The Unabashed Academic

Model vs mechanism -- trouble between the sheets

2012-02-24T05:54:00.002-05:00

As many of you who know me are aware, during this academicyear 2011-12, I’m deeply involved in an attempt to reinvent intro physics forLife Science students. Our traditional approach for this population – whichincludes prospective biology researchers and pre-health-care professionals –tends to look like a traditional class for mechanical engineers cut down in themath somewhat with a few “spherical cow” problems tossed in. Our group istrying to see how to make it feel more “biologically authentic” for our biomajors and pre-meds.*

I’m teaching a small (N=20) trial class and I’m teaching itin what’s being referred to this year as a “flipped class”. Instead of thestudents getting the basic material presented to them in a lecture and thengoing out to do problems on their own, they are expected to get the materialthemselves (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 webinstruction – in live interaction.**

I’m not preparing lecture videos – rather, our team ispreparing materials as a wiki-book. The students read two or three web pagesthe night before class and write a paragraph summary of each page and ask aquestion. I answer their questions on-line before class and use the bestquestions to open discussions in the next morning’s class. I think this has anadvantage over video in that they are learning to read scientific text – andnot just read it but evaluate and question it.

This experience is reminiscent of the famous old EDS superbowlcommercial, “Building anairplane in flight.” The material for each class is only ready for thestudents 36 hours before class (if I’m lucky). Each evening before class I haveto come up with at least half a dozen clicker problems to guide us through thecritical elements. Then during each class I have to be on my toes, looking atthe students’ answers, listening to what they are saying –and actually hearingit! And I have to be able to respond in the moment. No more, “That’s a veryinteresting question. I’ll bring you the answer tomorrow.” Rather, I now try torespond. “Great question. What do you all think?” This definitely started out beingout of my comfort zone. Although I’ve been giving clicker lectures for 10 yearsto classes of 200, I’ve always told them not to bother reading a text, andspent much of my class time giving them clearly outlined core principles andthen doing a few examples with 3-4 clicker questions. This new class is anadrenaline high and I think I’m getting addicted to it. In any case, my newteaching is not the point of this entry – what I learned from last week’sclasses is.

Two of the broad threads that run through all of physics arethe following. First, some advice I give to my students on day 1:

Physicsis about something real. Whenever you think about a physics example, start froma mental image of a physical situation and refer everything back to it.

In physics our equations, our graphs, our diagrams, are all aboutsome physical situation and are intended to inform us about some aspect orrelationship of that situation, building a richer and more multi-dimensionalview of it. In my long experience as a physics teacher (40 years), this elementis what’s missing for most intro students. They want to “answer-make” ratherthan “sense-make” and fail to build that underlying physical picture. This isreally what we are asking them to do when we advise them, “First, draw apicture”, but they often do it and then don’t use it for anything. Having aphysical model can (and should) guide them in understanding the mechanism of what’shappening, in eciding what they have to pay attention to and what they canignore, and in figuring out what principles are relevant with whatrestrictions.

Second, I tell my students:

The“style” of physics is to simplify. We always try to find the simplest examplethat illustrates a principle so we can understand it fully. We then use that exampleas the core of our thinking to elaborate into more realistic situations.

This is what I take to be the essence of the famous Einsteinquote – “Physics should be as simple as possible, but not simpler.” It’s ourmotivation for doing point masses, flat-earth gravity, the ideal pendulum andthe perfect Hooke’s law spring. (And the spherical cow.) It’s why we spend somuch time, both in our instruction and in our research, in what we sometimesrefer to in a self-derogatory but affectionate way as “toy models.” It’s ourway of getting a foothold that we can make sense of to imbed into organizingand “finding the physics” in a complex situation.

These two characteristics of physics lead me to twofundamental goals of my physics instruction: (1) help students put thatphysical picture into their every analysis of a physics problem, and (2) helpstudents understand the simple toy models and learn how to use them to organizetheir physics thinking.

Last week I had a dramatic example of how there is a dynamictension between these two aspects of physics thinking. We were studyingelectric fields and potential. A standard example is the “infinite flat sheetof uniform charge.”*** This is a nice example since the math simplifiesdramatically. Because of Coulomb’s law, any electric field has to look like acharge divided by the square of a distance (times a universal constant chosento 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 alreadyhas units of charge divided by length squared, so there is no room for anyother distance in a formula for the electric field. The result is that thefield has to be a constant, independent of the distance from the sheet.

This seems strange, but it actually makes sense. [Thisargument may be difficult without a picture. To see details with figures, go toour draftwebpage on the field of an infinite sheet, but I’ll outline the argumenthere briefly. You can skip to the next paragraph if you don’t want thetechnical details. They’re cute but not essential to my point.]

To get the total effect of the infinite sheet you have toadd up the Coulomb’s law contributions from each of the bits of charge in thesheet to the field at the point you are sitting at. Each bit of charge contributesa field vector that points along the line to your point from the charge that isproportional to one over the distance to that charge squared. As you go fartheraway to the more remote charges, they contribute less and less. Also, eachdistance charge is paired with another distant charge equally far away on theother side and these contributions tend to cancel – and cancel more and morethe farther away you get. The result is that for the entire infinite sheet, ifyou are a distance s from the sheet, only the circle right beneath you ofradius about 5s contributes significantly to the field you detect. So althoughwe say we have “an infinite sheet” that’s not what we mean. We mean: we have aflat sheet and the edges are far enough away that we don’t have to worry aboutthem. 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 theamount of charge you see grows like s². These two effects cancel toresult in a constant field.

The result of having a constant E field simplifies a lot ofthe math. The potential that goes with a constant field is just linear (sincethe derivative of the potential is the E field) so the math is really simple- 9^thgrade algebra. All those complex “curvy 1/r²” functions and vectorintegrals add up to give straight lines. It looks just like the same math forflat-earth gravity – where we take the gravitational field to be constantalways 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 20of the students). Then in class I asked the following clicker questions:

Iftwo uniform sheets of equal and opposite charge can be treated as if they wereinfinitely large, which of the following graphs might serve as a graph of (A) thex-component of the electric field and (B) the electrostatic potential as afunction of the coordinate x along the dotted line?

Since the fields from each sheet are constant and since thecharges are equal and opposite, outside of the sheet the two fields cancel, andbetween them they add. The result is that the E field (x component) looks likegraph 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 thepotential. That’s OK since for the electric potential (as for height when weare 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 looklike – the results were gratifying. Of the 19 students present, 16 chose answer8. But one student complained. He said, “I didn’t like any of them.” When Iasked why, he responded, “Because when you get near to the sheets you’ll seethe individual charges and the field has to go to infinity.” I brushed him offwith a brief comment about resolution – that it would only happen really reallyclose and we wouldn’t see it on this scale and anyway we were ignoringindividual 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 allover the lot. But one student (a different one) said, “It can’t be any ofthem.” When I asked why, he responded, “Because if you are sitting exactlybetween the plates the result has to be zero. If you are sitting there, forevery positive charge on one sheet there is a negative charge on the othersheet an equal distance away that will cancel. Also, it has to eventually asymptoteto zero for large values.”

Well! I was stunned. This was absolutely top quality physicsreasoning. He was using a physical picture and using it with a correct symmetryargument – another strong tool in the quiver of good physics thinking. (Thiswas something I had been trying to model in the class, but not fussing too muchabout.) Finally, he was focusing on limiting cases, another standard tool wetry to get our students to use. My first internal response was – sign this guyup as a physics major! My second was. Gak! I seem to want them to be looking atthis model example NOT in terms of the basic physical elements but as a toymodel that suppressed the underlying physical picture. Since a third major goalof my class is to teach my students to seekconsistency, what was I doing?

I don’t know that I handled my little epiphany well on thespot. I tried to be encouraging and praise the students’ thinking but still bringus back to being able to using the simplified model while “suspending ourdisbelief” – using it even though we know that “when we go to infinity thesheets that look infinite when we are close will turn into looking like pointcharges when we are far away”, and “we’ll ignore the fact that the charges areactually quantized and treat the chunky peanut butter as if it were perfectlysmooth since we don’t see the individual charges until we are really close andthen we’ll just ‘close our eyes’ for a moment while we are passing through thesheet”.

Really what it’s about is modeling.**** When we have complexsituations – like lots and lots of charges – it’s great to have “a stake in theground”; an example where we know an “exact” answer that we can refer to thatserves as a starting point for further examples and elaborations. But in introphysics we tend to focus on the simplicity of the models and not on thecomplexity of thinking about where they come from, what their validity is, andwhat value we can make of them.

Increasingly, I want to open up this Pandora’s box for mystudents. Trying to pretend that the physics is simple by hiding the deepstructure, both ontological and epistemological (i.e., what is it we areactually talking about and how is it we decide we know), is beginning to seemto me to be unfair to our students and not the best way to start students onlearning physics.

* For more info on this, check out the NEXUS physicsclass website.

** If you haven’t already listened to Emily Hanford’s audiodocumentary, “Don’t Lecture Me”, check it out at AmericanRadioworks.

*** I’m referring to the example as “sheets” rather than“plates” as is more common since my examples for biologists will be membranesrather than parallel plate capacitors with metal disks.

