Month: September 2018

Total 2 Posts

Big Online Courses Have a Problem. Here’s How We Tried to Fix It.

The Problem

Here is some personal prejudice: I don’t love online courses.

I love learning in community, even in online communities, but online courses rarely feel like community.

To be clear, by online courses I mean the kind that have been around almost since the start of the internet, the kind that were amplified into the “Future of Education™” in the form of MOOCs, and which continue today in a structure that would be easily recognized by someone defrosted after three decades in cold storage.

These courses are divided into modules. Each module has a resource like a video or a conversation prompt. Students are then told to respond to the resource or prompt in threaded comments. You’re often told to make sure you respond to a couple of other people’s responses. This is community in online courses.

The reality is that your comment falls quickly down a long list as other people comment, a problem that grows in proportion to the number of students in the course. The more people who enroll, the less attention your ideas receive and consequently you’re less interested in contributing your ideas, a negative feedback loop which offers some insight into the question, “Why doesn’t anybody finish these online courses?

I don’t love online courses but maybe that’s just me. Two years ago, the ShadowCon organizers and myself, created four online courses to extend the community and ideas around four 10-minute talks from the NCTM annual conference. We hosted the courses using some of the most popular online course software.

The talks were really good. The assignments were really good. There’s always room for improvement but the facilitators would have had to quit their day jobs to increase the quality even 10%.

And still retention was terrible. 3% of participants finished the fourth week’s assignment who finished the first week’s.

Low retention from Week 1 to Week 4 in the course.

The organizers and I had two hypotheses:

  • The size of the course enrollment inhibited community formation and consequently retention.
  • Teachers had to remember another login and website in order to participate in the course, creating friction that decreased retention.

Our Solution

For the following year’s online conference extensions, we wanted smaller groups and we wanted to go to the people, to whatever software they were already using, rather than make the people come to us.

So we used technology that’s even older than online course software, technology that is woven tightly into every teacher’s daily routine: email.

Teachers signed up for the courses. They signed up in affinity groups – coaches, K-5 teachers, or 6-12 teachers.

The assignments and resources they would have received in a forum posting, they received in an email CC’d to two or three other participants, as well as the instructor. They had their conversation in that small group rather than in a massive forum.

Of course this meant that participants wouldn’t see all their classmates’ responses in the massive forum, including potentially helpful insights.

So the role of the instructors in this work wasn’t to respond to every email but rather to keep an eye out for interesting questions and helpful insights from participants. Then they’d preface the next email assignment with a digest of interesting responses from course participants.

The Results

To be clear, the two trials featured different content, different instructors, different participants, and different grouping strategies. They took place in different years and different calendar months in those years. Both courses were free and about math, but there are plenty of variables that confound a direct comparison of the media.

So consider it merely interesting that average course retention was nearly 5x when the medium was email rather than online course software.

Retention was nearly five times greater in the email course than LMS.

It’s also just interesting, and still not dispositive, that the length of the responses in emails were 2x the length of the responses in the online course software.

Double the word count.

People wrote more and stuck around longer for email than for the online course software. That says nothing about the quality of their responses, just the quantity. It says nothing about the degree to which participants in either medium were building on each other’s ideas rather than simply speaking their own truth into the void.

But it does make me wonder, again, if large online courses are the right medium for creating an accessible community around important ideas in our field, or in any field.

What do you notice about this data? What does it make you wonder?

Featured Comments

Leigh Notaro:

By the way, the Global Math Department has a similar issue with sign-ups versus attendance. Our attendance rate is typically 5%-10% of those who sign up. Of course, we do have the videos and the transcript of the chat. So, we have made it easy for people to participate in their own time. Partipating in PD by watching a video though is never the same thing as collaborating during a live event – virtually or face-to-face. It’s like learning in a flipped classroom. Sure, you can learn something, but you miss out on the richness of the learning that really can only happen in a face-to-face classroom of collaboration.

William Carey:

At our school now, when we try out new parent-teacher communication methods, we center them in e-mail, not our student information system. It’s more personal and more deeply woven into the teachers’ lives. It affords the opportunity for response and conversation in a way that a form-sent e-mail doesn’t.

Cathy Yenca:

At the risk of sounding cliché or boastful about reaching “that one student”, how does one represent a “data point” like this one within that tiny 3%? For me, it became 100% of the reason and reward for all of the work involved. I know, I know, I’m a sappy teacher :-)

Justin Reich is extremely thoughtful about MOOCs and online education and offered an excellent summary of some recent work.

2018 Oct 5. Definitely check out the perspective of Audrey, who was a participant in the email group and said she wouldn’t participate again.

2018 Oct 12. Rivka Kugelman had a much more positive experience in the email course than Audrey, one which seemed to hinge on her sense that her emails were actually getting read. Both she and Audrey speak to the challenge of cultivating community online.

What Does Fluency Without Understanding Look Like?

In the wake of Barbara Oakley’s op-ed in the New York Times arguing that we overemphasize conceptual understanding in math class, it’s become clear to me that our national conversation about math instruction is missing at least one crucial element: nobody knows what anybody means by “conceptual understanding.”

For example, in a blog comment here, Oakley compares conceptual understanding to knowing the definition of a word in a foreign language. Also, Oakley frequently cites a study by Paul Morgan that attempts to discredit conceptual understanding by linking it to “movement and music” (p. 186) in math class.

