Unit Conversions as Gauge Transformations?

I recently read through a very nice blog post, references of which led me to reread Redish’s 2006 paper about how math is used in physics.  The following quote jumped out at me:

In introducing the concept of units to my introductory classes, I tell them that a quantity has a unit when we have an operational definition for assigning a number to it and that the number that results depends on the choice of an arbitrary standard. Since the standard is arbitrary, we may only equate quantities (or add them) if they change in the same way when we change our standard. Otherwise, a numerical equality that we obtain might be true for one choice of a standard but not for another. An equation that has physical validity ought to retain its correctness independent of our arbitrary choices.

Many physicists will recognize in the above the “scent of Einstein” – the idea that “a difference that makes no difference should make no difference.” Einstein, Poincaré, and others around the turn of the last century introduced the idea of analyzing how physical measurements change when you change your perspective on them – rotate your coordinates (in the case of identifying vectors and tensors), hop onto a uniformly moving frame (in the case of special relativity), or make a general non-linear coordinate transformation (in the case of general relativity).

Of course, the transformations that we use with units do not depend on spatial coordinates, but considering the symmetry transformation explicitly punched me in the gut because I realized that we don’t teach this explicitly to students in science classes.  Students trying to to convert units typically try either of two strategies where I work.  The first (used with metric conversions) is to consider how many factors of 10 there are between the units (like km and cm) and then move the decimal place that many times.  If students remember this procedure, they as often move the decimal point the wrong direction.  The second is to use the unit-factor method, wherein they often try “fun” factors like \frac{100\,\textrm{m}}{1\,\textrm{cm}}.

At least for proportional units—let’s leave affinely related units for another time—we can regard a quantity as a pair (n,u), where n lives in some mathematical space, and u is our unit, a physical “arbitrary standard”, as Redish calls it, which lives in some torsor for the multiplicative real numbers \mathbb{R}^\times.  On this space, we can define an equivalence relation for physical equivalence, (n,u)\equiv(\Lambda^{-1}\cdot n,\Lambda\cdot u), where \Lambda\in\mathbb{R}^\times is a scale-factor.  In my mind this leads to the following exercise:

The width of a desktop is measured in units of cm and found to be 134.4 multiples of 1 cm, or in other words, 134.4 cm.  If, instead, one had measured it in a different unit, how many multiples of the new unit would the length be?  Try for as many different units as you can.

This gets at two ideas.  Students don’t always think in terms of 3 cm being 3 multiples of a centimeter length.  They also don’t often get to choose what measurements they use, or the teacher has them convert in predictable ways.  Here they can see that (134.4, cm)≡(134.4/10,cm*10)≡(134.4*10,cm/10).  Would this help them build a chain of understanding of units?  I realize that when I convert, I explicitly play the epistemic game in my head, “m is a bigger unit than cm, so it takes less of that length”.  Do all students have that understanding?

The other epistemic game I play in my head is to think, “134.4 cm is about 100 cm.  I know 100 cm is the same as a meter, so 134.4 cm is a little bigger than a meter.  It’s definitely less than 2 meters, since 2 meters is the same length as 200 cm.  I’m pretty sure that 134.4 cm works out to 1.344 m.”

How do you teach students to understand units in elementary, middle, and high school?  I’d love to know what conceptual tools or epistemic games you give students.

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Mathematics Teaching Practices That Support Science Learning

Kate Owens recently threw down the gauntlet in an #iteachphysics chat for physics teachers to enumerate what mathematics teachers can do to support students in physics.

Since I’m licensed to teach both, I’ll try.  My invented audience here is all math teachers, from pre-K through post-secondary.  I’m only too happy to revise this list from your feedback or new evidence about how students learn math and science.