****David Hestenes, "Modeling games in the NewtonianWorld", Am. J. Phys. 60 (1992) 732-748.

Lose the lecture

2012-01-04T11:02:00.003-05:00

On New Year’s Day, my favorite radio program, AllThings Considered, aired a clip from Emily Hanford’s American Radioworksaudio documentary, “Don’tLecture Me.” It focused on thephysics (and the Physics Education Research) part of the documentary andfeatured Eric Mazur, David Hestenes, and me. And they led with my quote, “Withmodern 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 advocatingfiring faculty and turning universities into a sort of glorified YouTube.Rather, what I’m saying – to the faculty – is, “If you think lecturing is goodenough, you’re putting your own job at risk!” And that it’s up to us toreinvent ourselves so that we add value beyond what a student can find at theKahn Academy or at the University of Phoenix.

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

For those students who want to do research, the answer isobvious. Furthermore, the large research universities are the bulwark ofresearch and development in the US and the engine that has driven the worldeconomy for the past half century. But despite large blocks of governmentfunding, the support that enables the research arm is still the education ofour students. Now students who areinterested in research as a career is a(n important but) very small fraction ofthe population we currently serve.

For me the answer is clear. Learning is better, deeper, andmore effective when you do it with people -- through interactions with mentors,journeymen (TAs, grad students, and senior undergrads), and other students. Welearn best in a community of learners and – so far – on line interactions don’tlive up to the rich experience of a real interpersonal exchange.

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

I expect that the next ten to twenty years will produce amajor shakeout in the university system in the US akin to what began whenAmazon took shopping online and that is is still going on. Now, 10 yearslater, 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 onlineoriented businesses can't -- are doomed. How many of us bought Amazon stock in2002? It's now the second largest retailer -- behind WalMart -- and reliesentirely on internet sales.

Those academic Chairs, Deans, and Provosts who think that the newtechnology will make it cheaper to deliver their product with fewer faculty(and larger classes) are undermining the future of their own universities. Weshould be moving in the opposite direction, providing students with morefaculty interaction, more group learning environments, and more hands-onactivities. We need to make good use of digital technology, but we need to use iteffectively and go beyond what can be done by a student working alone with acomputer. 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

2011-12-27T09:23:00.000-05:00

Those of you who have been following this blog for a whileknow I have been somewhat obsessed with units. You know, those things wephysicists like students to stick on to nice clean mathematical variables tomake them messier and more complicated? And that students would much prefer toignore? Well, if you’ve been reading from the beginning you also are beginningto get an inkling of why I care about units so much. My evolving view ofscience 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 sciencephilosophically as a bridge between cognitive science and the epistemology ofreality – a “how do we know” about what we presume actually exists. And this iswhere 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 thatunits are a nice example of the bridge between cognitive modeling and math; thatwe use our everyday experience with distance, time, and mass to allow us to dosome mathematically very sophisticated things at an introductory level withoutactually bringing in that math. (What units are doing is tracking theirreducible representations of a transformation group in our equations!) Butthe implications of this are that our “map of the real world onto amathematical system” (see Units and stoichiometry) becomes limited. There arethings 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 choosingcoordinate systems and operational definitions, but while space and time can benegative as well as positive, mass cannot.

I also made the point that multiplication is allowed on thereal 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 correlateswith 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 physicsto medicine and biology. These illustrate nicely how these combinations correlatethe math cognitively to the physical world in a different way from thestraightforward sense we have of area or volume.

Of course the simplest and prototypical example is the squareof time. Although we have no conceptual correlate of “square time”, when wedefine an acceleration we are chaining the idea of rate. The first time weapply the idea of rate of change to position we get a velocity: length dividedby time has dimensions L/T. When we do it again, we get an acceleration: velocitydivided 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 “perunit time per unit time”.

This conceptual chaining is very reminiscent of the waymodern linguists and semanticists build up complex and abstract concepts, bychaining of metaphor (Lakoff and Johnson [1]) and by cognitive blending(Fauconnier and Turner [2]).

My biological explorations have led me to two differentexamples of distance to the fourth and sixth powers – hypervolumes. Figuringout the conceptual correlates for these unit combinations is interesting.

The first example is medical. Various glands in your bodysecrete a variety of chemicals. The health of those glands can be probed bychecking how much of that chemical is circulating in your blood. A simple bloodtest can measure the amount of chemical found in a particular sample andcalculate a density by dividing the mass found by the size of the sample. On myannual 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 SIunits, often in micrograms per deciliter. But we shouldn’t be surprised. Bloodpressure is still measured in “millimeters of mercury”!) Now if you want toknow about the health of the gland, to find out how effective the cells in thegland are in producing the chemical you might want to divide the density of thechemical found in the blood by the volume of the gland – measurable witha 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 makesense 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 pipethat has a continuous steady state flow is equal to the rate of flow times theresistance; this is analogous to the more familiar electrical law that thevoltage drop across a resistor is equal to the electric current times theresistance. 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 inverselyproportional to the area of the resistance perpendicular to the flow. Thisphysics of this is that Ohm’s law is basically about the balance of forces: theelectric force due to the potential drop pushes the charge through the resistoragainst the drag, which is proportional to the velocity. Since the charges aremoving at a uniform velocity (on the average) there is no net force. The factthat the electrical push balances the drag is Ohm’s law. The area arisesbecause the drag is proportional to the velocity and we want to express the lawin terms of electric current – basically velocity times area (times chargedensity). We introduce the inverse area to change velocity into current.

In the Hagen-Poisseuille law, we have a similar bit ofphysics: the forces due to the pressure drop pushes the fluid through the pipeagainst the viscous drag, which is proportional to the velocity. Since thefluid moves at a constant velocity, there is no net force. (There is anadditional complication in this case since the fluid doesn’t flow at the samerate of speed in all parts of the pipe, moving fastest at the center of thepipe, but we’ll ignore that here.) The fact that these two forces balance isthe H-P law. One factor of the area arises because the drag is proportional tothe velocity and we want to express the law in terms of current – velocitytimes area (if we use volume current; add a factor of mass density if we usematter 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 theelectrical case), pressure is related to force by a factor of area. Thisintroduces a second factor of area into the resistance of the H-P equation. (Tosee 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 ourprevious example, the higher powers of length (the square of a volume) camebecause we were using two different volumes – sort of a double density. In thiscase, both of our areas are the same but they have two sources. One from the forceof the fluid on itself into pressure, the other from converting the fluidvelocity into current.

In both of these examples the conceptual correlate of thecomplex unit is a product of different conceptual objects – two differentvolumes in the density of density example, and the same area for two differentpurposes in the HP case. These examples suggest to me that when we havedimensions that don’t have a direct conceptual correlate with a physicalconcept, understanding the conceptual blends that lead to the combined dimensionalstructure can help us make better sense of why a complex quantity looks the way itdoes.

[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 HiddenComplexities (Basic Books, 2003).

“Reliability of the FCI supports resources theory”

2011-10-09T08:50:00.002-04:00

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

2011-08-12T17:40:00.003-04:00

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

2011-07-24T11:26:00.004-04:00

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

2011-07-20T10:48:00.003-04:00

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 = mc²on 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.

[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

Species

2011-07-03T21:31:00.004-04:00

The idea of “species” is clear and simple in our everyday folk biology. In the Hebrew bible, to save the animals from the flood, Noah brought “two from all” (“shnaim mi-col” = שְׁנַיִם מִכֹּל), male and female onto the ark. (An “amah” = אַמָּה – translated today as a “cubit” – must have been quite a bit bigger than we think, given what we now know about the “all”, even just of all animals.) We have a pretty good idea what is intended by the story. Everyone can tell the difference between a tiger and a lion, even my three-year-old granddaughter. (But what’s a “liger” or a “tiglon”?) Darwin’s On the Origin of Species caused a furor, in part because he suggested that species were malleable.

But from the point of view of a physicist – especially one like me who has studied a little cog sci – the idea of “species” looks more to me like the standard categorization that our brain does. We define two things as the same if they are similar – equivalent for whatever limited purpose we might have in mind at the time. (See “One and the Same”.) Categorization seems to be based largely on our early experience, cultural environment, and practical considerations. It does not seem as if things have a “true essence”, a la Plato.

In physics, over the past century we have built up pretty good evidence that the particles of which atoms are made (and photons) do need to be thought of as identical because of the structure of quantum mechanics (about which more in a later post). But when it comes to living organisms, it’s pretty clear that while some are similar, they are not “identical” in the sense that if you exchanged them there would be no experiment you could do that would tell you that the exchange has taken place.

So if all we really have is a lot of distinct individuals, what do we mean by a “species”? One thing we like to do in physics, especially when we are getting started in studying a field and don’t have a strong theoretical framework, is to define things operationally – by some measurement process. That’s effectively how a species is defined in biology today – as a breeding population. Saying two organisms are of the same species doesn’t mean that two individuals can breed – they may be of the same sex, for example – but it means that two individuals are of the same species if there is a chain of individuals that can breed and produce fertile young. So Scooby-Doo (a Great Dane) and Ren (a Chihuahua) are of the same species even though a Chihuahua female could not carry a Great Dane’s pup to term and survive. Note that the production of a viable zygote, fetus, or even adult is not sufficient (viz. the liger or tiglon). It has to be able to survive a natural birth and be able to grow to adulthood and be fertile. You can “measure” whether two organisms are of the same species by doing a series of breeding experiments.

