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

Analyzing velocity-vs-timer-reading graphs

Update 2013-06-07: These are some notes I wrote over a month ago but never published. Since this work got me thinking about a how to align my instruction with SBG objectives and a later post I’m about to publish, it seemed like now or never.

In trying to get better at teaching graphical representations of uniformly accelerated motion, I tried something similar to Kelly O’Shea’s paradigm laboratory for the Constant Acceleration Particle Model (CAPM). Being not so brilliant at teaching myself, I think I injected a little too much of myself into it, but it was clear from a “pre-test” (review of Constant Velocity Particle Model graphs going into this unit) that students were confusing velocity and position, didn’t remember much about how to analyze velocity-vs-timer-reading graphs, so we needed to do some review. Part of this is the insanely long time between CVPM and CAPM, since we went ETM⟶CVPM⟶MTM⟶BFPM⟶CAPM⟶UBFPM. If I do this again, I will need to include more model-based reasoning problems that incorporate CVPM throughout the previous units. Thus, I found myself trying to come up with a list for us to make to summarize our large number of different scenarios in the CAPM paradigm (class) lab. For instance, in one of my periods, we had (among other things):

Velocity-vs-timer-reading graph for a cart on a ramp

Velocity-vs-timer-reading graph for a cart on a ramp

I had the students annotate each segment of each graph with how the speed was changing (“v↑” or “speeding up”, “v↓” or “slowing down”), which direction it was going (“↑r” or “up the ramp”, “↓r” or “down the ramp”), and what was going on (“” for nothing, “pushed”, “stopped”, etc.). We made important notes besides the graphs that students drew in their lab notebooks, such as “☆The mass doesn’t seem to affect the graph very much.” Then we tried to summarize what we could tell with a table like:

Student-driven exploration of the velocity-vs-timer-reading representation
Feature of the \vec{\boldsymbol v}\text{-}t graph Feature of the motion
point on the graph given by a pair (t,\vec{\boldsymbol v}) a data point, i.e. snapshot, of an object moving with a velocity \vec{\boldsymbol v} at a timer reading t
horizontal position of a point on the graph timer reading t
vertical position of a point on the graph velocity \vec{\boldsymbol v}
vertical distance of a point from the timer reading (t) axis (the \vec{\boldsymbol v}=\vec0 line) how fast it is moving (speed v)
position of a point above(+)/on(0)/below(-) the timer reading (t) axis (the \vec{\boldsymbol v}=\vec0 line) which direction it is going
steepness of the \vec{\boldsymbol v}\text{-}t graph’s slope in the neighborhood of a point how fast the velocity changes
sign of the \vec{\boldsymbol v}\text{-}t graph’s slope in the neighborhood of a point which direction the velocity is changing (somewhat artificial)
the \vec{\boldsymbol v}\text{-}t graph’s slope in the neighborhood of a point is moving [away from (+), parallel to (0), or toward (-)] the timer reading (t) axis (the \vec{\boldsymbol v}=\vec0 line) speeding up (+), maintaining a constant velocity (0), or slowing down (-)
point on the timer reading (t) axis (the \vec{\boldsymbol v}=\vec0 line) where the slope crosses the axis from negative to positive or vice versa. changing direction
? ?

Some classes were able to come up with their own entries. In others, I had to debase myself by suggestion them. In one class, it worked rather well to express my frustration that no one was saying anything, put a student in charge, and tell them that I would be silent while they figured it out. They put much wrong on the board, but they were jumping back to correct things they realized were wrong when trying to identify how to tell some of the other kinematic features; then the bell rang! I tried a similar approach in another class but didn’t give them enough awkward silence before going into silent mode myself. That class didn’t bother checking whether they hypothesized connections actually worked and weren’t given enough time to find out. I jumped into it with a few minutes to go and proceeded to ask them questions to test their statements, destroying all of them. I felt like rain on their parade. I had a hard time even convincing them that their statements were wrong because they could not tell me, given two points on the graph, which was moving faster. I intend to ask more questions like this next year during CVPM. I wanted to cry for them, and I was angry at all (including partly myself) who failed teaching them how to read a graph. Every year we say, “These are smart kids. They should be doing better on the science part of the ACT.” Now we know why. (Thus, I added the first three lines of the table above.)

