• The A to Z's of Hands on Math


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    I remember the first time I encounteredCuisenaire rods in a graduate workshop. “Be sure you allow time forkids to play with them,” began the instructor, looking around at aroom full of educators turning the tiny blocks into towers andpatterns of stripes. As we knocked over towers and tried to payattention to the instructions on how to use these colorful littlethings with students, we laughed. Even the adults were drawn toplaying with their math.

    I've since learned that there are amillion ways to play with your math and hold it in your hands. It'snot a necessary step for absolutely every student, but for most, itmakes math more fun, more tactile, and easier to understand. Mathmanipulatives can be a lifeline for some math strugglers, a shortcutto understanding for some thinkers, and a means to get to a deeperunderstanding for others. There are dozens of different products outthere for both arithmetic and geometry and even an array of productsfor algebra. There are also ways to make math hands on by bringing itinto the real world in other ways.

    The simplest math manipulatives areprobably counters. Nearly all of us were born with a nice set of tenof them attached directly to our hands. You can get fancy, of course,and buy a set or you can use what you have around the house. For thesmallest math learners, going from the symbolic abstraction ofnumbers to the reality of a quantity of things sitting in front ofyou is a big step in early learning. In fact, it was a big step inour development as a species to be able to do this. If you and your kids ever want to think about just how big a step, the Terry Jones documentary The Story of 1 is a fun look at the history of how numbers became numbers. We're so used to being able to think of four chairs or four apples or four fingers as just “4” that it's easy to underestimate how hard it is for the brand new learner.

    Next comes the even harder in counting up past your fingers. Going beyond ten means beginning to see a pattern in the way we structure our numbers and understanding placevalue. This is where counters can be useful to start, but can quicklybecome tedious. Different possibilities begin to really open up. Kids can place numbers on a hundred board or put manipulatives together tosee bigger numbers. Some materials, like Unifix cubes, take the counter and turn it into a piece in the puzzle. Join them togetherand you have a rod that represents a larger number. (Note that Legos are a good substitute for these.) Other manipulatives, like Montessori beads, Cuisenaire rods, or base ten blocks let kids seethe patterns in the numbers by putting them together to make staircases, squares, and other patterns.

    One of the oldest and best known mathmanipulatives is the abacus. There are lots of different versions ofthe abacus out there. The most popular one for homeschoolers is someversion of the abacus used by the Right Start curriculum where eachline has ten beads that are two colors to help students see thehalfway point and each level represents another place value. Movingthe beads means adding or subtracting numbers. If you become fluentwith an abacus, it can calculate large numbers extremely quickly. InJapan, some people practice the mental abacus whereby they use theimage of the abacus in their minds to do the calculations incompetitions, moving the beads back and forth mentally rather thanphysically.

    There are also lots of great toolsspecific to various hands on tasks. Many people have clocks forlearning to tell time or play money for learning to count. You don'tneed play money though. Real coins are great tools for place value aswell once a child understands that the penny is ones, the dime istens and the dollar is hundreds. Fraction wheels are especiallyuseful for younger students first learning about fractions. Spinnersand dice from games help kids learn about probability. There arealgebra tiles are a great way to introduce the abstract ideas behindalgebra. The Hands on Equations program helps explain basic conceptsin algebra, including integers and variables using a scale, pegs, andnumber cubes.

    My favorite tool for arithmetic isstill the Cuisenaire rods, simply because I believe they're the mostversatile single manipulative you can have. Cuisenaire rods don'thave markings to allow kids to count up to their value the way thatmost other manipulative blocks do. Instead, students are encouragedto see the rod as a whole, which helps take the symbolicrepresentation a step further. After playing with them for a fewdays, most kids will internalize their values. More importantly, the“six” rod or the “three” rod can be used to representsomething more than 6 or 3. They can show fractions, ratios, negativenumbers, or help kids see why certain triangle measurements won'twork. They are an incredibly flexible tool. Every time I think we'refinished with Cuisenaire rods in our math journey, they seem to comeout again for a lesson. I suspect I won't feel really comfortablepassing our set on until both my kids are past high school levelgeometry.

    Geometry has special manipulatives, aswell. While there are some which are mostly just practical, most ofthese particularly invite play. Like a box of Cuisenaire rods, abucket of pattern tiles is almost impossible not to play with. Evenolder kids tend to get draw into making patterns and exploring shapesby tessellating them out onto a table. Pentominoes and tangram tilesare made for doing puzzles. However, some of the best hands onactivities with geometry involve making your own discoveries orbuilding things yourselves. What better way to learn that all theangles in a triangle add up to 180 degrees than to make and cut uptriangles to show the angles form a line? Or what better way to showhow shapes look flattened into nets than to actually make shapes?

    In general, that's the step beyondmanipulatives and into the real world. We start by helping young kidslearn about the abstraction of numbers. Manipulatives help guidestudents through these abstractions, but by the time kids are inupper elementary and middle school, we've often moved so far awayfrom those counters that it's important to remember to dip back intothe real world with math labs and projects or simply using tools toexplore math in the real world so it doesn't become only a conceptualabstraction. Of course, word problems can do this too, but just likefor younger kids who are holding their first Cuisenaire rods ormoving their first beads across an abacus, there is something specialabout physical interaction for most students. Some ways we've takenconcepts to the real world has been building models with ratio,playing with origami, making hexaflexagons, playing store, and makingIncan counting ropes.

    * I've said what my favoritemanipulatives for math are. What are yours?

    * Do you use manipulatives? Why or whynot?

    * Do you agree that most kids needmanipulatives?

    * What else do you do to help kids movebetween the abstract and the concrete in math?
    This article was originally published in forum thread: The A to Z's of Hands on Math started by farrarwilliams View original post
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