Living in a Mathematical World: The Case for Calculators in the Mathematics Classroom

Unless you've been off the grid the last six months, you've seen the existential, hand-wringing headlines about the coming AI apocalypse. Artificial intelligence is coming for everything from our jobs to our news. But there's been a special kind of dread swirling around its potential impact on education. When the first (mainstream) generative AI tool, Open AI's ChatGPT, went public in November 2022, it sent teachers and parents into a panic about upended classrooms and curricula and students outsourcing their homework to a machine while using chatbots to write essays.

Time and experience with these tools (which now also includes Google's Bard and Microsoft's Bing AI) has cooled the rhetoric a bit. In the lead article from MIT Technology Review’s May/June 2023 Education Issue, “The Education of ChatGPT,” Will Douglas Heaven shares insights from teachers who believe that AI can have a positive impact on learning. That includes changing what educators teach, how they structure lessons, the work they assign, and the outcomes they expect from students—starting with the way they use a tool like ChatGPT.

To be clear, there are still an untold number of hazards posed by AI in education to go with the more immediate benefits. But isn't that the case with all new technology? Case in point: calculators, and the massive threat they posed to math educa...

What, you don't remember the Great Calculator Panic of the 1980s and '90s?

It seems strange from our 21st century vantage, where supercomputers are in our pockets and on our wrists and are as ubiquitous in schools as anywhere, that the humble calculator—which has been uber-normalized—would prove so threatening. But some saw them as such, not just for use by students taking standardized tests but by their very presence in the vicinity of a classroom. It was such a cause célèbre that the College Board Review made it the cover package of its Summer 1991 issue with the headline "The Case for Calculators."

The lead article, by Joan Countryman, assistant head for academic planning at Philadelphia's Germantown Friends School, and Friends School math teacher Elizabeth Wilson, is an impassioned, evidence-based case for letting students use calculators, whether they're of the simple or graphing variety. It's also highly readable because so much of their argument feels so familiar to our current AI moment.

"Calculators are transforming the way we teach, and the way our students learn, mathematics...[by] freeing ourselves of the arithmetic-driven curriculum," Countryman and Wilson wrote. "Calculators have changed what is possible and encouraged us to think more carefully about what is important for students to learn. ... We see the calculator as a tool, rather than a crutch. The calculator takes away some of the drudgery and allows our students to explore and to think about ideas."

You could swap "AI" or "ChatGPT" out for every instance of "calculator" in that last paragraph and it could easily appear in a defense-of-AI piece published today. In that way, it speaks to how little the ways our debates about technology change over time. But it's also a good reminder that it's so easy to retreat to an obtuse, closed-minded defensiveness about retaining the status quo whenever a new technology is introduced.

But, as Countryman and Wilson wrote more than 30 years ago, "All students need to learn to use calculators, if only to prepare to live in an increasingly mathematical world."

It bears repeating: AI is still so new that its full impact is impossible to discern. There will be positives and negatives and they will likely coexist. (See: smartphones, computers, the printing press.) But it's also worth remembering that we will have to live with the technology. And just as educators eventually embraced coding and computer science—and, yes, calculators—as necessary skills for students' future success, comfort and expertise with AI will one day (soon) become just as important in the work of prepping young people to live and thrive in an AI-powered world.

Technology—AI, computers, calculators—is a tool. Its utility and potential, for good and ill, rests in how we wield it. And as Countryman and Wilson passionately argue, pretending it doesn't exist or is so dangerous as to warrant total prohibition is the surest path to the worst outcomes—for students, educators, and society.

Carrie, a first grader practicing new skills, wrote a brief letter at the end of math class. "Dear Mrs. Countryman, I love calculators. They let you no [sic] big numbers." Like a mathematician using technology to investigate remote and difficult problems, Carrie had found delight in exploring what was heretofore unreachable. The calculator helped her reckon the sums, differences, and products of numbers that had been suggested by her friends.

In recent years, classroom teachers have seen significant changes in school mathematics brought about by introducing calculators into the classroom. Some examples will illustrate what we mean.

- Six-year-olds, after hearing Judith Viorst's Alexander, Who Used to Be Rich Last Sunday, ¹ used calculators to verify that a boy had used up his allowance. "Let's see," one child began. "He started with one dollar and spent fifteen cents. I'll put in 100 and take away 15."

