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# Rethinking How We Use Calculators

May 16, 2013 | José VilsonIf ever you come across a set of math teachers, whether at a common planning meeting or a bar during happy hour, bring up the conversation of calculators and watch the sparks fly. The arguments for and against calculators have the spirited vigor of a Red Sox vs. Yankees game without the animus. One side argues for the use of efficient and available technology in the classroom, while the other argues for numeracy and fluency to the highest order.

In other words, are you old school or new school?

## Two Schools of Thought

The old school suggests that, in order to develop a rich sense of numbers and fluency, we shouldn't allow students to use calculators. In a world over-equipped with technology tools, students must be able to do operations without the calculator there. In this school of thought, calculators strip students of curiosity about how numbers work because they can arrive at the answer just by pressing a few buttons, not by going through the long-established procedure of finding the answer. Calculators already spit out everything from long multiplication and division to graphs and solutions to simultaneous equations. The old school crowd isn't completely anti-technology; many of them stand by The Geometer's Sketchpad. They just wouldn't want all the mysteries and intricacies of math unlocked so quickly.

The new school suggests that we take a different outlook on the calculator issue. If we can so readily solve problems with a calculator, then why not give one to our students? The old way of writing out the multiplication table in both a table and list form is antiquated and tiresome. The calculator is also much more efficient, reducing the amount of time one spends on a problem. For instance, imagine trying to divide a seven digit number by a two digit number higher than 12? Such a task seems tedious when a calculator can do it in a fraction of the time. Even operations with fractions can be simplified with calculators, so finding things like the least common denominator or remainders feels pointless if the screen just told the student the answer.

My response to this one lies somewhere in the middle. I'm not totally against calculators. I use them frequently enough when creating answer keys for my exams -- after I take time to do the problems myself. I use them when working on my taxes, and I've used one when trying to get a new couch for my mom to fit in her apartment. (Thank you, Pythagorean Theorem!)

## The Right Tool Used the Right Way

Where I start siding with the old school mathematicians is in this: how do we*know*that the calculator is telling us the truth? Numbers don't lie, but humans make meaning of these numbers and hope to ascertain how they apply in the context given. If we rely solely on calculators without giving much thought to the number we've put down, or simply assuming the calculator is always right, then we end up with everything from wrong answers to the financial collapse of 2008.

Calculators are tools to help solve problems, not the solver of the problem itself. Our students need to develop a sense of numeracy that allows them to estimate the distance between two items, for example. We can't underestimate the usefulness of looking at numbers and making quick calculations by comparing those numbers without having to pull out a smartphone.

The calculations we make on the fly matter more than the ones we make in math or science class, yet these classes are where students get the most explicit reasons for using them. In larger math problems, I can see the usefulness of a calculator. For instance, when finding the differences between planets in scientific notation, we shouldn't have to plug away at doing operations to the first factors. However, if we insert the notation wrong in our calculator, we could end up with a number bigger than we bargained for.

We need to think about the way we use calculators, or any piece of technology. Assuming that the tool is doing exactly what we're asking it to do -- or actually has the answer to the question we've asked -- is a dangerous proposition for the non-thinker. We need a healthy balance of working within the number system and doing more complex problems. We need to let calculators serve their purpose in moderation.

What do you think?

## Comments (14)

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## I believe that students

I believe that students should not be able to use calculators in the beginning stages of building their foundation skills in mathematics.

Students need to build the confidence in doing simple arithmetic and calculations so when they achieve higher levels of mathematics they can then focus their learning on more complex concepts.

Students can then check their work via the calculator.

## Again: I don't understand why

Again: I don't understand why this serious debate about calculators still rages on: This problem has already been solved by the QAMA calculator- retaining the indispensable benefits of calculators but without the damage they do: This calculator shows the result only after the user also inputted a suitably reasonable mental estimate. See

www.QAMAcalculator,con

## I began teaching 6th-8th

I began teaching 6th-8th grade 4 years ago. I had a number of students in 6th grade very weak in math facts and they

transferred that weakness to their overall math skills. After a few months I insisted they use the calculator and would penalize if they did not. Over the course of the next three years their math skills grew, their confidence grew and their reliance on the calculator decreased. If I noticed a student using the calculator in place of their brain as this point I would mimic the old cowboy quick drawing his gun in a shoot-out. They would laugh and use their brain. I have noticed some students have excellent recall of math facts and that helps with computation but it does not also transfer to high order thinking skills and vice versa. So basically I say get to know your students strengths and weaknesses and give the tools that will allow them to grow in both areas.

