Questions tagged [undergraduate-education]

For questions about teaching students at the undergraduate (university) level.

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1answer
73 views

How to intuitively convince the students that a strip with two full twists is homeomorphic to the standard annulus?

Intuitively speaking, one space is homemorphic to another if one can be deformed continuously to another without tearing and gluing. It is more or less easy to convince the students that a square is ...
5
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3answers
228 views

What's the point of exercises without answers?

What is the point of exercises for which answers aren't provided? (That is to say, what is the pedagogical justification for such exercises? - Edit by someone other than original poster.) Commentary ...
6
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2answers
138 views

MacLane-Birkhoff's “Algebra” vs Jacobson's “Basic Algebra I,II” vs Lang's “Algebra”

(Cross-posted at Math.Stackexchange) I'm searching for an apt textbook(s) on Abstract Algebra for a very advanced undergraduate/graduate level course in Algebra, and would be grateful for any help. ...
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0answers
62 views

A KinderGarten of binomial coefficients [closed]

I tell the beginners that $(-n)!= \pm \infty$, if $n \in N$. Next, I tell them that the definition of ${\nu \choose k}$ as $$ \frac{\nu (\nu-1) (\nu-2) (\nu -3) ...(\nu-k+1)}{k!}~~~~(1)$$ is the most ...
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12answers
7k views

Is there a simple example that empirical evidence is misleading?

Suppose that I want to show a student that empirical evidence in mathematics is not enough and we do need proofs, what kind of examples can I use? By empirical evidence, I mean that (most of the time)...
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3answers
223 views

How to justify that students should come to class?

Nowadays, a student should be able to learn the course material at home through reading the textbook or follow one of the many free online courses. Some universities record video or audio of lectures ...
3
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6answers
248 views

What is an interesting high-school level topic to discuss using Mathematica or Geogebra?

I have to choose a topic to give a presentation. The topic should be high-school level or at most Linear Algebra 1 and Calculus 1. Conics and polygons in the Euclidean geometry are some fine topics ...
2
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1answer
131 views

Naming arbitrary constants: subscripts, primes, or just more letters?

When choosing names for arbitrary constants either during a lesson or while working with a single student, should one use$\{n_1,n_2,n_3,\dotsc\}$ or $\{n, n', n'', \dotsc\}$ or $\{a,b,c,\dotsc\}$? Is ...
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0answers
187 views

Use of Lockhart's *Measurement* in a course?

I greatly admire Paul Lockhart's Measurement (Harvard Press). Many of you know him through A Mathematician's Lament. One review of Measurement said, “Here Lockhart offers the positive side of the ...
4
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1answer
104 views

What is the notation for polynomial long division in Norway?

I will be teaching a calculus-type course in Norwegian. Our textbook is unfortunately in English (the curse of a small language), but any custom exercises should be and all exams have to be in ...
5
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0answers
163 views

Mathematical undergraduate education in Syria

I'd like to learn some things about undergraduate mathematical education in Syria EDIT: In particular I'm interested in students 15-16 years old. What are the main differences from the European ...
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1answer
99 views

What should I say about elementary number theory?

I need to give an option talk about elementary number theory module. I will discuss how it is study of positive integers particularly the primes and give some cryptography applications. What is a good ...
6
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3answers
199 views

Why is it difficult to freely change between points and vectors?

I have noticed working with bright undergraduates that it is not uncommon for them to have difficulty easily converting between a point—say, a point $p$ on a surface $S \subset \mathbb{R}^3$&...
7
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1answer
141 views

Best practices in teaching math to future elementary teachers

This question is about references in current best practices in teaching math to future elementary teachers at a university level. I am asking it because I do not see any question so far on this site ...
1
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2answers
102 views

Lower-division complex analysis textbook

I'm looking for recommendations for a good textbook to use for a hypothetical lower-division course in complex analysis, at a level of sophistication comparable to a second or third semester course in ...
0
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1answer
172 views

Future in mathematics

My sibling is done with high school and has always scored A in Maths and am not in position to advise her on the future in line of her niche. She's not yet in university and she's in her vacation but ...
8
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3answers
204 views

How to motivate students to do proofs?

