The Unabashed Academic

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

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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 http://nexusphysics.umd.edu. They are currently very much in a state of flux, so if you have suggestions or find problems with them please let me know. (redish-at-umd.edu)


[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.

27 December 2011

A higher power -- units again


Those of you who have been following this blog for a while know I have been somewhat obsessed with units. You know, those things we physicists like students to stick on to nice clean mathematical variables to make them messier and more complicated? And that students would much prefer to ignore? Well, if you’ve been reading from the beginning you also are beginning to get an inkling of why I care about units so much. My evolving view of science is rooted by the phrase, “Science is not about how the world works; it’s about how we can think about how the world works.” This puts science philosophically as a bridge between cognitive science and the epistemology of reality – a “how do we know” about what we presume actually exists. And this is where my interests and research have been living for some years now.
My previous posts on units (Units and stoichiometry, Cutting mathematicians some slack, and Teaching units) begin to make the case that units are a nice example of the bridge between cognitive modeling and math; that we use our everyday experience with distance, time, and mass to allow us to do some mathematically very sophisticated things at an introductory level without actually bringing in that math. (What units are doing is tracking the irreducible representations of a transformation group in our equations!) But the implications of this are that our “map of the real world onto a mathematical system” (see Units and stoichiometry) becomes limited. There are things we can do with the math that don’t make sense physically. For example, space, time, and mass get mapped onto the real number line by choosing coordinate systems and operational definitions, but while space and time can be negative as well as positive, mass cannot.
I also made the point that multiplication is allowed on the real line, but as we multiply we begin to construct new “conceptual correlates” such as area and volume, but run out quickly. We have no conceptual correlates with L4, though we might build four-dimensional volumes by analogy.
Well, I’ve found two nice examples of good physical uses of length to a higher power in my recent work looking for applications of physics to medicine and biology. These illustrate nicely how these combinations correlate the math cognitively to the physical world in a different way from the straightforward sense we have of area or volume.
Of course the simplest and prototypical example is the square of time. Although we have no conceptual correlate of “square time”, when we define an acceleration we are chaining the idea of rate. The first time we apply the idea of rate of change to position we get a velocity: length divided by time has dimensions L/T. When we do it again, we get an acceleration: velocity divided by time has dimensions, (L/T)/T, which by the rules of math is L/T2. So while we don’t have a concept of square time, we do have a concept of “per unit time per unit time”.
This conceptual chaining is very reminiscent of the way modern linguists and semanticists build up complex and abstract concepts, by chaining of metaphor (Lakoff and Johnson [1]) and by cognitive blending (Fauconnier and Turner [2]).
My biological explorations have led me to two different examples of distance to the fourth and sixth powers – hypervolumes. Figuring out the conceptual correlates for these unit combinations is interesting.
The first example is medical. Various glands in your body secrete a variety of chemicals. The health of those glands can be probed by checking how much of that chemical is circulating in your blood. A simple blood test can measure the amount of chemical found in a particular sample and calculate a density by dividing the mass found by the size of the sample. On my annual blood tests lots of these are measured in micrograms per milliliter (µg/ml). (Unfortunately for those of us who like to push the conventions of SI units, often in micrograms per deciliter. But we shouldn’t be surprised. Blood pressure is still measured in “millimeters of mercury”!) Now if you want to know about the health of the gland, to find out how effective the cells in the gland are in producing the chemical you might want to divide the density of the chemical found in the blood by the volume of the gland – measurable with a sonogram. The result is a “mass per volume per volume” – reported as (µg/ml)/ml but recognizable to a physicist as M/L6. In this case, we make sense of a sixth power of a length as a “per volume per volume” -- a density of a density.
A second interesting example is the Hagen-Poisseuille equation – “Ohm’s law for the pipe.” The pressure drop along a length of pipe that has a continuous steady state flow is equal to the rate of flow times the resistance; this is analogous to the more familiar electrical law that the voltage drop across a resistor is equal to the electric current times the resistance. This law has a lot of important biological implications in situations ranging from the motion of sap in trees to the motion of the blood in animals.
In the familiar Ohm’s law rule, the resistance is inversely proportional to the area of the resistance perpendicular to the flow. This physics of this is that Ohm’s law is basically about the balance of forces: the electric force due to the potential drop pushes the charge through the resistor against the drag, which is proportional to the velocity. Since the charges are moving at a uniform velocity (on the average) there is no net force. The fact that the electrical push balances the drag is Ohm’s law. The area arises because the drag is proportional to the velocity and we want to express the law in terms of electric current – basically velocity times area (times charge density). We introduce the inverse area to change velocity into current.
In the Hagen-Poisseuille law, we have a similar bit of physics: the forces due to the pressure drop pushes the fluid through the pipe against the viscous drag, which is proportional to the velocity. Since the fluid moves at a constant velocity, there is no net force. (There is an additional complication in this case since the fluid doesn’t flow at the same rate of speed in all parts of the pipe, moving fastest at the center of the pipe, but we’ll ignore that here.) The fact that these two forces balance is the H-P law. One factor of the area arises because the drag is proportional to the velocity and we want to express the law in terms of current – velocity times area (if we use volume current; add a factor of mass density if we use matter current). We introduce the inverse area to change velocity into current. But we also want to use pressure rather than force. In this case (not in the electrical case), pressure is related to force by a factor of area. This introduces a second factor of area into the resistance of the H-P equation. (To see this with equations, check out our text on it for the NEXUS Physics class.) This corresponds to the fourth power of the radius and can have powerful medical implications as well. (See our homework problem, Hold the mayo.)
So in the HP equation, we get a factor of L4. In our previous example, the higher powers of length (the square of a volume) came because we were using two different volumes – sort of a double density. In this case, both of our areas are the same but they have two sources. One from the force of the fluid on itself into pressure, the other from converting the fluid velocity into current.
In both of these examples the conceptual correlate of the complex unit is a product of different conceptual objects – two different volumes in the density of density example, and the same area for two different purposes in the HP case. These examples suggest to me that when we have dimensions that don’t have a direct conceptual correlate with a physical concept, understanding the conceptual blends that lead to the combined dimensional structure can help us make better sense of why a complex quantity looks the way it does.
[1] G. Lakoff and M. Johnson, Metaphors We Live By (U. of Chicago Press, Chicago, 1980).
[2] G. Fauconnier and M. Turner, The Way We Think: Conceptual Blending and the Mind’s Hidden Complexities (Basic Books, 2003).

