Certainty and Vulnerability: Learning and Teaching Science

I expect (and expect with excitement) that I will forever find that I have ideas about how the world works that are problematic in some manner or another. I have no illusion of the final certainty in my own knowledge of the world nor the scientific community's knowledge of the world.

• I know that I have ideas about how the world works that are at odds with others ideas I have. Sometimes "at odds" means logical inconsistency. Sometimes "at odds" means an ontological inconsistency. Sometimes "at odds" means an emotional incongruence. Some of these I am aware of and have ways of reconciling them. Some of them I am aware of and have not yet reconciled. Others, I am not even aware of.

• Importantly, I also have ideas that are at odds with some of core ideas that are central to contemporary scientific understandings. And with these, too, some of these I am aware of and some not.
  

In other words, I am a lot like my students.

Of course, I have fewer inconsistencies than my students–both in terms of internal consistency with myself and external consistency with core scientific knowledge. But the major difference between me and my students is that I know that the nature of the game is to continually work at locating sources of inconsistency and working through them. I know that this is the primary activity of doing and learning science, and I enjoy it.

This growing sense of science has changed me and how I interact with those around me. For the first part, I am much less concerned with maintaining an appearance of being knowledgeable. In fact, I spend a lot more time seeking out people to share the things I don't understand. I also spend more time seeking out people who challenge me and often point out things I don't understand. I am much more interested in exposing my knowledge vulnerabilities than my knowledge certainties.

Of course, there are times where I get roped into caring about my external appearance of being scientifically knowledgeable and acting in ways that are more about exuding knowledge than exposing and sharing my own uncertainty. But those moments are fewer and farther between. I hope to become less and less prone to such moments.

I wrote previously about the damage that school science had on my enjoyment and participation in science. In this way as well, school was not and is probably still not the place for these new habits of mine to be nourished. In fact, school tends to nourish the opposite. School typically pressures students into masking and hiding any and all forms of not understanding. We often take off points for students being "wrong", even when that being wrong comes with a sense of maturity, awareness, and propensity for future learning. We secretly (or not so secretly) cringe whens students exhibit misconceptions, as opposed to celebrating the possibility for exploring and coming to better know current ways of thinking and knowing. We often present ourselves in ways that stress that we are science knowledge experts rather than science learning experts, and students tend to model their own science identities based on this presentation.

In the coming years, I will have more and more opportunities to grow as a science learner, a science teacher, and as a mentor for future science teachers. I hope I can eventually live up to my own growing expectations. What I do know is this–achieving this will involve continually trying to expose my own teaching and mentoring vulnerabilities. It will involve seeking out those individual and communities that challenge me. It will involve locating and pressing through the inconsistencies I exhibit in my own ideas and practices of teaching.

Hopefully, the tenure process will not be the same negative force on my growth as an educator and researcher as school was on my growth as a scientist. I guess, we'll see.

Is there a vi hidden in that t?

In a previous post, I asked readers to consider how a student might end up with the following equation to describe a block sliding down a ramp of height h: vf = vi + sqrt(2gh)

I suggested there were two student approaches–both observed two years ago in an algebra-based physics course. The two approaches that were identified in the comments were exactly what I observed in real life:

Scott suggested that a student might have correctly approached the problem using energy principles and then simplified the algebra wrong–turning the square root of a sum of squares into a sum. Sure enough, one student in office hours last year made this mistake.

Chris suggested that a student might have been started thinking from the idea that final velocity is equal the initial velocity plus the change (i.e., vf = vi + dv). Sure enough a different student took this approach: having seen lots of energy problems where vf = dv = sqrt(2gh), he thought it would be reasonable to just add this change to the initial velocity.

I actually like this second mistake. Why? I think it's a reasonable mistake to make, it reflects underlying practices that are sophisticated, and it provides a generative opportunity for learning. Hear me out on this.

Despite the fact that we'd like students to start from first principles (i.e., energy conservation), many intro physics students simply start problems by following some mindless routines we've taught that mostly involve plugging numbers into equations. This student, however, seemed to be thinking about how the world and the math might work, and he was trying to figure it out without an equation from the textbook. He was going to construct his own equation. More specifically, he was thinking about how quantities change, which is one of the most important things for physics students to think about. He wasn't treating math as stuff to plug numbers into; he was thinking about math as a way to think about and express quantities and how they change.

