About the effectiveness of self-studying maths (compared with other subjects)

An important feature of mathematics is that it is relatively easy (compare to many other subjects) to know whether or not one's understanding is correct. There are plenty of ways to check: one can simply do exercises and check the answers. Even if a student has some false beliefs or misunderstanding, it is quite likely that after looking for proofs/counterexamples, the student will realise the problem.

This seems to make mathematics more suitable for self-study than many other subjects. A student need to receive regular tutorials on how to play the piano or to write stories/essays because they need to be sure they are on the right track, but for mathematics, a student could probably learn by themselves for a long time. (This is, of course, under the assumption that they understand the material from which they learn.) And indeed, there are many self-taught mathematicians in the world.

So it is natural to ask the following question: as a result of what I have described above, how is the role of students and teachers in the learning of mathematics different from the role of teachers and students in subjects such as Literature, Music, (or even science subjects such as Chemistry and Biology)?

This question applies, of course, to both school level and university level learning.

• Anything can be learned outside of class setting as long as one can read and has access to required books and tools. Math is no different than literature or music. But if you want to, say, dissect a human corpse, then you need a lab with a reliable supply of cadavers. Heck, even if you want to learn how to hammer a nail, you need a hammer and a bunch of nails to practice. Mar 2 '20 at 18:20
• If your criteria is "I can easily tell when I am wrong," I would suggest that piano ought to be easier to learn than mathematics. I mean, I can listen to the music I am making, and know if it sounds good or not. Jun 28 '20 at 20:33
• @XanderHenderson: there are innumerably many failed musicians out there seeming to provide evidence against your point. Have you ever watched shows like The X-Factor or American Idol? Lots of people who get thrown out in the preliminary stages genuinely believe that they are good singers. The same applies to, say, the restaurant business (see "Ramsay's Kitchen Nightmares", for example). The difference appears to me to be that in mathematics it is easy to compare answers. In other areas, comparison is perhaps not so easy as it seems (regardless of whether it could theoretically be automated). Jul 2 '20 at 21:17
• @WillR Similarly, there are innumerable "proofs" of the Riemann hypothesis, the Colllatz conjecture, and the twin prime conjecture. The website viXra is a good place to look for these things. I think that most people are pretty good judges of their own musical or culinary prowess, and that reality shows intentionally include an over-representative sample of those who lack that kind of self-awareness. Jul 2 '20 at 21:26

The question is extremely broad, since it talks about "both school level and university level learning," and doesn't make any distinctions between students with different levels of intellectual development and maturity. The brain of a kid at age 7 is doing really different things intellectually than the brain of an adult.

Certainly there are people who can self-learn math. For example, Abraham Lincoln rode around the court circuit, staying at lodging houses at night and studying Euclid, which he thought was a perfect model for presenting a logical legal argument. But most people aren't as smart or motivated as Abraham Lincoln, and most people learn most of their math as kids and teenagers, so their brains aren't as developed as Lincoln's was at that point.

I spend a lot of time every week working with my community college students in my office hours on mathematical problem solving. Most of these students need a lot of guidance.

An important feature of mathematics is that it is relatively easy (compare to many other subjects) to know whether or not one's understanding is correct.

For the students I interact with, this is mostly not true at all. Figuring out how to check your own answer is actually a very high-level skill. Few of them have this skill. As a typical real-world example, I have a student this semester who has had two years of calculus, but does computations like this:

$$\left(\frac{3}{2}\right)^2=\frac{6}{4}.$$

Are there ways to check this answer and realize that it's wrong? Of course. For example, one could simplify the result and observe that it's the same size as the number that was being squared, but that doesn't make sense, because we expect $$x^2>x$$ when $$x>1$$. Or they could compute $$1.5^2$$ on a calculator and compare. But this student doesn't have the kind of high-level understanding that would be required in order to think of these checks.

In my experience, the vast majority of people who learn math in school are pretty similar to the student I've described above in terms of their conceptual understanding. Their fluency with computation may be better, but they fundamentally imagine math as a black box, not as something that they can make sense out of. You could call this type of student "concept-blind." Concept-blind students are not going to succeed with self-study.

What is actually probably needed is more interaction with teachers who are themselves competent in math, who give proper attention to conceptual understanding, and who don't let students slide through to the next level without understanding what they're doing. How to make this happen, I don't know. At the K-6 level, there are far too many teachers who are themselves concept-blind. In the community college system here in LA, students tend to shop around on web sites like ratemyprofessor to look for professors who are "easy," which means that they're systematically avoiding the kind of appropriate guidance they need.

• "the vast majority of people who learn math in school are pretty similar to the student I've described" — it is not the students, it is the schooling. Mar 2 '20 at 18:23

The way I view my teaching is that I'm more of a guide and organizer. I can help guide students through the various topics in math but it's up for the student to figure things out for themselves. I can also organize material so that students are can figure things out in a systematic way. At the end of the day, I can't do the thinking for the students, but I can make (or do my best to make) the environment most effective for self-study learning. This way, I can be confident the student will do well without me.

Some of my techniques in the classroom are:

• Going over the course outline, so students know when to expect things
• Briefly talking about the textbook and its topics that will be used, so students know what they'll be learning. (I do my best to describe the course in ways someone who hasn't done the course understands)
• Talk about learning objectives and what the student has to do, which is basically do readings before class, self-mark homework, come to class with questions, etc

If everything goes well, classes shouldn't be lectures where I'm speaking to the student about what material they need to learn. Rather, classes are discussions about the material where I'm talking with the student who has self-studied the material.

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I can't honestly answer how math education is different than that of other subjects (since I don't teach them) but I'd suppose it would be harder to "build up" the content step-by-step without a more involved navigator. Also, where they'd navigate would be different depending on student.

