# Mathematics and Geometry

To Handle The Daily Math Problems in Life—And To Sharpen Thinking Skills

How Montessori Math Moves From Concretes to Abstractions: The Example of Long Multiplication

Children first encounter multiplication in our Montessori Primary program. They learn that it is a special form of addition—that is, putting the same quantity together multiple times. They use the Colored Bead Bars for this: these bars are made of different colored beads according to the numerical value of the bar. The 10-bead bars consist of 10 golden beads strung together; the 9-bead bars have 9 dark blue beads; the 8-bead bars are brown, and so on. To do a multiplication problem, let’s say 7 x 4, the student would take four of the white 7 bead bars, and count all the beads to get the result, 28. He would then convert the result into two golden ten bars, and a brown eight bar to symbolize 28.

Gradually, in late Primary and into Elementary, students are introduced to more advanced multiplication problems and strategies to solve them more quickly. They also move from very concrete presentations to more abstract ones, and finally graduate to solving long multiplication problems with pencil and paper alone. Throughout, our teachers have a wide variety of materials at their disposal—materials which build upon each other, and are integrated by a consistent, systematic use of colors (which helps as a memory aid).

- Students learn about the decimal system (place value) with the Golden Beads materials—units of individual beads, bars of ten beads, squares of 100 beads and cubes of 1,000 beads. Our elementary students also get the unique experience of working with the Wooden Hierarchical Material, which demonstrates, in concrete terms, the proportionate difference in size between a single unit and a million!
- They learn skip-counting with the Bead Chains that repeat the colors of the smaller bead bars: for example, a short 5-chain has five light blue 5 bead bars hooked together, and will make a square of 25 when folded together.
- They use the Multiplication Finger Board to understand and begin to memorize the times table. On this board, children set up and develop their own multiplication tables, which they often bind into little booklets and use to memorize their multiplication facts.
- They are introduced to multiplying larger numbers using the Golden Beads—exchanging units of beads to tens for carrying, and tens to hundreds. (Of course, they have first learned to add, and are now simply adding the same quantity several times.)
- They learn to multiply more abstractly with the Stamp Game, where units, tens, hundreds and thousands are represented by color-coded number squares, instead of beads. The photo show 2,321 x 3 set up with the Stamp Game.
- They work with the Small Bead Frame, and then the Large Bead Frame, the Montessori version of an abacus, where place value is indicated by bead position, and where students need to apply math facts to move the right number of units, tens, hundreds and so on.
- They are introduced to long multiplication (where the multiplier has two or more digits) with Montessori’s unique Multiplication Checkerboard material. Here the place value is indicated by the bead’s position on the board, and partial products are made visible. After setting the problem up with numerical tiles placed around the checkerboard, the child places the designated number of colored bead bars in the correct spots. For our example, he would take three light blue 5-bead bars, and place them in the unit square; three green 2-bead bars in the tens square, three yellow 4-bead bars in the hundreds square, and three lavender 6-bead bars in the thousands square.
- Later on, they multiply decimals using the Decimal Checkerboard, applying the same ideas and principles they learned with the Multiplication Checkerboard for integers to decimals.

Throughout, we introduce our students to increasingly more complex multiplication problems and ever larger numbers; we also guide them to apply math facts to work faster:

**More complex problems.**The multiplicand will grow to two digits, then three. The materials help to visualize what that means. For example, the differently colored squares in the rows of the checkerboard indicate the decimal places for the results.**Larger numbers.**Our students are fascinated by and eagerly do problems into the millions and beyond. With the Large Bead Frame, students can do math into the millions—and the Checkerboard can generate results up into the billion range. Not only do these large quantities challenge our students’ skills, they are inherently motivating to youngsters who are enthusiastic about digging into big work.**Using memorized math facts.**Instead of counting out multiple bead bars, then exchanging with the checkerboard, we guide our students to do the math facts in their heads. For instance, to solve 6 x 8, instead of putting eight 6-bead bars on the checkerboard, they arrive at 48 in their heads, and then place an 8-bead bar in the units, and a 4-bead bar in the tens. This shows students how knowing the facts makes them more efficient, and provides motivation to learn the facts. It’s also necessary to solve problems on the Bead Frames—an example of how mastery at one stage in the sequence opens the door to the next stage.**Writing the problem on graph paper.**We teach students how to write down the problems on paper using the correct place values, and how to document partial products. This facilitates cross-checking and identifying the source of errors.

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