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# Use Appropriate Tools Strategically

Mathematically proficient students consider the available tools when solving a mathematical problem. In the early grades, these tools might include pencil, paper, concrete models (manipulatives), a ruler, a calculator, or applets or apps.

Manipulatives are defined as “objects that appeal to several senses and that can be touched, moved about, rearranged, and otherwise handled by children” (Kennedy, 1986, p. 6). Using manipulatives in the early grades is one way of making mathematics learning more meaningful to students (Stein & Bovalino, 2001) as they are used to make abstract ideas more concrete (Moyer, 2001). Manipulatives and modeling are the first step in creating an environment where students can begin to understand abstract mathematical concepts in a variety of contexts. Decomposing and recomposing numbers should be done with manipulatives and models until it becomes something students can do mentally. However, Ball (1992) asserted that a manipulative does not, by itself, carry the intended meaning and does not guarantee that mathematical understanding will result from the use of manipulatives. It is actually the expertise of the teacher in the use of manipulatives (Raphael & Wahlstrom, 1989: Sowell, 1989) and the amount of time students are given to interact with the manipulatives that leads to increased achievement (Moyer & Jones, 2004).

Calculators should be used with care and caution in elementary school. Calculators should be available as computational tools, particularly when many or cumbersome computations are needed to solve problems. However, when the focus of the lesson is on developing computational skills or algorithms, the calculator should not be used. Today, the calculator is a commonly used computational tool outside the classroom, and the environment inside the classroom should reflect this reality.

All mental math instruction should ideally be focused on letting children find ways to calculate so they will concentrate on numbers and operations instead of on routines. Estimation should allow various reasonable responses which require thinking about the meaning of numbers (Sowder & Schappelle, 1994). Proficient students are sufficiently familiar with tools appropriate for their grade to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations.