This document discusses challenges that life science students face in learning to use mathematics effectively in physics. It argues that some difficulties are due to epistemological misconceptions about the nature of knowledge in physics and ontological misconceptions about mathematical representations of physical concepts. The document proposes teaching students explicit "tools" and "core principles" to help them blend physical meaning with mathematical symbols. It provides examples of how chemical bonds and electric fields can be taught using appropriate ontological metaphors to help students understand abstract mathematical definitions.

1. Albert
Donut boy
Schrödinger’s cat
Macavity
Emily
Cal Tech groupWill
Sheldon
KatherineTeaching IPLS students
to use math in science
Edward F. Redish
Department of Physics (Emeritus)
University of Maryland
AAPT Summer Meeting
July 20, 2020
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2. The NEXUS/Physics
“Gang of 5”
7/20/20 AAPT Summer Meeting 2
Ben Geller
Vashti Sawtelle
Chandra Turpen
Julia Gouvea
Joe Redish
Ben Dreyfus
Todd Cooke
The “Understanding & overcoming barriers
to using math in science” project team
Mark Eichenlaub
Deborah Hemingway
Joe Redish
Many of the ideas expressed here were developed as part of the NEXUS/Physics project
and the Under/Over projects supported by NSF and HHMI.
Work supported in part
by NSF grants 11-22818,
15-04366, 16-24478,
and a grant from HHMI.

3. Learning to use math in science
is important for life science students
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4. Learning to think with math
• The Scientific Foundations for Future Physicians
study (SFFP) lists a number of competencies
for students entering medical school
that their undergraduate experience
should prepare them for. This is the first.
• The Vision & Change study from the biology
research community has a similar goal for
biology majors.
Competency E1
Apply quantitative reasoning and appropriate mathematics
to describe or explain phenomena in the natural world. (SFFP p. 22)
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5. Let us add to our learning goals for an IPLS class
• Students will learn to use math effectively
in science, including being able to:
• Calculate the value of simple expressions and
solve simple equations when the equations are given.
• Work with equations in symbolic form including manipulating
them and interpreting them for physical implications.
• Generate appropriate symbolic equations describing physical
situations for which no explicit numbers are given.
• Reason both qualitiatively and quantitatively about
physical situations using symbolic mathematics.
• ….
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How often do our
IPLS classes ask
students to do
more than the
first?

6. IPLS students often struggle with the math
• IPLS students are well know for having problems
using math effectively in their physics classes.
• There is often active resistance
to using symbolic math.
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These are the
reasons we
sometimes give up
and settle for
only trying to
achieve the first
learning goal.
• This is sometimes assumed to be
due to a lack of math skills
(and sometimes it is*),
but often, something subtler
(and more interesting) is going on.
*D. Meltzer et al.,lots of AAPT talks

7. Why is using math in physics hard
for so many life science students?
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8. Meta-misconceptions
• While some students may indeed 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 that are not easy to remedy
by a review class 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)
7/20/20 AAPT Summer Meeting 8*D. Meltzer et al.,lots of AAPT talks

