Using math in physics is not the same as using math in math classes

Do you ever feel that your students don’t seem to know the pre-requisite math for your classes, even though they did well in their math classes? Math in physics is different from pure math and learning that can be challenging for students. They have to learn to develop physical concepts and blend them with symbolic math.1 This is harder than you might think.

In my 8 short articles in The Physics Teacher,2 I identify 7 tools students need to learn to build the blend — dimensional analysis, estimation, anchor equations (reading the physics in an equation), toy models, functional dependence, reading the physics in a graph, and telling the story. I explain both why these are so important and why so many students have difficulty with them in physics.

In each article, I suggest a variety of instructional methods you can use to help students learn to use these tools, and, as a result, develop a strong physical intuition and learn to “think with math”. The supplementary materials for each article and the associated submissions to the Living Physics Portal3 offer lots of relevant problems you can use in class and for homework.

1. Using math in physics
Edward Redish
Professor Emeritus
Department of Physics
University of Maryland
July 8, 2024
AAPT Summer Meeting Boston

2. Do we have a problem?
In my 50 years of teaching, I have often found that
my students had difficulty using math in physics.
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3. A problem my life-science students
have trouble with
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4. A problem my life science/pre meds
have trouble with
•What is 3½ divided by ¼?
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5. A problem they don’t have trouble with
•For the faculty-student get together you ordered 4 small pizzas
divided into quarters. The faculty ate two slices.
How many students can have a slice?
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6. A problem my engineering students
have trouble with
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7. A problem my engineers have trouble with
•The Electric field is defined by
"=$/&
where & is the charge on the test (probe) charge.
•If the probe charge & is replaced by a probe with
charge &/3 what happens to E?
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8. A problem my physics majors
have trouble with
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9. A problem my physics majors
have trouble with*
•A square loop of wire is centered
on the origin and oriented as in
the figure. There is a space-
dependent magnetic field
•If the wire carries a current, I,
what is the net force on the wire?
7/8/24 AAPT Summer Meeting Boston

!
B=B
0
y
ˆ
k
* Griffiths, Introduction to Electrodynamics (Addison-Wesley, 1999).9

10. Two paths to a solution
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nStudent B
nI’m pretty sure they
want us to do the
vector line integral
around the loop.
nIt’s pretty
straightforward.
nThe sides do cancel,
but I get the top and
bottom do too, so the
answer is zero.

!
F=Id
!
L×
!
B
"
#∫
nStudent A
nHuh! Looks pretty
simple – like a
physics 1 problem.
nThe sides cancel
so I can just do
on the top and bottom
where B is constant.
nGonna get

!
F=I
!
L×
!
B
What do you think happened next?
From a video of
two physics majors
working together
to solve a problem
in a third-year
E&M course.
10
⃗"=2%&!'"̂)

11. No argument!
•Student A immediately folded their cards
in response to student B’s more mathematically
sophisticated reason and agreed they must be right.
•Both students valued (complex) mathematical reasoning
(where they could easily make a mistake)
over a simple (and compelling) argument
that blended math and physics reasoning.
•The students’ expectations that the knowledge
in the class was about learning to do complex math
was supported by many class activities.
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12. What’s missing in all three examples
•In each case, students fail to “make the blend”
— combining information
about mathematical symbols
with knowledge about
what those symbols represent.
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13. What’s going on?
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14. Math in science is not the same as math in math*
•Math in math classes tends to be about numbers
(and the structure of abstract relationships).
Math in science is not.
•Math in math classes tends to use a small number of symbols
in constrained ways. Math in physics uses lots of symbols
in different ways.
•The symbols in science classes often carry meaning
that changes the way we interpret the quantity.
•In introductory math, equations
are almost always about solving and calculating.
In physics it’s often about
explaining and making meaning.
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When we do a derivation
of an equation we are
giving an explanation.
Intro physics students
rarely understand that
this is what we are doing.* Redish and Kuo, Phys. Sci. & Educ,, 24:5-6 (2007) 561.

