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Have you ever visited a new city and felt confused about finding your way –– even if you had a map and directions? After a few days you probably got a feel for the area, and even if you became lost from time to time, you could count on familiar landmarks and maps to help you on your way. With enough exploring, you ventured with more confidence.
We can think of the value of firsthand experiences for learning mathematics in a similar way. Math has many standards –– number sense, patterns, measurement, geometry, statistics/probability, computation –– and they're often unfamiliar, abstract, and confusing to students. We need to help children develop the ability and confidence to find their way around in each of these areas, see how they connect, and know what to do should they forget a fact or procedure.
Manipulatives are concrete objects that are used to symbolize values in math. Examples of manipulatives are Base Ten Blocks (rods, blocks and cubes), Tangrams and Pentominoes (geometrical shapes), colored beans (probability), dice, and lots of tools used for math strategy games. Math is largely an abstract science, one that is done through invisible lines of thought in our minds. Using manipulatives gives us a hands-on, visible way to show math processes. It also allows us to double check values by re-tracing our steps. For instance, as we learn to add, it is helpful to use beans, teddy bears or other objects to SHOW the value we are looking for. Then we can use words or numerals to RECORD the value. Finally, words can be used to EXPLAIN how a problem was solved. Using manipulatives gives learners a solid, concrete foundation on which to build advanced math concepts.
- Manipulatives help make abstract ideas concrete. A picture may be worth a thousand words, but there's no substitute for firsthand experience. Children learn by doing and touching. Tangible experiences help children retain their learning. Manipulatives give students ways to construct physical models of abstract mathematical ideas.
- Manipulatives lift math off textbook pages. While we want students to become comfortable and proficient with the language of math –– everything from the plus sign to the notations of algebra –– words and symbols only represent ideas. Ideas exist in children's minds, and manipulatives help them construct an understanding of ideas that they can then connect to mathematical vocabulary and symbols. Math situations come alive!
- One goal of the National Council for the Teachers of Mathematics Standards is to build students' confidence with mathematics. Manipulatives build students' confidence by giving them a way to test and confirm their reasoning. If students have physical evidence of how their thinking works, their understanding is more robust.
- Manipulatives are useful tools for solving problems. In searching for solutions, architects construct models of buildings, engineers build prototypes of equipment, and doctors use tools and computers to predict the impact of medical procedures. In the same way, manipulative materials serve as concrete models for students to use to solve problems.
- Manipulatives make learning math interesting and enjoyable. Give students the choice of working on a page of problems or solving a problem with colorful and interestingly shaped blocks, and there's no contest. Manipulatives intrigue and motivate while helping students learn.
From day one, we talk with students about how manipulatives help them learn math. These discussions are essential for first-time users and useful refreshers to refocus from time to time. We precede guided math learning by giving children time to explore each manipulative. Teacher demonstrations alone are like eating a papaya in front of the class and expecting children to know how it tastes. Free exploration time also allows students to satisfy their curiosity so they don't become distracted from the assigned tasks. We expect children to see if tiles can fall like dominoes; build tall towers with rods; or construct rockets out of cubes. After children have explored a material, we ask what they've discovered and record their observations on a chart so their classmates can get insights from their ideas.
Ground rules are also set for using the materials. We talk about the similarities and differences between using manipulatives in class and playing with toys or games. With toys or games, children can make up their own rules. With manipulatives, they are given specific problems and activities. We make clear, however, that they're free to make discoveries and explore new ideas. We have set up a system for storing materials and familiarize students with it. It's important for students to know where and how to store materials. A clear system makes the materials more accessible.
Once this process is in place, we can use the manipulatives for problem solving. After problem solving sessions, students often journal about their math learning. Children can use a combination of pictures and words to explain their math thinking and record solutions. Manipulatives support this writing component and provide concrete objects for children to describe. Periodically throughout the year we will have children take home materials and activities to do with their families. This gives the kids a chance to “show what you know”.
Because many of our math lessons involve class resources, there won’t be as much paper/pencil work coming home. It is also why it is so important that students are present for each math lesson so they have a chance to use the materials. Make-up work just won’t be the same learning experience.
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