All Questions
3,559
questions
43
votes
28
answers
11k
views
Good, simple examples of induction?
Many examples of induction are silly, in that there are more natural methods available. Could you please post examples of induction, where it is required, and which are simple enough as examples in a ...
- 12.2k
14
votes
3
answers
624
views
Why teach back substitution with row reduction?
Many linear algebra books include two versions of row reduction for solving systems of linear equations:
(1) Reduce to echelon form, and then use back substitution.
(2) Reduce to reduced echelon ...
- 8,079
7
votes
2
answers
157
views
When is it more advantaged for students to stop attending lectures? What to do then?
Under what situations is it more productive for math students to stop attending lectures and learn the material alternatively? What are some remedies?
I originally posed this as an Example Question.
user155
14
votes
6
answers
228
views
Books/(auto)biographies/references on how mathematicians study/studied (as students)?
As Geoff Pointer commented:
[...] As a composer I've learnt a lot from studying famous composers why wouldn't that also apply to studying maths and mathematicians of note as well? [...]
Are there ...
user155
4
votes
1
answer
79
views
What are ideas and strategies on improving at discovering counterexamples? [closed]
What are ideas and strategies on improving at discovering counterexamples?
I originally posed this as an Example Question.
user155
2
votes
0
answers
79
views
Where to find resources about study skills and strategies of immediate impact? [closed]
Where can research and resources be found, about study skills and strategies for university math of immediate impact? To wit, how do I efficiently find categorical, pragmatic advice about learning/...
user155
28
votes
6
answers
905
views
How can we help students who are very anxious about math?
In many parts of the world, the majority of the population is uncomfortable with math. In a few countries this is not the case. We would do well to change our education systems to promote a healthier ...
- 19.2k
14
votes
2
answers
596
views
Splitting the students by abilities
Disclaimer: This question will use as an example a computer-science university program, because it is much easier for me to describe and give some statistics, but it applies to math as well. The ...
- 8,807
29
votes
7
answers
2k
views
Good definition for introducing real numbers?
In the first lectures about calculus/analysis, you should introduce real numbers. Let's assume students know that rational numbers are.
What are the advantages or disadvantages in the different "...
- 9,122
20
votes
2
answers
273
views
Pressure vs. Laissez-faire: Literature dealing with balance in university-classes
I am seeking for some pedagogical literature dealing with the following question: Imaging you have an average class in college/university: What is a good balance between Laissez-faire and pressure?
...
- 9,122
23
votes
4
answers
2k
views
Non-answerable questions on exam: What to do?
What is a good strategy when you realize (e.g. while grading the exam) that a question on an exam was incomplete/wrong?
More concretely:
If it is decided that additional points should be given:
How ...
- 9,122
18
votes
5
answers
8k
views
Best textbooks to introduce measure theory and Lebesgue integration?
What are the best textbooks to introduce measure theory and Lebesgue integration to undergraduate math majors? Many students in such a class will go on to graduate school and be required to take a ...
- 1,376
45
votes
9
answers
3k
views
Knowing mathematics does not translate to knowing to teach mathematics. Why?
Many brilliant mathematicians seem to make average or even poor classroom teachers. Is this an accurate assessment? Has there been any research to explain the phenomena?
What is the difference ...
- 878
18
votes
5
answers
381
views
Is required reading of the text effective, and how can it be assessed?
This will likely depend on the class, of course. But I've asked calculus students in the past if (a) they regularly read the textbook and (b) whether this is helpful for them and (c) whether they like ...
- 10.5k
26
votes
2
answers
1k
views
Students who know high-level math before going to college
There is a high school in the city I live in which has some high-level math courses in their curriculum. It's a special math class mentored by some university lecturers, and the children basically do ...
- 8,807
25
votes
7
answers
3k
views
What arguments can I give a high school student why mathematics is important?
In almost all countries all over the world, mathematics is a main subject in school. Maybe the subject bringing trouble to families with kids.
It is clear that scientist, engineers, etc. need ...
- 9,122
40
votes
6
answers
14k
views
How can I estimate the length of an exam?
Background: I am fairly new at teaching, and in every subject I have taught, I have had difficulty estimating the length and difficulty of an exam. I need to write an exam for a university senior-...
