Count Me In!
By Michael Naylor

Count buttons as a class, using numbers and symbols.

Counting, it's often said, is the basis for all mathematics, and while this argument leaves out other important parts of mathematics, there's little doubt that counting is fundamental to understanding and communicating number ideas.

When young children are first learning to count objects, we pay attention to several components of the process: a) do the children know the words of the counting sequence in the correct order? b) do they count exactly one object each time they say exactly one word? c) do they have a method for keeping track of what they've counted, such as moving the object? Or do they count some objects twice or miss some objects? Being aware of these sub-skills can help us guide children in developing counting skill.

The challenge of counting continues through all grade levels, as children learn more sophisticated methods. This month's activities will engage your students in some great counting exercises!

Button count (Grades K-2)
Have each child count the number of buttons on his or her clothing, and record the individual results on the board. After each numeral, draw the appropriate number of circles to represent the buttons. As a class, count all the circles to get a total. Now draw lines to connect numerals on the list that add up to fives and tens, then count by groups. Point out that the total is the same.

For more practice, collect data from your students on the number of kids in each child's family and provide a written copy of the data for each student. Divide the class into small groups and have students try different methods of finding totals.

Tape the strips together. How high can the kids count?

Number strips (Grades K-2)
Cut paper lengthwise into strips two centimeters wide and have students write the counting sequence, starting at one and writing as many numerals as will fit on the strip. Students can work on the strips all year and tape them together to make one long strip. Kids love to see how far they can count with the strips, and they'll be practicing their numerals.

Skip-counting patterns (Grades 3-5)
Have your students write down a starting number from zero to 10 and choose a "skip number" from two to 12. They should add the skip number to the starting number repeatedly (until they reach 100 or higher), writing down the result each time so as to generate a list of "skip counts." Have students find at least two patterns in their list, and share them as a class. Their patterns might include patterns with the first or last digit, or sums of the digits. Do some patterns include all digits as the last digit, or are some missing? Why?

Big counting (Grades 4-8)
How many bricks are in your classroom? How many tiles are on the floor in the entire school? How many screws are in all the desks in the whole school? Look around and find a few objects that are so numerous, it would be almost impossible to count them. Then ask your students to do it! They can use estimation strategies, but they must be able to defend those strategies. A good strategy might be: "We counted one row of bricks and there are about 40 in a row. The wall is 35 rows tall, so that's about 1,400 bricks in one wall. Each room has four walls…" etc. An unacceptable method might be: "Everyone in our group thought there must be about a million, so we figured that must be right."

Ask your students to come up with their own large quantities of things that seem "impossible" to count, such as the number of hairs on their heads or the number of times their hearts will beat in their lifetimes. How might they go about counting such things? Invite them to explain their strategies.

Counting the bricks in a wall can be easier than it seems.

Square counting (Grades 5-8)
Ask your students how many squares there are on a checkerboard. There are 64 small squares, but what about squares of other sizes? Divide your students into pairs or small groups and have them count all the squares of all different sizes. They'll find interesting patterns in this problem-solving activity. There's one 8 x 8 square, four 7 x 7 squares, nine 6 x 6, sixteen 5 x 5 – in other words, the number of squares of each size is a square number!

If a group finishes early, challenge them to find all the rectangles that are one unit longer than they are wide (that is, 1 x 2, 2 x 3, 3 x 4, etc.). If they finish that, give them the biggest challenge of all – count all rectangles of all sizes on a checkerboard!

As students tackle these problems, they'll be learning to organize their results and they'll also be developing algebraic reasoning skills. You can count on it!

Michael Naylor is a professor of math education at Western Washington University, Bellingham, WA.

January, 2004, Vol.34, No.4