MOOCs and TOOCs and the role of problem solving in maths education

A quick quiz: which of the following four words doesn’t fit with the others??


We are going to muse about MOOCs today, a hot and highly debated topic in higher education circles. Are these ambitious new approaches to delivering free high quality education through online videos and interactive participation over the web going to put traditional universities out of business, or are they just one in a long historical line of hyped technologies that get everyone excited, and then fail to deliver the goods? (Think of the radio, TV, correspondence courses, movies, the tape recorder, the computer; all of which held out some promise for getting us to learn more and learn better, mostly to little avail, although the jury is still out on the computer.)

It’s fun to speculate on future trends, because of the potential—indeed likelihood—0f embarrassment for false predictions. Here is the summary of my argument today: MOOCs in mathematics are destined to fail essentially because the word Massive is intrinsically unrelated to the other words Open, Online and Courses. But, a more refined and grammatically cohesive concept: that of a TOOC, or Targeted Open Online Course, is indeed going to have a very major impact.

When we are teaching mathematics at any level, there are really two halves to the job. The first half is the one that traditionally get’s the lion’s share of attention and work: creating a good syllabus with coherently laid-out content, which is then clearly articulated to the students. The other half, which is almost always short-changed, and sometimes even avoided altogether, is to create a good set of exercises which allow students to practice and develop further their understanding of the material, as well as their problem-solving skills. In my opinion, really effective teaching involves about equal effort towards both halves; again this is rarely done, but when it is, the result usually stands well out above the fray.

Here are some examples of mathematics textbooks in which creating the problem sets probably occupied the authors as much as did the writing of the text: first and foremost Schaum’s Outlines (on pretty well any mathematics subject), which are arguably the most successful maths textbooks of the 20th century, and deservedly so, in my opinion. Then come to mind Spivak’s Calculus, Knuth’s The Art of Computer Programming, Stanley’s Enumerative Combinatorics, and no doubt you can think of others.

Good problems teach us and challenge us at the same time. They are the first and foremost example of Gamification in action. Good problems force us to review what we have learnt, give us a chance to practice mundane skills, but also give us an opportunity to artfully apply these skills in more subtle and refined ways.  They provide examples of connections which the lecture material does not have a chance to cover, they give students a chance to fill in gaps that the lectures may have left. When combined with a good and comprehensive set of solutions, problems are the best way for students to become active in their learning of mathematics, a critically important aspect. When further combined with a skilled tutor/marker who can point out both effective thinking and errors in student’s work, make corrections, and advise on gaps in our understanding, we have a really powerful learning situation.

Here is where the Massive in MOOCs largely kills effective learning. It is the same situation as in most large first year Calculus or Linear Algebra classes around the world. Officially there may be problem sets which students are exhorted to attempt, but in the absence of required work to be handed in and marked, students will inevitably cut down to a minimum the amount of written work they attempt. In the absence of good tutors who can mark and make comments on their written work as they progress through the course, students don’t get the feedback that is so vital for effective learning.

Once you have thousands of students taking your online maths courses, it becomes very challenging to get them to do problem sets and have these marked in a reasonable way. The currently fashionable multiple choice (MC) question and answer formats that people are flocking to can go some small way down this road, but rarely far enough. Students need to be given problems which require more than picking a likely answer from a,b,c or d. They need to define, to compute, to evaluate, to organize, to find a logical structure and to explain it all clearly. This is practice doing mathematics, not going through the motions!

When we are planning an open course for possibly tens of thousands of students from all manner of backgrounds, the possibility to craft really good problems accessible to all diminishes markedly. There is no hope of giving feedback to so many students for their solutions, so all we can aspire to are MC questions that inevitably ride on the surface of things and don’t effectively support the crucial practice of writing. Learning slips into a lower gear. Such an approach cannot be the future of mathematics education. Tens of thousands of students going through the motions? They will find something more worthwhile to do with their time, like just watching YouTube maths videos!

