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- 9^th 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.

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.

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).

“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).

[2] Perticles: http://www.citeulike.org/group/10888/

[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.

[6] http://www.physics.umd.edu/perg/tools/ResourcesReferences.pdf

[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).

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”.

Cutting mathematicians some slack

I think I finally get it – why my mathematical colleagues who teach service courses for science students don’t like to include units. For years I have been trying to convince my mathematician friends to be more careful with units.

Occasionally, I would run into an irritating example. Once I walked into the lecture hall before my class to set up a demo. On the board, left over from a calculus class in the previous hour, was an integral of “rho + 1” over an area of the plane. Next to it, the lecturer had written “rho = mass density.” If “rho” is a mass density, you can’t add a pure number to it. It’s like saying “I stayed for a certain amount of time – and, oh yes, for one more.”  One more what? Hour? Minute? Second? You have to specify units, and then you wouldn’t write just the number “1”.

I’ve written about a second example [1] that I found on a calculus exam in a class for scientists.

The population density of trout in a stream is

        

where r is measured in trout per mile and x is measured in miles. x runs from 0 to 10.

(a) Write an expression for the total number of trout in the stream. Do not compute it.

(b)…

If “x is measured in miles” then the “1” in the numerator of the fraction is a distance (1 mile) and the “1” in the denominator of the fraction is an area (1 square mile). Using the same symbol for two different kinds of quantities is not good scientific practice. We do it sometimes – consider all the different things that k can mean in physics – but to do it with just numbers is particularly bad -- and having two different "1"s seems particularly egregious.

For a number of years now, I’ve thought that the reason that mathematicians didn’t want to use units was because they wanted to be consistent about the level of the mathematics they used. Unit checks are in a sense “advanced” mathematics corresponding to group theory – a course many science students never take and that mathematicians and mathematical physicists typically only take as advanced students. The idea of group theory is to classify how mathematical structures change when something else is changed. For example, what happens to geometrical objects if you look at them in a mirror? Do they stay the same? If not, can the result be rotated so it’s the same as the original? Another example is, “What happens if you rotate your coordinate axes about the origin?” This analysis of this last explains why we use vectors and what we mean by a vector. And it’s group theory that underlies the properties of angular momentum in quantum mechanics, and therefore is responsible for the structure of the periodic table.

The reason that unit checks have to do with group theory is because units are about how a quantity changes when the standard that is used to assign a number to a physical quantity changes. If I measure a length with the unit of inches, what I mean is that I see how many times the standard size of “inch” fits into my length. I get some number. If I measure it with centimeters (a different and smaller standard size), I get a different (and a bigger) number. The same thing happens with area, but the number assigned to length increases by a factor of 2.54 while the number assigned to area increases by a factor (2.54)² = 6.45.

So making something into “not just a number” but into a quantity that may transform into something else is dealing with a kind of quantity that is more complex than might be appropriate to talk about in an intro math class.

But thanks to a discussion I had last week (see the post “Units and stoichiometry”), I now get that it’s really even worse than that.

What I discussed there is that when we are measuring something with units, what we are doing is mapping some aspect of the physical world into a mathematical structure – the real number line (or in the case of stoichiometry, the positive integers). What I pointed out was that we were actually modifying the math by blending it with our physical concept. That although it looks like we’re mapping something into the real number line, we limit the math that we keep. With distances, if we define a fixed origin, we can add and subtract our numbers freely, getting positives and negatives – and being able to interpret them with a conceptual correlate – something that makes sense to us and we can interpret in the world in which we live. For lengths, this works for multiplying two lengths together – or even three, as we connect that result to the mental concept of an area or a volume. We can go beyond three powers – sort of – by using metaphor and analogy, talking about multiple dimensions and hypervolumes.

With masses, we have to be more circumspect. Negative numbers are meaningless for mass and we typically don’t use them. (Modulo some issues in quantum field theory.) We sometimes multiply masses – for example when considering the gravitational force between two masses. But in most other cases where the product of two masses appears, the result can be rearranged to be a mass times a function of the ratio of masses. We don’t have a conceptual correlate for a “square mass”. (Though I wonder whether we couldn’t come up with one.) “Square times” appear in physics calculations, such as in accelerations or kinetic energy, but again, we don’t have a direct conceptual correlate.

