Montessori math alone is worth the price of a Montessori education. That may sound like an outrageous claim, until you consider how many children, how many adults are held back in life by an inability to solve mathematical every-day problems (e.g., evaluating whether they can afford a home), how many students avoid profitable, fulfilling careers in science, technology and engineering because of a fear of numbers.
Montessori math is the best possible foundation your child can get for being comfortable and competent in the world of numbers.
Here are four factors that make this Montessori math miracle possible:
- An ingenious set of materials that progress gradually from the concrete to the abstract. In contrast to most other programs, which introduce math in a rather abstract way, Montessori math begins with concrete materials, and then slowly progresses toward abstraction. For example, Children’s House students learn about the decimal system with the Golden Bead material, which contains single beads for units, ten-bars of beads for tens, a square of a hundred beads for one hundred, and a cube of a thousand beads for one thousand. Initially, children add or subtract big numbers with these beads, but over time other materials – the Stamp Game, the Dot Game, the Small and Large Bead frame – enable children to progress to abstract, paper-and-pencil math. Other materials concretize fractions, algebraic formulas, and geometric concepts, such as the Pythagorean theorem.
- An instructional sequence that recognizes and harnesses the interest by young elementary children in very large numbers and very small numbers. In most math programs, children start with small numbers and slowly work their way up to bigger numbers—up to 30 in Kindergarten, up to 100 in 1st grade, up to 1,000 in 2nd grade and so on. In contrast, in Montessori, we begin by introducing large numbers, all the way to 10,000 by age five or six, and all the way into the millions and beyond in the early elementary years. As Dr. Montessori observed, children are interested in big numbers, big problems—and with the unique materials, we can harness that interest into advanced, early math accomplishments.
- Individualization and mastery as a fundamental to the approach, not an after-thought. In a traditional program, a certain time is allocated in the instructional calendar for all students in a class to learn a given math concept—how to add and subtract to 30, how to reduce fractions, how to set up and solve a basic word problem. The teacher usually teaches to the "average" student, with some enrichment for the advanced ones, and remedial help for those struggling. But even with these adjustments, the calendar and government-set standards dictate that a teacher must move on after a certain time—even if several students haven’t mastered a concept. In math, this is deadly: as children move on without understanding foundational concepts or automatizing key skills, they become less and less confident in math, and may ultimately fear the subject. In Montessori, we avoid this "swiss cheese" teaching that leaves holes in students’ math skills by making all of our instruction 100% individualized: we only introduce a student to more advanced math concepts after the earlier ones have truly been mastered. With our mixed-age classrooms, it doesn’t matter how long a child take, or how quick she may be: we will group her with other at the same level, and provide as much additional support as needed to ensure she reaches her full math potential.
- An equal emphasis on conceptual understanding of mathematical ideas and hard math facts skills and knowledge. Often, parents face a touch choice: they can opt for a traditional school environment, with an emphasis on hard facts (e.g., learning the multiplication tables, and getting quick at basic arithmetic)—or they can opt for a progressive program, which emphasizes "mathematical thinking", but which, more often than not, shows a disdain for "mere memorization." In Montessori, we recognize the irrefutable fact that you cannot be good at math and comfortable with numbers without both together. That’s why our students have daily math facts practice—and time to thoroughly explore mathematical concepts with the Montessori materials. It’s why we motivate math fact learning—by showing students how they can improve their speed and efficiency with the materials having recall of facts ("7×6 = 42", instead of counting bead bars all the way up to 42.) It’s why we have a multitude of materials that make learning math facts fun, rather than relying on stickers or other extrinsic motivators.
Not surprisingly, Montessori children often are much advanced in their grasp of math. More importantly, however, they usually love it—something that will open doors for them not just in school, but also in life.
How this works in practice: multiplication in Montessori math
Content reproduced with permission from LePort Schools
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 Bead 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 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.) The photo above shows the quantity of 6,425 set up with the Golden Beads, a set-up that makes it very clear what large numbers the children are dealing with, as there are thousands of beads in this set-up.
- 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 are introduced to long multiplication (where the multiplier has two or more digits) with Montessori’s unique 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 checker board, 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.
This YouTube Video provides an excellent demonstration of using the checkerboard to solve a single-digit multiplication problem, a point of familiarity for students just being introduced to the Checkerboard: http://www.youtube.com/watch?v=XqlD6P3ogJc
- 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.
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 six 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 them 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.