Second, rather than treating every problem as completely new and different, he was trying to draw on previous results to simplify his approach to a new problem. Now maybe this drives you insane--once again, maybe you want all of your students to start from first principles every time. But the point is, this student was looking for possible connections between different situations and problems, and trying to generalize results. This is something we do all the time and it is quite sophisticated. Sure, when things go wrong, we go back to first principles. But to me, what makes the student approach even more sophisticated is that the student came to office hours, knowing that the equation was wrong. In fact, he had gone back to derive the right equation separately from first principles, but he still couldn't figure out why his first approach didn't work. It made sense to him that the final velocity should equal initial velocity + the change. This student knew the right answer, but wanted to know why a different approach–one he had invented and made sense to him–didn't work.

Where we went from there?

What I ended up telling the student was this: He was right. The final velocity does equal the initial velocity plus the change, but that we had to carefully consider what the change should be. I asked the student to recall what he had learned earlier in the semester.

vf = vi + at.

We discussed what this equation meant in terms of changing quantities, which was easy for him--because he was already in the mindset of thinking of mathematics this way. From this equation, we concluded the more time the ball spent on the ramp, the more the final speed it would have by the time it reached the bottom. So I started asking him how we could get the ball to spend more or less time on the ramp. We discussed several possibilities:

We could make the ramp longer or shorter
We could make the ramp more or less steep.
We could make the ball start with more or less speed

I focused our conversation on this last one, and offered: "That's weird. The equation is vf = vi + at . Vi only appears in one place to represent how much initial speed the ball has, but the time t spent on the ramp is also influenced by the initial velocity vi. "

He pondered over this for a while, looking puzzled before and finally asking, "So is there a hidden vi in the t?"

I loved this question.

I'm not going to go into all the details. But from there we spent a long time discussing and working through various mathematics to show that the reason that dv could not equal sqrt(2gh) is because dv depended on t, which depended on vi, which made vf depend on vi in a weird (non-linear) way. This, I think, was a way better learning opportunity (for both of us) than what might have happened had I simply validated his approach using conservation of energy.

Children Are Born Investigators

A paragraph from "A Framework for K-12 Science Education: Practices, Crosscutting Concepts, and Core Ideas" Copyright © National Academy of Sciences. All rights reserved. PREPUBLICATION COPY---Uncorrected Proofs

"The research summarized in Taking Science to School [1] revealed that children entering kindergarten have surprisingly sophisticated ways of thinking about the world, based on in part on their direct experiences with the physical environment, such as watching objects fall or collide and observing plants and animals [11, 12, 13, 14, 15, 16]. They also learn about the world through everyday activities, such as talking with their families, pursuing hobbies, watching television, and playing with friends [3]. As children try to understand and influence the world around them, they develop ideas about their role in that world and how it works [17, 18, 19]. In
fact, the capacity of young children—from all backgrounds and socioeconomic levels—to reason in sophisticated ways is much greater than has long been assumed [1]. Although they may lack deep knowledge and extensive experience, they often engage in a wide range of subtle and complex reasoning about the world [20, 21, 22, 23]. Thus, before they even enter school, children have developed their own ideas about the physical, biological, and social worlds and how they work. By listening to and taking these ideas seriously, educators can build on what children already know and can do. Such initial ideas may be more or less cohesive and sometimes may be incorrect. However, some of children’s early intuitions about the world can be used as a foundation to build remarkable understanding, even in the earliest grades. Indeed, both building on and refining prior conceptions (which can include misconceptions) is important in teaching science at any grade level. The implication of these findings for the framework is that building progressively more sophisticated explanations of natural phenomena is central throughout K-5, as opposed to focusing only on description in the early grades and leaving explanation to the later grades. Similarly, students can engage in scientific and engineering practices beginning in the early grades."

Office Hours: Anticipating Student Thinking

Teaching Situation

A student comes into office hours and wants to know why the final velocity of a block sliding down a frictionless ramp of height h is not equal to v_{final =} v_initial + Sqrt(2gh)

Your Homework:

Describe two different lines of thinking that could have led a student to honestly arrive at this question.

Describe how your discussion with the students would likely need to differ depending on which line of thinking the student had taken.

* Note: I actually observed this situation–two times in fact.

Jumping through Reasoning Hoops with The Spinning Hulahoop

This is a situation that I've been sharing and discussing with colleagues over the past few months:

Imagine you and friend are holding a hula hoop. Your friend grasps one part of the hula hoop in his hand (not so tight that it won't move through his hand), and you start spinning the hula hoop around until it seems to reach a constant rate of rotation. At this point, I just want to consider what's going on while the hula hoop is and continues to rotate at a seeming constant rate.

Now, based on my description, the hula hoop system can be described as having a constant influx of energy (from you pushing). That rate of energy in to the hula hoops is equal to the rate of energy being lost into your friend's hands. The equality of inflow and outflow rates seems consistent with the idea that the hulahoop is moving with a constant speed, and thus has a constant kinetic energy.

