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The Not-So-Elementary Task of Preparing Elementary School Teachers to
Understand Elementary Mathematics
Yopp Distinguished Speaker Series University of North Carolina, Greensboro
September 8, 2014
Dr. Randy Philipp
San Diego State University [email protected]
Abstract !If elementary school mathematics is so elementary, then why do so many people struggle to understand? Using video of children's mathematical thinking, we will consider some of the complexities of seemingly simple topics of whole numbers and fractions, highlighting some of the issues related to mathematical understanding, and I will share how a focus on children’s mathematical thinking motivates prospective elementary school teachers to more deeply engage with mathematics. !
Thank You!
UNCG Mathematics Education Group in the School of Education The Graduate Students James D. and Johanna F. Yopp
Presentation Plan
• What does it mean to do mathematics? • Consider issues related to mathematics
content. • There is a problem with how PSTs are
learning mathematics, but we found one approach.
• I’ll share 4 principles of mathematics teaching and learning addressed by a focus on children’s mathematical thinking.
But first: A secondary question to keep in mind today
2014 AMTE Session organized with Rob Wieman
Is Video the New Manipulative in Education? -Maarten Dolk, Freudenthal Institute
Examples of Manipulatives
Seeing Fractional Relationships
!
!
Manipulatives are Tools
Ball, D. L. (1992). Magical Hopes: Manipulatives and the Reform of Math Education. American Educator(Summer), 14-18, 46-47.
Thompson, P. W. (1995). Notation, convention, and quantity in elementary mathematics. In J. T. Sowder & B. P. Schappelle (Eds.), Providing a foundation for teaching mathematics in the middle grades (pp. 199-221). New York: State University of New York Press.
Moyer, P. S. (2001). Are we having fun yet? How teachers use manipulatives to teach mathematics. Educational Studies in Mathematics, 47, 175-197.
• With manipulatives, do we assume that the mathematics is embedded in the representation, for all to see?
• What are students seeing? • With video, do we assume the pedagogical
principle is explicit for all to see? • What are students, teachers, or colleagues seeing?
Presentation Plan
• What does it mean to do mathematics? • Consider issues related to mathematics
content. • There is a problem with how PSTs are
learning mathematics, but we found one approach.
• I’ll share 4 principles of mathematics teaching and learning addressed by a focus on children’s mathematical thinking.
Define Mathematical Proficiency
• Concepts !
• Procedures !
• Problem Solving!
• Reasoning and Justifying!
• Positive Outlook!
Define Mathematical Proficiency
• Concepts (Conceptual Understanding)!
• Procedures (Procedural Fluency)!
• Problem Solving (Strategic Competence)!
• Reasoning and Justifying (Adaptive Reasoning)!
• Positive Outlook (Productive Disposition)!
National Research Council. (2001). Adding it Up: Helping Children Learn Mathematics. Washington, D.C.: National Academy Press.!
The Strands of ���Mathematical Proficiency
National Research Council. (2001). Adding it Up: Helping Children Learn Mathematics. Washington, D.C.: National Academy Press.!
The Strands of ���Mathematical Proficiency
The tendency to see sense in mathematics, to perceive it as both useful and worthwhile, to believe that steady effort in learning mathematics pays off, and to see oneself as an effective learner and doer of mathematics.!
The capacity to think logically about the relationships among concepts and situations, including the ability to justify one’s reasoning both formally and informally.!
Integrated and functional grasp of mathematical ideas!
Knowledge of procedures, knowledge of when and how to use them appropriately, and skill in performing them flexibly, accurately, and efficiently.!
The ability to formulate mathematical problems, represent them, and solve them.!
The Strands of ���Mathematical Proficiency
The tendency to see sense in mathematics, to perceive it as both useful and worthwhile, to believe that steady effort in learning mathematics pays off, and to see oneself as an effective learner and doer of mathematics.!
The capacity to think logically about the relationships among concepts and situations, including the ability to justify one’s reasoning both formally and informally.!
Integrated and functional grasp of mathematical ideas!
Knowledge of procedures, knowledge of when and how to use them appropriately, and skill in performing them flexibly, accurately, and efficiently.!
