Running Head: DO CALCULATORS IMPROVE PROBLEM SOLVING ABILITIES?
Does the Use of Calculators Significantly Improve Students’ Problem Solving Abilities in Math?
Rebekah M. Stewart
Kennesaw State University
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Running Head: DO CALCULATORS IMPROVE PROBLEM SOLVING ABILITIES?
Introduction
The debate of using calculators in math, especially in elementary and middle grades, is a
long-standing one. Teachers argue that students need to develop mathematical understanding in
relation to the basic operations. Once students are in 4th/5th grade, does it really matter that they
do all computation with pencil and paper or are there times in which using a calculator could
indeed prove helpful? This idea originated from the continual debate of whether or not to
incorporate calculators into the math setting. The purpose of this study is to measure student
growth in regards to problem solving abilities. Does the use of a calculator help students to focus
more on the problem solving aspects of math? Will students perform the same with or without a
calculator? If the study yields positive results in favor for calculator usage in problem solving
scenarios, my teaching practice could be improved my allowing students to focus on the problem
solving aspect of math rather than spending time, energy, and efforts focusing on solving basic
operational skills via pencil and paper.
Definition of Problem Solving in Math
How is problem solving defined in a mathematical context? Problem solving is a higher
order cognitive process that requires detecting steps or processes (Xin, Zhang, Park, Tom,
Whipple, & Si, 2011). According to Van de Walle, Karp, and Bay-Williams, “Word problem
solving requires students to apply previously learned mathematical skills and thus promotes the
understanding of basic operations and whole number arithmetic” (2010, p.8). Monague, Warger,
and Morgan define math problem solving to be similar to reading comprehension in that it is a
“complex skill that requires students not only to calculate an answer but also to comprehend and
integrate the problem information, generate and maintain mental images of the problem, and
develop a viable solution path” (Montague, Warger, & Morgan, 2000, p.113). Xin, Zhang, Park,
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Running Head: DO CALCULATORS IMPROVE PROBLEM SOLVING ABILITIES?
Tom, Whipple, & Si remind us that reading comprehension skills definitely contribute to word
problem solving skills (2011).
Students build more mathematical knowledge through solving problems, and in a
community environment, mathematical thinking is fostered though listening and reflecting upon
others’ ideas regarding problem solving. Engagement is promoted through problem solving as
students relate to various mathematical concepts, facts, procedures and forms of reasoning to
build learning with comprehension. Students must be actively involved in problem solving in
order to identify and communicate ideas through problem solving scenarios. (Barrera-Mora and
Reyes-Rodriguez, 2013). Santos-Trigo, (2010) stated that: “An overarching principle that
permeates the entire problem-solving process is that teachers and students should transform the
problem statement into a set of meaningful questions to be examined” (p.301).
Schoenfeld has created four categories as a way to analyze someone’s problem solving
attempt: the goals the individual is trying to achieve, the individual’s knowledge (and resources
available), the individual’s beliefs and orientations about themselves and the material, and the
individual’s decision-making processes (2013). Math is about sense-making, and simple objects
and observations become starting points for explorations. Schoenfeld encourages us to have
questions asked, data sought, models built, and inferences drawn every day in our classroom
(2013).
The United States’ “Race to the Top” initiative has called for education to create more
students with greater problem solving abilities, ready for the workforce and/or college. Common
Core, which 44 out of 50 states have committed to, has called for eight mathematical practices
which include: making sense of problems and persevere in solving them, reasoning abstractly
and quantitatively, constructing viable arguments and critiquing the reasoning of others,
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Running Head: DO CALCULATORS IMPROVE PROBLEM SOLVING ABILITIES?
modeling with math, using appropriate tools strategically, attending to precision, looking for and
making use of structure, and looking for and expressing regularity in repeated reasoning
(“Common Core State Standards for Mathematics”, 2011). Success in math problem solving is
strongly related to overall math achievement (Krawec, Huang, Montague, Kressler, Melia de
Alba, 2013).