- 11.3k
29
votes
8
answers
2k
views
Good motivation for the introduction of Lebesgue integral?
When students take a course on real analysis, they have likely learned about Riemann integrals.
What is a good motivation why they have to learn a new way to integrate?
A student don't want to hear ...
- 9,122
31
votes
6
answers
867
views
Alternatives to University Lectures: Non-lecture Mathematics Classes
I am looking for resources for designing undergraduate mathematics classes that are not lecture-based. (Bonus points if the design is for an introduction to proof course).
For example, Robert ...
- 3,741
36
votes
13
answers
3k
views
Examples why university education is important for future high school teachers
At my university, the students in math are mixed up (1/3-1/2 are bachelor/master students, the rest are future high school teachers). A problem arising very often is the discussion dramatically ...
- 9,122
19
votes
6
answers
9k
views
Answers in exact form (e.g. including radicals) vs. Decimal Approximations
I was tutoring a student on early trigonometry. Solving for the hypotenuse of a right triangle, but with sine, not Pythagoras.
The student went through getting the sine of a 45° triangle, and gave ...
16
votes
6
answers
4k
views
What is the ideal course sequence for an advanced student of mathematics?
Suppose that you meet a student who:
has a firm grasp of algebra and trigonometry and is at least moderately intelligent
has read a book such as Love and Math by Edward Frenkel so has some ...
- 10.4k
13
votes
3
answers
337
views
Any suggestions on how to approach recursion and induction?
Much mathematics is intimately tied to recursion, be it in definitions (like factorials and integer powers) and proofs by induction. This is also very relevant in computer science and programming. ...
- 12.2k
18
votes
2
answers
641
views
Comparison of different concepts of integral
As the following math stack exchange question (and answers) show:
https://math.stackexchange.com/questions/703212/is-dxdy-really-a-multiplication-of-dx-and-dy
There are a lot of different ways to ...
- 1,748
15
votes
2
answers
251
views
How to choose/test future tutors?
When I have to give a big lecture with a lot of exercise groups, I have to choose the tutors who are grading the student's homework and explaining the lecture to the students.
How can I find out if ...
- 9,122
18
votes
3
answers
1k
views
How to deal with very motivated students having "off-topic" interests?
There are some very motivated students who are very interested in math (in general), where the interest takes over most of their time. The problem is that they don't put enough time in the lecture ...
- 9,122
22
votes
9
answers
1k
views
Is it good to have solutions of homework published?
At a course at the university, the students have to do homeworks every week which will be graded and discussed in exercise groups.
Is it a good idea to put "official" solutions of the homework on ...
- 9,122
11
votes
1
answer
244
views
How to design exercise groups in very advanced course?
In very advanced courses, typically there are only a few students attending and sometimes they are not willing to put so much efford into the course to participate or to pass an exam.
So I made the ...
- 9,122
22
votes
9
answers
2k
views
Any tips in explaining the central limit theorem in statistics?
Many of my students don't have a mathematical background, and are not comfortable with concepts such as limits, random variables and distributions. Is there any intuitive way of explaining why the ...
- 323
9
votes
4
answers
302
views
What activities can enhance student comprehension of concepts involved in logarithmic and exponential integrals?
I will be teaching a calculus class, specifically, integration of common functions (e.g., polynomials, logarithms, exponentials and the like). It has been my experience that if an abstract concept ...
user106
20
votes
3
answers
499
views
When is it appropriate to lecture?
I take it that lecture is rarely the most effective way for a student to learn. Lecture is a case where, I believe, research on learning firmly backs up the common experience that lecture rarely helps ...
- 1,713
20
votes
4
answers
418
views
Multiple Solutions Methods vs. Encouraging a Particular Approach
It happens frequently in math that problems have multiple possible solutions. This might become troublesome, e.g. when students use some other approach, hence, not learning the current topic.
One ...
- 8,807
12
votes
2
answers
240
views
How do I adapt a MWF class into a TuTh class?
Both universities I have taught at have tightly-stuctured calculus courses that follow a pattern of 3 sections of Stewart Calculus every week, with some sections afforded two days.
I find it ...
- 11.3k
13
votes
1
answer
219
views
Teaching functions/mappings early
Functions and mappings are usually introduced late in the curriculum, and functions of arity two or more are considered "advanced" (many don't even see them before college).
