Developing Mathematical Ideas

  • Developing Mathematical Ideas

    Collaborative Professional Development for K–Grade 8 Teachers

    Developing Mathematical Ideas seminars are designed to help teachers think through the major concepts of K–grade 8 mathematics and examine how children develop their understanding of those ideas. To help teachers learn and deeply understand the mathematics content that they teach, Developing Mathematical Ideas seminars engage participants in a collaborative learning experience that uses classroom episodes to illustrate students' thinking as the basis for the activities and discussions of the seminar sessions. Participants have opportunities to explore mathematics with the guidance of facilitators, to share and discuss the work of their own students, to view and analyze videos of student interviews and learn to interview their own students, and to read and discuss relevant research. Through this work, participants learn how to orient their instruction to particular mathematical goals and to develop a mathematics pedagogy in which student understanding takes center stage. 

    Developing Mathematical Ideas seminars bring together teachers from kindergarten through grade 8 to—

    • learn to recognize the key mathematical ideas with which their students are grappling;
    • consider the types of classroom settings and teaching strategies that support the development of student understanding;
    • become aware of how core mathematical ideas develop across the grades;
    • work on mathematical concepts and gain better understanding of mathematical content; and
    • discover how to continue learning about children and mathematics.

    The Developing Mathematical Ideas Series emerged from decades of work in mathematics teacher professional development on the part of its creators. Building on the individual professional development sessions first offered in the mid-1980s and the structured program examining the big ideas in mathematics teaching of the early 2000s, Developing Mathematical Ideas continues to renew itself and remain relevant in this new online version by incorporating its connections to the Common Core State Standards for Mathematics, revising content on the basis of the observations and insights from the field, and keeping up to date with the latest research.

    The Developing Mathematical Ideas Series has been offered successfully by many users. Here are just a few comments:

    Our teachers love DMI seminars. The seminars provide them with an opportunity to dig deeply into important mathematics content, look analytically at what students are saying and thinking about that content, and think about what this means for the students in their own classrooms. Many teachers here say that it was what they learned in our DMI seminars that helped them transform their teaching. 

    —Linda Ruiz Davenport, Director of K–12 Mathematics, Boston Public Schools

    Developing Mathematical Ideas has provided our district an avenue by which to engage elementary teachers in thinking about mathematical practices as a means to support the teaching of math. The professional development series offers teachers an in-depth study of not only math content but also the thinking that students apply to solve problems. The ongoing support from DMI trainers, case studies, and videos has provided us with a foundation for changing the way we engage our teachers and students in math instruction.

    —Brian Shindorf, Director of Elementary Education, St. Joseph School District, St. Joseph, Missouri

    The Developing Mathematics Ideas series is a must-have for those who provide professional development for K–8 teachers. The cases are compelling as well as revealing about children's insights into mathematical ideas. Each case also includes actual examples of children's work that provide teachers with many opportunities to consider key ideas. We have had tremendous success using these materials with teachers in our K–8 Mathematics Specialist Program in Virginia. Teachers develop a deeper understanding of the mathematics that they teach. Often, they also make changes in how they think about teaching and learning mathematics.

    —Joy W. Whitenack, Associate Professor, Virginia Commonwealth University

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    Building a System of Tens: Calculating with Whole Numbers and Decimals focuses on place value and decomposition of numbers, concepts at the basis of whole number and decimal calculations, the heart of elementary and middle school mathematics curriculum. Participants explore the base-ten structure of the number system, consider how that structure is exploited in multidigit computational procedures, and examine how basic concepts of whole numbers reappear when working with decimals.




    Making Meaning for Operations: In the Domains of Whole Numbers and Fractions concentrates on the kinds of actions and situations that are modeled by addition, subtraction, multiplication, and division and how students come to understand them. Participants investigate the four basic operations on whole numbers, and then move to a study of the operations in the context of fractions.



    Measuring Space in One, Two, and Three Dimensions explores the conceptual bases of measurement and how children develop their understanding of these ideas. Participants examine different attributes of size, develop facility in composing and decomposing shapes, and apply these skills to make sense of formulas for area and volume. They also explore conceptual issues of length, area, and volume as well as their complex interrelationships.


    Examining Features of Shape includes a study of angle, similarity, congruence, and the relationships between three-dimensional objects and their two-dimensional representations. Participants examine aspects of two- and three-dimensional shapes, develop geometric vocabulary, and explore both the definitions and properties of geometric objects.



    Reasoning Algebraically About Operations: In the Domains of Whole Numbers and Integers extends the work of Building a System of Tens and Making Meaning for Operations to examine the generalizations students make about the operations and the reasoning entailed in addressing the question, "Does this always work?" Participants express these generalizations in common language and in algebraic notation, develop arguments based on representations of the operations, study what it means to prove a generalization, and extend their generalizations and arguments to the domain of integers.



    Modeling with Data focuses attention on issues of data representation and analysis and on how students' ideas on working with data develop over time. Participants work with the collection, representation, descriptions, and interpretation of data. They learn what various graphs and statistical measures show about features of data, study how to summarize data when comparing groups, and consider whether the data provide insight in the questions that led to the data collection.


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    Patterns, Functions, and Change explores a variety of examples of different types of functions. Participants discover how the study of repeating patterns and number sequences can lead to ideas of functions, learn how to read tables and graphs to interpret phenomena of change, and use algebraic notation to write function rules. They also explore linear, quadratic, and exponential functions, with a particular emphasis on linear functions, and examine how various features of a function are seen in graphs, tables, or rules.