Math Education

Many of the most brilliant people in our culture have spent their lives contributing to and extending the field of mathematics. Mathematics itself has grown over the last two centuries from a tool of physics to a vast conceptual landscape. Math is essential for our culture. But we have a problem that everyone acknowledges, there’s a math education crisis in our school system. As math is currently conceptualized and taught at the K-12 level of education, math is destructive to our students and to our nation. The problem is math itself. The math that is being taught, from K through 10, is a small and rather insignificant sliver of mathematics. And it is but a small and rather insignificant sliver of our lives. By expressing mathematical concepts in drawings and pictures rather than in symbols, we can return relevance to school mathematics since pictures look like what they mean.

The Image shows an illusion. The squares labelled A and B on the top image (er, difficult to read, look for the smudges) appear to be very different shades of gray. However, A and B are both exactly the same shade. Their context creates an illusion of difference but they are both cut from the same cloth. The second image on the left shows some of that context removed, permitting us to begin to see A and B in their true color. The third image shows A and B by themselves, out-of-context, visual proof that the original difference is perceptual rather than actual. In schools, math is taught out-of-context, unanchored to direct experience. It looses its character, it becomes monochromatic. People need context to make sense of things. We need mathematical knowledge to be applied, not to fake problems in a textbook, but to real interests in our actual lives. Education is not about learning the abstract structure of a subject, it is about connecting ourselves to that subject, it is about understanding a subject within our own context.

What kind of mathematical knowledge should our society expect of its citizens and teach in its public schools? One way to think about an answer is to focus on mathematics itself, rather than on what people do with math as professionals, teachers and students. We should teach math that is attached to physical reality and used on a weekly basis. Teach math only as it occurs naturally in society. We should teach math that is anchored to the intuitive capabilities of the body rather than to the vast undisciplined stretches of the mind. In our schools we should replace symbolic math by iconic math.

 

Making Math Real

• Stop teaching math as a separate class.

Abstract math should be left to math professionals. Math that is not directly connected to our bodies and to our daily experiential world should not be part of the K-12 curriculum.

• Leave all symbolic calculation to the computers.

Symbolic math is not designed for human consumption. Rather it is designed for computers. With the ubiquitous availability of computational aids, math education should focus on comprehension rather than computation.

• Math is a tool, not a skill and not a discipline.

The context of math education should be math application. We need to know when to use a tool, what the tool does, and how to use that tool to achieve our pragmatic objectives.

• Iconic math reconnects abstract concept to our physical experience.

All of the math that we need to know can be expressed in iconic, physical, and manipulative forms. The advantages of formal math education can be achieved using mathematical tools that are intuitive, visceral, and consistent with our cultural heritage.

Overarching statements necessarily have exceptions. To be clear, young mathematical talent should certainly have the opportunity to excel. I’m thinking of the 99% of us who rarely do computation and the 80% of us who are actually traumatized by our experiences in math classrooms. This is a cost/benefit analysis: the cost of alienating 4 out of 5 people from mathematical thought is not balanced by the 1 in 100 who thrive on symbolic math. With the added catch that only a very thin sliver of mathematical thought is taught in our classrooms, the vast majority of the curriculum is learning how to imitate a rather naive calculator. We should design educational opportunities for the exceptions, and design public education for the public. I am not calling for the abandonment of math. I am advocating that we learn math in the contexts it is actively used in, as a necessary tool rather than as a separate discipline. The value of a mathematical tool is in direct proportion to its everyday utility. If you do it less than once a week, then it is not a skill, it’s a call for resources. If you really need an accurate answer, use a calculator.

To be concrete, we as a population do not need to know how to do long division. We need to know how to use a calculator to do division, and how to understand the results. We do not need to know how to add fractions, or how to multiply large numbers, or how to subtract negative numbers. We do not need to care about exact answers, but we do need to know how to estimate. We do not need to memorize the times table, but we do need to have notes and tools to help us find important numerical facts quickly. There is no long-term or short-term intellectual benefit for children to know how to do algorithms with a pencil and paper. The situation is analogous to the fallacious arguments that supported teaching Latin in public schools in the first half of last century. “It’s good for you” is not good for you.

