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. (pages 5-6)
  2. Mitrea, Dorina (2009). “Mathematics Underlying the Physics First Curriculum: Implications for 8th and 9th Grade Mathematics”.

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 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 Then I could type in the

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

Lab whiteboarding Defend-the-Model Protocol poster

This is a work in progress, and I encourage feedback. The current version of the poster source code will be at GitHub. Here’s the current PDF (lab-whiteboarding-defend-the-model-beamer-poster-18×24-2013-14-2.pdf) if you don’t have LaTeX.

The idea for the poster comes from Frank Noschese, but blame me for the implementation. As a poster it’s not great yet. The example is just there with no prompts for how to use it. A graph would be helpful. These I’ll add later if the basic idea is sound. It could also do with a different example.

I’ll post images of the various incarnations below.

1st attempt

1st attempt

Update 2013-06-16: Based on feedback from Josh Gates, I’ve changed the last conceptual tool.

2nd attempt

2nd attempt

SBG and Multiple Representations in Kinematics

words, table, motion map, equation, postion-time graph, velocity-time graph, acceleration-time graph

Multiple representations in kinematics

This year when how I graded standards starting morphing from what they actually said, I realized that it was time to pick new standards. As I started to imagine how I would change them, I realized that I have no systematic way to do so. In teaching the subject of kinematics (in particular, uniform acceleration, i.e. CAPM), solving a problem involves motion concepts, multiple representations of motion (words, tables, Motion Maps, position-versus-timer-reading graphs, velocity-versus-timer-reading graphs, acceleration-versus-timer-reading graphs, equations, and even some weird ones like position-versus-velocity graphs, etc.), translations between representations, and features of representations that correspond to motion concepts. I can think of at least four broad ways to do this:

  1. Standards correspond to relevant concepts. (N)
  2. Standards correspond to the multiple representations. (n)
  3. Standards correspond to translations between representations. (n(n-1))
  4. Standards correspond to conceptual features of representations. (Nn)

The formulas in parentheses give a rough estimate of the number of possible standards for N motion concepts and n different representations. However, these are not the only considerations. I have more-or-less made the decision that the level of difficulty should increase with standards added and not by increasing the burden of proof or morphing the standards, but this year I occasionally evolved from a mainly qualitative understanding sufficing to requiring a quantitative understanding at the end of a unit. I’d much rather have this built into standards, which should be clear on whether they require a qualitative or quantitative understanding. Which types of standards should bifurcate into qualitative and quantitative flavors? Which overall method of choosing standards is most conducive to student morale and success? Is there any research about which kind of SBG standards are most conducive to learning?

So far I’ve found some prior work on SBG and PER/multiple representations:

  1. The Physics Problem and Standards-Based Grading—Although it’s not what I’m looking for, it does consider the question, “How does one assess solving problems?”
  2. Multiple Representations for NYS Grade 3-8 Common Core Mathematics Tests—Here’s how NY uses MR (multiple representations). The MRs appear as subtypes of standards, if I’m interpreting this correctly.
  3. Kentucky fifth-grade math standards—A few of these involve something close to my “Method 1”.
  4. NCTM’s Principles and Standards for School Mathematics: Representations—Though not about SBG, it illustrates NCTM’s thinking about what it means to represent something.
  5. Multiple Representations of Limits—Now the NSA is spying on teachers?! 😉 Not about SBG, but some included Maryland standards are loosely based on MR.
  6. Information About Standards-Based Grading—Mixes methods in an interesting way.

It looks like this area is a bit weakly developed, and it’s definitely not very researched.

Summary of content

The basic motion data:

  • TIME: The coordinate location is “timer reading” (my compromise between “time”, which I consider too confusing, and “clock reading”, which Arons prefers), and a difference between two points is called a “time interval” (which is typically not located anywhere…oh well).
  • SPACE: The coordinate location is “position”, and a difference between two points is a “displacement”. The length of a path between two points is called “distance”.

The derived data are:

  • RATES: The convention in rates is rates per unit of time. We term the rate at which position changes “velocity” and the rate at which distance changes “speed”. These can each be “average” or “instantaneous” depending on whether they apply to a finite of the path as a whole or the infinitessimal germ around the choosen timer reading, respectively.
  • RATES OF RATES: To keep things simple, we’ll just consider one kind of rate, the rate at which velocity changes, namely, “acceleration”.

