by Carl Seltzer (excerpted from Middle School Math Interventions)
As teachers build their students' mathematical understanding from the concrete to the semi-concrete to the abstract, manipulatives can be a powerful partner in the process. Working with well-chosen manipulatives helps students to discover the algorithms embedded in the concepts and ensures that they are truly making sense of what they are learning, not simply memorizing. As noted in one famous study, we remember 90% of what we both say and do.
The first step in learning mathematics is to understand the concept being presented. Usually this step requires the use of manipulatives. Certainly it does in the formative stages of a student's learning. If the student does not understand the concept, the learning will not occur or the student will simply memorize something that has no meaning whatsoever to him or her.
Unfortunately, as students move up the mathematical ladder, manipulatives are used less and less frequently to introduce or reinforce concept learning. The thinking is that some concepts are just too abstract for manipulatives.
This is a needless omission, in my view. For students who have failed to grasp the standard algorithms when first introduced, it is possible to reteach difficult concepts using manipulatives no matter what the grade level.
Manipulatives to Use
Many manipulative materials can be used in concept development and the type selected will depend on the concept targeted. However, five manipulatives in particular have been proven to help students learn various concepts and show the patterns that lead to algorithms. These are Unifix Cubes or Snap Cubes, Base 10 Blocks, Pattern Blocks, Geoboards, and Two-Color Counters. This is not to say that other manipulatives have no place in mathematics teaching, but if you have a limited budget, these are the essentials.
The manipulatives for middle school and high school are different. For these grade levels, Two-Color Counters, Algebra Tiles, Pattern Blocks, Geofix, and Omnifix Cubes are essential.
Demonstrating the Multiplication Facts with Color Tiles
Let's start with multiplication. Let's say we want to multiply 2 x 4. If we use color tiles to represent this problem, multiplication can be thought of as finding an area -- a two-dimensional model:
Two sets of 4 rectangles = 8 rectangles.
2 x 4 = 8
2 x 4 is a multiplication fact.
Practicing the placement of tiles in various rectangular arrangements soon demonstrates how the different multiplication facts are derived and how some numbers (prime numbers) can't produce more than two arrangements.
Suppose you have 12 tiles. Six rectangular arrangements can be made:
12 x 1 or 1 x 12, 3 x 4 or 4 x 3, 6 x 2 or 2 x 6
For 9 tiles, only three rectangular arrangements can be made:
3 x 3, 9 x 1, or 1 x 9
For 5 tiles (a prime number), only two rectangular arrangements can be made:
5 x 1 or 1 x 5
Notice that any one of these rectangular arrays can be rotated to give the same design. This demonstrates the commutative property of multiplication.
Demonstrating Division with Base 10 Blocks
Base 10 blocks are a wonderful way to model division. Because multiplication and division are inverse operations, division is simply the process of finding a missing factor in a multiplication problem. Say you want to find the answer to 143 divided by 13. Stated another way, you are looking for 13 x ? = 143.
143 can be modeled with one Hundred flat, four Ten rods, and three Unit cubes. You want to arrange these units into a rectangle so that one side is 13 units long. From there, it's a simple matter to find the dimension of the other side. The answer, 11, can be seen in the figure below:
Demonstrating Fractions with Pattern Blocks
Pattern blocks can be used at each stage of students' developing understanding of the concept of fractions, from modeling the equivalence of fractions (1/4 = 2/8) to multiplying and dividing fractions by other fractions.
For example, let's say we want to multiply 1/2 x 1/3.
Using pattern blocks not only helps students visualize and manipulate fractional relationships instead of simply trying to memorize formulas, but it also produces enormous benefits for a better understanding of algebraic ideas.
Demonstrating Positive and Negative Numbers with Two-Color Counters
One of the best manipulatives you can use to model positive and negative integers is two-color counters. The counters are red on one side and yellow on the other. Let's have yellow represent positive numbers and red represent negative numbers. The opposite of a yellow counter (positive) is a red counter (negative).
Now let's look at the problem (+6) - (-3).
Suppose we have 6 yellow counters (+6) and want to take away 3 red counters (-3), but we don't have any red counters to take away. We could add 3 red counters and 3 yellow counters instead, which in effect would be adding zero [+ 3 + - 3 = 0].
The answer is +9.
Demonstrating Polynomials with Algebra Tiles
It is absolutely vital that students have hands-on experiences with algebra tiles or blocks when learning algebraic concepts. Algebra tiles can help students to understand linear equations that have no exponents and quadratic equations that have exponents to the power of 2.
Most algebra tiles come in the following three forms: Unit tiles are yellow on one side and red on the other. Rods are green on one side and red on the other. Squares are blue on one side and red on the other. The tiles represent the following values:
Yellow unit = +1
Red unit = -1
Green rod = +
x
Red rod = -
x
Blue square = +
x2
Red square = -
x2
Yellow unit (+1) + red unit (-1) = 0
Green rod (
x) + red rod (-
x) = 0
Blue square (
x2) + red square (
-x2) = 0
Adding, subtracting, multiplying, and dividing polynomials can be remarkably simple with algebra tiles. To take one example:
The Joy of Discovery
Effective learning takes place when students are self-reliant, self-confident, willing to take risks, organized, and motivated. However, most teachers will agree that the most important of these is motivation. Motivation comes from many sources, but I believe one of the best sources is the sense of discovery that students experience when they grasp a new concept. To quote Alfred North Whitehead, "From the beginning of his education, the child should experience the joy of discovery."
Manipulatives are not an end in themselves, but a means for students to get excited about mathematics and, from that excitement, to discover what they need to know to build a solid foundation for future learning.
