Middle School Math and the Calculator

I am frequently given the impression that parents of middle-grade students have been led to think there are two separate choices: their students should be encouraged to memorize the basic facts for addition and multiplication and master the processes or they should be allowed to use the calculator when solving mathematical problems. Since I don’t think it should be a choice of one over the other, I have chosen rather to share some thoughts on two related issues. In this article, when should a calculator be used? In the next article, how can the basic facts be reinforced while learning new mathematical concepts?

When to Use a Calculator

The United States National Council of Teachers of Mathematics encourages the use of calculators in the middle grades. Adults frequently use calculators to figure their grocery bills or to balance the checkbook. Most middle-grade students have access to them. With these facts in mind, under what circumstances should teachers make the decision to use calculators with their students? The following guidelines are helpful in answering this question.

A student could be allowed to use the calculator in the following situations:

1. The goal of the activity is to determine a solution when the required computation is beyond the student’s ability.

2. The goal of an activity is not to determine a computational solution, but some computation is necessary to achieve the goal. This may be true when a teacher wishes to have a student develop problem-solving skills or when exploring mathematical patterns.
   
   

A student should not be allowed to use a calculator when the goal of the activity is to determine a computational solution, and the student is able to do the necessary computation in a reasonable length of time.

Let’s look at a few examples to illustrate these guidelines:

Allowing students to submit real-world problems that are important to them can help stimulate interest in further study of mathematics. A middle-grade student may wish to determine the best deal when purchasing a bicycle after the prices of his choices have been reduced by various percentages. If he has not yet mastered multiplying with percents, the use of a calculator may be necessary to calculate the correct answer. Student participation in the activity requires a calculator; the goal of the activity is to determine a solution involving computation.

One very important principle of learning and applying mathematics is being able to identify patterns. A student in elementary school may be interested in finding a pattern in a problem that requires a substantial amount of calculation beyond the knowledge or patience of the student. Sometimes the computation may just be too tedious and complex to sustain the interest of an intelligent student. Since the teacher’s goal at this moment is primarily to give the student experience in finding patterns, allowing the student to use a calculator may be necessary. The goal of the activity is not to determine a computational solution, and student participation in the activity requires a calculator.

Examples:

• What is the average of all three-digit numbers that can be created using each of the digits 1, 2, and 3 exactly once? (Example: 231) Can you explain why your answer is correct? What is the average of all three-digit numbers that can be created using each of the digits 3, 4, and 5 exactly once? Do you see a pattern? Can you give three more digits that follow your pattern? Can you give three digits that do not follow your pattern?

• Suppose x, y, and z are negative numbers and x < y and y < z. Are the following expressions negative or positive?

  1. x/y

  2. y-x/z2

  3. (-x)(y)/z

  4. y3/x5
     
     

Note: Student participation in this activity does not require a calculator; the goal of the activity is to determine a computational solution.

Most mathematics programs include problems designed to show how some mathematical computations or concepts can be used to solve real-world problems. The goal of the exercise is to make use of the mathematics previously learned. Using a calculator would not provide the intended practice. Student participation in the activity does not require a calculator; the goal of the activity is to determine a solution by using computation.

Being able to determine a suitable estimation is another important mathematical goal, but it won’t happen without practice. The calculator can be an effective tool in checking the aptness of a student’s estimation.

Consider the following problems:

• Johnny and his family just moved into a new house. His mother gave him $20 and asked him to go to the corner store to buy food for dinner to feed their family of four. The items he selected cost as follows: bread–$1.67, butter–$2.59, apples–$3.99, lettuce–$2.98, meat–$6.34 and potatoes–$3.14. Did he have enough money to pay for the items selected?

• About half of the 4,864 baseball fans at a game bought a hot dog for 75 cents. The total amount spent on hot dogs that day was about:

  1. $1800

  2. $2,600,

  3. $3,000

  4. $3,700

Note: The solution to these problems does not require an exact number. The goal is to estimate the amount spent. The calculator could be used to check the estimate while the estimation skill is being developed.

When a teacher is deciding whether or not to use a calculator with a lesson, the ability and character of the student is an important factor. A student who has not mastered the basic facts may be left out of a lesson if not allowed to use the calculator. Allowing her to use it for a particular lesson while still requiring her to practice the basic facts may motivate her to persist in a goal to learn, enjoy, and apply mathematics. Another student who knows the basic facts and has mastered the processes but is easily bored with too much repetition may also benefit by being allowed to use the calculator for some problems.

At times the calculator can encourage students to do worthwhile exploration of mathematics.

Consider the following problems:

• Extend the pattern:
  12 = 1
  112 = 121
  1112 = 12,321
  11112 = 1,234,321
  
  For many middle-grade students, the calculation required could subtract from their finding the patterns. The solution to these problems does require exact numbers. The goal is to find a pattern. The calculator could be used.

• Try these two decimals: 0.8 and 0.4
  Divide the second by the first: 0.4 ÷ 0.8 = 0.5.
  Continue the process as before: 0.5 ÷ 0.4 = 1.25
  1.25 ÷ 0.5 = ______
  ___ ÷ ___ = ______
  ___ ÷ ___ = ______
  ___ ÷ ___ = ______
  ___ ÷ ___ = ______ 4

Note: Do you see a pattern? If so, can you explain why the pattern works? 

 Many of the examples I have mentioned illustrate when I think a calculator could be used to an advantage in a middle-school mathematics program. Comfort and skill in using the calculator is desirable. Some middle-school math programs recommend that a scientific calculator be available to students as a problem-solving tool. In grades five to seven they need a scientific calculator, and in grade eight they need a graphing calculator.

It is important for a student to learn the capabilities of a calculator. Paper and pencil, estimation, mental arithmetic, and the calculator are all important tools for solving everyday problems. The ultimate goal is for students to know when to use a calculator and when to use their computation and estimation skills. The most powerful method is the one that is most efficient and effective for the problem situation.

Memorization of the basic facts and mastery of the addition, subtraction, multiplication, and division processes are also very important if the student is going to be able to use mathematics effectively. These are not two exclusive aspects of middle-school math, and one should not be ignored at the expense of the other.

Permission to copy, but not for commercial use.

1. Dr. Louise Johnson is a retired mathematics educator and former university dean.