The Unabashed Academic

02 December 2014

Leopold Bloom and the Ontology of Cognitive Dynamics

As a result of some traveling, I didn't have a chance to get to the library and fill up with my usual relaxation reading of trashy mystery novels. I find them diverting and totally non-memorable. That's great! In a few years I can read them again and not remember how they turned out. I often read four or five a week.
I found myself with a lot of work to do with nothing to read to take breaks with. You can only do so many Crosswords puzzles. (Being on sabbatical doesn't mean you don't work – it means you work on the stuff you want to work on!) So I started perusing our collection of more serious novels to find something I had always wanted to read but had missed. Something interesting, but not too engrossing. When I pick up the really good stuff, I often get involved and read for four hours or more, blowing off the work I intended to do. I needed something that I could put down after 20 to 30 minutes of break. So, what's it going to be? Infinite Jest? Or Ulysses?
It was recently the birthday of the woman who published Ulysses, first as a serial, then as a bound book. (The books were confiscated and burned.) I learned about this from The Writer's Almanac the other day, so I decided to try Joyce's masterpiece.
Ulysses certainly seems to meet my requirements. It's interesting, but challenging – and pretty easy to put down. Joyce was one of the first to do a true "stream of consciousness" novel and it's been some time since I read another one. (Virginia Woolf – some years ago.)
Chapter one is about poet and philosopher Stephen Daedalus, who I remember from Portrait of the Artist as a Young Man. The stream of his thoughts are difficult. It feels like half the words are made up, and the other half are ones that I think are real but don't know – many in French, Latin, or German, well above my limited capabilities in those languages. But he's interested in interesting things.
Chapter two switches to a more prosaic character, Leopold Bloom. His thoughts run more to living in the moment and reacting to his context than to musing on deep issues like the transmigration of souls (metempsychosis – one of Daedalus' interests). He thinks about food, interacting with his cat, the people he meets in the street, sex. (Daedalus is interested in sex too but at a more poetic level.)
After two chapters, I've already found a number of things interesting about Ulysses. First, how true the stream of consciousness seems. My own stream of consciousness includes both Daedalus and Bloom kinds of thinking, and when I analyze my own thoughts, they really do the sort of thing that Joyce is transcribing. But second, how false it feels. The thoughts of Daedalus or Bloom both feel (in my response to reading them) choppy, disconnected, scattered. The thoughts in my own head feel (mostly) natural, coherent, flowing, despite looking similar if transcripted. Why the difference?
My suspicion is the key is personal meaning making. The hard part of explaining this is the critical question: "what does 'meaning' mean." It's a bit tricky – besides being self-referential. The definition I like best comes from reading semanticists and cognitive linguistics (Langacker[1], Lakoff, Fauconnier). The idea is that our concepts and thoughts are interpreted in terms of a large web of encyclopedic knowledge about the world we each live in. Meaning is an aura of associations – a subset of our world knowledge that we each activate in the moment to interpret an idea or concept.
This has a lot of implications that help me make sense of the world I see. First, it suggests that the meaning a given individual gives to an utterance, observation, something they hear or read, can depend strongly on context. Our interpretation of the context we are in (framing) controls what of our huge store of encyclopedic knowledge is primed – not necessarily in our conscious mind at the moment, but sort of "first in line" to get activated when a chain of associations is generated to run through our limited working memory.
Let me now turn back to the question of stream of consciousness. Reading a transcript of what might be an accurate rendition of Leopold Bloom's conscious thoughts (OK, LB is a fictional character, but you know what I mean!), Joyce is providing a transcript, but Bloom is not only streaming what the transcript says. He has presumably activated a whole set of associations with each one – and those are often associations I don't have. (This is even worse for me with Stephen Daedalus, since he is a contemplative Catholic and religion plays a huge role in many of his interpretations. There's a lot I'm missing here.) Bloom's chain is constructed with invisible links that his aura of associations make with each term. They provide the glue that sticks the pieces together and makes them feel coherent. When I interpret Joyce's transcript, I do so with my own mental transcript making my own meanings – and my auras of association don't always overlap enough with Bloom's that his stream feels coherent.
One thing this says to me is that "stream of consciousness" as a literary device leaves much to be desired. In his recent book, The Sense of Style, Stephen Pinker has a marvelous chapter that gives beautiful advice about writing clearly for good communication. (I highly recommend Chapter 3 for teachers as well![2]) The key idea is to structure your writing so that readers are given sufficient information to activate their interior contexts to create the intended meaning from your text. In stream of consciousness writing this becomes almost impossible. In Daedalus' stream, it is clear that local politics and theological issues of interest around 1900 significantly inform meaning for him. Hard for me to make this out without a scholarly "Handbook to Ulysses," – and I don't have enough interest in those issues to get one.
I conclude that communicating well with stream of consciousness is exceedingly difficult – particularly if one wants one's work to be perceived as meaningful in later generations. Too much needs to be explicated for your reader to both create the meaning you want and to make the flow of thought seem natural.
Now, those of you who know me know I'm not a literary critic. If you made it this far, you've been patiently waiting for me to get to the point. Here it is, 1000 words in. (Don't do this in a research paper!)