Here’s my short summary of the basic features of constantly accelerated motion.
Taking data at uniform time intervals:

  • Position data are quadratic (change in change is constant).
  • Velocity data (from changes in position change) are linear (change is constant).
  • Acceleration data (from changes in changes in position) are constant.

I’ll try to point out common misconceptions as we go, but some relevant from the get-go are:

  • Zero velocity implies zero acceleration, even for an instant.
  • Velocity decreasing means something slows down. (Failure to account for vector character of velocity.)
  • Constant speed implies zero acceleration. (#VectorFail)
  • Distance and displacement are often confused.
  • Position and displacement are often confused when making graphs.
  • Position, velocity, and acceleration are “undifferentiated”.

In crafting standards, I’ll try to limit them to between 5 and 10 standards total per unit, and one of those will be something like:

  • CAPM. I can solve problems involving constant acceleration. (Evidence: Correct quantities found through correct work.)
  • CVPM. I can solve problems involving constant velocity. (Evidence: Correct quantities found through correct work.)

Method 1: Standards correspond to relevant concepts.

  1. CAPM. I can represent the direction of acceleration. (Evidence: Representation of acceleration (+/0/-) is correct across all representations.)
  2. CVPM. I demonstrate how position differs from velocity. (Evidence: Representation of the two are different across all representations.)
  3. CAPM. I can infer velocity data from acceleration data and vice versa. (Evidence: Velocity and acceleration coordinate across all representations.)
  4. CVPM. I can infer position data from velocity data and vice versa. (Evidence: Position and velocity data coordinate across all representations.)
  5. CAPM. I can state whether given motion data exhibits constant acceleration. (Evidence: Explicitly identifies relevant model in CA case.)
  6. CVPM. I can state whether given motion data exhibits constant velocity. (Evidence: Explicitly identifies relevant model in CV case.)
  7. CVPM. I can state whether given motion data exhibits constant position. (Evidence: Explicitly identifies relevant model in CP case.)
  8. CAPM. I can represent constant acceleration motion qualitatively using multiple representations, such as words, diagrams, graphs, tables, and equations. (Evidence: Consistency and correctness of representations.)
  9. CAPM. I can represent constant acceleration motion quantitatively using multiple representations, such as words, diagrams, graphs, tables, and equations. (Evidence: Consistency and correctness of representations.)
  10. CVPM. I can represent constant velocity motion qualitatively using multiple representations, such as words, diagrams, graphs, tables, and equations. (Evidence: Consistency and correctness of representations.)
  11. CVPM. I can represent constant velocity motion quantitatively using multiple representations, such as words, diagrams, graphs, tables, and equations. (Evidence: Consistency and correctness of representations.)

OK, I think I lied. I tried to make it about the concepts and relationships among the concepts, but I did it using representations. Without some sort of representation, it’s way too abstract to judge the student work. Do we just want them to write on a test, “The area under a velocity graph gives displacement”, or do we want them to use this idea to do something?

Pros: This is closer to the spirit of what I really want them to know. It’s not about learning the representations. In this method, those are just tools to communicate and refine one’s ideas while allowing one to make calculations. This method more directly targets misconceptions.

Cons: Grading means that the teacher’s eyes must jump around the page looking at all the representations. What if four are right but one is wrong? Mark it a “not yet”? What if I miss this tiny detail on another student’s paper? I grit my teeth and go back through all the grading to make sure I was consistent. Some of the standards are a little vague. The difficulty of the standards change depending on the type of data students are provided. These are the kind of standards that might make one’s head ache after grading 100 quizzes.

Method 2: Standards correspond to the multiple representations.

Interpretation of a representation is hard to assess without correct explanation or another representation. Therefore, I focus only on creating each representation. I also eschew the good grammar of “constantly accelerated motion” for “constant acceleration motion”. The former seems to me to connote non-zero acceleration (and I have ELL learners), but maybe that’s just me.

