Narrow Definitions Constrict Creativity: More on Basic Facts

For this blog, I’d like to debunk some common math misconceptions further. Specifically, you might be surprised to learn that elementary school math is not built with unrelated, stacked vertical blocks. Thanks to advances in cognitive science over the last 40 years, the current mathematics curriculum is designed to echo language development by returning to concepts, emphasizing them, and then elaborating on them further with added complexity — and cycling back again. This plays out quite clearly in discussing foundational concepts like addition, subtraction, multiplication, and division. Let me elaborate and then return.

The Powerful and the Primitive

Many of you may remember the order of operations of mathematics. That’s right. Those who just whispered PEMDAS (Parenthesis, Exponents, Multiplication, Division, Addition, Subtraction) have been indoctrinated well. Like much of mathematics, the order of operations did not begin as it is carried out today. PEMDAS came about mainly in the 1600s and 1700s when algebraic notation took shape through the work of mathematicians like Descartes, Viete, and Leibniz. The notation was meant to signify, through either implication or otherwise, the more powerful operations so that writing out long-hand mathematics became simpler. This was vital, as communication was usually only through the written word, and mathematics had to be communicated very clearly without the ability to jump on FaceTime and work through any confusion.

Florian Cajori, one of my favorite mathematics historians (yes, they do exist), stated that the order of operations was in disagreement up through the 1920s when mathematicians debated whether multiplication or division was the more powerful operation. PEMDAS didn’t satisfy that argument, as they are considered equals and performed as they appear left to right. This is similar to how addition and subtraction are considered equals, albeit the latter are considered less powerful than the former.?

Why tell you this? Well, as things are structured in our children’s mathematics curriculum, it is also true that the primitive is linked to the more powerful as students develop cycles of mathematical fluency. For example, multiplication is often defined as repeated addition. Similarly, division is defined as repeated subtraction.?

Some quotes help with the point here. Many of you have heard Sir Isaac Newton’s quote from a letter sent to fellow scientist Robert Hooke:

“If I have seen further, it is by standing on the shoulders of giants."

But you may not know that this comes from earlier works, most attributed to John of Salisbury, a theologian of the 12th century, who wrote,

“We are like dwarfs sitting on the shoulders of giants. We see more, and things that are more distant than they did, not because our sight is superior or because we are taller than they, but because they raise us up, and by their great stature add to ours."

The operations of elementary mathematics stand on the shoulders of other operations and work together to enhance your fluency for all of them. This does not mean that you should learn one, then the next, and then the next. They should be learned together as they synergistically empower one another and help build flexibility and fluency.?

However, the question remains: why does research suggest that students struggle more with subtraction and division than with addition and multiplication? The reasons for the struggle are important, and learning about them may bring empathy to your role in shaping new mathematical learners (or your own).

If You Define Something Only One Way?

I have shared the following with many students and professional educators in my career, and I hope this story resonates or further shapes your understanding. I believe defining something narrowly can cause significant issues in more complex situations. A mathematical habit of mind (MHM) focuses on this understanding, where you develop the ability to provide alternative representations for a problem you’re seeking to solve. This contradicts how parts of our brain work; as Robert Heinlein playfully stated in his book Revolt of 2100,

“You can sway a thousand men by appealing to their prejudices quicker than you can convince one man by logic.”

It's harder to define something broadly, leaving open the necessary flexibility to handle the complexity that is necessary to a lived experience. In mathematics and life, we like to say it is something (the answer) or it is the enemy (the wrong answer). This is part of the human experience, but something that, through mathematical habits of mind, we can avoid.

To humanize mathematics, it was helpful to think about a notion many of us seek to understand—the idea of human love. I would start a lecture or collaborative session with a simple yet innocuous question: Would you want the word love, defined by your 12-year-old self, to represent the love you expect to have during your adult relationships??

The murmurs were always loud, and often, there was a talker who would share how naive they were when thinking about love as a 12-year-old. I was careful, as 12-year-olds were sometimes my audience at the time. Still, the idea that a preconceived notion of love would define their future relationships was often comical, or at least something they hoped to avoid. I would then present the idea that the preconceived notions of mathematical operations were parallel to their 12-year-old understanding of the word love. Often, without fail, this would result in stares of borderline hostility or, maybe, curiosity. For example, I would continue, if you think about subtraction as only “cutting off” or “taking away,” you might miss an application of the operation when trying to understand the difference in sizes of two measurements or the distance between two numbers. If asked to compare the heights of two buildings, would you know to use subtraction if you only thought about subtraction as taking away?

So What Else Is There?

Returning to the MHM that I initially referenced, the idea of “take away” or repeated subtractions is limiting when thinking about subtraction and division. They are not wrong, just like your notion of love at 12 wasn’t wrong. They are just limited. Subtraction is harder to understand than addition because research suggests that learners build their understanding of subtraction based on addition. This leads well to understanding the difficulties young learners encounter with division. Many in the US must learn multiplication facts through rote memorization and often answer division questions by reverse engineering their multiplication facts.?

How can we think differently about subtraction, or at least define it more broadly as we work towards fluency? Perhaps surprisingly, the US subtraction algorithm we teach young students isn’t taught everywhere. Here is an example of thinking about the robustness of number sense and how to answer a subtraction task. I’m sampling from the body of work from one of my favorite authors on the subject, Dr. Karen C Fuson. Here is an example of how you may recognize the subtraction that you were taught or the children you are working with:

Does this look familiar? It's quite ingenious to understand that you can represent numbers by fluently taking groups of 10 and placing them in the one's place, enabling you to “take away” ones and then subsequently work with the other place values. Truly, it's brilliant, and that should not be lost on anyone. The brilliance does get lost, though, in focusing on the doing, which is often all that happens in US classrooms.

