. What is the role of conjectures in learning mathematics? How do conjectures demonstrate mathematical power? How do conjectures build understanding and encourage connections? These questions will be explored in the following sections. A

LEARNING COMMUNITY FOCUSED ON STUDENTS' THINKING, COLLABORATION, AND INQUIRY Valuing and respecting students' thinking, establishing a collaborative atmosphere, and fostering a spirit of inquiry are three important aspects of our third- and fourth-grade classroom. We must hold students' ideas in high regard as they construct meaning about mathematics. All questions, ideas, comments, connections, and conjectures by students are significant. Students learn to take risks with their thinking as they share their ideas. The ability and willingness to develop their own methods for solving problems, to disagree with another student's idea, and to put forth conjectures are all developed throughout the year.

The physical arrangement of our classroom enhances our efforts. The majority of students' time is spent in small-group or whole-group discussions. The tables are arranged to accommodate six groups of four students each. During small-group time, students are encouraged to confer with their tablemates. Many conjectures are created collaboratively in these small-group discussions. The groups are arranged around the overhead projector, which is the focal point for whole-group conversations. During those discussions, students can interact with ideas from a wider audience.

My students engage in these mathematical discussions in an inquiry-based environment. For us, inquiry-based teaching and learning mean that everyone is a creator of mathematics. Students are given opportunities to construct knowledge and to reason through problems in ways that make sense to them. The students want to understand mathematics, and when they are puzzled about an idea, they want to investigate it. They feel confident in their own abilities when solving problems, especially when they can work together.

THE ROLE OF CONJECTURES IN THE LEARNING COMMUNITY Conjectures come from students' ideas. Students create conjectures individually, with a partner, or with a small group. Often ideas that students hear or think about during discussions spark a conjecture. At the beginning of the year, students defined a conjecture as an idea that can be applied to more than one number. They also decided that conjectures must have examples. The students record their conjectures in their mathematics notebooks and write them on transparencies in preparation for whole-group discussion. I have found that the conjectures that students write become clearer over time and through revision. One of my responsibilities as a teacher is to determine the best time for students to share conjectures with the rest of the class.

The students read their conjectures, share examples, demonstrate with manipulatives, or explain the representations. The remaining students ask questions, agree or disagree, explain why, and copy the conjectures into their notebooks. The class decides whether a conjecture is valid by offering counterexamples, generalizing, or supporting the conjecture with further explanations. Once the class agrees that the conjecture is valid, we post it on our conjecture wall for the entire year, along with the student's name and date (fig. 1). The conjecture wall shows a cumulative historical account of the students' thinking.

CONJECTURING DEMONSTRATES MATHEMATICAL POWER Mathematical power includes personal self-confidence about mathematics, as well as the ability to reason through problems and communicate with others about ideas and solutions. It is important for students to communicate their ideas and reasoning clearly. It takes confidence for a student to stand before classmates and explain a conjecture.

Once the conjectures are posted, they are used for reference and revision. During discussions it is common to hear a student refer to a certain conjecture when making an argument. Used this way, conjectures help contribute to, and build on, students' ideas. Sometimes new evidence comes to light, and we revise a conjecture accordingly. Like rough drafts, conjectures are always open to change. For example, Ellen, Robin, and Sammy's conjecture stated, "In fractions, we noticed that the bigger the bottom number (denominator), the smaller the fraction is. The smaller the denominator is, the bigger the fraction is. Examples: 1/4 is smaller than 1/2, and 1/16 is smaller than 1/8." As we explored nonunit fractions, we revised the beginning of the conjecture to state, "For unit fractions only, ..." Students in my classroom are accustomed to revising both their written and their verbal ideas.

Students find it exciting and helpful to pose a conjecture to the class. Being able to teach other students and share ideas to help them learn is empowering. Students learn that it is acceptable to revise or even eliminate a conjecture without fearing any criticism from classmates. Making sense, which leads to mathematical power, is the main goal in our learning community.

CONJECTURING TO MAKE CONNECTIONS Students are empowered when they make connections within mathematical content, across content, or to their own lives (see fig. 2). Students construct knowledge by connecting new material to previous experiences and knowledge. Conjectures are one way for students to make connections. For example, Sabrina made a conjecture linking multiplication to division. She said, "When the dividend is a multiple of the divisor, then the quotient will be whole with no remainders." A connection across content occurred when Lenny posed, "In science, we measure liquids in a breaker and use fractions to be exact. If we don't use fractions, it won't be right and you will mess up." Connections to life outside the classroom were made when Scott wrote, "There are fractions everywhere. If you looked at a pack of gum, each piece would be 1/15, my shoes are size 3 1/2, or aces in a pack of cards are 4/52."

Through discussions involving conjectures, questions, and comments, students are expected to justify their ideas, explain their reasoning, and strive to make sense of the mathematics. Overall, these connections demonstrate the power of mathematics.

CONJECTURES DEVELOPING IN THE LEARNING COMMUNITY On the first day of a new six-week unit on fractions, I gave my twenty-one third and fourth graders three preassessment questions. Each student responded to these questions in their mathematics notebooks:

We reconvened as a group and spent the rest of our class time discussing the three questions. Cody shared his thinking about the second question. He represented fractions by writing 5 ÷ 10 = 1/2 and 20 ÷ 40 = 1/2.