  1. CAPM. I can explain constant acceleration motion in words. (Evidence: Correct use of direction words and speeding-up/slowing-down words and other kinematic terminology. Problem: Students trying to say more are more likely to botch things up.)
  2. CVPM. I can explain constant velocity motion in words. (Evidence: Correct use of direction words and speeding-up/slowing-down words and other kinematic terminology. Problem: Students trying to say more are more likely to botch things up.)
  3. CAPM. I can create a table representing constant acceleration motion. (Evidence: Correct quantities (labels, numbers, and units))
  4. CVPM. I can create a table representing constant acceleration motion. (Evidence: Correct quantities (labels, numbers, and units))
  5. CAPM. I can create a Motion Map representing constant acceleration motion.
  6. CVPM. I can create a Motion Map representing constant acceleration motion.
  7. CAPM. I can create a position-versus-timer-reading graph representing constant acceleration motion.
  8. CVPM. I can create a position-versus-timer-reading graph representing constant acceleration motion.
  9. CAPM. I can create a velocity-versus-timer-reading graph representing constant acceleration motion.
  10. CVPM. I can create a velocity-versus-timer-reading graph representing constant acceleration motion.
  11. CAPM. I can create an acceleration-versus-timer-reading graph representing constant acceleration motion.

Here I still have the problem of requiring these representations to be qualitative or quantitative. I want students to be able to do both, but on assessments, I typically want quantitative accuracy.

Pros: This is easy to grade by correctness.

Cons: It does penalize students when they mess up one representation, for they are likely to mess up the others. They would have to reassess to correct the problem, but this is probably OK.

Method 3: Standards correspond to translations between representations.

words, Motion Map, s-t graph, v-t graph, a-t graph, tables, and equations
I left out translation between tables and equations and everything else because we don’t often use them. That still leaves 5 representations for 20 permutations way too many! I left the ones I expect them to do. Since it gets tedious to read a bunch of the same language, I represented it as a digraph:

translations among words, table, motion map, equation, postion-time graph, velocity-time graph, acceleration-time graph

Assessed translations among kinematics multiple representations

The evidence is the faithfulness of the translation.

Pros: It’s very easy to grade this way for simple tasks. You gave students a description of motion and asked them to make a position-vs-timer-reading graph? Dead easy.

Cons: It’s harder to grade for more complex tasks. IF you gave students a description of motion and asked them to make a variety of graphs and a motion map, can you tell the order they used? Did they go from a motion map to a position-vs-timer-reading graph or vice versa? Can you tell? This is enough to disuade me from using this method. Also, propagation of error: If students mess up one translation but correctly translate that wrong representation, is it OK? Or, do you want them to recognize when what they’re doing is not CV or CA motion?

Method 4: Standards correspond to conceptual features of representations.

Some of the features in the digraph below are from a previous blog post on analyzing velocity-vs-timer-reading graphs. Each of the features may be specified as occurring given some other kinematic event, like position when the object has reached a given speed, for instance. Again I have had to restrict myself to what I will assess. I based the arrows between objects on my current practice. I expect students to be able to do more, but I only included the most cognitively demanding tasks for assessment.

Figuring out of the features of position, displacement, distance, timer reading, time interval, instantaneous velocity, average velocity, instantaneous speed, average speed, instantaneous acceleration, average acceleration, speeding up or slowing down from the representations of words, tables, motion maps, equations, position-vs-timer-reading graph, velocity-vs-timer-reading graph, and acceleration-vs-timer-reading graph

Features from Kinematics Multiple Representations

Note that if you were expecting, as I was, the concepts to match up to the features, you were sorely disappointed.

Pros: These are easy to grade when students show work near the representation or by annotating it, which I show my students how to do. It’s also very clear for students what they should be able to do.

Cons: If students don’t organize work well, it can be as hard as the previous method when students translate between representations in an unexpected manner. It also prescribes the method that the student must use. For instance, why do they need to find displacement from a velocity graph directly when they have already used that skill implicitly in creating a position graph? (Answer: For those students, I simply suggest being more explicit in annotating their velocity graphs.)

Questions for the reader

  1. How do you deal with the issue of qualitative versus quantitative correctness? Which do you require of 9th graders? 11th graders?
  2. Is mastering representations of motion tantamount to learning about motions? If not, is it an acceptable proxy, or where does it fall short?
  3. I worry that what I choose for students is consistent with awesomeness. Am I wasting time in thinking about this?