I talked about my use of goal-less problems on my physics semester exam. The essential idea is that the question is actually just a description of a situation. The student’s job is to model the situation as best they can using the physics they know. First step: say which models apply and why. Second step: draw (and annotate) the graphs/diagrams that go with those models to represent the situation. Third step: use the diagrams to analyze the situation.
“This problem doesn’t ask for anything.”
We start to plant the seed of this concept in the third unit (Constant Acceleration Particle Model) with “baby goal-less problems.” At that point, they can really only draw the kinematics graphs. And if there is enough information to solve the problem, they can probably only calculate about two pieces of information about the problem. They usually feel uncomfortable that there isn’t a specific question asked, but they also soon realize that there aren’t a lot of solvable questions to ask given the information. They just don’t know a lot of physics yet. I tell them that later in the year I will be able to give them a situation like these and they will fill up an entire page with physics, and that seems unlikely to them (at the time).
By the next unit (Unbalanced Forces Particle Model), they start using this tool to compare small changes in a situation. The honors class starts with a block sliding down a ramp with various descriptions given of the motion and amount of friction. The regular class starts with a block pushed then released along floors with varying amounts of friction.
At some point, I started making the background of the page a 50% opacity fine graph paper to try and really encourage them to use careful diagrams/graphs as part of their solution.
What do students actually write?
The second half of the semester exam presented students with 5 goal-less problems and asked them to choose 2. Here are a few of those problems. I chose some of the most thorough responses as well as some sparser ones.
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- Regular Physics exam problem. This student solved for the values of the forces in a more graphical manner.
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- Regular physics exam problem. This student modeled the situation more qualitatively.
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- Regular physics exam problem. This student used vector components rather than solving graphically for the forces.
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- Regular physics exam problem. This student very thoroughly analyzed the situation using multiple models.
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- Regular physics exam problem. This problem refers to a game that students often play at the end of lunch. The student thoughtfully analyzed both parts of the motion.
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- Honors Physics Exam Problem. This student did a ridiculously thorough job of analyzing the motion of the falling crate.
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- Regular physics exam problem. This student did not have the depth of mastery to analyze the entire situation quantitatively but was still able to represent the problem with several graphs and diagrams that describe how the motion changed.
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- Regular physics exam problem. This student represented the situation with additional graphs, calculated quantities, and wrote expressions.
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- Honors Physics Exam Problem. This student did a thoughtful job of analyzing a very complicated situation (more complicated than anything we had practiced in class because of the number of distinct states).
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- Honors Physics Exam Problem. Student uses multiple models and shows very clear thinking in modeling the situation.
A couple uses of goal-less problems
One use that I really like is starting a new unit with goal-less problems when the new model is really an application of old models (projectile motion is constant velocity and constant acceleration, uniform circular motion is a special case of unbalanced forces). It all but requires them to make connections between what they learned earlier and the new special case that we are considering, and it reinforces the idea that they can model new situations using the tools that they have already developed.
Here is the start of the Projectile Motion Particle Model packet. In Honors Physics, I let them dive in with this without trying to prime them for free fall at all. With the regular class, we spent around 30 minutes analyzing the first half of take 1 of Dan Meyer’s basketball videos, drawing graphs for the motion of the basketball, and generally defining free fall as the-only-force-acting-is-Fg. We kept our discussion very qualitative leaving the quantitative analysis to happen when they struggled through the first line-up of problems.
The next two questions are identical, but the ball is changed to an aluminum ball then to a ping pong ball. After that, the ball is rolled off of a 1.5 m tall table, then launched off the end of the same table. In both classes, when they started to calculate numbers on the various problems, they had lots of disagreements at their tables and started looking for objects to roll and drop to see if their answers could actually be true.
And later, in the Central Force Particle Model unit in my honors class, we started out with spirited discussions about how to model this situation (after we had played with vectors and geometry a bit to derive some relationships for objects going around a circle at a constant speed):
Another great use for goal-less problems this year has been as reassessments. Usually, I take a piece of paper, draw a picture onto it, label it with some numbers, and ask them to show me the skills the want to reassess. It usually keeps them from cherry-picking the skills because they generally have to show them in context. If they are prepared, they can demonstrate mastery on a lot of objectives at the same time (and I don’t need to carry around a ton of problems specifically tailored to each objective).
Kelly,
This is great. I’ve found goal-less problems to be similarly useful, and the kids love them. But your kids’ work is truly amazing.
Posted by quantumprogress | March 5, 2011, 8:57 PMThanks! I’ll pass that on to them when they get back from spring break in a couple of weeks.
Posted by kellyoshea | March 7, 2011, 6:42 AMI’ve been meaning to try to start this. How do you begin so that you get anything decent?
Posted by Bryan Battaglia | March 6, 2011, 8:40 PMHi Bryan,
For the first couple of units with these problems, the first goal-less problem starts with a lot of structure. It gives the statement, then has parts (a), (b), and (c) spaced out on the page. The (a) is “What models apply to this situation and why?” The (b) is “Draw at least four diagrams/graphs to illustrate the situation. Choose the diagrams and graphs that you find most useful.” The (c) is “Using the models you have chosen, solve for any unknown quantities. Use more than one method to find the same answers whenever possible. Show your work and use units. Alway start with a variable expression.”
We do a lot of whiteboarding. And we talk a lot about how everyone does a pretty cruddy job at everything the first few times they try it. After they’ve done a few, I usually do one on the board for/with them. We talk a lot about how they aren’t going to learn much physics by watching me do a problem, but that they might get some ideas about how they can organize their work when they are doing a problem.
Annotating the graphs is key. I need to add that explicitly into the early instructions next year. Annotate first with symbols (the “run” on a v-t graph could be called delta-t, the rise might be called vi or delta-v, depending on the shape, etc). Then add in any numbers that you know next to the symbols (so vi becomes vi=2 m/s). Then it becomes pretty easy to use the graphs. Oh, and drawing qualitative graphs that are annotated is also key. They always want to draw quantitative graphs for me at first, and if they don’t know enough values, they fall apart. Once they get into the habit of drawing qualitative annotated graphs they start to see how they can use the graph to find the information that they don’t know.
If you try some, let me know how it goes!
I base a lot of my packets off of Matt Greenwolfe’s More Models in Modeling materials (scroll down to #3). You can also easily create goal-less problems from book problems by just removing the question.
Good luck!
Posted by kellyoshea | March 7, 2011, 6:37 AMI’m kind of in love with this and sharing it with all the physics teachers I know
Thanks for sharing it with us, and I look forward to learning more!
Posted by grace | March 8, 2011, 5:26 PMLove the idea. I’ll actually be using a lot this in a college physics “studio” course next fall. I like it because it focuses on the fact that in science we are modeling the world to understand it, not just to get numerical answers to questions. The other things I’ve used is to give an equation or a free body diagram, and ask them to draw the situation. This is like reverse modeling – what situation could this equation model. Doing the forward and the reverse process does something to the brain.
Posted by Brian | March 13, 2011, 10:59 AMThis is pretty amazing stuff. I need to ponder whether something like this could be viable in a math course.
Posted by R. Wright | March 14, 2011, 10:45 PMI really like the idea of putting graph paper behind the problems. How exactly did you do that?
Posted by Nico Werps | February 2, 2013, 8:27 AMI have a PDF of graph paper and drag it into my Pages file. Then I make it almost the size of the page, set it to floating, and make it not wrap text around it. I set it to 50% opacity and make it the background (then sometimes make the characters of text have a white background so that they are more legible on the graph paper). Hope that helps!
Posted by Kelly O'Shea | February 3, 2013, 1:54 PM