The Estimation Calculator

This text is a text/video-based case study that dissects a math lesson on computational estimation. It features a free, web-based application called the Estimation Calculator, a resource specifically designed to develop students’ computational estimation strategies.

Computational Estimation

Computational estimation involves making reasonable approximations for arithmetic problems, and it differs from estimations of measurement and time. Computational estimation is a form of problem solving that calls on a variety of skills used daily by adults, yet many people have never received adequate instruction on effective estimation strategies.

Teachers can begin developing children’s computational estimation abilities and expanding their range of estimation strategies in elementary school. Students are generally capable of the abstraction required for computational estimation by the late elementary grades, but explicit instruction and practice are essential.

Strong estimation abilities enhance children’s number sense, place value recognition, and understanding of the relative magnitude of rational numbers. The Common Core State Standards for School Mathematics provide guidance on children’s learning of computational estimation.

Relevant Common Core standards for estimation include the following mathematical activities.

“Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.”

“Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.”

Additionally, the National Council for Teachers of Mathematics (NCTM) recommends that all elementary students learn how to effectively estimate both mentally and on paper.

Despite consensus among national organizations for flexible estimation approaches, students often learn only one strategy for estimating. A commonly employed strategy involves rounding to the largest place value without an understanding of how reasonable the estimate is in terms of the exact solution.

For example, students in the upper elementary grades often learn to estimate by rounding to the largest place value (77 x 18 becomes 80 x 20) and solve the problem (1,600). Although the answer is technically correct and the estimation strategy is valid, students rarely consider how close the estimate is to the correct solution: 1,386. Rounding to the largest place value produces an estimate that is more than 15% higher than the actual answer.

Students need to understand that other strategies may bring them closer to the actual answer. Here are some other estimation processes students can use to generate a more reasonable estimate:

TRANSLATION: Changing the operation or mathematical structure of a problem so that it is easier to estimate. For example, a translation strategy involves altering 18 + 22 + 24 to 20 x 3.

REFORMULATION: Altering the numbers in a problem so that it is easier to find an estimate. For example, a reformulation strategy involves altering 18 + 22 + 24 to 20 + 20 + 20.

Misconceptions About Estimation

For example, students sometimes think that the exact answer is the only solution that is important. It follows, naturally, that estimation is a waste of time when emphasis is placed on exact calculations.

Students need to know the value of computational estimation before they will put effort into developing their skills. They must realize that computational estimation saves time and effort in the real world when there is not access to a calculator or when inputting information is too time consuming.

In many situations an estimate provides a useful check on a supposedly exact answer, even when using a calculator. Students should be reminded that a calculator yields a correct answer only if they enter the problem correctly. When students can appropriately estimate whether a calculator’s answer is approximately correct, they can then go on to identify possible issues related to incorrectly entering the problem.

I am a gifted resource teacher at Crozet Elementary School in central Virginia. I have taught kindergarten through fifth grade for over forty years, both as a regular classroom teacher and gifted resource specialist. I am a Google Certified Teacher, an Apple Distinguished Educator, a STAR Discovery Educator, a Fablevision Ambassador, and a Computerworld Laureate. I also recently became a National Board Certified Teacher!

My philosophy of teaching mathematics involves a mixture of direct instruction, discourse, and hands-on activities. I think that the teacher needs to provide enough instruction so that students have a direction but not so much that the teacher is dominating the learning experience. To this end, I place great value on asking questions that elicit students’ thinking about content. Although not easy, questioning and listening are two keys to helping students move beyond a superficial understanding of mathematics in my opinion.

With regards to teaching estimation and estimation strategies, I believe that is important for students to be able to find a range of acceptable estimates that closely approximate the correct solution. Just teaching a singular approach to estimation, like rounding to the largest place value, insufficiently prepares students to judge the reasonableness of a slew of possible estimates! Students need a repertoire of strategies and time to practice them in order to be successful estimators.

The school where I teach, Crozet Elementary School, is a small, rural school with approximately 300 students in kindergarten through fifth grade. Of these students, 24% receive free and reduced meals through federal assistance. Most of the students are White (86%); only 2.4% are African-American, 4.2% are Hispanic, and 3.5% are limited English proficient. The classroom in this case study had more females than males, which is not typical of the school in general. Of the 20 students in the class, all were classified as high performing in mathematics, with a smaller subset of the total group considered to be academically gifted. Despite the advanced characteristic of the class, I believe that this lesson is appropriate for all fifth grade students.

The Lesson (According to Paula White)

My overarching objective for this lesson was to encourage students to use alternative estimation strategies based on the reasonableness of an initial estimate. I relied on the Estimation Calculator, a free web-based application, to guide students toward increased precision, and I explained to the students that verbalizing their mathematical thinking was extremely important.

