In my role in the Carl Wieman Science Education Initiative at the University of British Columbia, I am often “embedded” in an instructor’s course, providing resources, assistance and coaching throughout the term. This term, I’m working with an instructor in a final-year, undergraduate electromagnetism (E&M) course.

The instructor has already done the hard part: he recognized that students were not learning from his traditional lectures and committed to transforming his classes from instructor-centered to student-centered. Earlier, I wrote about how we introduced pre-reading assignment and in-class reading quizzes.

This course is heavy on the math. Not new math techniques but instead, math the students have learned in the previous 3 or 4 years applied to new situations. His vision, which he shared with the students on the first day, was to introduce some key concepts and then let them “do the heavy lifting.” And by heavy lifting, he means the algebra.

The vector for this heavy lifting is daily, in-class worksheets. The students work collaboratively on a sequence of questions, typically for 15-20 minutes, bookended by mini-lectures that summarize the results and introduce the next concept.

We’re making great strides, really. After some prompting by me, the instructor is getting quite good at “conducting” the class. There are no longer moments when the students look at each other thinking, “Uh, what are supposed to be doing right now? This worksheet?” It’s fine to be puzzled by the physics, that’s kind of the point, but we don’t want students wasting any of their precious cognitive load on divining what they should be doing.

With this choreography running smoothy and the students participating, we’re now able to look carefully at the content of the worksheets. Yes, I know, that’s something you should be planning from Day 1 but let’s face it, if the students don’t know when or how to do a worksheet, the best content in the World won’t help them learn. Last week’s worksheet showed we’ve got some work to do.

The instructor handed out the worksheet. Students huddled in pairs for a minute or two and them slumped back into their seats. You know those cartoons where someone gets smacked on the head and you see a ring of stars or canaries flying over them? You could almost see them, except the canaries were the library of equations the students are carrying in their heads. They’d grasp at a formula floating by, jam it onto the page, massage it for a minute or two, praying something would happen if they pushed the symbols in the right directions. *Is it working? What if I write it like….solve for….Damn. Grab another formula out of the air and try again…*

After 10 minutes, some students had answered the problem. Many others were still grasping at canaries. The instructor presented his solution on the document camera so he could “summarize the results and introduce the next concept.” The very first symbols at the top-left of his solution were exactly the correct relationship needed to solve this problem, magically plucked from his vast experience. With that relationship, and a clear picture of where the solution lay, he got there in a few lines. The problem was trivial. No surprise, the students didn’t react with “Oh, so that’s why physics concept A is related to physics concept B! I always wondered about that!” Instead, they responded with, “Oh, so that’s how you do it,” and snapped some pix of the screen with their phones.

### Scaffolding and Spoon-feeding

We want the worksheets to push the students a bit. A sequence of questions and problems in their grasp or just beyond, that guide them to the important result or concept of the day. Here’s what doesn’t work: A piece of paper with a nasty problem at the top and a big, blank space beneath. I’ve seen it, often enough. Students scan the question. The best students dig in. The good and not-so-good students scratch their heads. And then bang their heads until they’re seeing canaries.

There are (at least) 2 ways to solve the problem of students not knowing how to tackle the problem. One is to scaffold the problem, presenting a sequence of steps which activate, one by one, the concepts and skills needed to solve the nasty problem. The Lecture Tutorials used in many gen-ed “Astro 101” astronomy classes, and the Washington Tutorials upon which they’re modeled, do a masterful job of this scaffolding.

Another way, which looks the same on the surface, is to break the nasty problem into a sequence of steps. “First, find the relationship between A and B. Then, calculate B for the given value of A. Next, substitute A and B into C and solve for C in terms of A…” That’s a sequence of smaller problems that will lead to a solution of the nasty problem. But it’s not scaffolding: it’s spoon-feeding and it teaches none of the problem-solving skills we want the students to practice. I’ve heard from number of upper-level instructors declare they don’t want to baby the students. “By this stage in their undergraduate studies,” the instructors say, “physics students needs to know how to tackle a problem from scratch.”