I am both a teacher and an education researcher. A lot of my research is qualitative. My data are often transcripts of videos of interviews, group problem solving, and focus groups. I often have to try to interpret what students are saying. I want to know not just what they say, but to go beyond the transcript and infer what meaning they are making. (I would normally have said "if any", but given the definition of "meaning" above, my students are always "making meaning", just not necessarily the kind of meaning I want them to.) My colleagues and I draw on a variety of tools to infer this – gestures, word choice, tone of voice, etc. – together with our understanding of the context and our everyday communication skills. Of course one must also bring a theoretical perspective on how to interpret what one sees, to transform an observation into a measurement.
For the interpretation of student responses there are two extreme theoretical orientations: knowledge-in-pieces theory (KiP) and theory-theory (θ2). The former views students as having lots of bits of "irreducible" knowledge or "primitives". These are the places where any reasoning chain [3] of
Claimdatawarrant = claimdatawarrant = claim
ends. A primitive is something like "unsupported objects fall" or "push harder and it will move faster". Of course, in physics, we create complex reasoning for these, but in "folk-physics" models of the world, these are things you learned as an infant by watching and testing how the world worked. They form the core of lots of our everyday thinking.
The KiP approach starts by assuming students tend to bring up individual primitives (or resources) and try to get by with that; or that they bring up an easily generated story composed of a few simple primitives (as in Kahnemann's "fast thinking" [4]). KiP researchers then try to analyze more complex patterns of reasoning and build up an understanding of "knowledge structures".
The θ2 approach starts by assuming students have a coherent theory of a phenomenon, and analysis is informed by this assumption. But what we as scientists see as a single coherent phenomenon or set of phenomena may be seen by students as being governed by different coherent (but more local) theories.
These two approaches start from opposite ends and move towards each other. We might imagine a continuum between these two extremes. Some student responses could be more towards one end than the other, but empirical observation might let us determine where on that continuum a particular student's response on a particular subject belongs.
I suggest that the situation is more complex than can be described by a single continuum and that my ramblings on reading Ulysses are relevant to seeing how.
My stream of consciousness story says that in anyone's thoughts there is a continual chain of sequentially associated items popping up and that while these may appear incoherent to an outside observer, to the one experiencing the chain, local meaning creates a sense of coherence in the local flow. But in this picture, the self-perception of coherence is about how thoughts are changing moment to moment, not necessarily about the long-term constant activation of a coherent theory summing up and managing a multi-minute long argument.
One may feel that one's own thoughts are coherent – and they may be – but I suggest that a personal feeling of coherence depends on a derivative (information local to a moment) rather than an integral (information over a long time scale) and may be misleading. This could be why a number of educational theorists I have conversed with feel strongly that one must begin by assuming coherence. It just feels that way from inside!
Of course when we are teaching physics to students, one of our long-term goals is for them to learn to build large-scale coherent arguments, with reasoning that reaches over many minutes, not just a step or two. Often, it looks to me as a teacher that many a novice physics student can't put three steps together without forgetting the first one!
When my research activity turns to analyzing a transcript of a student solving a physics problem, I'm often interested in their fine-grained stream of consciousness and the particular association that drives them in the moment. 
In a problem about Newton's third law (two interacting objects exert equal and opposite forces on each other), have they recalled Newton's second law (a = Fnet/m) or its folk-physics equivalent (an object moves in proportion to the force acting on it) and focused only on the force, ignoring the effect of different masses? In a problem on pressure in a liquid, have they focused on one variable (the depth), failing to be coherent about the implications of their choice of coordinate system on the sign of g? Is their response affected by locally activated epistemological resources, such as "trust my physical intuition" or "the authorities must know what they are talking about"? By affective responses: "This is scary" or "My intuition always disagrees with physics"? There are lots of local questions that are deeply interesting. [5]
That's all very KiP driven. But we do all have long term coherences in our everyday thinking. There are patterns and regularities that last over very long time scales. 
At age four, my daughter was able to sit in one place with a game or coloring book for an hour or more, totally engaged. If I watched closely, she may have been jumping from one idea to the next with what looked like little coherence, but often she was building a story, shifting and changing it, trying one thing then another until it felt right in the moment. And there was a long term frame – the story telling and the very fact that the activity was about story telling being coherent and persistent over a long time scale. My students also have even longer term coherences, over an entire semester regularly activating "My intuition always disagrees with physics and I should ignore it" at the first sign of trouble. My own long-term highly stable coherences include "always start with an equation you can trust."
So what is an appropriate ontology for thinking about our student's thinking? Should we pay more attention to the fact that thinking is often local and driven by short term coherences explicable using a KiP-like analysis? Or to the long-term framing and average patterns that appear and look more like θ2 when you step back and look at a coarser grain size?
Of course my answer is that you have to do both to get a complete picture. A nice example of this kind of "two-scale-doublethink" is provided in many-body quantum physics. I'll explain that in my next post.