Concept of Quantity

  1. Use numbers as adjectives in addition to nouns.  When we teach children to count, we often let them recite 1, 2, 3, …  This is fine, but we also want them to understand the one-to-one mapping between numbers and objects, e.g. 1 penny, 2 pennies, 3 pennies, …  The answer to “What do we get when we have 3 apples and someone gives us 4 apples is not 7 but 7 apples.”  I think CIMM‘s multiple representations are really good for this.
  2. Measure all the time.  The key to reasoning facilely about quantities is having numerous concrete contexts to anchor abstract concepts.  Then, do math on these measurements.  This should start in elementary school; see the Benezet Experiment for details.  Measuring can often provide the impetus to define a new mathematical model or study a new differential equation.
  3. Use units within mathematical calculations.  Students who have never done calculations with units will generally suffer from Exploding Brain Syndrome when expected to do so in science.  The problem is that they can’t chunk “3 cm” all in one go, so it halves their working memory.  Example: If we want to distribute profits of $72 among 3 people, then \frac{72\,\$}{3\,\textrm{person}}=\frac{24\,\$}{1\,\textrm{person}}=24\,\$/\textrm{person} . (Note that units are written singular by convention.)  Rates like this are great for understanding the meaning of derivatives; \frac{\textrm{d}v}{\textrm{d}t}=3.4 doesn’t convey as much meaning as \frac{\textrm{d}v}{\textrm{d}t}=\frac{3.4\,\textrm{m/s}}{1\,\textrm{s}}.

Concept of Variable

  1. Let variables refer to quantities (numbers with units), not just numbers.  You don’t need to define m to be the length of string in meters when you can define \ell to be the length of string, i.e. \ell=3\textrm{ m} versus m=3.
  2. Use units in equations with variables so that students get used to the difference.  According to convention, variables at typeset in an italic font (e.g. m for mass), whereas units are typeset in an upright roman font (e.g. m for meters).  Variables can stand for different values.  Units are constant, which can help understanding of partial derivatives when doing calculus.
  3. Use variables other than x and y.  Whereas x and y are great for quick calculations, if that’s all you use, then students come down with Exploding Brain Syndrome when expected to use more meaningful variables.  Graph V versus d (that’s V(d) for the math teachers) when illustrating the dependence of volume on a distance parameter d.

Concept of Equation

  1. Encourage students to see patterns of expressions in equations as chunks.  The Transition to Algebra Project has an interesting Algebraic Habit of Mind called “Seeking and Using Structure”.  (Hat tip to @davidwees for his tweet that clued me in.)  Students often struggle when solving equations like \tan 53^\circ=\frac{20\,\textrm{cm}}{x} in which x is in the denominator.  They can learn to recognize the analogy with 7=\frac{56}{?} (in which one writes a different equation from this fact family, \frac{56}{7}=?) to rewrite the relationship as x=\frac{20\,\textrm{cm}}{\tan 53^\circ}.  This is far less tedious than always using the “balance” concept of equation solving in this case.  Yay for multiple representations!
  2. Banish the next-step equals sign.  Give middle and high school students a problem with multiple steps, and you’ll probably see some variation of this: 25+30=55\cdot10=550.  Students use the equals sign to signify the answer to a computation and don’t think about an equation as a statement as successive equalities.  I wish I knew how to fix this, but I have no idea.  It definitely makes following reasoning very different.  Is it as simple as teaching students how to organize problem solving?  Breaking a calculation into smaller pieces can introduce roundoff error for students not prepared to carry through multiple variables.

Concept of Function

Edit on 2015-03-13: I’ve added below based on suggestions from @BlackPhysicists to include Inverse Function Theorem, Implicit Function Theorem, injections, “bifurcations and transitions to chaos”, and hysteresis.

