One of the many great ideas that I use from the University of Colorado Physics department is to start my 3rd-year Quantum Mechanics course off with a review assignment. On the first day I gave them an assignment which was due on the second day and had questions addressing the major relevant things that they should know coming into this course based on their prerequisites.
It consisted of some fairly straight-forward review questions on topics such as complex numbers, matrix multiplication, dirac delta functions, the relationship between energy and frequency for light, orthogonal functions, the deBroglie wavelength and basic discrete probability.
But what really makes this work is that you ask them to include, along with their solutions, a rating for each question on the following scale (credit goes completely to the University of Colorado folks that put the original assignment of this type together for these ratings):
- I knew this material, it was fairly trivial for me.
- I knew this material, and didn’t need to look up anything or get help, but it was not what I would call “trivial” for me.
- I knew this material, but still need to look something up in a book/notes or on the internet.
- I knew this material, but still need to get help from a person.
- I did not know this, and had to learn it for this homework.
I really like this system for multiple reasons. It communicates to them that there are some things that they should know from their previous courses and we are not going to use class time to review it. And if for some reason they have completely forgotten that material or never learned it in the first place, that they are in a position to go out and use the resources at their disposal to learn it. It is also a bit of a reflection exercise in that there are students that probably don’t realize how much help they ask for with their assignments and having to rate how much help they got might be a bit enlightening to them. And, of course, their rating of each question gives me a much better understanding of where everybody in the class stands coming in compared to if I just gave them the review assignment without asking them to rate the questions. This is related to the common issue of never knowing how much a student’s written homework represents their true level of understanding.
As for the idea of the review assignment in general. It won’t work that great if every single class they are taking has one, but since they typically have very little homework in the first week, asking them to do this first assignment in two days was very reasonable. It also saves me some class time and sets us up to immediately challenge more difficult things in class instead of always having to go through some review first.
Note: The University of Colorado shares all their course materials for their intermediate and upper-division reformed courses (Classical Mechanics, Quantum, E&M), including the review homework assignment that I adapted slightly based on my students’ prerequisites coming in.
(This is not a criticism of Peer Instruction. It is just a tale of one of those times where the right idea failed to gain traction)
Yesterday we tackled a classic parabolic motion conceptual question in class, “three ships”, shown below.
They answered this question in their pre-class “reading” (actually I’m using smartPhysics so they watched a multimedia presentation). I am teaching two sections (35 students each) and will just combine their data here. On the first in-class vote, they were 51% correct. After discussion (5 minutes in one section, just about 10 minutes in the other), they revoted and were only 59% correct, which is a learning gain of 0.16 for those who like to use learning gains on clicker questions.
And the discussion was super animated. It seemed like it was so productive, but I guess it was really more about the two sides (B vs. C) digging their heels in.
And this does happen on occasion with Peer Instruction. You see 50% correct on the initial vote and smile to yourself thinking that the subsequent peer discussions is going to be a good one, but then sometimes the number of correct answers barely budges on the revote.
In the case of this question, it seems like I need some scaffolding clicker questions or other activities leading up to this question. Perhaps this scaffolding won’t improve the initial vote, but will perhaps give them more points of reference and examples to use in their discussions, helping the peer instruction really do its job.
Before coming to this question we spent 20-30 minutes developing position and velocity graphs in the horizontal and vertical directions starting from a motion diagram of a basketball shot, but did not talk explicitly about time at all during the sequence. Then we discussed the clicker question for the horizontal projectile vs ball drop demo and they were nearly unanimous in getting that question correct. But it seems like more scaffolding is still in order.
In my 3rd year (intro to ) Quantum Mechanics course the first homework assignment I gave them was meant to serve as a mostly gentle review of probability and modern physics as is relevant to the first chapter of Griffiths.
But I also asked them to read an 8-page section of some supplemental notes prepared by Michael Dubson and Steve Pollock at CU-Boulder (they can be found in the “Lecture Notes” folder of the “all course materials EXCEPT assessments” link on this page). These notes talk explicitly about the postulates of quantum mechanics (which Griffiths does not), about postulates in general and they compare and contrast classical and quantum mechanics.
As part of that first homework assignment I asked my students to read these notes and gave them the instructions:
Please write one or two questions that were burning in your brain after you read these pages.
