What if you had a teaching assistant?

I know how demanding teaching is, and I know that you need a break. Would you like a teaching assistant? I’m moving away from full-time teaching, but I’ll still miss it (except for the grading on the weekends). I always enjoyed creating educational artifacts, such as tables, graphs, diagrams, animations, and simulations. I’d like to keep these skills current, so I may have something to offer you. I will be your teaching assistant for a small fee ($5?) yet to be decided. I’ll create an educational artifact for you, provide any source files to you, and at a time we agree on, I’ll release the work under an open license so that all teachers can benefit.

The only catch is that I’m doing this for fun. I can’t take every project, so I might start with the ones I think are most interesting. If I can’t get to the project, maybe someone else could. The first thing I probably need to do is set up a website of wishes and wish-granters, but meanwhile, you can just reply to this post. Teachers, would you be interested in something like this? What kinds of educational projects can you envision? Interested bystanders, would you volunteer your skills?

What can I do for you?

Here’s a brief list of my skills.

Subject Matter

  • Physics
  • Mathematics

Diagrams and Tutorials

Need the perfect diagram and can’t find it? I’ll make it better.

  • (Xe)LaTeX (PDF/PNG)
  • PGF/TikZ (PDF/PNG)
  • Asymptote (PDF/PNG)
  • D3.js (HTML)


Need a small interactive website, simulation, or program?

  • HTML/CSS/JavaScript
  • Node.js
  • Meteor
  • Python
  • GlowScript
  • Haskell
  • Ruby
  • C++
  • GeoGebra
  • Easy Java/JavaScript Simulations
  • PhET simulations (HTML5 sims modified to support inquiry)

Data Science

Need to analyze that concept inventory or unit test? Scrub the student information (replaced with teacher-chosen student IDs), and I can find the story that your data whispers.

  • R
  • Pandas
  • Julia
  • Excel (Ugh!)
  • Google Sheets

Something else?

I have other skills too, probably. Just ask. I would love to hear from you.


Learning About Atoms

The Science PLC at our school is considering what students should know about atoms in 8th and 9th grade science classes (including Physics First).  Just recently, Amber (Strunk) Henry posted on Twitter:

This is my attempt to arrange the ideas.

Map of the Territory of Things to Know

Next Generation Science Standards (NGSS)

Here are the progressions found in Appendix E of the standards.  I do digress into talk of matter and substance when it supports later understanding of atoms.  I’ve expanded these to list ideas explicitly and separately.

ESS1.A The universe and its stars

  • (Grades 9-12) Light spectra from stars are used to determine their characteristics, processes, and lifecycles. Solar activity creates the elements through nuclear fusion. The development of technologies has provided the astronomical data that provide the empirical evidence for the Big Bang theory.
    • Excited atoms/molecules emit light of particular frequencies and wavelengths (collectively called the emission spectrum of an atom/molecule).
    • The frequencies and wavelengths of light emitted by atoms/molecules depend on the structure of the atom/molecules.
    • Atoms/molecules absorb light at particular frequencies and wavelengths (collectively called the absorption spectrum of an atom/molecule).

PS1.A Structure of matter (includes PS1.C Nuclear processes)

  • (Grades K-2) Matter exists as different substances that have observable different properties. Different properties are suited to different purposes. Objects can be built up from smaller parts.
    • Matter can be made of different substances.
    • Substances have many properties, each with their own uses.
    • Objects are made of smaller parts.
  • (Grades 3-5) Because matter exists as particles that are too small to see, matter is always conserved even if it seems to disappear. Measurements of a variety of observable properties can be used to identify particular materials.
    • Indivisible particles of matter are too small to see.
    • Measurements of properties characterize substances.
  • (Grades 6-8) The fact that matter is composed of atoms and molecules can be used to explain the properties of substances, diversity of materials, states of matter, phase changes, and conservation of matter.
    • Molecules are made of atoms.
    • Matter is made of atoms and molecules.
    • Different atoms and molecules explain different substances.
    • Atoms and molecules behave differently in different states of matter.
    • Atoms and molecules change their qualitative behavior at phase transitions.
    • Matter is conserved because atoms are not destroyed in physical and chemical processes.
  • (Grades 9-12) The sub-atomic structural model and interactions between electric charges at the atomic scale can be used to explain the structure and interactions of matter, including chemical reactions and nuclear processes. Repeating patterns of the periodic table reflect patterns of outer electrons. A stable molecule has less energy than the same set of atoms separated; one must provide at least this energy to take the molecule apart.
    • An individual atom has structure explained by electromagnetic and nuclear interactions.
    • The structure of the atom explains:
      • arrangement of atoms into molecules
      • chemical reactions
      • nuclear processes
      • trends in periodic table
    • Energy is required to remove electrons from an atom.
    • Energy is required to break molecular bonds.