Cognitive Development
The days of pencil-and-paper-only, algorithmic style of math learning are waning
quickly. Researchers have recognized “the importance of mental computation, and have been
exploring its effects on students’ success in mathematics and the factors that influence children’s
accuracy and flexibility” (Varol and Farran, pg. 89, 2007). Mental calculation requires
performing arithmetical processes and calculations without the use of pencil and paper or
calculators. Varol and Farran (2007) inform us that, “…mental computation for addition and
subtraction of multidigit numbers plays a significant role on teaching children how numbers
work, how to make decisions about procedures, and how to create different strategies to solve
math problems” (p. 90). If a classroom learning environment doesn’t include mental math and
instead relies on pencil and paper for calculations, it is probable students will learn the process of
the algorithm and not truly understand what is happening with the relationship of numbers.
Studies show that students with mental math capabilities are better able to make flexible use of
the structure of the number system (Varol and Farran, 2007).
Having a cognitive understanding of how mathematical relationships are formed and
function is crucial to using computational tools. Students must be able to make connections
between the mathematical problem and approaches to the various mathematical concepts and
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Running Head: DO CALCULATORS IMPROVE PROBLEM SOLVING ABILITIES?
relationships needed to solve the problem. If a student is unable to do so, they will not be able to
coordinate technology to work in their favor (Barrera-Mora and Reyes-Rodriguez, 2010).
Calculators
Research suggests that low math achievement is attributed to a lack of motivation,
traditional math curricula using rote procedures, history of unsuccessful learning which causes
math anxiety for many, and absence of engagement during learning activities and lessons.
Technology is viewed as a resource to help lessen math anxiety and motivate students to learn
(Leh and Jitendra, 2012). Calculators are most definitely a tool of technology, and although
primitive in comparison to most recent technology devices, they have their place in the math
learning environment. The question is: when is it developmentally appropriate to introduce
calculator usage to students? We know that in middle and high school, calculators are an
everyday sight and even a requirement in upper math courses, but is it acceptable to have
elementary students experiment and learn mathematical concepts with the aid of a calculator?
Clark (2006) states that “calculators can be used in primary schools in a number of
situations” (p.18). Steele (2007) is of the belief that with “the future emphasis on the use of
graphing calculators, it is important for elementary teachers to introduce technology early and
prepare students with the technology skills that they will need in later years”(p. 59).
Technological tools allow for student experimentation and can “enhance and complement paper
and pencil approaches, supporting mathematical learning by offering opportunities to expand
students’ capabilities to visualize, experiment, obtain feedback, and consider the need to prove
mathematical results” (Arcavi & Hadas, 2000, p.32). Calculators allow for operation, analyzing,
and reflection upon cognitive thoughts. A calculator can be considered an “amplifier” if used to
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Running Head: DO CALCULATORS IMPROVE PROBLEM SOLVING ABILITIES?
perform arithmetic computations that are more challenging without the aid of technology
(Barrera-Mora and Reyes-Rodriguez, 2010).
The use of calculators may actually promote that students pay attention on the structural
components of problem solving, “…by facilitating the performance of routine procedures,
opening the possibility of approaching problems which were difficult to discuss within paper and
pencil settings, and modifying the cognitive processes that they develop to construct or to
operate representations of mathematical objects” (Barrera-Mora and Reyes-Rodriguez, 2010,
p.113). Calculators may have a place when concerned with student anxiety in math. Worry and
anxiety “…decreases the availability of working memory for a cognitive task, particularly when
the task requires the use of a substantial amount of working memory” (Fritts and Marszalek,
2010, p. 443). Technology is here to change and transform thinking, not to replace and serve as
the new way of thinking and teaching. Teachers must discuss the extent in which tools help us to
explore math tasks to enhance and complement problem solving processes. Santos-Trigo and
Camacho-Machin (2013) advise, “thinking of the task in terms of the affordances provided by
the tools demands that problem solvers focus their attention on ways to take advantage of the
opportunities offered by the tool to represent and explore the problem” (p.300).
Cons of Calculator Use in Elementary Settings
Of course, as with discussion regarding educational tools and technology, there’s an
opposable side. Steele (2007) states within his study of teaching calculator skills to elementary
students with learning problems that there may be behaviors that interfere with calculator use
such as, “…social skills deficits, problems in focusing, excessive movement, and disruptive
behaviors” (p.60). Instead of using the tool appropriately, students may choose to play with the
buttons and keys, distracting themselves and others. In addition to behaviors that may interfere
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Running Head: DO CALCULATORS IMPROVE PROBLEM SOLVING ABILITIES?
with calculator use, Steele provides modifications for using calculators. Some modifications
include, specific lessons on the use of a calculator, creating reminder charts with pictures for
uses of certain keys, and teaching students to make appropriate decisions about when to use a
calculator. Checking work is crucial in math and for some younger students, it may prove helpful
to check their calculator work by checking what they are working to key in and making sure it
matches their process. Erroneous calculator usage is counterproductive. It is important to teach
students to match the method of electronic calculations to the pencil and paper method. With
younger students, it is necessary to model calculators with real life scenarios so they can relate to
the problems and make connections (Steele, 2007).