Consideration #1
Energy would seem to flow into system at your hand, and flows out at your friends hand. But your hand and your friend's hand are spatially separated. This leads us to question one: How would you explain how energy gets from one side of the hula hoop to the other?

Consideration #2
Once again, energy is being lost at your friend's hand. But the speed of the hoop seems to be the same everywhere. More specifically, the speed of hula hoop pieces would seem to be the same on both sides of the hand. This leads us to question two: How would you explain how energy is lost at your friend's hand, while, at the same time, the kinetic energy remains the same throughout the process of moving through the hand?

Insight #1
The hula hoop is not a rigid object. Every time you pass the hula past you, you compress a piece of the hula hoop. With your friends hand pushing back, one side of the hula hoop is actually in compression. (We'll ignore for the moment whether or not the other side is in tension or not)

Insight #2
The compressed pieces of the hula hoop act as a energy storage mechanism. Your hand does work on pieces of hula hoop and that work goes into increasing the potential energy stored in the hula hoop. Alternatively, as pieces of hula hoop move across your friends hand, this potential energy is released as those pieces decompress. Thus, the energy lost at the hand is not the kinetic energy of hula hoop; rather it is the potential energy that was stored in the compressed parts of the hula hoop.

Consideration #3
The compressed pieces of the hula hoop are necessarily more dense than the pieces that are uncompressed (i.e., the compression forces the atoms closer together). Since the mass of the hula hoop must be conserved at each point in the circle, this requires that the less dense pieces move faster than the dense pieces (which move slower). This leads us to this questions: If your friend's hand is pushing back on the hula hoop pieces that move through it, how would you explain how the hula hoop pieces end up moving faster on the other side?

Insight #3
The piece of hula hoop right in your friend's hand is actually sandwiched between two different regions with distinct mass densities. Behind your friend's hand, the hula hoop is squished up like a spring. This "spring" creates a force which accelerates the hula hoop piece through your friend's hand, leaving it with a faster speed than before. This faster speed is consistent with the fact that the atoms are more spaced out. The faster speed allows it to get further away from the pieces behind it, which are still moving at the slower speed.

Oddity #1
Intuitively, your friend's hand would seem to the agent slowing things down. On the other hand, as defined by the original problem, the hula hoop seemed to be moving at a constant speed through out the whole process. Through the reasoning we've walked through, we're concluded that pieces of the hula hoop actually speed up through this region.

Loose ends and questions:

#1: It only really makes sense to describe the hula hoop as having a single rotation rate if it is a rigid body. Given that we've concluded it can't be a rigid body, is there a single quantity which describes the flow rate. Is it momentum? Is it kinetic energy? Is it mass current? Does this necessitate a change to the chain of reasoning anywhere?

#2: Is the other side of the hula hoop in tension? Is there any reason to think the hula hoop arc length is longer than, shorter than, or the same as it's resting arc length?

#3: How quickly does energy propagate from your hand to your friend's hand? How does this compare to the rate at which hula hoop pieces make the same journey?

#4: What's going on during the initiation stage before and as the hulahoop reaches steady state? Is this consistent with our stead state solution?

#5: What does this have to do with an electric circuit with a bulb, battery, and wire?

#6: Could you explore the validity of my story experimentally? How would you do it? Could you explore the validity of my story with a simulation? How would you do it? With either, what assumptions or approximations would you need to make?

#7: What parts of my story seem wrong? What assumptions have I made? Are they reasonable assumptions? What aspects of the situation am I ignoring? Is it reasonable? Overall, is this a viable model? How could you tweek it or refine it?

#8: Typically, we use energy to tell stories about initial and final states. Have we gained anything by trying to tell a spatially and temporally continuous energy story? Why is it so hard to tell such stories?

Two views of Science

Quite a few months ago, I was engaged in a somewhat heated discussion with a visitor about the nature of science, and physics in particular. The debate tended to orbit around the issue of whether or not one could be engaged in learning or doing physics without mathematics. Of course one's answer depends on what one mean by mathematics and what one mean by physics, so there was much to discuss.

If you know me, I tend to espouse a view of physics (and science) in which explanation and argumentation are central to its practice. This visitor espoused a view in which mathematics was central to physics. In the abstract, of course, these two commitments are not contradictory, but it can help to discuss specific examples, because we disagreed on much.

At some point, I introduced an example. I asked him to imagine that two friends are walking out in a field and find that the grass seemed taller on one end then the other. Both friends decide to investigate.