So what I’ve realized as a result of this discussion is that we use units not just to keep track of “different kinds of numbers” as I previously thought. Rather, we use them as a warning to check for “physical reasonableness” in a way that permits some kinds of legitimate mathematical calculations and forbids others in a way that might depend on physical context. So if I’m calculating a kinetic energy, I don’t care that “time squared” occurs in the denominator. If I were calculating a time interval, I would not want to carry out a calculation that resulted in a “square time.”

The use of units is therefore subtle and requires a blending of physical and mathematical knowledge in a way that constrains mathematical manipulations that the mathematics by itself does not. This puts unit analysis outside the realm of what many mathematicians want to be doing in a math class – teaching an “honorable” description of the mathematics that remains true to the math.

I now have more sympathy for their objections so I’m willing to cut them some slack, but still insist that students need to know how to deal with units to use math in science. I have two resulting messages for the two groups:

Mathematicians – Don’t try to “paste in” units into your math classes in a casual or sloppy way. If you must do it (and I would like it if you would), be careful – and realize that you may have to get students to bring in knowledge that goes beyond math, building on their understanding of length, time, and mass from their everyday experience.

Scientists – Don’t assume that units are trivial or even simple or that the mathematicians have handled it for you in their classes. Set aside some time for a careful discussion of why you care and why it’s important – and use units carefully in your lectures and lecture notes.

I have found that even when I am Draconian about units in my teaching – taking off lots of points for wrong units on exams – students still don’t believe that it’s important and that I really care -- and they seem not to understand what's going on. In my future classes, I’m certainly going to try to share this more extended discussion and justification of units with my students. I’ll let you know if it works better than what I’ve done in the past.

[1] Introducing Students to the Culture of Physics: Explicating elements of the hidden curriculum, E. F. Redish, in Proceedings of the Physics Education Research Conference, Portland, OR, July 2010, AIP Conf. Proc. 1289 (2010) 49-52.

Units and stoichiometry

As part of my group’s current work for HHMI (Project NEXUS) to build a physics course for biology majors and pre-meds, physicists and biologists at the University of Maryland are holding extensive discussions. One of the things we like to talk about is the way we each look at the world – what feels like “real physics” or “real biology” to us and what just looks fake. (“Fake biology in physics” is just using biological organisms as the physical objects in a physics example, but not learning anything useful for biology – like using a spherical cow as a projectile.)  In that context, my colleague Wolfgang Losert and I had an interesting discussion about the nature of units and dimensional analysis.

In thinking about how we use math to model the world I like to use a little diagram that helps me make explicit some of the features we often take for granted.[1]  Here is it.

We start on the lower left with some physical system we want to model mathematically. We have some property in mind that we want to describe. We choose an abstract mathematical structure to map onto, creating a mathematical model of the system. From that math, we inherit lots of generative and processing tools that allow us to do things that we can’t easily do with purely conceptual thinking. Once we’re done processing, we have to interpret our results back in the world and then evaluate whether it works – or whether we have to modify our model. [Note this this is meant neither as a description of the cognitive processes of how we actually use math in science nor as a normative way of teaching how to model with math in science. We actually do things in a much more blended and integrated fashion. The modeling model is more of a philosophical analysis rather than a cognitive one.]

Applying this model to units and dimensional analysis is interesting. When we decide how to quantify something in the physical world via measurement, we are modeling. For example, if I decide that a length or distance can be assigned a number by counting the number of times a standard unit fits into that length, I have decided to model length and distance using positive real numbers. With this, I inherit addition, subtraction, multiplication, and division. Addition and subtraction lets me create coordinate systems and vectors and interpret negative distances. Division lets me create fractions of my standard and imagine – at least for a bit – that distance is “exactly” described by the line of real numbers. (We often forget that this is a model.) The fact that I inherit multiplication lets me generate numbers that I can assign to areas and volumes. We use the resulting model both to describe objects (their linear dimensions, areas, volumes) and positioning in space.

But we often forget the limitations of the model. First, lengths of objects are not perfectly modeled by real numbers when you look closely. No object has a perfectly sharp edge. Most are “fuzzy” – if you measured to high resolution you would get slightly different numbers by measuring at different places. At one level this comes from the process to construct the object (How carefully were the boards the table is made of sanded?), at another from the fundamental physics of the structure (Atoms and molecules are discrete – and quantum fuzzy.). If we use distances in space, inheriting the math of the real number line means that we are assuming we are measuring in a Euclidian line, plane, or volume. If that turns out not to be the case (e.g., 2-D maps on the surface of the earth), we need to choose a different mathematical model.

