Question: At district workshops many teachers are asking, What is the purpose of the activity, Counting Around the Class? What mathematics do children learn when they skip count? What is the role of the teacher when doing these activities? Where do teachers find support in the Investigations program that will help them understand how to make connections for children between the activity, and the mathematical ideas children should be learning? Where else in the program do children have to skip count?
Anne Marie Marshall
District Leader for Elementary Mathematics
Milwaukee, Wisconsin
Answer:
Counting Around the Class: The Activity
Students count around the class by a particular number (depending on the grade level this may be by 2, 4, 5, 10, 15, 25, 150 etc.). That is, if counting by 2, the first person says "2," the next person say "4," the next "6," and so forth. Before they count students discuss the relationships among the factors and their multiples.
Counting Around the Class is designed to give students practice with counting by many different numbers and to foster numerical reasoning about the relationships among factors and their multiples. Students focus on:
- becoming familiar with multiplication patterns
- relating factors to their multiples
- developing number sense about multiplication and division relationships
Counting Around the Class: The Mathematics
There are many opportunities in Counting Around the Room to focus on different aspects of mathematics that are embedded in the activity. For example in Kindergarten and Grade 1, students regularly count around the circle. This is a way to count and double check the number of students in a group. In this activity students are working on the sequence of numbers, understanding that the number you say represents the place you hold in the sequence (many students will say "I'm not 11, I'm 5!"), and knowing what the total should be (the total number of students in the class). There are many opportunities in this activity to discuss questions such as: Is everyone here today? How many students are not here? Is this number lower or higher than our count yesterday? What do you think will happen tomorrow?
In Grades 2 and 3 students begin to count around by the multiples they are beginning to explore at other times in math class. In some cases the purpose is to practice finding the next multiple of a number and gain familiarity with the patterns in the multiplication tables. At first students may be adding on by 1's to get to the next number but over time may be utilizing a pattern or remember the sequence because of repeatedly constructing and hearing it. Also, students begin to consider the relationships between numbers. That is, they count by 5's and then 10's and consider how the final number is related (that is, if you are counting by 5 with a group of 23 students, the total will be half as much as when you count around by 10's). Similarly, counting by 2 and then 4.
In grades 3-5 students begin to count by larger numbers, still learning the multiples, looking for patterns and, most importantly looking for numerical relationships. If a group of third graders is counting by 25 it is important to ask questions such as, "Four people have counted so far and we are at 100. What number will we be at when 8 people have counted? After 12 people have counted?" Counting by landmark numbers is a way to help students become comfortable with these important numbers in our number system. This is a way to support students in using landmark numbers to solve large multiplication problems (e.g. if a student is solving 29 x 4 they may easily solve 4 x 25 and add 4 x 4.) In fifth grade students count back to 0 -- if we start at 1500 and we count by 75's will we get to 0 by the time everyone has counted?
This activity clearly highlights the skills and understanding necessary to multiply and work with equal groups. But also, students can identify a clear relationship between multiplication and division. For example, if you are counting by 6 and have reached 72 you may ask students to figure out how many students have counted so far. Finding the answer to this question can be done by solving 6 x ? = 72 or 72 ÷ 6 = ?. This relationship can be pointed out to students as students share their strategies that represent the different operations.
Counting Around the Class: Questions to Ask
In most classrooms the total number of students in the class is established early on in the year. This number becomes a "landmark" in its own way (it is interesting to hear a class say something like: "I know that 23 x 4 = 92 because we counted by 4's yesterday so I know that 23 x 8 = 184 which is double 92"). Many questions can be asked in relation to this "classroom landmark number" such as: What would we count to if our class was doubled or tripled? What if there were 10 more students in our class? What about half our class? Yesterday we counted by 12 and got to 276 and today 2 people are missing so what will we get to if we again count by 12?
