Manipulatives

Manipulatives: Efficient & Effective for the Concept Taught

Manipulatives are essential learning tools for many students. They are appropriate for ALL students. However, they must be efficient and effective for the concept being taught.

The purpose of using manipulative objects is the explication of the concept. They should not be used as inefficient calculators. The instructor should use relatively easy numbers and the student should be able to "Prove by Construction" to solve problems and create visual imagery which will sustain applications when the manipulatives are removed.

Numeracy & Subitizing

Students can use a domino or dice template and use flat glass marbles to illustrate patterns.
They should use tally marks of craft sticks and bundle when they get two sets of tallies. Bundling with craft sticks and pony tail holders allows the student to construct quantities to ten, bundle a ten or tens on a place value mat to build our number system and name what quantities are made of.
Students can use ten frames - commercial or home made- to create the ordered pairs of ten: 5+5, 6+4, 7+3, 8+2, 9+1. But the composition and decomposition of quantities between 5 and 10 should be stressed with domino patterns. See the article by Douglas Clements, Subitizing. What is it? Why Teach it? In the article he articulates two levels of subitizing, perceptual and conceptual. I believe that teaching young students to "count on to add" and to "count back to subtract" without stressing the subitizing patterns for 6, 7, 8 and 9, is one reason our students cannot subtract or add across a ten. The decomposition of these numbers is central to higher levels of math such as integer operations and algebra. Teachers working with older students can re-investigate these patterns for intervention.

Place Value and Expanded Form

Craft sticks or objects for bundling are efficient methods of building the place value system. After the student understands composing and decomposing quantities, the student can transfer to using Base Ten Blocks. Educators should be aware though that trading is not the same as bundling conceptually and therefore students should begin by building tallies, bundling at ten, building tallies of tens and bundling at one hundred.
Students should "read their quantities with their hands" as they touch and name the place values. They should tell what it is "made of" as they explain the quantities in expanded form.
Place value and the composition of quantities it the second conceptual level in the hierarchy of skills and students must truly understand it before moving on.
Students may use the same manipulatives to prove addition and subtraction problems before transferring to the pictorial level and finally the algorithms. For many students it is beneficial to perform several levels simultaneously. They might build a quantity in craft sticks and move step by step from the concrete manipulative to the abstract level with only numbers as they solve problems involving composing a ten after addition, or decomposing a ten to subtract.

Multiplication and Division Concepts

Students need to see the accumulation of quantities in groups and the decomposition of quantities by repeated subtraction of groups of a quantity. Students can build multiplication models with Unifix cubes for the lower times tables and with shoe strings or craft strings with pony beads for the larger multiplication facts. They can move the beads on the string to skip count, to see the products and the groups as they calculate. It is always helpful to have a "near point reference" available when students are learning multiplication facts.
There is no time table for times tables! Students with disabilities may struggle to automatize the multiplication facts because they are word retrieval activities. There is ample neuropsychological evidence that when student begin learning multiplication facts, they use more of the language hemisphere of the brain whereas subitizing and estimation activities are largely centered in the non-language hemisphere. Students with language based learning disabilities (dyslexia) need to be taught multiplication with fewer facts at a time and practiced to automaticity. Word retrieval deficits for these students creates the perfect storm for difficulties in math. This is why the use of manipulatives is essential. They create visual and tactile memories to support the language of these operations.
Inverse operations should be modeled and taught simultaneously to reduce the load on working memory and language retrieval.
Students can model division with Unifix cubes including the meaning of remainders. At the fourth and fifth grades, they should be encouraged to state the remainder as a fraction of the divisor. Thus, a student breaking a solid color tower of 24 cubes into groups of 7 would say, "24 divided by 7 is 3 with a remainder of 2 of the seven I need to make a new group."

Fraction and Decimal Concepts

Fraction circles made out of cardstock are great early manipulatives, but later students need to see fractions modeled in a variety of representations. These include pattern blocks, bar models, strip models and fraction tiles.
Teachers can find good references for teaching fractions through The Rational Number Project.
My favorite manipulatives for older students include pattern block fraction activities in which we create a place value mat with only "ones" and "fractions of one" so that the students can practice regrouping- decomposing one to model subtraction and composing one from an improper fraction to demonstrate simplifying. The other manipulative I love is fraction tiles but with the numbers face down so that students practice proportional reasoning using colors. The numbers and procedural instruction can get in the way of students truly understanding what happens in fraction operations.
The language used for both place value and fraction concepts is among the most important in mathematics instruction. The conceptual foundation of naming the fraction, the denominator as naming the fractional part creates the foundation for later operations and algebra. When naming fractions initially, be specific in describing the meaning. Two thirds should be taught, for example, as two of three equal parts whose name is two thirds.
Additionally, one should always link fraction manipulatives to the number line so that the student learns that fractions are numbers just as whole numbers are and that they can be used in applications across different concepts.
Decimal fractions need to be introduced as forming place values with fractions based on divisions by ten. I prefer to initially teach fractions using clay and dental floss so that students construct decimal fractions to the thousandth place. Then through a "re-unitizing" activity we move to the more traditional decimal representation. This can only happen though if the students have previously worked through re-unitizing activities with fraction manipulatives.

Pre-Algebra and Algebra

Integer operations should be introduced with a vertical number line. They can be effectively modeled with Unifix or linking cubes. I also use cardboard rolls from wrapping paper to model money problems and bank balances. There is nothing about negative that intuitively says "go left." Students need to understand negative numbers in a context that is meaningful before they can move to the abstract.
Linear functions can also effectively be modeled with Unifix cubes when simple word problems are introduced. Coding the slope as the "constant rate of change" and the y-intercept as the "initial or starting value," students can build solutions to simple linear function word problems and create tables of values.
The four attributes of slope can be modeled with pipe cleaners as can the shapes of most parent functions. Students can also model transformations of functions around the coordinate plane as well as the effects of the leading coefficients on function graphs using pipe cleaners. The teacher can ask the entire class to model changes in the air and see which students understand the concept and which students do not.