Calculating the Death of Learning

A photo of a young girl gazing at math problems | Courtesy of Geralt on Pixabay

What do you think about when you think about math? Most likely, the image of a calculator begins to form in your mind. This is no coincidence: the calculator was created by mathematicians in order to help them compute long, tedious, and difficult algebraic equations or formulas. These systems are called Computer Algebra Systems (CAS).

A photo of a Casio fx-115ES | Courtesy of Wikimedia Commons

Today, calculators are much more than tools used exclusively by genius mathematicians and scientists. Indeed, few people can make it through the educational system without using one, and this fact is especially true regarding our children. Finding its origins in the early 1990s, the calculator controversy began to spread across the nation. All at once, students were saying goodbye to the by-hand symbolic manipulation procedures that had previously dominated school mathematics instruction for countless decades. Some scholars at the time, including the widely-published M. Kathleen Heid and Michael Todd Edwards, argued that hand-held calculators provided classrooms with a richer approach to mathematics.1 However, there were those who held that calculators could damage, rather than improve, general education. Edward M. Gray, for example, argued that focusing on the many different cumulative strategic methods of mental math is more important to nurture than simple calculation. He was wary of the widespread implementation of calculators, as he believed that they had the power to sway the attention and subsequently cause a fall in performance.2 In addition to these viewpoints, the topic of disabled accommodation was also a common point of academic contention. The few studies done by other scholars seemed to point to technology as an aiding tool for those with learning disabilities and found that they generally equalized the academic standard of the classroom. Before long, however, the arguments in favor of calculators faded as it became clear that the implementation of the mathematics tools in classrooms began to cause children to have a less conceptual understanding of mathematics.3

The best way to organize this debacle is to look at why the calculator was implemented into general education in the first place. The CAS systems are also known as symbolic manipulators. Since the early 1980s, symbolic manipulation programs, or computer-based mathematics packages that perform exact arithmetic and symbolic algebra calculations, have been available for use by secondary mathematics teachers and students. The introduction of these systems into general education was originally seen as an open door to a possible shift from the emphasis on the traditional means of solution to a more nuanced and conceptual understanding of real-life aspects. Instead of wasting time by attempting problems that were near impossible to do on paper, one could now reach the same conclusion in a clear cut and understood manner in a matter of seconds. Shocked by their potential, scholars like Heid and Edwards believed that these systems could help bring about the creation of a richer approach to mathematics.4 Having access to classroom-available technological power, they claimed, enabled teachers to provide students with a whole new approach to mathematics and more solution-based classes, such as algebra. With the addition of the calculator to the classroom, students could hypothetically use symbolic manipulation, sense, and disposition all to reach an accurate conclusion. The process of learning mathematics would become one of paper and computer and could end with a better understanding of the underlying concepts of the subject. Defenders of calculators in classrooms would also point to the fact that the CAS keeps getting better and more advanced with each passing year as a reason to keep them in the hands of children. Pairing these powerful machines with conceptual learning, they argued, becomes increasingly important as society continues to advance, and children need to learn and adapt to the changes in the real world. Heid and Edwards acknowledged that this technology had the ability to change the way math was taught, and they believed that the CAS represented nothing more than an enrichment to the already existing curriculum.5

An up-close shot of a child filling out their math homework | Courtesy of jackmac34 via Pixabay

In addressing this line of reasoning, Edward Gray argues that the main focus should be on the conceptual understanding of mathematics alone. In his eyes, the calculator isn’t needed unless one finds oneself attempting to solve a near-impossible question. Branching off of this idea, Gray went on to argue that the focus on “learning how to solve” is being disregarded entirely by the widespread use of calculators. In constructing his argument, Gray gave many examples of how the average student from ages 7 to 12+ computes arithmetic. In his study, he found that the main methods of problem-solving were count-all, count-on, and known/derived facts. The use of cumulative strategies to solve almost any combination of addition, subtraction, multiplication, and division, was common throughout the classroom and observed in every student.6 Teaching students to use already understood concepts, Gray claimed, proved to be the most impactful method for their education. It was clear that those who used the methods listed above had higher performance rates than those who didn’t. It was also clear that not many students knew how to use derived facts, which is basically knowing to call back to past information. This is clearly a problem that needs to be addressed head-on before we even think of touching a calculator. While graphic calculators may nurture the problem-product linkages, the arena for further consideration must focus more sharply on the different types of thought that are brought to bear on mathematical activity by the able and the less able alike.7

Our dependence on calculators is made eerily clear by a popular study done by Bridgeman, Harvey, and Braswell. The study consisted of about a hundred college-bound juniors and had multiple levels of testing. Out of the total students, over 60% stated that they used a calculator almost every day, with only one student claiming to never make use of a calculator! The question remains, however: does the use of the calculator actually have any bearing on our abilities? The study attempts to answer this important query by testing students to find their average SAT-M test scores. After acquiring the scores, they then compared their results with the official results from College Board for college-bound seniors in 1991.8

The historical averages of SAT scores for college-bound seniors | Courtesy of Wikimedia Commons

