The Great Miscalculation

Why is 6 afraid of 7? Because 7 eight 9!
If you want to pick a fight in a room full of math teachers, whip out your TI-83 Plus and state an opinion about why you love or hate it. The ensuing feud will last for hours and people will inevitably leave the meeting with less information but far more angst than they came with. 

As a tech enthusiast and math teacher for the better part of the last decade, I have spent a lot of time pondering the appropriateness of calculators in 9-12 classroom. Group conversations about this generate nearly as much animosity between teachers as the Hatfields had with the McCoys, but one-on-one questioning makes it easy to see that teachers essentially fall into one of two camps on the calculator's place in the classroom.

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The "Can't Leave Home Without It" Group 
These teachers insist that as long as calculators are permitted on state and national tests (which affect everything from teacher performance evaluations, to student promotion, to college scholarship and acceptance), teachers have a responsibility to teach students how to take advantage of the resources available to them. They argue that student test scores are often compared against other students' scores and those comparisons lend to inaccurate extrapolations if their students are not well-versed in using the tools. As a result, they have a tendency to teach students how to do all math concepts with a calculator slant in the instruction. Intentional or not, this presents obvious concern for the other group.

The "Those Kids Don't Really Learn Math" Group 
These teachers point to the unavoidable tendency for calculator inclusion to eventually dissolve into learning how to pass the test rather than internalizing a deep conceptual understanding of mathematical principles. This group does not necessarily deny the advantages of technological assistance and accuracy in routine computational processes, but they express a firm belief that students should develop a full understanding of mathematical concepts and rationale before leaning on the benefits offered through calculator technology.

Miscalculation
My stance is based on a fundamental belief that both groups are wrong because of presuppositions they have about calculator use in high school math classes. College math professors will tell you an alarming number of students arrive and simply cannot survive their math class when the calculator crutch is removed. While this certainly is cause for concern regarding college-readiness of high school graduates, I believe the problem has nothing to do with calculator inclusion in high school math. Instead I believe the problem is how calculators are included in high school math. Overwhelmingly, high school math teachers use graphing calculators (which other than the TI-Inspires have not changed significantly since the TI-83 was introduced nearly 20 years ago) strictly for their procedural capabilities. At their heart, calculators can only add, subtract, multiply, and divide. Computationally, most graphing calculators can only functionally do basic arithmetic. All other calculations done on a calculator are estimates or arithmetic equivalencies of algebraic, statistical, or calculus functions. For example, a TI-83 cannot algebraically identify the factors of a simple quadratic function. Instead, it uses a relatively simple algorithm (that a programmer wrote and installed on the device) that allows the user to type in the coefficients of a quadratic function, and then it displays the factors. As far as the user is concerned, the calculator is doing algebra.

The √ of the problem
This is where the problem takes root. Many high school math teachers have no idea how a calculator works. They may know how to use it to solve problems, but the vast majority cannot articulate how a calculator does advanced (or even simple) functions. As a result, a teacher, that begrudgingly adopts classroom calculator use to boost student performance on a test, ends up teaching students a series of steps to "plug in and get the answer." When this happens, it is obvious that authentic learning has not happened. If plugging in a formula and clicking Go is all it takes to do advanced math, than my 6-year who can read the buttons on a calculator is ready for calculus. Let's act like all teachers agree that this is ridiculous and should stop. We are still left with the problem that not every teacher will stop using calculators, so their students will have inflated test scores (maybe), and industry/business leaders are begging for students with technology savvy--suggesting that our students should indeed be learning how to use calculators since they are really just handheld computers.

Create Before You Consume
I had an epiphany. We know that when you teach something, it forces you to learn that thing better. You can't teach what you don't fully understand. If the goal of a math course is to teach students both procedural and conceptual principles, then the calculator should not be entirely removed...you know, since it does procedural things faster and with more accuracy than a human. However, the procedural advantages offered through technology should compliment, not cripple, the conceptual understanding of the students. It hit me. Students can use programming as the mechanism to "teach" a calculator how to use arithmetic to do algebra. They can literally write the recipe (algorithm) for a computer to solve problems it is not intrinsically capable of solving at its core. In this light, they should be creating calculator programs (even if they're not elegant and/or glitchy) before they are consuming the programs created by professionals.

For TI calculators, a slew of free wikis are available to teach students (and teachers) the BASIC skills needed to write calculator programs in TI-Basic. In a school that offered literally no computer classes (Microsoft productivity does not count...and should be removed from high school in my opinion), I gave my Algebra 1 students a handout from the internet that taught them how to make a "Hello World" program on the calculator (this is the common way of introducing students to a programming environment in computer science). After working through the handout in 15 minutes, I displayed a simple quadratic function on the board and walked them through the steps of factoring. After showing the steps, I wrote another quadratic on the board and told them their assignment for the day was to write a program that would ask the user to input the coefficients from a quadratic and would immediately display the factors. Most of the students only considered functions that were positive and where "a" equalled "1." As a result, their programs would not work every time but would only work in perfect conditions. This sparked a conversation about what if one of the terms was negative, and what if "a" is not "1."

What happened over the next two class periods made it clear that having students create "teach" their calculators how to do algebra was the ideal way to subtly incorporate inquiry based learning principles into the math class. Students began asking for harder equations to test their program and see if they had handled all possible exceptions. They began trying to "break" each other's programs and then state why a program would or wouldn't handle certain inputs. What we ended up with was a week-long, content rich discussion in which students actively engaged in dissecting everything there was to know about quadratics. In the end, I eventually showed them how to use the professionally written program that came preinstalled on their calculator, but that "trick" was only shared after the students had developed a deep understanding of what a quadratic function really represents, and an appreciation for the algorithm used to solve such functions.

In case you're curious, that was the last year I taught Algebra and was by far the highest state test scores my students received.