Learning to use math in science

+
Learning to use math in science
Edward Redish
Department of Physics
University of Maryland USA
October 19, 2015University of Washington

+ Outline
n  Mathematics is an essential competency
for learning physics
n  Math as the language of physics:
Mathematics in a physics context is not the
same language as it is in a pure math context
n  Making meaning with mathematics
n  Mathematics as a way of knowing in physics
n  Case study:
Implications for interdisciplinary instruction
n  Conclusions

+
Mathematics
A critical competency
for learning physics
October 19, 2015 University of Washington

+ Mathematics in physics
and scientific epistemology
n  Scientific thinking is all about epistemology
– deciding what we know and how we know it.
n  In physics, mathematics became closely tied
with our epistemology beginning in ~1700.
n  Mathematics plays a significant role in physics
instruction, even in introductory classes.
(Not always in a good way, however.)
n  We don’t just calculate with math, we “make
meaning” with it, think with it, and use it to create
new physics.

+ Unpacking ...
n  As physics students learn the culture of physics
and grow from novice to expert,
many have trouble bridging what they learn in math
with how we use math in physics .
n  Many instructors are distressed and confused
when our students succeed in math classes
but fail to use those same tools effectively in physics.
n  For those of us who practice physics,
either as teachers or researchers, our knowledge
of physics is deeply blended with mathematics.
n  We may find it hard to unpack our blended knowledge
and understand what students find difficult.

+... Using Physics Education Research
n  My research group has been studying maths
in physics at the university level for ~20 years
in many contexts.
n  Engineering students in introductory physics
n  Physics majors in advanced classes
n  Biology students in introductory classes, both with mixed populations
and in a specially designed class for bio and pre-med students.
n  Data (mostly qualitative)
n  Videos of problem-solving interviews
n  Ethnographic data of students solving real HW problems
in real classes, either alone or in groups.
n  Some multiple-choice questions on exams or with clickers.
n  Theory
n  Resources Framework* – built on ideas from education, psychology,
neuroscience, sociology, and linguistics research
* Redish, “How should we think about how our students think?”,
Am. J. Phys. 82(2014) 537-551.

+
Different languages
Math in physics is not the same
as math in math

+ We often say that “mathematics
is the language of physics”, but...
n  What physicists do with maths
is different from
what mathematicians do with it.
n  Mathematicians and physicists load
meaning onto symbols differently and
this has profound implications.*
* Redish & Kuo, “Language of physics, language of math”,
Sci. & Ed 25:5-6 (2015) 561-590.

+
Our processing of equations
is more complex than in a math class.
n  We link our equations with physical
systems — which adds information
on how to interpret the equation
n  We use symbols
that carry extra information
not otherwise present
in the mathematical structure
of the equation

+
Examples:
Units & SigFigs
n  What are we doing
when we specify
the “units” of a quantity?
n  We are identifying our symbol not just as a number but as
a measurement – that brings physical properties along
with it.
n  What about significant
figures?Why do we bother
talking about them now
that we have calculators?
n  When we multiply 5.42 x 8.73 in a 6th grade arithmetic
class we want something different from
what we want when we are measuring the area of a (5.42
cm) x (8.73 cm) sheet of silicon.
In math terms, we are determining
which irreducible representation
of the 3-parameter scaling group
SxSxS it transforms by.
Since every measurement has
an uncertainty, it propagates
to the product, leaving many
digits shown by the calculator
as “insignificant figures”.

+
Example: Functional dependence
A very small charge q0 is placed at a point somewhere
in space. Hidden in the region are a number of electrical
charges. The placing of the charge q0 does not result
in any change in the position of the hidden charges.
The charge q0 feels a force, F.
We conclude that there is an electric field at the point where
q0 is placed that has the value E0 = F/q0.
If the charge q were replaced by a charge –3q0,
then the electric field at the point would be
a) Equal to –E0
b) Equal to E0
c) Equal to –E0/3
d) Equal to E0/3
e) Equal to some other value not given here.
f) Cannot be determined from the information given.October 19, 2015University of Washington
Nearly half of 200 students
chose this answer.
Given in lecture in algebra-based physics .

