Tuesday’s Lesson — What Can You Do with a Broken Calculator?

by Joanna Lu and Judah Schwartz

Kelly Goorevich teaches fourth grade at the Hosmer public elementary school in Watertown, Massachusetts. The following “letters to Mom” are part of an activity Ms. Goorevich developed to accompany her math lessons using the Broken Calculator.

“Mom! Don’t throw away my broken calculator! ‘Why?’ you ask. Because it can still be used. ‘How?’ I’m so smart … say I can’t use the 5 key and I need to multiply 5 x 27, I know I can add 3 + 2 and then do (3 + 2) x 27 = 135!”

“Why did you throw away my favorite calculator? It helps you think out of the box. That means think of new and interesting ways to learn. It can help me learn more about numbers. It’s very fun to find out how to solve it and not to go for the answer.”

“Mom, don’t throw away my broken calculator! … if your multiplication sign is broken …but the problem you are working with has a multiplication sign in it … BIG WHOOP! You can use addition or you could use division or you could use subtraction. You can sort of fix the broken keys with other keys.”

Broken Calculator: a laboratory for exploration

Figure 1: The Broken Calculator, a Flash interactive.

Some critics fear that calculators undermine the teaching of elementary math. With a calculator at hand, why would anyone bother to memorize number facts or learn computation? If students use calculators, what will they do if the calculator, well, breaks?

The Seeing Math™ Elementary teacher professional development course, currently distributed by Teachscape, turns the technology around and uses this unhappy scenario to develop mathematical skill and insight. The original Broken Calculator program, created by Dr. Judah Schwartz of Tufts University took this simple concept, developed the broken calculator scenario, translated it into a versatile application, and made for students a calculator with selectively disabled keys. (This can be applied in the form of a computer program called “Broken Calculator,” which runs in Flash, or one may simply pretend that certain keys on a calculator no longer work.) Students must solve arithmetic problems without using the broken keys. For example, they must solve a division problem in spite of a “broken” or disabled division key.

Consequently, students must be far more inventive in how they solve problems. In standard algorithms, the computational steps are simple. All one needs is knowledge of single-digit addition or multiplication facts. But Broken Calculator requires reasoning beyond the standard mechanics. It challenges students to explore number relationships and use their mathematical knowledge to invent and test a wide variety of strategies. The intellectual responsibility for finding the answer is shifted from the calculator back to the students.

Figure 2: Disabling keys means they’re no longer available to the user.

Broken Calculator is, in effect, an inexpensive, widely available number laboratory for developing and assessing students’ computational fluency. As students develop strategies for solving problems, they:

• Learn to reason mathematically
• Explore place value and relationships between operations
• Apply basic math skills, such as "math facts"
• Develop the flexibility to apply their skills to unfamiliar problems

Let’s start!

Use the Seeing Math Broken Calculator on the CD or at our website.

The interface is simple, and works much like a standard handheld calculator. The number and operator keys are “pressed” by pointing and clicking with the mouse, or by using the computer keyboard numbers and operators (+, -, *, /).

In several important ways, it may, in fact, be better than a “real” computer (that is, the type cased in plastic, with a keyboard and all!). For instance, in doing a problem such as 53 + 39, one can see the 53, the + sign, and the 39 before pressing the equals key.

Figure 3: Set a goal as a number or calculation.

A unique feature of the software allows you to disable specific keys. Selecting numbers or operators from this screen disables them, graying them out.

You can also set a goal to pose a problem or enter a target number by pressing the “set a goal” button, then “ok.”

Finally, another feature of the Broken Calculator software is the leading digit mode, accessible by a toggle switch that alternates between leading digit and normal modes. In leading digit mode, only 0, the decimal point, and the four operators are active after entering a numeral from 1 to 9.

Familiarize yourself with the software before attempting the following activities.

Place value: Disable 2, 3, 4, 5, 6, 7, 8, 9, x, and /. Now get the calculator to display 4321 (or any other target number). How many steps did it take? The history window on the right will keep track of steps for you. Can you tell in advance how many steps it would take to display any number?

Turn on the leading digit mode, and leave all the keys enabled. Set a goal of 678.9 and see how many steps it takes to display that number. What is the minimum number of steps you can take to reach your target? How do you know?

Understanding addition and subtraction: Try addition problems with the plus (+) key disabled; for example, 53 + 39. How many ways can you do this problem? Try subtraction problems with the minus (-) key disabled.

Understanding multiplication and division: Do several multiplication problems with x disabled. Do division problems with / disabled.

Signed numbers: Keep all keys enabled. Explore problems such as (-m) + (-n) or (-m) - (-n). For example, what is the sum of –7 and –9?, that is (-7) + (-9) = ? What about (-7) – (-9)? Note that there is no way to enter a negative number directly. It must be constructed!

Estimation: Suppose only the 1, 2, 7, 9 keys and all the operation keys function. Get the calculator to display 2059.

Video cases

To see Kelly Goorevich using the Broken Calculator with her students, you may access a public version of the Number & Operations: Broken Calculator teacher professional development course on the Seeing Math website. Our commercial partner, Teachscape, distributes twelve Seeing Math™ Elementary courses developed by the Concord Consortium. Try them out in your district for professional development opportunities where kids can “break” the calculator and still learn math!

Joanna Lu (jlu@concord.org) is Managing Director of Seeing Math. Judah Schwartz (judah@gse.harvard.edu) is Emeritus Professor of Engineering Science & Education at MIT, Emeritus Professor of Education at Harvard, and Visiting Professor of Education and Research Professor of Physics & Astronomy at Tufts University.