Many animals in biology are considered to be different species because different behaviors or choice of habitat makes natural breeding impossible. For example, two species of fly seem to differ only by the sounds they make in courting. (Of course there is a hidden assumption here – that we are talking about beings with two sexes. Things get all messed up when an organism can basically clone itself by reproducing by parthenogenesis, or when, like bacteria, they may exchange genetic material with very different organisms via viruses. But we’ll leave that discussion for another time.)

The definition of species as a breeding population is interesting since it means that you might not be able to tell if two individuals are members of the same species by experiments on them alone. (You might … if they could actually breed directly.) It might seem the opposite might be more obvious – that we could show that they are not members of the same species, but this raises interesting questions.

In The Ancestor’sTale: A Pilgrimage to the Dawn of Evolution, Richard Dawkins tells the interesting tale of a “ring species.” (I gather from conversations with biologist friends that this is a standard example in intro bio classes.) Apparently there are two species of salamanders that live together amiably in a region in California near the south end of a lake. Let’s call them groups A and B. They are clearly different species since they look different and don’t interbreed. But … if you travel up along the side of the lake on one side, you will find apparent variants of salamander A that do breed successfully with it. Similarly, if you travel up along the other side of the lake you find variants of salamander B that breed with it. And at the north side of the lake? You find salamanders that breed with both variants establishing the chain that prove that salamanders A and B are actually of the same species! Although this is called a “ring species” in biology, I prefer to call it a horseshoe species – because it’s open at the bottom.

This is neat – but a bit creepy. Suppose Harry was a salamander of group A, and Sally was a salamander of group B. Since there is a chain of friends through whom they can exchange genes they are of the same species. (To get in the right frame of mind, listen to Tom Lehrer’s, “I got it from Agnes”.) But now suppose a mining company was granted mineral rights on the north side of the lake by an eco-unfriendly administration. They destroy the habitat on the north side of the lake and all of the northern salamanders are wiped out. The chain is broken. Harry and Sally can no longer share genes and therefore they are now members of different species – despite the fact that they haven’t done anything or changed in any way!

Now here’s where it gets interesting. Physicists like myself who have studied relativity and quantum field theory are used to space and time being considered variations of the same thing. To see the implications of doing this, let’s consider a map of the lake as a 2-D graph of the different members of our horseshoe species of salamanders. But that’s at a particular instant of time. If we want, we can stack up 2-D maps of space, each one representing a later (or earlier) time. This makes our 2-D maps into a 3-D space-time graph, with the vertical axis (perpendicular to the maps) representing time. Let’s now imagine rotating our 2-D horseshoe downward, keeping Harry and Sally fixed in place, but taking our north-side salamanders down into an earlier time. The joining point of the two legs of the horseshoe now represents the common ancestor of the two groups of salamanders A and B. To see the implications of this for the concept of species, let’s switch to dogs where it’s easier for most of us to get a picture.

Analogous statements are true of Scooby-Doo and Ren. A plague wiping out all dogs except Great Danes and Chihuahuas would have the same effect on them. We would no longer consider them to be of the same species than we would tigers and lions to be the same species.

Let’s start with a “generic” dog – something like a Tamaskan (picture below) that looks a lot like a wolf – and imagine it as the common ancestor of all the dogs today. Clearly this is an oversimplification, since we should think of a breeding population as the ancestor of a population, not a single animal. (Even in the Bible, Seth, the third son of Adam and Eve, had to find a wife.) The descendants of that population were selected for by people from the natural variation the original breeding population showed – and variations that developed later (by recombination and mutation). The result is the variety of current dogs we see today. Now the ancestors of various breeds produced a range of variations in their pups from which the ones with desired characteristics were selected.

(Source, Wikipedia Commons)

Suppose now that all of the ancestors had not only pups with the variations that eventually led to Chihuahuas and Great Danes, but also ones that bred true. We’d then have not just a continuous breeding population of current dogs connecting our two extremes, but a breeding population connecting all current dogs to dogs who are identical genetically to all their ancestors.

So this suggests that what looks to us to be a “species” is not defined by the individuals involved, but by the gaps – which ancestors and intermediates happened to die out before today. If the right ancestors had “bred true and through”, we would not separate lions from tigers nor horses from donkeys (nor humans from chimps). These are all pretty close, though. What about more significant gaps – like lizards from birds?

This is what makes The Ancestor’s Tale so interesting. Dawkins traces our ancestry back, considering close relatives and showing what our common ancestors might have looked like. He goes further and further back and attaching to our family tree species that seem farther and farther from us.

So if I take the time-rotated picture seriously, it suggests to me that we are all part of one grand continuum of life, with our definition of species being (certainly) a convenience to us, but one that we might view as accidental if we had a more complete historical record. It’s a definition that depends on which intermediates happened to die, and not on anything intrinsic to the individuals we are considering.

Local or Global Coherence?

2011-06-07T10:17:00.001-04:00

In my last post I wrote about Oliver Sacks’ essay, “Stereo Sue” in his book The Minds Eye where he told of a woman who first developed stereoscopic vision in her 40s. In the following essay, “The Persistence of Vision”, he talks about his own loss of stereopsis as a result of the growth of a melanoma behind his right eye. First, as a result of the tumor, and then as a result of the treatment to kill the tumor, he lost vision in the eye – slowly, over a period of four years.

Besides this being scary (Who knew you could get melanoma behind your eyeball?), it turned out to be fantastically informative about the complex way in which vision works. Sacks, ever the scientist, kept a careful journal, drawing what he saw and describing how his brain interpreted the changing data stream. I think that there’s a lot to learn from his experience, but two points struck me as telling us something interesting about how the brain works: filling in, and reinterpreting, both saying something about the way our brain build the virtual world we live in. (See my earlier post, “Axioms”.)

At one point in his treatment, he had lost vision in the foveal part of the eye – the highly accurate cluster of retinal cells that we do most of our direct looking at things with. He retained peripheral vision (and has interesting things to say about that), but developed a scotoma – a large blind spot in the middle of the vision of his right eye. It’s well known that the brain “fills in” the blind spot – up to a point. (You can try this yourself with your own smaller and normal blind spot.) I read about this in a psych book a couple of decades ago. I recall a story of a man with a scotoma looking at a person sitting on a couch. When he turned to put the person’s head in the hole in his vision the person’s head disappeared – but an image of the patterned wallpaper behind the couch filled in the space where the head should have been. Sacks explored what filled in and what didn’t.

“It was easy to fill in a simple repetitive pattern – I started with the carpet in my office – though a pattern took a bit longer than a color, perhaps needing ten or fifteen second to duplicate. It would fill in from the edges, like ice crystallizing on a pond…I found that movement could be also filled in to some extent. If I looked at the Hudson River, slowly swirling or rippling with small waves, these too were reproduced…But there were strict limits. I could not simulate a face, a person, a complex object.”

Interestingly, he found that if he cut off his foot with his blind spot but wiggled it,

“the stump seemed to grow a sort of translucent pink extension with a ghostly protoplasmic halo around it. As I continued wiggling my toes, this took on a more definite form, until, after a minute or so, I had a complete phantom foot…which seemed to move with the movements I was making.”

This, and other examples he cites, seem to indicate that the brain “recruits resources” – hunting for things that will bring together the various bits of information it has to make a coherent sense.

But a second observation leads in a different direction. Once he lost peripheral vision in his right eye he lost the ability to see anything to the right side of his nose and developed what neurologists call “unilateral neglect.”

“…whatever comes into my visual field from that side is unexpected and startling. I cannot overcome the sense of bewilderment, even shock, when people or objects appear suddenly from my right. A massive slice of space no longer exists for me…”

Particularly surprising is that his brain seems to treat objects that move into that space as vanishing – even people.

“Kate [his wife] and I finished our walk and headed back to my office, I walked ahead and got into the elevator – but Kate had vanished. I presumed she was talking to the doorman or checking the mail, and waited for her to catch up. Then a voice to my right – her voice – said, ‘What are we waiting for?’ I was dumbfounded – not just that I had failed to see her to my right, but that I had even failed to imagine here being there, because ‘there’ did not exist for me.”

This (and other such examples) are fascinating because he knows at some level that he cannot see on the right. Yet the changing in accustomed patterns of sensory data seems to undo a component of his sensory coherence. He cites a colleague (M.-M. Mesulam) as saying, “Patients with unilateral neglect behave not only as if nothing were actually happening in the [blind side] hemisphere, but also as if nothing of importance could be expected to occur there.”

This, and other examples seems to suggest that the brain builds a local coherence, but can neglect large blocks of data that it knows perfectly well in responding to an immediate environment.

One of the most controversial theoretical issues in science education today is the appropriate scale at which to expect coherence in student thinking. I remember a conversation at an international science education conference nearly two decades ago with a friend and colleague about this issue. He took the point of view, “People live in the world. They have to have a coherent view of it. If they don’t, they would go crazy. When students give answers that seem bizarre to us, it’s just that they have a different coherence from ours. As science educators, it’s our job to find out how to one might look at the world coherently and see it the way they do.”

This is an extreme view (and he may have moderated it by now), but many education researchers assume – sometimes tacitly – that students are fully coherent in their thinking about something, even if we, from our perspective, don’t see what the coherence is that they are seeing.

If something as fundamental and as evolutionarily vital to survival as our interpretation of the spatial world around us through vision shows only local but limited coherence, I suggest that we can’t take coherence of brain function for granted. Rather, we have to treat it as an empirical issue, to be explored under varying contexts and conditions.