I am finding it difficult to motivate students on why they should how to prove mathematical results. They learn them just to pass examinations but show no real interest or enthusiasm for this. How can ...
4
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3answers
195 views

Analogy for multiplying modulo N

Sometimes I want to explain to laymen/new students/laywomen how addition modulo N works. There are some instructive analogies: Addition on the clock (12), Addition on weekdays (7). They illustrate the ...
2
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2answers
197 views

Learning proofs in introductory analysis courses

I have browsed the website a lot and I encountered many similar questions but not a question that asks the same question as I intend to. In introductory undergraduate classes in Analysis, usually, ...
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2answers
259 views

Alternating group without $S_n$

I'm going to start introducing my abstract algebra class to a variety of groups soon. Dihedral groups $D_n$ arise out of symmetries on polygons. And the Symmetric group $S_n$ makes sense as the group ...
3
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1answer
83 views

In a typical 3rd-semester multivariate calculus course in the US, what kind of area integrals do students actually learn to do?

I teach mostly physics and a little math at a California community college. I've never taught the multivariate calculus course, but I have taught the electricity and magnetism course for which the ...
2
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3answers
204 views

Geometry textbook with an abstract algebra emphasis

I'm teaching a variety of undergraduate and graduate geometry classes (mostly for in-service teachers) which range from elementary axiomatic geometry to more advanced transformational geometry. I'm ...
7
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1answer
118 views

Difficulty in teaching the coordinates of a vector with respect to a basis $\{v_1,v_2,\ldots,v_n\}$

Let $V$ be a finite dimensional vector space and let $B=\{v_1,v_2,\cdots,v_n\}$ be a basis of $V$. If a vector $v$ can be written as $$v=a_1v_1+a_2v_2+\cdots+a_nv_n,$$ we call $(a_1,a_2,\cdots,a_n)$...
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2answers
128 views

Studies about group tutoring sessions

I’m not sure if this question belongs here, so I apologize if it doesn’t. I work in a tutoring center at my university where we tutor every subject. Mathematics is in high demand, and occasionally my ...
10
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8answers
414 views

How does knowing more about mathematics help one's teaching of lower level course, such as calculus?

A question has been asked about why great mathematicians are not necessarily great teachers. On the other hand, I am wondering if knowing more mathematics actually helps with one's teaching of lower ...
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6answers
5k views

How to deal with a “protest” assignment?

I just received one assignment (by email) from a student. Out of 6 questions, "I don't know" is the answer to 4 of them. There is also a comment at the end of the assignment which suggests my ...
5
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3answers
192 views

Calculus workbook suggestions

Context: I am an assistant professor of mathematics at a small institution in the US. Our department uses Stewart's Essential Calculus for our calculus sequence, but I find that my students and I are ...
8
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6answers
241 views

Book recommendations on mathematics education focusing on geometry

I will be teaching Euclidean geometry to future teachers, and I am feeling a bit lost (I know geometry, but I am not that familiar with mathematics education). Is there some recent (as concise as ...
6
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5answers
620 views

Is it a problem if a senior student majoring in mathematics could not prove the quadratic formula?

According to a recent experiment conducted by user Steven Gubkin, nearly one half of his students in a senior level Real Analysis course do not have any idea how to prove the quadratic formula. Is ...
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2answers
139 views

What is a good format of tutorial sessions?

At my university, traditionally a few lectures of a course should be tutorial sessions. The idea is that instead of covering new materials, the teacher should go over many exercises so that student ...
5
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2answers
123 views

Published papers for Intro Stat students to read

I am looking for studies and experiments in the literature that I can share with undergraduate students in an intro statistics course. I do not expect students to understand everything, and I plan to ...
5
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1answer
421 views

How to balance the difficulty level and speed of lectures for students of very different levels?

I noticed that in my undergraduate class a few students understand things quite fast and some times see the proof before I even explain things. But some of them also have trouble understanding quite ...
4
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2answers
128 views

How to explain linear approximation to an equation to calculus students?

I am, at the moment, teaching calculus to students whose majors are, for example, biology, biochemistry, chemistry and geology. The course book is Claudia Neuhauser's "Calculus for biology and ...
2
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2answers
107 views

Is there any alignment on what a maths grad should know?

