# Instructional Routine: Problem Solving

I’ve been trying to work on developing a instructional routine that I can pass on to the other instructors in our introductory college physics course. The instructional routine pertains to getting students engaged in problem-solving, and is intended to give them an alternative process to our Department’s typical routine.

So what’s our typical routine? Our typical routine, which certainly isn’t horrible, is “Example Problem –> Whiteboard Problem”. That is, the instructor poses a word problem, models how to solve the word problem, and then students are posed a similar word problem and are asked to solve it collaboratively on whiteboards, following the general process. This can go pretty well or pretty so-so depending on a lot of the details, but one of the biggest pitfalls can be this:

- The process can put students into “Monkey see, Monkey do” mode, where they expect to mindless follow the algorithm their teacher teaches them.

So here’s the basic routine I’ve been working on (for which there are many variants),

**First: Posing the Qualitative Scenario** (with or without a question)

An example might be, “If we go outside and throw a ball as hard as we can, will it go high enough to get on top of the roof?”

**Second:** **Making the Scenario a Problem**

The qualitative scenario begs the question, “It depends”, so in the routine you typically ask students to think about what it depends on. It helps me to explicitly ask, “What information would we want to know about the situation to help decide?”

For the situation with throwing the ball above, I have gotten the following:

- How tall the building is?
- How tall the person is?
- How fast the person throws the ball?
- What the weather conditions are?
- What type of ball is being thrown?

The instructor’s job is to “catch every contribution” and to either press the student connect it to disciplinary constructs or to offer a connection. One can ask follow ups, such as “Can you tell us why you think the height of the person matters?”, or weigh in yourself (e.g., “Yeah, that makes sense, a taller person might have an easier time getting). In the above scenario, we had just talked about the conditions for free-fall, so I helped to offer the last two as information that would help us to know whether the conditions were or were not consistent with free-fall assumption. Depending on how much scaffolding you want at this time, you might connect each of those to constructs you’ve talked about like, “initial velocity”, “initial position”, but you might not.

The next task is to decide how that information will be gathered. I suggest one of three things can be done (or even some combination)

- Estimating (e.g, how fast does a car go on a typical highway)
- Measuring (e.g., using photogate to measure launch velocity.”
- Researching (e.g., what are typical car braking accelerations.)

For the example above, I asked students to estimate the building height, person height, and initial velocity, and to also come up with a specific ball / weather that they think would be fairly consistent with free-fall. This is a good exercise in estimating but also in understanding the assumptions that go behind a particular model.

Students in my class, did a mix of estimate and measure. They measured the height of our room and used knowledge of how many floors to estimate the building height. Other groups used knowledge of baseball pitching speeds in mph to come up with a reasonable “normal person vertical speed.”, but others looked up information for “how fast can humans throw stuff.”

If students are to work the problem, the instructor can choose to have the class all work the same problem (get consensus on good estimates), or let groups work with different estimates. This day, we did a consensus problem: 1meter tall person, 20 meter tall building, 25 m/s throw, a baseball (not a ping pong ball), and no wind / no rain. I chose this day to do the unit converting for them from their speed estimate which was in mph, but I wouldn’t always do this. I also directed us to the specific question of, “How high above or below the roof height does the ball get?” so that it wasn’t just “yes or no”.

The truth is the whole setup process can be fairly quick (or stretched out) depending on needs. Even just giving students a 1-2 minutes to chat in groups, collecting their ideas at the board and quickly connecting it to disciplinary constructs, goes a long way for students to orient to the situation and understand the relevance of the information. You might quickly take measurements for students, or involve them in a process of measuring that information that takes significant time. It all depends. The point is to spend sometime turning the situation and question into a problem, and along the way practice some other skills such as estimation, measurement, making assumptions, etc. It need not be everything.

**Solving the Problem: Creating and Using a Model**

In my mind, the above steps play out well whether you are going to model how to solve a problem or have students solve the problem. In my class, students had already solved 1D acceleration problems, and we had just finished activities and discussion about free fall, so students were ready to jump into problem solving. In my class, they have to make motion diagrams, both qualitative position and velocity diagram, and then either write equations that describe those graphs, or use slope/area ideas.

There can be some advantages to all doing the same numbers, especially for a more novice instructor. You can more readily scan the room and compare students’ progress, and students can more readily compare their answers.

There are also good advantages to having different numbers out there. For example, if students choose different speeds, then you might be able to discuss how the speed seemed to effect the maximum height reached.

You can also deliberately seed some good comparisons — two groups both do 1m and 2m tall persons with the same speed, or 10 m/s, 20 m/s, 30 m/s throwing speeds.

**Evaluating and Re-Posing**

At the end, we want students to evaluate the reasonableness of their answer. This can be done in variety of ways. Depending on the type of scenario you are posing, you might be able to compare to experiment. One trick for doing this can be to ask students early on, maybe even before even posing the situation, “What do you think is highest height a ball can be thrown up to? What’s a height you know is too high? What’a a height you know is too low?” That way they can compare their final answer to the bounds they previously set as reasonable.

