41 votes

Can mathematics be learned by ONLY solving problems?

Such an approach seems designed to force (or at least, strongly encourage) students to learn by pattern-matching from examples. This is one of three modes of student learning in mathematics described ...
Daniel Hast's user avatar
  • 4,893
27 votes
Accepted

Students understand during course but can't solve exam

Do NOT give exam questions that are intentionally more challenging than homework or in-class problems. I would recommend precisely the opposite. The point of the exam is really a spot-check that ...
Daniel R. Collins's user avatar
15 votes

What will happen if someone doesn’t make it to the IMO team?

You have posted two questions on this website so far. Both of them show what is (to me) an unhealthy focus on Olympiad problems. Firstly I want to remind you that you have value as a living being. ...
Steven Gubkin's user avatar
14 votes

Can mathematics be learned by ONLY solving problems?

I have been teaching students for the past 6½ years- in all levels of college undergraduate math (decent bit of physics too). I have found that analyzing learning and all the ways to understand ...
Trevor's user avatar
  • 141
13 votes
Accepted

Is it possible to improve logical thinking and problem solving abilities?

First of all I want to laud you on your knowledge of programming. You know a lot more than I did when I was your age. I tried to learn Italian after watching The Godfather but lost interest after a ...
Saikat's user avatar
  • 398
12 votes

Can mathematics be learned by ONLY solving problems?

Daniel Hast's answer is great, but I want to add one thing: What kind of mathematical ability do you want your students to learn? Are you measuring that ability or something else? I have seen way too ...
Joonas Ilmavirta's user avatar
10 votes

Theory-practice order vs practice-theory

A go-to article for me is the following. Short overview article with extensive endnotes to further supporting literature: Clark, Richard, Paul A. Kirschner, and John Sweller. "Putting students on ...
Daniel R. Collins's user avatar
9 votes

Can mathematics be learned by ONLY solving problems?

I am an alumni of Fazekas Mihály Gimnázium (Budapest) and I can attest to the fact that we were educated in a problem solving manner -- although not exactly as OP describes. For four years, all we ...
chx's user avatar
  • 219
9 votes

Students understand during course but can't solve exam

I always make homework (from the textbook and online in WeBWorK) and written assignments MORE difficult than exam questions. I tell my students this, with the reason being “if you can run 10 miles in ...
BobaFret's user avatar
  • 826
9 votes
Accepted

More intermediate steps or check well-understanding

Part 1: Do they really understand? My first thought is that you are running into the limits of working memory. As students try hard to understand step 5, they are pushing previous thoughts about step ...
WeCanLearnAnything's user avatar
8 votes

Are there any more mathematical journals or websites with the "problems and solutions"?

Problems columns I found (in 2009) useful to challenge undergraduates. Some of them may no longer be current. But even those may be interesting to consult past issues in your school library! ...
Gerald Edgar's user avatar
  • 7,499
8 votes

Combinatorial problems which can be solved with polynomials

An interesting example is Sicherman dice: A pair of 6-sided dice, with positive number of eyes on each face that are not the classic 1..6 ones; if you throw them, the distribution of total eyes is the ...
vonbrand's user avatar
  • 12.3k
7 votes

Are some people unteachable at mathematical problem solving?

You might want to learn more about a learning disorder called dyscalculia. Students (or people in general), who are diagnosed with this learning disorder, are not "thick", nor "stupid". An analogy ...
amWhy's user avatar
  • 2,095
7 votes

Looking for a HIERARCHY of math subjects

(1) Here is Margie Hale's tree: (2) And here is Gaspard Sagot's hierarchy:
Joseph O'Rourke's user avatar
6 votes

Can mathematics be learned by ONLY solving problems?

This approach would be fundamentally a violation of the entire axiomatic idiom of mathematical understanding and proof. In particular: Mathematics starts with careful definitions of terms. The ...
Daniel R. Collins's user avatar
6 votes

How actually are prime numbers taught in elementary school in United States and how easily do students learn what they're being taught about them?

