Posts Tagged 'Inquiry-Based Learning'

How is it going this time?

My Mathematical Thinking MOOC is now starting its ninth week out of a possible ten. (The last two weeks are optional, for those wanting to get more heavily involved in the mathematics.)

At the start of the week, registrations were at 38,221, of whom 24,342 had visited the site at least once, with 2,818 logging on in the previous week. But none of those numbers is significant – by which I mean significant in terms of the course I am offering. (People drop in on MOOCs for a variety of reasons besides taking the course.)

The figure of most interest to me is the number of students who completed and submitted the weekly Problem Set. In my sense, those are the real course students. As of last week, they numbered 1,013, and all of them will almost certainly complete the course. That is a big class. The undergraduate class I taught at Princeton this past spring (using my MOOC as one of several resources) had just 9 students.

My MOOC has two main themes: understanding how mathematicians abstract formal counterparts to everyday notions, and how they make use of those abstractions to extend our cognitive understanding of our world.

For much of the time the focus is on language, since that is the mechanism used to formulate and define abstract concepts and prove results about them.

The heavy focus on language and its use in reasoning gives the course appeal to two different kinds of students: those looking to investigate some issues of language use and sharpen their reasoning skills, and those wanting to develop their analytic problem solving skills for mathematics, science, or engineering. (The latter are the ones who typically do the optional final two weeks of the course.)

The pedagogy underlying the course is Inquiry-Based Learning.

To make that approach work in a MOOC, where many students have no opportunity to interact directly with a mathematics expert, I have to design the course in a way that encourages interaction with other students, either on the course Discussion Forum on the course website or using social media or local meetings.

Early in the course, I identify a few students whose Forum posts indicate good metacognitive skills and appoint them “Community Teaching Assistants”. A badge against their name then tells other students that it is worthwhile paying attention to their posts. The CTAs, there are currently thirteen of them, and I also have a back-channel discussion forum to discuss any problematic issues before posting on the public channel.

It seems to work acceptably well. To date, there have been over 3,700 original posts (from 957 students) and 3,639 response comments on the course Discussion Forum.

Since the only practical form of regular performance evaluation in a MOOC involves machine grading – which boils down to some form of multiple choice questions – it’s not possible to ask students to construct mathematical proofs. The process is far too creative.

Instead, I ask them to evaluate proofs (more precisely, purported proofs). To help them do this, I provide a five-point rubric that requires them to view each argument from different perspectives, assigning a “grade” on a five-point numerical scale. See here for the current version of the evaluation rubric.

Notice that the rubric has a sixth category, where they have to summarize their five individual-category evaluations into a single, overall “grade” on the same five-point scale. How they perform the aggregation is up to them. The overall goal is to help the students come to appreciate the different features of proofs, as used in present-day mathematics. The rubric asks them first to look at the proof from the five different perspectives, then integrate those assessments into a single evaluation.

After the students have completed an evaluation of a purported proof, their (numerical) evaluations are machine graded (more about this in a moment), after which they view a video of me evaluating the same proof so they can compare their assessment to one expert.

The goal in comparing their evaluation to mine is not to learn to assign numerical evaluation marks the way I do. For one thing, evaluation of proofs is a very subjective, holistic thing. For another, having been evaluating proofs by both students and experts for many decades, I have achieved a level of expertise that no beginner could hope to match. Moreover, I almost never evaluate using a rubric.

Rather, the point of the exercise is to help the students come to understand what makes an argument (1) a proof, and (2) a good proof, by examining it from different perspectives. (For a discussion of the approach to proofs I take, see my most recent post on my other blog, profkeithdevlin.org.)

To facilitate this, the entire process is set up as a game with rules. (Of course, that is true for any organized educational process, but in the case of my MOOC the course design is strongly influenced by video games – see many of the previous posts in that blog for more on game-based learning, starting here.)