  1. Interpret the meaning of a function.  Why does a functional relationship C(r) exist for the circumference of a circle with respect to its radius?  Two possible interpretations: (1) There aren’t two circles with two different circumferences but the same radius.  (2) Knowing the radius fixes the circumference exactly.  Often in mathematics (as well as science), we can’t say, “Radius causes circumference,” any more than we can say, “Circumference causes radius,” and we may as well write $r(C)$, but there is a structure found in science…
  2. Give functions a causal relationship (when appropriate).  Relations don’t necessarily have a causal relationship.  It seems (to me, at least) that it makes more sense to think of acceleration as a function of net force and mass \boldsymbol{a}(\Sigma\boldsymbol{F},m)=\frac{\Sigma\boldsymbol{F}}{m} (and to say that having a given mass and an applied net force causes a system to accelerate) than it does to think of net force as a function of acceleration and mass \Sigma\boldsymbol{F}(\boldsymbol{a},m)=m\boldsymbol{a} (and to say that a given mass and acceleration cause net force), let alone to think of mass as a function of acceleration and net force m(\boldsymbol{a},\Sigma\boldsymbol{F}=\frac{\Sigma F_j}{a_j} (no sum on index j) (and to say that a given acceleration and net force cause mass).  This last one is particularly ridiculous, since net force and acceleration have to be collinear for this to be well-defined.
  3. Be less fixated on functions.  Functions are an amazing abstraction, but they aren’t always the most interesting structure in a problem.  Magnetization in a ferromagnetic material (like iron) is widely known to exhibit hysteresis, that its magnetization (the strength of the magnet) does not depend directly on the applied magnetic field but also on its history of magnetization.  Was it magnetized before, or was it unmagnetized iron?  Describing these effects requires new models.  Likewise, pseudorandom number generators functions as commonly conceived in many programming languages are not properly functions (unless adds another variable, a seed, to their arguments).  Possible models, like monads, get very interesting from the perspective of category theory.  Just as multivalued functions are useful in complex analysis, so too are bifurcations and other aspects of chaotic behavior.  What’s a good balance between classical and modern mathematics?  Science has yet to find the answer as we struggle to decide how to teach hundreds of years of science faster and faster.

Graphs and Mathematical Models

Secondary mathematics builds mathematical models (essentially parent functions) through the course of high school education.  Although some of these could go in the sections above, I wanted to collect them here.

  1. Stop calling the axes x and y.  The terms “horizontal intercept” and “vertical intercept” are perfectly intelligible over “x-intercept” and “y-intercept”.  If your graphing program doesn’t let users change what they call the axes, then shame on you!  <“GeoGebra!” he grumbles under his breath.>
  2. Vary descriptions of slope.  Call it “gradient”, “ratio of changes”, “rate of change of ____ with ____”, or “how much ____ changes for every ____”, or “change in ____ for every change in _____”.
  3. Stop writing slope as m.  Seriously, who came up with that?  If your variables are x and y, then \frac{\Delta y}{\Delta x} is more descriptive.
  4. Rethink slope-intercept form.  The equation y=mx+b doesn’t promote adequate transfer of algebra skills to physics.  This even seems to be true when students take algebra the same year as or the year previous to Physics First.  I’m not totally decided on the answer, but I would suggest something like y(x)=\frac{\Delta y}{\Delta x}\, x+y(0).  Then it’s clear that y(x) is the value of y for a given value x and that y(0) is the value of y when x=0.  Too often students just call b the “starting value”, meaning that the graph “starts” at (0,b), without realizing that y=b when x=0.  For instance, the equation for position as a function of time might become x(t)=\frac{\Delta x}{\Delta t}\,t+x(0), which, when particular values are plugged in, becomes x(t)=\frac{-3\,\textrm{m}}{2\,\textrm{s}}\,t+8\,\textrm{m} or x(t)=\frac{-1.5\,\textrm{m}}{1\,\textrm{s}}\,t+8\,\textrm{m} or even x(t)=(-1.5\,\textrm{m/s})\,t+8\,\textrm{m}.  In this way it’s much easier to identify the meaning of the coefficient of t (the slope) and the constant term (the vertical intercept).  If students were taught this way, then going from the equation of the line y(x)=\frac{\Delta y}{\Delta x}\cdot(x-x_*)+y(x_*) to understanding the equation of the tangent line y(x)=\frac{\textrm{d}y}{\textrm{d}x}(x_*)\cdot(x-x_*)+y(x_*) would be trivial.

