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 ...
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 ...
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.
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 ...
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.
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/...
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 ...
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 ...
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 "...
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? ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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-...
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 ...
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 ...
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 ...
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 ...
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 ...
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. ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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?
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 ...
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.
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 ...
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 ...
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"...
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 ...
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 ...
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 ...
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, ...