The Joy of Sacks -- Recruiting Resources

2011-06-04T23:10:00.000-04:00

While trolling the library the other day for a stack of new trashy mystery novels (my relaxation of choice these days), I was delighted to find a new case study collection by Oliver Sacks, The Mind’s Eye (Borzoi, 2010). Sacks is a neurologist who first became famous with the publication of a collection similar to this, The Man Who Mistook his Wife for a Hat (Touchstone, 1998). If you have any interest at all in how the mind works, I strongly recommend Sack’s work. These case studies tell stories of individuals who suffer some brain injury and as a result lose a piece of what they usually do automatically. These dissociations of functions we normally see as unitary gives real insight the functioning of the brain. The title case study of the 1998 book is about a music professor who lost the ability to pull together bits of visual data into recognized objects. He really did, at one time, reach over to grab his wife’s head because the curve of her scalp matched the curve of his hat.

What I find particularly meaningful – and hopeful – about Sacks’ case studies is that he often picks ones where the people who have suffered a brain injury and a resulting loss nonetheless do not fall into despair, as many must, but find a way to go forward with their lives. He tells of an athletic young woman in TMWMHWFAH who lost her ability to interpret the signals from her muscles telling her about the orientation of her body and limbs (her proprioception) and almost became effectively paralyzed as a result – she would flail around knocking into things. But she worked to learn to replace her muscular perceptions by visual ones and regained much of her ability to function normally. A painter (in TMW…) lost his color vision and, instead of giving up painting, developed a new (and, it turns out, more popular) style. A novelist (in ME) had a stroke and lost his ability to read – but not to write (alexia sine agraphia)! He could write but couldn’t read what he had written to revise or build coherency. He learned to dictate and have people read to him and he has continued to maintain his output (of mystery novels).

I know that these are selected stories and that many people’s illnesses and injuries are too severe to give them a path back to a life of satisfaction and joy, but as I get older and the probability of something bad happening increases, I get satisfaction in knowing that at least some people manage to get over even severe bumps in the road.

But I digress. (Hey! I’m an academic! That’s what I do. Get over it.) The point of this screed is not for me to get maudlin about aging; it’s to see what points these stories make about how the brain works. I’ve got two for now.

The story of the novelist makes me appreciate the role of external objects and symbology as components of our everyday cognition. (cf. some parts of activity theory) My best example of this is building a talk in PowerPoint. I know how to do lots of things with PowerPoint – but it’s me and PowerPoint that know it together; I don’t know it by myself. Occasionally someone will ask me how to do something in PPT. I might know, but not be able to tell them. What I know is which chain of items to choose from a series of menus when they are presented to me – but I don’t have the menus themselves stored in memory. When I am deriving something in physics or math that requires far more steps than the 7 ± 2 that fits easily in my working memory, it looks like I have much more in play since I write my equations down and by scanning up and down the page, I can quickly swap in and out what I need without having to reconstruct those that I can’t keep active. Writing a novel (or an academic paper) certainly must be dramatically simplified by our ability to have written stuff down and quickly look back at it.

But the case study in The Mind’s Eye that really excited my and is responsible for my pulling out the computer on a Saturday night is “Stereo Sue”. In this, Sacks tells of a woman born with strabismus (cross eyes). Despite a number of operations when she was a child, she never achieved true stereoscopic vision. Now lots of animals get along without stereoscopic vision. In fact most prey animals such as antelopes and deer (also horses) have eyes that are on the side of their head so they can scan closer to a full circle more quickly. Presumably they have much more awareness of what’s to the side and behind them, but we seem to be descended from tree-living animals that had much more need of depth perception to negotiate a complex 3-D environment.

Now Sue was able to get along just fine using the other parts of her vision that allows us to place ourselves in space, particularly motional parallax – how things appear to change when you move. She could drive and even play softball. But in her 40’s she began to develop problems with her vision. She found a careful and flexibly thinking doctor who discovered that not only had she been cross-eyed, but the vertical alignment of her eyes was off. By adding prisms to her glasses, the doctor was able to correct this. And, with a series of exercises, Sue was able to initiate stereoscopic vision.

I wish I could repeat the whole story here, but this is too long and I want to (eventually) make a point. Go read it for yourself. I actually have two more points to make, one about the practice of science, one about the brain.

First, for decades, Sue’s doctors told her that because she had missed the “critical period” as an infant, her brain would never be able to learn to interpret the data from her two eyes stereoscopically. Of course that’s the result that Hubel & Wiesel found with kittens (and won the Nobel prize for). It’s a great result, but it does not obviously carry over to other species (ferrets, for example) and the human brain seems way more plastic – at least potentially – than we often assume. There are stabilities, but we may be making a serious error when we ignore or under-appreciate the possibility of dynamic reorganization.

My second point about the brain is this. After Sue had pretty fully developed her stereoscopic vision, Sacks tested her with an interesting example. This was text presented in a stereoscopic viewer – strings of unrelated words. When viewed monocularly, it looked just like text on a page – flat. But viewed stereoscopically, it became obvious that the words were on different levels – as if they were printed on stacked panes of glass.

When Sue first looked at this she saw the text as flat. When Sacks pointed out the 3-D aspect, she said, “Oh, now I see” and was able to see it just fine. Sacks says

“Given enough time, Sue might have been able to see all seven levels on her own, but such “top-down” factors – knowing or having and idea of what one should see – are crucial in many aspects of perception.”

Later he says

“If a stereo photograph is flashed on a screen for as little as twenty milliseconds, a person with normal stereoscopy can perceive some stereo depth straightaway. But what one sees in a flash is not the full depth; the perception of this requires several seconds, even minutes, in which the picture seems to deepen as one continues to gaze as it – it is as if the stereo system takes a certain time to warm up…. The underlying cause for this is unknown, though it has been suggested that it entails the recruitment of additional binocular cells in the visual cortex.”

This just strikes me as so similar to what I see with my students operating at a much higher level of brain activity and complexity. I find it delightful to see framing and the need to recruit other resources before something clicks into making sense occurring at the direct perceptual level.

Neuroscience, Education, and Reductionism

2011-03-24T14:20:00.000-04:00

As an education researcher and a theoretical physicist, I’ve been very interested in the issue of how to think about education – how to develop a theoretical frame.* So one thing I’ve been doing is studying a wide variety of topics about human thinking and learning – psychology, linguistics, and neuroscience.

I’ve been told by a number of my experimental neuroscientist friends that their science “has nothing for yet me” in my attempts to develop a phenomenology of learning, and to “come back in 50 years.” They say, “a lot is really still up in the air.”

I think this attitude is based on a misunderstanding of the relation of reductionism to theory building in the natural sciences. First, we theorists don’t “wait for experimentalists to get it right” before building theory. The process of “getting it right” involves an intricate interplay of experiment and theory, a dance in which the leading partner can frequently switch roles. Furthermore, the interplay need not be on the same level of reductionism – micro, meso, or macro. Information flows across levels. Work at a large scale can inform work on the small.

For example, in the 1860’s, James Clark Maxwell began the development of statistical physics that is the basis for what is perhaps the most powerful reductionism we know – the way to create an understanding of the macroscopic properties of matter as it emerges from the behavior of molecules 10 orders of magnitude smaller in linear dimension.

But when Maxwell created his theoretical structure, very little was known about the properties of atoms and molecules. Indeed, many leading physicists of the day were not even sure of their existence. There was no convincing argument demonstrating molecular size (thought there were some hints), and chemical experiments suggested that their interaction properties were extremely complicated.

Maxwell succeeded in building a theory of gaseous matter using almost trivial assumptions about the nature of molecules.** He treated them as if they were hard spheres – obviously a ridiculous and unsophisticated assumption given what we knew about chemistry at the time! Even the simple fact that molecules form liquids and solids show clearly that there are attractive forces and that they are important.

Nonetheless, Maxwell was able to produce a beautiful and effective reductionism – and one that passed information back to the micro level. His explanation of transport phenomenon (e.g., viscosity, which moves momentum across parallel layers of a flowing gas) showed that measurable macro properties depended on a small number of microscopic parameters in a straightforward way, allowing him to infer molecular properties (such as average speed and mean free path) from macroscopic experiments. Indeed, the confirmed predictions from his transport theory produced some of the most convincing arguments for the existence of molecules in the second half of the nineteenth century. (Though that’s a topic for another post.)

Phenomena that are undetectable at the micro level can build up coherently when going to the macro level leading to clearer signals. For example, the small deformations of an atom in response to an electric field – the electrons pushed one way, the nuclei pushed the other – cannot be measured at the atomic level. But if every atom is pushed the same way, the resulting effect is observable at the macro level, thanks to the factor of 10²³ in Avogadro’s number. In this way quantum predictions of atomic polarizability can be confirmed via macro measurements.

Finally, there are complexities at the larger scales that are difficult, if not impossible, to deal with by reducing things to smaller levels. We can’t really deal with the complexity of organic chemistry by reducing everything to the simple facts that there are atoms that interact and bind – and that there are only about 100 different kinds of atoms. The fact that the thousands and thousands of chemicals that make up our every day experience are combinations of a small number of different kinds of atoms is a deep and powerful insight – but it doesn’t begin to let us understand the complexity of organic chemistry. For that, we need organizing principles at a higher level.

