A description of the analytical tools developed in Physics Education Research for understanding students use of and difficulties with mathematics as used in science.

1. Analyzing
the
role
of
math
in
scien3fic
thinking
Edward
F.
(Joe)
Redish
Department
of
Physics
University
of
Maryland
6/24/15
MathBench
Workshop,
College
Park
1

2. Outline
• Mee3ng
each
other
• The
structure
of
mathema3cal
modeling
• Analy3c
tools
for
studying
epistemology
• Mathema3cs
as
a
way
of
knowing:
The
epistemology
of
math
in
science
• Analyzing
epistemology:
Its
role
in
learning
science
– Epistemological
resources
– Epistemological
framing
– Epistemic
games
– Epistemological
stances
6/24/15
MathBench
Workshop,
College
Park
2

3. GeUng
to
know
the
group:
Some
ques3ons
1. Introduc3ons:
Who
are
we
and
what
classes
are
we
working
on?
(individual)
2. Why
do
we
think
math
is
important
for
biology?
(Discuss
in
groups,
summarize
on
flip
charts
in
A
FEW
SENTENCES)
3. What
are
our
goals
for
the
development
of
specific
mathema3cal
skills
in
our
classes?
(Discuss
in
groups,
summarize
on
flip
charts
–
AS
MANY
AS
POSSIBLE)
6/24/15
MathBench
Workshop,
College
Park
3

4. My
background
• Ph.D.
in
theore3cal
nuclear
physics
–
25
years
as
a
prac3cing
researcher
• Switched
fields
to
Physics
Educa3on
Research
–
25
years
as
a
prac3cing
researcher
• My
educa3on
research
has
focused
on
– Teaching
and
learning
scien3fic
reasoning
– Cogni3ve
modeling
of
student
thinking
– Epistemology
– Use
of
math
in
science
• Past
5
years:
Building
NEXUS/Physics
–
an
introductory
physics
class
designed
to
mesh
with
and
serve
the
curriculum
of
a
bio
or
pre-­‐med
student
6/24/15
MathBench
Workshop,
College
Park
4

5. A
two-­‐step
analy3c
approach
• The
structure
of
mathema3cal
modeling
• How
we
think
about
and
use
mathema3cal
modeling.
6/24/15
MathBench
Workshop,
College
Park
5

6. Modeling
mathema3cal
modeling
• Scien3fic
thinking
is
all
about
epistemology
–
deciding
what
we
know
and
how
we
know
it.*
• In
physics,
mathema3cs
has
become
3ghtly
3ed
with
our
epistemology
beginning
in
~1700.
• As
a
result,
physics
is
a
good
place
to
start
studying
the
role
of
math
in
science.
It
plays
a
significant
role
in
all
our
professional
instruc3on,
even
in
the
introductory
classes
(not
always
in
a
good
way,
however).
• We
don’t
just
calculate
with
math,
we
build
knowledge
with
it
and
think
with
it.
6/24/15
MathBench
Workshop,
College
Park
6
* Karsai & Kampis, BioScience 60:8 (2010) 632-638.

7. Mathema3cal
modeling:
A
structural
analysis
6/24/15
MathBench
Workshop,
College
Park
7
• Oien
these
all
happen
at
once
–
intertwined.
(not
meant
to
imply
an
algorithmic
process)
• In
physics
classes,
oien
the
top
element
is
stressed
and
the
remaining
elements
are
oien
shortchanged.

8. In
physics,
math
integrates
with
our
physics
knowledge
and
does
work
for
us
• Lets
us
carry
out
chains
of
reasoning
that
are
longer
than
we
can
easily
do
in
our
head
by
using
formal
and
logic
represented
symbolically
– Calcula3ons
– Predic3ons
– Summary
and
descrip3on
of
data
• Our
math
codes
for
conceptual
knowledge
– Func3onal
dependence
– Packing
concepts
– Epistemology
6/24/15
MathBench
Workshop,
College
Park
8

9. Func3onal
dependence
• Fick’s
law
of
diffusion
• The
Hagen-­‐Poiseuille
equa3on
for
fluid
flow
6/24/15
MathBench
Workshop,
College
Park
9
Δr2
= 6DΔt
ΔP =
8µL
πR4
Q