I know this can’t be the best way to teach this. Engagement was low, and the whole paradigm lab had a demonstrative feel.

Position-vs-timer-reading graph for a cart on a ramp

Position-vs-timer-reading graph for a cart on a ramp

Graphical Methods Summary

Linearizing and analyzing graphs are tricky skills for students to pick up when doing physics.  I enjoyed the discussion of a poster created by Paul Jebb, on the Modeling Physics listserve, but I wanted to create my own.  The result is here:

graph-analysis-beamer-poster-18x24
graph-analysis-beamer-poster-18×24

It would be nice to add the Hestenes’ descriptions of the change in the graph, but perhaps that deserves another poster.

Incidentally, I found the LaTeX beamerposter package quite easy to use.  As usual with LaTeX, the sizes of margins and things required tweaking, but it should work to print out as an 18″ by 24″ poster.

The source code is clearly cribbed from the beamerposter template.  PGF/TikZ is used for the graphs.

Update: You can now find source code for this and other posters in my GitHub repository.

  \documentclass[final]{beamer} % beamer 3.10: do NOT use option hyperref={pdfpagelabels=false} !
 %\documentclass[final,hyperref={pdfpagelabels=false}]{beamer} % beamer 3.07: get rid of beamer warnings
  	\mode {  %% check http://www-i6.informatik.rwth-aachen.de/~dreuw/latexbeamerposter.php for examples
	\usetheme{default}    %% you should define your own theme e.g. for big headlines using your own logos
	\beamertemplatenavigationsymbolsempty
	\definecolor{royalblue}{rgb}{0,0.13725490196078433,0.4}
	\definecolor{royalblueweb}{rgb}{0.25490196078431371,0.41176470588235292,0.88235294117647056}
  	\definecolor{burntorange}{rgb}{0.8,0.3333333333333333,0}
  	\setbeamercolor{frametitle}{fg=blue!80!black}
    	\setbeamertemplate{frametitle} {
	\begin{centering}
    		\vspace{-1.5cm}\textbf{\insertframetitle} \par
	\end{centering}
}
  }
  \usepackage[english]{babel}
  \usepackage[latin1]{inputenc}
  \usepackage{amsmath,amsthm, amssymb, latexsym}
  %\usepackage{times}\usefonttheme{professionalfonts}  % times is obsolete
  \usefonttheme[onlymath]{serif}
  \boldmath
  %\usepackage[orientation=portrait,size=a0,scale=1.4,debug]{beamerposter}                       % e.g. for DIN-A0 poster
  %\usepackage[orientation=portrait,size=a1,scale=1.4,grid,debug]{beamerposter}                  % e.g. for DIN-A1 poster, with optional grid and debug output
  \usepackage[size=custom,width=45.72,height=60.96,scale=1.8,debug]{beamerposter}                     % e.g. for custom size poster (18in x 24in w/ printable 17in x 23in)
  %\usepackage[orientation=portrait,size=a0,scale=1.0,printer=rwth-glossy-uv.df]{beamerposter}   % e.g. for DIN-A0 poster with rwth-glossy-uv printer check
  % ...
  %
  \geometry{margin=.5in}
  \usepackage{array}
  \usepackage{booktabs}
  \newcolumntype{P}[1]{>{\raggedright\large}p{#1}}
  \def\imagetop#1{\vtop{\vspace{-1.5cm}\null\hbox{#1}\vspace{-1.5cm}}}
  \usepackage{tikz}

  \newcommand{\xx}{\textcolor{variable}{x}}
  \newcommand{\yy}{\textcolor{variable}{y}}
  \newcommand{\versus}{vs\ }
  \newcommand{\plotscale}{1.5}
  \newcommand{\plotline}{6pt}
  \newcommand{\formatmm}[1]{\textcolor{royalblue}{\textbf{#1}}}
  \colorlet{plot}{burntorange}
  \colorlet{variable}{blue!80!black}