So, for building a behavioral phenomenology of learning, my expectation is that some elements of neuroscience – even what we know today – are going to be important, even though the details might not be fully worked out. But the interplay of knowledge between the macroscopic behavioral level to the microscopic is already important. [For example, see how neuroscientists use the implications of behavior associated with neural damage.***] Interactions between learning theory and neuroscience might wind up being of value both to the macro and micro researchers.

* A Theoretical Framework for Physics Education Research: Modeling student thinking, Edward F. Redish, in Proceedings of the International School of Physics, "Enrico Fermi" Course CLVI, E. F. Redish and M. Vicentini (eds.) (IOS Press, Amsterdam, 2004).

** in Kinetic Theory of Gases: An Anthology of Classic Papers With Historical Commentary, Stephen Brush(Imperial College Press, 2003).

*** From Neuropsychology to Mental Structure, Tim Shallice (Cambridge U. Press, 1988).

On Intergenerational Inertia

2011-03-24T09:54:00.002-04:00

Instruction seems to have an immense intergenerational inertia. Why else, in an age of ubiquitous calculators, do we insist on subjecting 5^th graders to the torture of trying to learn to add multiple columns of numbers with perfect reliability or learn the mechanics of long division?

I conjecture that this is because our instruction goes back to a curriculum developed in 1900 when three-quarters of Americans worked on farms. (Elementary school education didn’t become compulsory throughout the US until 1918.) An appropriate job for a grade-school educated person was an account keeper or salesperson in a shop – where all calculations had to be carried out by hand.

As that motivation decays over decades, instead of changing our educational practice, we seem to continually re-invent ex-post-facto justifications for continuing to do what we know how to do. Today, we might justify it on the basis of “helping students to develop a number sense.” But there are much better ways to do this. [My favorite involved a program Judah Schwartz created called “What do you do with a broken calculator?” In this, students had to figure out how to carry out calculations when one of the keys on their computer-screen calculator failed to work.]

This inertia may be in part responsible for our tendency to create overly broad introductory physics courses. As teachers, we know all the things we learned a generation ago. They must be important, since we learned them and we did OK – and anyway they seem much easier to us after years of practice.* We also know that modern instruction requires us to update our teaching by including new topics. So we try to squeeze them in without removing the old. Of course the result is an increasingly superficial treatment – a course a mile wide and an inch deep.

When I first taught physics I was assigned to teach E&M to a class of physics majors. They had serious trouble learning the basic concepts of electricity and magnetism in a 15-week semester devoted entirely to the subject. Why should I imagine 40 years later that biology majors – students less well motivated and prepared in physics and math than self-selected physics majors, and with other equally difficult courses competing for their study time – should be able to master these ideas in 6 weeks?

Figuring out what students need to learn now – and what they need to do to learn it – is a much harder task than we typically assume, especially at the university level where teaching is a second tier activity that you are supposed to be able to do in your spare time when you’re not working on your research. This approach guarantees that we won’t have the time to be as thoughtful as the task needs, that we will have to fall back on what we learned when we were young a generation ago. This does ill serves our students well who will still be working a generation from now.

*Reverse Engineering the Solution of a "Simple" Physics Problem: Why learning physics is harder than it looks, E. F. Redish, R. E. Scherr, and J. Tuminaro, The Physics Teacher, 44, 293-300 (May, 2006).

On service courses

2011-03-20T15:30:00.000-04:00

In physics departments, a lot of the students we teach are not going to be physics majors. They are going to be engineers, chemists, computer scientists, biologists, and doctors. Everybody (that is, all physicists) agrees that physics is good for all future scientists since physics is the basis of all other sciences – at least that’s the way it seems to physicists. We tend to be reductionists. That is, we build everything up from fundamental low-level principles. Thus, in principle, quantum mechanics is a consequence of quantum field theory, chemistry is a consequence of quantum mechanics, biology is a consequence of chemistry, and medicine and social science are a consequence of biology. Therefore, every science is based on physics. Of course none of those connections (even the first) are filled in yet – or are likely to be filled in for many years, if ever. But that’s a topic for another essay. If those “in principle” reductionist connections are not why we teach physics to other scientists, what is?

I think that there are still lots of good reasons for other scientists to take physics. The first and most obvious is that physics actually plays a role in the actual practice of all those sciences. Chemists need to understand electricity and quantum physics, biologists need to understand fluid flow and energy, and doctors need to understand forces and torques as parts of their knowledge collection. But more important than the specific bits of physics is the “physics way of thinking”.

The core of the approach that physics takes to science is that of finding and reasoning from a small core of fundamental principles. And the deeper and more advanced we go in physics, the fewer and more powerful the principles become. At the macroscopic level, we learn conservation of energy and momentum and they are fundamental principles that allow us to think about and solve many complex situations. At a more advanced level, the fundamental principles become that our laws of physics should not depend on where we are or when we apply them. This invariance with respect to space and time turns out to imply energy and momentum conservation.

One result of this “push for principle” in physics is that we tend to simplify. Einstein said, “Physics should be as simple as possible, but not simpler.” We seek the simplest possible problem that illustrates the application of the principle we are trying to teach. I call these touchstone or toehold problems. Once we understand these simple examples thoroughly, then more complex situations can be built around them. The toehold examples provide starting points, ways of going forward or of organizing what we know in order to make sense of the complexity.

We spend a lot of time working on the simple case of a mass attached to a spring since this provides us with a way of thinking about any oscillating system. The core ideas behind oscillation are restoring force and inertia. The is some stable equilibrium of the system such that when the system is deformed away from it, there is a force that attempts to pull it back to the equilibrium point. Typically, that force gets stronger the farther you get from equilibrium and vanishes at equilibrium. As that system is pulled back towards equilibrium, it starts going faster. When it reaches equilibrium, the restoring force vanishes. But at that point it is moving and has inertia. Since there is no more force to stop it, it keeps going, goes past its equilibrium and we start all over. This pattern can be working out in detail for the mass on a spring and provides the basis for describing damped oscillations, driven oscillations, coupled oscillations and resonance, as well as (perhaps surprisingly) a basis for the theory of motion of protons and neutrons in a nucleus or the theory of light in quantum field theory.

Besides this “drive for the simple”, physics thinking has a number of other characteristics such as looking for coherence, thinking about physical mechanism, taking multiple ways of looking at things, and making the connection to everyday experience. (This last sometimes needs some refinement of our everyday experience to reconcile it with the science we have developed.) There are valuable tools that we commonly use to achieve these ends such as doing dimensional analysis or considering limiting cases.

In an attempt to make physical thinking more explicit in our traditional service courses, my colleagues and I at the University of Maryland have spent the past decade learning about our biology clients in our algebra-based service course and refining the course to help them learn to strengthen their scientific reasoning skills. [The method and results are laid out in our paper, Reinventing College Physics for Biologists: Explicating an Epistemological Curriculum , E. F. Redish and D. Hammer, Am. J. Phys., 77, 629-642 (2009).] We had good success with this class. On standardized mechanics conceptual tests (FMCE) we get much better pre-post fraction of the possible gain ( ~ 0.45-0.50) than in traditional classes ( ~ 0.15-0.30), and we get strong gains on our Maryland Physics Expectations (MPEX) survey, something not seen in most traditional classes.

Anecdotal comments from students are also often positive and supportive (though they sometimes complain that there is too much hard work for a 100 level class). Many students have said that the course changed the way they think about science for the better. (Some anecdotes are reported in the above-cited paper.) But a recent comment from a student brought me up short.

The week before the beginning of the second term, a student who had been in my first semester class came by to tell me I would be seeing him and his two friends from the class again. He said they had enjoyed the class very much, worked together on the homework (as I had encouraged them to do), and felt they learned a lot. As a result, they signed up for the second semester right away, choosing my course over some competing courses in their major. He added that they wanted to take my course, despite the fact that they were biology majors and therefore it wasn’t of much relevance for them.

Well! Despite the fact that I had thought carefully about what might be useful for biologists in their future careers, and focused on developing deep scientific thinking skills, it suddenly became clear that I had failed in an important part of my goal. I had managed to teach some good knowledge and good thinking skills, but I had not made the connection for my students to the role of that knowledge or those skills in their future careers as biologists or medical professionals. The occasional problem I had included with a biological or medical context did not suffice.

I now see that often I taught my principles and simple examples without making a real connection to show how those principles imbed in complex situations. This can make my basic principles and the problems I give seem more like puzzles and games than ways of parsing a more complicated problem.

I therefore propose we who are delivering service courses for other scientists – and I mean mathematicians, chemists, and computer scientists as well as physicists – ought to measure our success not just by the scientific knowledge and skills that our students demonstrate, but by their perception of their value to themselves as future professionals. We can tell ourselves, “Well, they’ll see later how useful all this is,” and they might, but that is really wishful thinking on our part. If our students see that what we provide is valuable now, they will maintain and build on what they have learned in our classes. Otherwise, it is likely that what we have taught will fade and our efforts will have been largely in vain.

For years I have been asking my students an extra credit question on my final exams, “Have you learned anything in this class that you think will be of any use to you in five years? If so, what? If not, why not?” (5 points for any thoughtful answer) Up to now I have been looking at the results with interest and amusement. I now intend to take those answers much more seriously and will use them as a significant measure of what I see as overall success in my service courses.

Epistemology matters

2010-10-31T18:22:00.006-04:00

Yesterday the family went down to the Stewart/Colbert Sanity/Fear rally. (I was stuck at home with a sprained ankle but I watched it live online.) The mall was packed! The family brought home pictures. Here's one. Courtesy, A. Fripp.)