[1] R. Langacker, Foundations of Cognitive Grammar (1987).
[2] Unfortunately, his Chapter 4 proceeds to violate most of the precepts in Chapter 3 and is almost incomprehensible. Maybe he intends it to be an "exercise for the reader" to figure out how to fix it. Very "active learning"!
[3] This chain is based on Toulmin's analysis of reasoning. Every claim must be supported by data, and the reason the data supports the claim is a warrant. But every warrant is also a claim, so, like a four-year old, we can continue asking "why" (demanding data and warrants) forever. This chain stops at primitives: Things we know from our everyday experience that we have no reason for. They are "just the way things are."
[4] D. Kahneman, Thinking Fast and Slow (Farrar, Strauss, & Giroux, 2011).
[5] A. Gupta & A. Elby, Int. J. Sci. Ed. 33:18 (2011) 2463-2488.

27 August 2013

What should we tell a colleague about DBER?

I had an interesting conversation with a colleague in another science department yesterday. A student wants to do a PhD in their department with a dissertation in DBER (discipline-based education research). That department has never done such a radical thing and we talked about whether they could put together a committee with enough on- and off-campus expertise to make it work at a scholarly level that he would respect. But in the discussion he mentioned that, one problem he had with education research is that it was too dogmatic.
This took me aback a bit. My own take on what DBER teaches me about teaching is that I often don’t know what’s actually going on. Whatever I assume is happening with my students it's probably more complicatedt. But I suppose we come across as dogmatic after learning much about what doesn’t work even though we expected it to.
I was reminded of some of my first experiences in PER (physics education research). Having worked for a few years in the ‘80s on bringing the new personal computer into the physics classroom, I was intrigued by what I learned about the growing community of physics education researchers. I was  inspired by the thoughtful and insightful writings of Arnold Arons and the careful experimental research of Lillian McDermott. So when I decided to switch my research from Nuclear Theory to PER (in 1992), I did it by taking a sabbatical at the University of Washington.  Lillian’s research group there was changing the way we though about teaching and Arnold was still living there after having retired a few years earlier. I had already had some encounters with Arnold and discovered that we really enjoyed arguing with each other. (A story for another time.) He agreed to meet me for lunch at the UW Faculty Club (the one with the marvelous mountain view) at least once a month for conversations.
Well, when I got to UW I learned that although I enjoyed arguing with Arnold, not everyone else did. Having lunches with a few of my nuclear and particle physics colleagues, I found that just mentioning his name was enough to raise hackles around the table – and turn some people red in the face with anger. Arnold apparently went round yelling at other faculty about how they were doing everything wrong and ruining their students. Their response was that they shut down and stopped listening to him.
Now my colleague from the other department wasn’t yelled at. But he definitely got the message – from some of our seminars, writings, and workshops – that we felt we had the answers and we were saying that, “if he would just listen to us and do things our way his teaching would go much better.” He is an award winning lecturer, takes his teaching very seriously, and feels strongly that he has an effective personal style that he doesn’t want to give up.
Even as a PER person myself, I resonated more with his side than with the PER view he was reporting. I know that lecturing is not usually as effective as an engaging activity, yet I still often lecture (even about not lecturing). When I talk to my colleagues in a seminar or colloquium I do try to have some engaging activities and to open some discussion. But both with my students and colleagues, I often take a chunk of time to tell a story. I’m a bit of a storyteller and I know that people interact will with a well-told story. I have had both students and colleagues come up to me years later and remember (accurately!) a story I have told in a lecture that they heard. So even when I do a flipped class, I often spend some time “lecturing” – telling some personal story that links to the point (like the one you are reading now).
So what is it that I want a colleague to take away from what we have learned in PER and DBER? I try to tell them that we have learned a lot that is helpful but that we do not have a magic bullet. Here are four things I would say that we have learned that it is valuable for a teacher.
1.     Think carefully about what your real goals are for the particular population of students you are teaching.

