Decompressing Teachers’ Decompressing Teachers’ Mathematical Knowledge:Mathematical Knowledge:

The Case of The Case of DivisionDivision

Presented by:Presented by:DeAnn HuinkerDeAnn HuinkerMelissa HedgesMelissa HedgesKevin McLeodKevin McLeod

Jennifer Bay-WilliamsJennifer Bay-Williams

Association of Mathematics Teacher Educators

Friday 26 January 2006This material is based upon work supported by the National Science Foundation under Grant No. 0314898. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation (NSF).

Session GoalsSession Goals

Gain a better understanding of unpacking Gain a better understanding of unpacking the mathematical knowledge necessary for the mathematical knowledge necessary for teaching division of whole numbersteaching division of whole numbers

Understand what the key pieces of that Understand what the key pieces of that knowledge package are and how they knowledge package are and how they impact the decompression of that knowledgeimpact the decompression of that knowledge

Consider how to scaffold learning Consider how to scaffold learning experiences to support teachers’ experiences to support teachers’ decompression of mathematical decompression of mathematical knowledge packages so they knowledge packages so they become more accessible as a become more accessible as a mathematical resource for teaching.mathematical resource for teaching.

Our guiding Our guiding questionsquestions

What does one need to know and What does one need to know and understand to teach division of whole understand to teach division of whole numbers well?numbers well?

What comprises the package (Ball and What comprises the package (Ball and Bass, 2000) or bundle (Ma, 1999) of Bass, 2000) or bundle (Ma, 1999) of knowledge– the key ideas, knowledge– the key ideas, understandings, connections, and understandings, connections, and sensibilities – that teachers need to sensibilities – that teachers need to develop so that it is available for develop so that it is available for teaching? teaching?

How can we, as teacher educators, How can we, as teacher educators, surface the complexities of division to surface the complexities of division to support teacher learning?support teacher learning?

Indeed, although connecting a Indeed, although connecting a topic that is to be taught to topic that is to be taught to related topics may be a related topics may be a spontaneous intention of any spontaneous intention of any teaching person, a fully teaching person, a fully developed and well-organized developed and well-organized knowledge package about a topic knowledge package about a topic is a result of deliberate study. is a result of deliberate study.

--Ma (p. 22)--Ma (p. 22)

Core Task: 169 Core Task: 169 ÷ ÷ 1414

The purpose of this task it to begin The purpose of this task it to begin to think more deeply about division, to think more deeply about division, so put your thoughts, attempts, and so put your thoughts, attempts, and missteps all on paper. Solve the missteps all on paper. Solve the following division problem using two following division problem using two strategies other than the strategies other than the conventional algorithm. Then solve conventional algorithm. Then solve the problems using the algorithm. the problems using the algorithm. Explain and represent your thinking Explain and represent your thinking using symbols, words, and diagrams, using symbols, words, and diagrams, as appropriate for each strategy. as appropriate for each strategy.

Here is what we Here is what we thought we’d get: thought we’d get:

Use of multiples Use of multiples of 10of 10

Direct modelingDirect modeling Repeated Repeated subtractionsubtraction

Pre-instruction Pre-instruction DataData

Strategy Percent of Attempts No Attempt 27% Direct Modeling 25% Repeated Subtraction 15% Dealing Out 14% Use of Multiplication 10% Partitioning the Dividend 1% Other 8%

• 30% of the attempted strategies produced a wrong answer

The core task The core task provided provided

insight into:insight into:–Teachers’ grasp of number sense Teachers’ grasp of number sense –The teachers’ ability to The teachers’ ability to demonstrate the relationships demonstrate the relationships among the operationsamong the operations

–The level of teachers’ The level of teachers’ understanding the mathematics understanding the mathematics behind the standard long behind the standard long division algorithmdivision algorithm

Reflection questions Reflection questions for viewing student for viewing student

work:work: What skills or mathematical What skills or mathematical thinking are embedded in this thinking are embedded in this strategy? strategy?

How is this strategy related How is this strategy related to the original problem? to the original problem?

When does this student know When does this student know that they are done?that they are done?

Using prospective Using prospective teachers’ work as sites teachers’ work as sites for discussion and for discussion and learning…learning…helps prospective teachers internalize what it means to “understand.” allows teachers to further develop their mathematical knowledge so that it becomes more accessible as a mathematical resource for teaching.offers a model of how teachers can offers a model of how teachers can utilize number-oriented strategies utilize number-oriented strategies to promote conceptual understanding to promote conceptual understanding in their elementary classrooms.in their elementary classrooms.

……helps prospective teachers internalize what it means to “understand.”

