mathematical knowledge for pre-service teachers: ways of
TRANSCRIPT
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
Evidence for Mathematical Unsophistication
“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
Evidence for Mathematical Unsophistication
“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)
Evidence for Mathematical Unsophistication
Interconnected Knowledge of Mathematics
Mathematical Disposition & Beliefs
(i.e. Habits of Mind)
Components of Mathematical Knowledge
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)
Components of Mathematical Knowledge
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)
Components of Mathematical Knowledge
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
Mental Act of Predicting
Examples of Ways of Thinking
(Lim, 2006)
Mental Act of Predicting
Examples of Ways of Thinking
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)
Is there a value of x that will make this statement true?(6x – 8 – 15x) + 12 > (6x – 8 – 15x) + 6
(Lim, 2006)
Examples of Ways of Thinking
SpecificInterpretations
Characteristicof Interpreting Act
Interpreting
Mental Act of Interpreting
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.”
Mental Act of Interpreting
Examples of Ways of Thinking
Non-referential Symbolic ReasoningInequality as a symbol
(Lim, 2006)
Is there a value of x that will make this statement true?(6x – 8 – 15x) + 12 > (6x – 8 – 15x) + 6
Mental Act of Interpreting
Examples of Ways of Thinking
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)
Examples of Ways of Thinking
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
Examples of Ways of Thinking
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
Examples of Ways of Thinking
Desirable Ways of Thinking
Math Knowledge for Pre-service Teachers
Mathematical Knowledge for Teachers
Curricular Knowledge
Subject-matter
Knowledge
Pedagogical Content
Knowledge
(Shulman, 1986)
Math Knowledge for Pre-service Teachers
Mathematical Knowledge for Teachers
Pedagogical Content
Knowledge
Curricular Knowledge
Ways of Thinking
Ways of Understanding
Subject-matter
Knowledge
Math Knowledge for Pre-service Teachers
Mathematical Knowledge for Teachers
Curricular Knowledge
Knowledgepackage
(Ma, 1999)
Subject-matter
Knowledge
Effective examples, analogies, tasks,
models, str., appl.
Mathematics of Students
(Steffe & Thompson, 2000)
Pedagogical Content
Knowledge
Math Knowledge for Pre-service Teachers
Mathematical Knowledge for Teachers
Pedagogical Content
Knowledge
Standards, scope,
sequences
Subject-matter
Knowledge
Instructional materials
Alternative Curricula
Curricular Knowledge
Relating PCK, WoU, and WoT
Mathematical Knowledge for Teachers
Pedagogical Content
Knowledge
Curricular Knowledge
Ways of Thinking
Subject-matter
Knowledge
Ways of Understanding
Relating PCK, WoU, and WoT
Ways of Thinking
Ways of Understanding
“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
Relating PCK, WoU, and WoT
Ways of Thinking
Ways of Understanding
• 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
Relating PCK, WoU, and WoT
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)
Relating PCK, WoU, and WoT
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?
Relating PCK, WoU, and WoT
An activity: Which rectangle is most square?
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?
Relating PCK, WoU, and WoT
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?
Relating PCK, WoU, and WoT
An activity: Which rectangle is most square?
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)
Relating PCK, WoU, and WoT
Ways of Thinking
Ways of Understanding
• 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.
Relating PCK, WoU, and WoT
A Task to Deter Impulsiveness
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.
Identifying numbers and selecting operations (impulsive)
versus
Analyzing quantities and relationships (quantitative reasoning)
Relating PCK, WoU, and WoT
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.”
Sharon and Terri were comparing the size of their palms. Who do you think has a larger palm?
Relating PCK, WoU, and WoT
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?