The Pliability of Numbers

My sister told me in recent years that she doesn’t believe in subtraction – that there is no such thing. I thought she was crazy, and clearly, she had a different interpretation of math than what I was taught.  When I was growing up, numbers were viewed as rigid, set in stone. They were black stamps that etched the pages of math books in their perfectly organized rows. Hardened rules were taught, and if executed precisely, led to exacting calculations. Pages and pages of work droned on until it was declared that you had reached math proficiency and passed, or math despair and had failed. We became perfect little calculators (or not). There wasn’t much room for free thinking. It wasn’t really necessary; you just had to follow the rules. I am a rule follower, so I marched to the beat of the classes, claimed my A’s, and moved on, never reflecting upon mathematical principles until recent years. My sister on the other hand, well, she doesn’t believe in rules imposed by others, she would much rather create her own. Therefore, her supposition about subtraction should not have been a surprise.  In her world, there is no subtraction, only positive and negative numbers – you just have to keep the sign with the number. At first this may sound ludicrous, but in reality, she demonstrates a superior understanding of numbers and operations. It does not match what is taught in traditional mathematics, however, it rings true, even in higher level math. If you have a conversation with mathematicians, they see numbers and operations differently than often what is taught. The traditional algorithms used in the classroom today came from hundreds of years of mathematicians figuring out the shortcuts. In the classroom, we often teach the shortcut that uses the least amount of paper. No wonder students get so confused, they never get a chance to develop the understanding of how that shortcut developed.

I have a theory that there are four categories that most people fit in regarding their experiences with mathematics. The first have an innate understanding of numbers and principles. They see beyond what is taught in the traditional classroom. It just always made sense. The second is like myself, the rule follower/memorizer. They usually get good grades and are seen to be good in math, although they may not really have much understanding beyond following the prescribed steps. The third group understands how to work with numbers and operations, but it may not match how it was taught, which led to frustration and either a distaste for math courses or struggling grades. The last group just hate math. It never made sense and they couldn’t remember all the rules. I realized early on when I became a classroom teacher that the rules seemed to be conflicting, and at times, counterintuitive. For example, If you want to compare numbers, then start with the largest digit. If you want to add numbers, start in the ones place, which, as it turns out, is not how most students would naturally solve an addition sentence. We often feel tied to the way we were taught because it is what is familiar, regardless of what category of mathematician we associate ourselves with.

I am now trying to create a new mathematical path for my students. One where they can see connections and relationships, where numbers make sense. I now realize that numbers and operations are more like clay than rock. They can be molded and shaped into different forms. Their flexibility allows you to reorganize them and shape them into friendly equations that make mental math and everyday calculations simpler. Understanding of operations allows you to estimate and compute in diverse ways. I contend that the mantra in traditional mathematics echoed those of Nike, “Just Do It!” However, I have developed a new mantra, one that can only be used when we are taught to think about the pliability of numbers and operations – “Don’t work for the numbers, make the numbers work for you!”  This understanding took me time to develop. I had to retrain my brain and develop a deeper understanding of numbers and operations. I had to change my perception of math, and that it was more about understanding relationships than following the rules. When working with students now, I focus on how numbers can be represented, what the operations truly mean, and how numbers, strategies, and operations are related. Students are demanded to think, analyze, and make connections within mathematics. Flexibility is the norm rather than the outlier.

If you are anything like I was in my earlier perceptions of math, you may be wondering what this flexibility looks like. I know when I was first introduced to this idea, I could not see outside my box and had someone show me just a few ways of solving other than how I was taught. The rest of the ways, I learned later on either through my students teaching me or through research articles. You can download samples of strategies used by my students over the years. The names I give them are not the official names, but rather, what we have called them as a class. It can be uncomfortable to stretch beyond our own experiences, but once we do, we have an understanding that could never have been reached before.

In America, we live in a culture deeply entrenched in traditional algorithms, and it can be challenging to see beyond, or to help others understand why it is important to teach math in a different way. I spend a lot of time educating the community on how and why I am teaching this way. I often show parents articles on the importance of problem solving for today’s students, such as chapter two from Making Sense: Teaching and Learning Mathematics with Understanding by James Hiebert et al. I encourage you to read the article, The Harmful Effects of “Carrying” and “Borrowing” in Grades 1-4 by Constance Kamii, which I have also shared with parents during conferences. Reading these articles may feel uncomfortable, but change often is.