## I have this exact debate with

I have this exact debate with myself almost every day when I'm teaching. I'd say I'm a little bit of both old school and new school when dealing with the calculators. I allow my students to have the calculators on the desk, however, there are many times where I will make the students put the calculator cover on and use scratch work to solve the problem. For example, I refused in the beginning to allow students to use calculators for order of operations problems because they were putting something incorrectly into the calculator (whether it be forgetting a parenthesis or using 2 as a factor instead of an exponent), and I was trying to stress the correct order in which to solve the problems. Even still now that we're at the end of the year, when a student tries to quickly solve an order of operations problem on the calculator, and they don't see the answer there in their multiple choice, they know to try to solve it on paper with only their brains, and they then are able to get the correct answer. Now on the flip side, calculators are good at catching small errors in the work, and I think as long as my kids write down their work and the process they are taking to solve the problem (and WHY they're doing that such process) then I'm very okay with them using calculators. Ah! I'm just so back-and-forth with this every day!

One last thing: I used to play a game with my students back when I taught 5th grade to help them with times tables and division, but also to show them the power of their own brains. It was called "The Brain vs The Calculator" (fancy title ha), and one student had a calculator, and the other didn't. I would ask a multiplication fact, and The Brain was able to yell it out whenever they had the answer. The Calculator wasn't allowed to yell it out until they had the answer on their calculator screen. It most cases, The Brains were answering much quicker, while The Calculators were still trying to type it in. I was just trying to show my students that yes, you can rely on your own brain, and it can sometimes be quicker for the smaller math problems!

## I have yet to see secondary

I have yet to see secondary math curriculum that addresses decimal place accuracy and errors for numerical approximations made on the calculator. Round off, truncation, and discretization error should be addressed when we use the calculator to calculate irrationals, non-terminating non-repeating decimals, root approximations, or any of the numerical differential or integration methods. Additionally, I'm surprised that the debate about calculators hasn't been replaced by one about computer use in mathematics classrooms. Sadly and ironically, computers in math classrooms are rarely used to compute. Instead, they are largely used for direct delivery of content, assessment, and for word processing (Powerpoints for presentations). I look forward to conversations about programing and computational software use in our math classrooms.

## I see that it may have come

I see that it may have come across that way, but I'm proud to be old school, too. Thanks for bringing up this discussion. I can only hope lots of primary school educators think more deeply about this.

## Thank you.

Thank you for your comment, Cathy. I'm in the same camp, frankly. The last paragraph is the clue, really.

## Tools

Exactly right, Texas Tutor. It's a tool and not the actual math. Perfect analogy, really.

## Of course ...

There's an assumption on your behalf that being "old school" is a bad thing. Not necessarily in my book. Actually, I lean more towards that rationale than the latter.

## As an element of "old school"

As an element of "old school" classification, I take issue with your supposed reasons for being "old school." I'm not so worried about the richness, mystery, etc., I personally think the use of calculators in early math classes diminishes students' ability to do math at higher levels. Students prefer to use calculators because it's easier. Well, maybe so in grade school, but in high school students without number skills simply don't do the math--even though they have and are allowed to use calculators. Why? Because if you can't complete a trinomial square because you don't know the numbers, you're not going to be motivated to take out your calculator at every step. How many problems would you last doing algebra this way, decimal approximations and all? It's simply too cumbersome to be effective for algebra/calculus. (Of course, not if one is talking about graphing/solver calculators--but my thoughts are geared more toward elementary/middle school.)

Also, I'm amazed by this notion we've taught our kids that arriving at some "easy to understand" number as the answer is the most important thing. Why should a student desire to see an approximation of 1.41 instead of the exactness of the square root of 2. (And, why 2 decimal places? Maybe we should just call it 1. Then 1^2=2. Ha.) But seriously, in mathematics we have the rare opportunity to be exact. Where else in a person's life does this exist? In my opinion, mathematics hits the real world in measurement, science, statistics, economics, etc. Students can get plenty of exposure to this in other classes. When does 1.41 serve better than the square root of 2? Yup, science lab. I get that. Let the science teachers worry about calculators and approximations. Let the mathematics teachers teach mathematics. Then it's not a calculators vs. non-calculators debate--the student with the abilities to do it both ways can choose based on the situation.