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Download a Sequnce of Instruction Graphic Sequence of Instruction

Although it is not necessary to use every unit in the suggested order, they are designed with specific topics and grade levels in mind. For a graphic that provides a visual representation of the sequence and relationships of the components in Investigating Number Sense, Addition, and Subtraction click on the attached pdf.

Grades K–1

Bunk Beds and Apple Boxes: Early Number Sense
by Catherine Twomey Fosnot
Bunk Beds and Apple Boxes was designed to be used at the kindergarten level. Its focus is early number sense. Children explore various arrangements of the same quantity and are supported to develop compensation and equivalence within the context of a pajama party—where a babysitter is challenged to keep track of eight excitable girls who continually change places on the bunk beds—and also in a grocery store where a grocer arranges two kinds of apples in different-sized trays. The unit introduces the arithmetic rack (rekenrek in Dutch)—the calculating frame consisting of two rows of ten beads with two sets of five in each. The rekenrek was developed by Adri Treffers, a researcher at the Freudenthal Institute in the Netherlands, and it provides a powerful model for exploring the composing and decomposing of number (Treffers 1991). In later units, the arithmetic rack is used to help students automatize the basic facts.

Beads and Shoes, Making Twos: Extending Number Sense
by Madeline Chang and Catherine Twomey Fosnot
Beads and Shoes, Making Twos uses the context of walking hand in hand in two lines to explore one-to-one correspondence, doubles, and even and odd numbers. As the unit progresses, a story about designing necklace patterns using only two colors of beads creates a context for extending an exploration of even numbers to greater amounts and invites students to work with groups as units. This unit was designed to be used at the K-1 level.

The Double-Decker Bus: Early Addition and Subtraction
by Nina Liu, Maarten Dolk, and Catherine Twomey Fosnot
The Double-Decker Bus, with the companion story of a little girl's efforts to count the number of passengers on fast-moving double-decker buses, introduces students to the benefits of using the five-structure to calculate quantities quickly. To support and extend this story, the arithmetic rack, whose beads mirror the number and organization of seats on the bus, provides a powerful model for solving addition and subtraction problems. This unit supports the automatizing of the basic addition and subtraction facts and was designed to be used in grade 1.

Games for Early Number Sense: A Yearlong Resource
by Catherine Twomey Fosnot and Antonia Cameron
Games for Early Number Sense contains 24 games that you can choose from as you consider the needs of your students. The unit includes notes for each game describing the mathematical landscape—the possibilities and openings for learning that can occur as children play. Sample dialogues are interspersed throughout to help you anticipate what learners might say and do and to provide you with images of teachers and children at work. The games foster the development of early number sense and addition, including the basic facts, and are appropriate for K–1.

Grades 1–2

Organizing and Collecting: The Number System
by Nina Liu, Maarten Dolk, and Catherine Twomey Fosnot
Based on a story about the efforts of the hopelessly disorganized Masloppy family to keep track of their belongings, Organizing and Collecting explores the number system by inviting children to examine ways to determine and represent quantities. As children inventory and record quantities of items in their classroom, they are supported in using grouping strategies to keep track (such as making packs of five or ten). Class charts depict the number of packs and loose items, and place value patterns in the number system become the focus. The unit was developed to be used in grade 1.

Measuring for the Art Show: Addition on the Open Number Line
by Catherine Twomey Fosnot
Measuring for the Art Show builds on children's early addition and measurement strategies within the context of preparing a blueprint to be used in cutting labels for an art show. Designed for use at the end of grade 1 or the beginning of grade 2, this unit focuses on the development of the open number line model within the context of measurement. As the unit progresses, the number line is used as a model for double-digit addition strategies. It is helpful to use this unit before you begin work on double-digit addition.

Trades, Jumps, and Stops: Early Algebra
by Catherine Twomey Fosnot and Patricia Lent
In Trades, Jumps, and Stops, the Masloppy family creates another context for learning about mathematics. The equitable distribution of funds from the family piggy bank to finance a trip to New York City introduces big ideas and strategies in algebra, including using substitution to examine for equivalence and using the associative and commutative properties of addition to make equivalent expressions. The movement of passengers at subway stops creates another context to explore net change and functions. The double number line is used as a model for understanding exchange and equivalence. This unit was designed for grade 2.

