MathBench workshop

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

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


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)


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


A two-‐step analy3c approach

•  The structure of mathema3cal modeling

• How we think about and use mathema3cal modeling.


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.

Mathema3cal modeling: A structural analysis


•  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.

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


Func3onal dependence

•  Fick’s law of diffusion

•  The Hagen-‐Poiseuille equa3on for fluid flow


Δr2 = 6DΔt

ΔP = 8µLπR4

Q

Packing Concepts: Equa3ons as a conceptual organizer


aA =FAnet

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

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.


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)”.


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.


Analy3c tools for studying epistemology


•  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.

IntroPhysicscontextEpistemological 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.

Mathematics and mathematical manipulations

have a regularityand reliability and are

consistent across different situations.

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)

Mathematical consistency

(If the math is the same, the analogy is good.)

Value of toy models

IntroBiologycontextEpistemological 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) Teleology justifies

mechanism

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)

Epistemological Resources:


•  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).

1. Epistemological resources: Example from NEXUS/Physics – Recitation: Why do bilayers form?


Prompt: Which term wins?

Prompt: …explain how phospholipids can spontaneously self-‐

assemble into a lipid bilayer…why this par3cular shape?


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.

Disciplinary epistemologies


•  “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..”


IntroPhysicscontext

IntroBiologycontext



Teleology justifies

mechanismSatisfaction(smile,

fist pump)

Interdisciplinary coherence

seeking

“Interdisciplinary coherence” – •  Coordinated resources from

intro physics and biology •  Blended context •  Posi0ve affect

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


2. Epistemological Framing: Example from Biology


•  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”

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.

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

Ashley’s dynamic switch


“Biological authen0city” – •  Coordinated math and intui0on •  In a biological context •  Posi0ve affect •  Significant value for

understanding biology

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


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.

3. Example from NEXUS/Physics: Filling in missing epistemic games.


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 = 6πµRvFi = CρR

2v2

For a given organism (of radius R) is there ever a speed for which these two forces have the same magnitude?

Many students were seriously confused and didn’t know what to do next.


•  “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!”

A useful epistemic game



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.


d

0

�

p = p0 + ρgd

pceiling = p0

pfloor = p0 + ρgh

pfloor − pceiling = ρgh ≈ 1 kgm3

⎛ ⎝ ⎜

⎞ ⎠ ⎟ 10 N

kg⎛

⎝ ⎜

⎞

⎠ ⎟ 3 m( ) = 30 N

m2 = 30 P


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.

Recursive plug-‐and-‐chug


Epistemological stances: The “go-‐to” e-‐resource


•  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.

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


5. Epistemological stances: An example from NEXUS/Physics

How two different professors explained it when students got stuck.


•  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.

Wandering around the class while students were considering the problem, I got

a good response using a different approach.


•  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 = − dUdr

I conjecture that a conflict between the epistemological stances of instructor and student make things more difficult.


Calculationcan be trusted







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.

Teaching physics standing on your head


•  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.

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?


MathBench workshop

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