Simple Problem

This was my attempt today to create a problem that:

(1) Isolates a simple skill (in such a way that it might be useful for SBF)
(2) Tries to connect the reader to the problem in some personal way
(3) Has the potential for even a bit of learning to occur

On the Diversity of Student Thinking

Here is a question that I have been fortunate enough to observe many students pondering over:

"A stone is dropped from a cliff. One second later, an identical stone is dropped from the same height. Describe the distance between the stones as they fall."

The question is a little bit interesting because it has this compelling wrong answer: The two ball's keep the same distance. I have seen this answer arise as simply an intuitive reaction, but I have also seen this intuitive reaction argued for in terms of school knowledge that students can call upon.

"Gravity acts the same on all objects."

"Gravity acts the same and the initial position and velocity are the same"

"Both balls falls with the same speed of 9.8 m/s"

While each of these students has given the same answers, their thinking about the problem is different. The first student I might characterize as over-generalizing from a rule they've learned*. The second student I might characterize as conceptually equation hunting (e.g, "If I plug in all the same stuff, the number that comes out can't be different"). The third student I might characterize as not distinguishing acceleration from velocity.

While students' answers to the question don't tell me much, listening to them explain tells me quite a bit about how they are approaching the problem and what difficulties they might be having.

In general, the diversity of thinking is something that intrigues me. Partially because it is just plain interesting much in same the way that the diversity of life is interesting to biologists. But it also intrigues me because it places serious constraints on the possibility of "carefully sequenced instruction."

The diversity of thinking, of course, is not restricted to wrong thinking. Consider the three following arguments for the correct answer. I've seen each of these arguments in person.

Pulling Away Argument

After one second, the first stone will have picked up 10 m/s of speed. The second stone, however, hasn’t even started moving at all, at least not yet. So it’s speed is still 0 m/s. Since the first stone is moving and the second stone is only about to start moving (but not moving yet), the first stone will pull away from the first stone, making the distance between them greater. This trend of pulling away (and making the distance greater) will continue because the first stone will always be faster than the second stone by 10 m/s.

Shadowing Argument

We can think of the second stone as “shadowing” the first stone—it will be where the first stone was, exactly one second later. Since both the original stone and the shadow stone are falling faster and faster as they fall, the distance that each travels in that one second must be greater and greater as well. Thus the distance that the second stone covers (in order to “shadow” the first stone) becomes greater and greater, showing that the distance between them increases over time.

Parabola Argument

The position vs. time graph for free fall is a parabola. If you pick two points on a parabola that are one second apart near the origin, you will see that there will not be much separation on the y-axis. However, if you pick two points that are one second apart that are also far from the origin, there will be a big separation. This shows that distance increases as time increases.

Just as the wrong answer doesn't tell you much, knowing that a student gets the right answer doesn't tell you much. Each of the above arguments tells you something quite different about the student. The first student is thinking about acceleration as the accumulation of speed and about the consequences of relative velocities. The second student has invented a novel way of thinking about the problem in terms of speed-distance-time comparisons. The third student is bringing graphical tools to this problem in a successful way.

Like I said, I think the issue of diversity in thinking is interesting, but it is also important. So, what does the diversity of thinking imply? I'm going to be exploring this issue over the next week or so in blog posts. I hope to discuss its implications for assessment. It's implications for curriculum design. And it's implications for teacher preparation.

* Footnote: I've heard students use this argument to support all kinds of non-nonsensical answers. Students will say that a ball thrown straight down and a ball thrown straight out will hit the ground at the same time because gravity is the same. Students will also say that a ball thrown a mile into the air will take the same amount of time to reach the peak of its motion as a ball thrown an inch into the air because gravity acts the same. "Gravity acts the same" functions as a blanket statement to cover any problem where a student is asked to compare times. In fact, the only time these students get the question right is when they are asked, "If a bullet is shot from a rifle and another bullet is dropped at the same time from the same height, how will time compare for the two to hit the ground?"

Pseudoteaching on the Guided Inquiry Front

Last year, I spent about a week substitute teaching in a college physics course for pre-service elementary teachers.