And they gave me some wonderful questions which should provide us with some rich discussion on Monday. Here they are:
- I’m very confused how a wavefunction can change instantly after a measurement has been made on it’s position. (note: several variations of this question showed up)
- What reasoning did Schrodinger have for writing down his equation? (note: several variations of this question showed up and one student noted that it looked a lot like an equation he had seen in our PDEs course)
- Why is gravity proving to be so difficult to incorporate into quantum theory?
- Do quantum and classical mechanics agree with each other in predicting the outcomes of physical phenomena at a particular intermediate scale between the quantum scale and the macroscopic scale?
- Why does Planck’s constant have that specific value?
- Does the wavefunction ever reconstruct itself after being collapsed due to an observation?
- How come taking measurements changes the look of the wavefunction? It almost looks like a dirac delta function in the after measurement graph shown.
- (edited for brevity) Since real-world sized objects are made up of large quantities of microscopic particles shouldn’t the (quantum) laws and properties that govern the small not also govern the behavior of the large, which are really just big groups of the small? Why would we get different physics looking at many than looking at one?
- If x and p are not well-defined for a point particle, how does putting a group of them together suddenly make them defined for the group? how many does it take? Two? Three? Four billion trillion? At what point does a system become macroscopic?
- Where did the notion of wave-particle duality originate?
- How valid is string theory and a fundamental level?
- How does a measurement give the particle a definite position?
- How does Psi-squared represent the probability that a particle is at a specific location when we are told that Psi only “carries information about the particle, it is not ‘the particle’ or ‘the position of the particle'”?
Fantastic! Now they’re curious. And I’m not great at establishing a framework that ties together the ideas in a course, but I think that these questions mostly provide that framework and it was them that generated it instead of me. It’s their framework! I am thrilled.
One of my very first posts on this blog discussed the quiz correction assignments that I use in my intro calc-based physics courses. I have since renamed them to quiz feedback assignments to better represent what I want the students to get out of these assignments. I would actually prefer to name them reflection assignments, but I feel like too many of the first-year science majors would frown on a word which sounds as touchy-feely as reflection so I settled on the likely to be more palatable “feedback”.
Today Kelly O’Shea posted about her experience with test corrections before and after implementing SBG in her classroom. I’m still a year away from my own SBG implementation, but her post got me thinking about an overdue change in my quiz feedback assignments and about introducing this type of assignment for homework as well.
Not all quiz mistakes lead to productive reflections
My quizzes are dominated by clicker-like conceptual multiple-choice questions and short answer questions which require a combination of translation of representation and short answer questions which one could consider to be analogous to an important step in a longer problem. The issue on the short answer questions is that I see a decent number of clerical errors (missing units, silly arithmetic errors, etc) where the student gains no further physics understanding by completing the full process that I ask for the quiz reflection assignments. So I added a clerical error category to the assignment that asks them to correct their clerical error and show how it leads to the correct answer instead of doing the full-blown diagnosis and generalization that I ask for the conceptual errors. The handout that I provide for the students is included in the post if you want more details.
Reflection assignments for homework
For my 3rd year (first four chapters of Grffiths) quantum mechanics course, I am going to offer the reflection assignments to my students for their homework as well as their quizzes. My plan is to mark each part of each homework question according to the following simple scheme:
- Correct (full marks) – The solution and/or requested explanations are complete and correct. You will not be penalized for one or two small mechanical errors such as dropping a negative sign or a factor of 2pi.
- Complete (half marks) – The solution and/or requested explanations are complete, but not entirely correct.
- Incomplete (no marks) – The solution and/or requested explanations are not complete. Examples include a correct solution without the requested explanation, partially completed solutions, and solutions which jump over important steps.
Quiz Feedback Assignment Handout
Quiz Feedback Assignment (Version 2)
Last updated Sept 1, 2011 (Joss Ives)
Our quizzes are designed to be both a learning experience and an assessment of your current level of understanding of the material. For both these reasons, I offer you the opportunity to learn even more and to improve your quiz score by carefully reflecting on your performance to learn from it. Completing this assignment appropriately within two days of the quiz being returned will allow you to increase your quiz score by half of the points that you missed. This is an all-or-nothing assignment. It is intended only for those students who are interested in making a serious effort to improve their understanding. If it is incomplete or not done well, you will not receive any additional points. Late quiz feedback assignments can be submitted any time up until the date of the final exam, but will only earn back one quarter (instead of one half) of the points that you missed.