15. Meta-misconceptions
•While some students may lack mathematical skills,
many (in my experience) can “do the math”
when it’s expressed as math.* (At least after a brief review.)
•More serious problems
(not easy to remedy by a review or by math ”drill-and-kill” activities)
are students’ inappropriate expectations about
•the nature of the knowledge they are learning
(epistemological misconceptions)
•the nature of what mathematical symbols
are being used to represent
(ontological misconceptions)
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* Torigoe & Gladding, AJP 79 (2011) 133-140
Meltzer, King, & Jones, many AAPT talks (2016-2020)

16. The key
•The critical difference is that
our symbols do not
just represent numbers.
•In physics, symbols map physical meaning into mathematical symbols that blends together
•our conceptual knowledge of the physical content and
•our structural knowledge of mathematical relationships and processes.
•Making this blend is a non-trivial mental process
that is rarely taught explicitly. How can we teach it?
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17. What can we do about it?
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18. Teaching the blend
•One way to begin to approach teaching the blend
is to be more explicit about identifying and teaching
the core types of reasoning we use in the blend.
•I have created a math-in-science toolbelt
with specific analytic and problem solving strategies.
•I teach these tools (epistemic games*) explicitly .
•Every time I use them (in class or in the text)
students see an icon that appears
to remind them that we are using that tool.
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* Tuminaro and Redish, Phys. Rev. STPER, 3, 020101 (2007).

19. AAPT Summer Meeting Boston
We can teach the blend through
general purpose tools / strategies
Dimensional analysisReading the physics
in a graph
EstimationTelling the story
Functional dependence/
scalingDiagrams
Anchor equationsSpecial and extreme cases
Toy modelsBuilding equations
from the physics
7/8/24 19https://www.compadre.org/nexusph/course/Building_your_mathematical_toolbelt

20. Skills apply to all topics.
•Adding a focus
on developing math skills
doesn’t add new topics.
•It modifies the way we teach
topics we already include.
•It does require us to be “meta”
— to be explicit about tools
since students don’t easily
generalize.
7/8/24 AAPT Summer Meeting Boston20Redish, The Physics Teacher, 59 (2021) 314

21. These math tools are discussed in detail
in a series of 8 papers in The Physics Teacher
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22. Ontological tools:
What kinds of things are our symbols?
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L, r
Δt
m
Q
T
Not numbers!
Measurements!
Units matter!
Dimensional
analysis
Estimation
Without the skills of dimensional analysis and estimation,
the numbers in our problems have no physical meaning!
Redish, The Physics Teacher, 59 (2021) 397; and 525

23. Your white cells help protect you by ingesting harmful
bacteria and destroying them. The bacterium might contain molecules safe
for it, but harmful to the white cell. But the white cell is bigger, so the
density of the harmful molecules will be less.
Suppose the macrophage has a diameter of 20 μm and the bacterium has a
diameter of 1 μm. How will the density in the white cell compare to that in
the bacterium once it has been ingested, broken up,
and distributed throughout the white cell?
Suppose the harmful molecules are on the bacterium
surface (membrane) and migrate to the macrophage’s
surface (membrane) and distribute across it once the
bacterium is devoured. How will the density of molecules on the white cell’s
surface compare to the density on the pathogen's surface?
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24. Reading the physics in a graph
•One of the best ways to help students build the blend
is through interpreting graphs.
•Graphs are a powerful tool for understanding what’s
happening physically, but many students see a graph
as an answer — something the teacher assigned them
to do, not as a sense-making tool.
•Learning how to interpret graphs both physically
and mathematically is something they need to learn.
7/8/24 AAPT Summer Meeting Boston24Redish, The Physics Teacher, 61 (2023) 51

25. A tilted airtrack has a spring at one end. A cart on it is pressed
against the spring and released. The cart bounces up and down.
The graph represents the velocity of the cart as a function of time
starting at the moment of release. Positive is to the left of the diagram.
Which of the letters on the graph can
identify an instant of time
at which the cart is
1.instantaneously not moving
2.in contact with the spring
3.moving down the track
4.at its highest point on the track
5.has an acceleration of zero.
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26. Toy models
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•One of the most powerful tools we use in physics
in learning to make sense of the world with math are toy models.
•We consider the simplest possible example that shows a phenomenon
and beat it into submission — until we understand it completely.
•These are extremely valuable starting points for building more complex
models of realistic situations.
•But biologists and engineers
want to focus on realistic situations.
This leads them to see toy models
as bogus and irrelevant.
•We need to motivate them
and show their value as a starting
point for more realistic modeling.26Redish, The Physics Teacher, 59 (2021) 683

