Making Education Public

One of the ongoing divides (no pun intended) in math education is what to do with calculators. One school holds that it’s senseless, in this age of microelectronics, to have kids engage in the effort required to memorize math facts when a calculator can do arithmetic for them. The other school decries this simplistic reform on the grounds that kids deprived of math facts are unable to access higher level math because they’ll have “no number sense”. In the interest of fairness, I’ll confess to being a member of the latter school.

I’m old enough to have been educated in mathematics before there was a choice to be made. So my computations  were immersed in the mysteries of the log table and slide rule. Sometimes I struggle with my decision to be part of school number two on the fear that I may be exhibiting an unconscious desire to have all students go through the torture of log tables, slide rules and trig tables with endless interpolation to get to the fourth significant figure. Maybe I just don’t want students to get a free ride on the roller coaster I had to sweat for. But after observing  hundreds of middle school students who  can’t tell me the factors of 36, grab a pencil to multiply by 10, or who can’t independently observe that 200 times 3000 is not some insurmountable tsunami of mathematical horror, I’ve come to the conclusion that my calculator free education gave me something special, something these kids don’t have, something that keeps them from the insights necessary to access higher level mathematics.

Most people in school number two (myself included) simplisticly argue that computation fluency is the key to number sense and just leave it at that. This argument is flawed and incomplete. What the heck is number sense anyway? If you asked 10 ‘mathematical’ people this question you’d likely obtain 15 or 16 definitions. Asking it of ‘non-mathematical’ people is requiring nuance from people who have no basis to build an answer from. I won’t make that argument here. Instead, I’ll pose a non mathematical question wrapped in a, perhaps belabored, analogy.

What if there was a writeulator? Imagine such a device under development right now in a secret ELA lab deep in the bowels of the Department of Education in Washington D.C. It fits in the palm of your hand and it looks something like a Blackberry without the big screen. On its side there’s a USB port and on the front, a small lowercase-only keyboard with nothing but 26 letters and a writeulate button. To use it, you simply type in four sentence fragments; one for character development, one for causality, another for conflict, and the last for a bit of complication. Then you plug the device into a computer through its little port and push the writeulate button. Voila! On the computer you get a complete short story. All the nasty spelling, sentence structure, plot, and paragraph development is done for you by the writulator and the output is delivered nicely wrapped in Microsoft Word.

Far fetched? Sure! But, think for a moment what such a device would do for your writing skills. Even more frighteningly, think of what it would do for your ability to even speak a coherent sentence. If you had such a thing in elementary school, kids could write short stories much more quickly. What will happen though, when those kids reach college and have to produce original composition? Language is a hierarchy of document, chapter, paragraph, sentence, word, and alphabet. Every time you write or speak you’re building a repertoire that links that hierarchy into a cohesive whole. A writeulator would disrupt the connection between a thought and its alphabet.

Calculators provide a similar disconnect. Arguably, math is even more hierarchical than language and I realize now, in hindsight, that every single time I did anything in mathematics (sans caclulator) I was traipsing through the hierarchy, walking down the stairs back to counting. Trig tables make you an excellent practitioner of mental math. Slide rules perfect estimation skills. Fluency with multiplication facts makes you fluent in factors. Factors make you fluent in division. And so it goes, on and on through the magnificent and ancient hierarchy of mathematics. Every such trip, strenghens your lower level skills and builds insights that are cut short by a calculator.

You would never expect to create a great writer without having first created a great reader. And you would never expect to create a great reader without serious vocabulary development. Yet in math, we already have our writeulator. It facilitates a novice’s desire to get to an answer by snipping the link between the problem space and the count. How bad is the snipping? Well you can walk into any calculator centric classroom and see solutions like 22 x 3 = 25, or a $350 tip on a $35 dollar meal. These are easy mistakes on a calculator and impossible to ‘catch’ for a student disconnected from the hierarchy.

Before my career as a math teacher, I was an engineer. My entire adult life has been steeped in applied mathematics and I can honestly say that the vast majority of my mathematical pursuits have been accomplished while driving or lying in bed at night. Mostly, you sort and sift through problems using rough approximations. Only when your decision making is down to a few paths do you put anything to paper and only after further ripping and shredding do you grab a calculator to make sure you haven’t made some huge mistake.

Solving a math problem is a bit like creative writing. There are no magical methods to producing great prose. It comes from having read hundreds of books. It comes from having written thousands of bad sentences. It comes from endless practice . It does not come from a writeulator!

 Likewise, with math problems there is a mental process of sorting and sifting through possible solutions. Each may be ill formed and incomplete, like a thought you haven’t expressed yet. But each is built upon a solid foundation and, done correctly, you have a pretty good idea where you’re going before you hit the paper with it. Calculators can’t help with this. What’s happening in this pre-solution phase? It’s number sense but it’s also a facility to build a solution path that withstands a bit of subtle probing prior to a final commitment to it. Skilled practioners have developed a ’sense’ of where to take a problem and apply mental math, estimation, and prior knowledge to flesh out their thinking before they put pencil to paper. For lots of problems, it happens in a flash and if you’re fluent in basic skills it is a subconscious, iterative process, just like the process you use to formulate sentences.

I’m not convinced that this is a teachable process. Each person develops their own bag of tricks to get through this fleshing out process. If you’ve spent your early development with calculators, your bag is empty. If your early development was without calculators, you’re forced to walk the hierarchy. You’re forced to develop tricks for your tool box. Your insights increase with every problem set. Premature calculator introduction deprives students of this vital forcing mechanism. Instead of gettiing kids with insight you get kids who go for the buttons before they know what to push. Of course, with few exceptions, a calculator will always produce something on its dim witted screen and for most novices, any result will do.

Proponents of calculator use, argue that computational fluency is not essential to higher level math. They observe that higher level math is abstract, symbolic, and largely computation free. They hold that these abstractions can be taught without the necessity of memorizing math facts, and that forcing this memorization robs time from the classroom that is better spent on loftier pursuits. What they miss in this argument is distressingly plain to see. Abstraction only works when one knows what is being abstracted.

 For example,’boy’ works well as an abstraction only because it references a universally accepted and well known set of attirbutes. Asking a student to understand the abstraction of a/b without first mastering the meaning of  1/9 is to attempt an abstraction on top of a riddle. Any student with a calculator can tell you that 1/9 is 0.11111111. Unfortunately,  the same student will not be able to tell you how many ninths are in one! If you really want to see horror, ask how many ‘bths’ are in a. That question will have you tossing the electronics while your students toss their lunch.