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I was settling into my seat at the beginning of a cross-country plane trip when a chatty gentleman sat down next to me and immediately tried to engage me in conversation. I am not antisocial, but I had an enormous amount of work to do and six uninterrupted hours without a phone ringing was very appealing. After some small talk, my seatmate inquired as to what I do for a living. That was my pass. I knew exactly how to turn him off. "I'm a math person," I stated proudly. After recovering from the initial shock, he immediately hailed a flight attendant and tried to have his seat changed. The flight was full and he was out of luck. It didn't take long before he ran out of things to talk about because, after all, what do you actually say to a "math person"? In an attempt to be friendly, he asked me to help balance his checkbook and inquired as to how he might help his son with division. When the drinks were served, I decided to seize the moment and show him that he, too, could "do math."
With a can of soda in front of each us, I asked him, "What do you think is greater, the distance around the top of the can or the height of the can?"
I then posed the same question to all the people sitting around us. We took a vote and the group decided that the height of the can was greater than the distance around the top. "Just look at it," my neighbor said. Then, as a good math teacher, I asked everyone to justify their answer. My new friend immediately broke out in a cold sweat and stammered something about not knowing the formula. "Then, I guess you'll just have to prove your answer another way," I told him.
People immediately got to work. Some used their napkins as measuring devices while others used their necklaces to measure both distances. They rolled the can on a piece of paper and then compared that distance to the height of the can; they put their fingers around the top of the can and then measured the height of the can with the same fingers. They were astounded to find that the distance around the top of the can was significantly greater than the height. They were equally astounded that they could solve this problem, without knowing a formula and, moreover, that they actually enjoyed doing the math.
Posing this same problem to a group of middle schoolers elicited the same responses. They especially enjoyed working together, being actively involved in the learning process, and not having to memorize a formula to solve a problem. Further, the discussion that evolved about circumference, characteristics of circles, and pi was incredible. Goldsmith and Mark (1999) state that, "Mathematical thinking develops through engagement in mathematical work." When students become active participants in their learning, there is a higher degree of understanding involved. Memorization, on the other hand, is a meaningless activity without a cognitive foundation (Moyer and Jones 1998).
Although there is a dearth of research to support the use of manipulatives in the middle grades, research does show that effective teachers use a variety of teaching modes and cater to diverse learning styles. McCoy (1992) found that using math manipulatives was helpful in dealing with students who had math anxiety or an aversion to learning mathematics.
Manipulatives are hands-on, touchy-feely concrete objects that help children better understand the abstraction that we call numbers. Too often, students have memorized a formula rule and then cannot recall it to apply or use. For example, when teaching prime and composite numbers we ask children to memorize that a prime number has only two factors (one and itself) and that a composite number has more than two factors. This is information that they will have to recall time and again especially in dealing with fractions. Why is one neither prime nor composite? Why is two the only even prime number? What special properties do square numbers have?
Children who learn about prime and composite numbers using manipulatives will have a better understanding of their importance. Have students make as many different arrays as they can for each of the following number of tiles: 3, 5, 7, 12, and 15. After making all the arrays for each number, have them list the factors for each number. (Note that an array can run either horizontally or vertically).
Students can draw their own conclusions about the numbers modeled on the left (the prime numbers) and those modeled on the right (the composite numbers).
Numbers on left:
- Only two factors (one and itself)
- Only one array
- Even number of factors
Teachers can tell the children that numbers with these characteristics are called prime numbers.
Numbers on right:
- More than one array
- Even number of factors
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One and itself are always factors
- More than two factors
Numbers with these characteristics are called composite numbers.
Ask children the questions posed above: Why is one neither prime nor composite? Why is two the only even prime number? After learning with manipulatives they will be able to explain their answers to these questions.
Now have students make as many different arrays as they can with 1, 4, 9, 16, and 25 tiles. How do the models for these numbers differ from the models for other numbers?
- A square array can be modeled
- There are an odd number of factors
- One and itself are factors, but there are other factors as well
This process leaves little doubt as to why these numbers are called "square numbers" and how we arrive at a square root. When students learn a concept through rote memorization, there is little or no understanding involved. Oftentimes, because of this, they rely on adults to give them the answer.
Learning with manipulatives provides children with a method of problem solving when they forget a rule. For example, when adding and subtracting integers, there are rules to follow to determine if the answer is positive or negative. However, if children forget those rules, they can use models to solve the problem. For example, they can use positive and negative counters. We can say that • = 1 and o = -1. This means • and o together equal 0, or a zero pair.
If we use only positive counters to show 3 it would look like this: • • •
If we use positive and negative counters to show 3, it could look like this: • • • • o or this • • • • • • o o o
With this little bit of knowledge we can now add and subtract integers.
2 + 3 = • • and • • • = 5
2 + (-3) = • • and o o o = -1
-2 + 3 = o o and • • • = 1
The Principles and Standards for School Mathematics (2000) tells us that we must make math accessible for all children. For children who may have difficulty using positive and negative counters, another option is to present them with a number line.
Manipulatives must be grade-level appropriate for them to be effective. They encourage students to be active rather than passive learners. When teaching with manipulatives, teachers are encouraging understanding rather than memorization. There is an old Chinese Proverb that states:
I hear and I forget
I see and I remember
I do and I understand
Manipulatives are not just for primary students nor are they just for special education students. Manipulatives are for anyone who wants to better understand and, ultimately, apply abstract concepts.
References
Goldsmith, Lynn T., and June Mark. 1999. What is a standards-based mathematics curriculum? Educational Leadership 57, no. 3: 40–44.
McCoy, Leah P. 1992. Correlates of mathematics anxiety. Focus and Learning Problems in Mathematics 14, no. 2: 51–57.
Moyer, Patricia S., and Gail M. Jones. 1998. Tools for cognition: Student free access to manipulative materials in control-versus autonomy-oriented middle grades teachers' classrooms. Columbus, Ohio: Clearinghouse for Science, Mathematics, and Environmental Education.
National Council of Teachers of Mathematics. 2000. Overview of principles and standards for school mathematics. www.nctm.org/standards/principles.htm.
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