When the rules and procedures one is taught When the rules and procedures one is taught are not conceptually anchored, memorization are not conceptually anchored, memorization must pass for understanding, and mathematics must pass for understanding, and mathematics becomes an endless, senseless parade of becomes an endless, senseless parade of disparate facts, definitions, and procedures. disparate facts, definitions, and procedures. --MET report (2001)--MET report (2001)

“ “Exploring these other ways to “divide” has Exploring these other ways to “divide” has helped me see helped me see what we mean by understanding. what we mean by understanding. I see that my I see that my understanding for division was understanding for division was pretty shallow. I thought I pretty shallow. I thought I understood but understood but doing this work has helped me to see that doing this work has helped me to see that I I really only understood what to do not why it really only understood what to do not why it worked.”worked.”

--Prospective Teacher (2005)--Prospective Teacher (2005)

… …allows teachers to further develop their mathematical knowledge so that it becomes more accessible as a mathematical resource for teaching.

To re-enter the world of the young child, one To re-enter the world of the young child, one needs to be able to deconstruct one’s own needs to be able to deconstruct one’s own mathematical knowledge into less polished and mathematical knowledge into less polished and final form, where elemental components are final form, where elemental components are accessible and visible. We refer to this as accessible and visible. We refer to this as decompression. decompression. --Ball & Bass (2000)--Ball & Bass (2000)

I found out that these strategies can help kids I found out that these strategies can help kids develop an understanding of what happens to develop an understanding of what happens to numbers during division. This was hard but I’m numbers during division. This was hard but I’m glad we did it. By understanding this concept at glad we did it. By understanding this concept at a deeper level I can better understand the a deeper level I can better understand the thinking of my future students.thinking of my future students. --Prospective --Prospective Teacher (2005)Teacher (2005)

……offers a model of how teachers can offers a model of how teachers can utilize number-oriented strategies to utilize number-oriented strategies to promote conceptual understanding in promote conceptual understanding in their elementary classrooms.their elementary classrooms.

Just like the children they will someday Just like the children they will someday teach, prospective teachers must have teach, prospective teachers must have classroom experiences in which they become classroom experiences in which they become reasoners, conjectures, and problem solvers. reasoners, conjectures, and problem solvers. -- MET Report (2001)-- MET Report (2001)

I learned how important it is to remember that I learned how important it is to remember that there is more than one way to get an answer. there is more than one way to get an answer. I’m always quick to set it up the traditional I’m always quick to set it up the traditional way and demonstrate the steps. Now I feel way and demonstrate the steps. Now I feel more comfortable showing other ways and more comfortable showing other ways and pushing students to solve problems in more pushing students to solve problems in more than one way. This will help me see what they than one way. This will help me see what they understand or don’t. understand or don’t.

--Prospective Teacher--Prospective Teacher

What mathematical What mathematical knowledge do knowledge do prospective teachers prospective teachers need to have for need to have for division?division?

Package of Division Package of Division Knowledge for TeachersKnowledge for Teachers

Post-instruction DataPost-instruction Data

Generalizing our Generalizing our procedure for other procedure for other content areas:content areas: What is core task?What is core task? How are they used?How are they used? What do they help us What do they help us explore explore with with our prospective teachers?our prospective teachers?

“…“…the challenge is to work from what the challenge is to work from what teachers teachers dodo know – the mathematical know – the mathematical ideas they hold, the skills they ideas they hold, the skills they possess, and the contexts in which these possess, and the contexts in which these are understood – so they can move from are understood – so they can move from where they are to where they need to go. where they are to where they need to go. For their instructors, …this means For their instructors, …this means learning to understand how their learning to understand how their students think.” students think.”

--MET Report (2001)--MET Report (2001)

“Understanding number and operations and developing proficiency in computation have been and continue to be the core concerns of the elementary mathematics curriculum. Although almost all teachers remember traditional computation algorithms, their mathematical knowledge in this domain generally does not extend much further. Indeed, many equate the arithmetic operations with the algorithms and their associated notation. They have little inkling of how much more there is to know. In fact, in order to interpret and assess the reasoning of children to perform arithmetic operations, teachers must be able to call upon a richly integrated understanding of operations, place value, and computation in the domains of whole numbers, integers, and rationals.”

--MET Report (p. 58)

To re-enter the world of the young child, one needs to be able to deconstruct one’s own mathematical knowledge into less polished and final form, where elemental components are accessible and visible. We refer to this as decompression. Paradoxically, most personal knowledge of subject matter knowledge, which is desirably and usefully compressed, can be ironically inadequate for teaching. In fact, mathematics is a discipline in which compression is central. Indeed, its polished, compressed form can obscure one’s ability to discern how learners are thinking at the roots of that knowledge. Because teachers must be able to work with content for students in its growing, not finished state, they must be able to do something perverse: work backward from mature and compressed understanding of content to unpack its constituent elements.

--Ball and Bass (2000)