My estimation of how many people it took to fill that Mall is ¼ million. (I gave that estimation as an exam problem a few years ago.) In the Washington Post this morning, Robert McCartney derides the whole rationality idea quoting Frost, “A liberal is a man (sic) too broadminded to take his own side in a quarrel.” What McCartney – and apparently Robert Frost – don’t get (and John Stewart does), is that rationalist are not dualists (cf. Wm. G. Perry, Jr.) who believe that there are only two sides to a quarrel. Rather, those of us we consider ourselves rationalists believe that we have reasons for the points of view we take and are therefore willing to participate in a conversation with and perhaps be convinced by people on the other side – if they have better reasons than we do. Or if not, we are still willing to consider compromise. This, a point made clearly by Stuart and Colbert, is what is sorely missing from our current political dialog.

Now I am no longer a strong believer* in Perry’s “epistemological stages” of intellectual development. I was once, and summarized them briefly as “dualist”, “relativist”, and “constructivist”. The last is someone who understands that knowledge is tentative, but is willing to take a stand on the grounds that, “this is what seems best to me now given what I know now.” (Perry has 8 stages with substages.) A constructivist is amenable to evidence and change of their viewpoint based on evidence. I consider that helping students develop that understanding of the nature of knowledge is a major point of attending college in the first place. McCartney gives pretty good evidence that maybe we do have to view at least politics through the lens of Perry’s stages of epistemological development.

[*My current take on epistemological development is strongly influenced by Elby and Hammer’s “epistemological resources”, which takes a more dynamic view of how people use their knowledge of how we know something.]

Axioms

2010-06-27T09:25:00.001-04:00

A couple of young fresh-faced smiling missionaries knocked on my door the other day, wanting to know if I read the bible. I turned them away – but tried to be nice about it. I’m not sure why, since I actually enjoy having conversations about fundamental issues. Superficially I suppose I thought I was too busy. But I really wasn’t. I was recuperating from a vacation and taking a day off to get over my jet lag. A more plausible reason is that I knew a discussion wouldn’t get us anywhere. Having thought about this sort of stuff (whatever it is) for decades, I’m pretty confident of my basic principles and I suspect so are they – and our agreement on basic assumptions would be pretty small. On the other hand, by having discussions with persons we disagree with, we get the opportunity to probe and refine our own thinking in ways we can’t do well alone, even if nobody’s mind gets changed about anything.

When I talk to such folks I sometimes find that they claim that they take the bible literally. As a theoretical physicist, I have some sympathy for the idea that our thinking can be much improved by not treating life as just a collection of snippets of knowledge we have come to know; that placing some “stakes in the ground” – some organizing principles that we choose to trust under a wide variety of circumstances – helps us to “get things right” more often, not to be led astray by little bits of knowledge that might be more complex and less straightforward than we realize.

But what should we choose as our holdfasts? The principles of physics I was implicitly thinking of in the above paragraph organize limited realms of knowledge – Newton’s laws, the Schrödinger equation, quantum field theory. Further, I’m used to thinking of those principles as both really, really good, but also temporary and local. Newton’s laws of classical physics are not superceded by the laws of quantum physics or relativity, but learning quantum physics and relativity helps us understand the boundaries of applicability of Newton’s laws. We have both a stability in knowing the places where the laws work and to what accuracy, and a flexibility in knowing that new places might be found where new laws have to be generated. These are not our basic principle – our axioms. Axioms are starting points for reasoning: principles we take as true because they are “self-evident” to everybody. Well, we now know better. Even some of Euclid’s axioms about geometry are now known to be (useful) approximations to the physical world rather than exact. Even in the abstract unphysical world of mathematics we know we have a choice of axioms for geometry.

Identifying axioms may help us understand the extent that people have irretrievable philosophical differences and where they might be able to seek out common truths in systems that appear at first to be fundamentally at odds. Let’s explore the nature of our axioms and in particular, let’s contrast the difference in the approach to axioms in the religious and scientific communities.

Religious fundamentalists want to take as an axiom that the bible is literally true – to be taken as the direct word of god. This seems strange and untenable to me. I’ve read at least the Judeo-Christian bible and have even read much of Genesis in the original Hebrew. My reaction is, “that can’t be right.” The first problem occurs right away. We’re all supposed to be descended from Adam and Eve. Cain and Abel didn’t make it to be ancestors, so we all get to descend from Seth, Adam and Eve’s other son. But who was Seth’s wife? If you start from only a single pair there has to be incest to get you started – Eve as mother for a few generations and sisters marrying brothers. That’s creepy and you don’t hear it talked about much by the fundamentalists. But even worse, that assumption makes a firm prediction – one the authors of the bible clearly did not know they were making. If all men are directly descended from Adam we should all have the same Y chromosome. If all women are directly descended from Eve they should all have the same X chromosome – and we should all have identical mitochondria. If we don’t (and we don’t) then there has to have been enough time from Adam and Eve for those genetic patterns to drift a lot – and that would imply a much too rapid rate of evolutionary change. Given what we now know about the human genome, the slow rate of evolutionary change is actually far more problematic for young-earth-bible-literalists than it is for old-earth-scientific-evolutionalists.

The only way I see of getting around this is by inventing lots of creation stuff that is not in the bible. Once you accept the incompleteness (or inconsistency) of the bible as literal truth in one place, it seems to me very hard to identify it as a basic principle.

So if I’m not willing to accept either the bible or Newton’s laws as “basic principles that underlie all knowledge”, what do I have as my epistemological axioms, the things that I accept as being fundamental to deciding what I, as a scientific rationalist, think I know?

Let’s see if I can “open the hood” of my thinking and clarify what goes on.
I suppose my first and simplest axiom is:

Axiom 1: There is a real world that exists independent of human observation.

This one seems to me a very good bet. Especially as we learn more and more about the size and extent of the universe around us. We are a very, very small part of it. It seems to me incredibly arrogant to assume that we are the whole point of the universe. Of course, there could be another universe that has some other point and we could be living in “The Matrix” – a universe constructed just for us – but this seems to me not to be a useful assumption, at least right now. Some scientists think that the laws of quantum physics require “an observer” and therefore we are essential to reality; but given my axiom 1, I find this very hard to take seriously.

So I answer the question, “if a tree falls in the forest does it make a sound?” by “yes it does” without any hesitation. Of course you have to decide what you mean by “sound”; if you mean something that is heard by an observer, then the answer is no. But if you mean something that produces the physical oscillations that we take to be “sound” then I say yes. Why would you tie an observer to sound more tightly than you do to vision? This is like asking, “if no one is there to see a rock is it invisible?” The both seem to me to be silly questions if you accept Axiom 1. (Of course, in philosophy a lot of why you might want to discuss such a question is to question the validity of axiom 1.)

Axiom 2: We each live in our own virtual reality, which is an approximation to the real world, not identical with it.

This one also seems incontrovertible. Each of us only has a limited set of information about the real world that we collect through our sensory apparatus – eyes, ears, skin. We assemble that data to create a model of the real world that we live in.

One bit of evidence for this is that through the process of science we have discovered many, many things that are undetectable to us directly through our senses. These can have great importance to our lives and kill us or save us – things ranging from bacteria and asteroids, from UV-light and x-rays to atoms and chemistry. A second bit of evidence is that psychological research has demonstrated very clearly how the brain assembles an internal version the real world – and have created strikingly powerful illusions show how we often get it wrong. Two of my favorites are Ed Adelson’s checkershadow, and Daniel Simon’s Invisible Gorilla.

So Axiom 1 says reality exists, axiom 2 says we can’t know it very well by direct observation. Axiom 3 holds out the hope that we can know it better by working together and trying to figure it out using the best tools we’ve got.

Axiom 3: Science is possible.

What I mean here by “Science” is a community activity addressed at finding out more about reality through observation and attempting to build coherent descriptions of what we see. The nature of this activity is complex and not easily described by a “scientific method” of a few steps. The critical point is that although we may use a theoretical framework to guide our observations and predictions, at base science is fundamentally empirical: built on our observations of what actually happens and our ability to weave an increasingly consistent story about how things are, behave, and work from fewer and fewer principles. I’ll write again later about my view about the nature of scientific activity and the lemmas that tell me when I define something as “science” and when not.

These aren’t so radical. Reality exists; we don’t see it directly; but we can figure it out if we’re careful. Even at this simple level I butt heads against some of the fundamentalists who want to say, “this is not the real world – what’s real is what we will experience after death.” Others will accept my 3 axioms but not the way I think I should apply them. But there is another fundamental difference between me and many religiously-based thinkers. We both believe that we need to develop a broadly accepted code of human behavior. How are we to know right from wrong? Where is the “good”? For them, it must arise from axioms. For me, the axioms are about finding out what “is”. What “should be” must be consistent with what is. IMHO, separating these two is an essential first step

Psychological symmetry or the cold really creeps into your bones

2010-06-15T06:38:00.001-04:00

Last weekend, my wife and I attended the play at the Arena Stage in Washington. It was a one-man show, a monologue called “R. Buckminster Fuller: The History (and Mystery) of the Universe.” Now I’m not usually a great fan of one-man monologues, although as an academic I’m typically perfectly happy sitting all day in lectures at a conference f the topics are interesting. But I thought that Fuller was a thinker and his musings might be interesting. Well, I was right. The text was basically taken straight from Fuller’s writings, but it was implemented in a very dynamic way using videocameras and projection in interesting ways – showing different angles of what we were watching and supplementing it with images so it was theatrically fun as well. And the character (actor Rick Foucheux) even used some interactive pedagogy that I have been trying to use in my classroom. He sometimes turned up the lights and asked the audience questions, insisting on responses. I quite enjoyed it.