Re-posing is important, and there are couple ways this can go. Posing Questions First. You can ask students to pose different question about the same scenario (e.g., students asked in my class how much time the ball would be in the air when it hit the roof… how fast it would hit the roof). But you can also ask students to pose slightly new scenarios, like, “What’s the minimum speed needed to get it on the roof?” or “If you throw the ball with twice as much speed, will it go twice as high?” You can certainly pose questions to students. For example, after problem-solving I had a clicker question about on the issue how doubling the speed would effect the height.” But it’s important to get students in engaged in the process of asking question about the same scenario, posing questions about different scenarios. The last type of re-posing is a bit more school-ish but also helpful, “What’s a problem similar to this one that could be asked, but that’s more difficult to solve?”

This last step also gives quick students more and more places to go while other groups finish up. I also have one of my quick groups write simulations in desmos, as part of their modeling process.

**Summary of the Routine:**

Certainly, you could choose to invest a lot of time in having students fully engage in all the steps, but that’s not necessary. The point is to always engage the whole process, but students can carry more of the responsibility for some parts, you can carry more the responsibility for other parts. But here are the four main parts I see.

- Interest Making
- Problem Making
- Modeling Making
- Sense Making

I say the first one is “interest making” is because showing students actual situations or qualitative situations should be done in a manner as to possibly cultivate an inkling of engagement and curiosity about the situation. I’m not saying it needs to be magical.

The second one is “problem making”, because it’s about turning that situation into a problem by discerning relevant information, relating it to physics concepts. This is the core of the practice that I want students to get more practice with, and is the one that word problems are the worst at skipping.

The third one is to modeling making, because everyone is to develop and apply a model that can be used to solve the problem. This is what we typically think of as problem-solving, and this is going to look different in different classrooms, curricula, etc.

The last one is sense-making about returning to think about the outcome, the model, rethink the situation, and even new situations. In a sense, the end process puts you back at the beginning of posing situations and asking questions, so you come full circle. There are lots of different ways of sense-making, but it needs to be an explicit part of the process, and it needs to be more than, “Does your answer make sense?”

**Challenge: Make it Simple, Imaginable, and Flexible:**

The issue I see with any instructional routine is having it crafted as simple enough for an instructor to get started with it. They need to be able to not just follow the steps, but have a feel for the purpose of each step, and probably have a good image / sense of what it can look like. You don’t want to get bogged down in all the possible variations at first, but have it possible that flexibility will open up through practice. I got to figure out what’s the a good “touchstone” example for this routine is.

Sorry for the ramble, just getting down my thoughts.

First, don’t apologize for rambling, this stuff is gold!

This post made me think about a lot of things.

1) What’s better, doing this at the very beginning of the unit or when you did when they’d already done some work with 1D constant acceleration? I think I would only like the former if they’re curiosity carries them through the work.

2) I think most of us would appreciate this articulation of a process but I would also think that most of us would not consider this a wholly new approach. My question is whether we can learn from each other about a best practice (say, to the question of mine above in (1)) or whether it’s better for us all to read blogs like this to get into the mindset of constant reflection.

3) I will occasionally use a tool I first experienced in my own HS class where I ask the class to solve a problem together and get the answer on the board after, say, 15 minutes. I tell them that answer will be 80% of everyone’s grade for the day but that the last 20% will be from 4 questions I’ll ask 4 random people. They freak out about that, of course, and I’m not really sure that peer pressure is the right way to go, but I mention it because in the classes where I’ve done it they start to realize that nearly all of my questions are along the lines of the re-posing you’re mentioning (what happens if I double the initial speed? How does the answer answer change if I move this whole apparatus to the moon? What if I cut in half the mass but maintained the size?). If I do it enough they start to prep for that during the 15 minutes: He might ask about doubling this number, what happens? He might talk about moving it to the moon, does that matter?

Thanks for this great post!

Response to 1: I’m not sure what is universally best, we do “challenge labs” at the end of each unit, where students have to do the whole process by themselves from beginning to end, so right now my method has been: 1st time: I pose scenario and question, they help turn into a problem and estimate values, I model problem-solving, then they brain storm new questions about that situation, answer that question, and likely they will brain storm a new situation and solve that situation on their own. But I may give them a new situation. [This is most like the old routine] 2nd time: I pose a physical situation and question, they tell me what they want to know and brainstorm ways we could measure, I guided that process to something I can measure quickly, and they solve the problem. By the time the challenge lab comes along, they will be responsible for the whole process, even deciding what to measure and how, etc. Structured labs in between give them experience with equipment and such.

Response to 2: Yes, I agree it’s not a wholly different process for anyone, but I kind of want us (in my department) to think of it is as different, in the sense that we have articulated important steps in the process and why we do them. It’s deliberate rather than haphazard. This is important because when many instructors try new things, there is a tendency to gradually “slip” back into old way of doing it… and in an effort to make things easier for students, or take less time, to start skipping these steps, or doing them in such a way as to make them “going through motions” rather than really engaging students. But I also, agree, I don’t want it to come across like, “This is a completely new way that’s better than what you normally do.”… more like, “these are the important things our new curriculum wants students to learn, and this is how we’ve restructured problem-solving practice to make sure students get feedback with it.” Having the challenge labs, I think helps keep instructors accountable to the process, because without students practicing the process, they are more lost on what to do on labs, because they haven’t been practicing turning situations into problems.

Response to 3: I like the idea behind what you are doing here, in the sense of getting students to step back from the specific problem to “learn something” bigger than the answer. And I think it’s a good example of what I hope to accomplish in the last step… not just “does my answer make sense?” but “What else can I ask,learn from the modeling process I just did?” –> Can I ask new questions, answer new questions, see new relationships, etc. That’s the goal.