27. Real springs only follow the Hooke's law model
for small displacements around their rest length.
Which graph might represent T (tension –
pos for stretch, neg for squeeze)
vs L (length) for a real spring?
A real spring behaves as follows:
•stretching from its rest length,
it obeys Hooke's law for a small stretch.
•As you stretch it further, it gets stiffer
as the coils begin to bend.
•Eventually it straightens out into a long straight
wire which is very hard to stretch.
•If you keep pulling harder, the wire suddenly
stretches easily and breaks.
•If you try to compress it, the coils get pushed
together and you can squeeze very hard without
getting much compression.
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T
L
TT
LL
T
L
TT
LL

28. Anchor equations
•Anchor equations provide stable starting points
for thinking about whole blocks of physics content.
•They are the central principles that provide a foothold
— a starting point for organizing our understanding
of an entire topic.
•They code for conceptual knowledge
•They are a good starting point for unpacking
other relevant knowledge
•They are a good starting point for solving problems
7/8/24 AAPT Summer Meeting Boston28Redish, The Physics Teacher, 59 (2021) 599

29. An example: Kinematics concepts (blended)
•The average velocity is given by the change in position
(How far did you move?) divided by the time interval
(How long did it take to do it?).
•The average acceleration is given by the change in velocity
(How much did it change?) divided by the time interval
(How long did it take to do it?).
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<"> =∆'
∆( " =)'
)(
<*> =∆"
∆( * =)"
)(

30. Unpacking
the anchor equations
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Using these equations,
problems are solved by:
1.Identifying relevant quantities
in the physical problem
(mapping physics to symbol)
2.Calling on relevant math
concepts and matching them
to the identified physical values
3.Identifying knowns and
unknowns and manipulating
to get solvable equations
4.Solving the problem

31. Packing Concepts into Equations:
Equations as a conceptual organizer
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
a
A
=

F
A
net
m
A
Force is what
you have to pay
attention to when
considering motionWhat matters is
the sum of the forces
on the object
being considered
The total force
is “shared” to
all parts of
the object
These stand for 3 equations
that are independently
true for each direction.
You have to pick
an object to pay
attention to
Forces change
an object’s
velocity
Total force (shared over
the parts of the mass) causes
an object’s velocity to change
When we just write “F=ma”
our students often miss the rich
set of conceptual associations
hidden in the equations
and mis-use them.
31

32. Where can I get resources to
teach Using Math in Physics?
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33. 7/8/24 AAPT Summer Meeting Boston33
Lots of additional materials are available
at the Living Physics Portal
Each tool has a module
in the Portal with lots of
solved problems from
across the curriculum.

34. Problem collection
•A collection of problems
designed to help students
develop math-in-science skills
is at the NEXUS/Physics website.
(Solutions in the Living Physics
Portal and in autograded ExpertTA.)
•http://compadre.org/nexusph/
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35. Commercial Venues
•TopHat
•Living Physics — An online
introductory physics text (for life
science students) that stresses
building the math/physics blend.
•ExpertTA
•NEXUS Physics — The Problem
Collection imported into an
autograded online environment
Royalties from these products go to the AAPT to support the operation of the Living Physics Portal.
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36. •The slides for this talk are available
at my homepage.
•Search “Joe Redish Presentations”.
•Thanks for your attention.
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37. References
7/8/24 AAPT Summer Meeting Boston37
•Language of physics, language of math: Disciplinary culture and dynamic epistemology, E. F. Redish and E.
Kuo,Science & Education,24:5-6(2015-03-14) 561-590. doi:10.1007/s11191-015-9749-7
•Elements of a Cognitive Model of Physics Problem Solving: Epistemic Games, J. Tuminaro and Redish,
Phys. Rev. STPER, 3, 020101 (2007).
•Using math in physics - Overview, E. F. Redish,The Physics Teacher59(2021) 314-318.
•Using math in physics - 1. Dimensional analysis,E. F. Redish,The Physics Teacher59(2021)
397-400.
•Using math in physics - 2. Estimation,E. F. Redish,The Physics Teacher,59(2021) 525-528.
•Using math in physics - 3. Anchor equations,E. F. Redish,The Physics
Teacher,59(2021) 599-604.
•Using math in physics - 4. Toy models,E. F. Redish,The Physics Teacher,59 (2021) 683-688.
•Using math in physics - 5. Functional dependence,E. F. Redish,The Physics Teacher,60(2022)
18-21.
•Using math in physics - 6. Reading the physics in a graph, E. F. Redish,The Physics
Teacher,61(2023) 651-656.
•Using math in physics - 7. Telling the story, E. F. Redish.The Physics Teacher62(2024) 5-11.