Using Calculators to Explore Mathematical Thinking
By Kathleen Lynch-Davis, Posted March 16, 2015 –
For decades, calculators served as a source of debate for
many people who have a vested interest in education, including parents,
teachers, and students. Over time we’ve learned that when used in meaningful
ways, a calculator can be a valuable tool for learning rather than just a
computation device. This view is supported by the Common Core’s Standards for
Mathematical Practice (SMP) 5: Use appropriate tools strategically (CCSSI
2010, p. 7). I like to engage elementary school students, preservice teachers,
and classroom teachers in problems that give students opportunities to develop
conceptual understanding while debunking the myth that calculators are useful only
Below are three examples of problems that I like to use that
represent different grade levels and mathematics concepts. For these problems,
I typically have students use a TI 15 calculator or one that possesses similar
Press 0 + 1 =
into the calculator. This will now make the calculator repeat +1 with any
number entered. For example, a student can now enter 6, press the = key, and
the number 7 will appear, thus supplying
the student with the answer to 6 + 1. Another option is to have students do
several +1 problems, see the pattern that emerges, and conjecture as to what is
happening to any number when they enter +1.
All students from kindergarten through grade 2 can predict
what happens when they add 1 to 6; then they can use the calculator to check
their answer. They can then see patterns when a 1 is added to a number and make
generalizations about the relationship of a 1 added to any number. (Adapted
from Van de Walle, Karp, and Bay-Williams. 2013. A Calculator Two-More-Than Machine, p. 137)
Laura wanted to enter the number 8375 into her calculator.
By mistake, she entered the number 8275. Without clearing the calculator, how
could she correct her mistake?
(Source: National Assessment of Educational Progress. 1992.
Grade 4 Mathematics Assessment.)
Students attend to place value without specifically asking
what number is in which place. Using the calculator, students are able to
conjecture how to fix the mistake and use the calculator to verify or discount
their claim, connecting that a difference of 1 in the hundreds place is indeed
a difference of 100.
Making Sense of
A bus holds 46 children. If 500
students are going on a field trip, how many buses do you need?
Students notice that the answer is 10, remainder 40; or if
they fail to use the integer divide key, the answer will be approximately
10.87. By using a calculator, students are able to focus on the meaning of the
remainder and make sense of how the remainder affects the number of buses
rather than spending their mental energy on the computation. Students, then,
need to make sense of the answer the calculator provides. Do they need 10 buses
or 11 buses? Using the knowledge
that all students must go on the field trip, they can use the remainder to
determine that another bus is needed. Therefore, the answer is 11 buses.
Share your Calculator
Calculators can be used for a variety of purposes: facilitating
a search for patterns, creating problematic situations, supporting concept
development, promoting number sense, and encouraging creativity and
exploration. These are just a few examples of problems that can encourage student
thinking and conceptual development while using the calculator as a tool. How
do your students do on problems like these? I invite you to share your
students’ work and ideas on these problems or share your own calculator
problems. I look forward to engaging in continued dialogue about using
calculators meaningfully in elementary school classrooms.
want to hear from you. Post your comments below or share your thoughts on
Twitter @TCM_at_NCTM using #TCMtalk.
is a professor in the Department of Curriculum and Instruction at Appalachian
State University. She currently teaches mathematics education and curriculum courses
to elementary- and middle-grades preservice and in-service teachers. Her
research interests include preparing elementary school mathematics specialists
and online learning in mathematics education.