10. Packing
Concepts:
Equa3ons
as
a
conceptual
organizer
6/24/15
MathBench
Workshop,
College
Park
10

aA =

FA
net
mA
Force
is
what
you
have
to
pay
amen3on
to
when
considering
mo3on
What
mamers
is
the
sum
of
the
forces
on
the
object
being
considered
The
total
force
is
“shared”
to
all
parts
of
the
object
These
rela3ons
are
independently
true
for
each
direc3on.
You
have
to
pick
an
object
to
pay
amen3on
to
Forces
change
an
object’s
velocity
Total
force
(shared
over
the
parts
of
the
mass)
causes
an
object’s
velocity
to
change

11. Mathema3cs
as
a
way
of
knowing:
Epistemology
• Math
in
science
is
not
just
for
describing
what
we
see
in
a
compact
way.
• Math
is
epistemological
–
it’s
a
way
of
genera3ng
new
knowledge.
6/24/15
MathBench
Workshop,
College
Park
11

12. Analyzing
Epistemology:
Dissec3ng
its
role
in
learning
science
• Understanding
student
behavior
is
considerably
more
complex
than
figuring
out
“what
they
know
and
what
they
don’t.”
• When
we
pay
amen3on
to
the
combina3on
of
dynamic
mental
response
and
the
role
of
epistemology,
a
lot
of
student
responses
look
different
–
and
more
complex
–
than
just
"they
don't
get
it”
or
even
“they
have
a
wrong
mental
model
(misconcep3on)”.
6/24/15
MathBench
Workshop,
College
Park
12

13. A
lot
of
what
students
do
makes
more
sense
if
we
consider
the
epistemological
reasoning
they
use.
• The
resources
students
bring
to
bear
in
a
classroom
are
affected
by
their
epistemological
expecta0ons
What
is
the
nature
of
the
knowledge
that
we
are
learning
and
what
do
we
have
to
do
to
learn
it?
• Student
responses
are
complex
and
dynamic.
• The
key
is
understanding
what
epistemological
resources
they
have
and
expect
to
use.
6/24/15
MathBench
Workshop,
College
Park
13

14. Analy3c
tools
for
studying
epistemology
6/24/15
MathBench
Workshop,
College
Park
14
• Epistemological
resources
(e-­‐resources)*
– Generalized
categories
of
“How
do
we
know?”
warrants.
• Epistemological
framing*
– The
process
of
deciding
what
e-­‐resources
are
relevant
to
the
current
task.
(NOT
necessarily
a
conscious
process.)
• Epistemic
games**
– A
coherent
procedure
for
assis3ng
in
crea3ng
or
recovering
knowledge
in
par3cular
circumstances.
• Epistemological
stances
– A
coherent
set
of
e-­‐resources
ac3vated
together
*Bing & Redish, Phys. Rev. ST-PER 5 (2009) 020108; 8 (2012) 010105.
**Tuminaro & Redish, Phys. Rev. ST-PER 3 (2007) 020101.

15. Intro
Physics
contextEpistemological
resources
6/24/15
MathBench
Workshop,
College
Park
15
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.
Mathematics and
mathematical manipulations
have a regularity
and reliability and are
consistent across different
situations.
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)
Mathematical
consistency
(If the math is the same,
the analogy is good.)
Value of
toy models

16. Intro
Biology
contextEpistemological
resources
6/24/15
MathBench
Workshop,
College
Park
16
Knowledge
constructed
from experience
and perception (p-prims)
is trustworthy
Physical intuition
(experience & perception)
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) Teleology
justifies
mechanism
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)

17. Epistemological
Resources:
6/24/15
MathBench
Workshop,
College
Park
17
• These
groupings
of
resources
are
labeled
as
“Intro
Bio”
and
“Intro
Physics.”
• This
is
to
indicate
that
these
are
epistemological
resources
commonly
perceived
by
students
as
relevant
in
their
intro
classes
in
these
subjects.
• Professionals
in
both
fields
tend
to
use
both
of
these
sets
resources
(though
with
different
distribu3ons
and
depending
on
sub-­‐field).

18. 1. Epistemological resources:
Example from NEXUS/Physics –
Recitation: Why do bilayers form?
6/24/15
MathBench
Workshop,
College
Park
18
Prompt:
Which
term
wins?

