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 ...
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  • 4,755
27 votes
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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 ...
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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 ...
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  • 141
12 votes
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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 ...
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  • 368
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 ...
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11 votes

Name the heuristic: exploiting the legitimacy of the questioner

The heuristic described here is one manifestation of what Polya (1945) and others thereafter refer to as trying a special case. I do not know of a more specific term for the context that you have put ...
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9 votes

The interplay of memory and mathematical performance

Anecdotally, based on self-observation and observation of many faculty and grad students: "if it's not in your head in some form, you can't think about it". A funny point here is that it seems not ...
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  • 13.4k
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 ...
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  • 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 ...
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  • 776
9 votes
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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 ...
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8 votes

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

Nice question! Let me add one reference to your list: Silver, Edward A. "On mathematical problem posing." For the learning of mathematics (1994): 14(1) 19-28. (PDF download link.) Silver cites ...
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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! ...
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  • 6,171
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 ...
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  • 12k
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 ...
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  • 2,008
7 votes

Looking for a HIERARCHY of math subjects

(1) Here is Margie Hale's tree: (2) And here is Gaspard Sagot's hierarchy:
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6 votes

The interplay of memory and mathematical performance

In light of your edited "more general" question, I thought I would make a few remarks. Historically, an early treatment of the subject of reasoning by analogy can be found in work on Gestalt ...
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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 ...
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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 ...
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  • 1,234
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 ...
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  • 4,544
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 ...
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  • 896
5 votes

The interplay of memory and mathematical performance

The strongest students often tell me that they like taking my courses because they learn so much in doing the work for my course. I lecture and give traditional, human graded, homework. One thing I do ...
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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. ...
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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. ...
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  • 1,101
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 ...
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  • 617
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. ...
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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 ...
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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 ...
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4 votes

Ideas for math problem solving class for undergraduate students in university

'Thinking Mathematically' by Mason, Burton and Stacey sounds like a good match. It has a large collection of problems/investigations using high-school level maths, and discusses how to go about the ...
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  • 5,540
4 votes

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

I've just happened to come across this free online course on logical and critical thinking. Excerpt from the site's "About the course": We are constantly being given reasons to do and believe ...
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  • 5,540
4 votes

Polya's "Nearby Problem" Heuristic and Inquiry Based Learning

(This answer has two parts: The first one is about existing research, and probably relevant, but succinct; the second one is about a problem solved in practice, and possibly relevant, but definitely ...
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