# Math Curricula

Learn about the math curricula at Wingra:

- Cognitively Guided Instruction (CGI)
- Investigations in Number, Data and Space
- Mathematics in Context (MIC)
- Adding it Up: Helping Children Learn Mathematics

**Cognitively Guided Instruction (CGI)**

In the Nest and Pond, some mathematics is based on *Cognitively Guided Instruction*. CGI is an approach to teaching mathematics, rather than a set curriculum. CGI is based on research that shows that children come to school with rich informal systems of mathematical knowledge and problem-solving strategies that can serve as a basis for learning mathematics with understanding.

The teacher's role in CGI is to build from each student’s prior knowledge so that they can eventually make connections between situational experiences and the abstract symbols typically found to represent them in mathematical equations (+, -, x and so forth.) With CGI, teachers:

- Analyze story problems and number sentences to determine their mathematical demands and recognize student responses in terms of cognitive development.
- Assess students’ thinking and design problems that will develop students’ understanding of concepts and skills.
- Facilitate discussions that provide a window into children’s thinking, strengthen children’s ability to reason about arithmetic, and build their capacity for algebraic reasoning.
- Use open and true/false number sentences to develop students’ understanding of mathematical concepts and skills.

During CGI, children spend most of their time solving a problem that is related to a book the teacher has to read to them, a unit they are studying, or something going on in their lives. A variety of student-generated strategies are used to solve this problem such as using plastic cubes to model the problem, counting on fingers and using knowledge of number facts to figure out the answer. Children are expected to explain and justify their strategies, and the children, along with the teacher, take responsibility for deciding whether a strategy that is presented is correct.

This is intentionally different than rote instruction. For one thing, it brings in the student voice, activating and engaging their own background, experience, and thoughts. Rather than simply being asked to apply a formula to several virtually identical math problems, they are challenged to find their own solutions to an identified problem.

Second, students publicly explain and justify their reasoning to their friends and the teacher. Third, teachers are required to open up their instruction to students’ original ideas, and to guide each student according to his or her own developmental level and way of reasoning.

This interactive work has a major impact on students’ learning. Not only are students learning specific ways to solve problems, they are also increasing their knowledge of the fundamental principles of mathematics. For example, students who learn the standard addition algorithm often learn little more than a procedure to find the correct answer.

Students who develop strategies to solve addition problems are likely to intuitively use the commutative and associative properties of addition in their strategies. Students using their own strategies to solve problems and justifying these strategies also contributes to a positive disposition toward learning mathematics.

## Investigations in Number, Data, and Space

In the Nest, Pond, and Lake students regularly work with a curriculum called *Investigations in Number, Data and Space*. Teachers at these levels select units from this curriculum designed to help children understand the fundamental ideas of number and operations, geometry, data, measurement and early algebra.

Each Investigations unit offers from two to eight weeks of mathematical work on topics in number, data analysis, and geometry; the number of units per year varies by level. Because of the many interconnections among mathematical ideas, units may revolve around two or three related areas—for example, addition and subtraction or geometry and fractions.

In each unit, students explore the central topics in depth through a series of investigations, gradually encountering and using many important mathematical ideas. Investigations provide substantive work in important areas of mathematics—rational numbers, geometry, measurement, data, and early algebra—and connections among them

Rather than working through a textbook or workbook doing page-by-page exercises, students actively engage with materials and with their peers to solve larger mathematical problems. Students actively use mathematical tools and consult with peers as they find their own ways to solve the problems. The investigations allow significant time for students to think about the problems and to model, draw, write, and talk about their work. In addition to the investigations, the curriculum also includes games and classroom routines that support mathematical thinking.

Our intent is to present problems in which students with different needs, experiences, confidence, and levels of skill will all do significant mathematical work, perhaps in different ways. We don’t track students; the working groups we create are both intentional and fluid. The evidence from research is that tracking students does not benefit students’ learning.

With Investigations, classrooms discussions are energetic. Children solve problems then explain, justify, listen evaluate and reevaluate. Investigations stresses making sense of mathematics, rather than rote procedures. Students build on ideas they already have and learn about new concepts they have never encountered. They also use their mathematical ideas, applying what they already know to new situations, and thinking and reasoning about unfamiliar problems.

The focus of Investigations is not just on learning a set of skills and procedures and information, but is on developing the capacity and inclination to reason about mathematical ideas. That includes being able to describe one’s own mathematical ideas, make connections between one’s own ideas and somebody else’s ideas, look at whether what you’re saying is reasonable, and justify the mathematical statements that you’re making.

