Watching a young child use the golden bead material to complete math equations in the thousands is impressive to most people. Observers make comments such as, “Wow, I definitely was not adding numbers that large that young!” or “She’s not just doing the work, she seems to really understand what she’s doing.”
Then there are the more skeptical observers: “Ok, but how will he learn to do math traditionally, on pencil and paper?” The short answer is yes, when he is ready. Though that is not often satisfactory to a skeptical adult. Understandably so – it’s one thing to pull out your phone to do a quick calculation, but certainly not practical to whip out your set of golden beads to add those big numbers!
The goal of concrete Montessori materials is to give children a deep, sensorial experience of the subject matter. Giving them the hands on experiences ensures they will really understand what one is just as well as they can understand how much larger one thousand is. The wonder in a young child’s eyes as they first hold the heavy thousand cube is beautiful. As they move through the curriculum, this understanding stays with them.
Passage to Abstraction
Eventually a child outgrows the golden bead material. As he masters the operations with the golden beads, the materials becomes boring and tedious. To add numbers like 5,652 + 3,427 means a lot of beads, carrying heavy objects, and more cleaning up. It’s just a lot to do when you already know what 1,000 is.
After this comes the Montessori stamp game. Rather than many, many beads, children have a set of square “stamps” color coded green for units, blue for tens, red for hundreds, green for thousands. On each stamp is printed either a 1, 10, 100, or 1,000. It is still somewhat concrete, but more abstract. As the child outgrows this material, things like the dot board and bead frames will come next.
Eventually, children begin to outgrow these materials as well. The final step is traditional pencil and paper math. At this point, the concept is well understood by the child. Adding large numbers is something they have probably been doing for years at this point. Some children will just say, “Oh, that makes sense,” after they receive a lesson. They have felt the difference in numbers with their hands, giving them a deeper understanding of place value. They have counted ten and carried it over to the next place, so adding that little “1” on top of the equation is logical.
And the other math?
Other math follows similar progression. Simple addition up to 20 can be counted on bead bars, with the addition strip board, and finger charts. Multiplication, division, fractions, decimals, and geometry all have hands-on, concrete materials available for the first lessons. Even concepts like algebra can be explored with manipulatives before learning more traditional methods.
But are they really ready for all this math?
To some, this may seem unnecessary. Or they see young children in preschool, kindergarten, or lower elementary working on concepts they did not learn until much older and feel the children are being pushed or forced to do work above their developmental abilities.
The best argument I have is to encourage you to observe these materials in action. Children are excited to do this work and choose it of their own free will. Children are so curious about the world and are driven to understand it better. The Montessori materials give them a concrete way to do so. When children seem to feel like they are being pushed to do more than they are ready for or want to do, Montessori guides often step back, observe, and ask questions. Sometimes a child who has seemed ready was not quite ready to move to the next step. Or maybe they’ve truly mastered it and are feeling bored, which can result in mistakes.
When I first discovered the Montessori math materials, I felt (as do many!) envious. How I wished I had been taught to do math this way. As someone who often struggled with mathematical concepts, I feel I would have grasped concepts much quicker had I been able to see how they worked. The squaring and cubing chains make sense of squared and cubed numbers. Carrying numbers makes sense when you can physically do it a few times. Fractions and decimals are quite simple when you can hold 1/2 in your hand.
What are your thoughts on the Montessori math progression?
