On our first test, students had a two-stage constant acceleration problem. It was not terribly difficult (starting from rest and the quantities given were all integers), but two-stage problems always mess up students. Students were given the times for each stage and the acceleration in the first stage.
Here’s two interesting break downs
Break down by Correctness of Velocity vs. Time Graph
In knight’s book, they call a setup to 1D acceleration problem a “Visual Overview”. In the visual overview, students must include a pictorial representation and a list of values and may include a motion diagram or a graph sketch.
- If students had a correct velocity vs time sketch, they had a 90% success rate in finding the total distance.
- If students had no velocity vs time sketch, they had a 60% success rate in finding the total distance.
- If students had an incorrect velocity vs. time sketch, they had 0% success rate in finding the total distance.
Breakdown by Approach to Finding Distance
Knight’s book teaches both graphical methods and equations methods. In class, I really encouraged working the problem both ways.
- If students approached using graphical method, they had a 83% success rate.
- If students approached using equations, they had a 61% success rate
- If students approached using a table, they had a 50% success rate (only 2 students)
- If students approach was mostly math scribbles, they had a 0% success rate.
This particular problem was very amenable to graphical approaches, because time information was given. I’m not sure these two trends would hold for every problem, but Kelly O’Shea would definitely say here, “Graphical Approaches for the Win!”
Edit: Of course, students who solved in both with equations and graphs had a 100% success rate (4 students)
While not having a good “setup” was a common error, other common mistakes included
- Implicitly treating all or part of the problem as a single stage of constant acceleration (the trap in using equations)
- Using constant velocity approaches for constant acceleration (e.g., using v= d/t and/or treating acceleration as velocity)
- Confusing position and velocity in a graph or calculation
Overall the test offered to few challenges for some students (1/3 As, with three 100s), and offered two few opportunities for some students to display partial understanding (2-3 really low failing grades). Average was a high C, and Median was a low B, which seems about right for the first exam.