In particular, the points they are awarded (by machine grading) for how close they get to my numerical proof-evaluation score are, like all the points the Coursera platform gives out in my course, very much like the points awarded in a typical video game. They are important in the moment, but have no external significance. In particular, success in the course and the award of a certificate does not depend on a student’s points total. My course offers a learning experience, not a paper qualification. (The certificate attests that they had that experience.)

Overall, I’ve been pleased with the results of this way to handle mathematical argumentation in a MOOC. But it is not without difficulties. I’ll say more in my next post, where I will describe some of the observations I have made so far.

Stay tuned…

 

MathThink MOOC v4 – Part 3

In Part 3, I describe some aspects and origins of the basic course pedagogy, and how they relate to student expectations.

This post continues the previous two in this series.

Expectations. So far I’ve talked about two expectations many students bring to my MOOC that cause problems:

(1) a perception that learning is a cycle of

instruction –> worked examples –> student exercises

(a process that’s better described as training, not learning), and

(2) a belief that failure is something to be avoided (rather than the essential part of learning that it is).

A third problematic expectation many students bring is based on the assumption that mathematics is a body of knowledge to be absorbed, rather than a way of thinking that has to be learned/acquired/developed. That belief is what can lead to the erroneous, and educationally debilitating, perception of mathematics that what makes it hard to learn is the sheer number of different rules and tricks that have to be learned, as described in the article about Jo Boaler’s work I cited in Part 1 of this series.

The view of mathematics as a large collection of procedures can get you quite a way, which explains the huge success of Khan Academy, which shows you all those rules – thousands of them! But it won’t get you to the stage of thinking like a mathematician. Mastering an array of procedures is fine if you are (1) willing to invest the time to keep learning new tricks and (2) prepared to end up working for someone who can do the latter (i.e., think mathematically). Because, increasingly, in the western world, it is that latter that is the valuable commodity. (I wrote about this back in 2008.) My use of the term “mathematical thinking”, rather than just “mathematics”, to title my course was designed to highlight the distinction, but many students nevertheless come to my MOOC expecting a mathematics course (in the sense they have come to understand that term), and are disappointed to discover that it is nothing of the kind. (Some have even asked why I don’t make it more like Khan Academy, a bizarre request which leaves me wondering why they don’t just enter the KA URL in their browser rather than navigate to my MOOC.)

Based on the kinds of issues I’ve been discussing regarding mathematical thinking, in designing my MOOC (and the classroom course that came earlier), I drew on a number of established pedagogies. Most notably among them is Inquiry-Based Learning. For a general background on this powerful and effective learning method, check out this 21-minute video.

Do please watch this video. The focus of much of the video is producing professional mathematicians, and that reflects a common use of the IBL method in mathematics majors classes. In my course, however, with its focus on general mathematical thinking skills for use in many life situations, I don’t ask the students to act as those in a regular IBL class – that would be impossibly hard to pull off in a MOOC in any case. But I believe the general learning principles apply (perhaps even more so), and some of the comments in the video from people who pursued careers in industry address that aspect.

Another pedagogic strategy I adopt is one that has been used in mathematic education since the time of the ancients, which I usually refer to as the Mr Miyagi Method, after the Japanese martial arts expert in the hit 1984 movie The Karate Kid. Having promised to teach karate to the young American Daniel Larusso, Mr Myagi makes his young student paint a fence, wax a floor, and polish several cars. Only with great reluctance does Daniel acquiesce, but in due course he discovers the value of all that effort, as you see from this brief clip.

As I say, this form of teaching has been used in mathematics for centuries. The reason is that in many cases it is impossible to appreciate how mathematics can be applied in a particular situation until enough of the relevant mathematics has been learned. So you design small, self-contained exercises to develop the individual component abilities. Mathematics textbooks have been doing this since they were written on clay tablets five thousand years ago. It’s what most people experience as “mathematics education.”

An attractive alternative is project-based learning. (Again, please do watch this short video.) Unfortunately, whereas PBL is fine for a regular course, in a MOOC that is designed to be of value both to students working on their own, with few if any additional resources, and to students who just participate in a part of the course, it is not an option. That leaves the Miyagi Method as the only game in town.