This is the dilemma I’m facing. How do we scaffold without spoon-feeding? How do we get them solving nasty problems like a physicist without laying a nice, thick trail of bread crumbs?

Fortunately, I have smart colleagues. Colleagues who immediately understood my problem and knew a solution: Don’t scaffold the nasty problem, scaffold the problem-solving strategy. For a start, they say, get the instructor to model how an expert physicist might solve a problem. Instead of slapping down an elegant solution on the document cam, suppose the instructor answers like this:

**Determine what the problem is asking.***Alright, let’s see. What is this problem about? There’s A and B and their relationship to C. We’re asked to determine D in a particular situation.***Identify relevant physics.***A, B, C and D? That sounds like a problem about concept X.***Build a physics model.**Identify relevant mathematical relationships. Recognize assumptions, specific cases. Select the mathematical formula that will begin to solve the problem.**Execute the math**. Carry out the algebra and other manipulations and calculations.

(This is where the instructor has been starting his presentation of the solutions.)**Sense-making**.*Sure, we ended up with an expression or a number. Does it make sense? How does it compare the known cases when A=0 and B goes to infinity? How does the order of magnitude of the answer compare to other scenarios?*In other words, a few quick tests which will tell us our solution is incorrect.

Wouldn’t it be great if every student followed a sequence of expert-like steps to solve every problem? Let’s teach them the strategy, then, by posing each nasty problem as a sequence of 5 steps. “Yeah,” my colleagues say, “that didn’t work. The students jumped to step 4, push some symbols around and when a miracle occurred, they went back and filled in steps 1, 2, 3 and 5.” Students didn’t buy into the 5-step problem-solving scheme when it was forced upon them.

So instead, for now, I’m going to ask the instructor to model this approach, or his own expert problem-solving strategy, when he presents his solutions to the worksheet problems. When the students see him stop and think and ponder, they should realize this is an important part of problem-solving. The first thing you do isn’t scribbling down some symbols. It’s sitting back and thinking. Maybe even debating with your peers. Perhaps you have some insight you can teach to your friend. Peer instruct, that is.

Love this post! One of the most challenging things is to get students to generalize physics problems and not memorize solutions to specific problems. I think the method scaffolding is an excellent tool to teach this. :)

Thanks, Carrie. I swear I could see the canary-formulas circling around the students’ heads…

The unfortunate part is that I’m writing about a 4th-year physics class. Why isn’t this scaffolding approach, or some other problem-solving strategy, already taught and fine-tuned by the time these students reach this advanced stage. I guess too many instructors don’t realize how much thinking they do before putting pen to paper, or think they are doing a good job by breaking nasty problems into a sequence of small steps. Little do we realize that choosing the next step is just as important as carrying it out.

Interestingly enough, I apply a very similar model to public policy making. There are so many similarities amongst disciplines that it’s frustrating more people don’t speak interdisciplinarity. But you do, my friend, and that’s amazing. Great post.

That’s certainly a theme I’ve seen throughout my time in the CWSEI — the actual content of our courses accounts for only a fraction of what it takes to be an effective instructor. The skills required to create a classroom culture of learning, choreographing activities and interacting with students are skills that cross disciplines. Thank-you for the kind words, Raul.

Peter,

It’s interesting that your complaints of spoon feeding are the exact same as most people (me included) would have regarding traditional cookbook introductory labs.

I like to use clickers to do something similar to what you are discussing with the worksheets for problems/examples in class. They do tend to lean to the spoon feeding side because it is usually the first time the students have encountered the given concept in an exercise type of situation. But it seems worthwhile to put in some effort to make these “clicker-based examples” more scaffolded and less about spoon feeding. One of the things I consider important (but don’t always do very well) when spoon feeding is necessary is to step back afterward and recap (with their help) how we got from the beginning to the end of the problem/example.

I agree that sometime you just can’t avoid (or perhaps even want) the occasional spoon-feeding problem. I really like your practice of stepping back and the end and look at the big picture.