But a slight rethinking of the enterprise, together with some common sense, can perhaps orient us in a more profitable direction. An education system ought to make enough money to at least fractionally support itself. People are willing to pay for something if it has value to them, and they tend to work harder at an activity if they have committed to it monetarily. All good technical writing has a well-defined audience in mind. These are almost self-evident truths. What we need is to think about crafting smaller, targeted open online courses, that generate enough income to support some minimal but effective amount of feedback on students’ work on real problem sets. By real I mean: problems that require thinking, computation, explanation.

Can this be done? Yes it can, and it will be the big education game changer, in my humble opinion. We will want to stream people into appropriate courses at the right level. Entry should be limited to those who have enough interest and enthusiasm to fork out some—perhaps minimal, but definitely non-zero!—amount of money, which hopefully can be dependent on the participant’s region; and who can pass some pre-requisite test. Yes, testing for entry is an excellent, indeed necessary, idea that will save a lot of people from wasting their time. Having 300 people from 10,000 pass a course is not a successful outcome. Better to have targeted the course first to those 1000 who were eager and capable. Then you get a lot more satisfaction across the board, from both students and the educators involved.

A major challenge will be how to provide effective feedback for written work. Relying exclusively on MC exercises should be considered an admission of failure here. If and when this challenge is overcome, TOOCs will have the potential to radically transform our higher education landscape!

5 thoughts on “MOOCs and TOOCs and the role of problem solving in maths education

  1. leonig100

    I am learning Hyperbolic Geometry using your UHG series. They are of very high quality and you have obviously given them a lot of thought to make them accessible to a wide audience. I acknowledge that doing the exercises is a very useful way to clear one’s mind and understand the subject more deeply. However if there are no answers to the questions this tends to put people off because one cannot establish that the answer is correct and this can put one off. I am a retired engineer with an interest in understanding mathematics rather then applying formulae to solve worldly problems.

    1. njwildberger: tangential thoughts Post author

      You make a very good point that I also acknowledge in this post: having solutions to exercises is important. It is my hope to some time post solutions to the exercises in UHG. The issue is time, I seem to suffer a serious lack of it! Thanks for the suggestion though.

  2. extranosky

    I was greatly helped during my undergraduate years by my supplementary reading rather than by my textbooks. Having answers and hints to questions really is a great help in learning a topic.
    I sound off your point in the need for having solutions to exercises.


  3. adam

    On the subject of pre-testing. This is a great idea! Competancy tests should be mandatory, especially if money is involved. One idea is to divide the testing into competancy testing, and over-competancy testing. Competancy testing questioning would be on pre-requisites. Over-competancy testing would be a would test if one already knew what the course would teach. Further, testing in this way would allow for a more fine tuned callibration of student competancies (the spectrum of subjects would be wider/deeper then just pre-requisite testing). As an example, this callibration could be you looking at the result of a test, and determine how you are going to approach guidance towards filling in the competancy gaps and knowledge gaps.

    On the subject of limitations of automated teaching courses, and beyond, into fundamental limitations with teaching over the computer. Consider that when we interact with the computer, that there is much information at our fingertips, but that interaction over the computer can never be as rich as true physical interaction. I think with mathematics, this is less of a problem, as it is by its very nature oriented towards information as opposed to the tangible. This is distinct from a subject like chemistry, as there is going to be a point you would have to work with the tools of the chemist.

    “An education system ought to make enough money to at least fractionally support itself.”
    Very true. Part of the issue I think, is that you need an effective framework with which to callibrate a student to the gaps, and an effective teaching strategy. I can imagine you go to a website and pay to have a lesson callibrated for you. Value added might involve some kind of certificate for various hurdles (Not sure how that would work, but there are online degrees). I suppose having a certificate for passing a course in say, algebraic geometry, would be better that saying I know some algebraic geometry. It might impress someone anyway if you collect certificates of competancy, similar in some respect to say the Comp TIA certificates. Connections for mathematical competancy to solving technological problems that have a monetary value, would I think raise the value of a math certificate. Maybe this would be another job of the “institution” to catolog these connections and network accordingly (Does job placement sound far fetched?). Perhaps some kind of balance can be struck here, between teaching what you are engaged in and want to teach, making money doing it, and guiding people on a path that you think would be good for them, and good for our future.

    No definite solutions here, but maybe some stuff to consider.


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