The ability to formulate mathematical problems, represent them, and solve them.!
The Strands of ���Mathematical Proficiency
The tendency to see sense in mathematics, to perceive it as both useful and worthwhile, to believe that steady effort in learning mathematics pays off, and to see oneself as an effective learner and doer of mathematics.!
The capacity to think logically about the relationships among concepts and situations, including the ability to justify one’s reasoning both formally and informally.!
Integrated and functional grasp of mathematical ideas!
Knowledge of procedures, knowledge of when and how to use them appropriately, and skill in performing them flexibly, accurately, and efficiently.!
The ability to formulate mathematical problems, represent them, and solve them.!
1 Make sense of problems and persevere in solving them.
2 Reason abstractly and quan=ta=vely. 3 Construct viable arguments and cri=que the reasoning of others.
4 Model with mathema=cs. 5 Use appropriate tools strategically.
6 AFend to precision (in language and math) 7 Look for and make use of structure. 8 Look for and express regularity in repeated reasoning.
The CCSS Process Standards
1 Make sense of problems and persevere in solving them. 2 Reason abstractly and quan=ta=vely. 3 Construct viable arguments and cri=que the reasoning of others. 4 Model with mathema=cs. 5 Use appropriate tools strategically. 6 AFend to precision (in language and math) 7 Look for and make use of structure. 8 Look for and express regularity in repeated reasoning.
Two 5th-Graders ���(*This is video clip #9 on IMAP CD.)!
Circle the larger or write =. 4.7 4.70
Students Learning Mathematics
Which of these strands do your students develop?
Presentation Plan
• What does it mean to do mathematics? • Consider issues related to mathematics
content. • There is a problem with how PSTs are
learning mathematics, but we found one approach.
• I’ll share 4 principles of mathematics teaching and learning addressed by a focus on children’s mathematical thinking.
Below is the work of Terry, a second grader, who solved this addition problem and this subtraction problem in May.
Problem A Problem B
Ones Task
1) Does the 1 in each of these problems represent the same amount? Please explain your answer.
2) Explain why in addition (as in Problem A) the 1 is added to the 5, but in subtraction (as in Problem B) 10 is added to the 2.
Problem A Problem B
1) Does the 1 in each of these problems represent the same amount? Please explain your answer.
2) Explain why in addition (as in Problem A) the 1 is added to the 5, but in subtraction (as in Problem B) 10 is added to the 2.
Example of Score of 4
“No, in the addition problem it represents a ten because 9 + 8 = 17, but in the subt. problem it represents 100 or 10 tens.”
“In the addition problem it is a ten from the 17 so we add one more ten to the 50 + 30. In the subtraction problem it’s not 1 more ten it’s 10 tens so we are taking 3 tens away from 12 tens.”
Problem A Problem B
1) Does the 1 in each of these problems represent the same amount? Please explain your answer.
2) Explain why in addition (as in Problem A) the 1 is added to the 5, but in subtraction (as in Problem B) 10 is added to the 2.
Example of Score of 0
“In Problem a you adding one. In problem B you are adding 10.”
“Because in A you need to carry the 1 in 17 over. But in B you are taking the 1 from the 4.”
Principal Investigators Randy Philipp, PI Vicki Jacobs, co-PI
Faculty Associates Lisa Lamb, Jessica Pierson Research Associate Bonnie Schappelle Project Coordinators Candace Cabral Graduate Students John (Zig) Siegfried Others Chris Macias-Papierniak,
Courtney White Funded by the National Science Foundation, ESI-0455785
Prospective Teachers
Experienced Teachers Before
Professional Development
Experienced Teachers
w/2 years of Professional Development
Experienced Teachers
w/4 or more years of
Professional Development
Ones Task Group Means (Standard Deviations)
0–4 scale
Prospective Teachers
Experienced Teachers Before
Professional Development
Experienced Teachers
w/2 years of Professional Development
Experienced Teachers
w/4 or more years of
Professional Development
0.31 (0.80) 1.58 (1.26) 2.49 (1.22) 2.72 (1.05)
Ones Task Group Means (Standard Deviations)
0–4 scale
29 of 35 Prospective Teachers Scored 0
Statistically Significant Differences
The Strands of ���Mathematical Proficiency
The tendency to see sense in mathematics, to perceive it as both useful and worthwhile, to believe that steady effort in learning mathematics pays off, and to see oneself as an effective learner and doer of mathematics.!
