Montessori Rationale Melissa Plunkett Montessori has allowed for the development of a peaceful and whole child with her study and materials. She brought us the phrase follow the child, as that is how we might all be treated. Not being bent, broken and tamed, but allowed to flourish and develop in the natural and whole way as it were meant to be. With this method the mathematical mind was brought to light as being the forefront of the child s development. It has always been this Montessori math that intrigued me the most. Not just the materials but the way the abstract was suddenly made solid to me when I was learning the materials and their functions. this is how I wish I would ve been taught and the if only I had this chance tells me that as long as we follow the child, use the materials as they were meant to be used and feed this mathematical mind the number of children who could be blessed by the method is endless. Montessori begins with the most basic, recognizing size difference (bigger and smallerknobbed cylinders) and how one might look compared to two or three or ten through actually touching the numbers (red/blue rods) and one-to-one correspondence through so many materials (spindle boxes, cards and counters). Before moving on to newer concepts the child is allowed to practice and absorb one material and concept at a time at his/her own pace. There is no rushing through the process, no need to keep up or compete, just learning for the joy of learning. The concept of moving from concrete to abstract also lies at the heart of the Montessori curriculum. We give them real and concrete numbers that they can feel, touch, move and see.
We also supply materials that they will continue to see and use throughout all operations and for their entire schooling career. So we begin with introducing the Golden beads, one bead is called a unit and symbolizes the number one. You continue to count one, two, three, until you have a line of ten unit beads and then we ( tens never stay loose ) connect them all together to make a ten bar. These ten bars are much easier to manipulate, still giving a representation (actually having ten items) but can be moved around without making such a mess. We count these ten bars (one ten, two tens, three tens) until we come to ten tens and again ( tens never stay loose ) until we have ten, ten bars and hook them together to make a hundred square, Again these squares still give the representation of 100 items but are easier to manipulate. Continue counting the hundred squares until we reach ten hundred squares in which we convert to a thousand cube. The golden beads build a foundation for our math as well as give us a simple way to work with big, big numbers. We will come back to these golden beads at the beginning of introducing each new operation. You will also see the children move back to these materials when the more abstract ones begin to overwhelm them or return to them for fun. After golden beads we move on to the stamp game. The stamp game begins our basic passage to abstraction with color coding the place values and place value families. So we have green for units (as well as one thousand, one million, one billion), blue for tens (ten, ten thousands, ten millions, ten billions) and red for hundreds (hundreds, hundred thousands, hundred millions, hundred billions, etc.). In the stamp game each place is represented by a tile with that number on the tile. We begin by building numbers with the tiles and move forward to operations. As we build the numbers (using 9 green unit tiles to build the number nine) and reach ten we remember the phrase ten never stays loose and so we will never have ten tiles
in a row we would exchange (oh I think we might hear that word often in math) ten unit tiles for one ten tile. We can also borrow (another term we might hear often) a ten tile, to get ten units if we were to need it. Now we might introduce the child to our dot game (notice the use of the word gamethis doesn t mean that it isn t work but that it is work that is fun and as such we call it a game). With this material you continue the use of color-coded place values, using a grid and making dots to represent your numbers. Seems a little more abstract to me! So in our dot game you make the number 2,345, we place 5 green dots into the unit column grid, then we place 4 blue dots into our tens column grid, then we place 3 red dots into our hundred column grid and finally two green dots in our thousands column grid (keep in mind this is how I do it, the columns are color coded so that you could use a black dot in each column if you would like). You can then add a number to this, take some of these dots away (subtract) or make this number several times to multiply. The next step on our passage to abstraction is the bead frame; this material reminds me of the all too familiar and glorious abacus. Again our beads are still color coded (place value) and each bead row only has ten beads. Sliding beads from the bank (left side) to the right builds our numbers. What a simple and easy tool! There aren t a lot of items to move around, so the number 20,000 is simply represented by two blue beads in the ten thousands row and nothing in the others (representing zero as nothing), this is a very abstract concept and children will be well along in their path to abstraction when they begin using this material. It is also
wonderful for the child who has a difficult time with neatness and who may require less objects to make their work simpler. Finally before moving to paper we will learn about our golden mat; Which is a gold colored mat with columns sewn onto it, beginning with a mat only to the thousands, then higher and finally lower (decimals). The golden mat allows us to revisit those bead chain and bead stair color coded beads that we worked with in primary and sometimes even into elementary. We use these beads first to build numbers, then to put together and exchange for larger beads, when we get to 10 in a column we must move a one bead (or twenty a 2 bead) into the next column, thus allowing the child to see us carrying the one, two and so on into the next column when writing it on paper. This mat comes in handy for addition, addition with carrying, multiplication, subtraction, division and even in upper el; all these operations with decimals! We also have the materials specific to certain operations such as the strip board (one for addition and one for subtraction), for multiplication you have the Multiplication bead board, checkerboard(s) and flat bead frame, and for division you have the division bead board, racks & tubes and the pegboard. Although the pegboard has other functions too such as finding the lowest common multiples, greatest common factors. These materials may not be used for all operations but serve an important role in the operation performed on them. They also serve as a goal for those beginning students, when they see the older students using them they frequently want to know when it will be their turn!