+ Huh?
n  The topic had been discussed in lecture
and students had read text materials
showing a mathematical derivation.
n  When asked, most students could cite
the result, “The electric field is independent
of the test charge that measures it.”
!
E
!
r( )=
!
Fq0
E net
q0

+
What’s going on?
n  Many students treated the physics
as a pure math problem:
If A = B/C what happens to A
if C is replaced by -3C?
n  They ignored the fact that F here
is not a fixed constant, but represents
the force felt by charge q0 and
therefore depends on the value of q0.

+ Example: Lots of parameters
When a small organism is moving through a fluid,
it experiences both viscous and inertial drag.
The viscous drag is proportional to the speed
and the inertial drag to the square of the speed.
For small spherical objects, the magnitudes
of these two forces are given by the following equations:
Fv = 6πµRv
Fi = CρR2
v2
For an organism (of radius R) is there ever a speed
for which these two forces have the same magnitude?
Given as a discussion question in a class for introductory physics for
bio students. (A year of calculus was a pre-requisite for the class.)

+ Many students were seriously confused
and didn’t know what to do next.
n  “Should I see if I can find all the numbers
on the web?”
n  “I don’t know how to start.”
n  “Well, it says ‘Do they ever have the same magnitude?’
How do you think you ought to start?
n  “Set them equal?”
n  “OK. Do it.”
n  “I don’t know what all these symbols mean.”
n  “Well everything except the velocity are constants
for a particular object in a particular situation.”
n  ....[concentrating for almost a minute...]
“Oh! So if I write it .... Av = Bv2... Wow! Then it’s easy!”

+ Making meaning
with mathematics
It’s done differently
in physics and math!

+
The structure
n  Our examples suggest that the critical
difference in maths as pure
mathematics and maths in a physics
context is the blending of physical
and mathematical knowledge.
n  How does this work?

+ The structure of mathematical modeling:
•  Often these all happen at once – intertwined.
(the diagram is not meant to imply an algorithmic process)
•  In physics classes, processing is often stressed
and the remaining elements shortchanged or ignored.

+
In physics, math integrates with
our physics knowledge and does work for us
n  It lets us carry out chains of reasoning
that are longer than we can do in our head,
by using formal and logical reasoning
represented symbolically
n  Calculations
n  Predictions
n  Summary and description of data
n  Development of theorems and laws
n  Our math also codes for conceptual knowledge
n  Functional dependence
n  Packing concepts
n  Epistemology

+ Functional dependence
n  Fick’s law of diffusion
n  The Hagen-Poiseuille equation
for fluid flow in a cylindrical pipe
Δr2
= 6DΔt
ΔP =
8µL
πR4
⎛
⎝⎜
⎞
⎠⎟ Q
From a course in physics for biology and life-science students.
These functional dependences have profound implications for biology.

+ Packing Concepts into Equations:
Equations as a conceptual organizer

aA =

FA
net
mA
Force is what
you have to pay
attention to when
considering motion
What 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.

+
A theoretical structure
for analyzing these ideas:
n  In physics, we “make physical meaning”
with maths. How does that work?
n  In physics, maths are a critical piece
of how we decide we know something
(our epistemology). How does that work?

+ What does “meaning”
mean?
Some advice from cognitive science

+What does “meaning” mean?
n  Draw on cognitive semantics – the study of
the meaning of words in the intersection
of cognitive science and linguistics. Some key ideas:
1.  Embodied cognition –
Meaning is grounded in physical experience.
2.  Encyclopedic knowledge –
Webs of associations build meaning.
3.  Contextualization – Meaning is constructed
dynamically in response to perceived context.
4.  Blending – New knowledge can be created by
combining and integrating distinct mental spaces.

+ Mathematical meaning in math
n  One way embodiment allows math
to feel meaningful is with symbolic forms*:
associating symbol structure with relations
abstracted from (embodied) physical experience
n  Parts of a whole: ☐ = ☐ + ☐ + ☐ ...
n  Base + change: ☐ = ☐ + △
n  Balancing: ☐ = ☐
n  A second way maths build meaning is
through association via multiple
mathematical representations
n  Equations
n  Numbers
n  Graphs
* Sherin, Cog. & Instr, 19 (2001) 479-541.

+
Mathematical meaning in physics
n  Physicists tend to make meaning
of mathematical symbology by associating
symbols with physical measurements.
n  This allows connections to physical experience
and associations to real world knowledge.
n  Examples:
n  Symbolic quantities in physics often have units,
meaning they are different types of quantities
that cannot be added or equated. (time ≠ space)*
n  Quantities may be considered as variables or constants
depending on what problem is being considered.October 19, 2015University of Washington
* However, there is context dependence!
(How far is it from Seattle to Olympia?
About an hour.)

