Mathematical Knowledge for Pre-service Teachers: Ways of

Mathematical Knowledge for Pre-service Teachers:

Ways of Understanding and Ways of Thinking

Kien Lim

June 23, 2008

Outline of Presentation

Evidence of Mathematical Unsophistication

Components of Mathematical Knowledge

– Examples of WoU and WoT

Math Knowledge for Pre-service Teachers

– Relating PCK, WoU, and WoT

Potential Avenues for Research

Out of 64 pre-service teachers,only 47% recognized that the answer is 10 minutes

42% obtained 70 minutes, and 11% found other numbers.

Evidence for Mathematical Unsophistication

Direct-Proportional Item

The ratio of the amount of soda in the can to the amount of soda in the bottle

is 4:3. There are 12 fluid ounces of soda in the can, how many fluid ounces

of soda are in the bottle?

(a) 8 fluid ounces

(b) 9 fluid ounces

(c) 15 fluid ounces

(d) 16 fluid ounces

(e) None of the above

Inverse-Proportional Item

The ratio of the volume of a small glass to the volume of a large glass is 3:5.

If it takes 15 small glasses to fill the container, how many large glasses does

it take to fill the container?

(a) 9 glasses

(b) 13glasses

(c) 17glasses

(d) 25 glasses

(e) None of the above

Can Bottle

Small glass

Large glass Container

Pretest

3%

64%

6%

27%

1%

Pretest

53%

9%

4%

24%

10%

Posttest

6%

78%

3%

11%

2%

Postest

42%

13%

2%

40%

2%

138 students


“Many preservice elementary teachers are profoundly mathematically unsophisticated. In other words, they displayed a set of values and avenues for doing mathematics so different from that of the mathematical community, and so impoverished, that they found it difficult to create fundamental mathematical understandings.”

(Seaman & Szydlik, 2007, p. 167)

11 interviewees

2

1

0

3

3

1

5

4

1a. What is the greatest common factor of 60 and 105?

b. What is the least common multiple of 60 and 105?

2. Calculate 0.4 ÷ 0.05.

3. Three Brooke has a 3/4 pound (12 oz.) bag of

M&M’s. If she gives 1/3 of the bag to Taylor,

what fraction of a pound does Taylor receive?

http://www.math.com/homeworkhelp/EverydayMath.html


“Many preservice elementary teachers are profoundly mathematically unsophisticated. In other words, they displayed a set of values and avenues for doing mathematics so different from that of the mathematically community, and so impoverished, that they found it difficult to create fundamental mathematical understandings.”

(Seaman & Szydlik, 2007, p. 167)

“A mathematically sophisticated individual [is a person who] has taken as her own the values and ways of knowing of the mathematical community.” (Seaman & Szydlik, 2007, p. 167)


Interconnected Knowledge of Mathematics

Mathematical Disposition & Beliefs

(i.e. Habits of Mind)


Knowing & Doing Mathematics as a Discipline

“A mathematically sophisticated individual [is a person who] has taken as her own the values and ways of knowing of the mathematical community.” (Seaman & Szydlik, 2007, p. 167)

Two Complementary Components of Mathematical Sophistication

Ways of Understanding

Ways of Thinking

Mental Act

(Harel, 2007, in press)


Two Complementary Subsets of Mathematics

“The first subset is a collection, or structure, of structures consisting of particular axioms, definitions, theorems, proofs, problems, and solutions.

This subset consists of all the institutionalized ways of understanding in mathematics throughout history.”

“The second subset consists of all the ways of thinking, which characterize the mental acts whose products comprise the first set.”

(Harel, in press)


Two Complementary Subsets of Mathematics

Examples of Ways of Thinking

SpecificPredictions

Characteristicof Predicting Act

Predicting

Mental Act of Predicting

Is there a value of x that will make this statement true?(6x – 8 – 15x) + 12 > (6x – 8 – 15x) + 6

“Of course there is. … Let’s see, I was taught to combine like terms. I was taught this (>) is actually an equal sign ...” (11th grader, Calc)

:–9x + 4 = –9x – 2

6 = 06 > 0

“Is there a value for x … may be there isn’t.”

Association-based Prediction



(Lim, 2006)



Coordination-based Prediction

“I’m guessing [yes], just because this has … plus 12 and this has a plus 6, no matter what value of x I put in, this will always be true.”

(11th grade, Algebra 2)


(Lim, 2006)


SpecificInterpretations

Characteristicof Interpreting Act

Interpreting

Mental Act of Interpreting


“Of course there is. … Let’s see, I was taught to combine like terms. I was taught this (>) is actually an equal sign ...” (11th grader, Calc)

:–9x + 4 = –9x – 2

6 = 06 > 0

“Is there a value for x … may be there isn’t.”



Non-referential Symbolic ReasoningInequality as a symbol

(Lim, 2006)




Referential Symbolic Reasoning

Variable being unspecified

Inequality as a proposition

(Lim, 2006)

“I’m guessing [yes], just because this has … plus 12 and this has a plus 6, no matter what value of x I put in, this will always be true.”