Interestingly enough, I found myself agreeing with many aspects of Fuller’s philosophy, much more than I expected. But there was one place where I strongly disagreed with what he said. At one point he said something like, “The wind doesn’t blow, different places on the earth suck!” This got a good laugh from the audience, but it wrinkled my brow. That didn’t feel right to me and I’ve been thinking about why I had that reaction.

When we describe phenomena in physics (or in any science) we typically choose a level at which to make our description. If we’re talking about the wind, we talk about the density of the air and its velocity – macroscopic concepts that we can see, measure and feel. If we’re talking about hot and cold we talk about temperature and heat flow. If we’re talking about electric currents, we talk about the electric potential difference and the current – volts and amps. We can’t see or feel those (Well – actually we can, in terms of the hair on our skin rising or feeling shocks, but we don’t have a lot of experience with that and we don’t tend to interpret those feelings quantitatively the way we do wind speed or heat and cold.) but we can measure them directly with devices we can hold in our hand.

It’s a question of ontology. Wikipedia current defines that as the philosophical study of the nature of being and the categories of being. I like to think of ontology as answering the question, “What kinds of things are there?” – or perhaps better, “What kind of concepts should I use to describe a particular phenomenon?” If we are discussing the motion of the air, the thing we are talking about is “air” – a substance that we consider has the properties that it is continuous (no gaps), is everywhere, has a variable mass density (any volume of it has mass but the same volume can have different masses in different places) and it can move. If we want to describe the causes of its motion, it is natural to try to apply Newton’s laws, which give the general principles that successfully describe the motion of essentially all classical objects. This description relies on our being able to identify forces – pushes and pulls.

Fuller’s statement about the wind makes clear that, if we are talking only at the macroscopic ontological level, there is a possible symmetry in our description: we can say that there are forces in the air that push it from place to place, or there are forces in the air that pull it from place to place. We can describe the motion in terms of repulsive (pushing) forces or attractive (pulling) ones. Newton’s synthesis of the laws of motion tells us that forces on objects are caused by other objects, so we have to identify who is doing the pulling or pushing, but that’s not a big deal. It could be the air acting on itself or it could be bits of the earth pulling or pushing on the air (Fuller’s “some places suck!”).

Another example of this kind of symmetry is heat and cold. When I am teaching about thermodynamics I ask my students to place their hand first on the cloth part of their chair and then on the metal part of their chair. They all respond that the metal feels colder than the cloth, even though we have just had a discussion that says that concluded when things are left to stand together for a long time they tend to come to the same temperature. We resolve this by deciding that it is the rate at which they come to the same temperature that matters in what we feel. The temperature of your hand is higher than the temperature of the chair in my classroom, so when you touch either the cloth or the metal heat energy will tend to flow out of your hand into the chair. It flows into the metal faster so it feels colder.

In temperate climates, we tend to see the “hot” as the active agent that moves. But I suspect that if we lived in a climate that was extremely cold most of the time we would see the “cold” as the active agent. When you go out in a temperature of 40 below, you might feel that “the cold is sucking the heat right out of you” or that “the cold just seeps right into your bones.” This is an ontological symmetry: we could describe things equally well in terms of the motion of “heat” (better: “heat energy”) or of “cold”. We would just reverse the direction of the flow if we switched our description from heat flow to cold flow. Everything else would look the same and it would work fine. In my classes some student often makes this suggestion.

In physics, symmetries are of great importance. If we decide that we are going to create a theory in which the placement of the origin of our coordinates doesn’t matter, then the theory we create will necessarily conserve momentum. If we decide that our theory should not depend on the orientation of our coordinate system, then the theory we create will necessarily conserve angular momentum.

But the symmetries I’m discussing here are different. They’re not really physical symmetries; rather, they’re psychological symmetries. It’s a question of how we look at the system we’re describing. Whereas physical symmetries are about what we think we are able to explain with our theories. [Aristotle made no attempt to create a theory of gravity. The result of the earth’s gravity was imposed on the theory without physical explanation. The center of the earth was the center of the universe where everything tended to go. There was a “special point” in the theory, so theory was not independent of coordinate system and momentum did not emerge as a relevant concept.]

Another nice psychological symmetry is in the construction of Newton’s laws and contact forces. For a physicist or an engineer looking at a bowling ball being hit by a hammer it seems natural to look at the changing motion of the ball as responding to the forces it feels. But for a biologist looking at an active organism, it looks like it’s the intent of the organism that causes its motion to change. If you are riding a scooter and push your foot on the ground, your backward push is what sends you forward. Since Newton’s third law says that for every force that one object exerts on another, the other objects exerts an equal and opposite force backward on the first, we could write Newton’s second law to say that an object responds negatively to all the forces it exerts instead of positively to all the forces it feels. Since mostly in physics we are talking more about inanimate objects we tend to prefer the traditional choice. [This is discussed in detail in my paper with Rachel Scherr, Newton's zeroth law: Learning from listening to our students, The Physics Teacher, 43, pp. 41-45 (2005).]

So if these psychological symmetries are really symmetries and it doesn’t matter which way we look at it, why was I uncomfortable with Fuller’s statement?

The reason is that the current paradigm of science is to relate things across levels. Up until the 19th century, most of science was about learning to describe the regularities in the world as we saw it. But when, at the end of the 19th century, we began to understand the structure of matter in terms of atoms and molecules, another level became available. We could now describe the properties of matter we had observed phenomenologically in terms of the structure of matter and what is happening to its molecules. The flow of the air is naturally understood in terms of pressure – the air pushing on itself – and that pressure can now be interpreted in terms of the average momentum and the number density of the molecules moving around. The temperature of matter that controls heat flow can now be interpreted in terms of the average kinetic energy of a molecule. At the molecular level, the ideas of pressure and heat flow have natural and simple explanations; the concepts of “sucking” and “cold flow” do not. It’s not always useful or necessary to think down to the molecular level. An electrician can be perfectly competent thinking about volts and amps without ever considering electrons. But crossing levels changes the ontology of how we think about matter and enriches our view of what’s happening. And it provides a “psychological symmetry breaking” that chooses which of our symmetrical descriptions are more appropriate.

Now when I’m teaching these physical concepts I think not just about the entire package that I have learned through my many years of studying physics. I also think about what I can expect my students to know and what might appear natural to them that appears bizarre to me. For many of my university science students, although they know perfectly well about atoms and molecules, they don’t necessarily have the idea that their observations at the everyday level should in fact be consistent with what we know about the structure of matter. This has lots of implications that I will discuss in other posts.

One and the Same

2010-04-11T09:39:00.004-04:00

Some years ago when my daughter was in early middle school (or thereabouts) she learned about fractions and repeating decimals. She complained that the teacher said that the repeating decimal 0.9999… was the same as one. She wasn’t convinced. She said, “it couldn’t be because no matter how far you go there’s always a difference.” I smiled and said I would prove it to her. Here’s how my proof goes.

Since we are trying to figure out what 0.9999… is, let’s give it a name so we can talk about it. Let’s call it “A”. So we start with the definition

          A = 0.9999… .

Now let’s multiply A by 10. This just shifts the decimal, giving the result

          10A = 9.9999… .

Since the decimal goes on forever, we haven’t changed the part beyond the decimal at all. The right side is therefore what we started with +9, or:

          10A = 9 + A.

Since the equals sign means that the things on the two sides are different ways to represent the same thing, if we subtract the same thing from both sides, we will still get two things that are equal. Subtracting A from both sides gives

            10A – A = 9 + A – A
or
            9A = 9.

This is easily solved by dividing both sides by 9 to give A = 1 as her teacher claimed.
She was mollified, but not satisfied. Eventually she accepted it (I won’t say “figured it out” because of what I say below) and became a master of such things as calculus, differential equations, and how to calculate pi. But the incident stuck in my mind and now I’m the one not entirely satisfied.

This little exchange illustrates something that I have found increasingly interesting in the years since it happened: the relationship between mathematics and the way we construct it, and the way we apply it to our world. Math by itself is a self-contained structure with principles and rules so rigid that they can be carried out by a computer – sometimes. (The places where this fails are particularly interesting.) But when you think about how it actually works, there are often some hidden psychological elements. (Please don’t tell my mathematician friends that I think math rests on psychology. I might get thrown out of the club.) The critical issue is: When do we decide two things are “the same”?

The questions of identity – when two things can be thought of as “the same” – is a tricky one in everyday life as well as in math. In Star Trek, since he didn’t have the budget to show a spaceship landing and taking off, Gene Rodenberry created the “transporter” – a device that dissolved any object (including people) into their molecules, carefully taking note of where each one was, beaming the information down to another location and reassembling the object out of different molecules in that place in exactly (one hopes) the same configuration as the original.

[In one episode, the beaming is assisted by “Heisenberg compensators.” Legend has it that at a press conference at a Star Trek convention, someone asked Rodenberry, “How do those Heisenberg compensators work?” He responded, “Very well.”]