19. Prompt:
…explain
how
phospholipids
can
spontaneously
self-­‐
assemble
into
a
lipid
bilayer…why
this
par3cular
shape?
6/24/15
MathBench
Workshop,
College
Park
19
Hollis:
I
mean,
in
terms
of
like
bio
and
biochem,
the
reason
why
it
forms
a
bilayer
is
because
polar
molecules
need
to
get
from
the
outside
to
the
inside
...
so
you
need
a
polar
environment
inside
the
cell.
But
I
don't
know
how
that
makes
sense
in
terms
of
physics.
So...
Cindy:
So
like
what
I'm
saying
is,
you
have
to
have,
like
if
it's
hydrophobic
and
interac3ng
with
water,
then
it's
going
to
create
a
posi3ve
Gibb's
free
energy,
so
it
won't
be
spontaneous.
So,
in
this
case,
you
have
the
hydrophobic
tails
interac3ng
with
whatever's
on
the
inside
of
the
cell,
which
is
mostly
water,
right?
Hollis:
Or
other
polar
molecules.
Cindy:
Yeah,
other
polar
molecules.
It's
going
to
have,
and
that's
bad
...
That's
a
posi3ve
Gibb's
free
energy...[proceed
to
unpack
in
terms
of
posi3ve
(energe3c)
and
nega3ve
(entropic)
contribu3ons
to
the
equa3on.]
Hollis:
Yes.
See,
you
explained
it
perfectly
...
Cause
I
was
thinking
that,
but
I
wasn't
thinking
it
in
terms
of
physics.
And
you
said
it
in
terms
of
physics,
so,
it
matched
with
bio.

20. Disciplinary
epistemologies
6/24/15
MathBench
Workshop,
College
Park
20
• “in
terms
of
bio,
the
reason
why
it
forms
a
bilayer
is
because
polar
molecules
need
to
get
from
the
outside
to
the
inside”
• “
if
it’s
hydrophobic
and
interac3ng
with
water,
then
it's
going
to
create
a
posi3ve
Gibb's
free
energy,
so
it
won't
be
spontaneous
and
that’s
bad..”

21. 6/24/15
MathBench
Workshop,
College
Park
21
Intro
Physics
context
Intro
Biology
context
Physical mapping
to math
(Thinking with math)
Teleology
justifies
mechanismSatisfaction
(smile,
fist pump)
Interdisciplinary
coherence
seeking
“Interdisciplinary
coherence”
–
• Coordinated
resources
from
intro
physics
and
biology
• Blended
context
• Posi0ve
affect

22. Epistemological
framing
• Depending
on
how
students
interpret
the
situa3on
they
are
in
and
their
learned
expecta3ons,
they
may
not
think
to
call
on
resources
they
have
and
are
competent
with.
• This
can
take
many
forms:
– “I’m
not
allowed
to
use
a
calculator
on
this
exam.”
– “It’s
not
appropriate
to
include
diagrams
or
equa3ons
in
an
essay
ques3on.”
– “This
is
a
physics
class.
He
can’t
possibly
expect
me
to
know
any
chemistry.”
• This
can
coordinate
strongly
with
affec3ve
responses
6/24/15
MathBench
Workshop,
College
Park
22

23. 2.
Epistemological
Framing:
Example
from
Biology
6/24/15
MathBench
Workshop,
College
Park
23
• Biology
III:
Organismal
Biology
– A
principles-­‐based
class
that
structures
the
tradi3onal
“forced
march
through
the
phyla”
of
a
biological
diversity
class.
• Some
of
the
principles:
– Common
ancestry
(deep
molecular
homology)
– Individual
evolved
historical
path)
(divergent
structure-­‐func3on
rela3onships)
– Constrained
by
universal
chemical
and
physical
laws.
• Uses
Group
Ac3ve
Engagement
(GAE)
lessons
(including
math!)
“Todd the biologist”

24. Ashley’s
response
to
the
use
of
math
in
Org
Bio
6/8/14
Gordon
Conference
24
I
don’t
like
to
think
of
biology
in
terms
of
numbers
and
variables….
biology
is
supposed
to
be
tangible,
perceivable,
and
to
put
it
in
terms
of
lemers
and
variables
is
just
very
unappealing
to
me….
Come
3me
for
the
exam,
obviously
I’m
going
to
look
at
those
equa3ons
and
figure
them
out
and
memorize
them,
but
I
just
really
don’t
like
them.
I
think
of
it
as
it
would
happen
in
real
life.
Like
if
you
had
a
thick
membrane
and
tried
to
put
something
through
it,
the
thicker
it
is,
obviously
the
slower
it’s
going
to
go
through.
But
if
you
want
me
to
think
of
it
as
“this
is
x
and
that’s
D
and
this
is
t”,
I
can’t
do
it.
Discussing
the
use
of
Fick’s
Law
in
controlling
diffusion
through
a
membrane
of
different
thicknesses.

