Using Math Manipulatives to Aid Learning
The Need for Math Manipulatives
The main goal of today's mathematics instruction is to teach students to be problem solvers and flexible thinkers who can apply mathematical ideas to a wide variety of situations. This goal is much different from that of a few years ago when proficiency in basic paper/pencil computation was the main thrust. Making today's goal a part of your math lessons may require a renovation in the way you organize and teach your math classes.
In order for a student to be a problem solver, he must fully understand math concepts— not just memorize prescribed steps to get correct answers. Many students commonly view math as a collection of arbitrary rules to memorize, and hence they dislike the subject. We teachers must do all that we can to change that misconception. Math Manipulatives are the tools! Manipulatives are objects students use to actively learn a concept. For example, when a first-grade student is presented with the number 42 for the first time, he may see a 4 but not understand that it represents four tens. By using Unifix cubes, he can model 42 by making four trains of ten and two ones and begin to visualize what 42 really means.
Teaching a new math concept (regardless of the grade level) should always begin at the concrete stage (where the students are using math manipulatives), then move to the semi-concrete stage (where the students are watching teacher demonstrations or using pictures), and finally progress to the abstract stage (where the students are using only numerals). Each new concept introduced with manipulatives makes math come alive for your students and gives meaning to abstract ideas through experiences with real objects. This teaching makes the students active participants in the learning process. Through touching and moving objects, learning takes place and the "light bulb" goes on. The involvement of multiple senses in the learning process tends to make the learning more permanent. In this way learning becomes an active process of building knowledge and making sense.
Math Manipulatives in the Upper Grades
"Learning" math requires learners of all ages to actively participate. The skills and concepts introduced to older students are increasingly complex; "concrete" introductions often pave the way to true understanding. For example, while students in fifth and sixth grades would no longer need to use manipulatives for addition and subtraction of whole numbers (this is understood at the abstract level now), they would need concrete models for the introduction of percent, ratio, division of fractions, geometric formulas, and integers.
Have you ever mixed up which geometric formula to use to find the circumference of a circle and which one to use to find the area of a circle? It is d or r? Perhaps if you learned only by rote memorization, you never attached any meaning to these formulas. When students use manipulatives to solve and verify problems themselves, they begin to own the solutions.
For instance, when teaching students how to find the circumference of a circle, have them work with a partner and participate in an activity. Give each pair of students several cans (cylinders) of various sizes and a length of string. Direct the students to measure the diameter of each can with the string and then to find the distance around the can (circumference or perimeter), using the string. Ask the students whether they notice any relationship between the circumference and the diameter of each can. They should have discovered that the circumference is always a little more than three times the diameter. This special relationship then leads into your explanation of "pi"—equalling approximately 3.14—and the formula C = d. You could even tie in a biblical application here by having the students read II Chronicles 4:2, which describes a molten sea being made for temple worship that is ten cubits from brim to brim (diameter), and has a line of thirty cubits that did compass it round about (circumference).
Managing Math Manipulatives
Manipulatives can be a challenge to manage because they add more activity and more noise, and they require space and organization. But they can be implemented successfully with a little preplanning and thought.
Here are a few tips to keep in mind:
- Prepackage manipulative materials into individual sets, using resealable plastic bags, envelopes, or plastic containers with lids.
- Store math manipulatives in durable, labeled containers, such as plastic storage baskets or shoe boxes.
- Keep a "spare parts" container for those who will inevitably lose a piece.
- Plan a method of distribution that works for you (e.g., student helpers, teacher's aids).
- Make students responsible for care and cleanup of manipulatives.
- Cut down on the noise by using construction-paper work mats.
- To reduce costs, prepare some teacher-made manipulatives. They can be just as effective as commercial items. Enlist parent volunteers to help you make them.
You will soon be reaping the benefits of the seeds you have sown in purchasing, making, planning, and organizing your manipulatives for teaching. Your students will begin to enjoy math, and you will be relieving them of "math anxiety" by teaching them concepts they will truly understand. In the book of Proverbs, Solomon teaches that "knowledge is easy unto him that understandeth." (Proverbs 14:6)
Reprinted from Balance, a publication of the School of Education, Bob Jones University. Used with permission of Bob Jones University. Please write BJU Press, for permission to reproduce this article.
All of BJU Press's elementary math textbooks have corresponding math manipulatives and are designed to develop critical-thinking skills, a biblical worldview, and a love of learning. Check them out!