09 October 2011

“Reliability of the FCI supports resources theory”


It’s rare that reading a paper in PER (Physics Education Research) will take me through an emotional roller coaster, but I recently had that experience with a paper by Nathan Lasry and friends in the September AJP. [1] I came across the title and abstract on Michael Wittmann’s Perticles blog that gives early notice of papers in PER. [2] The title of the paper is “The puzzling reliability of the Force Concept Inventory.” By "reliability" they mean test-retest consistency. Since I’ve been giving my students the FCI [3] (or the similar FMCE [4]) as a pre-post test in my algebra-based physics class for many years (and then occasionally putting some of those questions on my final), I was interested to see what they had found. The abstract says that the authors gave the test twice to within a week to students in the second semester of an introductory physics class. The test-retest results were consistent – but students’ responses on individual items were much less so. They conclude the abstract with the sentence, “The puzzling conclusion is that although individual FCI responses are not reliable, the FCI total score is highly reliable.”
My first reaction to this was a smug satisfaction followed by irritation. “Well! You might find this result puzzling, Nathan, but I don’t.  If you had asked me, I would have predicted it.” I look at physics teaching and learning through the theoretical lens of the resource framework. This is a theory of student thinking based on careful educational research, teachers’ experience, and the growing understanding of cognition based in psychology and neuroscience. It began with Andy diSessa’s “knowledge in pieces” approach [5] and has been extended and elaborated by many researchers over the past two decades [6], including some members of my research group at Maryland. [7][8][9]
The heart of the resource idea is that student knowledge of physics (indeed, any knowledge) is made up of bits and pieces that are linked together in a structure whose activation is dynamic and highly context dependent. As students learn, they often make a transition from being highly confident about their answers (which, however, may be inconsistent when they are activated in different situations), to being confused, and finally to being more certain of answers that become more consistent and consistent with the physics they have been taught.
One of my former students, Lei Bao (now at the Ohio State University), developed a method for analyzing the FCI and other such tests by treating the state of student knowledge as a probability variable. Bao’s Model Analysis measures the state of student confusion by presenting an “expert equivalent set (EES)” of items. (Such items may not appear equivalent to confused students.) [10] His hypothesis is that the student has a probability for giving a particular answer to a particular item and that probability is what is being measured by the set of questions.
So in the theoretical framework I use it’s to be expected that a student may well be unstable enough in their knowledge to answer the same question differently on two successive tests. Bao’s model would be useful if there were a well-defined probability of answering questions in an EES correctly and that the probability were more stable than the answers to individual items. Lasry et al.’s result supports this idea.
But let’s go meta for a second. Why should Lasry et al. find this result puzzling? Many teachers who are not well versed in education theories know that students’ knowledge fluctuates. I suspect that part of the problem is that the context of “an exam” activates “measurement” in the researcher and this in turn activates, “measuring something that can be measured – uniquely.” This in turn leads to the activation of what might be called the binary pedagogical misconception – the idea that the student either knows something or doesn’t and that a test measures which.
I waited with interest for my copy of the AJP to appear in the mail. When it did, I read Lasry et al. with anticipation. It was very clear from the first half of the paper that they had done a very careful and well thought out experiment with excellent statistical analyses. But my next emotional state was delight. In the discussion section I found the following paragraph:
From the perspective of a resources model, the FCI questions provide a context that activates concept-of-force related schema or a related set of resources. Given that the context for the test and retest was similar, the resources activated should be similar, and hence the probability of selecting a given FCI response should be similar. This similarity means that the probability of choosing an answer will be the same every time, not that they will choose the same answer every time. Hence, although individual responses fluctuate, the overall time-averaged mean-score is unchanged. In retrospect, our data provide good empirical support for the resource model.