In England, the Government announced in 2012 that they would ban the use of
calculators in primary schools on the premises that, “…primary school children use them “too
much too soon”” (“Does the maths add up,” paragraph 1, 2012). “[The Parliament] found that the
number of children with a poor grasp of calculation has more than doubled over the last 30
years” (“Does the maths add up”, paragraph 6, 2012). Professor Celia Hoyles OBE, Director of
the National Centre for the Excellence in the Teaching of Mathematics, said, “It is important that
calculators are used appropriately: so children do not become dependent on them for arithmetic
but at the same time are able to use them as a tool to support their own problem solving” (“Does
the maths add up”, paragraph 8, 2007).
One study performed by Pyke and LeFevre compared students with the use of direct
recall, mental computation and calculator use. Self-generation versus calculator learning led to
more frequent reports of answer recall on the test and “there may be memory benefits distinctive
to self-computation” (Pyke and LeFevre, 2011, p.608). The results of the testing found that
calculator-only learning led to the poorest recall on the test. It shows that “people perform better
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Running Head: DO CALCULATORS IMPROVE PROBLEM SOLVING ABILITIES?
on arithmetic problems when answers are self-generated versus observed or obtained with a
calculator” (Pyke and LeFevre, 2011, p.613). Pyke and LeFevre go on to discuss that “arithmetic
fluency is often required to efficiently grasp real-world problems and more complex math
concepts…adults educated before calculator use was common have better direct access ability
than recently educated adults” (Pyke and LeFevre, 2011, p.615). Pyke and LeFevre suggest that
if students do use calculators that they learn how to self-enforce the when and where of using
calculators. This may be accomplished by routinely encouraging students to attempt to recall a
piece of information for a certain period of time before using a calculator.
Methodology
This action research project lasted seven weeks and included 37 students in fourth grade
math classes. There are about 400 students at the school where the research took place with a
varied population regarding gifted, special education services, and ELL. 63% of the student
population is White, followed by 27% Black, 7% Hispanic, and 3% Multiracial. Our county is
one of the most economically disadvantaged counties and our school is at a 73% rate for free and
reduced lunches. I think that being in a low SES environment makes a difference. Parents have
fewer resources available and oftentimes, their own education level makes it difficult to help
their students as they get older.
The two classes are close in make-up in regards to EIP, giftedness, gender, race, etc.
Students do not currently use calculators in my math class. My first class, Class A, will have the
opportunity to use the calculators with problem solving. My second class, Class B, will be doing
the same problem solving tasks without calculators and only using mental and paper
computation. All students will take the same pretest with Class A using calculators and Class B
using mental and paper-based computation. Students will be exposed to some problem solving
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Running Head: DO CALCULATORS IMPROVE PROBLEM SOLVING ABILITIES?
techniques prior to the study, but the introduction of calculators will be the new factor for Class
A. I will use compound problem solving that requires multiple steps and operations to solve.
A mixture of formative and summative assessments was used in compiling data and
forming results. The summative assessment was a post test version of the prior given pretest. As
students moved through the process of using a calculator vs. not using a calculator, I took
anecdotal notes on how the process wass going, student responses, and positive and negative
comments made by students as well as comparing results of the students’ problem solving tasks.
I also completed an engagement survey at a random point during the assessments. I ranked
students on a scale of 1 to 3, with one being not engaged, two being somewhat engaged, and
three being engaged. Students were measured on their accuracy in their problem solving tasks
and whether or not their incorrect understanding came from miscalculations or from not
understanding the content/assignment at hand. I have used independent-measures t test on
comparing the two classes’ results, boys’ and girls’ from both classes results, and results on the
EIP math students within each class. The results gained have provided insight as to whether or
not student use of calculators significantly impacts problem solving abilities.
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Running Head: DO CALCULATORS IMPROVE PROBLEM SOLVING ABILITIES?
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Running Head: DO CALCULATORS IMPROVE PROBLEM SOLVING ABILITIES?
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