# 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.

# 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

# Unifying types of interactions when teaching forces

Update: The code below is now in my GitHub repository.

This post is about a quick writeup on interactions that I made this weekend for students. Since modern physics so often loses to more concrete (and older) material when teaching physics, I wanted to give my students a flavor of the kind of particle physics done in the last 200 years. I thought that I’d let them read this and decide which interactions we should take into account when drawing system schema and force diagrams. I’m not completely satisfied with my treatment of the normal force and the Pauli Exclusion Principle within the article. My language is also a little overblown, but believe it or not, I did tone it down and cut out adjective chunks like “low-energy effective” and replace with “everyday” when describing interactions. Anyway, PDF and XeLaTeX source are below.

While I’m at it, I should also describe the way I teach system schemata. Because I taught energy first, I didn’t want to distract with forces. Also, interactions are not exactly forces; forces describe the effect of an interaction on a particular object. Hence, I felt content to label interactions with simple codes for the type of interaction. Last year I used $\vec{\boldsymbol F}$ with a superscript for the type of interaction (with the “dealer→feeler” as a subscript). I like it for uniformity and the fact that it emphasizes that each type of force is indeed a force, but unfortunately, few college-level textbooks do this. When the emphasis is solving problems, a more succinct notation wins, so I thought that this year I’d give students a choice of force symbols. I hope to learn more about how students approach physics with their choices and to notice which approach is more effective at making connections between interactions, interaction energies, and forces. Examples (In the “Force Symbol” column, the first symbol is my version from last year and the second is what I typically see in problem-solving-focused mechanics textbooks):

Interactions, Interaction energies, and Forces
Interaction Interaction symbol Interaction energy symbol Force symbol
gravitational g $U^{\textrm g}$ $\vec{\boldsymbol F}^{\textrm g}$ or $\vec{\boldsymbol W}$
normal n N/A $\vec{\boldsymbol F}^{\textrm n}$ or $\vec{\boldsymbol N}$
frictional f N/A $\vec{\boldsymbol F}^{\textrm f}$ or $\vec{\boldsymbol f}$
tensional t N/A $\vec{\boldsymbol F}^{\textrm t}$ or $\vec{\boldsymbol T}$
elastic el $U^{\textrm el}$ $\vec{\boldsymbol F}^{\textrm el}$ or ?
electric e $U^{\textrm e}$ $\vec{\boldsymbol F}^{\textrm e}$ or ?
magnetic m $U^{\textrm m}$ $\vec{\boldsymbol F}^{\textrm m}$ or ?

From where do our everyday interactions come? (interactions.pdf)

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\title{From where do our everyday interactions come?}
\author{Brian Vancil}
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\begin{figure}[!ht]
\caption{Known and guessed-at interactions in nature}
\label{intdiag}
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We know of only a few fundamental interactions in nature, and one human project in physics has been to explain everything that we observe in terms of simpler interactions.  If you look at Figure~\ref{intdiag}, as you move down the diagram, you will find interactions that describe more and more specific circumstances.  Each is useful, but on their own they explain very little.  In the history of science, humans have worked upward in Figure~\ref{intdiag}, taking scientific models that appeared to be different but were really different aspects of the same thing and unifying them into a single simpler theory that explained more.
For instance, the electric interaction is responsible for the following everyday interactions:
\begin{description}
\item[normal interaction]The repelling squishiness of matter as two objects push against each other is due to two sources: (1) electric interaction involving negatively charged electron clouds around positively charged nuclei and (2) the Pauli exclusion principle---not really an interaction!---between the electrons.  Something as simple as sitting in a chair involves a normal interaction.  So does air pressure and air resistance.
\item[tensional interaction]The electric bonding of negatively charged electron clouds to positively charged nuclei creates an attractive intermolecular interaction so that when a substance is stretched by a bit, it tends to pull back together.  Parts of a rope pull on other nearby parts of a rope through a tensional interaction.
\item[elastic interaction]An extreme form of the tensional interaction, in which matter changes its shape by a lot.  Springs and elastic are good examples.
\item[frictional interaction]Friction is not completely understood, but it involves the grinding together and shearing of irregular surfaces along each other.  Hydrocarbon molecules also play a role between the surfaces.  Something as simple as walking across a floor requires friction.
\item[electric interaction]Electrically charged objects attract or repel each other depending on whether their charges are opposite or the same, respectively.  If you have ever experienced static electricity, you know this well.
\end{description}
As an example of how the human project of searching for simpler explanations progressed, the \textbf{magnetic interaction} is familiar from magnets, but it was discovered in the 1800s that electric and magnetic interactions are really both part of a single \textbf{electromagnetic interaction}, which is responsible for the static electric interactions already mentioned, electricity, magnetism, and even light (really the entire spectrum of electromagnetic radiation).  In the second half of the 1900s, physicists learned to describe both electromagnetic interaction and the \textbf{weak interaction} (responsible for many forms of radioactive decay) by a single theory of the \textbf{electroweak interaction}.

Attempts have been made to unify the \textbf{strong interaction} (responsible for both the nuclear interaction that holds protons and neutrons together in the nucleus of an atom and for the interaction that holds quarks together within protons and neutrons) with the electroweak interaction into a single interaction.  These theories go by the name of Grand Unified Theories, but all of them predict types of matter that we haven't seen yet.

