WHY ARE MOORE METHOD COURSES EFFECTIVE?

byAnnie Selden

Arizona State University Tennessee Technological University

John SeldenMathematics Education

Resources Co.

Prepared for the R.L. Moore Legacy Conference

April 5-7, 2000

aselden@asu.edu, selden@tntech.edu

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• Are they effective?• In what ways?• What does effective mean?• Do more students succeed?

• Do students learn more mathematics?

• Do they understand the understand the concepts better?

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Do they learn • to make proofs?• to read proofs?• to make conjectures? • about the culture of mathematics?

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Outside of mathematics• do they think more clearly?

• understand themselves better?

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We will not answer these questions.

We will discuss:• research in mathematics education (which could get answers)

• “transition” or “bridge” courses (which might precede and resemble Moore Method courses).

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Research in mathematics education is

not what it was 30 years ago.

• Accretion view of knowledge

I tell you something. You add it

on to what you already know.• Behaviorism You only look at what can

be directly measured, i.e., non-mental things.

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• Agricultural comparison-type research

Two fields of wheat are compared. One

gets fertilizer. The growth (e.g., output

in number of bushels of wheat) is compared. The difference is

attributed to the fertilizer.

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When applied to two classes with

differing curricula, such studies are

less informative than in agriculture.

Students are not as uniform as wheat

plants.

Why an effect has occurred is

important.

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Current research in mathematics

education has three main perspectives

on knowledge:• It is constructed from old knowledge and developed by adaptation. (assimilation & accommodation -- Piaget)

• It is social before being developed individually. (Vygotsky)

• It is situated, i.e., consists of the way an individual acts in various situations. (Lave, et al)

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These perspectives reject the idea

that simply telling a person something

is always enough to cause her/him to

build useable knowledge (like adding a

page to a book).

They see learning as requiringreflection and are concerned with the

contents of students’ minds -- an idea

incompatible with strict behaviorism.

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Research in mathematics education

often investigates students, teachers, or

classes.• What do they know?• What do they believe about mathematics?

• How do they learn/teach various concepts? (e.g., function, limit, derivative)

• How do they solve (nonroutine) problems?

• How are any of these changed?

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Nowadays, interest tends to be in

• conceptual, rather than procedural, learning, and in

• (nonroutine) problem solving.

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Example of procedural, but not

conceptual, knowledge:• A student can graph y=x3,

but given just the graph, cannot estimate f(-1).

• A student can calculate derivatives of many functions, but cannot sketch the tangent line to f(x)=x2+1 at (1,2).

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Some ResultsTraditionally taught secondary students tend to see mathematics as

procedural and view correctness as

authority- , rather than reasoning-

based.

Proofs in geometry are not seen a

useful in solving out-of-class geometry

problems, and they occasionally ask if

proofs can contain words.

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Constructing new conceptions, e.g.,

function, often involves cycling

through several stages -- action,

process, object.

[Breidenbach, D, Dubinsky, E., Hawks, J., & Nichols,

D., “Development of the process conception of

function,” Educ. Studies in Math. 23 (1992), 247-285.]

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(Nonroutine) problem solving can be

analyzed in terms of one’s:• knowledge base• heuristics• monitoring and control• beliefs.

For example, effective control is acharacteristic of good problem solvers.

[Schoenfeld, A., Mathematical Problem Solving, New

York, NY: Academic Press, 1985.]

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Mid-level undergraduate cannot

validate (determine the correctness of)

simple proofs. E.g. If n2 is a multiple

of 3, then n is a multiple of 3.

[Selden, A. & Selden J., “Can you tell me whether this

is a proof?”, Proceedings of PME-23, Vol. 1, p. 364;

preprint available.]

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Mid-level undergraduates cannot

unpack the logical structure of

statements of calculus theorems, e.g.,

If g is continuous at c and f is continuous at

g(c), then f og is continuous at c.

[Selden, J. & Selden, A., "Unpacking the logic

mathematical statements,” Ed. Studies in Math. 29

(1995), 123-151.]

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Many senior math majors need help

understanding new definitions, e.g.,

A function is called fine if it has a

root (zero) at each integer.

[Dahlberg, R. P. & Housman, D. L., "Facilitating

learning events through example generation," Educ.

Studies in Math. 33, 283-299, 1997]

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Many successful calculus students

cannot solve moderately nonroutine

problems, even when they know the

required information, e.g, Find at least

one solution to the equation 4x3 - x4 = 30 or explain why no such solution exists.

[Selden, J., Selden, A. & Mason, A., “Even good

calculus students can’t solve nonroutine problems,” in

J. Kaput and E. Dubinsky (Eds.), Research Issues in

Undergraduate Mathematics Learning: PreliminaryAnalyses and Results, MAA Notes No. 33, 1994,

19-26.]

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What might one ask about Moore Method courses or students?

• How do Moore Method students compare with others on conceptual grasp or the ability to prove theorems?

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• What about their (nonroutine) problem-solving ability?

• Can they validate (determine the correctness of) someone else’s proof?

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• Do some Moore Method teachers help students by keeping them in Vygotsky’s zone of proximal development?

• Can the whole course be seen as a “slow motion” discussion?

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• To what degree can students be seen as apprentices?

• What can be said about the culture of a Moore Method classroom? Do special socio-mathematical norms develop, and does this lead to a distinctive view of mathematics?

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“Transition” or “bridge” courses

By transition courses we mean undergraduate

courses students often take between their

earlier procedural courses (e.g., calculus, diff.

eq.) and later proof-based courses (e.g.,

abstract algebra and real analysis).

Their purpose is to help students learn about

proofs. Occasionally, they include interesting

mathematics not covered in other courses.

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Transition courses often teach various kinds

of logic and set theory before doing much

with proofs. They also cover functions,

equivalence relations, and some other

accessible topics such as number theory.

The logic and set theory is taught abstractly,

not in the context of proofs. It is

decontextualized.

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Transition courses can be effective

in that students have less difficulty

making and reading proofs in later

courses.

But they are not as effective as one

might hope. Many students find them

difficult and do not gain confidence

about making proofs on their own.

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We conjecture that much of the effectiveness

of Moore Method teaching and much of the

difficulty in transition courses can be

explained by the same idea.

What students need to know in order to prove

theorems (aspects of logic, set theory, etc.) is

best learned in the context of proving

theorems.

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Teaching logic in an abstract,decontextualized way (like abstract algebra)

does not automatically provide the ability to

use it in proofs.

Being able to use truth tables to justify that

modus ponens is a valid argument is not the

same thing as the ability to notice, at line 8 of

an argument, that an inference can be drawn

from lines 3 and 6 (depending on modus

ponens).

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The logic component of most transition

courses will not help students know that the

proof of:The sum of two functions is continuous at

a point whenever they both are.should look like:Let a be a number and f and g be

functions. Suppose f and g are continuous at a.. . .Therefore, f+g is continuous at a.

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Annie has been moving her transition

course to a Moore Method type of

course (with group work) with good

results.

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We think such a transition course

would make a good precursor to a

Moore Method course and the two

(taken together) as a sequence would

greatly improve how students viewed

mathematics and what they could do.

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There seems to be no reason a

transition course (with a significant)

Moore Method component could not

be taught to first-year students.

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Teaching such a course very early

(perhaps as a precalculus course)

would probably deepen what

students’ learned in all of their

remaining undergraduate mathematics

courses.