One friend starts by wondering what causes this to happen: Does it have to do with sunlight? Does it have to do with water? Did someone cut it this way? Do animals graze on one side more than the other? So, he walks around the field, feeling the soil for moisture or color difference. He examines the grass to see if it's been bitten or cut. He tries see if there are different species of grass. He tries to imagine how the sun traverses over the sky, and wonder if the trees on the fields edge would shade one side more than the other. He tries as best he can to determine if the field is slanted one way or the other, thinking that water would flow differently. He goes to the edges of the field to look for any creeks or other water sources. He looks for evidence of animal tracks or tracks from a machine.

The other friend wonders if there's a discernible pattern in how the grass height varies: Does it really vary? Does it vary linearly? How quickly is the grass height changing across the field? So he gets out his ruler and starts measuring the heights. He carefully lays out a grid of measurements that spans the field and begins tabulating the data, making sure to get track of his uncertainties. He uses the table to generate various plots showing height of grass vs. various positions. He uses the graphs to draw in trend lines, and then starts modeling the data with various mathematical functions. He goes on to determine values (and bounds) for the parameters in his mathematical models, and even estimates the goodness of his fit.

Now granted, both kinds of activities have value in science: (1) exploring plausible causal mechanisms and looking for qualitative evidence to help define the possible space of explanations; and (2) carefully using measurement tools to quantify aspects of phenomena in order to look for mathematical structure and relations. Let's call the first kind of activity: "the pursuit of causal explanation" Let's call the second kind of activity: "the pursuit of quantitative structure"

Of course, as I mentioned above, these two need not be disjointed activities. Looking at mathematical structure can lead one to ask new and different questions requiring explanation. Seeking evidence to support or refute an explanation may require that one collect evidence that is quantitative in nature. But the visitor saw little value in the first activity. He likened it to the kind of science that took place with aristotle–loosey goosey ideas about how the world works with no mathematical structure. He also saw immense value in this second kind of activity–he likened it to how mathematical structure is used in quantum physics.

He went on to argue that the mathematical model for the grass was an explanation and that it could be a good explanation if it could predict with some degree of reliability the height of grass anywhere in the field. To him, it didn't matter whether the grass was cut by a lawn mower or whether it was a matter of water source. Of course, someone might care, but caring about those questions is not science but of agriculture. To him, science was about laying down the quantitative structure and coordinating that structure with quantitative data in increasingly precise and accurate ways. I felt like all the mathematics wasn't an explanation, but that it might serve the point of helping to better define the space of possible explanations or to point to new questions or inquiries about the phenomenon.

Our conversation went on for quite sometime, and it was never quite resolved. Afterwards, I came to think that the root of our different views of science were actually in much deeper differences in worldview. The visitor was a professor of physics from a developing country, and I am a physics education researcher who grew up in a US middle class home. Based on much more conversation with him, I have come to see that his views are much more rooted in pragmatic needs for technical training and economic development, while my views are much more rooted in some idealistic notions of the enlightenment. We are both a product of our personal, cultural, and nationalistic histories. I have tried for sometime to write up more about this difference, and how our views of science are embedded with deep issues of culture and history. It's something I need to think about more and write up carefully, so I'm going to get to it eventually. But for now I wanted to just write about the back story of our science conversation and foreshadow that later blog post.

I don't believe in physics

When I first met my wife, and she learned that I was studying to be a physicist, she said, "Oh. I don't believe in physics." I replied, "That's fine. I don't really believe in physics either." We both laughed.

While I do value and enjoy physics as a set of cultural activities to take part in , it's not something I believe in. I'm not sure I know what would it mean to believe in it. For example: I'm fine saying I believe that objects fall toward the earth. And I really like learning about how ideas about falling objects have changed over centuries: from it being about the natural tendencies of objects, to it being about forces acting between massive objects at a distance, to it being gravitational fields propagating through space and time, to it being about how energy curves space-time manifolds, to it being about graviton exchanges, etc.

But do I believe in forces? Do I believe in fields? Do I believe in space-time manifolds? Do I believe gravitons? No, not really. Do I think there is value in humans learning, thinking about, and exploring those ideas? Absolutely. Do I think that playing such physics games requires taking on stances of realism from time to time? Probably. Do I think one has to believe in the ontologies we create to model the world? Not really.

Over the years, I have told lots of people that my wife doesn't believe in physics. That usually gets some laughs as well. Just before we left Maine, a friend of mine trapped my wife into a public display of participating in physics. My wife was explaining to my physics friend something about cooking–how some eggs float and some eggs sink, and how that has something to do with whether the eggs have gone bad. My physics friendly slyly asked the question, "Why does that happen?" And my wife went on explain why... and not long after she was trapped. She was doing physics, explaining sinking and floating in terms of densities. My physics friend called her on it, and she was shamed into believing in physics, at least for a moment. We all laughed.