Moving Math

Moving Math

By Cheryl Allen

The Montessori math materials might be some of the most memorable materials in the curriculum. I can still feel the weight of the thousand cube from when I held it in my hand as a five- or six-year-old.

I am not young, so all our classroom beads were glass. The visual difference and tactile difference between the unit, ten bar, hundred square, and thousand cube brought numbers to life. I could spend (what seemed like) hours taking my tray over to the back corner of the classroom to get four ‘tens’ or five ‘thousands’ to build a number with my friend Blake. We could build and write numbers up to 9,999. We could add and subtract, bringing more thousand cubes or returning them to the shelf. We could make exchanges, carrying ten hundred squares over to the shelf to get that thousand cube. I was supposed to take the tray for those, but I loved carrying the thousand cube in my hands. It let me know that one thousand was a lot; it was heavy, much heavier than one unit. It helped me feel the differences in place value.

Montessori’s Golden Beads introduce place value. They show the place values through sight and weight; a unit is one golden bead, and a thousand cube is a thousand golden beads connected together into a cube. This attractive material primes the brain for much more advanced work later. Ten squared is one hundred and is a square shape, ten cubed is one thousand and makes a cube. When we added, we built one number, setting out our units, ten bars, hundred squares, and thousand cubes across the work rug as you would write it. We set out number cards next to the number, so we saw the written form of the number and saw it represented with the golden beads. Then we did it again for the next addend. When it was time to add, we joined all our units together and counted them. If we had more than nine, we needed to make an exchange by taking ten units back to the shelf and getting a ten bar. We repeated the process with each column and admired the sum. Blake and I were excited to go upstairs to the Lower Elementary classroom, where we would get to make even larger numbers. We were a bit disappointed to discover that there were not even more thousand cubes in that classroom. We were limited in our work by the number of materials, and our school was limited by the amount of space the materials took up.

Miss O’Connor, our lower elementary guide, showed us the Stamp Game. Each tile is about one-inch square. The tiles were in the same colors as the written numbers we had been using, and she showed us the comparison. The green tile with the ‘1’ written on it represented the unit; the blue tile with the ‘10’ written on it represented the tens; the red tile with the ‘100’ written on it represented the hundreds, and the green tile with the ‘1000’ written on it represented the thousands. As adults, we describe this as moving from the more concrete (when size and weight are involved in the material and their representation of each place value) to the more abstract, when each tile is the same size with different colors to distinguish the place value. The ‘thousands’ is the unit position of the thousand family, so it is the same color as units of the simple family (and units of the millions, billions, and so on for each family). Blake and I added and subtracted numbers with the Stamp Game, using only one work rug each, and we learned to multiply and divide using the same Stamp Game.

Years later, when I became a lower elementary guide myself, I came to appreciate the Stamp Game from a different perspective. This material can be used for so many lessons. Sometimes parents worry that their child is still using the Stamp Game, as if they are ‘stuck.’ The Stamp Game can be the material used for simple addition, subtraction, multiplication, and division (both grouping and sharing). When the child is ready, the Stamp Game will stick with them through double-digit multiplication and division, squaring numbers, and finding the square root of given numbers. Using the Stamp Game, older elementary students often become excited about creating squares and finding square roots. Creating a square number with the Stamp Game lets a child see the geometry of how a square is built, a concrete understanding of an advanced subject.

The Stamp Game tiles are housed in a box with four equal sections, one for each place value, and four smaller sections that hold skittles for multiplication and division work. To add and subtract with the Stamp Game, we set it up the same way we set up the Golden Beads. Students no longer need to go back to the shelf to get materials to build their number and then again to exchange them, the ‘bank’ comes to them as the box of tiles. As students age, they can focus for longer periods of time and do not require as much movement as a part of their lesson.

To multiply, we build the same number over and over again, as many times as our multiplier tells us to do so. To remember to build the correct number of times, we set the skittles along the side. For instance, if we were to multiply 3,427 by 6 we would set out six green skittles, build 3,427 by each skittle, then join them to get 20,562. What happens when we have ten thousands with a material that only goes through thousands? We just turn over a ‘10’ stamp in the ten thousands place and remember that it represents ten thousand. When we multiply with double digits, we discuss the fact that each ‘ten’ gets ten times what the unit gets. If we multiply 3,427 by 34, we would set out three blue skittles to represent the thirty and four green skittles to represent the units. Rather than building the number 34 times (although someone always tries it once), we build 3,427 four times, once for each unit, and 34,270 three times, once for each ten.

Again, we get to join all our units, tens, hundreds, thousands, and even ten thousands to find that our answer is 116,518. Guess how we show the hundred thousand?

Most classrooms have Racks and Tubes, or test-tube division which is so exciting — seven wooden racks, 70 tests tubes, 700 beads! Yet, using the Stamp Game for division before Racks and Tubes sets students up for success. To divide, we use our skittles again to represent our divisor and build our dividend in containers or small piles closer to the bottom of our rug. If we were to divide 5,838 by 7, we would set out seven green skittles and, beginning with our thousands, share equally to each skittle. Since we cannot share the thousands equally, we have a lot of exchanges to make, then we start sharing our hundreds among the skittles, exchange if needed, and keep going until we have shared as many stamps as we can. If we have any left over, we have a remainder. In this case, our quotient is 834, each skittle will have stamps representing 834. Just as with multiplication, we can use the Stamp Game for single-digit division; double-digit division with the blue skittles; triple-digit division with the red skittles; and quadruple-digit division with special larger green Skittles.

Using the Stamp Game, you can also see what happens when you multiply a number by itself. If you did not get the chance to discover that 27 taken 27 times makes a square when you set out the stamps and the skittles, you may have missed out on the magic of squaring numbers. There is a pattern of squares and rectangles when you create a square number, one you build if you are starting from the squared number, and one you begin with if you are finding the square root. Rather than square roots being a memorization of numbers or a formula, it becomes something of beauty, created and understood.

That small box containing multi-colored tiles and skittles is more like the wardrobe leading to Narnia, a doorway to a world of learning, beauty, and excitement. As much as the memory of the feel of Golden Beads stayed with me since I used them as a child, the Stamp Game is the one material that I would not want my classroom to be without as a guide. If your child is still doing the Stamp Game, the question you want to ask is, “What are you doing with the Stamp Game?” Here is an opportunity for your child to teach you. •