The crotchety medical officer, Bones McCoy, was very nervous about this – as well he might be! Suppose what really happens is that pulling you apart into your molecules simply kills you and assembling those molecules elsewhere simply (!) recreates a clone of you with exactly your memories who thinks he is you. We know from quantum physics that all atoms of a particular kind are truly identical (that is a story for another entry), so it shouldn’t matter. Since he has all your memories, he looks to everyone else as if he is you – and they don’t care that the “real” you has been destroyed. Is your consciousness continuous from the destroyed you to the recreated one? Or has the second consciousness been created anew?

Of course, this technology doesn’t exist (yet), so we don’t have to worry about it. Except of course we do. When we go to sleep, or become unconscious due to injury, when we wake up are we the same person? When you meet an old flame years later, they have had a range of experiences that have changed them. Are the still “the same person you knew”?

These issues go quite deep into our whole interpretation of our lives. When do we consider two different things “the same”? And things are always different. The laptop computer on which I am composing this is continually picking up and emitting individual atoms and molecules – gas, dust, etc. Its transistors and chips are continually changing as the carry electric currents, heat up, and cool down. The magnetic storage on the hard drive may deteriorate as it is used. In a few years, it “won’t be the machine it used to be” – but what will it be?

The basic principle seems to be, when it’s useful to treat something as a single thing over time, let’s just go ahead and do it. “A difference that makes no difference is no difference.”

Let’s return now to the issue of identity in math. With my daughter’s example of the repeating 9’s, we have to say, If we are going to use this in a system where we manipulate symbols in the systems of arithmetic and algebra, then we have to consider “1” and “0.9999…” as being two different representations of the same thing. Similar things happen with fractions. Math is a study of relationships and equality. Two principles that are critical to it are:

We are interested in knowing when two different ways of representing something are the same so the definition of equality (or identity) is crucial and,
We assume that if we do the same thing to two different representations of the same thing we still get two representations of the same thing.

These rules don’t always hold. For example, in arithmetic, you can easily generate nonsense by dividing by 0. Funny things can happen whenever infinities or things that in principle take an infinite number of steps are involved. (Think of Zeno’s paradox, for example.) There is a lot of fun math that studies how to handle these situations – math like calculus, the theory of complex variables, and functional analysis.

We are usually comfortable when we do arithmetic and algebra as long as no infinities are involved, but the issues of identity are still deep, especially when we are trying to apply math to the physical world.

An example of this is fractions. Our arithmetical system needs fractions to be complete and consistent. If we start with integers, it’s easy to figure out what adding means – just “counting on”. Five year olds usually get this. To be able to go backwards and solve equations with adding (like, what do you have to add to 5 to get 12, that is, 5 + x = 12), we have to invent the opposite of addition – subtraction. This then leads us to construct negative numbers. We can then add anything to anything and subtract anything from anything. Repeated addition leads to multiplication. To be able to go backwards and solve equations with multiplication (like, what do you have to multiply by 5 to get 30, that is, 5x = 30), we have to invent the opposite of multiplication – division. This then leads us to construct fractions. We can then multiply anything by anything and divide anything by anything (except 0) and solve all kinds of equations.

Our rules now lead us to identify certain fractions as “the same”. The fraction rules say that when you multiply fractions you multiply the tops and put the result on top and multiply the bottoms and put the result on the bottom. This accords well with our everyday sense that if I take ½ of something and divide it in two parts again, I will get ¼ of the original thing; that is, (½)x(½) = (1x1)/(2x2) = ¼. This also tells us that

2/4 = (1x2)/(2x2) = (1/2)x(2/2).

But since 2/2 = 1, we must identify 2/4 and 1/2 as the same number.
This is all very well and good, since it increases the number of ways we can represent the same thing and that leads to more things we can do with it. But it creates some problems with mapping our numbers into the real world.

As a young physicist, I was strongly tempted by the idea that the mathematics I was learning in physics was the real world – that the world was somehow number. As I have aged, I have become increasingly convinced that this is the wrong way to look at it. A better way now seems to me to be the following:

Mathematics is the abstract study of relationships and how things can be looked at in different ways without changing some essential essence (often quantitative – but not always). We look for patterns in the physical world that match these relationships and rules. When we find them, the tools developed for abstract relationships in math can be carried over and will tell us things about physical relationships that cannot be easily seen directly.

But we do have to carefully watch for cases where there is a shift in the “essential essence” of what we care about. If we are talking about figuring out how many bricks are needed to build a building, multiplying and dividing and adding and subtracting will work just fine. If we are talking about dividing a pizza into slices, the fact that 1000/1000 = 1 does not really tell me that if I ask for a pizza, that it is OK to take the pizza, cut it up into 1000 pieces, and give me all the pieces. When applying math to the real world in any way, we have to be careful about whether the “essential element of reality” that is correctly represented by the math is what we care about, or whether other essential elements have been missed.

83333

2010-04-08T05:45:00.001-04:00

The other day while driving to the office, the odometer of my car turned to 83,333. Pretty typical. My car is 8 years old and I commute nearly 20 miles each way to work, so I’m putting on about 10,000 miles per year. But I liked that number. It reminded me of something. What? Was it the cosine of some angle? I’m a physicist and I like to work with numbers, so I do run into cosines all the time. That didn’t seem to ring true. Was it some simple fraction? That probably was it. I decided to see if I could figure it out. Here’s my approach.

Consider the infinitely repeating decimal 0.8333333…. . Is it some simple fraction? Well I know that 0.333333…. is just 1/3 and this looks almost like it – at least the repeating part. Can I manipulate this to make it look like that? First there this “0.8” in front that doesn’t belong. So let me write my decimal at 0.8 + something else. Subtracting off the 0.8 I get 0.033333…. So my decimal = 0.8 + 0.03333…. The 0.8 is just 8/10 which equals 4/5 by cancelling the common factors of 2 at the top and bottom. The other part is almost the 1/3 but it has a 0 in front after the decimal. I know that you put extra zeros after the decimal by dividing by 10. So 0.03333…. = (0.3333….)/10 or (1/3)/10 = 1/30. So my fraction is 4/5 + 1/30. I can combine these together by putting them over a common denominator. If I multiply the first fraction by 1 = 6/6 (you can always multiply something by one without changing its value – but you can change the way it looks to help you) I get 4/5 = (4x6)/(5x6) = 24/30. I can now add this to the 1/30 to get 25/30. This is 25/30 = (5x5)/(5x6) so I can cancel the common factor of 5 to get 5/6. A nice simple fraction as I had expected (and should have remembered). Then I see that the complement of my decimal – it’s difference from one – is just 0.166666…. (You have to make the two of them add up to 0.99999….. which actually is just equal to 1 – but that’s a discussion for another entry.) If I had seen that I would have gotten it right away. I work a lot with fractions and I know that 1/6 is about 0.16 and is a repeating decimal.
I know, I know! DWDM (Driving while doing math) – or maybe it should be “DWA” (Driving while academic) – is probably as dangerous as talking on a cell phone while driving. Not a good idea! But the exercise illustrates some lovely principles about number and education.

First, one of the things I have been trying to do with my work recently is to teach non-physicists (biologists, actually) to think a bit “in the way that a physicist does.” They often tell me this when I ask what I should teach. But what does this mean? I think my example, although it’s just about number and not about physics itself, tells me something. As a physicist I see numbers as real things, not just abstract relations. They’re things with structures and connections and properties. I can work with them, turning them twisting them, multiplying by 1 in various forms (5/5 or 6/6 for example) or adding 0 in various forms to change the way they look and give insight into what they are. Being able to see them in different ways gives them a solidity and a reality that I suspect most of my students don’t feel. Seeing that I can do the problem in the two ways discussed – manipulating it into a fraction by taking it apart and looking at the complement and recognizing a familiar number – gives me the comfort that arithmetic and all the complicated stuff about fractions and decimals isn’t just something I have to remember. It’s something that makes sense. It’s reliable and consistent. I can look at things in a variety of ways and confirm my answer and catch my mistakes, though, luckily in this case, I didn’t make any. I often do, especially when doing math in my head, but having the multiple perspectives on number (and equations) usually helps me nail down the correct form pretty quickly.

I showed this analysis to one of my students who came in to my office for help on this week’s homework. She was flabbergasted. She said she had never seen anything like that. How sad! This is the sort of thing I would love to see taught to every 5th grader. We seem to be making a useful transition in our teaching of arithmetic – from doing tedious rote math (adding long columns of 4 and 5 digit numbers) to using calculators to eliminate the tedium. But learning to use a calculator shouldn’t just get rid of the tedium but also give students the sense that numbers are somehow magic and can’t really be thought about.

[If you can find it, I recommend Isaac Asimov’s 1939 short story, “The Weapon Too Dreadful to Use”, which describes a world in which number had been automated by computers so much that people didn’t realize you could figure things out. Asimov isn’t usually predictive, but this one hits the nail on the head.]

Calculators can in fact be used to help students develop this “sense of number” very effectively, though they often are used in exactly the opposite way. For some references to this, check out the literature review in my paper with my student Tom Bing, “Symbolic manipulators affect mathematical mindsets,” (Am. J. Phys. 76, 418-424 (2008)). Interesting books on the sense of number, how people develop it and build it into math include:

Stanislas Dehaene, The Number Sense – a very readable book about the neuroscience and psychology of basic math.

George Lakoff and Rafael Nunez, Where Mathematics Comes From: How the embodied mind brings mathematics into being – this one is quite a bit more technical and takes one deep into sophisticated math.