2.     Find ways to get sufficient feedback from the students that you can figure out, not just whether they have learned what you have taught, but how they have interpreted it and what knowledge and perspectives they bring to your class.

3.     Respect both the knowledge they are bringing and them as learners. “Impedance match”* your instruction to where they are and what they have to work with.

4.     Repeat. That is, go back and re-think your goals now that you know more about your students.
Sometimes we do sound too dogmatic. “Just make your class more interactive.” “Be the guide on the side instead of the sage on the stage.” “Flip your classes.” “Use this particular instructional method and follow the steps carefully.”
Of course this is exactly the sort of thing we tell them not to do with their students. If we want to change the way our colleagues teach, we have to engage them in the process, learn what they bring to the table, and let them construct their new way of teaching for themselves – with appropriate support, occasional guidance, and scaffolding.
This last (scaffolding) in this circumstance means providing them with or helping them develop the tools to learn for themselves. Accomplishing step 2 is decidedly non-trivial in our traditional large-lecture science classes. We often give few tests and rarely get sufficient feedback in lecture from enough students to get a sense of the class. It’s why clickers are potentially a game changer – but only if used to probe more deeply into student thinking and if that information is used to change what we do. Faculty may need some scaffolding to get them to both implement step 2 and see what one can learn from it.
One of the things I’ve taken to doing in the past few years in my large class is to give challenging (often multiple choice or short answer) quizzes once a week. I return them in the next class and present the results of how many people chose each answer. I then demand a discussion of why people chose the wrong answers, am respectful of those answers, and try to help us all understand why people might naturally choose those answers – and how we could all develop approaches to thinking about the problems that would help us catch those natural errors in reasoning. (For more discussion and an example, see the “Instructional Implications” section of my Oersted Lecture.)
So I want to raise the messages we have been sending about how to teach our students to a meta-level. If you are a DBER colleague who wants to help other faculty improve their teaching
1.     Think carefully about what you like them to actually do.
2.     Find ways to get feedback from them to understand where they are and what they bring to the table.
3.     Respect the knowledge and experience they bring in and work with them where they are.
4.     Repeat.
Thanks particularly to Renee-Michelle Goertzen and Chandra Turpen for insights and discussions that helped me in developing this perspective.
* For those who are not physicists, “impedance matching” is a term from signal theory. If you are sending a signal down some channel (“channel” means a signal path like a wire or a fiber optic cable) it has some resistance to the signal – it decreases the energy of the signal slightly as it travels down the channel. The parameter that measures the rate at which the signal loses is called the impedance. If you connect two cables and send a signal from one cable into the next, a lot of it will reflect back and not go through unless the impedances of the two cables are the same -- matched.

01 July 2013

The World is an Ill-posed Problem

I got into trouble the other day on one of my listserves. A physics teacher asked if anyone had a collection of electrostatics questions that might be appropriate for multiple-choice assessments in intro physics. I had just completed a first pass on putting together a set of clicker questions for intro physics, and many of them had been drawn from quiz and exam questions that had worked well. So I posted a link to them on the listserve.[1]

I then got severely taken to task by one of the listserve regulars because too many of my questions were ambiguous. The word he used was “ill-posed”. I looked at some of the questions and realized that in transitioning from exam or quiz question to a PowerPoint-displayed clicker question, I had been forced to remove a lot of text, including some specifications. Someone who wanted to use them as assessments would have to reconstruct the more complete context. My bad. This means that I should not only be posting our clicker questions, but our exam and quiz questions in their original form in order for them to be useful. I’ll get right on it.   

But I was interested in my reader’s reaction. I liked my clicker questions and had used some of them extensively in class. Many were very effective for generating engaging and useful discussions. Some of the best discussions took place when some student said, “Hey! Wait a minute. What if …” – and they noted an unspecified ambiguity. I would have lost those discussions if my questions were all “well posed”. What was I trying to do?

An explanation relates to my perception of how we ought to be teaching the mathematical content in our physics classes. This is represented by my “four-box” diagram show below.