For the week I was there, the class was well into learning about bulbs and batteries. In the course, groups of student work through materials in a somewhat self-paced manner. The instructor strolls around and discusses the students' investigations with them. Every once and a while the instructors do a formal "check out" before students can move on to another section.

One group was being "checked out" by me on a section about voltage, and I discussed with them the questions that the instructor had designated that I discuss with them. I don't remember the details. But I remember feeling that they had met some reasonable minimum standards for explaining why they had carried out the investigation, what they had learned, and what the results had to do with concepts they had learned prior.

As I was walking away, I kept listening (as I always do), and overheard them continue to discuss:

"So, is like current the flow of voltage? ...Like, you know what I mean? I'm trying to figure out what voltage is. And I was thinking that maybe voltage is like logs flowing down a river, and current is the flow of water moving those logs along."

"I think of it more like a train. Like, current is the train cars moving along, and the voltage is like the engine car, driving the train along."

"And so are the tracks, then, like the wires?"

...

"So what's resistance?"

The conversation went on for a bit like this. I was thinking to myself, "That's the problem with a preplanned checkout." The teacher gets the students to discuss what the teachers (or the curriculum) wants to talk about, but the students don't get to talk about what they really want to discuss.

The section of the curriculum was heavily focused on empirical observations and making sense of those empirical relations in terms of the model that the curriculum would have them develop. But these students wanted to talk about their own models–what is voltage like? Is voltage a thing like a log? Is current like a moving train? Is voltage like a engine car? How do I make sense of resistance?... They weren't intellectually concerned with what the curriculum was dishing out.

Later, at another table, I was doing another "check out". The group explained to me what they did and what they learned, but then I asked the students to explain to me what they had predicted would happen, and if the result had differed in anyway. They kind of shot some guilty glances around at each other, and one finally said, "Well, we stopped doing the predictions." Another one added, "It just confuses us to think about the predictions, especially if we are wrong."

Finally, at another table, I was walking and saw one of the students crossing out stuff from a prior page. I asked, "What are you doing?" The student said that they were erasing their prediction, because it was wrong. I asked them why they would go back and erase a prediction. The student responded that they didn't want to get confused with the wrong answer later when going over her notes or when studying for the test.

What does all of this tell me? Guided curriculum can be dangerous. These students were intended to be doing inquiry, but they we mostly just jumping through inquiry hoops. And I don't blame the students–they learned to do this in class because that's the hidden curriculum that was being taught:

The students in the first example had learned in class not to discuss certain aspects of their own ideas or models. In particular, they had learned not to talk about "What things are like?" This wasn't just because I was there. I actually came back over to this group and we talked about their models for quite a long time, and it was clear to me that they weren't having these kinds of conversations on a regular basis.

The students in my second and third examples had learned that their ideas were worthless (and confusing to think about).

The problem with (some) guided inquiry like this is the illusion of learning. Instructors doing these kinds of "check outs" can convince themselves that students are building powerful scientific models, but really students are just learning not to share any ideas that might be wrong, not to have conversations that they aren't supposed to have, and to hide interesting questions and insights that are outside the bounds of the "guided curriculum".

To me, this is pseudoteaching and pseudolearning at its worst, because students are not not learning. It's quite the opposite. They are learning that their ideas, questions, and curiosities have little to do with science and science learning (except that their ideas are usually wrong).

At the end of the day, if students are learning to avoid taking intellectual risks around the instructor, that instructor doesn't stand a chance of helping those students learn.

"Don't give 'em Components until they Beg"

"Don't give 'em Components until they Beg" ... is a quote from someone's blog I read recently. I can't seem to find it. If this is you, I'll update this post to link and reference you! That post made me reflect back on this story from the frontiers of physics help study:

Update: The original post that inspired me to share this story can be found at Quantum Progress.

Last semester in help study, it was hit-or-miss in terms of attendance. If students were assigned a difficult homework assignment that week, we might have 50 students show up. If it was easy, we might only have a handful. For that reason, we usually assigned 2 helpers to be in help study.

