Introduction: Standards, Goals and Objectives of the Project
As a student, I used calculators for many purposes. My first memory of using calculators was in first grade when our teacher had us complete a similar activity as described by Huinker (2002) where we started entered 1 + 1 = = = into our calculator and then would race to see who could get the highest number within a certain time frame (p. 318).. I remember marveling at the fact that the calculator knew to keep adding one more even though I wasn’t pressing + 1 each time. I felt as if the calculator were magical and somehow knew what I wanted, not really making the connection between what I typed and what the calculator did. I admit the game was always a bit stressful! Reflecting on this experience, it did not seem conducive to creating positive associations with calculators. In later years, calculators were used for simple computations within solving other problems. In high school I began using graphing calculators and I always felt as if it were speaking a different language. Sometimes even now, when working with students, I will become frustrated when for example, the student enters a function and it does not graph. My usual solution to this is to find another calculator that “works” since I’m not sure how to fix it! Calculators seem temperamental to me even to this day.
As a mathematics paraprofessional in a middle school, I work providing intervention to students who have scored in the 15th percentile or lower on benchmark tests. Calculators are often used for computations such as dividing three digit numbers by a two digit numbers, usually within a homework assignment. I have students solve problems on their own using paper or building from facts they know unless the teacher specifically states calculators can be used. If a teacher does allow the student to use calculators on their homework, they are mainly used for multiplication and division facts they do not feel comfortable with. Students in pre-algebra use graphing calculators for graphing functions and finding the slope as well as comparing the slopes of two functions. When calculators are used, the students focus more on the process of solving problems and less on the frustration of not knowing basic facts. At a separate time, students work on their multiplication facts through flash cards, Fastt Math, or other forms of quizzing. For students who are in middle school however, there is an overreliance on calculators for basic computations. I have seen students who are performing 7th and 8th grade skills but are getting incorrect answers and struggling to perform on the problems not because they do not understand the current math content, but because they are counting on their fingers to add and subtract numbers. My theory is that if students learn to see calculators as a tool for constructing meaning and not as a way to easily find computation answers, they will be forced to learn their computations and view calculators in a different light. Therefore these activities seek to do so.
Calculators play a more specific and clear cut role in mathematics at the secondary level than at the elementary level. In the NCTM Curriculum Focal Points, calculators themselves are never specifically mentioned. In the NCTM Principles and Standards for School Mathematics, calculators are mentioned within Numbers and Operations where students are expected to “use a variety of methods and tools to compute, including objects, mental computation, estimation, paper and pencil, and calculators” at the second grade level (NCTM, 2006, p. 23). Calculators are again mentioned in grade 3-5 content expectations in relation to selecting the proper tools for completing problems within Numbers and Operations expectations; and are mentioned twice in 6-8 Numbers and Operations expectations. Within the Common Core standards for mathematics, calculators are mentioned in the processes and proficiencies to be mastered by students. Within this section, it describes how students might use their graphing calculators to make sense of problems and persevere in solving them. It also describes how math proficient students use tools effectively. The tools include calculators and provide an example of high school students being proficient in using graphing calculators to analyze graphs of functions and solutions. Calculators are again mentioned under high school mathematics, number and number systems as a way for students to become acquainted with new number systems and their notations (CCSS, 2010, p. 6-7). The use of calculator is based on building conceptual knowledge but also as a tool to use when computing.
There has been much debate over whether calculators should be used at all in lower elementary classrooms. Some argue that the use of calculators in kindergarten to grade two creates a reliance on the calculator, where students :may not learn computational algorithms” and creates a “false sense of confidence” in their mathematics abilities (McCauliff, 2013, p. 5).In countries such as Great Britain, calculators are banned or viewed as being inappropriate for student use. Great Britain recently banned the use of calculators in primary schools “for most purposes” because students use them “too much too soon” (Editorial, 2002, p. 2). In Japan, there is a “virtual absence of calculator use in Japanese” classrooms, even “in eighth-grade” (McCauliff, 2013, p. 8).
The NCTM position on using calculators in elementary schools is “calculators have an important role in supporting and advancing elementary mathematics learning” but the calculator must be “selective and strategic.” Providing students with opportunities to work with calculators at a young age can be beneficial, as NCTM reasons calculators “promote the higher-order thinking and reasoning needed for problem solving in our information- and technology-based society, and they can also increase students’ understanding of and fluency with arithmetic operations, algorithms, and numerical relationships” (NCTM, 2011, p.1). With more rigorous mathematics curriculum with the Common Core, students are using graphing calculators to find the slope, compare slopes of functions, etc. at an earlier age, as early as six grade in the district where I work. Students therefore must have an understanding of calculators as tools of discovery rather than as a way to check ones work or complete what should be automatic computations. While the latter ends up developing the calculator as a crutch while the former helps develop students’ critical thinking skills and awareness of the calculators varied purposes beyond simple computation. The students must be provided with proper calculator instruction at a young age to have the foundational skills to make them proficient at using more complex calculators. The focus of these activities is on building number sense and number pattern awareness, not on building computational skills. One hindrance to implementing a strong use of calculators in the lower elementary grades is teachers are unsure of best practices and feel they do not have the proper resources (McCauliff, 2013, p. 11). These activities serve as a starting point to build the teachers confidence in introducing and using calculators with their students.
The purpose of this project is to explore effective use of calculators to teach Numbers and Operations in Base 10, including patterns and number relationships, in first grade. More specifically, the project has the following objectives:
2. Familiarize students with ideas of the base 10 system through the use of patterns, repeated addition, and studying number relationships.