25. Another
response
of
a
student
to
math
in
Org
Bio
6/8/14
Gordon
Conference
25
The
limle
one
and
the
big
one,
I
never
actually
fully
understood
why
that
was.
I
mean,
I
remember
watching
a
Bill
Nye
episode
about
that,
like
they
built
a
big
model
of
an
ant
and
it
couldn’t
even
stand.
But,
I
mean,
visually
I
knew
that
it
doesn’t
work
when
you
make
limle
things
big,
but
I
never
had
anyone
explain
to
me
that
there’s
a
mathema3cal
rela3onship
between
that,
and
that
was
really
helpful
to
just
my
general
understanding
of
the
world.
It
was,
like,
mind-­‐boggling.
The
small
wooden
horse
supported
on
dowels
stands
with
no
trouble.
When
all
dimensions
are
doubled,
however,
the
larger
dowels
break,
unable
to
support
the
weight.
Watkins & Elby, CBE-LSE. 12 (2013) 274-286

26. Ashley’s
dynamic
switch
6/24/15
MathBench
Workshop,
College
Park
26
“Biological
authen0city”
–
• Coordinated
math
and
intui0on
• In
a
biological
context
• Posi0ve
affect
• Significant
value
for
understanding
biology

27. Epistemic
games:
A
poten3ally
useful
tool
• Epistemic
game:
A
structured
ac3vity
usable
for
approaching
a
variety
of
knowledge
building
tasks
and
problems.
It
has
an
entry
point,
rules,
an
end
point.
– Making
a
list
– Compare
and
contrast
– Cost-­‐benefit
analysis
– Mechanism
analysis
(3me,
space,
rela3onships)
– Recursive
plug-­‐and-­‐chug
6/24/15
MathBench
Workshop,
College
Park
27
Collins & Ferguson, Educ. Psychol. 28 (1993) 25
Bing & Redish, Phys. Rev. ST-PER 5 (2009) 020108;
Bing & Redish, Phys. Rev. ST-PER 8 (2012) 010105
Tuminaro & Redish, Phys. Rev. ST-PER 3 (2007) 020101.

28. 3.
Example
from
NEXUS/Physics:
Filling
in
missing
epistemic
games.
6/24/15
MathBench
Workshop,
College
Park
28
When
a
small
organism
is
moving
through
a
fluid,
it
experiences
both
viscous
and
inerCal
drag.
The
viscous
drag
is
proporConal
to
the
speed
and
the
inerCal
drag
to
the
square
of
the
speed.
For
small
spherical
objects,
the
magnitudes
of
these
two
forces
are
given
by
the
following
equaCons:
Fv = 6πµRv
Fi = CρR2
v2
For
a
given
organism
(of
radius
R)
is
there
ever
a
speed
for
which
these
two
forces
have
the
same
magnitude?

29. Many
students
were
seriously
confused
and
didn’t
know
what
to
do
next.
6/24/15
MathBench
Workshop,
College
Park
29
• “Should
I
see
if
I
can
find
all
the
numbers
on
the
web?”
• “I
don’t
know
how
to
start.”
– “Well,
it
says
‘Do
they
ever
have
the
same
magnitude?’
How
do
you
think
you
ought
to
start?
• “Set
them
equal?”
– “OK.
Do
it.”
• “I
don’t
know
what
all
these
symbols
mean.”
– “Well
everything
except
the
velocity
are
constants
for
a
parCcular
object
in
a
parCcular
situaCon.”
• “Oh!
So
if
I
write
it
....
Av
=
Bv2...
Wow!
Then
it’s
easy!”

