41
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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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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 ...
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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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12
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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 ...
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12
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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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10
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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 ...
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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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9
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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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9
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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 ...
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8
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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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8
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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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7
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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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7
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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
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 ...
- 23.6k
6
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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 ...
- 1,527
6
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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 &...
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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 ...
- 6,129
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 ...
- 1,018
5
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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
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 ...
- 23.6k
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. ...
- 1,225
5
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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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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
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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, ...
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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 ...
- 11.5k
5
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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
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Can mathematics be learned by ONLY solving problems?
TL;DR
Can one learn math only by solving problems? Technically yes, practically it depends. Is it a good idea? That cannot be answered without considering why one wants to learn math.
Babbling ...
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4
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How to retain the key points of an exercise?
Any exercise is just one example of a family of similar exercises. You can explore those similar exercises by writing and solving your own variations of the problem.
To be more concrete, I assume ...
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4
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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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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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