What if you were to understand conceptually that a task like this asks the distance between the two numbers on a number line as measured by the unit, in this case, 1s? This approach is how many other countries represent this task to young children. If you understood that you could move the numbers the same distance, you could find numerically easier numbers to compute and solve the task differently. For example, you could add ten to both numbers. To make this clear, add 10 to the ones place in 53 and add 10 to the tens place in 28:

Efficient Calculation Is Not the Point

Helping learners become efficient calculators is not the point of learning mathematics or this exercise I’m sharing. So, if you are whispering to yourself, the other way makes sense; I just want to use that way; why are you showing me another way!? This is about what it feels like to broaden your definitions. The best math teachers always remember what it feels like to be asked to do something mathematical.

With the inventions of technology, calculators, and even more sophisticated tools like AI assistants, the need for humans to be effective calculators is long gone. Still, human creativity and problem-solving are needed more than ever. Mathematician Conrad Wolfram , co-founder of Wolfram|Alpha , recently posted about this notion in his blog.

“The curriculum of school (and early college) math education is hugely out of line with an optimized, modernized core computational subject that’s needed. That misalignment is chiefly because computers mechanized calculating beyond previous imagination. In turn, this has massively broadened, deepened, and made math or computational thinking more critical to everything in life over the last few decades.”

The point is to create a circumstance where we can think about tasks, not be overwhelmed by them, and creatively play with the increasingly more robust conceptions of numbers, operation, and problem-solving. Only through experimentation can we broaden our definitions of concepts, and often, we remove the narrowing effect by making mistakes.

Returning to our “love” example, it may have only come as a mistake with love that you were able to broaden your definition of it. Speaking for myself, I will admit that I made many to get to where I am today with that word. If you understand the above subtraction approach, try to develop a different way of thinking about the task. Feel free to create your own; give yourself space to play. That is the essence of being mathematical—give yourself the space to be what you are and make mistakes toward a less narrow version of yourself.

What about Division?

Research suggests many approaches to teaching division, but we’ll elaborate on two approaches in elementary mathematics to help model good mathematical habits of mind (MHMs). Quite a bit has been written on this topic since the early 90s, including by some of our favorite researchers: Lyn English , Dr. Joanne Mulligan, Dr. Renata Carvalho, and Dr. Jo?o Pedro da Ponte.

The types of division in elementary mathematics are fair sharing (partitive division) and measurement (quotative division). You are likely most familiar with “fair sharing,” which US teachers tend to utilize the most when representing division for their students. The task for fair sharing (partitive) is that the learner is presented with a given quantity of something (usually sugar-infused food) and is asked to share it equally with a certain number of their peers.

Measurement division expands the definition, allowing more flexibility in problem-solving and application of division operations. It is repeated subtraction, basically how many parts of the same amount you can cut off until you run out of stuff from the original amount.

If you think about long division as an algorithm, it is often equated with spiders or something that lurks in the night and causes terror. There is a good reason: if you use the long division algorithm often taught in US schools, you must use division, multiplication, and subtraction operations to perform the division calculation task. Take a look below. Yikes!

What if we applied the less familiar concept of measurement division to a long division task, as is typically presented in elementary mathematics? Let’s see how this feels.

If we are working in a creative process rather than an algorithmic process, the question above (648/2) asks how many 2s you can cut off of 648 until you run out of stuff. It also asks what half of 648 is, but that is a different way of thinking about division. With how many 2s, we want to know what number (of 2s) you can cut off of 648. You could choose 10 or 100, it doesn’t matter, start with something you are comfortable with. So, if we cut off two hundred groups of 2s, we have used up 400 of our total of 648, which means we still need to cut off 2s from the remaining 248. We can cut off one hundred 2s from 248, which uses another 200 of our stuff; now, we have 48 remaining. From there, you can cut off ten 2s, and then ten more 2s, which uses another 40 of our stuff, leaving us with just 8. 4 groups of 2s can be cut off of 8—and then we are out of stuff.

Count the 2s you cut off: 200+100+10+10+4, or 324 groups of 2s can be cut off. So your answer to the division question is 324, which means how many 2s you can cut off or how many groups of 2s fit into 648.

Returning to Conrad Wolfram’s opinions shared in his blog piece; you could just ask Siri to calculate this with your phone. Go ahead and ask her: what is 648 divided by 2? She will answer in whatever voice you have set her to (mine is Irish) and respond quickly and correctly. If we change our perspective on the purpose of mathematics to broaden our understanding of the world and to creatively develop mathematical habits of mind, then this exercise makes sense. If your child is struggling with homework at the end of the day and you are tired, you may unintentionally inhibit the growth of mathematical thinking. “Just do it this way” cements the myth, though it does help with ending your day without further friction. In the long run, you are exposing yourself and your children to a choice of aspirin or vitamins. Is this an immediate solution you are communicating to your children without addressing the long-term concern? Like all things, also give yourself a break when you make a mistake. Learn from your mistakes; consistency over time builds habits of mind.

Avoid Narrowing Your Definitions with MathTrack Institute

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