When all these steps are mastered we begin the process of paper only. Keep in mind that along the way each of these materials has included working the material or equation on paper along with the material. But once a child can do an operation on paper only without the use of material that is how we know it has truly been mastered. Now this doesn t mean they aren t allowed to or never go back to the materials, because they are such beautiful and simple materials, it just means if they choose to complete their problems on paper only that they are fully able to do and understand that totally abstract concept. By learning and practicing the equations first, it helps to understand the why along with the how of math on paper. As a child I loved math and understood how, but didn t realize until I was introduced to Montessori as an adult the why of math. I have discussed the materials and their progression with the order of introduction along the passage to abstraction, now I would like to talk a little about the order of operations. In Montessori we teach addition, multiplication, subtraction and then division. I believe not only is multiplication an extension of addition and so it only makes sense to do that next, but that along the passage to more difficult equations multiplication is a more simple operation to understand then subtraction and certainly division is the most complex. After these operations have been learned to paper the student then begins to explore fractions (all operations and equivalencies as well as mixed numbers) with our wonderful fraction material including the fraction skittles for a basic introduction to fractions then onto the fraction insets for labeling and bookmaking to get a feel for fractions to one tenth. Before moving on to operations with the fraction boxes we use the insets for the introduction to each operation as well as the introduction to equivalences. Shortly following work with fractions we would introduce the
decimal system using the decimal mat (or board) and checkerboard with all the operations, then onto percentages and how they all relate to one another. Things then come full circle back to golden beads, stamp game then the peg board for beginning practice on Squaring, Cubing and square rooting. After these materials the children will move onto the cubing material and then finally to paper. Some other math materials that are covered during this 6 year cycle would include measuring of all sorts; volume, time, distance, liquid and weight with both Metric and English systems. Also money systems, value, Making Change, history, currency (rates of exchange & interest). We would also cover integers (positive and negative on a number line) and charts and graphs through teacher made materials that apply to the students he/she is guiding. The use of all the materials in each operation is so important for our children. I believe that all children must be introduced to all materials with all operations and allowed to practice with each material. This allows for repetition and practice of the concepts without boredom, each time you get out a new material, to the child it appears they are practicing something new and fresh. It also allows for a child to find a comfort area, maybe they only use the bead frame a few times and it is too difficult so they move straight to paper or they choose to practice with the stamp game until they are ready for paper. It allows for individualized learning of each concept. Some children may choose a favorite and others may use all of them or use different materials for different operations. It also allows for our Montessori children to travel from one
Montessori school to the next and have access to the same materials and methods they have been taught with in other areas. It is amazing to see how the Montessori materials bring to life these abstract concepts of algebra, squaring, cubing, etc. for the upper the elementary. A Montessori child will leave the environment not only knowing how to cube, but understanding the process of creating a cube from a number; A REAL cube, a geometric solid cube, not just the number 3 times. With Montessori s real life and hands on approach to math we and the children cannot just know (or memorize the rules to) Math but we can really understand it. We can see it unfold in front of us day after day. We can experience true equations in solid form and then really understand them when they come to us as abstract. Maria let us see math in our lives and in our worlds and for that we will be forever grateful. The teacher, when she begins work in our school, must have a kind of faith that the child will reveal himself through work (Montessori, 1967, 276). She helps us to recognize that each child has a sensitive period for learning; the guide must be forever on watch, observing the child and ready to guide them when they reach their sensitive periods and be ready when they are. Be prepared; that is the Montessori way. Prepared Environment, prepared teacher and prepared child. In Montessori Math sequence, repetition, consistency and the passage to abstraction are the building blocks of the math curriculum. The Child s mind is not prepared for numbers, by certain preliminary ideas; given in haste by the teacher, but it has been prepared for it by a process of formation, by a slow building up of itself (Montessori, 1914, 165). We don t feel you need to learn subtraction and this age and not before and not after, you learn it once you have
mastered the other operations before it at your own pace. We follow the child and his mathematical mind. In our work, therefore, we have given a name to this part of the mind which is built up with exactitude and we call it the mathematical mind (Montessori, 1967, 185).