Please make sure to attach your quiz paper so I know what you are talking about. You can write or type your quiz corrections, but please put them on a separate sheet from the original quiz.
Types of Errors
For the purpose of this assignment I have divided the common types of errors into conceptual errors and clerical errors. These require slightly different correction processes, and these process are explained below. If your error seems to lie in some grey area between conceptual and clerical error, treat it as a conceptual error.
These errors represent mistakes in your thinking, mistakes in setting up the problem, mistakes in translation of representation or incomplete understanding of concepts. Translation of representation is when you need to take information from one representation (such as word descriptions, graphs, motion diagrams, symbolic equations) and translate this information to another one of these representations. Incorrectly labeling a negative value as positive, grossly misreading a value off a graph or accidentally swapping your initial and final conditions are all considered conceptual errors in the translation of representation category.
To receive credit for your conceptual error feedback, you need to address the following two phases for each question or problem for which you did not receive full credit. See detailed description of each below:
1) Diagnosis Phase (DP) – Identify what went wrong.
In this phase you need to correctly identify your errors, and diagnose the nature of your difficulties as they relate to specific physics principles or concepts, a problem solving procedure, or beliefs about the nature of science and learning science.
Please note that an incorrect diagnosis or a merely descriptive work (such as simply noting the places where you made mistakes) is unacceptable. You need to analyze your thinking behind your mistakes, and explain the nature of these difficulties. Hence, in this phase you need to identify why you answered the way you did, where your understanding might have been weak, what you found difficult, what knowledge or skills you were missing that prevented you from correctly completing the solution, etc.
Poor Diagnosis – No description of thinking behind difficulty
- “I was confused.”
- “I thought it would be 5 N.”
- “I picked the wrong equation.”
- “I didn’t remember to use F=ma.”
Good Diagnosis – Focuses on reasons for actions
- “I thought that the larger velocity would mean the larger force.”
- “I knew it was angular momentum, but I didn’t apply it correctly – I neglected the angular momentum of the ball about the pivot point of the rod.”
2) Generalization Phase (GP) – Learn from your mistakes by generalizing beyond the specific problem.
In this phase you need to identify what deeper physics understanding you have gained from your diagnosis. By carefully thinking about the particular aspects that were problematic to you in approaching the question/problem, and correlating them with the correct solution, you should develop a better understanding of the basic physics principles. In your writing you should identify this new understanding and describe how it will prevent you from having similar problems in the future. Please note that merely stating the correct solution, by copying or paraphrasing another student’s solution for a question is unacceptable. You are expected to generalize beyond the specific problem to discuss the general principles of physics.
In your writing you are very welcome to identify not only your understanding of your mistakes, but also your appreciation for the aspects of your thinking that were already correct and successful in your original attempt. It is hoped that you will hold on to the good elements you already have and add new good ones by completing the feedback.
Poor Generalization – Focuses on generic activity
- “I learned to read the question carefully”
- “I learned to pick the right equations before solving a problem”
Poor Generalization – Focuses on the specific problem
- “I learned that the amount of work from A to B is the same as the amount of work from B to C.”
Good Generalization – Generalizes beyond the specific problem
- “I learned that the acceleration does not depend on the velocity. This is consistent with Newton’s Second Law which says that the acceleration depends only on the net force and the total mass.”
Clerical errors are those where you answered the question incorrectly in a way that was not due to a lack of physics understanding and where it is not reasonable to expect that you would be able to improve your physics understanding or mathematical fluency by learning from your mistakes. Examples of clerical errors include: your answer being incorrect due to a silly math error (accidental extra factor of 10 from a unit conversion, obvious arithmetic errors), forgetting to include units on your final answer, or making a mistake due to not reading the question carefully. These are errors where completing the Generalization Phase seems unproductive because the only thing sensible to write would be something along the lines of “I learned to read the question carefully” or “I need to be more careful of my arithmetic and always double-check my solutions.” Remember that errors in translation of representation are not considered to be clerical errors.
For clerical errors I ask you only to do one phase, the Correction Phase (CP). In this phase you identify your clerical error, how it led to your incorrect answer and how the corrected clerical error leads to a correct answer.
Acknowledgements: The quiz feedback assignment was originally developed by Charles Henderson (Western Michigan University) and Kathleen Harper (The Ohio State University) and much of the wording is theirs or is based on theirs.