Nurturing the Voices of Young Mathematicians with Dyads and Group Discussions.
To the casual observer, Sarah would most likely not have appeared to be a mathematician. She quietly went about her work, never bothering anyone and always in the background. Through the use of a structure called a dyad (Weissglass 1997), however, Sarah's mathematical abilities became apparent and appreciated by all. In a dyad, students are paired, and each receives an equal amount of time to talk while the other listens without interruption or judgment. Dyads are not conversations; rather, they offer an opportunity for students to try out new ideas. Dyads are considered confidential. For these reasons, I do not listen in. Students are encouraged to share their own thoughts later in a group discussion or writing assignment, but not their partner's thoughts. On this day, a dyad session followed by a group discussion helped Sarah find her voice and express her thoughts. She provided us all with a wonderful learning opportunity that was not originally in my lesson plan--an opportunity for a young mathematician to shine.
Using Dyads in a Lesson
Sarah, a fourth grader in my grade 3--4 class, had been involved with using dyads as part of the general classroom culture for about a year. Our class was engaged in the study of patterns. I had shared a book, Dinner at the Panda Palace (Calmenson 1991), with the students to establish a context for an exploration of patterns. (For a more complete explanation of the lesson, see Wickett [1997b].) The story recounts the arrival at a restaurant of one elephant, then two lions, three pigs, and so on, with the number of animals increasing up to ten chicks. "How many animals have come to dinner?" I asked the class. The quiet of the room was soon replaced by the hum of students engaged in dyads, sharing their ideas with partners who listened quietly without interruption. After one minute, I asked the partners to switch roles so that the listeners now became the speakers and the first speakers now became the listeners. I chose one minute because it is a reasonable amount of time for students of this age to share their thinking when they are not interrupted. It is also a reasonable amount of time to expect another student to listen attentively. After the pairs of children had their turns sharing and listening to each other, I asked for their attention once again. "How many animals have come to dinner?" served both as the focus for the dyad and a starting point for the subsequent class discussion.
Several children shared their ideas with the class about how many animals had come to the restaurant for dinner. One child stated that the problem could be solved by adding 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10, which would equal 55 animals. Another youngster pointed out that the process could be reversed, beginning with 10, then adding 9, 8, and so on back to 1. This student said that it was easier for her to add the bigger numbers first rather than last, as in the first explanation. The students knew that problems could be done in different ways, and it was important to share these ways with classmates.
Sarah's Thinking after the First Dyad
Sarah quietly raised her hand and shared a most unusual and sophisticated idea with the class:
I was wondering something: What if I just added 5 ten times? It seems like it should work, but it comes Out to 50 instead of 55, and I am wondering why. I added 5 because it is a friendly number for me to add and it's in the middle of 1 and 10. I thought about the middle of 1 to 10 because 1 was the least amount of animals and 10 was the most, 50 5 was in the middle. Instead of adding the numbers in order, 1 + 2 + 3 and so on, I thought I could pair them, like adding 10 and 1 to make 11, and 5 would be the middle. It seems like it should work, It seems like adding the number in the middle ten times, because there were ten groups of animals, should work.
The class sat motionless, trying to follow Sarah's thinking. Quickly and without prompting, the children again began working in dyads to try to make sense of what Sarah was thinking. Again, after one minute, I reminded the students to switch roles so that the listener became the speaker and the speaker became the listener. A short period followed the dyads, during which students worked together in pairs to explore Sarah's idea and their own thinking.
Later, Sam, a special education student, shared his thoughts with the class (all students' names other than Sarah's have been changed): "At first, I was confused about adding 1 and 10 to make 11. But then I thought real, real hard! She could add 1 and 10 to make 11, and then 2 and 9 to make 11, and 3 and 8 to make 11, and so on, which would come out to five pairs of numbers that make 11. Then because there are five pairs, she could add 11 five times to get 55. It's a pattern! It's just adding the numbers in pairs in a different order. Then I think Sarah thought 5 was in the middle." According to legend (Hall 1970), the famous mathematician Carl Friedrich Gauss used a similar method at the age of ten when his class was asked to find the sum of all whole numbers from 1 to 100.
"But 5 and 5 makes 10, not 11. So 5 isn't half of 11. That's sort of where I got stuck," interjected Maria.
