Revisiting the Powers of 2
Curcio, Frances R., Zarnowski, Myra, Teaching Children Mathematics
Concept development is not an all-or-nothing affair. Instead, understanding builds slowly through repeated, similar, and refined experiences requiring observation, problem solving, and reflection. Because we recognize that building concepts takes time, we were pleased to be invited back to Public School 146Q in Howard Beach, New York, to work with fourth graders who had participated as third graders in our mathematics and poetry activities (Curcio, Zarnowski, and Vigliarolo 1995). We decided to revisit an idea we had explored with the children the previous year so that students would have the opportunity to consolidate their learning related to a nontraditional mathematics topic for elementary school mathematics - the powers of 2.
One of the mathematics concepts that had surfaced during our discussions with the children when they were in grade 3 was the explosive nature of doubling. Children were surprised to learn how quickly numbers increased when doubled. In fact, when presented with a choice of paying a fine for an overdue library book of $0.10 per day over the course of forty-two years or a fine of $0.01 doubled each day for thirty days, they unanimously chose the latter doubled amount, expecting to pay less money.
To revisit the topic of doubling, or the powers of 2, with the fourth graders, we presented several ways for the students to use language to explore the concept. Our approach promotes discourse in mathematics instruction through discussion, drama, drafting, and drawing and illustrates the teaching envisioned in the Professional Standards for Teaching Mathematics (NCTM 1991).
Promoting Discourse in Mathematics
We began by reminding the children of a tale we had shared the previous year about an overdue library book, in a poem called "Overdues" by Shel Silverstein (1981). Many of the children not only recalled the poem but enthusiastically recited it. After enjoying the poem, we constructed a table comparing fines of $0.10 per day with fines beginning with $0.01 doubled each day over ten days (see table 1). As the column "Doubling Fine from Previous Day" was constructed, several children confused "adding 2" with "multiplying by 2." Since the fines of the previous day were doubled, the children realized that "adding two" was inappropriate. This initial discussion aroused the children's interest as they listened to a related story dealing with the powers of 2.
The Rajah's Rice: A Mathematical Folktale from India (Barry 1994) tells how a young girl named Chandra cures the Rajah's sick elephants and earns the right to claim the reward of her choice. Cleverly, she chooses a reward that multiplies. Since this book deals explicitly with the powers of 2, it was particularly suitable for building on the children's past experiences with this topic. The book became a springboard for discussing, dramatizing, drafting, and drawing children's interpretations of the powers of 2. Other related "multiplying" stories are listed in the appendix.
Table created during a discussion of "Overdues"
"Overdues" Comparison of Fines
Doubling Fine Day $0.10 per Day from Previous Day
1 $0.10 $0.01 2 $0.20 $0.02 3 $0.30 $0.04 4 $0.40 $0.08 5 $0.50 $0.16 6 $0.60 $0.32 7 $0.70 $0.64 8 $0.80 $1.28 9 $0.90 $2.56 10 $1.00 $5.12
To focus the children's attention on the story, we used the Directed Reading-Thinking Activity (Haggard 1988; Stauffer 1969, 1980). …