30. A
useful
epistemic
game
6/24/15
MathBench
Workshop,
College
Park
30

31. 6/24/15
MathBench
Workshop,
College
Park
31
4.
Example
from
Algebra-­‐Based
Physics
showing
how
e-­‐games
interact
with
framing.
• The
following
problem
was
given
at
the
end
of
the
first
semester
of
an
introductory
class.
– EsCmate
the
difference
in
air
pressure
between
the
floor
and
the
ceiling
in
your
dorm
room.
(Note:
you
may
take
the
density
of
air
to
be
1
kg/m3.)
• A
student
working
on
this
problem
framed
the
task
incorrectly
and
got
trapped
playing
the
wrong
game.

32. 6/24/15
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d
0
p = p0 + ρgd
pceiling = p0
pfloor = p0 + ρgh
pfloor − pceiling = ρgh ≈ 1
kg
m3
⎛
⎝
⎜
⎞
⎠
⎟ 10
N
kg
⎛
⎝
⎜
⎞
⎠
⎟ 3 m( ) = 30
N
m2
= 30 P

33. 6/24/15
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An
inappropriate
game
• One
student
decided
she
needed
an
equa3on
for
pressure:
She
chose
PV
=
nRT.
• She
decided
she
needed
the
volume
for
the
room.
• She
decided
it
must
be
1
m3.
(?!)
• She
maintained
that,
despite
TA’s
hint,
“I
think
you’ll
agree
with
me
this
is
an
es3ma3on
problem.”
• She
decided
if
it
wasn’t
1
m3,
then
the
prof
probably
gave
the
value
in
a
previous
HW.

34. Recursive
plug-­‐and-­‐chug
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35. Epistemological
stances:
The
“go-­‐to”
e-­‐resource
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• Both
students
and
faculty
may
have
developed
a
pamern
of
choosing
par3cular
combina3ons
of
e-­‐resources.
• The
epistemological
stances
naturally
taken
by
physics
instructors
and
biology
students
may
be
drama3cally
different
–
even
in
the
common
context
of
a
physics
class.

36. 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
6/24/15
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5.
Epistemological
stances:
An
example
from
NEXUS/Physics

37. How
two
different
professors
explained
it
when
students
got
stuck.
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Workshop,
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• Remember!
(or
here)
• At
C,
the
slope
of
the
U
graph
is
posi3ve.
• Therefore
the
force
is
nega3ve
–
towards
smaller
r.
• So
the
poten3al
represents
an
amrac3ve
force
when
the
atoms
are
at
separa3on
C.

F = −

∇U F = −
dU
dr
This figure was
not actually drawn
on the board
by either instructor.

38. Wandering
around
the
class
while
students
were
considering
the
problem,
I
got
a
good
response
using
a
different
approach.
6/24/15
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Workshop,
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• Think
about
it
as
if
it
were
a
ball
on
a
hill.
Which
way
would
it
roll?
Why?
• What’s
the
slope
at
that
point?
• What’s
the
force?
• How
does
this
relate
to
the
equa3on
F = −
dU
dr

39. I
conjecture
that
a
conflict
between
the
epistemological
stances
of
instructor
and
student
make
things
more
difficult.
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Calculation
can be trusted
By trusted
authority
Physical mapping
to math
(Thinking with math)
Physical intuition
(experience & perception)
Physical mapping
to math
(Thinking with math)
Mathematical
consistency
(If the math is the same,
the analogy is good.)
Physics
instructors
seem
more
comfortable
beginning
with
familiar
equa3ons
–
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
intui3ons.

40. Teaching
physics
standing
on
your
head
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• For
physicists,
math
is
the
“go
to”
epistemological
resource
–
the
one
ac3vated
first
and
the
one
brought
in
to
support
intui3ons
and
results
developed
in
other
ways.
• For
biology
students,
the
math
is
decidedly
secondary.
Structure/func3on
rela3onships
tend
to
be
the
“go
to”
resource.
• Part
of
our
goal
in
teaching
physics
to
second
year
biologists
is
to
improve
their
understanding
of
the
poten3al
value
of
mathema3cal
modeling.
This
means
teaching
it
rather
than
assuming
it.

41. Conclusion
/Discussion
• Considering
the
way
we
teach
math
and
how
students
respond
using
our
four
analy3c
tools
(e-­‐resources,
e-­‐framing,
e-­‐games,
&
e-­‐stances)
appears
to
help
and
give
us
insight
into
teaching
math
to
biology
students
in
a
physics
class.
• Might
such
analyses
be
of
any
use
for
using
math
in
biology?
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