Note that this diagram is intended to represent the philosophical imbedding of math into physics and NOT as a process that should be followed and NOT as a model for how the brain interprets physics. The processes of actually doing physics or thinking about math in physics are much more intricate and contain lots of feedback loops.

The diagram IS intended as a way of thinking about the role of math in physics – and of guiding us as instructors to be sure that we are including some activities for each linking line. In this picture, the idea is as follows: Math provides a set of symbolic structures that have been tested for consistency (as much as possible – pace Gödel) and that provide well-developed and tested procedures. We use these mathematical structures by first selecting a set of physical phenomena that we want to describe. We then create ways of making mappings of physical properties onto elements of a mathematical structure – a mathematical model of the physics system. From this, we inherit the processing and solution structures of the math, allowing us to develop relationship and solve complicated problems, well-beyond what we can manipulate or store in our heads at any single instant. This is extremely powerful and works well. It is a major part of our success in physics.

But the math is not the whole story by a long shot. The other parts of the diagram are just as important. Choosing what bits of the great blooming and buzzing world we are going to pay attention to and model mathematically is definitely the hard part – and where the art is in doing science. [2] Interpreting and evaluating whether the model works – and for what – play a critical role in establishing and refining our mathematical model.

Now to get back to the point. “Ill-posed” is a mathematical term. It means that the problem has not been stated in such a way that there is an unambiguous and unique solution. But this lives firmly on the top side of our diagram. Stating that a problem is ill-posed or well-posed means that you are evaluating it as to how well framed it is as a mathematical problem – how well it is situated in the upper-left box.

In my teaching, however, my focus is much less on whether my students can turn the crank on a well-posed mathematical problem. I am much more interested in whether they can “see the physics in the math” – model physical systems mathematically and evaluate the math they generate in terms of the physics. The physics is NOT identical to the math. The world is an ill-posed problem. A major part of what I want my students to do is consider what it means for a problem to be ill or well posed. What has to be specified? What is the physical system like? What do I have to ignore to enable the construction of a mathematical problem that is well structured? Is it OK to ignore these things? Just for now? Always?

We often are so concerned with making sure that our students can “handle the math” that we ignore – worse, suppress – the essential issue of matching the physics to the math. In my experience, this appears to be one of our habits that lead so many of our students to reject physics as “irrelevant to real life.” [3] We might argue that it’s a part of our selection process for physics majors. Physics majors have to be strong in the mathematical skills and we can claim that they will learn modeling and evaluating approximations later in their careers.

But I even object to this for physics majors. I did not really learn that the math wasn’t the physics until I was well into research. I would have been a stronger and more effective physicist earlier had I understood the difference between the math I was being taught and the physics that it was attempting to describe. In my advising of graduate students, I have seen many who had trouble with Jackson (Electricity & Magnetism) or Sakurai (Quantum Mechanics) because they focused too strongly on the math and failed to blend their physics knowledge with it. (For a more technical discussion of this, see my paper with Tom Bing in Phys Rev. [4])

What’s even worse is that most physics students are not going on to be physicists. They are going to be engineers, biologists, and doctors. Teaching only “well-posed” physics to these students tends to turn them off. These students tend to care more about reality than about math and they want to see the connection of the physics to the real world. You might say, “Well that’s what well-posing does for them. It states what has to be specified in order to use the math.” But until a student understands what’s going on it doesn’t look like that to them. What they see is “You’re just giving me unrealistic [and, for them, by implication uninteresting] problems.” They find learning to pose and evaluate problems from a real-world physical situation (and then solving them) more satisfying.

If we go back to the original argument – “this is for an assessment, a quiz or exam, so you can’t give them ill-posed problems” – I would concur, in a way. You don’t want your exam questions to be too ambiguous. But if you only test them on mathematical manipulation, they will ignore the critical other parts of my diagram. 

Therefore, I explicitly try to include in my exams questions that test the students on the other parts of the diagram. I might assign a problem that starts with a real-world example, then propose a set of approximations, ask the student to solve it, but then ask, “of the proposed approximations, which would you want to put back first in order to get a more realistic result, if you had more time.” 

Or I might give a multiple-choice question in which an experiment is proposed and ask the students, “From this experiment what can you conclude?” If the case is that the result being sought is known to the students, but the experiment is insufficient to show it, the correct answer may be not “the correct answer” (that is, what really happens), but “nothing can be concluded from this experiment.” Or, I may ask them to construct a problem or constrain a problem through order-of-magnitude estimations.