One week, it wasn't particularly busy, and I was watching a group of two students working on a 2d-collision problem with the help of a graduate student. The problem was something like this

A 2kg mass moves east at 2 m/s. Another 2 kg mass is moving 30 degrees south of west at 1m/s. If the two collide and stick together, what will their speed be?

The students and helper were approaching the problem like this:

(a) Set up the conservation of momentum equations in the x and y directions

(b) Find the x- and y-components for the velocity for the mass moving off-axis

(c) Plug in numbers and solve for unknown

(d) Use those unknowns to find the speed and the direction.

Standard solution path, but along the way, they made an error. Can you spot it?

2kg 2m/s + 2kg 1m/s cos (30) = 4kg vx

2 kg 1m/s sin (30) = 4kg vy

The mistake they made was not taking into account the direction of the ball. There should be a minus sign somewhere above to reflect the fact that one ball is moving eastward and the other is moving westward.

Now, neither the students, nor the graduate student, knew what mistake they made. But since the students had the answer at the back of the book, they knew they had made some mistake.

Now, I didn't know what mistake they had made either. They had been working on pieces of paper that kept the details of their work somewhat hidden from me. So I just let them keep working on it, while I went to another board to work out the problem quickly myself.

Here's what I did on the blackboard:

As you can see, my approach was quite different from theirs:

• It emphasizes geometry
• It emphasizes the vector nature of momentum
• It emphasizes that system momentum is the sum of individual particle momentums
• It emphasizes conservation in a simple way (p is both initial and final momentum)
• It emphasizes momentum (as a singular physical quantity) rather than velocity and mass
• It solves the problem through geometrical ideas (here law of cosines, but you could just as easily use a ruler and protractor)

And it does all of this in one diagram and one equation.

I think the students' approach does this:

• It emphasizes equations and algebra
• It hides vector nature of momentum (in signs), also where students make their mistake
• It distributes the concept of conservation across many terms and many equations (masking the fundamental principle from the exercise)
• It emphasizes mass and velocity, not momentum (which is king!)
• It solves the problem through too many unnecessary steps: break into components only to have to combine them back again.

I'm not saying students shouldn't ever learn to solve problems by using components. I'm just saying that "jumping the gun" by teaching components first can't be a good thing.

I suggest (as others have) doing it this way, if at all possible:

(a) Start with protractors and rulers–make 'em do it the old fashioned way

(b) Then, maybe introduce law of cosines and law of sines–give 'em a trick or two

(c) Then, then maybe, think about introducing components

We ended up finding the error in their work (together), and discussed a little bit about the solution I took. They were really interested in what I had done, because it looked like less work. But the graduate student stuck around even longer to discuss this solution, because he was intrigued by the fact that he had never learned to think about momentum problems this way.

Learning to Teach

Quote from UMaineprofessor (quoting someone else):

"The first time you teach a course, you teach yourself. Then second time you teach a course, you teach the course. And (maybe) the third time you teach a course, you teach the students."

My Frustration over Pseudo-"Something" Problems

I organize and volunteer for a physics help study at my university. We get a range of students, including those just looking for quick answers and help on homework, those really looking for meaningful help in learning physics, and even those who love just hanging out to talk physics.

I don't mind that some students are there to only "get' quick help and aren't in it for some deep understanding. Most of the courses aren't structured in a way to help them learn. I am happy to guide them a long a little bit, and make them a little less frustrated by the crap they have to put up with. By helping them out (sometimes a little bit too much), I gain their trust and they actually get to learn something meaningful from time to time. Never underestimate trust.

This week, it seems, students have been learning about electric potential. They are assigned a lot of "exercises". I refuse to call them problems, because there is nothing problematic about them. Take for example, this problem, for which several students called me over to discuss.

The question just asked students to find the potential at a point in space due to three charges. The students had to do a bit of geometry to find some unknown lengths, but otherwise it was a simple straightforward calculation. Each of the student I met had correctly done the calculation. They had called me over because they had gotten the right answer that v= 0. These students must have been surprised by the answer, because they each called me over to discuss, "Does this make sense?"