This collection of data would represent the control or the base data that the researchers would utilize. After finding the control, they began adding stipulations to the test and subsequently observed how their actions affected the scores. They found that the use of calculators resulted in a modest score increase on a test composed of the type of mathematical reasoning found on the SAT, although effects on individual items ranged from positive through neutral to negative. The test also consisted of many different ethnicities and backgrounds, which caused the data to be a bit more skewed and less reliable as an overall conclusion. From the base data, the addition of calculators did not seem to drastically influence the overall scores of the experimental group of students. Instead, they were merely proven effective on conceptual problems by turning them into simple calculation routines and by benefiting low-scoring students. However, the removal of calculators altogether would actually benefit the high-scoring students. Thus, the study ultimately proved that it is impossible to make generalizing statements such as “calculators always improve scores,” because every student has a different level of understanding.9

Despite the presence of controversy that surrounds them, calculators will remain useful tools that can sometimes aid those with less conceptual understanding of mathematics. Many people are born with learning disabilities that cause math to be extremely difficult to understand. With that in mind, it shouldn’t be easy to say that learning math should be based on conceptual understanding and connections with derived knowledge. Attention to calculator use and students with disabilities is necessary as a combined result of the general controversy of calculator use in mathematics education, the lack of evidence-based value on calculators for modern society, and the specific controversy regarding calculators for students with disabilities. The largest of these controversies has to do with the underlying idea of fairness. Many parents think that it would be unfair to limit calculator use to special accommodations for disabled students because they believe that it will give them a supposed advantage. The use of calculators by students with disabilities for computation, however, has been likened to common tools to assist students with other types of disabilities. You would never deny a physically disabled child mobile technology, so the idea that giving children with learning disabilities calculators is unfair is simply discriminatory in nature.10 Nevertheless, to prove these allegations wrong, three scholars, Emily C. Bouck, Gauri S. Joshi, and Linley Johnson, conducted a test to see the difference in calculator use among students with and without disabilities. The test they set up involved 146 sixth-graders and 149 seventh-graders, of which about 17% had learning disabilities. The children were given both a multiple-choice and an open-ended exam. At the end of the test, students self-reported using calculators more often on multiple-choice rather than open-ended problem-solving assessments across the board, with the exception of sixth-grade students without disabilities, who received traditional curriculum materials. Students without disabilities self-reported using a calculator more frequently on both types of assessments as compared to students with disabilities. The data from this study found that students with disabilities answered more questions correctly when they used a calculator, and went on to prove that they were more likely to answer questions correctly when using this tool. This observation was also true for students without disabilities.11

The controversy of calculator use in pre-college math has been rampant since the first mention of the computer systems. As a result, there have been countless studies on the effectiveness of calculators and how they affect our overall understanding of mathematics. Although the idea of widespread calculator use is steeped in subjectivity, it must be affirmed that mental math is enhanced by knowledge of how to use calculators correctly. The calculator is supposed to encourage learning by bettering one’s general understanding of mathematical concepts—It makes concepts that are difficult to understand easier and quicker to learn. Regardless of one’s stance on the matter, each side agrees that mental math is a very important and crucial subject to learn as a young child. Without mental math, basic mathematics is very difficult and the child may even suffer in the long run. Calculators may be given out as accommodation for those with disabilities, but because they have an innate tendency to discourage the memorization of basic mathematical theory, they should be left out of the required supplies for the remainder of students. As evident in the study by Bridgeman, Harvey, and Braswell, the use of a calculator barely changed the overall scores of a variety of children in different grade levels. In fact, it may have made conceptual problems harder to solve because of common calculator errors. The best way to integrate calculators is to only include them in high schools so that prior knowledge is gained before students are given access to the tools. Looking through multiple studies, it is clear that calculators are not to blame. The educational system needs to regain its curriculum and use calculators as tools and not as crutches. Calculators are not the death of learning today, but they could become it if this blatant misuse continues.12

