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I am trying to improve how I learn in general but specifically in math and a common suggestion I keep coming across is the difference between active learning and passive learning. The problem is, most sources don't exactly say what examples of "active learning" are, specifically when dealing with math.

My questions are:

  • What are examples of active learning in math? What are some ways in which I could learn math in an "active" way?


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"Don't just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?" --- Paul R. Halmos –  littleO Feb 15 '14 at 21:56
Unfortunately my active learning in math will lead me down some rabbit holes where I'll not actually study what I'm supposed to. But at that point, you'd have already fallen in love with math, so it's really just not a big deal. –  Kainui Feb 16 '14 at 5:47
Everyone interested in this type of question should consider voting on questions in the Math Teaching stackexchange proposal. It only needs 7 people to vote on questions to get to the next phase: area51.stackexchange.com/proposals/64216/… –  Brian Rushton Feb 17 '14 at 1:13
Question everything you read and learn: terrytao.wordpress.com/career-advice/… –  Holden Lee Feb 18 '14 at 22:15
One way to learn math in an active way is writing lecture notes. I've tried this the last term with "geometry and topology" (in German - see result. There is an A4 PDF and an A5 version in 'other formats'). This was awefuly time consuming, but now I have a document which I really know. I know where I can find stuff in case I forget something. And I get lots of feedback from other students who read my definitions, proofs and examples. And it certainly makes it easier for other students to learn about the topic. –  moose Feb 22 '14 at 8:26

8 Answers 8

I will try to answer your question using an answer that I posted on the related Math Educators' site.

IMHO, your goal (and mine), should be to strive for "mathematical maturity". That is the ability to see the interrelationships between math concepts rather than just the concepts in isolation.

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There are many many theories of learning and teaching and understanding and I find none of them helpful. This includes the theory of passive and active learning as well as the odd notions of intuitive vs. formal, constructionism, learning styles, collaboration, APOS, and a great many other oddities. I have studied this question assiduously from the point of view of a student for sixty years and my conclusion is that what is missing in mathematics education is explanation. Unfortunately we have no science of explanation, no professors of explanation, no degrees in explanation, no journals of explanation. If fact the very concept of explanation is interpreted in a variety of imaginative but non-helpful ways. The values of scholars do not include explanation. In fact, some famous mathematicians expressed distain at questions involving how they arrived at their discoveries. I believe it was Newton who said that when a magnificent building is finished, the scaffolding is removed. Or, in the words of an old saying, "It's for me to know and you to find out." My best nutshell advice is to always ask why, why define this, why define it this way, what led you to look at that, what led you to make that discovery, what sense does this make, etc. Everything inherently makes sense and we should never stop looking for it.

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I think the other answers here are great, but I thought I would provide one more perspective: "the best way to learn is to teach someone else". When you are learning passively by reading or listening to lectures, you are guided by the hand and it is easy to convince yourself that you understood everything. A good way to test this, is to try to explain what you learned to someone else or apply your knowledge. Here are three ways I recommend:

  1. Explain what you are learning to an interested friend, one that has enough background to ask you questions. Of course, sometimes your friends don't have the time, so
  2. Explain what you are learning by writing a blog. Even if nobody reads your blog, it is better that private notes because even the though someone might read makes many write more carefully and prod their own understanding, it works magics for me. It also helps you to learn how to communicate to the mathematical community, which is an essential skill if you go on to graduate school and one that is seldom explicitly taught. Finally,
  3. Participate on Q&A sites like Math.SE. Don't simply write your homework problems here, that is useless, but as you read, if something doesn't make sense, try to restate your confusion in your own words. Sometimes formulating a clear question can be incredibly enlightening to your own understanding. I have solved many of my problems by simply trying to write down a precise SE question and thus solving my own confusion without even asking. On top of this, as you become more comfortable, you can try your hand at answering questions. On a site like this, you effectively have a whole community of tutors that will give you positive reinforcement when you do well (+1 votes) and correct you when you are wrong. Of course, as you do this, respect the community norms and try to learn something from every answer you give.
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Polya's How to Solve It contains excellent advice. Especially pertinent to your question is his Fourth Principle: once you've solved a problem, don't just heave a sigh of relief. (It's OK to heave a sigh of relief, but don't stop there.) Can you simplify the solution? Can your solution be adapted to solve similar problems? Is there a problem you tried previously, but couldn't solve, which this problem throws light on? Etc.

Feynman described two of his approaches. First, he always carried around in his head 5 problems he was stuck on. Anytime he learned a new technique, he'd run through his problems to see if it would help with any of them. And he also carried around 5 techniques. Anytime he heard of a new problem, he'd run through the techniques to see if any of them would help.

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I want to emphasise something that was briefly touched in other answers, but which I think is the core of active learning.

Taking notes and reading on your own is, in my opinion, still quite passive. In fact, mindlessly taking notes during lectures can actually be a disengaging process. The only way to really learn actively is by actually struggling with hard problems.

Now, this is something many people end up passing on. I have many colleagues who just go to the lectures, do a couple of mandatory assignments, nod at the teacher, but when it gets hard they just give up (and often come ask questions here...).

Now, learning by struggling with hard exercises is awfully time consuming. You can easily spend 10+ hours more per chapter in a book like Munkres or Rudin, but at least you'll be able to actually use what you've learned actively in the future.

