We have all witnessed a major shift in how educators approach the subject of mathematics in US schools. The focus is now on developing a “deep understanding” rather than teaching procedures. The common core math standards ask teachers to delve deeper into fewer topics. Instead of learning one standard algorithm for each arithmetic operation, students are exposed to multiple approaches and asked to explain why each algorithm works. By both positioning the teacher as guide rather than an instructor and empowering students to explore math topics for themselves, math is no longer a series of facts to be memorized but a continual process of reasoning and discovery. Even with this major conceptual shift sweeping the country, the term “math fact” is still alive and well.
In my opinion, the very idea of a “math fact” undermines the entire movement. In George Orwell’s book “1984”, Orwell uses the “math fact” 2+2=4 as the ultimate truth. While Big Brother has the omnipotent power to rewrite history and reduce truth to a intricate web of lies, He has no control over this one, last, dogmatic truth. Orwell chose 2+2=4 as the ultimate truth because this mathematical expression transcends cultural barriers and is seemingly beyond reproach. The very acceptance of 2+2=4 as the archetypal “math fact” illustrates how deeply ingrained a procedural approach to math is in our society. It is as if school-taught math is the true Big Brother.
What if I told you that I could easily make 2+2 equal something other than 4? I don’t need to use torture or a long mathematical proof. I won’t even have to lie to you. 2+2 = 11 in base 3. For many people this statement is utterly jarring. In America, the concept of a base is hardly touched upon on in grade school. Children are taught to unquestionably accept and memorize that we count to 10 by saying: “zero, one, two, three, four, …”. This rhythmic and almost mindless sequence is seared into our brains from the very beginning. But why does 10 come after 9? How can we blame children for approaching math as a set of facts when their first introduction to the subject is in the form of a hard and fast rule?
The English Language Arts also begins in a similar fashion with the alphabet song. However children don’t view the alphabet as a supreme truth. Very early on, children are exposed to the idea of other languages. The alphabet doesn’t necessarily have to begin with “a” and end with “z”. What is important to understand is that letters, or symbolic characters, can be strung together to create words. These words can then be used to represent or describe ideas. Young children soon learn that while the letters and sounds change, this underlying pattern does not. We all freely accept that a “house” is also a “casa”. Just as international languages are variations on a fundamental concept, our base 10 number system is one of many possible numerical schemes following the fundamental pattern of a base.
The purpose of a base is to allow us to encode any quantity with a set number of digits. Without the invention of a base, we would have to invent a new character for each increasingly larger quantity. While we use language to convey ideas and feelings, the numerical system gives us the ability to represent any quantity quickly and efficiently. In base 10, the base we commonly use, an infinite number of quantities can be sufficiently described using only ten digits. In base 2, the base that governs computers, an infinite number of quantities can be described using only two digits! Just like how you can greet people by saying “Hello” or “Hola”, you can represent a set of nineteen things using the numerical representation “19” (base10) or “1011” (base2). When you understand the pattern behind our number system, these possibilities feel natural. When you only know the facts, this alternate language feels viscerally wrong.
The term “Math Fact” is most prevalent when children are learning multiplication. As a child, I could recall my multiplication facts faster than any other student in the class. However, I never understood the idea of partial products. Unless I could solve a multiplication equation using the written algorithm or from memory, there was no way I was going to find the answer. I did not know that the written algorithm for multiplication was based on one of many ways to manipulate quantities. I only knew how to bring down a zero and carry a one.
Even during this time of change in mathematics, I still hear about teachers who ask students to memorize their multiplication facts before they have even introduced the idea of joining equal groups. A popular twitter post ridiculed a school for enacting a strict fact memorization policy. The elementary school defended the policy by stating: “in order for our students to be successful with higher level mathematical concepts, we first need to ensure that they can perform basic calculations fluently.” This statement shows a complete lack of understanding of what “higher-level” mathematics entails. Advanced mathematics is not the ability to reduce and solve longer equations. It is a more advanced form of pattern manipulation and critical thinking. In fact, Keith Devlin, a professor of mathematics at Stanford, describes all of mathematics as the science of patterns.
Schools focus on “fact fluency” because it’s easy to mandate and easier still to assess. Before the age of computers, it was a useful job skill to be able to quickly compute sums in one’s head. Today, companies need critical thinkers who can solve multifaceted and open-ended problems. Just like how children will never learn to read by memorizing word after word, they will never understand math by memorizing equation after equation. Children have to sound out each letter, hear the new sound they made and slowly start to recognize and remember those patterns. Think twice next time someone tells you 2+2=4.