  1. M. Kathleen Heid and Michael Todd Edwards, “Computer Algebra Systems: Revolution of Retrofit for Today’s Mathematics Classrooms?,” Theory Into Practice 40, no. 2 (2001): 128-129.
  2. Edward M. Gray, “An Analysis of Diverging Approaches to Simple Arithmetic: Preference and Its Consequences,” Educational Studies in Mathematics 22, no. 6 (1991): 572-573.
  3. Emily C. Bouck, Guari S. Joshi, and Linley Johnson, “Examining calculator use among students with and without disabilities educated with different mathematical curricula,” Educational Studies in Mathematics 83, no. 3 (2013): 382-383.
  4. M. Kathleen Heid and Michael Todd Edwards, “Computer Algebra Systems: Revolution of Retrofit for Today’s Mathematics Classrooms?,” Theory Into Practice 40, no. 2 (2001): 128-129.
  5. M. Kathleen Heid and Michael Todd Edwards, “Computer Algebra Systems: Revolution of Retrofit for Today’s Mathematics Classrooms?,” Theory Into Practice 40, no. 2 (2001): 132-134.
  6. Edward M. Gray, “An Analysis of Diverging Approaches to Simple Arithmetic: Preference and Its Consequences,” Educational Studies in Mathematics 22, no. 6 (1991): 567-569.
  7. Edward M. Gray, “An Analysis of Diverging Approaches to Simple Arithmetic: Preference and Its Consequences,” Educational Studies in Mathematics 22, no. 6 (1991): 572-573.
  8. Brent Bridgeman, Anne Harvey, and James Braswell, “Effects of Calculator Use on Scores on a Test of Mathematical Reasoning,” Journal of Educational Measurement 32, no. 4 (1995): 328-329.
  9. Brent Bridgeman, Anne Harvey, and James Braswell, “Effects of Calculator Use on Scores on a Test of Mathematical Reasoning,” Journal of Educational Measurement 32, no. 4 (1995): 339.
  10. Emily C. Bouck, Guari S. Joshi, and Linley Johnson, “Examining calculator use among students with and without disabilities educated with different mathematical curricula,” Educational Studies in Mathematics 83, no. 3 (2013): 370-371.
  11. Emily C. Bouck, Guari S. Joshi, and Linley Johnson, “Examining calculator use among students with and without disabilities educated with different mathematical curricula,” Educational Studies in Mathematics 83, no. 3 (2013): 381-382.
  12. Brent Bridgeman, Anne Harvey, and James Braswell, “Effects of Calculator Use on Scores on a Test of Mathematical Reasoning,” Journal of Educational Measurement 32, no. 4 (1995): 330-331.

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38 Responses

  1. This article was a very interesting read, I never thought that the use of calculators was a topic of debate, I guess I was just accustomed to having one in my math classes, and after reading the arguments that were presented in the article, I think the use of calculators should be limited but not restricted. Taking classes in high school we were always reminded and encouraged to use calculators because it was more efficient and faster, but I never really considered that there could be mental drawbacks to having calculators because they always just felt more convenient.

  2. This is an interesting article, I have never thought of calculators as tools that can harm our learning, it might be because I grew up with this tools. Instead, I think that this devices are really helpful and can be very beneficial since we can use calculators to check our work in a fast way. I think that without the use of calculators during high school and middle school, we would have never finished a complete chapter, since using a calculator can increasingly reduce the time of a person performing an algebraic and calculus problem.

  3. From experience I know that the use of calculators has helped and hindered some aspects of learning. I have been taught to memorize certain equations and other such things to perform written calculations, but for things like trigonometry it can be a little harder which is where a calculator can come into play. Whether you choose to write out or calculate an answer there is always room for error and one may be needed to verify the other, which isn’t a bad thing.

  4. I can tell that calculators have affected the way that I have learned as well as other students. Such as easy computation is done on a fancy calculator has effects to a higher level of mathematics to where mathematics calculators become obsolete. As well as teachers in my high school, seeing the effects of calculators settings benefiting the student just by memorizing key algorithms to get the answer, then using mathematical methods to solving the problem

  5. The use of calculators in the classroom has always toed the line between being beneficial and being disadvantageous for students. While it remains a useful aid for completing more complicated problems, the very presence of calculators in a student’s life, in the classroom and beyond, greatly impacts the mental process when mathematics becomes involved, for better or for worse. Often, many would resort to the easily accessible calculator that smartphones come equipped with nowadays to answer mathematical questions that would appear in their daily lives. Now whether this dependency solely came from the introduction of calculators in the classroom can be further discussed and studied, but one cannot deny that they are part of the catalyst for this change. And it may be easier to blame the decreased mental mathematical abilities of students on technology, but a look at the grand picture is required and as Doyon mentioned, the curriculum and teaching methods implemented in today’s education may need to be reassessed and reformed to allow calculators to be used to its full potential in helping students learn mathematical concepts in the long run.

  6. Calculators have caused great advances in the world of mathematics, but also great debacles. From my own experience I can say that calculators create a certain conformism when it comes to solving problems and make the student or apprentice become dependent on it. Our calculation teacher does not allow us to use the calculator, and it seems to me a fair exercise, the calculators should be used as a tool to solve large-scale problems and not simple operations. Excellent article.

  7. I never really considered how calculators have affected the mathematics world. This article shows how calculators might have been more harmful than helpful to mathematics. Calculators have been around since I was born so it’s always just been normal to use calculators for anything a little more difficult than easy math. This article was really interesting and gave me a new point of view on how calculators can make us complacent.

  8. Back in my High School days, the use of calculators was not only allowed, but welcomed since my teacher oftentimes made us use the devices given to us by the district. The most fundamental truth in mathematics is that everything can be done by hand, as long as one has the knowledge to do so. However, doing this all the time takes up too much time; and since most standardized tests nowadays are critically time, the use of a calculator is pretty much mandatory if you want to make the time limit. Personally. Having been in both Calculus I and II in high school, I found that using a calculator is extremely beneficial when it comes to checking one’s work along with speeding up your math.

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