# All Questions

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In an exam we have Question (5points) find $\lim_{x\to\infty}(x-\sqrt{x})$. A student answered: $\lim_{x\to\infty}(x-\sqrt{x}) =\lim \sqrt{x}(\sqrt{x}-1)=\infty \cdot\infty=\infty$. My question is:...
10k views

### Should we stop differentiating between ln and log?

In many U.S. middle schools and high schools, $\ln$ and $\log$ are treated differently, with the intent that $\log$ is equivalent to $\log_{10}$. However, in undergraduate courses and in the academic ...
15k views

### Given a 3 4 5 triangle, how do you know that it is a right triangle?

Without knowing the Pythagorean theorem, and in presenting reasons why the theorem might be true (without giving a full proof), is there any way to give examples of triangles that are intuitively ...
3k views

### Impressive examples where a “proof by picture” goes wrong

There are many proofs where the whole idea can be expressed by a picture and often naturally translated into a correct formal proof. Often one has to argue with students that a picture is not a proof ...
3k views

### Should we teach functions as sets of ordered pairs?

The context of this question is an "introduction to proofs and mathematics" class for freshman/sophomore math majors. Most textbooks for such a class say something about functions between arbitrary ...
2k views

### How to solve the problem of Wolfram Alpha?

I teach to predominantly non-majors in college algebra, precalculus, and calculus. How can one possibly incentivize or rationalize assigning practice problems outside of class when this software is ...
2k views

### Unusual applications of integration

I am trying to teach my calculus students to apply integration by thinking about what they are integrating rather than just applying formulas. Calculus books are full of formulas like "to find the ...
683 views

### Would taking 5 minutes to explain the history behind a mathematical idea help stimulate learning the idea?

I read a paper in my "Research Issues in Mathematical Education" class that I have applied to the Undergraduate Calculus I and Calculus II class that I teach. I take five minutes to explain the ...
5k views

### What websites allow students to purchase solutions to problems?

I am a college instructor who's just had an outbreak of academic dishonesty connected to students posting take-home exam problems on a platform called Chegg. Chegg collects a membership fee from ...
863 views

### Counterexamples in first year calculus

Many believe (I think rightly so) that the presentation of counterexamples should play an important role in the teaching upper level mathematics courses such as real analysis and topology. ...
639 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 ...
2k views

### How should LaTeX be taught to university students?

There are several groups of people that would benefit from learning LaTeX in college. Future teachers can use it to write exams, scientists and mathematicians can write papers, and everyone can write ...
8k views

### How to justify teaching students to rationalize denominators?

I'm teaching an "intermediate algebra" college course ($\approx$ junior high school or beginning high school algebra) and we have a bunch of problems on rationalizing denominators. How do I motivate ...
3k views

### How can I teach my students that other disciplines are important too?

I'm sorry if this is the wrong place to ask this but I don't know any other place to ask. I'm a research professor, but I enjoy teaching and put a lot of time into my classes. I mainly teach ...
7k views

### How does a teacher come up with plausible wrong answers for multiple choice tests?

When taking a MOOC in calculus the exercises contain 5 options to select from. I then solve the question and select the option that matches my answer. Obviously only one of the options is correct. But ...
2k 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 ...
2k views

### What is the quantitative data on effectiveness of “modern” teaching methods?

What research has been done on how much and in what circumstances various non-lecture types of teching are effective with regards to student knowledge and performance? Meta/review studies preferred ...
3k views

### Common Core, threat or menace? Or maybe ok after all?

I have a background in math but no contact with secondary education or kids. I hear all sorts of stories ... horror stories mostly ... about the Common Core math curriculum in the USA. Then I hear ...
3k views

### How do you go about writing your own lecture notes for a new course?

In advanced graduate courses, there are often no textbooks on the material being covered, and sometimes no good introductory material at all. Even when there are textbooks, many professors prefer to ...
1k views

### How to write like a mathematician?

I learned to do math proofs in college. But recently I have begun studying more advanced math books and I've noticed some mathematicians frequently make assumptions that I don't. When I write proofs, ...
2k views

### How to Teach Adults Elementary Concepts

I've recently taken on the task of helping out in my school's Math Center. The courses I assist in range from Algebra to Calculus. While I'm younger (in my 20's), most of the students at the school ...
2k views

### Teaching students to find and correct their own errors

Many students have a fairly good grasp of the topics they are learning but fall down because they miss fatal errors in their work. Some don't check for errors at all, while many simply can't find them....
4k views

### When did US mathematics programs start failing to prepare incoming students for books like “Baby” Rudin?

I've seen in a lot of questions about "which textbook to use for intro analysis", and inevitably Rudin's Principles of Mathematical Analysis comes up, with the (almost cliche) rejoinder that "today's ...
1k views

### What methods successfully identify and eliminate severe math anxiety?

What methods are effective in identifying and eliminating severe math anxiety, this most terrible and unfortunate part of modern mathematics education? This question is not about ordinary math anxiety ...
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 ...
1k views

### What do you do when you realize mid-lecture that your lesson plan is not working?

I had this happen today, and several times before. I had had difficulty preparing the material, because I knew my students strengths and it didn't seem suited to them. However, it seemed the right ...
6k views

### Why would you teach Calculus before teaching Real Analysis?

Let's assume our students are actual aspiring mathematicians. Why would we introduce our students to Calculus rather than Real Analysis? After all, "Calculus is a subset of Real Analysis". He will ...
3k views

### A Non-Unique Factorization of Integers!

I'm going to introduce my students to the fundamental theorem of arithmetic (uniqueness of integer factorization to prime factors), and I don't want them to take the uniqueness for granted! To make my ...