(11th grade, Algebra 2)


SpecificProofs

ProofSchemes

Proving

Mental Act of Proving

(Harel & Sowder, 1998)

1 + 3 = 4 even1 + 5 = 6 even7 + 9 = 16 even13 + 25 = 47 even

odd + odd = even

Empirical proof scheme

Deductive proof schemeodd + odd = (even + 1) + (even + 1)

1 + 1 become even.Even plus even is even.

Why is sum of two odd numbers even?

“My teacher said so.”“It’s stated in the text book.”

Authoritative proof scheme


Mental Act of Proving

(Harel & Sowder, 1998)

A Goal for Pre-service Teachers

Undesirable Ways of Thinking

• Deductive proof scheme

• Non-referential symbolic reasoning • Referential symbolic reasoning

• Association-based reasoning • Coordination-based reasoning

• Empirical proof schemeAuthoritative proof scheme


Desirable Ways of Thinking


Mathematical Knowledge for Teachers

Curricular Knowledge

Subject-matter

Knowledge

Pedagogical Content

Knowledge

(Shulman, 1986)



Pedagogical Content

Knowledge


Ways of Thinking


Subject-matter

Knowledge




Knowledgepackage

(Ma, 1999)

Subject-matter

Knowledge

Effective examples, analogies, tasks,

models, str., appl.

Mathematics of Students

(Steffe & Thompson, 2000)

Pedagogical Content

Knowledge



Pedagogical Content

Knowledge

Standards, scope,

sequences

Subject-matter

Knowledge

Instructional materials

Alternative Curricula


Relating PCK, WoU, and WoT


Pedagogical Content

Knowledge


Ways of Thinking

Subject-matter

Knowledge



Ways of Thinking


“Students develop ways of thinking only through the construction of ways of understanding,and the ways of understanding they produce are determined by the ways of thinking they possess.”

(Harel, in press)

The Duality Principle


Ways of Thinking


• Identify target WoU & WoT

Teaching with the Duality Principle in Mind

Target WoU: Ratio as a Measure

Target WoT: New quantity can be formulated in terms of two or more other quantities

Target WoT: Referential symbolic reasoning

Target WoU: Ratio as a Multiplicative Comparison


An activity: Which rectangle is most square?75 feet

114 feet 455 feet

508 feet

185 feet

245 feet

A BC

Diff = 39

Diff = 53

Diff = 60 Ratio 0.66

Ratio 0.90

Ratio 0.76

Question: What does 39 mean? What does 0.90 mean?(Adapted from Simon & Blume, 1994)


75 feet

114 feet 455 feet

508 feet

185 feet

245 feet

A BC

Diff = 39

Diff = 53

Diff = 60

An activity: Which rectangle is most square?

75 feet

455 feet

185 feet

Question: What does 39 mean?



Ratio 0.66 Ratio 0.90 Ratio 0.76

75 feet

455 feet

185 feet

A BC

100 feet

100 feet

100 feet

Question: What does 0.90 mean?


An activity: Which rectangle is most square?75 feet

455 feet

185 feet

A BC

100 feet

100 feet

100 feet

76

90

66

Ratio 0.66 Ratio 0.90 Ratio 0.76

Question: What does 0.90 mean?



What WoU can this activity potentially foster?

• Unit ratio strategy for comparing ratios

What WoT can this activity potentially foster?

• Changing the form without changing the attribute (e.g. common denominator to compare fractions)

• Ratio as a means to measure “squareness”

• Ratio as a multiplicative comparison

• Difference as an additive comparison

• New quantity can be formulated in terms of two or more other quantities

• Referential symbolic reasoning (attend to meaning)


Ways of Thinking


• Identify target WoU & WoT

Teaching with the Duality Principle in Mind

• Consider students’ existing WoU & WoT

Pedagogical Content

Knowledge

• Select, construct, analyze task/activities

Terri’s hand

105 mm

70 mm

90 mm

84 mm

Sharon’s hand

Sharon and Terri were comparing the size of their palms. Who do you think has a larger palm?

9 students compared ratios.

3 students compared differences.

10 students compared areas.


A Task to Deter Impulsiveness

Terri’s hand

105 mm

70 mm

90 mm

84 mm

Sharon’s hand


9 students compared ratios.

3 students compared differences.

10 students compared areas.

Identifying numbers and selecting operations (impulsive)

versus

Analyzing quantities and relationships (quantitative reasoning)


A Task to Deter Impulsiveness

Terri’s hand

105 mm

70 mm

90 mm

84 mm

Sharon’s hand

A Student’s Written Comment:

“Dr. Lim had the great art of using awesome little tricks that would make us think you used, ratios, for example, when in fact it was multiplication! This was a great tactic, because often I would rush right into what I had just been taught, not even looking into the problem.”



A Task to Deter Impulsive Reasoning

Potential Avenues for Research

Research Questions

• How do pre-service teachers’ existing WoT affect their development of WoU?

• What specific WoT should a teacher preparation program address? What tasks are effective?

• What tasks are effective for accessing students’ WoU and WoT?

• What are the differences between U.S. and Chinese pre-service teachers’ WoT and WoUbefore and after teacher preparation program?

Thank You

Documents

Mathematical Knowledge for Pre-service Teachers: Ways of