+ Example:
A vector line integral*
n  A square loop of wire is
centered on the origin and
oriented as in the figure.
There is a space-dependent
magnetic field
n  If the wire carries a current, I,
what is the net force on the
wire?
!
B = B0yˆk
* Griffiths, Introduction to Electrodynamics (Addison-Wesley, 1999).
From a video of two physics majors working together
to solve a problem in a third-year E&M course.

+ Two paths to a solution
n  Student B
n  I’m pretty sure they
want us to do the
vector line integral
around the loop.
n  It’s pretty
straightforward.
n  The sides do cancel,
but I get the top and
bottom do too, so the
answer is zero.
!
F = I d
!
L ×
!
B
"
#∫
n  Student A
n  Huh! Looks pretty
simple – like a
physics 1 problem.
n  The sides cancel
so I can just do
on the top and bottom
where B is constant.
n  Gonna get
!
F = I
!
L ×
!
B
!
F = IL2
B0
ˆj
What do you think happened next?

+
No argument!
n  Student A immediately folded his cards
in response to student B’s more mathematically
sophisticated reason and agreed she must be right.
n  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.
n  The students expectations that the knowledge
in the class was about learning to do complex math
was supported by many class activities.

+ Analyzing mathematics
as a way of knowing
Epistemological resources

+ Example 3:
A rocket is taken from a point A
to a point B near a mass m.
Consider two(unrealistic) paths
1 and 2 as shown. Calculate
the work done by the mass
on the rocket on each path.
Use the fundamental definition of the work
not potential energy. Mathematica may or may not be
helpful. Feel free to use it if you choose (though it is not
necessary for the calculations required).
From a video of three physics majors working together to solve a problem in
a third-year Math Methods course.The problem is intended to show how the
path independence of work comes about for conservative forces.

+
What’s happening?
n  During this discussion three students
are talking at cross purposes.
n  They are each looking for different kinds
of “proofs” than the others are offering.
n  They use different kinds of reasons
(warrants*) to support their arguments.
n  Eventually, they find mutual agreement
– after about 15 minutes of discussion!
October 19, 2015University of Washington* S. Toulmin, The Uses of Argument (Cambridge UP, 1958)

+ S1: what’s the problem? You should get a different
answer from here for this... (Points to each path on
diagram)
S2: No no no
S1: They should be equal?
S2: They should be equal
S1: Why should they be equal?
This path is longer if you think about it.
(Points to two-part path)
S2: Because force, err, because
work is path independent.
S1: Well, OK, well is this— what was the answer
to this right here?( Points to equation)
S2: Yeah, solve each integral numerically
S1: Yeah, what was that answer? ...
S1: Matching
physical
intuition
with the
math
S2: Relying
on a
remembered
theorem

+ I’ll compare it to the number of...OK,
the y-one is point one five.
S1: I, just give me the, just sum those up... I just want the whole
total... this total quantity there... (Points to integrals again)
S2: Oh, it was point four.
S3: No, that’s the other one [direct path].
S1: you gave it to me before,
I just didn’t write it down.
S3: Oh I see, point, what, point six one eight
S1: See, point six one eight, which is what I said, the work done
here should be larger
S2: No, no no, no no no
S3: the path where the x is changing
S2: Work is path independent.
S1: How is it path independent?
S2: by definition
S3: Somebody apparently proved this before we did...
1
r2
dr
2
3 2
∫ =
1
y2
+ 9
dy
1
3
∫ +
1
x2
+1
dx
1
3
∫
S3: Trusting
the
mathematical
calculation

+
Analytic tools for studying
epistemology
n  Epistemological resources*
n  Generalized categories
of “How do we know?” warrants.
n  Epistemological framing**
n  The process of deciding what e-resources
are relevant to the current task.
(NOT necessarily a conscious process.)
n  Epistemological stances
n  A coherent set of e-resources
often activated together
*Bing & Redish, Phys. Rev. ST-PER 5 (2009) 020108; 8 (2012) 010105.
** Hammer, Elby, Scherr & Redish, in Transfer of Learning (IAP, 2004)

+ Careful!
n  These are NOT intended to describe distinct mental
structures. Rather, we use them to emphasize different
aspects of what may be a unitary process: activating a
subset of the knowledge you have to a particular situation.
n  Warrant – focuses on a specific argument, typically using
particular elements of the current context. (“Since the
path integral of a conservative force is path independent,
these two integrals will have the same value.”)
n  Resource – focuses on the general class of warrant
being used. (“You can trust the results in a reliable
source such as a textbook.”)
n  Framing – focuses attention on the interaction between
cue and response. (You decide you need to carry out a
calculation.)