3. Build students familiarity with calculators and understanding of how to best use calculators as a tool for learning.
Overview of the Calculator Project
This project is a series of activities to build students familiarity with using calculators as tools as well as building their understanding of the base 10 number system. It is aimed at the first grade level and is based on the article by DeAnn Huinker (2002) entitled “Calculators as learning tools for young children’s explorations of number” where two teachers used the constant addition tool on a calculator to practice counting and to skip count. The examples of using calculators as a pedagogical tool provides students with a means of using the calculator other than for checking ones work or for practicing computation. The activity from this article will be modified in that students will be using constant addition and subtraction tools to study the base 10 system, as well as added several other activities that build on the constant addition activity. This project would be intended as an activity for a whole first grade class to do independently or in small groups. Students will be expected to already be familiar with the concept of 10, of what represents 10 and how to create the number 10 on their calculator as a combination of the 1 and 0 button. This project will build students’ knowledge of a calculator as a resource in the classroom. Also, as explained by the NCTM, “the strategic use of calculators enables elementary students to engage in mathematically rich problems that involve recognizing and extending patterns, testing ideas, and exploring relationships, without getting caught up in the mechanics of rote computation” (NCTM, 2011, p.1). This activity will benefit students as they will be able to build their understanding of the Base 10 system and how numbers can be combined or taken apart to represent to create 10s and build on each 10 to create larger numbers. Students are learning to “recogniz[e] the relative magnitude of numbers including being able to compare and order numbers” through these activities (Huinker, 2002, p.319). Students are exploring and developing ideas of numbers through the use of the calculator.
In this activity, first grade level students will complete Worksheet 1 using their calculator to explore how many ones equal one ten. The standard being met is The project meets the following first grade CCSS Standards 1.NBT.2 “Understand that the two digits of a two-digit number represent amounts of tens and ones” as well as the subsets of this standard a, b, and c: ” Understand the following as special cases: (a) 10 can be thought of as a bundle of ten ones — called a “ten”
The activity can be adjusted to meet standards above and below first grade to meet students varying needs. The correlated kindergarten standard is K.NBT.1: “Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g., by using objects or drawings, and record each composition or decomposition by a drawing or equation (e.g., 18 = 10 + 8); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones” (CCSS, 2010, p. 15). The lower level students’ needs will be met through worksheet 2 in which the students in which it requests students to decompose numbers into their ten’s and ones. Students will use the calculator by beginning at 10, thinking how many they have to add to get to a specific number such as 14. The second grade standard is 2.NBT.2: “Count within 1000; skip-count by 5s, 10s, and 100s” (CCSS, 2010, p. 19). This standard will be met through Worksheet 3, in which students will follow instructions to use the constant addition capability to add 5 (5 + 5 = =) observe the pattern, and record the pattern on their worksheet. Students will repeat the process for skip counting with the number 10 and 100, skip counting up to 1000.
The type of calculator needed for this activity is any basic calculator found in elementary classrooms. I would recommend a Kids Big Calculator (SL200BLU) from Casio calculator as this calculator has easy to use keys and easy to read screen as well as a straight forward design. In the case of my activity the specific type of calculator is not an issue as long it is student friendly and age appropriate. A calculator with two lines of text on the screen would be helpful in that student’s would be able to visually see what they have typed in as well as the answer, which would be especially helpful in Worksheet 1. I would also emphasize when completing this activity with a class to have a classroom set of identical calculators to ensure that all students are using calculators with the same capabilities.
Students will be assessed by the completion of their worksheets and by informal conferences. The teacher will, based on student answers, ask individual students to explain what they did to arrive at their answer to allow the teacher to see into the students reasoning. The teacher will also ask follow up questions, for example with Worksheet 3, what the second number to the right is and what the first number to the right is to see if students are understanding the concept that is being brought across by the worksheet. Students will be expected to answer all the questions on the worksheets accurately and explain their reasoning to demonstrate their increased knowledge.
I have never actually taught this concept or used this activity in the classroom. However, I think this concept will be a worthwhile one for the classroom as it serves dual purposes: one, it exposes students to using calculators as a tool and not as an answer giver; and two, it builds students’ knowledge of the Base 10 system. One concern that I have is I feel the worksheets will be too hard for students to understand. I want to be able to portray my meaning through graphics more, but I am afraid taking away the words will lead to the worksheet not making sense. This is the first time I have created worksheets for younger grade levels where the students are more visual and less able to independently work with text based instructions.
McCauliffe, E. (2003. The Calculator in the Elementary Classroom: Making a Useful Tool out of an Ineffective Crutch, Top of Form
Bottom of Form
Common Core State Standards Initiative. (2010). Common core state standards for mathematics. Available from http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf
Editorial. (2012). Does the maths add up? Education Journal, (145), 2.
Huinker D. (2002). Calculators as learning tools for young children’s explorations of number. Teaching Children Mathematics, 8 (6), 316-321.
McCauliff, E. (2013). The Calculator in the Elementary Classroom: Making a Useful Tool Out of an Ineffective Crutch. Concept. 27 (11) 1-13.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author. Available from http://standards.nctm.org/.
National Council of Teachers of Mathematics. (2006). Curriculum focal points for prekindergarten through grade 8 mathematics: Reston, VA: Author. Available at http://www.nctm.org/standards/default.aspx?id=58.
National Council of Teachers of Mathematics. (2011). Calculator Use in Elementary Grades. Available from http://www.nctm.org/about/content.aspx?id=8854