I’m not as happy with this… not as coherently packaged as first, trying to do too much, etc. But here it is.

Question 1:

In class we discussed that 60 mph means that you travel 60 miles in each and every 1 hour of travel. What does it mean for a car to be traveling at 60 mph if it hasn’t yet gone for an hour? What does the “60” mean, for example when you’ve only traveled for 30 minutes, or 15 minutes, or 1 minute?

Question 2:

You and a friend are driving along the highway at 60 mph, exactly 9 miles away from your exit. Your friend announces, “It should take about 9 minutes to travel those 9 miles, because every mile takes one minute.”

How do you think your friend arrived at this conclusion? Is he right? Explain.

Question 3:

Your friend is back at it again, keeping track of where you are one the highway, but this time he jots down the mile-marker every ten minutes.

Are you guys driving with constant speed, increasing speed, or decreasing speed? Explain how you can tell from the data above.

Question 4:

What’s your best estimate for where your car will be at 7:00am? Explain how you came to your answer and why it makes sense to you.

Question 5:

During your drive, you and your friend start arguing about whose car is better at speeding up. Your car can go from 0-60 mph in 4 seconds. Your friend’s car that can go from 0-80 mph in 5 seconds.

Whose car would you say is better at speeding up? Explain how you reached your conclusion.

I’m drafting my pre-class reading quizzes for the summer session. Here is day one. Please comment.

Question 1:

Imagine you are tutoring an 8th grade student in physics. You notice that the student is a little confused about the difference between what is meant by 3:00 pm and what is meant by 3 hours. How would you explain to this student the difference between 3:00 pm and 3 hours?

Question 2:

You are still tutoring this same student, and notice the student is having a similar problem, except now they are confused about the difference between being at mile-marker 72 and having traveled 72 miles. How would you explain to them the difference between being at mile-marker 72 and having traveled 72 miles?

Question 3:

Here is a physics problem! Imagine you are driving a car along a long straight. Your friend in the passenger seat is recording where you are every hour of the trip.

Explain why your friend might reach the conclusion that the speed of the car at 3:00 PM was 24 miles/hour.

Then explain why that answer is probably not correct.

Question 4:

You might have noticed that the table above does not have information about where the car was at 5:00PM. This information is missing because your friend was asleep. What do you think is a good estimate for the mile-marker of the car at 5:00PM? Explain how you determined your answer and why you think this is a good estimate.

 Question 5:

Here is one final question to consider. How much total distance did you and your friend travel between 2:00PM and 5:00PM? Why is this answer different than your answer to question 4?

A paper that an undergraduate researcher and I were reading this week is this:

” Are All Wrong FCI Answers Equivalent?” by Helena Dedic, Steven Rosenfield, and Nathaniel Lasry, published in the 2010 Proceeding of the Physics Education Research Conference.

Essentially the paper examines patterns of student responses to the 4 Newton’s law questions in the FCI in order to identify different classes of students. Using a statistical method called Latent Markov Chain Modeling, they group students together based on how they respond to the four questions–not just in terms of correctness, but which answer they pick specifically. Their analysis identified 7 groups of students. One surprising thing from their analysis is that the groups form a natural hierarchy.

Basically, the authors looked at how students transition from one group to another, by calculating the probability of transitions occurring between classes. The cool result is that there is a strong directional bias to the transitions. In other words, there exists an ordering of the groups (e.g., C1-C7), such that transitions are always toward C1. After instruction, students are likely to either stay put or move up the hierarchy, but they are very unlikely to move down the hierarchy.

What’s even more interesting is that the classes do not form developmental stages. This means that C7-C1 is not a progression of learning. Certain forward transitions do not happen, or are very unlikely to happen.

It’s been helpful for me to think of the situation quantum mechanically. There are discrete cognitive states which can be measured, and there are certain probabilities of transitioning into different cognitive states; some transitions are very unlikely to happen and can perhaps even be forbidden. The fact that they fall in a hierarchy also reminds me of quantum mechanics, in that we can order QM states by there energy levels, and this ordering implies a bias in how perturbed and unperturbed systems such as the hydrogen atom will transition.

