Using Calculators to Explore Mathematical Thinking

  • Using Calculators to Explore Mathematical Thinking

    2015_03_16_LynchDavis_art_a

    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 for computation.

    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 functionality.

    One-More-Than Machine

    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’s Calculator Correction

    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 Remainders

    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 Tasks

    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.

    Your Turn

    We want to hear from you. Post your comments below or share your thoughts on Twitter @TCM_at_NCTM using #TCMtalk.

    2015_03_16_LynchDavis_aupic_aKathleen Lynch-Davis, lynchrk@appstate.edu, 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.

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    Amanda Johnson - 11/8/2015 3:13:40 PM
    I think using a calculator in the classroom when needed is completely appropriate. Calculators should not be relied on for simple tasks, however, when a math task is taking too long just because a student cannot do the figure in their heads, I think a calculator becomes necessary. The calculator can help the student find the final answer after doing their own work, and I do not think this takes anything away from the task.

    Drew Polly - 3/19/2015 4:15:53 PM
    Chris- Nice ideas. I have used #1 before with fourth graders and they love it. The ideas of your tasks and the ones that Kathleen mention allow the calculator to be used as a tool to support students' thinking, but do not do all of the thinking and work for the students. I believe that the idea of better supporting understanding comes through conducting research about the influence and benefit of using the technology, providing examples for teachers, and spreading the message that technology uses such as these are beneficial.