Brendan W. Sullivan
  • Member for 7 years, 10 months
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How can I give feedback that is not demotivating?
7 votes

I'm not sure how well this suggestion would fit into your specific course set up, but I want to share something that has been helpful for me in other contexts: I like to set a general theme/...

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Weekly quizzes as an alternative for midterms? What is this called?
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3 votes

This sounds like a method described in the following paper as second chance grading. The logistics are slightly different (less frequent exams with groups of topics, as opposed to weekly quizzes) but ...

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What topics could be covered in a course on fractals?
6 votes

Background: In my senior year of undergrad, I was a TA for our school's "Fractal Geometry" course, having worked with the professor before on a research project in fractals. We followed some ...

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What are some good ways to motivate and introduce reasoning abstractly about abstract algebra?
6 votes

In general, issues with abstraction and axioms can boil down to losing the ability to visualize what's going on. I'd suggest appealing to visual/tactile intuition as much as possible, especially in ...

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Content for a 40-minute lecture on graph theory for high schoolers
12 votes

I study pursuit-evasion games on graphs, so I will recommend using the cops & robbers game as a way to introduce graph theoretic terminology, concepts, and examples. It should also keep the tone ...

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Most popular setups for recording video lectures
4 votes

I will share what I have been doing. I cannot claim this is popular, nor that it is necessarily the best, feasible option for everyone. But I know that when I started searching online for suggestions, ...

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Patterns that unexpectedly fall apart at large $n$
9 votes

I second Sue Van Hattum's suggestion that you should not be so concerned with how large the $n$ is where the pattern eventually fails. I'll go one step further and recommend an example where that $n$ ...

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How can I teach my students the difference between a sequence and a series?
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41 votes

I have no evidence to back this up, nor do I know how I could obtain such a thing. But I strongly believe this is a vocabulary issue, and I would like to see the term "series" phased out of usage in ...

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Writing up a proof that assumes what is to be proven?
10 votes

I always provide the following example whenever a student assumes what they want to prove: Suppose 0=1. Then 1=0 must be true. Then we can add both equations to deduce that 1=1. This is a true ...

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What are some common ways students get confused about finding an inverse of a function?
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6 votes

I've noticed a few issues when students solve problems of the form, "Find the inverse of this function", and not all of the issues are necessarily because of the students' misunderstanding of what an ...

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'Low-algebra' examples of induction
2 votes

How about the Tower of Hanoi puzzle and finding the optimal number of moves? This link describes the recursive solution procedure and a proof of optimality using induction. https://proofwiki.org/...

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What are some good low-prerequisite examples for the heuristic advice "If you cannot prove it, prove something stronger."?
2 votes

I think the Squeeze Theorem can provide a good opportunity for this phenomenon. The example I describe below arose in a Real Analysis course for upper-level mathematics majors that I taught this ...

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Drawing vs Constructing
7 votes

I would emphasize the algorithmic nature of a construction over the ad hoc nature of a sketch/drawing. Tell your students that a construction must be accompanied by a specific sequence of steps (a "...

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Comparison Tests in Calculus
6 votes

I believe option #2 (teach both and methods for choosing) is best. I'll try to illustrate with examples. $\displaystyle{\sum_{n=1}^\infty \frac{n}{n^2+1}}$ The intuition (which most students see ...

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Motivation for uniform continuity
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3 votes

First, I think this is a great question for this site because it made me question how I teach uniform continuity in my own classes and sent me on an exploration for better ideas. So, thanks! Now, to ...

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Neat topics or problems to include in a probability class
3 votes

I suggest the probabilistic method in combinatorics and graph theory, for a few reasons: This will introduce the students to other mathematical concepts that they may not be aware of already, but you ...

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An application of the Cauchy criterion for undergraduates?
1 votes

One application is infinite series, since they are characterized by their sequences of partial sums. First, use the Cauchy Criterion for Sequences to establish a similar result for series. Then, use ...

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Interesting but very easy epsilon-delta problems?
Accepted answer
6 votes

I suggest using rational functions. Students are used to evaluating limits of rational functions because such examples are prevalent in most calculus courses. Moreover, I think the work required to ...

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Teaching undergraduates who expect a high-school-like learning environment
2 votes

This may seem strange to offer an answer to my own question after just a week, but I did find a useful resource that may be helpful in addressing the issues described in my question. This week, I have ...

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Problems which require interpreting definitions
2 votes

I'm trying to find more problems suitable for early college students (students who know algebra and calculus) that involve translating words into mathematical notions, [...] problems which seem ...

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Teaching the proper syntax of “such that”
2 votes

I suggest having a class discussion about formal logical phrases and symbols, and how to translate those into natural English. For example, your student wrote this For a and b to be relatively ...

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Ideas for a 2 weeks project focused in polynomial functions
0 votes

Since a precalculus is, by its nature, designed to prepare students for calculus, I recommend gently introducing students to the concepts of calculus using the tools they already have. In this case, ...

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Is there a simple real-world problem I can use to motivate a formula for $\displaystyle \sum_{i=1}^n i $?
9 votes

Perhaps you can make the topic "applicable" by comparing it to some popular social media posts of the form, "How many triangles are in this figure? Only geniuses get this right!" Take a $5\times 1$ ...

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Resources for Inquiry-based Projects with Undergraduates
3 votes

Student Research Projects in Calculus Cameos For Calculus I particularly like the first one because the authors include with each project a description of how long it may take a student, any issues ...

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Good metaphor to explain the difference between pointwise and uniform convergence
3 votes

Other answers suggest sticking to the definitions and just teasing out their logic carefully. I agree that this is a good method and you should mostly stick to this. @Behzad's answer with the picture ...

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Topics for a general education course
2 votes

I teach, essentially, a "math course for non-math students who need to fulfill a college requirement" and have found great success with topics that don't seem like math to the students but certainly ...

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Do iPhones help students in their math class?
6 votes

I teach undergraduate students (ages ~17-22), with class sizes ranging from 10 to 30. I have come to realize that students will have their phones with them no matter what, so it would be unnecessarily ...

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Do students confuse $\log_ab$ and $\log a^b$?
9 votes

I don't believe students subconsciously misread the notation as an exponent with a word in front. I think they are not sure at all how to read the notation! When using logarithms (whether in a basic, ...

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Entertaining examples of multiply quantified statements
1 votes

This one might take some extra massaging to make it "mathy" (re: "half" and "part" and "some"), but I like it as a fun twist on Abe Lincoln's classic (which @...

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Explanation of Counter-example?
9 votes

I believe this depends on how clearly the counterexample is stated. Consider this claim: $f:\mathbb{R}\to\mathbb{R}$ is continuous $\implies f$ is differentiable. Imagine a student, let's call him ...

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