?
Example of ���Mathematical Disposition
70 - 23 53
76 - 23 53
“Yes, math is like that sometimes.”
Mathematical Integrity "An education isn't how much you have
committed to memory, or even how much you know. It's being able to differentiate between what you do know and what you don't."!
!(Anatole France, Nobel Prize-winning author)!
Presentation Plan
• What does it mean to do mathematics? • Consider issues related to mathematics
content. • There is a problem with how PSTs are
learning mathematics, but we found one approach.
• I’ll share 4 principles of mathematics teaching and learning addressed by a focus on children’s mathematical thinking.
The Problem Even when students attend a thoughtfully planned mathematics course for prospective elementary school teachers (PSTs), too many go through the course in a perfunctory manner. (Ball, 1990; Ma, 1999; Sowder, Philipp, Armstrong, & Schappelle, 1998) Many PSTs do not seem to value the experience.
Why? Many PSTs hold a complex system of beliefs that supports their not valuing the learning of more mathematics. For example, consider the effect of believing the following:
“If I, a college student, do not know something, then children would not be expected to learn it. And if I do know something, then I certainly don’t need to learn it again.”
In other words…
We, mathematics instructors of PSTs, are answering questions that PSTs have yet to ask.
Should we try to get PSTs to care more about mathematics for mathematics sake?���
This is one starting point. We took a different starting point, building on the work of Nel Noddings (1984).
Why not start with what PSTs care about?
Children! -Darling-Hammond & Sclan, 1996
Start with that about which prospec=ve teachers care.
Circles of Caring
Children
When PSTs engage children in mathema=cal problem solving, their circles of caring expand to include children’s mathema=cal thinking.
Circles of Caring
Children
Children's Mathematical Thinking
And when PSTs engage with children’s mathema=cal thinking, they realize that, to be prepared to support children, they must themselves grapple with the mathema=cs, and their circles of caring extend.
Children
Children's Mathematical Thinking
Mathematics
Circles of Caring
This idea is at least 100 years old.
John Dewey, 1902
Every subject has two aspects, “one for the scien=st as a scien=st; the other for the teacher as a teacher” (p. 351).
“[The teacher] is concerned, not with the subject-‐maFer as such, but with the subject-‐maFer as a related factor in a total and growing experience [of the child]” (p. 352).
!!Dewey, J. (1902/1990). The child and the curriculum. Chicago: The University of Chicago Press. !
Looking at mathematics through the lens of children’s mathematical thinking helps PSTs come to care about mathematics, not as mathematicians, but as teachers. ���
Mathematics Children
We tested this theory.
When compared to PSTs who did not learn about children’s mathematical thinking, PSTs who
did learn about children’s mathematical thinking
a) improved their mathematical content knowledge and
b) developed more sophisticated beliefs.
Major Result*
*Philipp, R. A., Ambrose, R., Lamb, L. C., Sowder, J. T., Schappelle, B. P., Sowder, L., et al. (2007). Effects of early field experiences on the mathematical content knowledge and beliefs of prospective elementary school teachers: An experimental study. Journal for Research in Mathematics Education, 38(5), 438 - 476."
Where Might We Infuse Children’s Mathematical Thinking?
• Education Courses • Mathematics Courses • Hybrid Courses (CMTE)
– Link Content to Children – Help PSTs Consider Future (e.g., Ali)
How did the focus on children’s thinking help the PSTs?
A PST Speaks for Herself Why study about children’s mathematical thinking while learning mathematics? CMTE Clip, 0:23 - 0:56
Presentation Plan
• What does it mean to do mathematics? • Consider issues related to mathematics
content. • There is a problem with how PSTs are
learning mathematics, but we found one approach.
