# Tag Archives: WeBWorK

## Targeting the inverted classroom approach

Image via Wikipedia

A while back I wondered out loud whether it was possible to implement the inverted or “flipped” classroom in a targeted way. Can you invert the classroom for some portions of a course and keep it “normal” for others? Or does inverting the classroom have to be all-or-nothing if it is to work at all? After reading the comments on that piece, I began to think that the targeted approach could work if you handled it right. So I gave it a shot in my linear algebra class (that is coming to a close this week).

The grades in the class come primarily from in-class assessments and take-home assessments. The former are like regular tests and the latter are more like take-home tests with limited collaboration. We had online homework through WeBWorK but otherwise I assigned practice exercises from the book but didn’t take them up. The mix of timed and untimed assessments worked well enough, but the lack of collected homework was not giving us good results. I think the students tended to see the take-home assessments as being the homework, and the WeBWorK and practice problems were just something to look at.

What seemed true to me was that, in order for a targeted inverted classroom approach to work, it has to be packaged differently and carry the weight of significant credit or points in the class. I’ve tried this approach before in other classes but just giving students reading or videos to watch and telling them we’d be doing activities in class rather than a lecture — even assigning  minor credit value to the in-class activity — and you can guess what happened: nobody watched the videos or read the material. The inverted approach didn’t seem different enough to the students to warrant any change in their behaviors toward the class.

So in the linear algebra class, I looked ahead at the course schedule and saw there were at least three points in the class where we were dealing with material that seemed very well-suited to an inverted approach: determinants, eigenvalues and eigenvectors, and inner products. These work well because they start very algorithmically but lead to fairly deep conceptual ideas once the algorithms are over. The out-of-class portions of the inverted approach, where the ball is in the students’ court, can focus on getting the algorithm figured out and getting a taste of the bigger ideas; then the in-class portion can focus on the big ideas. This seems to put the different pieces of the material in the right context — algorithmic stuff in the hands of students, where it plays to their strengths (doing calculations) and conceptual stuff neither in a lecture nor in isolated homework experiences but rather in collaborative work guided by the professor.

To solve the problem of making this approach seem different enough to students, I just stole a page from the sciences and called them “workshops“. In preparation for these three workshops, students needed to watch some videos or read portions of their textbooks and then work through several guided practice exercises to help them meet some baseline competencies they will need before the class meeting. Then, in the class meeting, there would be a five-point quiz taken using clickers over the basic competencies, followed by a set of in-class problems that were done in pairs. A rough draft of work on each of the in-class problems was required at the end of the class meeting, and students were given a couple of days to finish off the final drafts outside of class. The whole package — guided practice, quiz, rough draft, and final draft — counted as a fairly large in-class assessment.

Of course this is precisely what I did every week in the MATLAB course. The only difference is that this is the only way we did things in the MATLAB course. In linear algebra this accounted for three days of class total.

Here are the materials for the workshops we did. The “overview” for each contains a synopsis of the workshop, a list of videos and reading to be done before class, and the guided practice exercises.

The results were really positive. Students really enjoyed doing things this way — it’s way more engaging than a lecture and there is a lot more support than just turning the students out of class to do homework on their own. As you can see, many of the guided practice exercises were just exercises from the textbook — the things I had assigned before but not taken up, only to have them not done at all. Performance on the in-class and take-home assessments went up significantly after introducing workshops.
Additionally, we have three mastery exams that students have to pass with 100% during the course — one on row-reduction, another on matrix operations, and another on determinants. Although determinants form the newest and in some ways the most complex material of these exams, right now that exam has the highest passing rate of the three, and I credit a lot of that to the workshop experience.
So I think the answer to the question “Can the inverted classroom be done in a targeted way?” is YES, provided that:
• The inverted approach is used in distinct graded assignments that are made to look and feel very distinct from other elements of the course.
• Teachers make the expectations for out-of-class student work clear by giving an unambiguous list of competencies prior to the out-of-class work.
• Quality video or reading material is found and used, and not too much of it is assigned. Here, the importance of choosing a textbook — if you must do so — is very important. You have to be able to trust that students can read their books for comprehension on their own outside of class. If not, don’t get the book. I used David Lay’s excellent textbook, plus a mix of Khan Academy videos and my own screencasts.
• Guided practice exercises are selected so that students experience early success when grappling with the material out of class. Again, textbook selection should be made along those lines.
• In-class problems are interesting, tied directly to the competency lists and the guided practice, and are doable within a reasonable time frame.
These would serve as guidelines for any inverted classroom approach, but they are especially important for making sure that student learning is as great or greater than the traditional approach — and again, the idea of distinctness seems to be the key for doing this in a targeted way.
What are your suggestions or experiences about using the inverted or “flipped” classroom in a targeted way like this?

