**In Part 1 of a series, I focus on the distinction between high school math and university-level mathematics, suggesting they are effectively different subjects that are best learned in different ways.**

One of the biggest obstacles in giving an online course on mathematical thinking, which my MOOC is, is coping with the expectations students bring to the course – expectations based in large part on their previous experience of mathematics classes. To be sure, prior expectations are often an issue for regular, physical classes. But there the students have an opportunity to interact directly with the instructor on a regular basis. They also have the benefit of a co-present support group of others taking the same class.

But in a massive open online class, apart from locally configured support groups and text-based discussions on the MOOC platform discussion forum, each student is pretty much on her or his own.

The situation is particularly bad for a course like mine, designed to help students transition from high school mathematics to university-level mathematics. For one thing, the two are so different as to be in many ways completely distinct subjects.

School mathematics tends to be almost exclusively procedural, mastering established methods to solve artificially constructed problems designed to be amenable to such an approach. The student who best masters all the techniques in the syllabus and becomes skillful in pattern-matching problems to solution methods, does well. (I know that first hand; it’s how I got to university to study mathematics!)

In contrast, university mathematics is about learning how to deal with a novel situation of a kind you have not encountered before. (If no one else has encountered it, we call it mathematics research.) Though it certainly can involve pattern matching and the application of established, standard procedures, it usually does so only as components of a novel solution you develop to deal with that particular situation. Moreover, at university level, the problems are typically of a “prove that this is true (or false)” variety, rather than “solve this equation” or “compute the value of that formula.”

What is more, while a school math problem typically has a right answer, university mathematics generally involves much more than mere correctness. Indeed, there may not be a unique “right answer.”

Not only is the subject matter different, so too is the pedagogy. Almost all students’ experience of mathematics learning in school is teacher instruction. The teacher describes a method, does a few worked examples, and then asks the students to do a few similar ones. Rinse and repeat.

It’s a very efficient way to cover a lot of ground when the goal is pattern matching and procedure application. It works for school mathematics. Unfortunately, it does not prepare the graduates for the other kind of mathematics. (It also leaves them without ever having a satisfactory answer to their question “What is this good for?”, a question that leaves anyone versed in mathematics astounded. “What is it not good for?” is a more interesting question. It does not have a simple answer, by the way. It’s a very nuanced question.)

It’s like teaching someone the elements of bricklaying, carpentry, plumbing, and electrical wiring, and then asking them to go out and design and build a house. You need all of those skills to build a house, but on their own they are not enough. Not even close.

In deciding, almost two years ago now (before the New York Times had heard of MOOCs) to develop a MOOC to help people learn the other kind of mathematics, what I call *mathematical thinking*, I knew I was taking on a big challenge. I’d found it hard to teach that kind of course in a physical classroom with just 25, carefully selected students at elite colleges and universities.

On the other hand, most people go through their entire mathematics education without ever encountering what I and my colleagues would call “real mathematics,” and many of them eventually find they need to be able to handle novel situations that involve – or may involve – or could productively be made to involve – mathematical thinking. So I felt there was a need to have a resource publicly available to help them acquire this valuable ability.

The huge dropout rates in MOOCs did not really bother me. For a mathematical thinking course, it’s possible to gain value from dropping into the course for just a few days – and to keep coming back at future times if required. The focus was not on credentialing, it was developing a valuable mental ability – a powerful way of thinking that our ancestors have developed over three thousand years.

That way of thinking can be utilized profitably in many other courses that do yield a certified credential, so students could approach the course as a low-stress, no-risk way of preparing for subsequent learning.

The course *is* structured as course for those students who seek an encapsulated experience, and in many ways that yields the greatest benefits, in large part because of the interactions with other students working on the same stuff. But the majority of students who have taken it the three times I have offered it have just taken a part of the course.

Each time I gave the course, I changed it, based on what I had learned. When it launches again in February, it will be different again. This time, in some fairly significant ways. In the coming days, I’ll describe those changes and why I made them.

First out of the gate, I’ll describe what exactly were the problems caused by those expectations many students brought to the course, and how did I try to deal with them. Also, what am I changing in the coming version of the course to try to help more people make what is a very difficult transition: from being *taught* (i.e., *instructed*) to being able to *learn*. The reward for making that one transition is huge. It opens up all of mathematics, and in the process makes it much, much easier.

The traditional, instructional way of teaching procedural mathematics frequently leaves students with the impression (dramatically documented by my Stanford colleague Jo Boaler) that mathematics consists of a large number of rules to be learned. But at the risk of sounding like those weird web advertisements (you know, the ones with a drawing or photo of a strange looking person) promising to teach that “one great trick” that will change your life, let me leave you by telling you the one great trick that all mathematicians learn:

You just have to master, *once*, a particular way of thinking, and you no longer need all those rules.

That’s what my course focuses on. Stay tuned.

What were your enrollment and real finishers with certificate in your previous courses .

Total enrollment: 58,300 (a meaningless figure)

Total active at some time: 44,141 (includes the many who never intended to do more than dabble)

Total active in final week: 4,961 (the “real class,” those taking the full course

as a course)Total that completed the course (more than minimal or “passive activity”): 3,900

Total submitting Final Exam: 978

Number of students receiving a SoA: 3,167

Number of students awarded a SoA with Distinction: 676

I have taken several MOOCS now, and my experience is in line with yours. MOOCs are great for a refresher or to learn new stuff, once you have already learned how to learn. I saw a statistic that 85% of people taking MOOCs have a degree (BA/BS ?) already. I would imagine that these people’s experience is similar to mine.

I think that your course is about the best that can be done for a MOOC. The pedagogy and technology is designed to help the student at every step. Judging by some comments in the forums, a sizable fraction of students who had not seen this material before have learned the basics of mathematical thinking. Whether this is sufficient is an open question.

Judging from what I have read, learning the rudiments of mathematical thinking (or just logical, rational, evidence based thinking) should be taught from a very early age, and not left for the transition from high school to college. Doing so would certainly help people to understand what mathematics *really* is, rather than the present misunderstanding. It would also change the way education is viewed, from memorizing ‘received wisdom’ to learning how to think critically about anything — plus a set of critical ‘data’, which includes history, science, arts and literature.

I agree entirely that this kind of thinking should be developed much earlier. Particularly as we have ubiquitous devices that do arithmetic and other basic computational mathematics for us, whereas no device built is capable of good mathematical thinking – an ability of real importance it today’s worlds.

Conrad Wolfram has a good sound-bite: We need students to be first-rate problem solvers, not third rate computers. One problem — political and social — seems to be that it easier to train and measure computation than it is to teach and measure critical thinking. This is not a new problem, and anecdotally, it seems that the only thing that makes a real difference is an inspiring teacher or mentor. I think that a lot of people hate math because they encountered an anti-mentor in their early education. Once that path is set, it is very hard to change.