• I’ll share 4 principles of mathematics teaching and learning addressed by a focus on children’s mathematical thinking.
Four principles of mathematics and mathematics teaching and learning addressed by focusing upon
children’s mathematical thinking.���
Principle 1 The way most students are learning mathematics in the United States is problematic. Principle 2 Learning concepts is more powerful and more generative than learning procedures. Principle 3 Students’ reasoning is varied and complex, and generally it is different from adults’ thinking. Principle 4 Elementary mathematics is not elementary.
Four principles of mathematics and mathematics teaching and learning addressed by focusing upon
children’s mathematical thinking.���
Principle 1 The way most students are learning mathematics in the United States is problematic. Principle 2 Learning concepts is more powerful and more generative than learning procedures. Principle 3 Students’ reasoning is varied and complex, and generally it is different from adults’ thinking. Principle 4 Elementary mathematics is not elementary.
Ally, End of Grade 5!
An average 5th-grade student at a high- performing local school
Ally, VC #11, 1:35 - 2:30
Ally, End of Grade 5!
An average 5th-grade student at a high- performing local school
Questions to Consider How did Ally come to hold these conceptions? What do teachers need to know to be able to hear these conceptions in their students and support them as they expand their reasoning to other number domains?
NAEP ITEM: 13-year-olds
Estimate the answer to . You will not have time to solve the problem using paper and pencil.
€
1213
+78
a)1 b) 2 c)19 d) 21 e) I don’t know 14% 24% 28% 27% 7%
€
1213
+78
Is thought of as 12 7
13 8
Four principles of mathematics and mathematics teaching and learning addressed by focusing upon
children’s mathematical thinking.���
Principle 1 The way most students are learning mathematics in the United States is problematic. Principle 2 Learning concepts is more powerful and more generative than learning procedures. Principle 3 Students’ reasoning is varied and complex, and generally it is different from adults’ thinking. Principle 4 Elementary mathematics is not elementary.
Felisha, End of Grade 2 Felisha learned fraction concepts using equal-sharing tasks in a small group over 14 sessions (7 days). She had not been taught any procedures for operating on fractions. Felisha, VC #15, 1:39
Felisha, End of Grade 2
Questions to Consider What do teachers need to know to help students make sense of fractions as Felisha has done? What do teachers need to know to be able to pose a question that might help Felisha expand her thinking?
Consider two problems
• A problem for secondary-school algebra students • A problem for primary-grade students (K-2)
A Secondary-School Algebra Problem 19 children are taking a mini-bus to the zoo. They will have to sit either 2 or 3 to a seat. The bus has 7 seats. How many children will have to sit three to a seat, and how many can sit two to a seat?
x = # of seats with 2 children; y = # of seats with 3 children
x + y = 7 and 2x + 3y = 19
10
8
6
4
2
- 2
- 2 2 4 6 8
(2, 5)2x+3y=19
x+y=72x + 3y = 19 and y = 7 - x
2x + 3(7 - x) = 19
2x + 21 - 3x = 19
-x = -2
x = 2 and y = 5
A Primary-Grade Problem 19 children are taking a mini-bus to the zoo. They will have to sit either 2 or 3 to a seat. The bus has 7 seats. How many children will have to sit three to a seat, and how many can sit two to a seat?
51% of kindergarten students in six classes (36/70) correctly solved this problem in May.
How do you think they did that? “Throughout the year children solved a variety of different problems. The teachers generally presented the problems and provided the children with counters. . . but the teachers typically did not show the children how to solve a particular problem. Children regularly shared their strategies... —Carpenter et al., 1993, p. 433. (JRME v.24, #5)
Four principles of mathematics and mathematics teaching and learning addressed by focusing upon
children’s mathematical thinking.���
Principle 1 The way most students are learning mathematics in the United States is problematic. Principle 2 Learning concepts is more powerful and more generative than learning procedures. Principle 3 Students’ reasoning is varied and complex, and generally it is different from adults’ thinking. Principle 4 Elementary mathematics is not elementary.
