Balanced Math Program
“It is only when you build from within that you really understand something. If children don’t build from within and you just try to explain it to a child then it’s not really learned. It is only rote, and that’s not really understanding.”
Ann Badeau, Second Grade Teacher
We believe that math needs to be taught conceptually, beginning with the contrete at all grade levels and that mathematical understanding is deepened when students learn the concepts in the context of problem solving. To enable students to make sense of math concepts, a wide variety of manipulatives need to be available. Opportunities to explore and discuss mathematical thinking should be an integral part of instruction throughout the grades. After students have developed conceptual understanding about numbers and operations, procedures are introduced. This means that numerical relationships should be developed before we ask students to learn computational algorithms or basic facts. We want our students to be numerically powerful. We want them to be able to develop meaning for numbers and operations, look for relationships among numbers and operations, understand computational strategies, use them appropriately and efficiently, and make sense of numerical situations. We also want our students to develop grade-appropriate conceptual understanding of core ideas in all the strands of mathematics.
To create students who are both mathematically powerful and competent learners of mathematics, we need a balanced math program that emphasizes number sense and problem solving while teaching the core concepts of all the strands. A balanced math program includes the following elements:
- Number Sense Activities: Mental Math, Estimation, Number Talks
- Student-Centered Concept Development: Opportunities for students to discover concepts using hands-on or problem-based learning activities
- Learning Through Communication: Students explain their thinking and justify their answers
- Rich Problem Solving Experiences: Students develop and connect their problem solving skills through real-life problem solving experiences
- Performance-Based and Standards Assessments: Students are assessed in both performance assessments as well as classroom and state standardized tests
TEACHING METHODS AND MATERIALS
State adopted math textbooks provide one resource tool for teaching math; however, the math text should not drive the curriculum and instruction of a well-balanced math program. These books provide structure, but shouldn’t be the main method of teaching. A well-balanced math program should be driven first, by the content standards, and second, by a variety of instructional strategies that support student learning of concepts and the development of number sense. Teachers should be well versed in what we refer to as “essential math standards” so they are able to cover all the essential standards by the end of the year. The following are teaching methods that should be in place in each classroom:
Number Talks or Mental Math Sessions
These are brief, daily sessions that provide students with opportunities to practice mental math and to solve arithmetic problems in a variety of ways. Typically, this involves the teacher posting an equation and having the students share their individual strategies with the class. The teacher acts as a facilitator who encourages a variety of ways to solve problems and helps model efficient thinking when necessary.
Student-Centered Concept Development
Many state-adopted textbooks do not provide a thorough and developmentally appropriate method of teaching concepts to the students. This requires teachers to use resources beyond the adopted textbook in order for students to experience a rich and discovery-based method of understanding key concepts for their grade level. For grades K-5, we recommend using the most important 2-3 units from the Investigations series*, as well as other resources such as Nancy Norman’s Math Games for the Primary Grades, Kathy Richardson’s Developing Number Concept series, Marilyn Burns’ About Teaching Mathematics and Into Arithmetic series, and Elizabeth and D. Miller’s Read it, Draw it, Solve it series. (*Specific recommended materials list available upon request through our Curriculum Department.)
In middle school, grades 5-8, instructional activities should focus on deepening conceptual understanding of the mathematics in all the strands: Data Analysis, Number, Algebra, Geometry, Measurement, and Probability. Instructional materials used may be CPM (College Preparatory Mathematics) or a standards-based textbook, but conceptual understanding or sense-making should be the primary focus of all instruction. The goal of these activities is to prepare students for the abstract thinking necessary for success in Algebra and Geometry.
Learning Through Communication
This is a classroom expectation that students will consistently explain their thinking as they solve equations and problems. Teachers should ensure that the question, “How do you know?” is a part of the daily vocabulary so that students have practice articulating their thinking. This type of communication cements student learning and can be done verbally or in writing, including pictures and words. Students should also communicate their thinking in problem solving activities so teachers and students have an understanding of how the students solved the problem.
Rich Problem Solving Experiences
Problem solving involves using a strategy or strategies to make sense of a problem. The ten key strategies that should be taught in every classroom are: Act Out or Use Objects; Make a Picture, Diagram or Model; Use or Make a Table, Chart, or Graph; Make an Organized List; Guess and Check; Use or Look for a Pattern; Work Backwards; Use Logical Reasoning; Make it Simpler, and Brainstorm. These strategies are introduced in many textbooks, but they need to be taught using a more developmentally appropriate approach. We believe that the best teaching occurs when teachers are able to choose the materials that best match their students’ developmental and conceptual needs–materials should not drive the instruction. Each teacher should consider how to best teach these strategies for their grade level so that students are taught these strategies K-8. In addition, real-life problem solving experiences should be used whenever possible. Finally, school sites will incorporate Problem of the Month activities to help illustrate the importance of problem solving strategies and to create a community of learners at each school, as well as to guide instruction.
- Each teacher will administer his/her own classroom assessments to determine student progress on math standards. These can be derived from current textbooks or other developmentally appropriate assessments that the teacher deems useful.
- Each teacher will administer the state standardized STAR Mathematics test (Grades 2-8) in the spring of the year.
- Teachers in grades 2-7 will administer the performance-based MARS assessment in March of each year. This assessment is a series of multi-step problem solving tasks that require the students to not only solve the problem correctly, but to explain his/her thinking. To prepare students for this assessment, the district will provide three pre-assessments (two mandatory, one optional) for each grade level. Teachers can score and analyze these pre-assessments and use the information to not only inform their instruction, but to communicate to their students about their progress.
Written by Administrative Council with additional drafts by Pam Jasso, Dina Flanagan, Sarah Orton, Patty Wool and Marge Trainer.