Learning to use math in science

+

Learning to use math in science Edward Redish

Department of Physics University of Maryland USA

October 19, 2015 University 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


+ 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

October 19, 2015 University of Washington * 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, 2015 University 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( ) =

!Fq0Enet

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 = 6πµRvFi = CρR

2v2

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 = 8µLπR4

⎛⎝⎜

⎞⎠⎟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 =FAnet

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

October 19, 2015 University of Washington * 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.


* 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 = B0yk

* 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 = IL2B0 j

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, 2015 University 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...


1r2dr

2

3 2

∫ = 1y2 + 9

dy1

3

∫ + 1x2 +1

dx1

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


Knowledgeconstructed

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)

Calculationcan 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

IntroPhysicscontext

Except for the first, each of these often involve math.

+ An a meta-epistemological result: Coherence


CoherenceMultiple 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










+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


Knowledgeconstructed

from experience and perception (p-prims)

is trustworthy


Information from an authoritative

source can be trusted


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 situationsHeuristics

Living organisms are complex and require multiple

related processes to maintain life

Life is complex(system thinking)

Function implies structure

IntroBiologycontext

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)


+ 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.




Satisfaction(smile,

fist pump)

+


IntroPhysicscontext

IntroBiologycontext



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

B A 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.


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 = − dUdr

+ A conflict between the epistemological stances of instructor and student can make teaching more difficult.









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.


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