Andy picked up on what Maria said and offered his ideas: "Well, I thought the same thing as Maria, and then I tried 6, but two 6s make 12, which is too big. So Joaquin got out a piece of paper and drew two circles. He put an X in one circle and then the next X, he put in the second circle, until we did that with ten Xs. The circles both had five Xs and we still had one more X, so we put a half of an X in each circle and that made 5 1/2 Xs in a circle, or 11 altogether for two circles" (see fig. 1).
"And if you use repeated addition, ten groups of 5 1/2 make 55!" exclaimed Holly. "Isn't that amazing, Sarah? Your idea does work; you just needed to find the right middle number!"
This classroom dialogue shows how using dyads followed by group discussion develops and brings out the mathematical promise of all students. Through the use of dyads, children were motivated to think about a complex idea, their dialogue helped them believe in themselves and believe that, mathematics should make sense, and they had the opportunity to explore a rich and interesting idea generated by one of their classmates. The students responded thoughtfully and even congratulated Sarah on her idea. Sarah's confidence and ability to articulate her unusual idea were outcomes of her experience using dyads in class. Sarah's written reflection about her feelings at the beginning of third grade highlights this change in her level of confidence: "At the beginning of third grade I felt scared because there were new kids and I was very shy. I was afraid to raise my hand or talk and I was afraid the teacher would call on me and I wouldn't have the answer."
My Experience with Dyads
At the same time that Sarah began third grade and felt these emotions, I began my participation in the Equity in Mathematics Education Project, funded by the California Mathematics Project. Giving all children access to all aspects of the learning process was a constant focus of the program. During my involvement with the project, dyads proved to be an effective means of access for me. Dyads offer an opportunity for participants to be heard without judgment or interruption. The structure allowed me to clarify my thoughts about project issues with one other person before sharing them with the group. This process helped build my confidence, thereby giving me greater access to group discussions. I decided that this structure might help children find the mathematician within, and I began regularly using dyads in my classroom. Sarah's written reaction to the idea reinforced my own experience:
Sharing my thinking with a partner helped. Lots of times in the beginning I wouldn't raise my hand because I was afraid of getting the answer wrong. By checking my answer with a partner or a small group first, I could try my idea first and think and we would all have the same answer and we wouldn't be so shy. When I worked with a partner, I would not be so shy with them because everyone else couldn't hear. Soon I wasn't getting the wrong answer anymore. And besides I wasn't getting a lot of answers wrong and I got a lot more right. And sometimes when I had to listen and think about what my partner said, my partner had stuff to say that helped me understand.
All children have the potential to learn mathematics and display the characteristics of mathematicians. Holly, Joaquin, and Andy were high achievers in mathematics and have benefited and grown through the use of dyads. Also showing evidence of "mathematical promise" are Sarah, Maria, and Sam, students who might have been over-looked as mathematicians. Dyads provided the structure and opportunity for all to develop their potential. As educators, it is our professional responsibility to encourage the mathematician in all children, offering them a challenging, stimulating curriculum that enables them to succeed at the highest levels possible. "Mathematics is a right, not a privilege." This challenge is one to which we must rise with passion, persistence, and caring for the good of all our children. It is their right and our responsibility.
Maryann Wickett, teaches third and fourth grades at Carrollo Elementary School in San Marcos, California. Her interests include learning and writing about issues of equity and access to ensure the best education for all students.
Calmenson, Stephanie. Dinner at the Panda Palace. New York: HarperCollins Publishers, 1991
Hall, Tord. Carl Friedrich Gauss: A Biography, Cambridge, Mass.: MIT Press, 1970.
National Council of Teachers of Mathematics (NCTM). Curriculum and Evaluation Standards for School Mathematics. Reston, Va.: NCTM, 1989.
Weissglass, Julian. Ripples of Hope: Building Relationships for Educational Change. Santa Barbara, Calif.: Center for Educational Change in Mathematics and Science, 1997.
Wickett, Maryann. Building Supportive Classroom Communities: Realizing All Students' Mathematical Promise--Sarah's Story. Santa Barbara, Calif.: Center for Educational Change in Mathematics and Science, 1997a.
-----. "Serving Up Number Sense and Problem Solving: Dinner at the Panda Palace." Teaching Children Mathematics 3 (May 1997b): 476-80.
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|Author:||Wickett, Maryann S.|
|Publication:||Teaching Children Mathematics|
|Date:||Feb 1, 2000|
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