Even is a regular classroom, and for sure in a MOOC, I would however strongly recommend not adopting Mr Miyagi’s method of delivery. It would surely have been better (as an educational strategy, though not as a movie scene) if he had first explained to Daniel what those chores had to do with learning karate. If a student has to ask, “Why am I learning this?”, the teaching has failed. Why not tell the student from the start?

But remember, times change, and skills and abilities that were valuable in one era sometimes become far less significant, as we are reminded by another Hollywood blockbuster character, Indiana Jones. So you’d better be sure that when you tell a student why a particular topic is important, the reason you give is plausible. (Note: In today’s world, no one balances checkbooks any more – heavens, most people no longer have a checkbook – and no householder uses geometry to figure out how much carpet to order for a room.)

Turning the failing-as-part-of-learning meme on my own journey of learning how to design and give a MOOC, I think that so far I have definitely failed to make sufficiently clear to my MOOC students (1) the basic goals of the course, (2) the approach I am taking to try to achieve those goals, and (3) how those goals lead to adopting the methods I have just outlined above.

To be sure, I laid everything out in detail in the guide-notes I posted on the course website, and in some of my earlier posts to this blog, that I link to from the course site. The problem was, many students never read everything on the site; indeed, some appear not to have read any of the site information.

Now, you might say, they had an obligation to do so. It’s their education, after all, not mine. But MOOCs are about taking learning to a much wider audience than is reached by traditional higher education, and if a MOOC instructor does not manage to connect to that audience, then that is a failure of mission.

As a result, one change I am making with the new version of the course in February is that one of the first things the students will encounter is a video of me explaining the course pedagogy.

[From the very first offering of the course, I posted video discussions between me and my then course TAs, in which we discussed the course design, but those discussions really only made sense after a student had spent some time in the course. So from the second run onwards I cut them into short segments that were released on the site throughout the course. I suspect those discussions were perceived more as “Charlie Rose type” television conversations, rather than providing key information about how to take the course. (In the second of the two discussions, I even asked a professional television and radio host I know to moderate the discussion.) In any case, they did not have the effect I hope will be achieved by a face-to-face explanation by me, as the instructor, of the course goals and structure, given before the course starts. You can view those two earlier videos at: Team Discussion (8mins), What’s New in Number Two (10min 45sec).]

Will my new introductory video solve the problem? I don’t know. For sure, MOOC students do watch (almost) all the videos. Indeed, if there is a problem, it is that some seem to perceive the videos as the most important component of the course, a perception the news media seem to share. Why is that a problem? Because video instruction (i.e., direct instruction) in fact-based, science disciplines does not work. Indeed, instructional videos do actual harm by re-enforcing any prior-held false beliefs, as Derek Muller explains in this video. (Yup, putting math and science education out as a MOOC is hard!)

My guess is that my new introductory video will have an effect, but it will be limited, and many will still be left feeling confused as the course moves ahead. Unfortunately, since the only tools we have at our disposal in a MOOC are video, text, and social media, I don’t see what more I can do, so my gut feeling at this stage is that with the new video I will have gone as far as the medium allows. Nothing works for everyone. All we can do is design for a feasible maximum.

I’ll say (yet) more on this theme of recognizing, anticipating, and dealing with student expectations in my next post. Based on giving three successive versions of my MOOC now, I think the student expectations issue is much more significant in a MOOC than in a regular class. The reason is that in a MOOC, because you have no direct contact with the students, you have very limited ability to counter or correct or allow for those expectations. Your only real strategy is to identify them, and pre-emptively try to lessen their impact on the student.

great-expectations-posterTRAILER (LOOKS GOOD)


I'm Dr. Keith Devlin, a mathematician at Stanford University. I gave my first free, open, online math course in fall 2012, and have been offering it twice a year since then. This blog chronicles my experiences as they happen.

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