  \title[Graph Methods]{Graphical Methods Summary}
  \author[Vancil]{Brian Vancil}
  \institute[Sumner]{Sumner Academy of Arts & Sciences}
  \date{2012-04-07}

  \begin{document}
  \begin{frame}{Graphical Methods Summary}
    \defaultaddspace=.25em
    \vspace{-2cm}
    \begin{tabular}{P{.25\linewidth}P{.16\linewidth}P{.30\linewidth}@{\quad}>{\arraybackslash}P{.21\linewidth}} \toprule[.1em]
    \normalsize Mathematical model & \normalsize Graph shape & \normalsize Written relationship & \normalsize To linearize, graph\ldots \\ \midrule[.1em] \addlinespace

    \formatmm{constant}\ \ \  $\yy=b$ &
    \imagetop{\begin{tikzpicture}[scale=\plotscale,domain=0:4,line width=\plotline,smooth]
    \draw[color=plot] plot (\x,3);
    \draw[] (0,4) -- (0,0) -- (4,0);
    \end{tikzpicture}}
    & $\yy$ is constant. & \\ \addlinespace \midrule \addlinespace

    \formatmm{proportional} $\yy=m\xx$ &
    \imagetop{\begin{tikzpicture}[scale=\plotscale,domain=0:4,line width=\plotline,smooth]
    \draw[color=plot] plot (\x,.8*\x);
    \draw[] (0,4) -- (0,0) -- (4,0);
    \end{tikzpicture}}
    & $\yy$ is directly proportional to $\xx$.  & \\ \addlinespace  \midrule \addlinespace

    \formatmm{linear} $\yy=m\xx+b$ &
    \imagetop{\begin{tikzpicture}[scale=\plotscale,domain=0:4,line width=\plotline,smooth]
    \draw[color=plot] plot (\x,.6*\x+1);
    \draw[] (0,4) -- (0,0) -- (4,0);
    \end{tikzpicture}}
    & $\yy$ is linear in $\xx$.  & \\ \addlinespace \midrule \addlinespace

    \formatmm{inversely proportional} $\yy=\frac{a}{\xx}$ &
    \imagetop{\begin{tikzpicture}[scale=\plotscale,domain=.1:4,line width=\plotline,smooth,samples=40]
    \draw[color=plot] plot (\x,.4/\x);
    \draw[] (0,4) -- (0,0) -- (4,0);
    \end{tikzpicture}}
    & $\yy$ is inversely proportional to $\xx$.  & $\yy$ \versus $\frac{1}{\xx}$ \\ \addlinespace \midrule \addlinespace

    \formatmm{power law} $\yy=a\xx^{n}$ &
    \imagetop{\begin{tikzpicture}[scale=\plotscale,domain=0:4,line width=\plotline,smooth,samples=40]
    \draw[color=plot] plot (\x,.25*\x*\x);
    \draw[] (0,4) -- (0,0) -- (4,0);
    \end{tikzpicture}}
    & $\yy$ is proportional to $\xx^{n}$.  & $\yy$ \versus $\xx^{n}$ \\ \addlinespace \midrule \addlinespace

    \formatmm{square root} $\yy=a\sqrt{\xx}$ &
    \imagetop{\begin{tikzpicture}[scale=\plotscale,domain=0:4,line width=\plotline,smooth,samples=40]
    \draw[color=plot] plot (.25*\x*\x,\x);
    \draw[] (0,4) -- (0,0) -- (4,0);
    \end{tikzpicture}}
    %& $\yy^{2}$ is proportional to $\xx$. & Graph $\yy^{2}$ \versus $\xx$ \\ \addlinespace \midrule \addlinespace
    & $\yy$ is proportional to the square root of $\xx$. & $\yy^{2}$ \versus $\xx$ \\ \addlinespace \midrule \addlinespace

    \formatmm{exponential} $\yy=ab^{\xx}$ &
    \imagetop{\begin{tikzpicture}[scale=\plotscale,domain=0:4,line width=\plotline,smooth,samples=40]
    \draw[color=plot] plot (\x,{pow(pow(4,.25),\x)});
    \draw[] (0,4) -- (0,0) -- (4,0);
    \end{tikzpicture}}
    & $\yy$ is exponential in $\xx$. & $\log \yy$ \versus $\xx$ \\  \addlinespace

    \bottomrule[.1em]
    \end{tabular}
  \end{frame}
  \end{document}