Philipp pm slides

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 RPhilipp@mail.sdsu.edu

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.