On the other hand, the ...
- 8,807
14
votes
1
answer
162
views
How do online practicing websites decide how much practice / repetition is 'enough'?
I've worked with a few online programs that present students with drills / exercises / etc. for a period of time before deciding that they've 'done enough' for the day (e.g. Reflex Math). How do these ...
- 483
8
votes
4
answers
291
views
Multidimensional differentials for students with poor spatial imaging
When teaching multidimensional differentials (I'm assuming the students grasped the one-dimensional case), there are many useful parallels relating to spatial imagination.
For example, when ...
- 8,807
30
votes
7
answers
16k
views
What are the comparative advantages of open-book versus closed-book exams?
I would like to know the advantages and disadvantages of open-book exams as compared to closed-book exams, particularly in standard undergraduate courses like calculus or linear algebra.
My practice ...
- 4,036
20
votes
4
answers
224
views
How do you handle a wide ability range when delivering a 50 min tutorial with lots of material to get through?
So far I've tried building my presentation from elements, each of which is differently paced.
A lot of tutorials are given just by rapidly writing fully worked solutions on the board, thereby leaving ...
- 303
23
votes
2
answers
1k
views
Should geometric algebra be presented early on in undergraduate education?
The Cambridge University GA Research Group’s website along with the “Geometric Calculus R & D Home Page” should serve as a good introductions to geometric algebra, along with the Wikipedia ...
user89
23
votes
4
answers
605
views
Keeping quicker students engaged and interested throughout a course
In a college math course one is bound to find a fairly broad range of students in terms of their quickness in understanding the material. This is due to many reasons, including differing mathematical ...
- 2,093
83
votes
19
answers
28k
views
What is a good handwriting font for mathematics?
My students frequently mix up my $t$'s with my $+$'s and my $y$'s with my $4$'s.
What is a good handwriting font for distinguishing these and other easily confused symbols?
- 11.3k
13
votes
2
answers
425
views
What prerequisites would college students need for a course based primarily on Euclid's elements?
I love Euclid's elements, and would like to base a course around them. Before I can pitch it to my supervisors, I need to know where it would fit in the curriculum. While it begins from elementary ...
- 11.3k
23
votes
13
answers
5k
views
What is a free and simple 3D plot software for students?
I need any plot software on Linux or Windows that my students should use it for plotting 3D functions.
I want introduce any software that be free and useful for bachelor students.
- 633
14
votes
3
answers
268
views
According to Common Core standards, what math skills are beginning Kindergarteners supposed to have?
I remember looking once at what chikdren in Kindergarten were expected to know, and it was quite a bit. I have a young son, and would like to know:
What is a Kindergartener expected to know about ...
- 11.3k
14
votes
2
answers
358
views
Engaging students in computer lab
I am currently teaching the workshop for a class on chaos and fractals in a computer lab. The class is predominantly first year, first semester university students.
Worksheets have been developed for ...
- 796
19
votes
1
answer
474
views
The "water triangle" proportional reasoning task
(I previously asked this at The Mathematics Teaching Community, but I'm hoping it would attract further answers here.)
The Wikipedia page on proportional reasoning mentions a "water triangle"...
- 10.7k
13
votes
2
answers
125
views
How to bring an undergraduate researcher up to speed on a brand new topic
I am doing most of my research in a relatively new area of mathematics called finite subdivision rules. I have an undergraduate student who has begun doing research with me on some of the properties ...
- 11.3k
15
votes
8
answers
7k
views
Visual Pythagorean demonstration
I know that there is a visual demonstration of $a^2+b^2=c^2$ using a smalĺ piece of paper, but there are also a lot of variations.
Which visual or drawing demonstration of the Pythagorean theorem can ...
- 261
27
votes
5
answers
3k
views
What are some good examples to motivate the implicit function theorem?
I always had problems teaching the implicit function theorem in advanced analysis courses. This result is motivated by later applications, but it would be great to provide easily accessible examples ...
- 4,109
40
votes
11
answers
10k
views
Formula sheets and books during tests and exams
Some teachers make memorizing formulas, definitions and others things obligatory, and forbid "aids" in any form during tests and exams. Other allow for writing down more complicated expressions, ...
- 8,807