+Some physics e-resources
Knowledge
constructed
from experience
and perception (p-prims)
is trustworthy
Algorithmic
computational steps
lead to a trustable
result
Information from
an authoritative
source
can be trusted
A mathematical symbolic
representation faithfully
characterizes some feature
of the physical or geometric
system it is intended
to represent.
Highly simplified
examples can yield
insight into complex
mathematical
representations
Physical intuition
(experience & perception)
Calculation
can be trusted
By trusted
authority
Physical mapping
to math
(Thinking with math)
Value of
toy models
There are powerful
principles that can be
trusted in all
situations
Fundamental
laws
Intro
Physics
context
Except for the first, each of these often involve math.

+
An a meta-epistemological
result: Coherence
Coherence
Multiple ways of
knowing applied to
the same situation
should yield the same
result

+
Consider previous examples
in this language
E = F/q
q à -3q
Calculation
can be trusted
Physical mapping
to math
Calculation
can be trusted
Physical mapping
to math
By trusted
authority
Calculation
can be trusted

+Epistemological framing
n  Depending on how students interpret
the situation they are in, and on their learned
expectations, they may not think to call on
resources they have and are competent with.
n  This can take many forms:
n  “I’m not allowed to use a calculator on this exam.”
n  “It’s not appropriate to include diagrams or equations
in an essay question.”
n  “This is a physics class. He can’t possibly expect me
to know any chemistry.”
n  This can coordinate strongly
with affective responses.
n  This becomes particularly important when students
and faculty choose different ways of knowing.

+ The language of epistemology
n  This language provides
nice classifications of reasoning
– both what we are trying to teach
and what students actually do.
n  But can it provide any guidance
for instructional design?

+ Case Study: Implications
for interdisciplinary instruction
Lessons from NEXUS/Physics

+ NEXUS/Physics: An introductory
course for life science majors
n  Create prototype materials
n  An inventory of open-source
instructional modules that
can be shared nationally .
n  Interdisciplinary
n  Coordinate instruction
in biology, chemistry, physics, and maths.
n  Competency based
n  Teach generalized scientific skills so that
it supports instruction in the other disciplines.
* Redish et al., NEXUS Physics: An interdisciplinary repurposing of
physics for biologists, Am. J. Phys. 82:5 (2014) 368-377.
http://www.nexusphysics.umd.edu

+Epistemological resources
Knowledge
constructed
from experience
and perception (p-prims)
is trustworthy
Physical intuition
Information from
an authoritative
source
can be trusted
By trusted
authority
The historical fact of
natural selection leads
to strong structure-
function relationships
in living organisms
Many distinct
components of
organisms need to be
identiﬁed
Comparison of related
organisms yields
insight
Learning a
large vocabulary
is useful
Categorization
and classiﬁcation
(phylogeny)
There are broad
principles that govern
multiple situations
Heuristics
Living organisms
are complex and
require multiple
related processes to
maintain life
Life is complex
(system thinking)
Function implies
structure
Intro
Biology
context
In intro bio, typically none of these often involve math.
Redish & Cooke, Learning each other’s ropes, CBE-LSE. 12 (2013) 175-186.

+ Missing!
n  These are critical components
woven deeply into every physics class!
n  These are not only weak or missing
in many bio students, they see them
as contradicting resources they value.
Value of
toy models
Fundamental
laws
Life is complex
(system thinking)
Function implies
structure

+
This demands
some dramatic changes!
n  We cannot take for granted that students
will value toy models. We have to justify their use.
n  We cannot take for granted that students
will understand or appreciate the power of principles
like conservation laws (energy, momentum, charge).
We have to teach the idea explicitly.
n  We have to create situations in which students learn
to see the value of bringing in physics-style thinking
with biology-style thinking in order to gain
biological insights. (“Biologically authentic” examples)

+Disciplinary epistemological framing:
Discussion – Why do bilayers form?
Prompt:
How can phospholipids
spontaneously self-assemble
into a lipid bilayer?