No other discipline embeds its abstract theoretical constructs into pre-college education.

I did a lot of math as a student. After toting around a footlocker of math notes for over a decade, never once looking at them, I decided to throw them away. When I was a carpenter building a house I used math extensively as a tool. As the saying goes: Measure twice and cut once. When I worked on optimization of silicon circuitry I also used math extensively, but in the form of binary decision systems. I rarely used numbers in that work. When I was a programmer, I used common math every so often, but it was usually much better to conceptualize programs using esoteric recursive algorithms and relational databases. When I write about math education, I almost never do any math. I’m mostly trying to understand how math education manages to ignore people and psychology and history and utility and physiology and comprehension. I often play with math, doing Sudoku or other puzzles. Every now and then I need to make an estimate of the cost of the groceries in my basket at the supermarket. Every now and then I need to check that a bill is correct. I deal with numbers when I do my income tax, although I could easily avoid that by giving the task to a professional. I teach math and so I’m constantly using it in the classroom as the content being taught. I study and develop iconic math concepts, so I do a lot of specialized work that I know to be math but that the mathematical community does not recognize as math. So I can sum up my own experiences by saying that math is quite irrelevant to my personal life, and quite important to my professional life. None of the math I use professionally is taught in schools, and very little of the math I learned in school is useful to my professional or my personal life. That is, I don’t really use the math I was taught, and the math I do use I learned electively, by choice, when I was ready to learn it.

 

Things Change

Symbolic math was a great innovation about a century ago. It eliminated many confusions and clarified what mathematicians meant when they communicated with one another. Symbolic math was prescient, it provided the technical foundation for the conceptual development and growth of computers. Symbolic math is wonderfully well-suited for automated computation, and it is horribly suited for human learners and for those of us who are not mathematically inclined. Symbolic math served humanity well until about 1980, when personal computers and handheld calculators became widely available. Over the decade of the 1980s the work of doing math was transfered from pencil and paper to electronic machines. The turning point for me was 1988, when Steven Wolfram introduced Mathematica, a symbolic computing program that could do just about any mathematical computation you could specify. By the 1990s, there was very little justification for people to do calculation by hand. If the result was important, not using a calculator would be irresponsible. If the result was not important, then a quick estimate would suffice. By 2010, everyone had a calculator built into their cellular phone. For anything more complicated, Wolfram Alpha and many other web-based computing applications are available for free.

During the course of a century, symbolic math migrated from a highly refined technical tool for mathematicians to a powerful and essential technique used within silicon computers. Over the same period, the teaching of symbolic math in schools has evolved from a useful and limited technical tool for figuring out the value of a bale of hay to an elaborate form of mental torture that has alienated most of the US population. People really don’t like symbolic math, and for good reason, symbolic math is not designed for human use. Our unfortunate school system requires coursework in symbolic math from elementary school through college. The Sputnik reaction in the early 1960s cemented math into the curriculum. (School systems thrive with feet in concrete. School is the social institution most resistant to change.) Math education has been completely unable to adapt to the changes of the information age that were facilitated by symbolic math.

Schools need to stop teaching symbolic math! One way for this to happen is for elementary and secondary schools just to stop teaching math as a separate class. Our brains no longer need to do the work of electronic calculators. We should limit the language of symbolic computation to its rightful place inside silicon. We should teach math as a tool, in context, to solve particular problems. We should teach difficult math when it is needed, but we should not teach irrelevant math at all. An alternative solution is to redesign math so that it is adapted to humans, so that it reflects how we think and what we do. The objective of iconic math is to return math to its natural pre-twentieth century origin as part of human physical activity. If we must teach the processes of math, then we should at least teach using iconic forms that show these actual processes.

Math is not an arduous mental exercise, it is not a forced march of memorization and high-stakes testing. Except for a small number of professionals, math is not even a discipline. Math is a convenient tool, a hammer that is particularly useful only when we encounter a needy nail. Math education is learning the appropriate use of a particular set of tools. These tools always come equipped with a machine that takes care of the raw computation (i.e. an abacus, book of tables, slide rule, calculator, computer, iphone, etc.). All we need to understand is how to use the tool and what to do when the tool provides results.