I asked him where he got his idea, and he said that it was from Bill's conjecture during our previous unit on division. Bill's conjecture was that "If you divide a number by a higher number, you get a fraction, except zero." He gave many examples, including 12 ÷ 24 = 1/2, 12 ÷ 96 = 1/8, 12 ÷ 48 = 1/4, 3 ÷ 6 = 1/2, and 3 ÷ 12 = 1/4.

Cody showed his idea by drawing figure 3 on a transparency. As class was ending, Cody showed me a conjecture that he had written in his notebook. It read, "Any number divided by twice itself equals 1/2, except 0." He gave the examples 5 ÷ 10 = 1/2, 10 ÷ 20 = 1/2, and 20 ÷ 40 = 1/2.

The following day, I asked Cody to read his conjecture from a transparency on the overhead projector. After some initial comments and questions, I was curious to see what the class thought of Cody's idea. I asked the students to write in their notebooks why they disagreed or agreed with Cody's conjecture and to explain their reasoning.

As I walked around the room, I noticed that the class agreed with Cody's idea. Many students used examples to see if his conjecture worked. Shawn had discovered a pattern and recorded the following in his notebook: 2 ÷ 4 = 1/2 3 ÷ 6 = 1/2 4 ÷ 8 = 1/2 5 ÷ 10 = 1/2

Some students generalized Cody's conjecture to denominators other than 2 and were very excited about this connection. For example, Bob and Nick tried 3 as the denominator and wrote, "Any number divided by 3 times itself would equal 1/3, except for 0. 4 ÷ 12 = 1/3, 5 ÷ 15 = 1/3, and 12 ÷ 36 = 1/3."

I also observed several students making patterns in their notebooks as they used 2, 3, 4, 5, and 6 for the multiplicand and corresponding divisor (fig. 4). After fifteen minutes, we came together for group discussion. Most students were anxious to share their "new" conjectures, which were Cody's conjecture using numbers other than 2. I asked them how we could revise Cody's conjecture to include all their numbers. Megan suggested we use "algebra." Many students were puzzled by her suggestion. She explained that algebra uses letters for numbers. With a little help she proposed, "Any number divided by x times itself equals 1/x, except for 0." Megan elaborated by saying, "x can mean any number because we don't have the space to write every single different number into a conjecture." Abby connected this idea to the notion of infinity as she explained, "You can't make all of the numbers its own separate conjecture because numbers go on forever." Some of the students understood the idea of x and what it represents. Most understood the concept of infinity and its relationship to this class conjecture. The class decided it would be fair to post Cody's original conjecture on our conjecture wall and also put up a class conjecture using Megan's algebraic representation.

We also had some discussion about zero and why it is an exception. Kelly said, "If there was nothing there, then you couldn't make it a fraction because you didn't have anything to start with." In her mind, zero by itself meant that nothing was there. Sammy reminded the class, "With multiplication and division, when you have '0,' it means there are no sets, or nothing." Kim ended this rich discussion by posing a question to the class, "Does Cody's conjecture work with negative numbers?"

REFLECTIONS During group discussions, I jot down interesting ideas in my own notebook. I use my comments to assess students' learning, pose questions, and reflect. My notes help to capture what occurs during discussions.

I was pleased with the students' connections, generalizations, and questions in our discussions about fractions. The students made connections among the content areas of fractions, division, algebra, and integers and with important mathematical ideas like infinity and zero.

Their connections demonstrated that they had remembered and applied some of the content learned prior to our new fractions unit. Listening to their discussion reminded me how valuable it is for students to share their reasoning and thinking. Their interactions illustrated how important and powerful students' ideas can be for their classmates' understanding.

Some students generalized Cody's conjecture to other numbers. On their own, they checked to see whether the conjecture held true for numbers other than 2. I was excited to see how some students used patterns as a strategy for extending Cody's conjecture. For example, one student started with 2, then experimented with 3, 4, 5, and 6. Using patterns helped students to generalize beyond the given information. Patterns also helped students organize their thoughts and provided opportunities for counterexamples to surface.

CONCLUSION Conjectures play three important roles in the development of mathematical power. First, they empower students with a feeling of ownership because the students see their ideas as important and as contributing to the understandings of their classmates. Knowledge and understanding are developed collaboratively. Second, conjectures enable students to discover and construct "new" mathematical knowledge by connecting what they are trying to learn to previous experiences and knowledge. Third, conjectures are a vehicle for students to make connections within mathematical content, across content, and to their daily lives. In summary, conjectures help students make sense of the mathematics they are learning.

AUTHOR: Danise Cantlon

Danise Cantlon, dcantlon@scnc.holt.k12.mi.us, teaches grades 4 and 5 in a multiage classroom at Elliott Elementary Professional Development Center, Holt, MI 48842. She has been involved with various mathematics and technology projects for over seven years. The author expresses her appreciation to Michael Hynes for his helpful comments in preparing this article for publication. Photograph by Danise Cantlon; all rights reserved

SOURCE: Teaching Children Mathematics 5 no2 108-12 O '98. Reproduced with permission from Teaching Children Mathematics, copyright 1998 by the National Council of Teachers of Mathematics. All rights reserved.