For example, if I was teaching Euclidean Geometry using The Elements, virtually everything would build up step-by-step and I, as teacher, would just be there for clarification or elaboration. I could imagine a gifted student who would seemingly recreate Euclid Elements on their own, as if I wasn't needed. Further, since that math has a universality to it, differences between the student, the teacher, and translations of the text, will probably not make too much difference.

In contrast, if I was teaching, say, gender, political, or religious studies, it would be extremely different in terms of my personal involvement, and the outcomes of the students.

For me, teaching has some universal principles which are the same for all fields, be it maths, physics, music or literature. More precisely, two core values are,

• assisting students in an investigative learning procedure is a common ground for teaching in general, since, knowledge that is doscovered (partially or as a whole) by a student is more persistent and stable than knwoledge that is simply revealed to them by a "source of knowledge" - teacher, book, internet etc.
• diversifying the lesson, according to the needs of each student. In the 21st century it is mandatory to take into consideration the special characteristics of each student in order to effectively teach. Students mhave access to so many sources and come from vastly different backgrounds making it indeed hard and ineffective to try to conform all of them to a "universal" way of teaching.

As for the differences, as a math and music teacher (theory and practice) I have to say that they are lying on the differences of each field. For instance, teaching music theory one often needs to explain why is something that way in e.g. classical music theory. To do so, a discussion in the class about the era of the under discussion music genre and a relevant research on the web may help students detect the reasons behind the phenomenon at hand. All the abovementioned procedure demands and develops high level congitive skills such as synthetic, analytic as well as critical thinking.

While the above skills are also necessary in and trained by studying mathematics, mathematics pose many more challenges. One reason behind it is the fact that mathematical formalism, when compared e.g. to musical one, is often far more distant from the initial "stimulating phenomenon" - in terms of the abstraction needed to arrive at it. For example, the emergence of a certain rule in classical music theory can be explained, most of the times, by looking at the ethics and aesthetics of the time, while the notion of "number" being simultaneously a ground notion in school maths and a result of a high level abstraction is more hard to deeply understand.

Thus, while mathematics seems like the perfect field for self-study - the algorithm of self-study is indeed described easily and seems "computationally" efficient - this is not the case for me. The intuition behind the key notions of mathematics and the consequtive abstraction in order to arrive to them consist a constant challenge for the learner. Hence, a teacher is needed in most of the cases so as to channel the student's efforts towards the right directions and let them see the intuition behind it.

Let me close this with an example of how difficult could self-learning be. Imagine a student that has arrived at the notion of the "tangent line of a circle" (N1) and is about to be introduced to differential calculus and, more specifically, to the notion od the "tangent line of a function's graph" (N2) - having, of course, knowledge of limits and all the prerequisite knowledge. The transition for N1 to N2 requires an abstraction from the circle's tangent line to that of a more general curve. However, letting the student find the most adequate property of N1 to generalize so as to arrive to N2 may lead to several problematic generalizations - e.g. considering a tangent line as the one that cuts the curve at exactly one point or meddling with the graphs convexity and so on. At this point, a teacher who can orientate the student is indeed needed.

The assumption is that math is more suitable for self study because the learner can know when he has gotten something right. In fact there is a wide range of material with answers, but it is not always clear which problems to do. Doing all of them will need too much time. Doing some runs the risk of missing a subtle concept. A good teacher will help the learner figure out which problems to do.

If a learner does a problem and gets it wrong, the learner can't always find out why, leaving them unable to learn from their mistakes as well as stuck and frustrated. At times the learner may get the right answer by a method that worked in this case but not in all cases, causing them to learn the wrong things.

Many students who study on their own, may learn the algorithms without the higher understanding that would be useful as they continue.

A good teacher, will help a student who is stuck by pointing them to the next step. Those who look at a solution will never get just the step that they are stuck on, but more than that and will miss out on valuable practice.

A student may think they understand something because they got a few problems right, but not do enough practice to cement their learning.

I have addressed why I think a teacher is useful in mathematics given it's uniqueness in problem solving and available answers.

Ok firstly I need to debunk your misconception regarding mathematics being easier to self learn than other subjects.

Firstly, mathematics does not have an experimental component for the most part, therefore peer review for content online is a natural outcome.

Secondly, you seem to be of the generation that takes the internet for granted in some respects. As a follow on lemma of my first point, this technology enabled this outcome to occur in a historically rapid fashion.If you wish to have my point proven in the most impactful way possible, disconnect your internet connection, and continue your pursuit. If tertiary education is free in your country, drop out. It shouldn't be long before you are exhausted and drunk in a public bath, much like the best of the best millennia ago. We are so fortunate in modern times, it's hard for me to find words to describe it.

Thirdly, the entire concept of self-taught is a fallacy. Abandon a child in an unpopulated area, and granted he or she survives a few months, when you find them they will be very much hunter gather in psychology.

Thanks for your input in the community.

• I disagree with your opinion, and I don't think you have made any points in this whole answer that substantiates it. Many subjects in mathematics can be learnt simply by using well-written textbooks, regardless of whether or not the internet is available (although, of course, the internet does indeed help). Jul 5 '20 at 11:37
• Yes, well written text books made vastly more available by online purchases, the existence of quality printed textbooks is not a counter argument for the points I have made. Jul 10 '20 at 8:43
• Of course it is. "Thirdly, the entire concept of self-taught is a fallacy."---What I said disproves your point. Unless, of course, you define "self-taught" to mean not with any external aid, even textbooks or online references, which is a frankly ridiculous way to interpret the question. Jul 10 '20 at 8:54
• well what do you mean by external aid? the efforts of an entity other than yourself being useful hence teaching you a lesson of sorts? Anyway I don't know what else I can say here Jul 16 '20 at 12:49