How many times have you wondered or come across perplexed adults inquiring about the fact that why Montessori focuses on teaching multiplication after addition and not subtraction? I have been asked this question from parents, traditional education teachers and even some montessorians who are either not yet trained or still under training.
Before I get into the real talk of the connection and reasoning, let’s highlight upon some of the magical math materials in Montessori Elementary Classroom. The child gets to experience the journey from concrete to abstract with the manipulation of all these math materials.
The first lesson of addition is presented with the golden beads after the child has worked with single digit addition problems with two or more addends with the colored beads. They work simultaneously on addition of larger (4 digit) addends with the help of the golden beads. Once the guide notices the mastery is attained for static addition, the child is introduced to the exchange game.
The Exchange game teaches children the concept of 10 or more. The children learn in a concrete way how 10 units change into 1 ten, 10 tens into a hundred, 10 hundreds into a thousand and so forth. The child is also given an opportunity to work with (4 digit) addends moving from the golden beads to the stamp game. In the stamp game, the colored squares are now substituting for the golden beads, ten bar, hundred square and one thousand cube.
As the children attain mastery of the stamp game, they are introduced to the small bead frame. This material prepares the child for abstract addition. The frame has four horizontal bars representing the units, tens, hundreds and thousands with 10 beads on each, with the same colors (units-green, tens-blue, hundreds-red, thousands-green) as of the stamps. With the help of the small bead frame the child is able to make the largest sum up to 10,000.
After the mastery of the small bead frame, the child is introduced to the hierarchical cubes. This material is a visual representation of the whole decimal system that the 6-12 year olds will use for years. The children truly see the connection with the large bead frame, which is the next material that they will be working with. The large bead frame helps the child to make quantities up to one million (1,000,000) and the hierarchical cubes are a preparation and introduction to the place holders.
Simultaneously, the child also gets to work with the Dot Board that is a complete representation of abstract. The dots are used instead of any concrete material. By this stage, mostly the children are either already comfortable with working on addition in abstract (without material) or are almost there.
The guide can now say that the child has attained mastery in addition!
Multiplication is usually introduced simultaneously with the help of the multiplication board, once the child is comfortable with exchanging quantities and has also mastered dynamic addition with the stamp game.
Are you wondering now, why are the children being introduced to multiplication before subtraction? Multiplication is actually repeated addition. It is very similar to addition in rule application. For instance, the placement of two or more addends does not affect the total sum. Say 3 + 7 + 5 will be 15 and 7 + 5 + 3 will also give us 15.
In Multiplication if we change the placement of the multiplicand and the multiplier, the product with still be the same either way. Lets say 7 x 3= 21 and 3 x 7= 21. In both equations, our product is 21. Multiplication snake game is one of my favorite materials to reinforce this concept and to prove that it is repeated addition.
With the snake game, the child works on making the snake with the colored beads and the task card represents the addition symbols in order to construct the snake game. The child then makes multiplication equations by putting like quantity beads together. This step is the reinforcement of the connection of addition and multiplication.
Addition prepares the child for multiplication, where as subtraction helps to prepare the child for division lessons.