Concept of Scalar Versus Vector Versus Bivector Versus Pseudovector Versus Matrix Versus Tensor Versus Spinor Versus…

Quantities in physics can be classified by how they behave under rotation.  This is a richly beautiful subject called group representations.  What do we want students to be able to do?  We hope that they can reason using each of these types of quantities.  Graphical representations of vectors have not been as useful to students as we would have hoped (Heckler and Scaife, 2015).  Algebraic representations seem to be easier to grasp, but that doesn’t mean that we should ditch graphical representations.  It does mean that we should choose algebraic and graphical representations carefully, so I hope in this section to begin a conversation about approaches to vector analysis.

  1. Teach integer operations using vectors (and call them vectors).  Yes, one-dimensional vectors are still vectors.  This is isomorphic to number-line methods for integers.  You know, the ones with all the cute arrows.  The sign tells us direction.  Please don’t let students think that vector is just an evil villain.  Vectors are our friends!
  2. Drop \boldsymbol{\hat{\imath}}, \boldsymbol{\hat{\jmath}},\boldsymbol{\hat{k}}.  This may be controversial, and I really don’t mind them personally, but it adds another brain transcription layer to go between “x-direction” and \boldsymbol{\hat{\imath}}, and that’s not good notation.  I would propose \boldsymbol{\hat{x}}, \boldsymbol{\hat{y}},\boldsymbol{\hat{z}} for low dimensions for younger students, but the problem with doing so is that it’s confusing to then use x, y, z as coordinates to make a general vector, like x\boldsymbol{\hat{x}}+y\boldsymbol{\hat{y}}+z\boldsymbol{\hat{z}}.  Do you think that you or students will find this too confusing? Physicists sometimes use \boldsymbol{\hat{e}}_1, \boldsymbol{\hat{e}}_2, \boldsymbol{\hat{e}}_3, etc., and I tend to stick with the hat for unit vectors, though I don’t mind dropping it to \boldsymbol{e}_1, \boldsymbol{e}_2, \boldsymbol{e}_3, etc. when the symbols are so well-known.  How is this an improvement over \boldsymbol{\hat{\imath}}, \boldsymbol{\hat{\jmath}},\boldsymbol{\hat{k}}?  In more advanced contexts, it’s easy to write \boldsymbol{v}=v^i \boldsymbol{e}_i using the Einstein summation convention with proper contravariant and covariant indices.  Hopefully doing this consistently will help build facility with vectors as in the Heckler and Scaife article linked above.
  3. Try Geometric Algebra instead of the Vector Analysis you learned in school.  (Note: Useful resources come from University of Cambridge, including the very readable paper Imaginary Numbers are not Real — the Geometric Algebra of Spacetime.)  Even if you don’t start teaching vector multiplication using concepts from geometric algebra, the insights you get might change the way you think about vectors, and recovering the square root of -1 as i=\boldsymbol{e}_2\boldsymbol{e}_1 is just plain cool.  Claiming that geometric algebra simplifies all physics equations may be slight hype, but maybe that’s just because I’m used to the way they used to look.

Related Work

  1. Hake, R.R. (2002). “Physics First: Opening Battle in the War on Science/Math Illiteracy?” Submitted to the American Journal of Physics on 27 June 2002. Preprint. http://www.physics.indiana.edu/~hake/PhysFirst-AJP-6.pdf (pages 5-6)
  2. Mitrea, Dorina (2009). “Mathematics Underlying the Physics First Curriculum: Implications for 8th and 9th Grade Mathematics”. http://www.physicsfirstmo.org/participants/MathConnections.php

Numeral systems, Or Until I get a proper math blog…

I have a young daughter and have been thinking for the past few years about how we count in the United States.  While there are many interesting alternative number systems, not all of these are useful for teaching a young child to count.  Many commentators on mathematics education have cited the? Chinese system for its monosyllabic digits and ease of composing them into the numbers 0-99, and others have pointed to other interesting mnemonic? features.  However, what I have in mind is inspired by what the late Tom O’Brien has done (paywalled).  I think that for older students, we can use the positional system without the extra letters, but I’ll leave them in for now. So, here’s my version of his system.