I’m a big fan of integrating peer instruction (via i>clicker, in my case) and worksheets. It’s good to give students feedback during the worksheet so they can see if they’re on-track and keep up their confidence. However, going over the answer to Question 1, then 2, then 3 allows the students to sit back and wait for the solution as soon as they get stuck. Instead, we try to pose a clicker question at a time when most students have finished, say, Question 3. The clicker assesses the concepts up to that point in the worksheet. If they get the question, they know they’re doing fine. If they don’t get it, their peers get them back on track. In this way, we give them feedback without simply giving the answer to the question. Well, that’s the plan, anyway :)

It really varies a lot how I time the presentation and voting of these questions. I have never used worksheets in this context (examples/problems), but that is something I would really like to try out in the future. That, and less spoon-feeding.

For worksheets that are more in the style of the University of Washington Tutorials, they tend to build toward a “do you really get it?” type of question at the end of the sub-section. These seem to be well suited for use as a clicker question, and then after the peer instruction, they can go back and clean up the stuff that they didn’t get correct leading up to that question.

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Our physics education group RELATE at MIT very much concur with with Peter’s observations about the value of scaffolding problem solving (which, it should be noted, harks back to and extends Polya’s 1940s classic “How to Solve It” approach to math problems).

We have developed a pedagogical approach called Modeling Applied to Problem Solving (MAPS), which we are using in our free Mechanics Online course, just launched on March 1. The emphasis in our course is teaching students to approach challenging multi-concept problems systematically, to concentrate on planning and understanding the solution rather than focusing on obtaining the answer.

We published some evidence of the effectiveness of MAPS when used in a three-week review course for students who struggled with first semester mechanics.

Thanks for sharing those links to your group’s work. The CLASS results you report in the compadre paper are really encouraging. Their largest positive shifts are in the problem-solving categories so it certainly suggests students find the approach useful. That you also see positive shifts in conceptual understanding — that’s a very nice consequence!

Interesting, this problem-solving strategy is similar to George Polya’s How To Solve It (as I mentioned the other day during my seminar). Okay, the book does focus on thinking about mathematics, but it really is about problem-solving. Polya tried to dissect the art of problem-solving into concrete steps of “mental operations” and questions that can be applied to map new problems onto old ones that you’ve seen before.

Thanks for making that connection, Dhavide. Sometimes I feel I might be wasting my time, writing about things like this. After all, Polya already wrote about it so well. But then I reassure myself that it’s useful to struggle and think about things for myself first. You should see the awesome wheel I reinvented last week…

Speaking of your seminar last week, I really like the way you took Polya’s algorithm…

1. Understand the problem.

2. Devise a plan.

3. Carry out the plan.

4. Check your solution

…and applied to the problem of teaching a course. You’re absolutely right that many of us are guilty of diving in at Step 3 without Steps 1 and 2. And rarely do we carry out Step 4 after the term is over.

What do you think about the research by Kapur on no scaffolding aka productive failure?

http://www.tandfonline.com/doi/abs/10.1080/07370000802212669#preview

Thanks for the link, Matt. I certainly wasn’t thinking about productive failure when I watched the students struggling in this physics class. Honestly, I haven’t read the research about productive failure but it seems to have some similarities to the “invention activities” developed by Dan Schwartz and colleagues. In those activities, students are given just enough to get them started and they’re asked to invent a result, for example, an algorithm for determining standard deviation. They’re not expected to get the right answer. But by struggling with the idea and the inputs and variables that contribute to the answer, they are (in Schwartz’ words) “prepared to learn.” And they do.

I know from working with invention activities, the is a lot of scaffolding going on. But it’s not the concept that’s scaffolded, it’s the activity. The scenarios, which information is revealed, what question or goal gets the students thinking about the right things: these are all carefully designed and choreographed. So is the content of the presentation that occurs during the “time for telling.” I imagine it’s the same with activities designed around productive failure. It’s certainly not just letting the students flounder, grasping at canaries.