Also in the 1900s, physicists worked to unify the gravitational interaction with the other types of interactions to create a so-called Theory of Everything.  Most of those attempts failed, but we humans have learned a lot from the failures, and we are still at it.  In addition to the human project of unifying interactions, there is also an opposite human project of using these interactions to describe more and more complex systems of particles, everything from neutron stars to superconductors to everyday materials.  Will there ever be an end to the human drive to organize and explain the universe?
\end{document}


It’s been a while since I took thermodynamics and statistical mechanics, so I’m using this post partly as a way to refresh my memory—please tell me I’m wrong! It’s also a way for me to start a conversation on definitions of temperature, some of which are abysmally bad. Because science has a misconception problem, we ought to think of a good way to progress toward more sophisticated notions of temperature throughout one’s educational timeline and to make clear when we break from the naïve notions and why.

One has only to google “definition of temperature” to realize the problem: Once one skips the dictionary definitions about temperature being a scale of hotness (which, as I will argue, is arguably better than what follows), one gets to definitions that say something about how temperature is a measure of the average translational kinetic energy of the atoms in a substance.  (I won’t link to these because I don’t want to increase their Google PageRank.)

### Bad definition 1: Temperature is a measure of the average translational kinetic energy of the atoms in a system.

Compare to NGSS draft 2: DCI PS3.A: “Temperature is a measure of the average kinetic energy of particles of matter. The relationship between temperature and the total energy of a system depends on the types, states, and amounts of matter present.”
It’s a nice picture.  It’s just not hard to break.  For instance:

• What would happen to the temperature if you threw the system?

### Bad definition 2: Temperature is a measure of the average translational kinetic energy of the atoms in the rest frame of a system.

Even if you try to correct for this naïvely by specifying that the kinetic energy be measured in the center of mass frame:

• What would happen to the temperature if you split the system in two pieces and threw them in opposite directions?

Or even worse:

• What would happen to the temperature if you spun the system?

### Bad definition 3: Temperature is a “measure of the average translational kinetic energy associated with the disordered microscopic motion of atoms and molecules”.

HyperPhysics has the best bad definition of temperature.  However, I wouldn’t go as far as they do and call it an “operational definition”.  What is “kinetic energy associated with disordered…motion”?  What counts as disordered motion?  Is a sound wave (phonon) disordered?  It’s what we might think of as a vibration (which many texts cite when they talk about temperature).  How do we measure disorder for the purpose of calculating the “associated” kinetic energy?  Why doesn’t rotational energy count?  What do we mean by “measure” (another problem with calling it an operational definition)—even if I account for all the “disordered” kinetic energy, what value do I put down for temperature?

What’s the simplest system we could envision that breaks this definition? Note that to break the measure part, there must be a non-monotonic relationship between temperature and kinetic energy.  Candidates include:

• photon gas (Many thermal systems include EM radiation.  How do we measure the temperature of light when it isn’t made of atoms?)
• low-temp quantum solids???
• diatomic gases???

### Good definition 1: Temperature is the rate of change of energy with entropy (TODO: specifics on how this is defined) (with volume and the number of particles fixed)

HyperPhysics also gets credit for this good definition.  However, the authors don’t give any examples where the behavior of temperature differs drastically from the simplistic $T\propto U$ idea.
The standard definition in thermodynamics is:

$T=\frac{Q_\textrm{\scriptsize reversible}}{\Delta S}$

However, the standard definition of temperature in statistical mechanics is:

$T=\left(\frac{\partial S}{\partial U}\right)_{N,V}^{-1}$

Does this always work?  What does it mean? Without teaching entropy well, this might be hopeless. However, I recently read an interesting paper on a simple model to introduce the need for entropy as a thermodynamic variable:

• Abreu, Rodrigo de, and Vasco Guerra. “Introducing Thermodynamics Through Energy and Entropy.” American Journal of Physics 80.7 (2012) : 627–637. 4 Jan. 2013. <http://link.aip.org/link/?AJP/80/627/1>.

## How could we make the teaching of temperature better?

I envision a tripartite system of teleological, conceptual, and operational definitions, where we scale up the complexity of the conceptual definitions (i.e. develop new models for temperature) as a student progresses through the system. This has to be explicit, or students won’t understand either the nature of science or why their old ideas are not quite right.

### Teleological definition: Temperature is a scale that tells us the direction of heating when two systems (both in equilibrium) come into contact with each other.

Even this isn’t obvious and needs some justification. If we have three systems in equilibrium, A, B, and C, why couldn’t A heat B, B heat C, and C heat A if they were brought into pairwise contact?
It does get the point across why we care about temperature, which helps to ground our other definitions and provide continuity in the notion of temperature.

### Operational definition 2 (later elementary): Temperature is what a thermometer measures.

This should be connected to the idea of heating and reference temperatures (from special systems with understood behavior). It can later include the idea of different scales. The idea of absolute zero should come from an extrapolation of gas law data from a student experiment.

### Conceptual definition 1: Temperature is a measure of how frenetic and disordered the motion of atoms/molecules in a substance is.

Note that we’re pretty close to a bad definition but that it’s always augmented by our teleological and operational definitions. This is not worth getting to until students understand that matter is made of molecules. Students should see simulations of matter at different temperatures to get a feel for what we’re talking about. It’s simplistic and qualitative.

### Conceptual definition 2: Temperature is a measure of the distribution of energies in a substance given by $T=\left(\frac{\partial S}{\partial U}\right)_{N,V}^{-1}$.

That is, roughly, it’s the ratio of added energy to the change in entropy that results. We’re moving past our bad definitions.

### Conceptual definition 3: We should define a new “temperature” as an inverse energy scale $\tau=-\beta=-\frac{1}{k_B T}=-\frac{1}{k_B}\frac{\partial S}{\partial E}$

Regular absolute temperature fails our teleological definition at negative absolute temperatures. This new definition gets the direction of thermal transfer right when two systems come into contact. See, for instance:

• Braun, S. et al. “Negative Absolute Temperature for Motional Degrees of Freedom.” Science 339.6115 (2013) : 52–55. <http://arxiv.org/abs/1211.0545>