Digits 0-9 (monosyllabic versions)
Symbol Name
0 rho (as in zee-rho)
1 one
2 two
3 three
4 four
5 five
6 six
7 ven (as in seh-ven)
8 eight
9 nine
Base units (monosyllabic versions). If the SI prefix for ten were just one syllable, I might consider using it instead of “T”. I also don’t like the mixed distribution of uppercase and lowercase prefixes in SI. I would like to switch up the long vowel sounds (for powers of 10 greater than 0) and short vowel sounds (for powers of 10 less than 0), but I’ll save that for later. In particular, regard anything beyond ty=T as provisional.
Place value Symbol Name
10-3 k kū (pronounced koo)
10-2 h ta (pronounced tah)
10-1 t te (pronounced teh/oak)
100 U (not normally used) tu (not normally but pronounced tuh)
101 T ty (pronounced tee
102 H to (pronounced toh/toh)
103 K ku (pronounced kuh)
104 TK ky (pronounced kee
105 HK ko (pronounced koh)
106 M mu (pronounced muh)
107 TM my (pronounced mee
108 HM mo (pronounced moh)
109 B bu (pronounced buh)

Putting it all together

  • To get a sense of where this meets our current system, 43 becomes 4T3 and is read “fourty three”.
  • To get a sense of where this enforces uniformity over tradition, 37 becomes 3T7 and is read “threety ven”.
  • Try counting from 0 (rho) to 3T (threety).
  • For a more complicated example, 73029 becomes 7TK3K7T8 and is read “venky threeku twoty nine, but we would probably leave off the base units and write it in the traditional way with 0 (rho) as a placeholder.
  • For an even more complicated example, 3.142 becomes 31t4h2k and is read “three onete fourta twokū”, but we would probably also leave off the base units and write it in the traditional way.

Mathematical Notation: Summation

In response to David Wees’ post about summation notation, I’d like to suggest that the terseness of mathematical notation is a godsend when working on long calculations, but it should, perhaps, be collapsible for experts and expandable for beginners.  What I mean is that while

\sum\limits_{i=1}^{6} i^2 or

\sum\limits_{i\in1\ldots6} i^2

may suffice for the expert, beginners may prefer

\mathop{\Sigma\text{um}}\limits_{i\text{ from }1}^{\text{to }6} i^2 or

\mathop{\text{Sum}}\limits_{i\text{ from }1\text{ to }6} i^2 .

I hate the linear form that David Wees mentions,

Summation (i, 3, 6, i2) = 32 + 42 + 52 + 62 = 86,

for it loses the spatial memory aspect of the original summation convention.  The real problem seems to be that this last expression serializes for computers well, but the other mathematics is hard to type.  I would prefer “smaller bits” of mathematics, like Sum and Sequence:

Sum[Sequence[Lambda[i,i^2], 1..6]] or even Sum[(i->i^2)[1..6]]

It would be nice if computers would do f[A] if f[a] is defined for every a in A without some kind of function like Map or Apply.  Here’s a longer version of the last expression:

Sum[Apply[Lambda[i,i^2],1..6]]

In fancy LaTeX form that might be:

\sum (i\mapsto i^2)[1..6] or

\sum \left[(i\mapsto i^2)[1..6]\right]

if we want to make the operator precedence completely clear.  Here, “1..6” is some Ruby-like syntactic sugar to mean the set (really: sequence) {1,2,3,4,5,6}.