- Fourth graders counted by 29. "We have 391 students in our school. If a school bus holds 29 kids," a student figured, "and we have 13 buses, how many kids can fit? Press 29, press plus, press equals, and repeat thirteen times."

- Sixth graders used a graphing calculator to make a scatter plot of data they had on the height of skyscrapers in the cities of Philadelphia and San Francisco.

- Seventh and eighth graders, graphing functions like y = tan 10X + sin 10X with a graphing calculator, speculated on how the number 10 affected the graph.

- Tenth graders wrote calculator programs that found the largest perfect square that is a factor of a number; did synthetic division; used the law of sines and the law of cosines to solve for the angles of triangles; used the change of base procedure to find the log of a number to any base.

Calculators are transforming the way we teach, and the way our students learn, mathematics. We are freeing ourselves of the arithmetic-driven curriculum. We no longer require students to practice computation endlessly. We no longer need to construct problems so that the numbers involved do not interfere with the mathematical principles being taught. For example, in textbooks, problems involving cubes and cube roots often use the numbers 8, 27, 64, or 125. Square and square-root problems use 16, 25, 64. If applying the law of cosines is part of a solution, either the numbers have to be "cooked," or the students spend more time on arithmetic than on thinking about how to solve triangle problems.

Calculators and computers are tools that support student work. Searching for patterns, solving problems, even learning basic facts, are all enhanced when we use this technology. Our students are engaged and enthusiastic about the fundamentals of mathematics. Learning basic mathematics involves exploring ideas about patterns, about quantity, about dimension, and about shape.² We feel that overemphasis on computation skills hinders students' learning. Calculators have changed what is possible and encouraged us to think more carefully about what is important for students to learn. Our approach to teaching mathematics is supported by the National Council of Teachers of Mathematics Curriculum and Evaluation Standards, a report that calls for making calculators available to all students, from kindergarten on, at all times.

School mathematics has never involved just the acquisition of arithmetic and algebraic skills. Critical thinking, problem solving, and reasoning were always listed as significant parts of the curriculum. Yet, for too many students, computation—the different branches of arithmetic, operations on whole numbers, fractions, decimals—was all that they ever saw of mathematics.

The rules of computation can be daunting. Consider the school bus question posed earlier. "How many 29 passenger buses do we need for 391 students?" As a long division exercise this question requires at least eight steps set up the problem; guess; multiply, subtract; bring down the next digit; guess, multiply, subtract. There are lots of chances for error in computation—wrong guesses, multiplication or subtraction mistakes, lining up the digits incorrectly. There is also the conceptual error of producing a right answer, say 13.48, that makes no sense in the context of the problem.

An Opportunity for Increased Understanding
 

We see the calculator as a tool, rather than a crutch. The calculator takes away some of the drudgery and allows our students to explore and to think about ideas. The newest technology, the handheld graphing calculator, allows students to sketch graphs of simple or complex mathematical functions with a few keystrokes. While we still think they should be able to give reasonably accurate sketches of common functions by hand to demonstrate their understanding, exploring graphs on their calculators is a good way to learn about them. It is no longer necessary to do the calculations, make a table, then plot enough points to see the graph of an equation. Examining the behavior of a graph now requires simply writing the equation on the calculator and watching as the graph develops on the screen.

Without a graphing calculator, it is not difficult to teach linear functions, and certainly we taught quadratic and trigonometric functions for many years before we had graphing calculators. Now, however, using this new tool, it is easier for students to explore the behavior of parabolas under many different conditions, to get a more solid understanding of the relationship of changes in a quadratic equation to changes in its graph, to look at the enlargement of a small bit of the parabola near its vertex and compare it to a larger section of another parabola. This is also true for more complicated functions. Exploring the properties of higher degree polynomials and rational functions is so difficult when all the graphing is done by hand that the topic is given less emphasis in some classrooms. With a graphing calculator, students can observe, investigate, and come to understand the behavior of functions like y = (x³ - 10x² + x + 50)/ (x - 2), or y = 18x⁶ + 3x⁵ - 25x⁴ - 41x³ - 15x².