What I especially like about this paper is the meaning they give to each of the classes (in terms of student schemas), and how those interpretations build on much of what we know about student thinking about Newton’s 3rd Law. Each of the classes is generated from the data, and are more complexly defined than what I write below, but the gist is this

Class 1: They get all the questions right, essentially. This class is intended to represent a class that is highly likely to be a Newtonian Thinker about these Questions

Class 2: This class essentially gets the all the questions right except for the the question about the Newton’s third law when two objects are speeding up together. The interpretation given here is that these students are thinking of the Newton’s third law pairs within a Net Force (or competing forces) schema.  ** In my teaching physics class, we had long arguments about this question **

Class 3: What I can make of this class, is that the students essentially get either 1 question wrong (but not the speeding up question), or two questions wrong (but not in a pattern similar to Class 4 or Class 5). I wasn’t really quite sure what to make of this class, exactly. Students had about 65-75% of answering any question correctly, implying that there is an assortment of 1-2 answers wrong. But the Modal answer for each question is the correct answer, where as Class 2 the Modal answer for one of the question is wrong.

Class 4/5:  These two classes are almost the same. They both answer consistently that the more “dominant” object exerts more force. Class 4 says that when maintaining speed neither object is more dominant (so forces are same); whereas Class 5 says that the one of the object is more dominant (forces not same).

Class 6: I think this less strongly defined (with lots of scatter), but I gathered they included students who said that objects can be obstacles that are merely in the way without exerting a force.

Some cool comments about transition probabilities: I think a really cool gem in this paper is that there is a fairly low transition probability from Class 2 to Class 1. In other words, students who start in Class 2 only have a 37% of transitioning up to Class 1. This is in contrast with Class 3, where students have a 65% of transition up to Class 1 (and low probability of going to Class 2). This means that the cognitive state of thinking about Newton’s 3rd Law Pairs in terms of Net Force is a fairly stable state. This mean that being high in hierarchy doesn’t necessarily imply you’ll ever get to the top! It’s like there a trajectory of learning that gets you nearly to the top, but never quite there. Other things are this: Students with a dominance schema (class 4 and 5) have a high probability of basically staying put, and also true of Class 6, for which students are 58% likely to stay put.

It’s kind of weird, but only Class 3 makes significant movement to Class 1. Classes 2, 4, 5, and 6 have limited mobility, at least within the time frames they are looking.

Anyway, I’m curious what other people think, and what they see in their FCI data.

I’ve been thinking about teacher preparation, and keep coming back to Duckworth:

“First, teachers themselves must learn in the way that the children in their classes will be learning… Second, the teachers work with one or two children at a time so they can observe them closely enough to realize what is involved for the children. Last,  it seems valuable for teachers to see films or live demonstrations of a class of children learning in this way, so that they can begin to think that it really is possible to run their class in such a way. The fourth aspect is of a slightly different nature. Except for the rare teacher who will take this leap all on his or her own on the basis of a single course and some written teachers’ guides, most teachers need the support of at least some nearby co-workers who are trying to do the same thing, and with whom they can share notes. An even better help is the presence of an experienced teacher to whom they can go with questions and problems.”

From the Essay ‘The Having of Wonderful Ideas’ by Eleanor Duckworth

One question I asked a lot of the future physics teachers during their oral assessments was to discuss the following expression in terms of what it could mean (i.e., the story it tells) and if it seems to be a valid statement or not.

Δx = vf Δt – 1/2 a (Δt)²

Follow up question were targeted at asking students to identify correspondences between various terms and features of graphs.

With some students, this led to a conversation about all three of equations below–how they were similar and different etc, and how they corresponded to features of graphs.