9. 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 - and the same symbol may have different
meanings.
• 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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10. The blend
• The critical difference in the way we use math in science
is that our symbols do not just represent numbers.
• In physics, symbols represent a mapping of physical
meaning into mathematical symbol 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.
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11. The blend has powerful implications
• Epistemological:
Having to do with the nature of scientific knowledge
• Physical meaning can be represented by and organized
with symbolic equations.
• We can reason about physical systems using abstract mathematical
manipulations.
• If the starting point is physically correct and the math
is done correctly, the conclusion should be physically correct.
• Ontological:
Having to do with what we identify as “things” in science
• The physical world can be most conveniently
described with quantities that have
no direct sensory correlate but are
best defined through mathematical structures.
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12. Can the blend be taught?
• Most professional physicists have never explicitly been
taught about the blend. We learned it over many years —
often (maybe especially for theorists) not until after they
have completed their undergraduate training.
• Is it possible to teach this to our “I-only-have-to-take-
one-year-of-physics” students in other disciplines?
• To do so, we probably need to have a detailed
understanding of what “living in this blend”
involves.
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13. Teaching the blend:
Epistemological issues
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14. An epistemological misconception
• Not only do life science students have little experience
with “reasoning using mathematics”,
they often also have little experience with
“reasoning from powerful principles.”
• In introductory physics there are only a small number
of ideas to learn and these can be deployed
in a wide variety of problems.
• Life science students expect (prefer) to memorize
large numbers of equations and examples rather
than synthesize down to a few powerful methods.
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15. Teaching the blend
• One way to begin to approach teaching the blend
is to be more explicit about identifying and teaching
“the tools” and “the core principles”.
• 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)
they have an icon that appears
to remind them that we are using that tool.
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16. Some tools in the toolbelt
• Dimensional analysis
• Estimation
• Scaling and functional
dependence
• Anchor equations
• Repackaging
• Extreme and
special cases
• Toy models
• Telling the story
• Reading the
physics in a graph
• Building equations
from the physics7/20/20 AAPT Summer Meeting 16

17. Epistemic games (e-games)
• Strategies, not algorithms!
• Students have to learn the rules of the game
(e.g., in “Estimation” you have to be able to justify
why you chose the numbers that you made up and you
can’t keep more than 2 sig figs anywhere in the calculation)
• They have to practice in multiple contexts
to learn how to use and adapt the tool
to different situations.
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18. A thread: Not adding
new content
but changing
how we do
the existing content
• I’m not proposing to add
new content to an already
packed curriculum.
• The idea is to change how
we teach the content and
what we expect the
students to learn.
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19. Negotiating a new
epistemological stance
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20. Epistemologies: The nature
of scientific knowledge
• Knowledge in science
is not just a collection
of facts and processes.
• Much of the value of
scientific knowledge is in
understanding mechanism
(chained causal reasoning)
• — and in synthesis
(how powerful principles
integrate large blocks
of knowledge).
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• Symbolic mathematical equations
help to organize and connect these
different knowledge structures.

21. 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.
• Coding for conceptual knowledge
• A starting point for unpacking
other relevant knowledge
• A starting point for solving problems
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22. An example: Newton’s 2nd law organizes conceptual
knowledge about how force affects motion
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23. 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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< 𝑣 > =
∆𝑥
∆𝑡
𝑣 =
𝑑𝑥
𝑑𝑡
< 𝑎 > =
∆𝑣
∆𝑡
𝑎 =
𝑑𝑣
𝑑𝑡

24. 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.

25. Starting with the anchor equation
• Helps students learn to match the physical situation
with math symbology.
• Get used to the idea that
they can do simple algebraic manipulations.
• Learn that they can set up (create) equations
starting from a physical situation:
they don’t only have to use equations
that someone else gave them
(or that they searched on the web).
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26. Teaching the role of math
in physical ontology
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27. Ontologies
• Complex concepts may require dynamic
or blended ontologies (e.g. wave/particle duality)
• Example 1: Metaphors for energy and chemical bonds*
• New ontologies may be defined mathematically.
Students may not be used to quantities where
the fundamental structure of “what it is” is
mathematical
• Example 2: The conceptual shift
from Electric Force to Electric Fields
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* B. W. Dreyfus, et al., Phys. Rev, ST-PER 10 (2014) 020108

28. Ontological example 1:
The chemical bond
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29. Example: The chemical bond
Many students bring a “piñata” model of a chemical bond.
"But like the way that I was thinking of it, I don't know why,
but whenever chemistry taught us like exothermic, endothermic,
like what she said, I always imagined
like the breaking of the bonds
has like these little [energy]
molecules that float out,
but like I know it’s wrong.
But that's just how
I pictured it from the beginning."
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30. Huh! If it takes energy to break
a chemical bond, where DOES
the energy come from?
(Like we get from food)
Isn’t it chemical energy?
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But the chemists have told them:
It takes an input of energy to break a chemical bond!