This more specific question relates to a more general question of what is a maths degree aiming for. Do any universities define a high level goal for pure mathematics degrees at all? If so, are there ...
8
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3answers
2k views

Are students majoring in pure mathematics expected to know classical results in mathematics very well by their graduation?

For example, I am confident that very few students majoring in pure mathematics can write a complete proof to the Abel–Ruffini theorem (there is no algebraic solution to general polynomial equations ...
6
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2answers
165 views

Vector calculus texts that are free-as-in-speech?

I'm looking around for a text that covers vector calculus and multivariable calculus, and that is also "free as in speech," not just "free as in beer." In other words, I'm looking for texts that are ...
2
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0answers
85 views

Tactile Learning Activities in Mathematics

Julie Barnes, Jessica M. Libertini. Tactile Learning Activities in Mathematics: A Recipe Book for the Undergraduate Classroom. 2018. MAA Press. AMS Bookstore link. Can anyone comment / review ...
15
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7answers
385 views

Should theorems be proved to students who are not majoring in mathematics?

My impression to students majoring in mathematics is, whenever we teach them a theorem, a proof should be given in the class, or at least as a reading assignment. However, how about students not ...
4
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0answers
65 views

Long-form, multi-step, skills-integrating applied mathematics problems in calculus I, II, III

When recently teaching Calculus II to college students, I instructed my students to read and be ready to work through the first 8 or so questions of James Walsh's climate modeling differential ...
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4answers
291 views

How to deal with students who object to me teaching material that won't be in the exam?

I sometimes encounter students who ask questions like 'Why are we learning this if it won't be on the exam?' If there is time to spare I like to teach interesting applications or natural extensions of ...
6
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1answer
189 views

Measure of Improvement in Math Skills from Remediation with ALEKS

I am analyzing some data on ALEKS for my home institution (ALEKS, an acronym that stands for "Assessment and Learning in Knowledge Spaces", is an online tutoring and assessment program that includes ...
4
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4answers
271 views

Recommended list of things calculus students should be required to memorise?

I am seeking a list of topics that students taking calculus should memorise. Some topics from Calculus I might include: $\varepsilon-\delta$ definition of limit; Definition of the derivative of a ...
0
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1answer
155 views

Grades in a university course on category theory, curving, and how they reflect on the students and/or teacher [closed]

I originally posted this on the Mathematics Stack Exchange, thinking that the best place to post it, but the question quickly accumulated a bunch of close votes since it was not quite within the scope ...
3
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2answers
87 views

Quadratic modeling project with upward-facing parabola

I'm teaching a college algebra course and I'm trying to design a few projects that involve modeling with quadratic functions. So far I have two ideas that involve downward-facing parabolas (projectile ...
4
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2answers
76 views

A Markov chain demonstration that doesn't require computers

I have a large probability class and would like to do some memorable demonstrations of Markov chains for them. Does anyone have any recommendations for a "low-tech" demo that doesn't involve ...
1
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1answer
124 views

When self teaching, should I learn set theory before continuing ap calculus?

I am studying ap calculus now, before I move onto differential equations etc., but the thing I am unsure of is, should I learn set theory before continuing on my ap calculus sections?
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7answers
2k views

In which course should we teach solving general cubic and quartic equations?

I am guessing solving general cubic and quartic equations should be taught in a course somewhere between precalculus and Galois theory, though personally I do not recall learning this topic ever in ...
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2answers
177 views

Topics in Mathematics for a 15 minute demonstration

I need to appear for an interview for the post of Assistant Professor in Mathematics in an undergraduate college. My Backgorund : I have studied topics like Algebra comprising of Group Theory,Ring ...
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3answers
1k views

Is it a bad idea to use an old textbook such as Differential and integral calculus, with examples and applications for calculus course?

I am wondering if it is a bad idea to use an old textbook, such as Differential and integral calculus, with examples and applications by George A. Osborne. This book was published in 1906 and there ...
5
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3answers
171 views

How to introduce Group Theory to a general audience in 15 minutes?

How to introduce Group Theory to a general audience in 15 minutes? I know that it will be quite tough to introduce Groups to a general audience in such a short time. So what will be a good way to ...