OK. Stop. Hold the phones. Students spontaneously calling me over to talk about whether an answers makes sense. NOT, "do I have right answer?" NOT, "how do I do this?" They wanted to talk about whether or not an answer made sense.

And this is where I had to sigh. Because saying that the v =0 doesn't mean much. Really, you have to know what the potential is at nearby points to say anything. Because knowing differences in potential tells you something about electric fields and/or where charges are likely to move. Sure, sure, maybe you can make some argument about how it tells you that the potential there is the same as the potential at infinity. And then we can talk about how much net work it would take to bring in a particular from infinity. Sure, sure, sure. But what does v =0 mean? Not much.

So, here these students are, and they have been asked to do this "exercise" by performing some rote calculation using the formula k*q/r. They do it well. And they are puzzled? And I have to be the one to tell them that answer is meaningless, pretty much anyway. Not that their answers is meaningless, but that any answers would be meaningless. (Note: We did talk about why it was meaningless; and what else you would need to know for it be meaningful. But I digress)

Here's why I am so frustrated. We complain so much about students not stopping at the end of the problem to ask, 'Does this make sense?' But I'm the one who has to look them in the face and say, well, your professor has assigned you a problem that isn't about making sense of anything. He just wanted you to do some push-ups. I have to say that the best we can do is to check your work and make sure you calculated it correctly, and maybe to offer a mathematical explanation for why it seems plausible (based on geometry) that it could work out to zero. But I can't offer them an answer to the question, "Does it make sense?" from a physical sense without really twisting things around and bringing in a lot of baggage about potential energy, work, infinity, etc.

Am I wrong? Can I make meaning out of this stupid calculation students have been asked to do?

On the Perils of Pushing for Correctness

Last semester, I gave intro physics students a series of easy, commonsense questions about force and motion. Before instruction, students were doing between 80% and 95% correct on the questions. After instruction, students were doing fairly bad, between 25 and 50% correct. It seemed from the data that students had learned to ignore their productive intuitions, and were simply trying to apply rules that made no sense to them. While we may have eradicated their misconceptions (they no longer say a truck exerts more force on a car than vice versa), we eradicated any sensibility toward the world as well.

It reminded me of an excerpt described below:

Eleanor Duckworth (1996) describes an experiment in which children were taught about density as an explanation for sinking and floating (in a "cognitive acceleration" experiment):

"Among the rules Englemann taught the children, the principle on was : 'An object floats because it is lighter than a piece of water the same size; An objects sinks because it is heavier than a piece of water the same size. Kamii and Derman describe fascinating instances of conflicts between the rules children were taught and their own intuitions–their common sense." In addition to the rules they often gave other explanations, typical of school children their age: 'because it's heavy', 'because it's little', 'because it has cracks in it', 'because I pushed it'; or simply, 'I don't know why.'

"In other instances, the rules seemed to come between the children and their intuitions in ways that led to nonsense not normally encountered in children their age. One child hefted a large candle in one hand and a birthday candle in the other, but having seen that the both floated, maintained, 'they weigh the same.' Another child said that a tiny piece of aluminum that sank weighed more than a large sheet that floated on the surface. Clearly these children were trying to apply rules rather than coming to terms with objects. A typical 6-year-old's reaction to the aluminum foil, for example, might be to say that the tiny piece sank because it was too tiny, and the large piece floated because it was flat.

"In another part of the Kamii and Derman assessment, no longer dealing with sinking and floating, the children were asked why the water level rose in a glass when an object was immersed in it. Two of the four replied, 'Because it is heavier than a piece of water the same size.' The other two children, who tended generally to remain true to their intuitions, answered that the object pushed the water out of the way."

Duckworth (1996) On the Having of Wonderful Ideas.

Englemann (1971). Does the Piagetian approach imply instruction? In Green, Ford, & Flamer (Eds.) Measurement and Piaget. New York: Macgraw Hill

Kamii and Derman (1971) Comments in Englemann's paper. In Green, Ford, & Flamer (Eds.) Measurement and Piaget. New York: Macgraw Hill