**Mathematics in Context**

Mathematics in Context (MIC) is a comprehensive middle school mathematics curriculum for students in grades five through eight. The intent of MIC is to engage students in learning and applying mathematics in the context of interesting, real-life problems.

MIC has been highly rated by Project 2061 of the American Association for the Advancement of Science (AAAS). MIC content emerged from research conducted through National Center for Research in Mathematical Sciences Education (NCRMSE), and the curriculum was funded by the National Science Foundation.

The complete MIC program contains 40 units, 10 at each grade level. The units are organized into four content strands: number, algebra, geometry, and statistics (which also includes probability). Each year, levels teachers from the Lake and Sky select a number of units from each strand to work with as part of their mathematics curriculum.

In a typical unit, students (1) are introduced to mathematical objectives along with engaging questions in a letter, (2) work through a thematic series of activities and problems, each of which lasts several days and involves group and independent work, (3) complete summary questions at the end of each section, intended to facilitate the integration and consolidation of the concepts and skills they have been studying, and (4) work on a series of assessment activities designed to evaluate major goals of the unit as they are applied to real-world contexts.

MIC consists of mathematical tasks and questions designed to stimulate mathematical thinking and to promote discussion among students. Students explore mathematical relationships; develop and explain their own reasoning and strategies for solving problems; use problem-solving tools appropriately; and listen to, understand, and value each other’s strategies.

Connections are a key feature of the program—connections among topics, connections to other disciplines, and connections between mathematics and meaningful problems in the real world. Mathematics in Context emphasizes the dynamic, active nature of mathematics and the way mathematics enables students to make sense of their world.

In traditional mathematics curricula, the sequence of teaching often proceeds from a generalization, to specific examples, and to applications in context. MIC reverses this sequence; mathematics originates from real problems. The program introduces concepts within realistic contexts that support mathematical abstraction.

Students exiting MIC understand and are able to solve non-routine problems in nearly any mathematical situation they might encounter in their daily lives. In addition, they will have gained powerful mathematical practices, regarding the interconnectedness of mathematical ideas, that they can apply to most new problems typically requiring multiple modes of representation, abstraction, and communication.

This knowledge base will serve as a springboard for students to continue in any endeavor they choose, whether it be further mathematical study in high school and college, technical training in some vocation, or the mere appreciation of mathematical patterns they encounter in their future lives.

Learn more about the MIC curriculum (.pdf).

## Adding it Up: Helping Children Learn Mathematics

In January of 2001, the National Research Council released, “Adding it Up: Helping Children Learn Mathematics” a report on preK-8 math education. The National Research Council (NRC) functions under the auspices of the National Academy of Sciences (NAS), the National Academy of Engineering (NAE), and the Institute of Medicine (IOM). Below is a summary of their findings. (from www.infocusmagazine.org/1.1)

- Students emerging from traditional elementary school arithmetic have developed habits that make the study of algebra more difficult. For example, they have an orientation to execute operations rather than to use them to represent relationships, which leads to the use of the equal sign to announce a result rather than to signify an equality. They also have trouble moving from an addition statement written horizontally to its equivalent subtraction statement (e.g., writing 35 + 42 = 77 as 35 = 77 - 42, or writing x + 42 = 77 as x = 77 - 42).
- The most effective sets of activities for teaching about rational numbers spend time at the outset helping students develop meaning for the different forms of representation. Students work with multiple physical models for rational numbers as well as pictures, realistic contexts, and verbal descriptions. Time is spent helping students connect these supports with the written symbols for rational numbers.
- When class norms allow for students to feel comfortable doing mathematics and sharing their ideas with others, students see themselves as capable of understanding.
- In the brain, the way that new knowledge is organized and connected to previous knowledge is critical to the ability to retrieve and apply that knowledge. Learning with understanding leads to better organization and connections in the brain than does memorizing.
- Knowledge learned with understanding provides a foundation for generating new knowledge and for solving unfamiliar problems.
- A good conceptual understanding of place value in the base-ten system supports multi-digit computational fluency, accurate mental arithmetic, and flexibility with numbers.
- Justifying and explaining ideas improves students' reasoning skills and their conceptual understanding.
- Teachers need to expand the study of data beyond just graphing data. Four key processes are describing, organizing, representing, and analyzing data.
- Elementary students are capable of learning more geometry than is usually taught. Given enough early opportunities to learn about geometric figures, by the end of second grade they should be able to identify a wide range of examples and non-examples of geometric figures; classify, describe, draw, and visualize shapes; and describe and compare shapes based on their attributes.