A major component of learning physics is learning to look at the real world and generate a well-posed problem; not only to see what that means but why you might want to do so and what are the implications of doing it. In my current class, I often use ill-posed problems in the hope of generating discussion and encourage my colleagues to consider moving in this direction as well.

[1] These materials are being developed for a new physics course for life-science majors. The goal of the course is NOT to introduce the students to every physics concept they might ever see in their lives; rather, it is designed to provide support for difficult concepts they encounter and use is biology and chemistry, and to prepare them for upper division biology classes. To see these materials, go to They are currently very much in a state of flux, so if you have suggestions or find problems with them please let me know. (

[3] E. F. Redish, J. M. Saul, & R. N. Steinberg, Student expectations in introductory physics, Am. J. Phys. 66, 212-224 (1998).

[4] T. Bing & E. F. Redish, Epistemic complexity and the journeyman-expert transition, Phys. Rev. ST Phys. Educ. Res., Vol. 8 (Feb 2012), 010105. doi:10.1103/PhysRevSTPER.8.010105. 

24 February 2012

Model vs mechanism -- trouble between the sheets

As many of you who know me are aware, during this academic year 2011-12, I’m deeply involved in an attempt to reinvent intro physics for Life Science students. Our traditional approach for this population – which includes prospective biology researchers and pre-health-care professionals – tends to look like a traditional class for mechanical engineers cut down in the math somewhat with a few “spherical cow” problems tossed in. Our group is trying to see how to make it feel more “biologically authentic” for our bio majors and pre-meds.*
I’m teaching a small (N=20) trial class and I’m teaching it in what’s being referred to this year as a “flipped class”. Instead of the students getting the basic material presented to them in a lecture and then going out to do problems on their own, they are expected to get the material themselves (typically, watch a lecture video) before class and then go to an “active engagement” class with problems and discussions. I like this idea, since it puts the effort into where on-site education adds value over web instruction – in live interaction.**
I’m not preparing lecture videos – rather, our team is preparing materials as a wiki-book. The students read two or three web pages the night before class and write a paragraph summary of each page and ask a question. I answer their questions on-line before class and use the best questions to open discussions in the next morning’s class. I think this has an advantage over video in that they are learning to read scientific text – and not just read it but evaluate and question it.
This experience is reminiscent of the famous old EDS superbowl commercial, “Building an airplane in flight.” The material for each class is only ready for the students 36 hours before class (if I’m lucky). Each evening before class I have to come up with at least half a dozen clicker problems to guide us through the critical elements. Then during each class I have to be on my toes, looking at the students’ answers, listening to what they are saying –and actually hearing it! And I have to be able to respond in the moment. No more, “That’s a very interesting question. I’ll bring you the answer tomorrow.” Rather, I now try to respond. “Great question. What do you all think?” This definitely started out being out of my comfort zone. Although I’ve been giving clicker lectures for 10 years to classes of 200, I’ve always told them not to bother reading a text, and spent much of my class time giving them clearly outlined core principles and then doing a few examples with 3-4 clicker questions. This new class is an adrenaline high and I think I’m getting addicted to it. In any case, my new teaching is not the point of this entry – what I learned from last week’s classes is.
Two of the broad threads that run through all of physics are the following. First, some advice I give to my students on day 1:
Physics is about something real. Whenever you think about a physics example, start from a mental image of a physical situation and refer everything back to it.
In physics our equations, our graphs, our diagrams, are all about some physical situation and are intended to inform us about some aspect or relationship of that situation, building a richer and more multi-dimensional view of it. In my long experience as a physics teacher (40 years), this element is what’s missing for most intro students. They want to “answer-make” rather than “sense-make” and fail to build that underlying physical picture. This is really what we are asking them to do when we advise them, “First, draw a picture”, but they often do it and then don’t use it for anything. Having a physical model can (and should) guide them in understanding the mechanism of what’s happening, in eciding what they have to pay attention to and what they can ignore, and in figuring out what principles are relevant with what restrictions.
Second, I tell my students:
The “style” of physics is to simplify. We always try to find the simplest example that illustrates a principle so we can understand it fully. We then use that example as the core of our thinking to elaborate into more realistic situations.
This is what I take to be the essence of the famous Einstein quote – “Physics should be as simple as possible, but not simpler.” It’s our motivation for doing point masses, flat-earth gravity, the ideal pendulum and the perfect Hooke’s law spring. (And the spherical cow.) It’s why we spend so much time, both in our instruction and in our research, in what we sometimes refer to in a self-derogatory but affectionate way as “toy models.” It’s our way of getting a foothold that we can make sense of to imbed into organizing and “finding the physics” in a complex situation.
These two characteristics of physics lead me to two fundamental goals of my physics instruction: (1) help students put that physical picture into their every analysis of a physics problem, and (2) help students understand the simple toy models and learn how to use them to organize their physics thinking.
Last week I had a dramatic example of how there is a dynamic tension between these two aspects of physics thinking. We were studying electric fields and potential. A standard example is the “infinite flat sheet of uniform charge.”*** This is a nice example since the math simplifies dramatically. Because of Coulomb’s law, any electric field has to look like a charge divided by the square of a distance (times a universal constant chosen to set the measurement scale). When we have an infinite sheet, we have no “charge” we can use – it’s infinite –we only have the charge per unit area. This already has units of charge divided by length squared, so there is no room for any other distance in a formula for the electric field. The result is that the field has to be a constant, independent of the distance from the sheet.
This seems strange, but it actually makes sense. [This argument may be difficult without a picture. To see details with figures, go to our draft webpage on the field of an infinite sheet, but I’ll outline the argument here briefly. You can skip to the next paragraph if you don’t want the technical details. They’re cute but not essential to my point.]
To get the total effect of the infinite sheet you have to add up the Coulomb’s law contributions from each of the bits of charge in the sheet to the field at the point you are sitting at. Each bit of charge contributes a field vector that points along the line to your point from the charge that is proportional to one over the distance to that charge squared. As you go farther away to the more remote charges, they contribute less and less. Also, each distance charge is paired with another distant charge equally far away on the other side and these contributions tend to cancel – and cancel more and more the farther away you get. The result is that for the entire infinite sheet, if you are a distance s from the sheet, only the circle right beneath you of radius about 5s contributes significantly to the field you detect. So although we say we have “an infinite sheet” that’s not what we mean. We mean: we have a flat sheet and the edges are far enough away that we don’t have to worry about them. So the result is: only a circle of radius 5s matters. As you increase s, the effect of each charge on the sheet falls off like 1/s2, but the amount of charge you see grows like s2. These two effects cancel to result in a constant field.
The result of having a constant E field simplifies a lot of the math. The potential that goes with a constant field is just linear (since the derivative of the potential is the E field) so the math is really simple- 9th grade algebra. All those complex “curvy 1/r2” functions and vector integrals add up to give straight lines. It looks just like the same math for flat-earth gravity – where we take the gravitational field to be constant always pointing in the same direction.
So here’s where I ran into something interesting last week. Students read my draft webpage (and it was read and summarized by 20 out of 20 of the students). Then in class I asked the following clicker questions:
If two uniform sheets of equal and opposite charge can be treated as if they were infinitely large, which of the following graphs might serve as a graph of (A) the x-component of the electric field and (B) the electrostatic potential as a function of the coordinate x along the dotted line?