+Disciplinary epistemologies
n  Hollis: “in terms of bio, the reason why it forms
a bilayer is because polar molecules need to get
from the outside to the inside”
n  Cindy: “if it’s hydrophobic and interacting with water, then
it's going to create a positive Gibb's free energy, so it
won't be spontaneous and that’s bad..[proceeds to
unpack in terms of positive (energetic) and negative
(entropic) contributions to the
Gibbs free energy equation.]”
n  Hollis: I wasn't thinking it
in terms of physics.
And you said it in terms
of physics,
so it matched with biology.
Physical mapping
to math
Function implies
structure
Satisfaction
(smile,
ﬁst pump)

+
Intro
Physics
context
Intro
Biology
context
Physical mapping
to math
Teleology
justiﬁes
mechanismSatisfaction
(smile,
ﬁst pump)
Interdisciplinary
coherence
seeking
“Interdisciplinary coherence” –
•  Coordinated resources from
intro physics and biology
•  Blended context
•  Positive affect

+ Epistemological stances –
“Go-to” e-framings
n  Both students and faculty may have
developed a pattern of choosing
particular combinations of e-resources.
n  The epistemological stances first
chosen by physics instructors and
physics students may be dramatically
different – even in the common context
of a physics class.

+
The figure shows the PE of two interacting atoms as a function
of their relative separation. If they have the total energy shown
by the red line, is the force between the atoms when they are
at the separation marked C attractive or repulsive?
C
BA
Total energy
r
Potential
Energy
Example:
Epistemological stances
Given as a discussion question in a class for introductory
physics for bio students. (A year of calculus was a pre-
requisite for the class.)

+ How two different professors
explained it when students got stuck.
October 19, 2015
University of Washington
n  Remember! (or here)
n  At C, the slope of the U graph
is positive.
n  Therefore the force is negative –
towards smaller r.
n  So the potential represents
an attractive force when
the atoms are
at separation C.

F = −

∇U F = −
dU
dr
This figure was
not actually drawn
on the board
by either instructor.

+Wandering around the class while students
were considering the problem, I got
a good response using a different approach.
n  Think about it as if it were a ball on a hill.
Which way would it roll? Why?
n  What’s the slope at that point?
n  What’s the force?
n  How does this relate
to the equation
F = −
dU
dr

+ A conflict between the epistemological
stances of instructor and student
can make teaching more difficult.
Calculation
can be trusted
By trusted
authority
Physical mapping
to math
Physical intuition
Physical mapping
to math
Mathematical
consistency
(If the math is the same,
the analogy is good.)
Physics instructors
seem most comfortable
beginning with familiar
equations – which we
use not only
to calculate
with, but to code
and remind us
of conceptual
knowledge.
Most biology students
lack the experience
blending math and
conceptual knowledge,
so they are more
comfortable
beginning with
physical intuitions.

+ Teaching physics
standing on your head
n  For physicists, math is the “go to”
epistemological resource – the one activated first
and the one brought in to support intuitions
and results developed in other ways.
n  For biology students, the math is decidedly
secondary. Structure/function relationships tend
to be the “go to” resource.
n  Part of our goal in teaching physics to second
year biologists is to improve their understanding
of the potential value of mathematical modeling.
This means teaching it rather than assuming it.

+ Mathematics
as a way of knowing
Epistemological resources

+Analytic tools for studying
math in physics
n  The structure of mathematical modeling
n  The conceptual components of blending physical
and mathematical knowledge.
n  Epistemological resources
n  Generalized categories of “How do we know?” warrants.
n  Epistemological framing
n  The process of deciding what e-resources are relevant to
the current task. (NOT necessarily a conscious process.)
n  Epistemological stances
n  A coherent set of e-resources often activated together

+
Conclusion
n  An analysis of how math is used
in physics, including both an unpacking
of what professionals do and an analysis of
how students respond, can give insight into
student difficulties reasoning with math.
n  Such an analysis has implications
for how we understand what our students are
doing, what we are actually
trying to get them to learn, and (potentially)
how to better design our instruction
to achieve our goals.

Learning to use math in science

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