There is so much to know of mathematics. Creating longer and longer lists of what teachers should know does not seem promising. What are some of the big ideas in mathematics that would seem to have a lot of leverage in practice? How do teachers need to understand these ideas—for example, what does it mean to be able to “unpack” ideas, as well as to connect them?

Leader: *Nadine Bezuk;* Members: *Dick Askey, David Dennis, Shelly Ferguson, Jill Bodner Lester, Leigh* *Peake, Erick Smith*

In an effort to address this question, we discussed what it means to be a “big idea” in mathematics, and how “big ideas” differ from “average ideas.” We also discussed the meaning of “unpacking ideas.” We often had to remind each other that our task was to focus on the big ideas and not to make another list. During our discussion, group members would often say to each other, “Yes, that's an important concept, but is it a *big idea*?” Our working definition of a “big idea” was a concept or topic that is foundational to many other topics. This is compared to an important idea, which, though significant, is a topic on which relatively fewer other topics are dependent. Throughout our discussion, we attempted to keep this distinction in mind.

We found the topic of teacher development to be intertwined with our group's question. Group members shared their experiences in working with teachers, which seemed to help them think about the notion of “unpacking ideas.” This also highlights the importance of teacher development in improving mathematics learning.

We addressed our group's question in several ways. We began our discussion by identifying what we believed are the “big ideas” of mathematics that instruction needs to include. We also discussed other points we felt important for teachers to understand, recognize, or address in their teaching, and what teachers should be able to do and understand to be able to “unpack ideas.”

Several workshop sessions spurred our thinking. Liping Ma's work (1999), and particularly her notion of profound understanding of fundamental mathematics, made a significant contribution to our discussions. Ma's work highlighted for us the importance of a few fundamental concepts, or “big ideas,” to further understanding of mathematics.

Dick Askey also intrigued us with

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Knowing and Learning Mathematics for Teaching
Question #3
DISCUSSION GROUP #3
There is so much to know of mathematics. Creating longer and longer lists of what teachers should know does not seem promising. What are some of the big ideas in mathematics that would seem to have a lot of leverage in practice? How do teachers need to understand these ideas—for example, what does it mean to be able to “unpack” ideas, as well as to connect them?
Leader: Nadine Bezuk; Members: Dick Askey, David Dennis, Shelly Ferguson, Jill Bodner Lester, Leigh Peake, Erick Smith
ASSUMPTIONS MADE IN FRAMING THE DISCUSSION
In an effort to address this question, we discussed what it means to be a “big idea” in mathematics, and how “big ideas” differ from “average ideas.” We also discussed the meaning of “unpacking ideas.” We often had to remind each other that our task was to focus on the big ideas and not to make another list. During our discussion, group members would often say to each other, “Yes, that's an important concept, but is it a big idea?” Our working definition of a “big idea” was a concept or topic that is foundational to many other topics. This is compared to an important idea, which, though significant, is a topic on which relatively fewer other topics are dependent. Throughout our discussion, we attempted to keep this distinction in mind.
We found the topic of teacher development to be intertwined with our group's question. Group members shared their experiences in working with teachers, which seemed to help them think about the notion of “unpacking ideas.” This also highlights the importance of teacher development in improving mathematics learning.
SUMMARY OF THE MAIN POINTS OF DISCUSSION
We addressed our group's question in several ways. We began our discussion by identifying what we believed are the “big ideas” of mathematics that instruction needs to include. We also discussed other points we felt important for teachers to understand, recognize, or address in their teaching, and what teachers should be able to do and understand to be able to “unpack ideas.”
Several workshop sessions spurred our thinking. Liping Ma's work (1999), and particularly her notion of profound understanding of fundamental mathematics, made a significant contribution to our discussions. Ma's work highlighted for us the importance of a few fundamental concepts, or “big ideas,” to further understanding of mathematics.
Dick Askey also intrigued us with

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Knowing and Learning Mathematics for Teaching
alternative models and formats by sharing information about and examples of mathematics textbooks used in other countries.
Deborah Ball's session on the use of classroom videos again highlighted for us the interconnection of “big ideas” and teacher development, as well as the large number of decisions teachers must make during every mathematics lesson.
The big ideas of mathematics. Our group identified the following topics as “big ideas” of mathematics:
multiplicative structures, including multiplication, division, fractions, decimals, percents, ratio and proportion, connection between rational numbers and repeating decimals;
structure of the number system, number sense, decomposition/place value, using benchmarks to estimate numbers;
systems of operations, operation sense, structure of operations, properties of operation systems (i.e., operations as an interconnected system rather than experiencing operations individually or in isolation);
similarity, scaling with respect to dimension;
geometric measure, including area, finding area by dissection of figures, comparison of polygonal areas, perimeter, volume;
uses and concepts of variables;
the notion of proof.
Other points. Our group also discussed other points that we felt are important for teachers to understand, recognize, or address in their teaching. They include the following:
Teachers need to recognize the importance of generality.
Teachers need to recognize and take advantage of opportunities for making connections in the curriculum, to help children understand interrelationships between concepts.
Teachers need to decide what to pursue (and what not to pursue) and seize opportunities.
Teachers need to know what concepts students will be encountering in a certain school year and in the next few years. This will help teachers shape students' strategies to highlight those that will lead to important concepts and de-emphasize strategies that will not be widely used in mathematics in upper grades.
Teachers need a rich understanding of concepts and related topics and of the linkages between topics.
Teachers need to understand the order of difficulty of concepts and how problem contexts may make problems easier or harder.
Teachers need to recognize that powerful mathematical ideas evolve through connecting multiple representations.
Teachers need to recognize that powerful mathematical ideas can be expressed with precision and clarity.
Teachers need to develop flexibility about ideas, recognizing what is important and how it connects with other topics.
Instruction for teachers and prospective teachers needs to start by identifying the mathematics that we want children to understand. Instruction should start with those topics, so that prospective teachers really understand those concepts and then understand where those topics are usually taught and what comes next.
In order to “unpack ideas,” teachers should be able to:

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Knowing and Learning Mathematics for Teaching
effectively judge the validity of any strategy offered by a child (e.g., so that teachers are not telling students who are using less familiar strategies that they are wrong);
recognize parts of students' understandings that are valid and can be built upon, even if answers or processes are incorrect;
use language correctly, recognize limitations in students' incorrect language, and make connections between everyday, less formal language and formal mathematical language;
switch flexibly between different interpretations of concepts.

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Knowing and Learning Mathematics for Teaching
Question #3
DISCUSSION GROUP #8
There is so much to know of mathematics. Creating longer and longer lists of what teachers should know does not seem promising. What are some of the big ideas in mathematics that would seem to have a lot of leverage in practice? How do teachers need to understand these ideas—for example, what does it mean to be able to “unpack” ideas, as well as to connect them?
Leader: Ruhama Even; Members: Rachel Collopy, Alice Gill, Genevieve Knight, Neil Portnoy, Marty Simon
ASSUMPTIONS MADE IN FRAMING THE DISCUSSION
The initial response of the group to the above question was to state important topics that elementary school teachers need to know and understand well to teach mathematics. As listing topics did not seem useful, the group decided to choose two different mathematical constructs and try to unpack what it means for teachers to understand them, anchoring the analysis in specific concrete situations. The two constructs chosen were proportional reasoning and generalization, as both are central in mathematics but of different natures. Time did not allow for significant progress on both topics. Later, the group expanded the discussion of the question and focused on the nature of teacher content knowledge and its relation to teaching.
SUMMARY OF THE MAIN POINTS OF DISCUSSION
The first topic mentioned as an important topic that elementary school teachers need to know and understand well in order to teach mathematics was proportional reasoning. Instantly, it became clear that such a title is only the tip of an iceberg. Struggling with the question, What are teachers' understandings of proportional reasoning, group members emphasized the ability to distinguish between additive and multiplicative reasoning, mentioned the need for conceptual knowledge of the four basic operations, and suggested different examples of ratio as measurement (mixture, slope, fractions).
At the first plenary panel discussion, Mark Saul presented a list regarding teachers' understanding of fundamental mathematics. Proportional reasoning was not included in this list. This omission puzzled group members, as proportional reasoning was the first topic mentioned in our group, and it occupied a great deal of the first day discussion.
The proportional reasoning discussion

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Knowing and Learning Mathematics for Teaching
led to more general questions, such as these: What understandings can we reasonably expect teachers to have? What is important to know about mathematics? What kinds of understandings are needed? Examination of these questions gave rise to and highlighted the importance of processes, such as abstraction and generalization in mathematics. For example, group members emphasized that teachers should know what is generalizable and know important generalizations that cut across different topic areas.
The rather long, still quite in its initial stages, examination of what understanding teachers should have about proportional reasoning, convinced the group that mentioning more topics would not advance the group work. It was clear that a thorough analysis of what it means for teachers to know mathematical topics, concepts, and processes is needed.
In addition to an initial unpacking of teacher understanding of specific mathematical topics, concepts, and processes, the group discussed the nature of teacher content knowledge. Group members debated the idea that it comprises two kinds. One kind is of the same nature as K-12 students ' knowledge of mathematics. The other kind goes beyond students' knowledge and has to do with the specific act of teaching, an anticipatory level of mathematical knowledge (e.g., knowing the difference between partitive and quantitative division). The group also felt that teacher content knowledge should be viewed in a broad sense to include understandings, conceptions, subject-matter related beliefs, knowledge about the nature of the discipline, and so forth.
Teacher knowledge in relation to teaching was considered in two ways, although briefly, because of time constraints. It was emphasized that one's thinking about teacher knowledge of mathematics is connected to one's vision of good teaching of mathematics, and therefore advancement in the first requires more articulation of the latter. Another idea which the group planned to work on was to analyze teacher content knowledge in a specific teaching situation. The group felt that analysis of teacher content knowledge in the context of teaching has the potential to provide insight into and advance understanding of this issue. We intended to analyze the video excerpt from Ball's third-grade class where students were asked to write number sentences equal to 10. The group was intrigued by the teacher's decision to introduce algebraic symbols in the third-grade discussion. Group members felt, for example, that such a teacher decision requires teachers to know, not only that letters can be used to generalize a pattern observed in a series of number sentences but also that letters in algebra are used in different ways (e.g., generalized number, unknown, variable, parameter).
ISSUES
In conclusion, the group felt that to advance in its response to the question presented, it is important to further examine the following:
teacher content knowledge of specific mathematical topics, concepts, and processes;
different parts of teacher knowledge of mathematics: The knowledge expected from students, specific-teacher knowledge, teacher knowledge of mathematics in the context of teaching.
vision and models of good teaching of mathematics and connections to teachers' knowledge of mathematics.