31. The student difficulty in this case
in fundamentally ontological
• Energy is an abstract concept, so sense is made of it
through metaphor and analogy.*
• Since from early days, they learn that energy is
conserved, the most natural analogy is matter / stuff.
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* G. Lakoff and M. Johnson, Metaphors We Live By (U. Chicago Press, 1980/2003)
B. W. Dreyfus, et al., Phys. Rev, ST-PER 10 (2014) 020108

32. Ontological metaphors for energy: 1
It works!
Energy can be transformed
Energy is never lost
Energy as a substance
Scherr et al. 2012 science.howstuffworks.com
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Comfort zone issue:
Students
can be uncomfortable
with negative numbers ,
not having a good
physical metaphor
for them.
OK for KE,
but not for PE!

33. phet.colorado.edu
Energy as a vertical location
Objects are at energies
Objects go to higher/lower energy
This works too!
Ground state
Excited states
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Ontological metaphors for energy: 2
For potential you
need a different
metaphor —
now energy
can be negative
and you have
a good blend.
Negative
numbers can
make sense here.

34. 347/20/20 AAPT Summer Meeting

35. 357/20/20 AAPT Summer Meeting

36. The energy comes from the reaction,
not the bond.
• The “bond energy” of a chemical bond
(a positive number) represents
the magnitude of the potential depth
holding the atoms together
(a negative number).
• Energy is released in a chemical
reaction when a shallow bond
(a weak bond)
is replaced by a deeper bond
(a strong bond).
The energy balance
in this reaction is
-4 ⇿ -10 + 6
Amount released in ⇾ rection
Amount needed for ⇽ reaction
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37. Distinct disciplinary perspectives
• Physicists and biologists (and chemists)
make different tacit assumptions.
• Physicists tend to isolate a system to focus
on a particular physical phenomenon
and mechanism.
• Biologists (and chemists) tend to assume the natural
and universal context of life – a fluid environment
(air and water taken for granted).
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Sources of energy
in biological systems:
Glucose and ATP

38. Ontological example:
Electric concepts
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39. Example:
Electric quantities defined mathematically
• Often in physics we define “what something is”
(its ontological classification) by what it is mathematically.
• How it transforms under changes
• Is it a vector or scalar?
• What are its dimensions (units)?
• What is its mathematical character?
• Is it a value or a function?
• Is it a parameter or a variable?
• Many of our constructs are like this
— hard to build a physical mental construct
if you’re not used to doing that sort of thing.
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40. There are four crucial electric concepts
of differing mathematical character
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* Based on “The Big Square”,
a lesson by Vashti Sawtelle and
Ben Geller, adapted from
The Modeling Collaboration, 2012

41. Diagrams that
we use
to construct
each, illustrate
the different
characters
of the quantities.
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42. 7/20/20 AAPT Summer Meeting 42
These equations are not just showing ways
to calculate a new quantity. Each represents
a significant ontological shift — and making
sense of this is a conceptual challenge.
Connection via equations

43. Concluding remarks
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44. Using math in science: Take away messages
• Using math in science is not just using math.
There are a lot of new skills
(blending physical meaning with math)
that students need to learn.
• Some student difficulties are epistemological —
inappropriate expectations about
the nature of knowledge in physics.
• Some student difficulties are ontological —
difficulties with seeing mathematical structure
as carrying physical meaning.
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45. This is the tip of the iceberg!
• There remains much room
for research, both empirical
and theoretical.
• There is much room for
new kinds of curriculum
focused on these issues.
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46. Resources
• I am preparing a series
of modules on
“Using Math in Science”
for the Living Physics Portal
with overview papers on each
e-game and collections of sample
problems for teaching them.
• These should begin to be
available by the middle of August.
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The Living Physics Portal
https://www.livingphysicsportal.org

47. The talk is now open for
questions and comments.
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