From Concrete To Abstract

by Barbara O’Brien, Ph.D.

Lately, my two-year-old has been singing, “8-9-10” throughout the day. Now, I know she does not yet grasp the meaning behind these numbers: that 8 is less than 9, which is less than 10 and so forth. This cute song serves as a reminder that much of how she learns involves transferring what she knows concretely into something that is abstract.

My daughter seems to have first learned the language of numbers through awareness of how we reference numbers (e.g., “please wait one minute,” “we have to leave at 4pm”). But, for her to fully understand that these abstract symbols represent quantity, she needs experience with concrete materials (e.g., blocks, counting beads) to fully understand that the numbers 1-10 represent specific amounts. And, preparing children for Mathematics is what a Montessori education does well. It first offers a concrete foundational experience, which then leads to an overall appreciation and mastery of addition, subtraction, multiplication, division, geometry, and algebra.

I have enjoyed watching my two older daughters, ages 4 and 5 1/2, learn Mathematics at Chesterfield Montessori School. One favorite activity of theirs was the Sprindle Boxes. After receiving their lesson (i.e., one-on-one instruction from the teacher) they patiently and systematically place the correct number of wooden rods in a box corresponding to number labels. The most important aspect of this lesson, and all lessons, is that they are allowed to learn at their own rhythm – they can return to this work as often as they like throughout the weeks so that the concept underlying the work becomes wholly integrated. My daughters do not come home and announce that they learned about the number “5” today – but rather, I see evidence of such learning when they set the dinner table with the appropriate number of plates for our family.

Another area that has prepared my daughters for Mathematics is Montessori’s Sensorial work, which involves Visual, Tactile, Auditory, and Stereognostic senses. The purpose of such work is for the child to gain clear information about her environment, which helps her build an organizational framework of her intelligence. Within the Visual domain, for example, my daughters enjoyed work with the Pink Tower. In their words:

“They are little cubes [which range from 1 cm to 10 cm]. You get out a rug and unroll it on the floor. You take the pink tower off the shelf [one cube at a time] and then you build the tower: Big block, medium, keep building and the littlest one goes on top [each cube is 1 cm larger that the previous one]. Then you put the tower away and roll up the rug.”

There is much more to this task than simple block play. First, there is Planning – the child must prepare for her work as she rolls out the rug and brings the blocks over one at a time. Second, this task requires Working Memory as she has to determine if there is another block in between the size of the one she just chose and the size of the one she just placed on the tower. Third, the child has to Inhibit the desire to merely choose the block that is closest to her. Finally, because part of the task involves dismantling the tower, putting it away, and rolling up the rug, the Pink Tower gives an additional opportunity to practice Attentional Task Switching. These cognitive processing skills correspond to Executive Functioning, which is related not only to important outcomes in mathematics, but also to overall academic performance and social functioning. While counting ability and understanding number magnitude is certainly important, these Executive Functioning skills actually enable learning to occur.

I have realized that each task in my daughters’ Montessori classroom is layered with extensive opportunities to plan, concentrate, attend, execute, and understand. Each task has much more depth than what I had expected. I know that my daughters are not merely ‘playing with blocks;’ but they are practicing and developing an intrinsic, self-motivating love of learning. They learn that their work has purpose and through that work, they discover their purpose.

Barbara O’Brien is a Developmental Psychologist and has three daughters. To learn more about Montessori education at Chesterfield Montessori School, please call 314-469-7150 or visit www.Chesterfieldmontessori.org.