I first went into math education 15 years ago, because I was convinced that the purpose for teaching mathematics in schools is changing, and I am fascinated by that whole process.
From a psychological perspective, understanding the purpose of your activities is the key to getting information into long-term memory. Purpose is the thread that weaves together the various definitions, techniques, and procedures that students learn. Without a clear concept of purpose, mathematics becomes a series of mnemonic tricks and memorization games.
The purpose of an activity comes from one of two places: 1. from the results 2. from the process.
The real-world economy emphasizes results, whereas education should strive to emphasize process. The word “process” here does not mean “procedure”. “Process-oriented” means there’s an openness to think about an activity from multiple perspectives, instead of a narrow focus on getting to a result. Of course, education is arguably more results-focused than ever -- as teachers today know from the past 24 years of standardized testing mandates -- yet I am optimistic that major changes are coming to math education.
Before the computer age, it made sense that mathematics was more focused on results than other subjects, because it was an economically vital skill. Every business needed accountants and clerks who could efficiently add and multiply numbers to make sure the books were correct and that no one had been stealing. In fact, the word “computer” originally was a white-collar professional job where people would solve equations all day and that paid about the same as entry-level teaching jobs today! But while that job no longer exists, since machines can handle all the calculations, the math curriculum has not changed at all. So our math classrooms are essentially training students for these now non-existent jobs!
I don’t believe the content of the math curriculum needs to change, actually. What needs to change is its purpose. Math education needs to transition from being the most results-oriented subject to instead becoming the most process-oriented subject.
Memorization without purpose is discarded by our brains, unless we go to great lengths to reinforce and continually practice the information. Before the computer era, when lots of good-paying jobs required math skills, it made sense to assume that many students would benefit economically from the procedures they memorized in math class. This is no longer true. The purpose of math today, I believe, is to develop students’ intuitions about information. In other words, math teachers need to stop telling students “it’s important because you’ll need it for the next class” and start telling students “numbers are the basic building blocks of information--mastering mathematics prepares you to correctly interpret information.”
Of course there is a need for result-oriented instruction in education, as well. But to me, that should be the role of science today, more so than mathematics. Let science be presented as the parade of rational progress and increasingly-accurate models of reality. Mathematics should represent the opposite side of the brain -- the intuition-based “reality check” that insists on grounding abstract rationality back to its origin in our sense perceptions.
Historically, mathematics was valued more for its process than its results. Number can be defined as “organized perception.” Babies as young as 6 months old, and many animal species as well, can distinguish numbers up to 13. You don’t need to know the symbol “3” to recognize three birds sitting on a telephone line. Numbers, and the geometric patterns associated with them, are the pinnacle of objective, perceptual truth. Historically, philosophers across the world recognized mathematics for this unique process of delineating objective truth.
For a truth to be completely objective, it must be a necessary truth, without any arbitrarily-chosen element.
The sum of the angles of a triangle in two dimensions, for example, will always equal a straight line. Contingent truths, however, while still considered “objective” in a general sense, do have a subjective element within. The number 360 was chosen to represent the complete revolution of a circle partly for subjective aesthetic reasons, (it’s close to 365, and it has lots of factors, including 60 which was already associated with time and circles). So we still say it’s objectively true that there are 360 degrees in a circle, but this is contingent on the choice of the number 360, which is the subjective element.
There is no subjective element in the facts of addition, multiplication, fractions, and exponents. Until you add in negatives, that is. The negative sign represents a direction that is arbitrarily chosen, not an immutable necessity. It is a choice that negative should represent “down” or “left” and not “up” or “right”.
It’s actually fascinating that in the year 1600, the words “positive” and “negative” were technical terms of logic, and had no meaning in mathematics. They only acquired their usage in math, as well as their present-day connotation of “good” and “bad”, after the invention of paper money (in the English colony of Massachusetts!) eventually led bankers to use the negative and positive symbols to represent credit and debt. So paper money and credit-based banking created the conditions for the negative symbol, along with its subjective element, to become associated with numbers.
Before 1700, most mathematicians still regarded numbers as necessary truths grounded in geometry. When negatives were put on equal footing as other types of numbers, however, it allowed a contingent truth, with its subjective element, into numbers’ previously immutable realm.
This marked a clear turning point in the purpose of mathematics. Mathematics was no longer the process-oriented delineator between subjective and objective truth. Instead, mathematics became the results-oriented predictor of economic value. Today’s global economy, driven principally by bank-created debt money, can be said to be “algebratized” in this way.
Every algebra equation could also be represented as a geometric diagram, which would help students understand why rules work. But the point of Algebra isn’t for students understand why the rules work. Algebra is mathematics presented in its most results-oriented form. The purpose of algebra is to represent the problem as a list of symbolic rules to follow in order to most efficiently get to the result. Algebra works even when you don’t fully understand why, because its rules already encode the logic of the symbolic relationships.
Even though algebra itself is results-oriented, algebra still can be taught in a process-oriented way. To truly understand algebra, students should be introduced to two deeper habits of mind:
Abstraction — stripping away the visual details of something, keeping only the relationship that matters.
Analysis — breaking a whole into its parts to study how they connect.
In order to understand how algebra abstracts number, students must first be able to visualize numbers and number patterns. Without this intuitive understanding that numbers are how we organize our perceptions of the world, algebra will become pure memorization for students.
In the 19th and early 20th-centuries, algebra drove the field of mathematics away from its process-oriented roots and into a results-oriented economic function. After World War II, the entire education system in America can be said to have been similarly “algebratized”, away from its process-oriented roots (promoted by John Dewey) and towards a “more scientific” results-oriented model.
In the same way that 18th-century mathematics led the results-oriented movement in society, math education today I believe can lead the return to a more process-oriented educational future.