Each time we choose a different concept to measure (and typically an operational definition to go with it), we have a new set of numbers to decide about. If, for example, we are thinking about mass, we map to the positive real numbers since (so far) we don’t have an interpretation for what negative mass would be. (To fit into our current mathematical structures consistently, it would have to have anti-gravity when interacting with positive masses.) We inherit addition and subtraction, but multiplication doesn’t do us much good (except when calculating a gravitational force). We don’t have any physical quantity that we map onto “square mass” so we have to treat the unit of mass differently than we treat a unit of length. (See Solomon Golomb’s cute little problem, “Proving a penny equals a dollar”.) Also, when we put two masses together, they don’t quite add. Their gravitational potential energy reduces their joint mass somewhat (using E = mc² on the negative gravitational potential energy) yielding a “gravitational binding energy” that can be significant for astronomical sized objects. It's usually small and we can ignore it in almost all of physics, but it speaks to the limitations of the “real number” model of mass.

I find it particularly interesting that although each of these two models maps properties in the real world into real numbers, they behave differently in the way the real numbers can be used correctly (squares are meaningful in one and not in the other, negative values are meaningful in one and not in the other). This is one of the points students often don’t get. They think math in science is just about assigning numbers and they prefer to drop the units until the end. They miss that the way you can use those numbers changes depending on which physical dimension you are talking about. And we almost never tell them.

This is why we physicists think that dimensional analysis and units belong to us and why learning to think with units is so much to the essence of “thinking like a physicist;” because with the units, you are supposed to bring along your knowledge of the physical nature of the measurement being considered and constrain your use of the number assignments appropriately.

But the comment that started me thinking about this again and how it fits with my model for analyzing modeling, was Wolfgang’s comment that “Stoichiometry is chemistry’s dimensional analysis.” I thought that was a really exciting insight.

Now I think many of my readers are probably physicists and, if you are like me, you might not have studied much chemistry. (I only studied chemistry in high school when there were a lot fewer elements in the periodic table!) I had to ask him to remind me what stoichiometry was. What he meant was what Wikipedia calls “reaction stoichiometry” – the way the ratio of the numbers of atoms in a chemical reaction need to be adjusted so they react together properly. Basically it’s the statement that in a chemical reaction the number of each kind of atom doesn’t change – they just rearrange. If I’m building water molecules out of hydrogen and oxygen molecules, since hydrogen and oxygen atoms come in pairs (usually, as H₂ and O₂), and since water is H₂O, you need to combine two hydrogen molecules with one oxygen molecule and you wind up with two water molecules:

2H₂ + O₂ ⟷ 2H₂O.

From the point of view of my model of modeling, the chemists are mapping each kind of atom onto the set of positive integers. Wolfgang's insight is that this is like introducing a “unit check” for every distinct atom – using integers rather than the real numbers. We inherit addition and subtraction (you can move things from one side of the reaction arrow to the other) but multiplication, division, and fractions don’t associate with new physical quantities.

It’s interesting to evaluate this integer model and think about why it works. It’s not that the atoms remain inert when they react. The electronic structure changes as atoms undergo chemical reactions. In some sense, the reason you get away with stoichiometry in chemistry is that chemical reactions don’t affect atomic nuclei at all. So what you are counting are the different nuclei and saying you have the same number of each kind throughout a chemical reaction. Despite the fact that we talk as if the atoms were staying fixed in the chemical reaction, it’s really the nuclei that remain conserved. The electron states of the “atom” are shifted around, so although it’s a reasonable approximation to treat the atoms as fixed in a chemical reaction, it’s not quite correct. The stoichiometric equations and the “ball-and-stick” models chemists use are highly symbolic and rely on applying the mathematical counting model described here. The chemists’ “space filling models” are a little more realistic, showing the atoms overlapping and deforming a little. But for some circumstances it must be necessary to keep in mind that the electrons are shared and no longer may belong to a particular atom. We might have to think of some of the electrons are really shared over an entire molecule, producing a kind of “band structure” like we have in conducting crystals.

So even some of the simplest mathematical ideas in physics and chemistry – unit checks and counting of atoms – reveal themselves to be (very) useful models, but to only tell a part of the story.

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[1] Problem Solving and the Use of Math in Physics Courses, E. F. Redish, in Proceedings of the Conference, World View on Physics Education in 2005: Focusing on Change, Delhi, August 21-26, 2005