One consequence of graphing with a calculator is that students necessarily become more comfortable with the ideas of domain and range. Probably everyone at some time writes an equation, pushes the graph button, and then sees that nothing happens. Often the students say that something is wrong with the calculator. A graphing screen can have large boundaries or tiny ones. Thinking about where on those possible screens the graph will exist, and which parts of the graph are more useful to see, are important questions more easily explored with this powerful tool.

Having such a tool at hand at all times is a privilege. Not to take advantage of these capabilities seems to us to deny students the opportunity to increase their understanding of mathematics. Our students need time for mathematics: to explore, discuss, describe, interpret, organize, collect, predict, solve. They need to learn to understand, represent, and solve mathematical problems. They need the experience of selecting and using appropriate tools and methods. They need practice in applying a variety of mathematical techniques in the solution of real-world problems. They need to use the language and notation of mathematics to express quantitative ideas and spatial relationships. They need practice in constructing valid arguments. Using an inexpensive calculator to enhance their learning of arithmetic will give students more time to develop real mathematical power.

A Calculator for Everyone
 

Yet some people still question whether it is wise to use calculators in mathematics classes. Years ago, we barred calculator use from some classes and from parts of some tests. We no longer do so. There were practical difficulties. Monitoring calculator use is difficult, e.g., some students have calculators on their wrist watches. But more important, the calculator is a tool that should be used. Research suggests that students who use calculators maintain or improve their pencil-and-paper skills. More important, "students using calculators seem better able to focus on correct analysis of problem situations."³

Students are impressed when they see that it is possible to calculate a square root to any degree of accuracy with a pencil and paper. But no one would advise teaching all students to calculate square roots by hand before allowing them to use the square root button on a calculator. If a student needs a pencil and paper to do a calculation, she might as well use a calculator. Everyone should know how to do simple calculations in her head. These seem to us essential: mental arithmetic, including single digit computation, multiplying by 10, dividing by 2, and adding 10 + 10 + 10; decimal equivalents for simple fractions; and estimation skills.

It is unwise to require students to demonstrate proficiency in these tasks before using a calculator. Instead, we try to teach students to use the calculator wisely, to examine calculator results to judge if they are reasonable, to answer questions for themselves by using the calculator, to look for patterns, to examine graphs, to wonder how the calculator does its work, and thus to learn still more mathematics. At the same time, we ask students to view mathematics as an interesting field of study in which new ideas are emerging at an incredible rate, a field of study in which there is not always one correct answer, where reflection is needed to decide upon appropriate procedures, where some proficiency is required in order to make intelligent decisions as a citizen. Mathematics is a field of study that is open to everybody, not just a select few who happen to have natural aptitude.

There should be no opportunities for anyone not to use a calculator. Requiring calculator use, and this means including calculator-active questions on standardized tests, will more likely promote equity than allowing their optional use. At the moment, those who can afford to buy scientific and graphing calculators clearly have an advantage over those who cannot. When calculators are mandatory for college entrance examinations, however, school districts will have to provide them, just as they now provide safe sports equipment. All students need to learn to use calculators, if only to prepare to live in an increasingly mathematical world.

Everybody Counts, a report on the future of mathematics education published by the Mathematical Sciences Education Board, states:

Only in the United States do people believe that learning mathematics depends on special ability. In other countries, students, parents, and teachers all expect that most students can master mathematics if only they work hard enough.⁴

All students can learn mathematics. The appropriate use of calculators will only deepen their understanding of the discipline. Technology is changing the definition of what is basic in mathematics, and it's a change for the better.

References

1. Judith Viorst, Alexander, Who Used to Be Rich Last Sunday (New York, N.Y.: Macmillan, 1989).
2. See, for example, Lynn Arthur Steen, editor, On the Shoulders of Giants, New Approaches to Numeracy (Washington, D.C.: National Academy Press, 1990) for a discussion of what is fundamental in mathematics.
3. Mathematical Sciences Education Board, Reshaping School Mathematics (Washington, D.C.: National Research Council, 1990), 23.
4. Mathematical Sciences Education Board, Everybody Counts (Washington, D.C.: National Research Council, 1989), 10.