Can mathematics be learned by ONLY solving problems?

Question

Here is the concept:

1. Student is presented with a problem. He/she may not even understand what is being asked, or may attempt.

2. Students reads a solution to the problem. In it there may be explanations about concepts - so there is a "content" part, but as part of an actual problem being solved, at the level the student is currently at.

3. If answer is correct, moves on to the next problem.

4. When a set of problems of a given topic is consistently correct (say 10 out of 10 consecutive questions), student is given the next topic, or the next difficulty level.

Point being: At no point is there a proper "lesson". What are your views about such an approach, compared with lesson-solve, lesson-solve, etc.

Answer 1

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 in this article by Frank Quinn; it is the least powerful, most fragile, and most error-prone of the three. To quote the relevant passage:

There are (roughly) three levels of student work in mathematics:

1. Follow patterns inferred from examples;
2. Use systematic methods and algorithms that, among other things, account for the patterns in examples; and
3. Exploit the mathematical structures that lie behind algorithms.

Each level provides substantially more flexibility, range and accuracy in applications than the one before it. Learning at each level is faster, more powerful, and more transferrable than at the one before it.

As Daniel Collins points out in their answer, a pattern-in-examples approach misses the fundamental way mathematics is structured and often facilitates deep conceptual misunderstandings. I find the "fundamental principles of mathematics" in Hung-Hsi Wu's article "The Miseducation of Mathematics Teachers" to be a helpful way to think about this:

1. Every concept is precisely defined, and definitions furnish the basis for logical deductions. [...]
2. Mathematical statements are precise. At any moment, it is clear what is known and what is not known. [...]
3. Every assertion can be backed by logical reasoning. [...]
4. Mathematics is coherent; it is a tapestry in which all the concepts and skills are logically interwoven to form a single piece. [...]
5. Mathematics is goal-oriented, and every concept or skill in the standard curriculum is there for a purpose.

The approach you describe would probably violate the first three of these fundamental principles: problems would be assigned that use terms that haven't been precisely defined, it would be unclear what theorems were considered "known" and available for students to use at any given time, and the solutions might make assertions whose reasoning isn't fully explained.

To avoid this, one could fully define all terms in the problems, include full proofs of all assertions in the solutions, and require students to give full proofs of all claims (allowing citations to previous solutions' proven theorems) in their own work. However, this would significantly increase the difficulty of the course, as students would have to develop all the theorems for newly stated definitions on their own. Unless the questions are of a very basic nature, this would essentially put the students in the position of researchers.

Now that I think about it, this is starting to sound pretty similar to the Moore method, where the students are given lists of definitions and theorems and expected to prove them on their own and present proofs to the class. An advantage of this method is that it can lead to much deeper student learning of the content, and builds mathematical maturity by teaching research skills. Closely related difficulties are that it requires much more mathematical maturity and discipline (including on the instructor's part), and it's not practical to cover nearly as much content unless the students are very advanced and willing to put a whole lot of time into it (and maybe not even then).

More broadly, inquiry-based learning (IBL) methods are becoming increasingly widespread and have been used in many classrooms with great success. It might be worth looking into various IBL models to see if one of those could accomplish what you're going for here without drawing students into a "pattern-in-examples" mindset.

(As an unfortunate footnote, I feel I shouldn't mention Moore's method without noting that its namesake, Robert Lee Moore, was a virulent racist who abused his position of power to overtly and systematically discriminate against Black students.)

Answer 2

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 mathematics, in particular, is a very necessary first step. (See above posts!)

To the OP: I believe your conclusion is correct, the process you outlined alone doesn't constitute a proper lesson. One commonality I see that most students crave but don't always realize, is a narrative. Some do, and ask: how do we know these things?, why are we doing this?, etc. Narrative may facilitate a crucial step in teaching after all the analyzing of how we present mathematics is done. After we have given the types of lectures, assignments and one-on-one advice that we think are the most effective, we may have missed packaging it all up in a narrative.

Depending on the context; giving an arc of where we have been and why we must do it this way, to get us to where we are going next, is a good way to motivate a desire to delve deeper.

More explicitly, you want students to be able to do the algorithmic process and also understand how or why it must be done that way. That way ultimately they may view mathematics more axiomatically as they see a need for consistent definitions and theorems. It is then that they will understand this is necessary in being able to know if one is correct or not logically. I am often quite up-front in saying this is how you will learn this material for now. I tell algebra, trig, and calculus students the things they should simply memorize for the time. Usually with the backing: [Of course these things must not be taken for granted, but can be proven rigorously using only logic. We will not be doing that for this course or even in your later courses. However, at a certain level, after you have learned the how you must treat with the why. This is where you will go in math.]

When there is particular confusion or curiosity I will engage in a small discussion about abstract ideas. The further students are in math the more I think they will feel comfortable hearing that what they are doing now is just a very small piece of the whole of mathematics. Often these abstract connections help students and serve as glue to get them engaged in what they are doing now.

Answer 3

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 many students who can do some formal manipulations (solving equations, differentiating functions, or the like) but who do not understand what any of it means. Sure, it is useful to be able to differentiate $x^{\sin(\log(x))}$, but that without any understanding of what it means to differentiate something with respect to something has little value — although it is often enough to pass exams with decent grades.

The ability to produce correct solutions is not exactly the same things as understanding. If a student can explain their answer so that every symbol has a meaning and every intermediate step is a meaningful statement, then understanding is likely. But if only the final number or formula or something is correct and nothing else is evaluated or checked, be careful about your conclusions. The key question is: What is a sufficient or correct solution?