Δx = vi Δt + 1/2 a (Δt)²

Δx = vavg Δt

Δx = vf Δt – 1/2 a (Δt)²

The growth that I’ve seen in many of the students is this. At the beginning of the semester, most students would have either said that the equation was wrong by pointing out that the wrong velocity and the minus sign, or for some, their only way of determining its validity would have been to substitute another equation into that equation until it looked like an equation from a textbook that they already knew. Thus, by the authority of the textbook, the statement must be true as well.

Most students by the end of the semester were able to look at this brand new equation–one they have never seen or considered before–and tell both a verbal story and a graphical story that explains and validates the equation without need for algebraic substitutions. I am impressed.

In other areas, growth has come in fits and starts, and there were many relapses. Many students during oral assessments this last week exhibited one of more of the following difficulties:

• Confusing instantaneous velocity with constant or average velocity (e.g., claiming that an object thrown upward with a speed of 20 m/s goes 20 meter in the first second)
• Confusing change in velocity for average velocity–the two are subtly related in constant acceleration case. Interestingly, this mistake came up both mathematically and conceptually for the same student in the same assessment.
• Confusing acceleration, velocity, and displacement–claiming that when an object falls 10m, it will take 1s to do so, and be up to 10 m/s of speed.
• Confusing change in speed with change in velocity–claiming that a bouncy ball experiences no change in velocity upon rebounding off the floor.

To be fair, these relapses often came up within much more complex physical situations and tasks that were more cognitively demanding–bouncing balls, oral assessments, deriving a result, etc. It’s good to know that, distinguishing these concepts, is not stable and automatic for them. It requires specific cognitive attention to maintaining the distinctions. They can do it, but are prone to making these mistakes when there are many balls to juggle all at once. I discussed with students how the mistakes they are prone to making here are similar to the mistakes we encountered all semester in looking at student work and how they are similar to the ones they are going to encounter in the classroom, so it is best to understand these mistakes well by making them yourself in as many different situations as possible, and doing the hard work of finding your way out of them.

Doing oral assessments for standards makes the assessments spontaneously grow into interesting conversations. Sometimes, what I thought I was initially assessing became something else entirely. Sometimes during assessments, I would respond to wrong things student said by saying, “OK. Tell me about that again, and then tell me why what you just said can’t be right.” Some students would be able to recover, and others would need more time, and I’d give it to them. Other times, I would send a student off, saying, “There’s a mistake in here. I want you to go figure out what the mistake is. When you come back, I don’t just want to know what the right answer is, I want to know why you made that mistake and why it’s a big deal.”

It’s been interesting to watch the varied resiliency of students in the face of my refusal to accept their work as proficient. The only times I know I was doing a less-than-good job was when they would come back and say, “Is this what you want?” Fortunately, it wasn’t often, but it did happen a few times. Most of the times they would come back saying, “OK. I got it now,” or, “I still don’t get this, can we talk about it some more?”

Next year, I’m putting a 2-week rule on my standards, so that students have to turn in an initial draft within 2 weeks. I had many students flying in last minute trying to work through all the standards.

An email I received today:
First of all let me introduce myself… I am just finishing up my first year as a Physics teacher. I have come across a situation that I am a little bit stuck on trying to explain. I have been going through Electricity and Magnetism. Well, I should say that I have already taught this, but am now reviewing for the final exam, and I am coming back up to explaining what causes magnetism. I just don’t feel like I explained it clearly enough. When I try to explain Orbital magnets as well as Spin magnets I seem to struggle being clear in my explanation; how would you go about trying to explain this to High school seniors? If you could offer an explanation I would appreciate it. I am still trying to wrap my head around being clear in my explanations. I did not go to school for Physics; I did take a couple of classes and that qualified me to teach it. I want to be the best teacher I can be so that is why I am asking for some help. Once again I appreciate you taking the time to answer my questions.

My *way too long* of a response:

Thanks for the email. Congratulations on making your way through your first year teaching physics!

