive Learning
It's a really stressful situation. I can achieve but not retain expertise in maths problems.
History
6 months back, I studied integration in Calculus at college. I learnt it all ...
6
votes
4
answers
590
views
Ideas for the introduction of the derivative?
I want to introduce to my class to the derivative, but I am still searching for a good, realistic context that isn't too hard to understand, without seeming to be contrived. Do you have an ideas for ...
2
votes
2
answers
140
views
Are questions on overlapping solids of revolutions without prior definitions and instructions fair given that there are divided interpretations?
If words of command are not clear and distinct, if orders are not thoroughly understood, the general is to blame. But if his orders are clear, and the soldiers nevertheless disobey, then it is the ...
10
votes
4
answers
500
views
Surrounding a subject and strangling it to death versus concentrating on the main point
Standard calculus textbooks begin by introducing limits, including
limits of a fraction as the numerator and denominator approach $0,$
limits of a fraction as the numerator and denominator approach $\...
6
votes
3
answers
493
views
What is the ULTIMATE Calculus syllabus
After such amazing answers I got here for a related question (link at the end if someone still wants to share with me their views)...
Here is the concept:
If you were to create the ULTIMATE Calculus ...
2
votes
0
answers
99
views
what is the standard subdivision or classification of calculus related rates problems?
I am working on a project where I have to group/classify calculus problems. Now with most the calculus topics, it's usually obvious how it's divided in various textbooks, but when it comes to related ...
8
votes
1
answer
193
views
How can I deal with the time pressure of teaching a short course?
I am an undergraduate applied math student. In about a month, I will be teaching two nine-hour math courses (one precalculus, one calculus) to a small group of motivated high school students. My broad ...
7
votes
2
answers
261
views
How to catch students from different subjects' interest to math?
I have just started to teach Calculus to freshmans and sophomores who study non-mathematical subjects, e.g., international relations, psychology. They have to take few mathematics classes -including ...
28
votes
4
answers
1k
views
The Undergraduate Responsibility Gradient
We tell undergraduate students that they should study two to three hours for every hour they spend in class. We know that many students don't follow through with this nearly to the degree that they ...
14
votes
4
answers
744
views
Is there research for or against such an approach in teaching calculus?
Copying from Calculus Made Easy by Silvanus Thompson (2nd ed., 1914):
CHAPTER I:TO DELIVER YOU FROM THE PRELIMINARY TERRORS
The preliminary terror, which chokes off most fifth-form boys from ...
4
votes
4
answers
2k
views
How are the basic trigonometric functions introduced to students?
The fundamental trigonometric functions $\sin(x)$ and $\cos(x)$ are used throughout the sciences, but I believe students are often introduced to a very limited initial understanding where it is ...
23
votes
2
answers
1k
views
Is Knuth's suggestion on teaching calculus a good idea?
Note: I myself am not a math educator, though I plan to be one someday.
In this letter, Donald Knuth suggests an alternate way of teaching calculus, based on big-O (introduced via a related big-A ...
16
votes
2
answers
530
views
Nontraditional calculus recitations
I'm a math grad student, and next semester I start TAing a calculus class for the first time. We all know about the standard recitations: instructor gives short lecture on some more difficult topic ...
18
votes
5
answers
2k
views
How to convince students of the integral identity $\int_0^af(x)dx=\int_0^af(a-x)dx$?
A common identity in integration is $\int_0^af(x)dx=\int_0^af(a-x)dx$.
The steps to prove it (algebraically, ignoring the geometric method) are as follows.
Let $u=a-x$ so $dx=-du$.
$\int_0^af(a-x)...
11
votes
6
answers
1k
views
How can I convince students that Fourier series are useful?
Main question: Calculating the coefficients of a Fourier series can be difficult and time-consuming. How might a student be motivated/convinced to go through these (potentially tedious) details? Are ...
10
votes
1
answer
701
views
Language to Distinguish Between Variables and Arbitrary Constants
Today in second semester calculus, I found myself stumbling a bit to provide a natural-sounding explanation for all the letters involved in the expression
$$
\lim_{t \rightarrow \infty} \int_1^t \frac{...
13
votes
7
answers
2k
views
When should we get into limits in introductory calculus courses?
All of the calculus textbooks I've used (teaching at community colleges) start with the first chapter covering limits. (Perhaps after a review chapter.) I think this order is wrong.
Historically, ...
26
votes
7
answers
4k
views
Why are we so careful in saying that dy/dx is not a fraction?
Calculus instructors are mostly very careful to explain that $\frac{\mathrm{d}y}{\mathrm{d}x}$ is not a fraction, and multiplying both sides of an equation by $\mathrm{d}x$ is nonsense, wrong, or evil....
3
votes
5
answers
396
views
Average Rate of Change isn't/is Statistics
I have the common misconception in my business calculus classes that the Average Rate of Change, say from $x=1$ to $x=5$, is the statistical average of the rates on the four unit intervals $1$ to $2$, ...
3
votes
2
answers
416
views
Significant digits in high-school Calculus?
Significant digits are in the curriculum for both my Chemistry and my Physics class, and all my Calculus students have taken at least Chemistry. It also pops up in the school's Precalculus class (...
20
votes
3
answers
2k
views
What to do when a majority of students have insufficient mastery of the prerequisite material?
I'm currently teaching a second-semester calculus course in which a significant percentage of my students (over half) either tested into the course just out of high school, took a much easier first ...
20
votes
4
answers
1k
views
Evaluating the reception of (epsilon, delta) definitions
Both education researchers and mathematicians discuss the challenge of (epsilon, delta) type definitions in real analysis and the student reception of them. My impression has been that mathematicians ...
15
votes
6
answers
3k
views
Is there any difference between teaching calculus for math and engineering students?
In our university both math and engineering students attend in the same calculus classes. There are arguments in our department about the possible influences of this approach on students. It seems ...
4
votes
1
answer
132
views
Would chapter mini-projects be more beneficial than weekly homework for large classes n>100
I am trying something new this semester in my Calculus classes. I currently have over 400 students, and there is no way I can find the time to grade and give constructive criticism on all the ...
27
votes
6
answers
795
views
Would taking 5 minutes to explain the history behind a mathematical idea help stimulate learning the idea?
I read a paper in my "Research Issues in Mathematical Education" class that I have applied to the Undergraduate Calculus I and Calculus II class that I teach. I take five minutes to explain the ...
33
votes
14
answers
2k
views
Revisiting topics from previous courses [closed]
I teach calculus to students who have almost all taken calculus before. (Primarily first-year college students who took calculus in high school but didn't perform well enough to skip the course.)
...
27
votes
3
answers
1k
views
Counterexamples in first year calculus
Many believe (I think rightly so) that the presentation of counterexamples should play an important role in the teaching upper level mathematics courses such as real analysis and topology. ...