Often What We Think We are Teaching is Not What Students are Learning After students learn to apply the distributive property to algebraic expressions, for example,
2(x + y) = 2x + 2y…students often overgeneralize, for example,
(x + y)2 = x2 + y2
Often What We Think We are Teaching is Not What Students are Learning
Often What We Find Students are Thinking is Not What We Have Been Teaching!
At the time of this interview, Javier, in grade 5, had been in the United States about one year, and he did not speak English before coming to this country. Javier, VC #6, 0:00 - 1:10
Javier*, Grade 5
How many eggs are in six cartons (of one dozen)?
I: How did you get that?
I: “How did you know 12 x 5 is 60?”
J: “Because 12 times 10 equals 120. If I take the [sic] half of 120, that would be 60.”
Javier thinks for 15 seconds, and says, “72.”
Javier writes and says, “5 times 12 is 60, and 12 more is 72.”
*At the time of this interview, Javier had been in the United States about one year, and he did not speak English before coming to this country.
6 x 12
= (5 x 12) + (1 x 12)
= [(12 x 10) x 12] + 12
= [12 x (10 x 12)] + 12
= [12 x (120)] + 12
= 60 + 12
= 72
One Representation of Javier’s Thinking
What would a teacher need to know to understand Javier’s thinking, and where do teachers learn this?
Place value
(Distributive prop. of x over +)
(Substitution property)
(Associative property of x )
How do we, as teachers, respond when faced with a child who has a more
mathematically creative or innovative solution than we have?���
Four principles of mathematics and mathematics teaching and learning addressed by focusing upon
children’s mathematical thinking.���
Principle 1 The way most students are learning mathematics in the United States is problematic. Principle 2 Learning concepts is more powerful and more generative than learning procedures. Principle 3 Students’ reasoning is varied and complex, and generally it is different from adults’ thinking. Principle 4 Elementary mathematics is not elementary.
Elliot, Grade 6: 1 ÷ 1/3?*
(*This is video clip #16 on IMAP CD.)
“One third goes into 1 three times because there is three pieces in one whole.”
Elliot, Grade 6: 1 ÷ 1/3?*
(*This is video clip #16 on IMAP CD.)
Elliot does not realize that the remaining 1/6 needs to be seen as 1/2 of 1/3.
Follow-up: 1 1/2 ÷ 1/3? “There are three one-thirds in 1, and another 1/3 in 1/2, and there is 1/6 left over.”
Elliot, Grade 6 Question for PSTs
Explain Elliot’s reasoning. What does one need to understand to be in the position to support Elliot?
Issue Understanding is seldom black and white. Everybody understands something about anything, and no one understands everything about anything.
What knowledge do teachers need to teach mathematics?
What knowledge do teachers need to teach mathematics?
Common Content Knowledge
Specialized Content Knowledge
the mathematical knowledge teachers are responsible for developing in students
mathematical knowledge that is used in teaching but not directly taught to students
Pedagogical Content Knowledge “the ways of representing and formulating the subject that make it comprehensible to others” —(Shulman, 1986)
Common Content Knowledge
Specialized Content Knowledge
the mathematical knowledge teachers are responsible for developing in students
mathematical knowledge that is used in teaching but not directly taught to students
Pedagogical Content Knowledge “the ways of representing and formulating the subject that make it comprehensible to others” —(Shulman, 1986)
Evaluate and understand the meaning of 12 ÷ 3.
Write a real-life story problem that could be represented by the expression 12 ÷ 3.
How might children think about the problem you wrote?
Our Approach
MCK (CCK and SCK)
PCK (KCS)
“Those who can, do. Those who understand, teach.”
Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4–14."
“Those who can, do. Those who understand, teach.”
Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4–14."
MCK
PCK
“Those who can, do. Those who understand, teach.”
Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4–14."
MCK
PCK
Which of these strands do your students develop?
Is Video the New Manipulative in Education?
• With manipulatives, do we assume that the mathematics is embedded in the representation, for all to see?
• What are students seeing? • With video, do we assume the pedagogical
principle is explicit for all to see? • What are students, teachers, or colleagues
seeing?
!
Discussion
Thank you.