Since the fields from each sheet are constant and since the charges are equal and opposite, outside of the sheet the two fields cancel, and between them they add. The result is that the E field (x component) looks like graph 8; constant between the sheets and zero outside. Looking for graphs whose (negative) derivative looks like 8, we see that both 3 and 9 work for the potential. That’s OK since for the electric potential (as for height when we are talking about gravity) we can choose any reference point as zero potential. It’s only the shape that matters physically.
On the first question – what does the electric field look like – the results were gratifying. Of the 19 students present, 16 chose answer 8. But one student complained. He said, “I didn’t like any of them.” When I asked why, he responded, “Because when you get near to the sheets you’ll see the individual charges and the field has to go to infinity.” I brushed him off with a brief comment about resolution – that it would only happen really really close and we wouldn’t see it on this scale and anyway we were ignoring individual charges in this simple model.
On the second question – what does the potential look like – the results were more mixed. About half chose answer 3, but the rest were all over the lot. But one student (a different one) said, “It can’t be any of them.” When I asked why, he responded, “Because if you are sitting exactly between the plates the result has to be zero. If you are sitting there, for every positive charge on one sheet there is a negative charge on the other sheet an equal distance away that will cancel. Also, it has to eventually asymptote to zero for large values.”
Well! I was stunned. This was absolutely top quality physics reasoning. He was using a physical picture and using it with a correct symmetry argument – another strong tool in the quiver of good physics thinking. (This was something I had been trying to model in the class, but not fussing too much about.) Finally, he was focusing on limiting cases, another standard tool we try to get our students to use. My first internal response was – sign this guy up as a physics major! My second was. Gak! I seem to want them to be looking at this model example NOT in terms of the basic physical elements but as a toy model that suppressed the underlying physical picture. Since a third major goal of my class is to teach my students to seek consistency, what was I doing?
I don’t know that I handled my little epiphany well on the spot. I tried to be encouraging and praise the students’ thinking but still bring us back to being able to using the simplified model while “suspending our disbelief” – using it even though we know that “when we go to infinity the sheets that look infinite when we are close will turn into looking like point charges when we are far away”, and “we’ll ignore the fact that the charges are actually quantized and treat the chunky peanut butter as if it were perfectly smooth since we don’t see the individual charges until we are really close and then we’ll just ‘close our eyes’ for a moment while we are passing through the sheet”.
Really what it’s about is modeling.**** When we have complex situations – like lots and lots of charges – it’s great to have “a stake in the ground”; an example where we know an “exact” answer that we can refer to that serves as a starting point for further examples and elaborations. But in intro physics we tend to focus on the simplicity of the models and not on the complexity of thinking about where they come from, what their validity is, and what value we can make of them.
Increasingly, I want to open up this Pandora’s box for my students. Trying to pretend that the physics is simple by hiding the deep structure, both ontological and epistemological (i.e., what is it we are actually talking about and how is it we decide we know), is beginning to seem to me to be unfair to our students and not the best way to start students on learning physics.