Filed under Clickers, Inverted classroom, Linear algebra, MATLAB, Screencasts

## Boxplots: Curiouser and curiouser

The calculus class took their third (and last) hour-long assessment yesterday. In the spirit of data analytics ala the previous post here, I made boxplots for the different sections of the test (Conceptual Knowledge (CK), Computation (C), and Problem Solving (PS)) as well as the overall scores. Here are the boxplots for this assessment — put side-by-side with the boxplots for the same sections on the previous assessments. “A2” and “A3” mean Assessments 2 and 3.

Obviously there is still a great deal of improvement to be had here — the fact that the class average is still below passing remains unacceptable to me — but there have been some definite gains, particularly in the conceptual knowledge department.

What changed between Assessment 2 and Assessment 3? At least three things:

• The content changed. Assessment 2 was over derivative rules and applications; Assessment 3 covered integration.
• The way I treated the content in class changed. Based on the results of Assessment 2, I realized I needed to make conceptual work a much greater part of the class meetings. Previously the class meetings had been about half lecture, with time set aside to work “problems” — meaning, exercises, such as “find the critical numbers of $y = xe^{-x}$. Those are not really problems that assess conceptual knowledge. So I began to fold in more group work problems that ask students to reason from something other than a calculation. I stressed these problems from the textbook more in class. I tried to include more such problems in WeBWorK assignments — though there are precious few of them to be had.
• The level of lip service I gave to conceptual problems went up hugely. Every day I was reminding the students of the low scores on Conceptual Knowledge on the test and that the simplest way to boost their grades in the class would be to improve their conceptual knowledge. I did not let their attention leave this issue.

Somewhere in a combination of these three things we have the real reason those scores went up. I tend to think the first point had little to do with it. Integration doesn’t seem inherently any easier to understand conceptually than differentiation, particularly at this stage in the course when differentiation is relatively familiar and integration is brand new. So I think that simply doing more conceptual problems in class and stressing the importance of conceptual knowledge in class were the main cause of the improvements.

Quite interestingly, the students’ scores on computation also improved — despite the reduced presence of computation in class because of the ramped-up levels of conceptual problems. We did fewer computational problems on the board and in group work, and yet their performance on raw computation improved! Again, I don’t think integration is easier than differentiation at this stage in the course, so I don’t think this improvement was because the material got easier. Maybe the last test put the fear of God into them and they started working outside of class more. I don’t know. But this does indicate to me that skill in computation is not strictly proportional to the amount of computation I do, or anybody else does, in class.

To overgeneralize for a second: Increased repetition on conceptual problems improves performance on those problems dramatically, while the corresponding reduction in time spent on computational exercises not only does not harm students’ performance on computation but might actually have something to do with improving it. If we math teachers can understand the implications of this possibility (or at least understand the extent to which this statement is true) we might be on to something big.

The scores on problem solving went two different directions. On the one hand, the median went up; but on the other hand the mean went down. And the middle 50% didn’t get any better on the top end and got worse on the bottom end. I’m still parsing that out. It could be the content itself this time; most of the actual problems in integration tend to take place near the end of the chapter, after the Fundamental Theorem and u-substitution, so the kinds of problems in this section were less than a week old for these students. But quite possibly the improvement in conceptual knowledge brought the median up on problem solving, despite the newness of the problems. Or maybe the differences aren’t even statistically significant.

What I take away from this is that if you want students to do well on non-routine problems, those problems have to occupy a central place in the class, and they have to be done not outside of class where there’s no domain expert to guide the students through them but in class. And likewise, we need not worry so much that we are “wasting precious class time” on group work on conceptual problems at the expense of individual computation skill. Students might do just fine on that stuff regardless, perhaps even better if they have enhanced conceptual understanding to support their computational skills.

It all goes back to support the inverted classroom model which I’ve been using in the MATLAB course, and now I’m wondering about its potential in calculus as well.