You’ve really opened a big issue, not only about magnetism, but about the nature of explanation and learning. My first suggestion in making progress through the big issue would be to watch and mull over this commentary by Richard Feynman, which is both about the nature of explanation and the nature of magnetic forces: http://www.youtube.com/watch?v=MO0r930Sn_8

I’d love to talk about the video. For me, the biggest issue in teaching is not having the best explanation, but in understanding the ways in which my students are going to come to learn new things from the things they already know. That means I need to know a lot about how they already think, understand things, and even how they come to settle upon things as being true or not. Only in this way can I know what the best explanation might look like for them. And at the end of the day, we have to pursue that explanation together.

Magnetism is notoriously funny. Some teaching physics friends of mine refer to “magnetism” as “turtles all the way down”. It’s a joke, but also true. Big magnets are made up of mid-size magnets, and those mid-size magnets are made of magnets that are tiny… and those made up of ones even tinier…and those magnets are made up of particles that are magnetic, those are made up of sub atomic particles that are magnetic. Starts to sound ridiculous, that our best explanation of magnets is that magnets are made up of magnets… but that’s all we have really.

Of course, when tiny magnets work together, they can produce magnetic effects that are larger, or when not working together, they produce tiny or no magnetic effects. Why are those tiny things magnetic? I don’t really have good explanations for those things. But understanding that there are magnetic things, and that they can push and pull and twist on each other, and that they can arrange themselves in ways that amplify those pushes and pulls is pretty much everything to know. Then the devil’s in the details of figuring out what arrangements lead to what kind of effects. The really specific details are the tough work of undergraduate and graduate physics.

If you watched the video, you’ll see that Feyman’s point is that also that magnetism is not meant to be explained -it is meant to do the explaining. That’s a stance that he is taking–magnetism is a powerful idea to explain other things. The story of how we have come to accept magnetism as a true and worthwhile idea is really curious.  I know I probably haven’t satisfied your question, but it’s the best I can do.

What would you have written?

A colleague of mine, Warren Christensen, posted a link to this article, about some research going on at North Dakota State University. The article is titled, “Child speech experts say don’t worry if your toddler’s language regresses.” Warren linked to the article because a colleague of his, Erin Conwell, carries out the research and because his son Owen has participated research study. But something else about the article has stuck with me since reading it.

In the article, they discuss a moment in which a child says, “Daddy tumble monkey on the mat.” The errant statement is described as a causative overgeneralization, but they go on to discuss the statement in the following light:

It was an exciting time – a huge step in her daughter’s development. “It meant she was opening it up; she was ready to go and was starting to play with language on her own.”

[The child] had just moved into a whole new phase of language processing, going from a mere mimicking of the speech patterns she’d been hearing to applying “rules” of language she’d learned by listening to others to form her own word structure.

It made me wonder about what kinds of “mistakes” I should be really excited to see my students make as they are learning math or science. What kinds of mistakes would suggest that students are moving from merely regurgitating facts and procedures to applying and playing with and rules to form new ideas?

I’m really curious to hear from others about specific examples they might have seen in their teaching, in which student mistakes could be seen as a reason to celebrate their transitioning from passive mimicry to productive play with important ideas and rules.

In my teaching physics class, students are expected to complete some content standards, which they have to either write up ahead of time and explain to me in person or prepare for writing in real-time on a board while they explain.

Some common places that students are struggling in the kinematics standards include:

• Justifying when, why, and how Vavg = (vi + vf) / 2 *
• Using graphs analytically in order to do something, not just to merely describe what is already known **
• Reasoning about the value and direction of acceleration in non-standard cases (e.g., not freefall, circular motion, etc) ***
• Providing mathematical or physical argument to justify one’s statement, not merely substituting equations ****

—————-

* I’m thinking of re-naming the course, “Where do all these 1/2’s come from anyway?”

** Next semester, I’m going to model more of this explicitly and have them practice

*** In this problem, they have draw x vs.t, v vs t, and a vs t, for a bouncy ball. Then, in the next set of standards they have to take data to figure out a rule that describes how much energy is lost in each bounce, and then go back and adjust their drawings accordingly.

**** Lots of physics majors get by with strong algebra skills