Decompressing Teachers’ Mathematical Knowledge: The Case of Division Presented by: DeAnn Huinker Melissa Hedges Kevin McLeod Jennifer Bay-Williams Association of Mathematics Teacher Educators Friday 26 January 2006 This material is based upon work supported by the National Science Foundation under Grant No 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 Goals Gain a better understanding of unpacking the mathematical knowledge necessary for teaching division of whole numbers Understand what the key pieces of that knowledge package are and how they impact the decompression of that knowledge Consider how to scaffold learning experiences to support teachers’ decompression of mathematical knowledge packages so they become more accessible as a mathematical resource for teaching.

Our guiding questions  What does one need to know and understand to teach division of whole numbers well?  What comprises the package (Ball and Bass, 2000) or bundle (Ma, 1999) of knowledge– the key ideas, understandings, connections, and sensibilities – that teachers need to develop so that it is available for teaching?  How can we, as teacher educators, surface the complexities of division to support teacher learning?

Indeed, although connecting a topic that is to be taught to related topics may be a spontaneous intention of any teaching person, a fully developed and well- organized knowledge package about a topic is a result of deliberate study. --Ma (p. 22)

Core Task: 169 ÷ 14 The purpose of this task it to begin to think more deeply about division, so put your thoughts, attempts, and missteps all on paper. Solve the following division problem using two strategies other than the conventional algorithm. Then solve the problems using the algorithm. Explain and represent your thinking using symbols, words, and diagrams, as appropriate for each strategy.

Here is what we thought we’d get: Use of multiples of 10 Use of multiples of 10 Direct modeling Direct modeling Repeated subtraction Repeated subtraction

Pre-instruction Data

The core task provided insight into: –Teachers’ grasp of number sense –The teachers’ ability to demonstrate the relationships among the operations –The level of teachers’ understanding the mathematics behind the standard long division algorithm

Reflection questions for viewing student work: What skills or mathematical thinking are embedded in this strategy? What skills or mathematical thinking are embedded in this strategy? How is this strategy related to the original problem? How is this strategy related to the original problem? When does this student know that they are done? When does this student know that they are done?

Using prospective teachers’ work as sites for discussion and 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 utilize number-oriented strategies to promote conceptual understanding in their elementary classrooms.

… …helps prospective teachers internalize what it means to “understand.” When the rules and procedures one is taught are not conceptually anchored, memorization must pass for understanding, and mathematics becomes an endless, senseless parade of disparate facts, definitions, and procedures. --MET report (2001) “Exploring these other ways to “divide” has helped me see what we mean by understanding. I see that my understanding for division was pretty shallow. I thought I understood but doing this work has helped me to see that I really only understood what to do not why it worked.” “Exploring these other ways to “divide” has helped me see what we mean by understanding. I see that my understanding for division was pretty shallow. I thought I understood but doing this work has helped me to see that I really only understood what to do not why it worked.” --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 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. --Ball & Bass (2000) I found out that these strategies can help kids develop an understanding of what happens to numbers during division. This was hard but I’m glad we did it. By understanding this concept at a deeper level I can better understand the thinking of my future students. --Prospective Teacher (2005)

…offers a model of how teachers can utilize number-oriented strategies to promote conceptual understanding in their elementary classrooms. Just like the children they will someday teach, prospective teachers must have classroom experiences in which they become reasoners, conjectures, and problem solvers. -- MET Report (2001) I learned how important it is to remember that there is more than one way to get an answer. I’m always quick to set it up the traditional way and demonstrate the steps. Now I feel more comfortable showing other ways and pushing students to solve problems in more than one way. This will help me see what they understand or don’t. --Prospective Teacher

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

Package of Division Knowledge for Teachers

Post-instruction Data

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

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