# Questions tagged [problem-design]

For questions how to design and create questions, exercises and examples.

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### Are there any benefits to having an entire course's homework problems available from day one?

I am designing a course for the upcoming fall semester, and I am tossing around an idea in my head. While planning which topics to cover each week and how to set the pacing of the course, I figured I ...
1k views

### Good examples of functions defined as definite integrals of elementary functions?

I am writing some Calculus content, and I would like a "big list" of useful functions which are defined by definite integrals, but are not elementary functions. Two examples of such functions are  ...
8k views

### How does one create “good” math problems?

As lifelong students of mathematics, we find problem-solving to be absolutely essential to enhance our understanding of the subject. Teaching others what we know serves to reinforce our existing ...
6k views

### Examples and applications of the pigeonhole principle

The Pigeonhole Principle (or Dirichlet's box principle) is a method introduced usually quite early in the mathematical curriculum. The examples where it is usually introduced are (in my humble ...
2k views

### “Correct the following mistake”-style questions?

Does anyone have any experience giving students incorrectly "solved" math problems and asking them to identify this error? Being self-critical is one of the skills that I would like my students to ...
987 views

### Good problems that uncover difficult points in a theory

There is a great quote of Yitz Herstein: The value of a problem is not so much coming up with the answer as in the ideas and attempted ideas it forces on the would-be solver." A number of such ...
2k views

### Good examples of Lagrange multiplier problems

I've noticed that most Lagrange multiplier problems I've seen can be solved with other methods. Often the method of Lagrange multipliers takes longer than the other available methods. I don't like ...
319 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 ...
229 views

### Problems which require interpreting definitions

I'm trying to find more problems suitable for early college students (students who know algebra and calculus) that involve translating words into mathematical notions. A nice example is this one: ...
908 views

### Good Examples of Questions to have Students Ponder Over Without Paper

A few times now I've found myself in a situation where I want to give a precalculus-level undergraduate student something to think about that's kinda fun and that's well outside of their coursework. ...
318 views

### Is there a framework to study the mathematical competence in problem-posing?

I've read some articles about it and those papers show us that there are some facts to understand the mathematical competence in problem-posing. Besides that, those investigations show that there are ...
606 views

### How to use false theorems or proofs?

I would like students to be critical and not believe that every proof they see is correct. Lecturers make mistakes and students should not think: "That must be a valid argument/proof/syntax because it ...
197 views

### Question about the process of creation of problems and exercises in Mathematics

I have a question about formulating problems and exercises in Mathematics. When attempting to create a problem of Number Theory or Real Analysis, for example, in this process, is the problem first ...
1k views

### Challenge questions for extremely bright kids

I suppose this is the place for my questions as much as any place is: I'm a math student coming on my 3rd year of undergrad, and I am working as a counselor at a Summer math camp. The camp is for 12-...
898 views

### Name the heuristic: exploiting the legitimacy of the questioner

As a child, I made frequent use of a particular 'trick' in order to make short work of many different problems. The general form is to be presented a question which wants a definite (numerical) answer,...
287 views

### Can number theory help me create equations with nice solutions?

I'm teaching a remedial algebra class, and I recently put a radical equation on a quiz. At this point, the students had only solved polynomial equations by factoring, so the equation had to turn out ...
1k views

### How does Project Euler come up with such good problems so rapidly?

Ever since I learned about Project Euler, I have been astonished and wondering about how Colin Hughes (the creator of Project Euler) manages to come up with such problems at such a rapid pace (once a ...
236 views

### Problems Without Posting Available Solutions

Is there evidence in the education literature to support or refute the claim that students who are given problems without solutions to work out on their own do better on different problems in the ...
410 views

### Grade 8 “Question of the Day” Ideas

I've begun a "Question of the Day" (QOTD) contest in my grade 8 class where I post a problem solving question on the wall each day and students are challenged to answer it in their own time. I need ...
2k views

### Applications of Calculus 2 to Physics

I'm teaching a section of Calculus 2 (integration techniques, arc length, surface area, improper integrals, parametric & polar functions, sequences, and series ) next semester and would like to ...
612 views

I am currently in a course titled "teaching reading in a content area." While there are plenty of examples of different strategies that can be used in different subject areas, there are little ...
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### Mathematical Task with Various Solutions

I need to come up with a mathematical task for middle school (9th grade), which involves either algebra, functions, probability or statistics (anything but geometry actually). My problem is, that the ...
451 views

### Examples where it easier to prove more than less [duplicate]

Especially (but not only) in the case of induction proofs, it happens that a stronger claim $B$ is easier to prove than the intended claim $A$ since the induction hypothesis gives you more information....
883 views

### Interesting but very easy epsilon-delta problems?

I am teaching a real analysis class. Students in the class have inconsistent high school algebra skills. They now have a complete but tenuous understanding of $\varepsilon$-$\delta$ limits. I want to ...
364 views

### Is there a repository with K-12 math exercises and problems written in LaTeX (English or Spanish)?

I've been searching for a repository or a database with math problems (K-12 levels) written in LaTeX, but with no luck. Neither in Spanish (my student's language) ...
178 views

In a traditional exam, there is a strong focus on facts and techniques. For instance, in a course on linear algebra, students are asked to diagonalize matrices and they have to check whether a given ...
194 views

### Topics for Discovery-based Projects

Are there any existing sources for topics or projects which are: Suitable for first and second year undergraduates taking introductory math classes (possibly with some prompting or a couple pages of ...
1k views

### Arithmetic / Algebra problem generation software?

I'm struggling to find any good "problem generation" software or sites. I want a program that will do the following: Generate typical problems at all sorts of levels (from toddler age through to teen,...
107 views

### What are the fundamentals of writing practice questions for early teen students?

I have to write some practice questions, or toy problems for some students in the age range 11 to 14. This is for a workshop I will be running. The students are encouraged to work on these questions ...
359 views

### A question about Vector Analysis problems

Why is it difficult to find really challenging vector analysis problems (problems about Green's, Stokes' and Gauss' theorems in a Calculus 3 course) in Calculus books? Most of the problems are ...
147 views

### The art of designing of problem sets

For proof-based math courses, the gist of the learning happens in problem sets and so it is essential to design them well. We would appreciate responses containing references (eg. from active learning)...
210 views

### Where to find good exercises for term operations?

I'm searching for exercises for practising operations with terms. They should involve working with decimal numbers and fractions (ideally one should convert decimal numbers to simple fractions like ...
52 views

### Project Based Learning or Applied Math involving modular arithmetics?

I am searching for ideas to construct Project Based Learning type that involves Modular arithmetics with eventually geometry and could fit in High School level What could a nice research question ...