PS1.B Chemical reactions

  • (Grades K-2) Heating and cooling substances cause changes that are sometimes reversible and sometimes not.
  • (Grades 3-5) Chemical reactions that occur when substances are mixed can be identified by the emergence of substances with different properties; the total mass remains the same.
    • Mass is conserved in chemical reactions.
    • Measurement of properties of substances identifies when chemical reactions have taken place.
  • (Grades 6-8) Reacting substances rearrange to form different molecules, but the number of atoms is conserved. Some reactions release energy and others absorb energy.
    • Chemical reactions result in different molecular arrangements of atoms.
  • (Grades 9-12) Chemical processes are understood in terms of collisions of molecules, rearrangement of atoms, and changes in energy as determined by properties of elements involved.
    • Chemical reactions occur when molecules collide and atoms rearrange.
    • Changes in energy during a chemical reaction depend on properties of the atoms involved.

Let me know if you think I’ve forgotten anything here!

AAAS Science Assessment

The AAAS has a great website under the auspices of Project 2061 that lists ideas and misconceptions related to Atoms, Molecules, and States of Matter.

Arnold B. Arons. Teaching Introductory Physics

Arons identifies four lines of evidence necessary to build an early quantum model of the atom:

  1. Bright line spectra of gases.  This requires understanding of how accelerated charged particles can emit light, how charged particles can absorb light.  It should include the Balmer-Rydberg formulae for hydrogen.
  2. Radioactivity
  3. Size of atoms (electron cloud and nuclear).  Evidence from multiple sources.
  4. Photoelectric effect and photon concept

How should this knowledge be arranged?

TODO: I’d like to work on a Learning Landscape, Knowledge Packet, or Learning Progression synthesizing these sources, but that will have to be added later.

Models of Atoms

  1. BB Model of Atoms and Molecules (hard, indivisible balls)
    • Needed to explain phases of matter.
  2. Dalton Model of Atoms (hard, indivisible balls that can combine)
    • Needed to explain chemical reactions in integer ratios.
  3. Plum Pudding Model / Thompson Model (negatively charged electrons embedded in a positively charged medium)
    • Needed to explain static electricity.
  4. Planetary Model / Rutherford Model
    • Needed to explain the Geiger-Marsden gold foil experiments.
  5. Bohr Model / Rutherford-Bohr Model
    • Needed to explain why electrons don’t fall into the nucleus after radiating EM waves.
    • Needed to explain the Rydberg formula.
  6. Bohr-Summerfeld Model
    • Needed to allow elliptical orbits
  7. Schrödinger Model / Electron Cloud Model
    • Needed to explain more satisfactorily why electrons don’t fall into the nucleus after radiating EM waves
    • Needed to explain atoms with more than one electron
    • Needed to explain periodic table trends
    • Needed to explain spectra of large Z atoms
    • Needed to explain intensities of spectral lines
    • Needed to explain Zeeman effect from magnetic fields
    • Needed to explain spectral splittings (fine, although this could be done with Klein-Gordan equation and is really a hack onto the non-relativisticSchrödinger equation, and hyperfine) Note: I need to go back to my QM books on this one.
  8. Swirles/Dirac Model
    • Needed to explain spectra of large Z atoms better
    • Needed to explain the color of gold and cesium
    • Needed to explain chemical and physical property differences between the 5th and 6th periods
  9. Quantum Field Theory Model
    • Needed for ???
  10. Nuclear Shell Model / Goeppert-Mayer et al. Model
    • Needed to explain radioactivity

About the biggest controversy is disagreement over the need to teach pseudo-historically these models.  This is leaving out all the really bad ones.  However, the terrible picture that society has adopted as the meme for atom (see below) affects student perceptions of the atom.

Stylised Lithium Atom by Indolences, Rainer Klute on Wikimedia Commons.  Note that this is only a model, based loosely on the Bohr model.  Also, these 3 electrons couldn’t all occupy the same circular orbit.

It would be nicer if students came into classrooms with the following conception of an atom.

Helium Atom QM by Yzmo from Wikimedia Commons. This is a much better rendition of the electron cloud but might be as bad for the nucleus. However, it is nice that it shows scale.

Physics teachers tend to like the Bohr model in that it can quickly (although magically) explain the Rydberg formula.  However, there are many reasons to dislike the Bohr model.

Classroom Experiments

TODO: What classroom experiments or simulations could help students to progress in their knowledge of atoms?

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

How I got the Scribbler 2 robot with a Fluke 2 board working on Mac OS X



  1. Start-up Guide (shipped with Scribbler II) – Play around with the demo modes. I couldn’t try the USB interface because I didn’t buy the USB-to-serial converter…oops. However, the Fluke 2 board obviates the need for it.
  2. Follow the instructions at the IPRE wiki to download and install Calico (a framework for programming).
  3. Follow the instructions at http://quantumprogress.wordpress.com/2013/06/19/modeling-and-robots-on-a-mac/ to set up bluetooth.
  4. Do a firmware upgrade.
    I had trouble creating a custom tty port to use, so I left my original “/dev/tty.Fluke2-05FB-Fluke2” and in the terminal typed:

    cd /dev
    sudo ln -s tty.Fluke2-05FB-Fluke2 tty.scribbler

    You will need to replace “05FB” with whatever you see when you type ls -l /dev/tty.* from Terminal.app. Then I could type in the StartCalico.app:

    from firmwareupgrade import *
    upgrade("fluke", port="/dev/tty.scribbler")
  5. Initialize the robot (from the StartCalico.app using Myro):
    from Myro import *

Now, according to Matt Greenwolfe, the next step is to write low-level movement routines that use the wheel encoders.