They got it! They even cited us! Excellent!
But then my third emotion kicked in: dismay. One paragraph hidden in the middle of the discussion section, Nathan? No comment in the abstract or conclusion that points out that you came in with a theoretical expectation (even if it wasn’t explicitly stated) and your result strongly supported a competing theory? Why isn’t that the main point of the paper? Why isn’t the title something like, “Reliability of the FCI supports resources theory”? I suspect that the paragraph was put in as an afterthought in response to a comment from a referee (not me).
We in PER often make the claim that we are “applying the methods of science to the question of student learning.”  One of those methods that is fundamental to science is developing hypotheses and testing them; and coordinating validated hypotheses into theories. We don’t do nearly enough of this in PER. Isn’t it time we education researchers began to take ourselves seriously as scientists?
[1] N. Lasry et al., “The puzzling reliability of the Force Concept Inventory,” Am. J. Phys. 79(9), 909-912 (September, 2011).
[3] D. Hestenes, M. Wells and G. Swackhamer, “Force Concept Inventory,” Phys. Teach. 30, 141-158 (1992).
[4] R.K. Thornton and D.R. Sokoloff, “Assessing student learning of Newton’s laws: The Force and Motion Conceptual Evaluation,” Am. J. Phys. 66(4), 228-351 (1998).
[5] A. A. diSessa, “Knowledge in Pieces,” in Constructivism in the Computer Age, G. Foreman and P. B. Putall, eds. (Lawrence Earlbaum, 1988) 49-70.
[7] E. F. Redish, “A Theoretical Framework for Physics Education Research: Modeling student thinking,” in Proceedings of the International School of Physics, "Enrico Fermi" Course CLVI, E. F. Redish and M. Vicentini (eds.) (IOS Press, Amsterdam, 2004).
[8] D. Hammer, A. Elby, R. E. Scherr, & E. F. Redish, “Resources, framing, and transfer,” in Transfer of Learning: Research and Perspectives, J. Mestre, ed. (Information Age Publishing, 2004).
[9] M. Sabella and E. F. Redish, "Knowledge organization and activation in physics problem solving," Am. J. Phys. 75, 1017-1029 (2007).
[10] L. Bao and E. F. Redish, “Model analysis: Representing and assessing the dynamics of student learning,” Phys. Rev. ST-PER 2, 010103, 1-16 (2006).

12 August 2011

Teaching units

I’ve had some response from a few of my mathematical colleagues to my posts on units (Units and Stoichiometry and Cutting Mathematicians Some Slack). One said “2 apples + 2 apples = 4 apples. What does that tell us about 2 oranges + 2 oranges? Nothing. So instead we teach 2 + 2 = 4.” A second objected to my characterization, and said that he did use units when he taught calculus but only insisted that his students explicate them at the beginning and end of the calculation and during a calculation to “keep the mathematics clean.” A third commented that he expected that students already should have been taught about units before they came to a calculus class and that “It's hard enough fitting in the calculus, let alone all the other stuff students SHOULD have been taught but weren't.”