I also wanted students to have a contextual hook for why estimation was necessary, so I incorporated word problems that applied to the students’ real lives. I made informal observations of students’ conversations. At the end of the lesson, I assessed student learning formally through their written answers to questions.

The lesson occurred in three parts: Introduction, Activity, and Conclusion.

Introduction

Activity

For each problem, one partner was assigned the role of “worker,” and this person made an estimate and verbalized the strategy that was used. The other student’s role was to take notes and write down the worker’s strategy.

Conclusion

Primary Technology: The Estimation Calculator

In this lesson, Paula chose to use the Estimation Calculator, a software application developed specifically to develop students’ skills in computational estimation, estimation strategies, and making reasonable estimates.

The Estimation Calculator provides a visual indicator of the reasonableness of an estimate within a certain percentage above or below the exact answer. Paula used this tool to encourage students to implement different estimation strategies for addition and multiplication problems involving whole numbers. The Estimation Calculator provides feedback on the reasonableness of estimates to mathematical problems.

RESEARCH & INFORMATION FLUENCY: Students apply digital tools to gather, evaluate, and use information.

Classroom In Action

Throughout the activity portion of the lesson, receiving feedback on the reasonableness of an estimate became an impetus for students’ use of various estimation strategies.

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Student Work

Assessing Written Answers (According to Paula White)

Students were scattered throughout the room during the activity portion of the lesson. Some were sitting on the floor beneath the whiteboard, others at desks, and a few lounging on the classroom couch.

As I walked around and observed, I saw that most students were busily discussing their estimation strategies with their partner. I immediately noticed that students were having rich discussions about the problems. They were checking their estimates on the Estimation Calculator and, in many cases, revising their strategies. However, it was quite apparent that the conversations did not correlate with what was being written on their worksheets.

For example, one pair had a rich dialogue about how to estimate 26 x 59. The student making the estimate for this problem explained that rounding to the tens place for both numbers (30 x 60) produced an estimate of 1,800, which was outside of the range of a reasonable approximation based on the Estimation Calculator. The student then rounded 26 to 25 and explained that rounding down would produce a more accurate estimate because the rounded number was closer to the original number.

I reviewed all of the worksheets and saw a lack of written clarity and specificity much like the previous example. It was almost as if the conversations were too complex and too long for the students to clearly capture in the space provided. Yet, I wanted the students working in pairs and talking to each other because I believed that verbally sharing encouraged students to expand their repertoire of strategies and approaches. In my opinion, mathematical discourse was a powerful way to strengthen conceptual knowledge of estimation.

Cumulative Assessment (According to Paula White)

I gave all of the students a short written assessment on the day after the lesson. The assessment included questions for the upcoming unit as well as two questions pertaining to estimation.

The first estimation question was a 4-digit by 4-digit subtraction problem, and the second question was a 2-digit by 2-digit multiplication problem. I asked the students to estimate the answer and described the estimation strategy that they used to arrive at the approximation. Students wrote their answers on paper and did not use the Estimation Calculator to check for reasonableness. I chose not to use the Estimation Calculator in the assessment because I wanted to see whether or not students would use alternative strategies that did not involve rounding to the largest place value without being prompted by the tool’s visual feedback.

The results were quite interesting! All of the students used a variety of strategies to solve both of the problems even when not using the Estimation Calculator. The subtraction problem produced the most diverse set of answers (9,644 – 5,788). One student articulated that he initially rounded both numbers to the hundreds place (9,600 – 5,800). He then realized that the rounded numbers were not easy to mentally subtract, so he rounded 9,644 to 10,000 and then subtracted 5,800 from 10,000 to arrive at his estimate (4,200).

Although rounding to the largest place value would have resulted in a reasonable estimate on the Estimation Calculator had it been used, this student chose to use a more precise estimation strategy that involved compatible numbers. I am fairly certain that this student would not have answered the question in the same way had he not practiced problems using the Estimation Calculator.

Teacher Reflection

Estimation Strategies

I honestly think that incorporating the Estimation Calculator into my estimation lessons forced students to think about using different strategies.

Students were less reliant on basic rounding and more adept at examining the original numbers as clues for creating an estimate after the lesson. I believe that this change was a result of the visual feedback that the Estimation Calculator provided.

Why The Lesson Worked

Two key components in this lesson’s success were paired collaboration and probing questions.

Students listened to their partner’s estimation strategies and, in many instances, offered their personal approach without being prompted. This was especially true when the Estimation Calculator showed that an estimate was unreasonable and outside of the 15% range. The exchange of ideas in a one-on-one, student-to-student situation enhanced the applicability of various strategies in ways that a more teacher-centered, directive manner would fail to achieve.

The Technology

I did not experience any major classroom management issues with the technology. I am certain that this was a result of the cumulative experiences that the students have using various tools on a daily basis. The students knew my expectations of their technology use, and they acted accordingly.

However, that does not mean that everything went smoothly!