* For more info on this, check out the NEXUS physics class website.
** If you haven’t already listened to Emily Hanford’s audio documentary, “Don’t Lecture Me”, check it out at American Radioworks.
*** I’m referring to the example as “sheets” rather than “plates” as is more common since my examples for biologists will be membranes rather than parallel plate capacitors with metal disks.
****David Hestenes, "Modeling games in the Newtonian World", Am. J. Phys. 60 (1992) 732-748.

04 January 2012

Lose the lecture

On New Year’s Day, my favorite radio program, All Things Considered, aired a clip from Emily Hanford’s American Radioworks audio documentary, “Don’t Lecture Me.”  It focused on the physics (and the Physics Education Research) part of the documentary and featured Eric Mazur, David Hestenes, and me. And they led with my quote, “With modern technology, if all there is is lectures, we don’t need faculty to do it. Get ‘em to do it once, put it on the Web, and fire the faculty.” 

Now this is slightly embarrassing. I’m not really advocating firing faculty and turning universities into a sort of glorified YouTube. Rather, what I’m saying – to the faculty – is, “If you think lecturing is good enough, you’re putting your own job at risk!” And that it’s up to us to reinvent ourselves so that we add value beyond what a student can find at the Kahn Academy or at the University of Phoenix.

The issue really is a broad one and one that universities mostly have their head in the sand about. Namely, when huge amounts of class materials are available on line, and on-line universities exist to give certification on the basis of those materials, what value do the brick-and-mortar universities add that is worth their extra cost?

For those students who want to do research, the answer is obvious. Furthermore, the large research universities are the bulwark of research and development in the US and the engine that has driven the world economy for the past half century. But despite large blocks of government funding, the support that enables the research arm is still the education of our students.  Now students who are interested in research as a career is a(n important but) very small fraction of the population we currently serve.

For me the answer is clear. Learning is better, deeper, and more effective when you do it with people -- through interactions with mentors, journeymen (TAs, grad students, and senior undergrads), and other students. We learn best in a community of learners and – so far – on line interactions don’t live up to the rich experience of a real interpersonal exchange.

But the transformations that are increasingly pressed on us by Deans and Provost focused on this year’s bottom line, take us further from that value that only we can add and move us towards delivering education that is increasingly equivalent to what the on-line colleges can deliver. This is a recipe for disaster. Brick-and-mortars can't compete financially with online institutions on their own turf. We have too many maintenance costs.

I expect that the next ten to twenty years will produce a major shakeout in the university system in the US akin to what began when Amazon took shopping online and that is is still going on.  Now, 10 years later, many brick-and-mortar stores are out of business with more to come. Those who can't deliver online -- or learn to provide value that online oriented businesses can't -- are doomed. How many of us bought Amazon stock in 2002? It's now the second largest retailer -- behind WalMart -- and relies entirely on internet sales. 

Those academic Chairs, Deans, and Provosts who think that the new technology will make it cheaper to deliver their product with fewer faculty (and larger classes) are undermining the future of their own universities. We should be moving in the opposite direction, providing students with more faculty interaction, more group learning environments, and more hands-on activities. We need to make good use of digital technology, but we need to use it effectively and go beyond what can be done by a student working alone with a computer. If we are to survive as a major component of the educational system, we need to seriously begin reinventing universities for the digital age.