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.
You have been warned, proceed at your own risk.
Here is another counterpoint to the (currently) highest voted answers. It should come as no surprise that I am heavily biased in favor of the problem-based approach, because that was how I learned math as a kid. Please adjust the size of your "pinch of salt" accordingly.
I was mulling over the current answers and I have to admit (I guess mostly due to my bias) that a few rub me the wrong way. Someone is bad-mouthing good ol' problem-approach that pushed forward math understanding for hundreds of years – I had to do something. Still, when I tried to find any concrete data that would concradict them, nothing came up. But then, why I am so confident that they are wrong? Then I realized that perhaps we understood the OP's question differently and that ambiguity is the source of disagreement. The rest is not actually an answer, but a number of related comments that the reader (I hope) will still find useful in the context.
Two kinds of math work:
This is a gross oversimplification, but that distinction will be useful here, so let us split the mathematicians into two groups:
- Mathematicians that apply already-known math to some applications (accountants, engineers).
- Mathematicians that invent new math results (math researchers).
Of course, realiy is more complex and one kind could not exist without the other, but I am doing this to emphasize difference.
On the meaning of progress:
Let me ask a question:
What is our ultimate goal of teaching math?
There are lots of possible answers, to give a few:
- To make the world better (or worse).
- To push human civilization forward (backward).
- To make people happier (or more miserable).
- To improve probability of human survival (or delay our inevitable demise).
- To undersand the world better (or just the theory rather than the world).
- To create something beatiful (or too complicated/abstract/meta).
- To earn a living (or take some jobs away from people).
Whatever our personal goal is, the best proportion of math work (math application vs. math research) differs. We can't answer which learning approach is better until we know what our ultimate goal is, and depending on that goal the problem-based approach of learning could be good or completly bad.
On "lectures" in problem-based learning:
To confuse the matter even more, let me observe that even in problem-based learning there are "content-giving" phases like explaining the solution, etc., which in extreme sense might become plain lectures.
On the other hand, many math courses end with exams and so the whole thing could be viewed as problem-based approach (the problems were on the exam) with too long explanation.
In other words, the distinction is far from precise and some common sense is needed.
On problem-based learning:
Since I remember math for me was always a tool to do something – this is how I learned math. Behind everything I have learned there was some problem to solve, even in normal highschool math classes I would solve problems from math circle. Only at the university level I got some non-problem-based education in form of lectures. Many friends I had at the university or from math circle and various competitions had similar experience. This is only anecdotal data, but I will state something stronger: every single person I know that produces good math research (and I know anything of how that person learned) had learned math mostly on problems. Furthermore, it might be the best way to teach theoretical computer science (at least it got me the best results of everything I've tried).
Please observe that with math research things rarely are precisely stated, many concepts may lack definitions (creating good definitions is a crucial step) and instead of reasoning we rely on intuition. First drafts of the proofs are usually not coherent and lots of research starts without clearly stated goals. Also, in math research using known algorithms not always yields results and mathematicians have to infer patterns from (sometimes abstract or only partially-formed) examples. These and similar soft skills are essential in math research and should be trained. For me they are more important than just knowledge, which can be picked-up along the way with a deeper understanding than non-problem-based approach.
What I want to say is that problem-based learning seems excellent if we want to train a research mathematician. It probably is one of the things that caused me to do math research. It might be worse for non-research math, but I have yet to see a conclusive results in this matter.
On math metrics:
One problem with evaluating teaching method quality is what qualiy metric to use. First, it depends on our "ultimate math-teaching goal", and even if that was agreed on, the right metric might not exist or be impossible to measure. Do we take imperfect test scores? Or try to gaguage deep understanding? Math fluency after 10 years? Number of succesful math applications? Number of theorems our society deem great in the next century?
I see lots of reasons why non-problem-based approach could be potentially better for people who will not become research mathematicians.
However, gathering any data that would truly be convincing seems impossible. Similarly, proving superiority of problem-based teaching also looks hopeless.
On some drawbacks of problem-based teaching:
It might be obvious, but let me emphasize it here: a good sequence of problems is crucial for the problem-based teaching to work. In particular each student may need a different sequence of problems to reach deep understanding or even succeed. Frequently I found that I have to adjust problems on-the-fly to make them both interesting and managable.
Furthermore, one has to anticipate students' mental models to provide them with tasks that will help them learn, which is hard without years of experience.
Problem-based approach can also be really frustrating and exhausting for both the student and the teacher – strong motivation (or encouragement) is necessary.
Problem-based teaching works really well for students that are naturally curious esp. if they reached a higher level of mathematical maturity. Unfortunately, according to my experience, without curiosity (or if attention is on something else) it works much worse.
Taking that into account, any research that would evaluate problem-based approach needs to get all the above issues right, but then it becomes even more impossible.
My own experience suggest that problem-based teaching is excellent in producing research mathematicians. Still:
- It is hard to teach using problems well (probably not every teacher can do it).
- It does not work well for every student.
- Teaching people to be research mathematicians when they don't need it seems like a waste of time and effort.
Can math be learned only by solving problems? Technically yes, but that might be a really bad choice. Among others that depends on:
- How rigid your "only" is and how efficient with your learning you want to be.
- Your ultimate goal for learning math and whether you want to become a research mathematitian.
- Whether you have patience and strong motivation.
- Whether you have a problem set that is tailored to you and dynamically adapts with your growth (read: whether you have a teacher that is willing to invest a lot of time and effort in teaching you).
I hope this helps $\ddot\smile$