I would love to get an answer by a teacher who is trying to teach prime numbers to elementary school students about what's happening with their attempt to teach them prime numbers. I would like them ...
Thierry's user avatar
  • 1,527
6 votes

What are some ways that one can progress from stage 2 to stage 3 of the rigor stages that Terry Tao has described?

This is not as much to answer the original question (to which the answer is just that you develop any skill by trying to practice it and evaluating the results) but to tell what my understanding of a &...
fedja's user avatar
  • 3,831
5 votes

How to retain the key points of an exercise?

For me, the process is as follows: Do the exercise. Do the exercise again. This is probably faster than the first time, since I have a vague feeling of what I should be doing, and maybe remember some ...
Tommi's user avatar
  • 7,019
5 votes

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

I will take a stab at an answer though clarifying what level of education we are talking about would help. I have never created problems for things like Qual exams (essentially masters exams) so I ...
DRF's user avatar
  • 1,018
5 votes

Native language, writing, and mathematical problem solving

Based on a quick Google Scholar search, it seems like there was a flurry of activity in this area ("education literature on writing to learn mathematics") circa 1990, with at least 3 book-sized ...
Daniel R. Collins's user avatar
5 votes

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

The answer to your question is yes. Check out the recent textbook: Singer, F. M., Ellerton, N. F., & Cai, J. (Eds.). (2015). Mathematical problem posing: From research to effective practice. ...
Benjamin Dickman's user avatar
5 votes

Can mathematics be learned by ONLY solving problems?

What looks to be missing is teacher interaction. The student is interacting with a workbook. So where is the learning occurring? The student may learn something while exploring the problem. ...
Cort Ammon's user avatar
  • 1,225
5 votes

Drumming up interest in journal-problem-solving and competition prep

Our MAA section (North Central) has an annual team math competition for undergraduates. It has proven more attractive to the students than the Putnams (which we haven't tried for many years). The ...
paw88789's user avatar
  • 655
5 votes
Accepted

Stating identity is not the same as knowing value

I had a similar interaction with a student this weekend. I tend to walk it back to talk about these objects are just ways we invented to talk about number and quantity. They translate into a language. ...
Jeffery Thompson's user avatar
5 votes
Accepted

Can we solve math just by speaking and not using any other extra devices?

Mathematics has a highly optimized writing style. It is like shorthand, except that there are widely agreed standards for how to read and write it. This means that hand-writing numbers, formulas, ...
Jasper's user avatar
  • 3,158
5 votes

Are some people unteachable at mathematical problem solving?

As written, this seems overly caught up in your personal experience, but I think there's a real question here. A lot of people make it to various stages of education having concluded that they're ...
Henry Towsner's user avatar
5 votes
Accepted

Practical case for solving with system of 2 equations

Here's one that students always enjoy. A bottle and a cork cost 1 dollar and ten cents. The bottle costs $1.00 more than the cork. How much does each cost? Student often think that it is one ...
Amy B's user avatar
  • 8,037
5 votes

How important is it to come up with or learn an elementary solution?

Sharing the impressions of a person who earned 2 IMO bronze medals in his youth, but whose dreams of a successful research career were never truly fulfilled :-) Mathematics is, indeed, not only about ...
Jyrki Lahtonen's user avatar
5 votes
Accepted

What will happen if someone doesn’t make it to the IMO team?

Based on this question and your previous question, my impression (correct me if I'm wrong) is that you are driven to become a hardcore problem-solver, and you've had fun participating in Math Olympiad,...
Justin Skycak's user avatar
4 votes

Drumming up interest in journal-problem-solving and competition prep

Not a high-quality answer, but based on some decades of observation: among the kids' "peers" (and you have to figure out what that means), participation in the program needs to be positioned to ...
paul garrett's user avatar
  • 14.3k

Only top scored, non community-wiki answers of a minimum length are eligible