For me the best reference for training this kind of learning are books like Putnam and Beyond, which pose very challenging questions related to most major undergraduate topics. Unless you're extremely gifted, working through the first chapter in 'basic techniques' like induction and proof by contradiction will leave you feeling sorry for your own stupidity. But when the solution does come it is usually very rewarding.

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Another wonderful book focusing on problems, although fairly specialized, is Steele's Cauchy-Schwarz Master Class. –  J W Feb 15 '14 at 22:50
I just looked at Putnam and Beyond. And I do feel stupid. –  Neikos Feb 16 '14 at 5:18
Engel's Problem-solving Strategies is also a great reference for that matter. –  LeGrandDODOM Feb 23 '14 at 16:53

The issue is not particularly specific to mathematics:

  • passive learning is being taught, writing down what you have been taught, and learning what you have written down
  • active learning is answering questions which require you to apply what you have been taught

So in preparation for an exam, a passive learner will spend most time going through notes while an active learner will spend most time practising with previous editions of the exam. Most people do both.

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Would you consider 'deliberate learning' to be the same as 'active learning'? –  John H Feb 16 '14 at 23:41
@John H: No - I would regard 'deliberate learning' as setting out to learn something, while 'incidental learning' would be a positive side-effect of an activity which I undertook for other reasons, such doing my usual job, or reading a novel, or playing a game. –  Henry Feb 16 '14 at 23:51
That actually makes sense, thanks. –  John H Feb 16 '14 at 23:52

I'd say

  • Take notes in lectures. Even if a pdf is provided, if you write your own notes, you will have to process the information, you will figure out what you don't understand, and you will remember it better. Ironically, I remember taking notes in some lectures and never looking at them afterwards because I remembered what was said.

    • I now find that when I am reading textbooks at a more advanced level I have a pen and a pad of paper to scribble down the more important equations as I read them, and to fill in the missing steps in the mathematical arguments. If I don't "interact" with the material in this way, I rapidly get lost and tune out.
  • Practise problems. Math is a subject where you learn by doing. In an academic setting, ideally there will be handouts, and your work will be marked and/or solutions will be provided. If you are self-studying, it is worth looking out for textbooks where solutions to exercises are provided, and paying extra if necessary.

  • Find a community that you can interact with. In an academic setting, this could be a tutorial group, meeting fellow students after lectures etc. Outside of academia, resources like this website are invaluable.

Those are the most important points I can think of off the top of my head. But the concept extends more generally. Whatever you want to achieve, how can you do it interactively? For example, if you're revising a set of lecture notes, rather than just reading them through, set yourself the task of extracting the most important points, putting them onto flip cards (or latex them up into a pdf if you choose), and then test yourself on those points a couple of times before the exam.

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Quoting my own Professor here regarding this topic. He does research into the most effective way to learn mathematics, and also effective teaching strategies:

Your lecture notes will be your most valuable resource. You will refer to them when you do homework, or prepare for a test or an exam. So:

  • during a lecture, take notes

  • later, read the notes; make sure that you have correct statements of all definitions, theorems, and other important facts; make sure that all formulas and algorithms are correct, and illustrated by examples

  • fill in the gaps in your notes, fix mistakes; supplement with additional examples, if needed

  • add your comments, interpret definitions in your own words; restate theorems in your own words and pick exercises that illustrate their use

  • write down your questions, and attempts at answering them; discuss your questions with your colleague, lecturer or teaching assistant, write down the answers

It is a waste of time to try again and yet again to understand the same concept. so, when you are sure that you understand a particular definition, theorem, algorithm, etc., write it down correctly, in a way that you will be able to understand later. This way, studying for an exam consists of re-calling and not re-learning; re-calling takes less time, and is easier than re-learning.

Keep your notes for future reference: you might need to recall a formula, an algorithm or a definition in another mathematics course.

Attend lectures regularly! Concentrate and follow lectures, take notes, and fix them later (how?). Ask questions, respond to questions from the instructor, discuss material with a colleague.

for each section that is covered in class:

  • study solved examples from the textbook ... don't just read through a solution! Hide the solution, work on your own; if you get stuck, first understand why you got stuck, i.e., what's the problem. Only then look at the solution in the book to see how that problem was resolved. If you just read through a solved example, most probably you will miss the point. of the exercise/example; when you encounter similar situation, you will not know what to do.

  • work on suggested practice questions; do as many as you need in order to feel that you know the material; mark the exercises that you are not certain about, or do not understand, discuss these with your colleagues and/or with your TA

  • work on homework problems to see what homework you are supposed to do; solutions to homework will be posted regularly on the course web page.

Weekly tutorials Be prepared for your tutorial: study sections that were covered in class; ask questions, discuss theory or assignments with your colleagues, ask your instructor to explain things you are unclear about. http://ms.mcmaster.ca/lovric/lovric.html

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This feels very cookie-cutter and not very inspiring. Although repetition is crucial to memorization, I think it leaves out the best part, understanding. A prime example is that there are literally thousands of people who can calculate an integral and will tell you it's the area under a curve, but can't explain anything else about them other than that they are apparently magic. –  Kainui Feb 16 '14 at 5:55
My lecture notes were very rarely my most valuable resource. Lecture notes are a 60-minute sketch of what actually happened in class: the lecture. And what is the lecture? It's a 60-minute stand-up routine of some small nugget of what's written in the text. It is the text that is your most valuable resource, not your notes. –  Tac-Tics Feb 18 '14 at 23:32

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