These comments from three mathematicians (all of whom I know to be first-rate teachers of math) support my view that we need to continue to work on building an interdisciplinary dialog with mathematicians concerning service courses for science – and those of us who teach science need to better understand our students’ thinking on these issues. Over the last 25 years of teaching physics majors, engineers, and biology students, I have seen many of my students display serious difficulties about how they think -- or don't think -- about units.

My previous two posts suggested some theoretical reasons why students might get confused about units. We also need an articulation of our learning goals – what we want our students to learn about units – and some substantial research on how students interact with units and what their difficulties are. Physics Education Researchers – including me – unfortunately seem to have let this one slide. In any case, here's my summary of what I want my students to learn about units.

1. Many of the quantities we deal with in science are measurements, not numbers. They are intended to correspond to something physical and the particular number assigned to them depends on the choice of unit made.  This means that equations like "1 inch = 2.54 cm" are correct and legitimate in science.

2. Because of point 1, not all mathematical operations or results that might be perfectly legitimate for numbers are allowed for physical quantities. So, for example, getting a negative result for a mass is a red flag, warning you that either something has been set up incorrectly or there is an error in the calculation.* While for a position coordinate, it is perfectly legitimate (usually -- though I give examples in which it is not** because of physical situations in order to get students thinking about these issues).

3. The "unit" should be treated as a part of the value of a symbol.  It behaves like a mathematical symbol but one whose value is fixed and never changes.  (Kind of analogous to the square root of minus one in complex numbers.) It should be manipulated in the same way as algebraic symbols. (Modulo grammatical equivalences -- "meter" in the numerator cancels "meters" in the denominator.)

4. It is preferable in learning to reason scientifically with mathematics not to put numbers in too soon, but to use symbols for both variables and constants and to do most of the manipulations with those. (One's choice of symbol should give a clue to the kind of quantity being described -- the dimension rather than the unit.) This permits students to learn a variety of useful "checking" tools to make sure one has not made a manipulation error -- dimension checks and taking limits as the constants get large or small. (It's hard to take the limit "as 4 goes to infinity.")

5. When actually putting numbers in, it is valuable to keep units, since often problems will often be phrased in mixed units. Although it is possible to change everything to common units first, failing to do this is a very frequent source of error for students and keeping the unit symbols in the calculations activates their awareness of the need to convert.  

6. Note that in equations when we write something equals 0, that zero often has a unit that we tend not to express. 

The mathematicians legitimately want the mathematics to proceed ”uncluttered by the units." I like to do that as well. But the scientifically correct and consistent way to get problems to be "uncluttered with units" is either to keep it as symbols rather than numbers, or to create natural scales and divide through so as to make equations dimensionless. To see what I mean by this last, check out my notes on how to do it for a Math Physics class: <http://www.physics.umd.edu/perg/MathPhys/content/2/pstruc/dimsDE.htm>.

In addition to suppressing units when we do pure math, we might also note that when writing computer code we often suppress units. (This led to a multi-million dollar failure at NASA.) But any object oriented programming language can correctly retain (and even check) units. I doubt that classes in computation, either in computer science, engineering, or physics departments, teach objects with units – despite the fact that it would be incredibly useful for scientists.

I don't object to mathematicians leaving out units entirely. I would encourage them to introduce and talk about units, or even require their students to identify the proper unit at the end of a calculation. What I do object to is when they ignore units in a way that leads to equations that are incorrect if you fail to hold in your mind things like "the 1 in the numerator is a length and the 1 in the denominator is an area". Students' working memories are cluttered enough with trying to handle knowledge that they haven't yet organized. Having to keep track of the units of their numbers puts an additional burden on them. It leads them to ignore keeping track of dimensions in the middle of a calculation – and that becomes a habit that’s hard to break.  And then in my class they write equations that both confuse them and hurt them on their science exams.

This is not a trivial point and it is not just one of "we do things in different ways." Service course should serve. Can't we work this out to find a set of approaches that is comfortable for both mathematicians and the scientists who need and value their classes?

* Or that you need to rethink some fundamental assumptions! Negative mass and square masses occur naturally in quantum field theory and have led to interesting new ways to look at the physical world.
** Position data taken by a sonic ranger is always positive since it can’t see “behind itself”.