It is important for students to consider what they predict the total will be when they start counting. "Today we will be counting around the room by 15. We know there are 23 students in our class so what number do you think we will end on?" This question offers the opportunity for students to make a reasonable estimate. If you have several estimates on the board ranging from 250 to 1000 it is important to ask "Which of these are reasonable and why?" Students should explain their estimate in a way that shows they have an understanding of the number being counted by, how many is being counted, and how large the answer should be. For example a student might say, "I know that 15 x 10 = 150. I can double that to find out 15 x 20 = 300. Three more kids need to count and I know that 2 x 15 = 30 so the answer will be 15 more than 330." Or, "I know that when we count around the room by 10 we get to 230. The 5 from the 15 is half of 10 so I can add 115 (half of 230) + 230 to get 345." Both these answer demonstrate a developing sense of important relationships among numbers.
It is also useful to stop the group in the middle of counting and ask: "We are counting by 6's and we are at 72. How many people have counted so far?" This allows for students to solve a smaller multiplication or division problem than the original one posed to the class. (If we count by 6, what number will be end up with after everyone has counted?) Some students may actually count the number of students who counted; others may remember a number at a certain earlier point and add on (I remember that when the 8th person counted they said 48 and I can count from there); and some may use numerical reasoning (I know 72 ÷ 6 = 12 so 12 people have counted, or I know that 6 x 10 = 60 and 2 more 6's get me to 72 so 12 people have counted).
Another aspect of this activity is exploring the patterns of multiples once they have been established. Sometimes you may want to record the multiples as students say them, both to emphasize the patterns and to support students who are having difficulty keeping track of where they are. Once the counting is complete you can ask, "What do you notice about the pattern of 15? Why do you think the ones place in the multiples alternates between 5 and 0? What do you notice about the 10's place? How many multiples are in each group of 100?" These questions will push students to do more than simply notice patterns but to discuss the mathematics that is behind the pattern. For example: they may be pushed to notice that the structure of 15 as 10 + 5 results in the alternating 5 and 0 in the 10's place. With each new multiple you are adding a 10 plus half of a 10.
The Structure of Counting Around the Class
This activity can be done while students are sitting in a circle -- the count begins in one place and moves around the circle until everyone has counted. Since you will doing this activity over time it is likely that students will change places frequently so they will not always be responsible for a low or high number. It may be, though, that you have a weaker student who is working on lower numbers or may be intimidated by the "public" aspect of this activity. You may want to seat that person at the early end of the counting to decrease any anxiety. Or you may want to allow that student time to work on the multiple before the activity so s/he will already be familiar with the patterns.
Another way to structure the activity is to allow students to raise their hands to say the next multiple. In this case it needs to be clear that there is no calling out until you are called on and you can only say a multiple once. You can leave as much wait time as you want in between numbers to allow all students to figure out what is next. And you can also call on students based on what you want their challenge to be.
In either case it is important (as in every area of mathematics) that students be clear that we are all practicing and learning together and all mistakes and miscalculations are an expected and necessary part of learning mathematics.
Where to Find Counting Around the Class
Attendance is a routine in Kindergarten and Grade 1, where it is called Counting, and can be found written up at the end of any Kindergarten or Grade 1 unit.
In the second grade unit, Coins, Coupons, and Combinations, students are introduced to Counting Around the Class when they are working on counting by 5s. In this version of the activity, pairs of students add a train of 5 cubes to a container as the whole class keeps track of the count.
Counting Around the Class first appears as a Ten-Minute Math activity in the third grade unit, Things That Come in Groups. Thereafter it appears as a Ten-Minute Math in Landmarks in the Hundreds (Grade 3), Arrays and Shares (Grade 4), Landmarks in the Thousands (Grade 4), Containers and Cubes (Grade 5).
An activity related to Counting Around the Class, called Skip-Counting on the 100 Chart, appears in two third grade units, Things That Come in Groups and Landmarks in the Hundreds, and in a fourth grade unit, Landmarks in the Thousands.
Lucy Wittenberg and Katie Bloomfield, TERC
Summer 2001
This information was reprinted with permission of CESAME, Northeastern Univ., and the Educational Alliance, Brown University.
