# 4.3.1 Tools for Each Grade Level Span Select the links below for an overview of appropriate tools at each grade level span:

Grade Span K–2

Educational research indicates that the most valuable learning occurs when students actively construct their own mathematical understanding. One way to facilitate this is to provide opportunities for children to explore, develop, test, discuss, and apply ideas. Extensive and thoughtful use of physical materials, particularly in the primary grades, is conducive to the concrete kinds of learning that lay a satisfactory foundation for the development of this mathematical understanding."

Johnson, 2012

Grade Span 3–5

Manipulative materials in teaching mathematics to students hold the promise that manipulatives will help students understand mathematics. At the same time, as with any "cure", manipulatives hold potential for harm if they are used poorly. (No matter what the grade level of the students, the sole use of manipulatives should not be for modeling procedures; instead manipulatives should be made available as tools for problem solving). Manipulatives that are improperly used will convince students that two mathematical worlds exist - manipulative and symbolic. All mathematics comes from the real world. Then the real situation must be translated into the symbolism of mathematics for calculating. For example, putting three goats with five goats to get eight goats is the real world situation, but on the mathematics level we say 3 + 5 = 8 (read three add five equals eight). These are not two different worlds but they are in the same world expressing the concepts in different ways."

Teaching Today, 2012

3–5 Tools

Grade Spans 6–8 and 9–12

The effectiveness of hands-on learning does not end in 5th grade. Research indicates that students of all ages benefit by being introduced to mathematical concepts through physical exploration. Planning lessons that proceed from concrete, to pictorial, to abstract representations of concepts makes content mastery accessible to students of all ages. With concrete exploration (through touching, seeing, and doing), students can gain deeper and longer-lasting understandings of math concepts (Teaching Today, 2012).

6–8 and 9–12 Tools