Questions tagged [undergraduate-education]

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

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1answer
55 views

Calculus book for basic calculus and repetition from videregående

Does there exist a textbook in Norwegian (bokmål and nynorsk should both be fine) that covers the more advanced mathematics (needed for calculus) from videregående (secondary school, includes both ...
2
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1answer
67 views

Applied ODEs for Numerical Methods

I am looking for a list of ODEs to use as examples in the teaching of a numerical methods course for engineers. I am looking for first and second order examples - the more applied (to engineering) ...
4
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2answers
1k views

Asking students to define “unique”

Context: This is for introductory linear algebra course, near the beginning. As a sort of "exit survey" after one of my lectures, I would like to ask my students to try and define what "unique" is ...
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6answers
291 views

How should students say in words the notation for a limit?

$$\lim_{x\rightarrow a} f(x)=L$$ Which way should students best get in the habit of? The limit of $f(x)$, as $x$ approaches $a$, equals $L$ The limit of $f(x)$ equals $L$, as $x$ approaches $a$ The ...
3
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1answer
200 views

The most transparent exposition of Bayes' Theorem

I am seeking the most transparent exposition of Bayes' Theorem (for undergraduates). I would prefer to avoid mentioning "prior" and "posterior," and instead focus on frequencies. The Wikipedia entry ...
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4answers
364 views

Why is absolute value difficult?

My understanding is that students find absolute value to be challenging to learn or understand. Off the top of my head, I can come up with two possible reasons for this. Absolute value is a piecewise ...
16
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5answers
8k views

What is the most difficult concept to grasp in Calculus 1?

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 ...
12
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6answers
3k views

Is it a bad idea to offer variants of a final exam based on the type of allowed calculators?

Background/rant: I am in charge of teaching our single quarter course on vector calculus (don't ask me why the department head thinks the area can be covered in half a semester). The two biggest ...
10
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2answers
276 views

Introductory real analysis before or after introductory abstract algebra?

What are the pros and cons for students of taking introductory real analysis before or after introductory abstract algebra, assuming they are going to take both? I recognize that the overlap between ...
11
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4answers
353 views

Why are the contents of contest maths so different from contents of degree-level maths?

I wonder why topics examined in high school math contests are so different from the maths learned by those who are seriously studying a math major at a university. Firstly, contests like IMO, ARML, ...
8
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1answer
135 views

Motivation vs. Rigor

This is such a vague topic that I hesitate to post. I constantly struggle between the time-tradeoff between motivating a topic, and delving into the rigorous details necessary to fully "grok" the ...
9
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1answer
1k views

Diagram of Methods to Solve Differential Equations

I am currently trying to build a flow chart to visualize all tests there are to tell whether an ordinary differential equation is solvable and how to solve it. This is for tutoring purposes. The ...
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3answers
336 views

“Out of fashion” topics in degree level math

I just had a look at the curriculum of a university's math faculty 100 years ago. Most of the topics there are the same as the topics taught today, including complex analysis, differential equations, ...
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4answers
233 views

Inefficient methods

I see many teachers use slow methods to solve a given problem where there's another faster methods that doesn't demand much more effort. I'm not looking for mistakes like saying that $a$ is the slope ...
9
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4answers
204 views

Delivering mathematics lectures via tablet and projector

Owing to an injury, I need an alternative solution for delivering a semester-long mathematics course that I was originally going to teach using chalk and blackboard. It seems that a good option might ...
4
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2answers
144 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 ...
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3answers
417 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 ...
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3answers
261 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. ...
16
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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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2answers
239 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
252 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
140 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
224 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 ...
3
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1answer
117 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
166 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
102 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
216 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 $S \subset \mathbb{R}^3$&...
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1answer
225 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 ...
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2answers
109 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
180 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
217 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
198 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
202 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, ...
11
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2answers
261 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
90 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 ...
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3answers
223 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
122 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
137 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 ...
11
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8answers
438 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 ...
16
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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
209 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
262 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
635 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
149 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
133 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
426 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
130 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
113 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 ...
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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
177 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 ...