Minilessons for Early Addition and Subtraction: A Yearlong Resource
by Catherine Twomey Fosnot and Willem Uittenbogaard
Minilessons for Early Addition and Subtraction is a resource of 78 minilessons that you can choose from throughout the year. The minilessons in this guide were designed to be used in grades 1-2. Some of the minilessons make use of quick images with pictures. We call these "billboards." They are carefully designed pictures that support the development of important strategies for addition and subtraction by building in potentially realizable suggestions or constraints. Flashed for only a few seconds, they encourage children to give up trying to count each item and instead to use their natural ability to subitize—to perceive small amounts (such as 2, 3, or 4) as units and use them. In this way, children are supported to count on, skip-count, use doubles, and make use of the fivestructure. Other minilessons in this resource unit make use of the arithmetic rack. Still others use ten-frames and coins (such as dimes and quarters) to encourage children to recognize and use landmark numbers (tens and twenty-fives) as they calculate. No matter which tool is used—billboards, the arithmetic rack, ten-frames, or coins—each minilesson is crafted as a tightly structured series, or "string," of computation problems designed to encourage children to look to the numbers first, before they decide on a computation strategy. The strings are likely to generate discussion of certain strategies or big ideas underlying an understanding of early number sense, addition, and subtraction.

Grades 2–3

Ages and Timelines: Subtraction on the Open Number Line
by Catherine Twomey Fosnot
Ages and Timelines begins with the story of eightyear- old Carlos's visit with his great-grandfather. Carlos's fascination with his great-grandfather's beautiful silver hair, and the time it will take him to grow a similar mane, initiates a family-wide investigation into age differences. The context of age differences and the open number line model support the development of efficient subtraction strategies. The unit also supports the generalization of subtraction as removal, as difference, and as an operation to find a missing addend. It was designed to be used at the end of grade 2 or the beginning of grade 3.

The T-Shirt Factory: Place Value, Addition, and Subtraction
by Catherine Twomey Fosnot and Patricia Lent
Students revisit the Masloppy family in The T-Shirt Factory as Grandma Masloppy begins a lucrative T-shirt business. This context and follow-up simulations highlight place value, regrouping, equivalence, and systematic exchange as the T-shirts are packed in rolls of ten and boxes that each hold ten rolls. As the unit progresses, standard addition and subtraction algorithms are explored in the context of recording in company ledgers what is left in the warehouse after various sales and replenishing. The values of the sales and remaining inventories are also tabulated (each T-shirt sells for $10), and the concept of place value is developed to four places. This unit was developed to be used at the end of grade 2 or the beginning of grade 3.

Minilessons for Extending Addition and Subtraction: A Yearlong Resource
by Catherine Twomey Fosnot and Willem Uittenbogaard
Minilessons for Extending Addition and Subtraction can be helpful in grades 2–3 as you work with addition and subtraction beyond the basic facts. These minilessons are crafted with computation problems that, when placed together, are likely to generate discussion of certain strategies or big ideas that are landmarks on the landscape of learning for addition and subtraction, particularly with two and three digits. The guide contains approximately 75 minilessons, structured as strings of related problems. Although the emphasis is on the development of mental arithmetic strategies, this does not mean learners have to solve the problems in their heads-but it is important for them to do the problem with their heads! In other words, as you use this guide, encourage students to examine the numbers in each problem and think about clever, efficient ways to solve it. The relationships between the problems in the minilesson will support them in doing this. The open number line is used throughout to represent student strategies.

Grades K–3

– Overview Book
– Unit Books
– Read-Aloud Books
– Resource Guides
  • Sequence of Instruction
  • Landscape of Learning
  • Samples
  • Buy Now
  • Young Mathematicians At Work book and CDs

    Grades 3–5
    Grades 4–6