You might be able to embed lectures into your questions or your answers. But trying to avoid any explicit lecturing and using only exercises does not seem easy or worthwhile. I have never seen any material written in such a coy way, and I doubt it's possible. (I'll be gladly proven wrong, though!) There is also the issue that students may neglect the "story" and proceed with their exercises; if the answer stated on the last line is the same as they got, why read what's in between? If you want all students to focus on theory/lecture/(whatever you call it), I strongly recommend reserving time for it explicitly.

And please make your students not only solve problems, but also explain their solutions. To themselves, to each other, and to you. If you can include that in your concept, it sounds far better to me. Problem solving in any kind of isolation sounds dangerous and inefficient.

Answer 4

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 did we solved problems. We did nothing else. There were no lectures as such. The teacher provided guidance or stated definitions and theorems based on the problems we just solved. But problem solving was the backbone of all we did. On the grade 9-12 level we are talking about this worked perfectly. I know anecdotally this was extended later to grade 5-12. To emphasize, OP said "He/she may not even understand what is being asked, or may attempt." -- this never was the case. The problems were definitely solvable -- provided, of course, you remembered everything you learned so far and managed to apply in just the correct way. By no means was this easy but it was doable.

As an example, you could posit a problem where you have a right triangle ABC with A having the right angle and draw the altitude AD. Now provide some lengths and ask about others. This will lead pupils to recognize the triangles ABC, DBA, and DAC are similar and in turn, this will lead to proof #6 of the Pythagoras theorem. Disclaimer: this is not to say this was how we have proven Pythagoras for the first time -- I don't remember exactly as it was more than 25 years ago now. But it could've been this one. And we definitely had one more than proof.

Every other theorem can be formulated as a problem in a similar fashion. I have four A4 sized several hundred pages thick exercise books to show it can be done :) If I ever wanted to go into teaching (I have a math teacher masters but I do not teach) I would use these as an example to create my lessons. I treasure these so much that I FedExed them over at considerable cost when I immigrated (with just two suitcases so these didn't fit).

For more, see Bruce R Vogeli: Special Secondary Schools for the Mathematically Talented: An International Panorama which has the 1995-1996 curriculum from our school in Appendix A1.

Answer 5

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 proposed learning process contains no mention of definitions. I tell my students all the time that one of the essential strengths and advantages of the mathematics discipline is the use of unambiguous definitions for words.

If a person doesn't have that, then they'll be forced into piecing together a relational, approximate understanding of terminology. Like in natural languages. Which is essentially un-mathematical.

Plus theorems and concepts will not be proven. Essentially you'd be promoting not real mathematics, but cargo-cult mathematics. Faith-based math. Fake math.

Answer 6

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. However, learning the difficult things this way is pretty difficult. Few people "discover calculus" on their own because some problems were put in their way. So that means most of the learning is occurring during the phase where they are reading the solution and related material. For this to be a "good" learning phase, you would have to somehow have constructed the material presented to cover the things the student needs to know, despite not having interacted with the student before that part of the workbook was written.

The approach does remind me greatly of the Mr. Miyagi approach to teaching:

• Wax some cars I needed waxed
• Paint a fence I needed painted
• Stain a deck I needed stained
• Now I'll show you that what you did was Karate

However, the Eastern approach to teaching martial arts is very different from that of Western school teaching! The concepts that are taught in school don't always transfer well to an alien form of teaching.

Answer 7

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 warning

You have been warned, proceed at your own risk.

Disclaimer I:

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.

Disclaimer II:

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:

1. Mathematicians that apply already-known math to some applications (accountants, engineers).
2. 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.

Summary:

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 $\ddot\smile$

Answer 8

I'm a little surprised not to see any reference to the 'Harkness' approach in the replies so far. I won't write at great length, but will include some references. Can students learn ONLY from solving problems? Almost, but there are crucial discussion-based lessons taking place in between each problem set.

Harkness teaching began at Philips Exeter and they share their mathematics problem sets online: https://www.exeter.edu/mathproblems

Wellington College in the UK have successfully implemented this approach for A level Maths (equivalent to US grades 11/12), among other subjects. They share their mathematics resources here: http://maths.wellingtoncollege.org.uk/resources/

Answer 9

I'm seeing a drawback in this kind of approach: imagine you are teaching quadratic equations to a class, and you give them following examples:

• $x^2-5x+6=0$ (solution 2 and 3)
• $x^2-4x+4=0$ (solution 2 appears twice)
• $x^2-4x+5=0$ (no real solutions)

Keep out: the second one is an equation with two solutions (the multiplicity of the solution equals 2).

I believe that students will think that a quadratic equation has either zero real solutions, either one, either two, but in fact, quadratic equations have either zero real solutions or two, but the two solutions might coincide.

I don't think that students are capable of inventing the notion of solution multiplicity by themselves (although this becomes very important while working with higher level polynomial equations and complex numbers), therefore I'd say that a minimum level of theory is always required.

Answer 10

You have two buckets: one of two liters and one of three liters, and the question is: you are only allowed to completely full and empty a bucket. How will you achieve getting one liter?

The answer is simple: full the bucket with two liters, empty this in the three liter bucket, fill the two-liter bucket again and try to empty in the three-liter bucket until that latter one is full => you'll end up with one liter in your two-liter bucket.

Then you ask that question with the following values:

• $2$ liter and $5$ liter.
• $2$ liter and $11$ liter.
• $2$ liter and $21$ liter.
• $3$ liter and $8$ liter.
• $3$ liter and $14$ liter.
• $4$ liter and $23$ liter.
• $17$ liter and $50$ liter.

Your student will get the impression always needing to fill the first bucket and empty it in the second one until the result is reached.

However, the values have always been chosen as follows: $\mod(\text{second}, \text{first})=-1$.

So, from the moment your student gets the question with values like $3$ and $7$ (possible but more difficult) or even $4$ and $22$ (no solution possible), the chances are great that your student will be lost.