These are people publishing their thoughts about math education in national publications and tier-one research journals. Yet you’d struggle to find a single math education researcher who’d agree with either of their characterizations of one of the most important strands of mathematical proficiency.

Here are two useful steps forward.

First, Adding It Up is old enough to vote. It was published by the National Research Council. It’s free. You have no excuse not to read its brief chapter on procedural fluency. Then critique that definition.

Conceptual understanding refers to an integrated and functional grasp of mathematical ideas. Students with conceptual understanding know more than isolated facts and methods. They understand why a mathematical idea is important and the kinds of contexts in which is it useful. They have organized their knowledge into a coherent whole, which enables them to learn new ideas by connecting those ideas to what they already know. Conceptual understanding also supports retention. Because facts and methods learned with understanding are connected, they are easier to remember and use, and they can be reconstructed when forgotten. (pp. 118-119.)

If you’re going to engage with the ideas of a complex field, engage with its best. That’s good practice for all of us and it’s especially good practice for people who are commenting from outside the field like Oakley (trained in engineering) and Morgan (trained in education policy).

Second, math education professionals need to continually articulate a precise and practical definition of “conceptual understanding.” In conversations with people in my field, I find the term tossed around so casually so often that everyone in the conversation assumes a convergent understanding when I get the sense we’re all picturing it rather differently.

To that end, I think it would be especially helpful to compile examples of fluency without understanding. Here are three and I’d love to add more from your contributions on Twitter and in the comments.

A student who has procedural fluency but lacks conceptual understanding …

  • Can accurately subtract 2018-1999 using a standard algorithm, but doesn’t recognize that counting up would be more efficient.
  • Can accurately compute the area of a triangle, but doesn’t recognize how its formula was derived or how it can be extended to other shapes. (eg. trapezoids, parallelograms, etc.)
  • Can accurately calculate the discriminant of y = x2 + 2 to determine that it doesn’t have any real roots, but couldn’t draw a quick sketch of the parabola to figure that out more efficiently.

This is what worries the people in one part of this discussion. Not that students wouldn’t experience delirious fun in every minute of math class but that they’d become mathematical zombies, plodding functionally through procedures with no sense of what’s even one degree outside their immediate field of vision.

Please offer other examples in the comments from your area of content expertise and I’ll add them to the post.

BTW. I’m also enormously worried by people who assume that students can’t or shouldn’t engage creatively in the concepts without first developing procedural fluency. Ask students how they’d calculate that expression before helping them with an algorithm. Ask students to slice up a parallelogram and rearrange it into a more familiar shape before offering them guidance. Ask students to sketch a parabola with zero, one, or two roots before helping them with the discriminant. This is a view I thought Emma Gargroetzi effectively critiqued in her recent post.

BTW. I’m happy to read a similar post on “conceptual understanding without procedural fluency” on your blog. I’m not writing it because a) I find myself and others much less confused about the definition of procedural fluency than conceptual understanding (oh hi, Adding It Up!) and b) I find it easier to help students develop procedural fluency than conceptual understanding by, like, several orders of magnitude.

2018 Sep 05: The Khan Academy Long-Term Research team saw lots of students who could calculate the area of a kite but wrote variations on “idk” when asked to defend their answer.

2018 Sep 09: Here’s an interesting post on practice from Mark Chubb.

2018 Sep 27: Useful post from Henri Picciotto.

Featured Tweets

Featured Comments

Karen Campe:

Can find zeros of factored quadratic that equals zero, but uses same approach when doesn’t equal zero. E.g. can solve (x-3)(x-2) = 0 but also answers 3 and 2 for (x-3)(x-2) = 6.

Ben Orlin:

The big, weird thing about math education is that most pupils have no experience of what mastery looks like. They’ve heard language spoken; they’ve watched basketball; they’ve eaten meals; but they probably haven’t seen creative mathematical problem-solving. This makes it extra important that they have *some* experience of this, as early as possible. Otherwise math education feels like running passing drills when you’ve never seen a game of basketball.

Mike:

Today a student correctly solved -5=7-4x but then argued that -4x +7=-5 was a different equation that had to have a different answer.

Michael Pershan:

This has definitely not been my experience, and I don’t think this is consistent with the idea that conceptual and procedural fluency co-develop — an idea rooted in research.

William Carey:

I really like that way of talking about it. The way I think of it is a bit like exploration of an unknown continent. One the one hand, you have to spend time venturing boldly out into the unknown jungle, full of danger and mistakes and discovery. But if you venture too far, you can’t get food, water, and supplies up to the party. Tigers eat you in the night. So you spend time consolidating, building fortified places, roads, wells, &c. Eventually, the territory feels safe, and that prepares you to head into the unknown again.

Jane Taylor:

A student who can calculate slope but has no idea what it means as the rate of change in a real context.

Kim Morrow-Leong:

An example of procedural fluency without conceptual understanding is adding up a series of integers one by one instead of finding additive inverses (no need to even call it an additive inverse – calling it “canceling” would even be ok.) Example: -4 + 5 + -9 + -5 + 4 + 9

2018 Oct 13 NCTM offers their own definition of procedural fluency in mathematics.