Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

## Reasoning

The goals of the NCTM's reasoning process standard are that "in grades K-4, the study of mathematics should emphasize reasoning so that students can-

• draw logical conclusions about mathematics;
• use models, known facts, properties, and relationships to explain their thinking;
• justify their answers and solution processes;
• use patterns and relationships to analyze mathematical situations;
• believe that mathematics makes sense."
(NCTM, Curriculum and Evaluation Standards for School Mathematics, p. 29) Video Overview
This video profiles classroom excerpts that illustrate the central role of reasoning in mathematics. As students explain and justify their thinking and solution processes throughout the excerpts, teachers emphasize that how a problem is solved is as important as its answer. For example, students are observed in the following contexts:
• reaching consensus on estimates of the number of cranberries in a jar
• determining how to measure from a second-floor classroom window to the playground below
• recording estimates on a number line
• devising a process for estimating the number of buffalo that could fit on a playground
• finding patterns and relationships in different operations with the same number
• explaining how they computed equations in a window puzzle
• using geoboards to discover relationships among fractions
• identifying mathematics in a story
• creating methods to find the number of times that valentines are exchanged in a classroom
• discovering the relationships between different geometric shapes
• sorting pattern blocks by color
• recognizing patterns in probability
• determining how many marshmallows to purchase for a field trip
• analyzing sampling representations of elements in a woodpecker's habitat
• discovering the benefits of different methods of whole number computation
• discussing their analysis of graphs in newspapers
An Exploration
For Teacher workshops

Who Am I? What Am I?
This investigation focuses on reasoning in mathematics. Each pair will need these items:

• paper
• pencil or pen

1. Pose the following riddles to the teachers. As each clue is revealed, teachers should record possible solutions. After all the clues for each riddle have been revealed and teachers think they have a solution or several solutions, go back over the clues to check the accuracy of their reasoning.

Who Am I?
I am between 50 and 100. The sum of my digits is 9. I am a multiple of 15.

What Am I?
All my angles are equal. I am a three-dimensional object. I have six faces. People like to use me to make dice.

2. Have the teachers work in pairs to create at least two riddles. The riddles may involve numbers, geometric shapes, or other mathematical ideas.

3. Have the teachers take turns posing their riddles to the whole group. The teachers should not verbalize the possible solutions as each clue is revealed but instead write them on paper. After all the clues have been disclosed, have the teachers discuss their proposed solutions, strategies used, reasoning, and justifications.

4. As a whole group, examine the use of logical reasoning to pose and solve the riddles. Then discuss ways to use riddles with elementary school students to help develop mathematical reasoning.
Topics for Discussion
The following areas provide a focus for discussion after you view the video. You may want to customize these areas or focus on your own discussion ideas.

Emphasizing Mathematical Reasoning

1. These classroom episodes showed students making sense of mathematics. Describe the conditions that allowed this emphasis on mathematical reasoning. Identify some of the mathematical concepts and ideas that students were making sense of in the lessons.
2. Students were often asked to justify or explain their solutions and ideas. Compare the emphasis placed on reasoning processes in the lessons with an emphasis on finding correct solutions.
3. Cite examples of the students' enthusiasm and engagement in the various lessons. How does an emphasis on reasoning promote interest and intrinsic motivation in learning mathematics?
4. How does a focus on reasoning promote confidence in students as learners? Specify examples from the classroom episodes in which students displayed confidence.

Facilitating Student Reasoning

1. Select your favorite task from all those profiled in the video. Why did you select this particular task? How did this task promote students' reasoning? Brainstorm a list of task characteristics that place an emphasis on mathematical reasoning.
2. Investigate the teacher's role in a classroom. State examples of the interactions of the teachers and their students in the various lesson episodes. Identify the use of probing questions and wait time to facilitate reasoning.
3. In some of the lessons, students were allowed to use materials of their choice. How does this practice facilitate students' reasoning? Cite examples from the lesson episodes.
4. Describe ways to allow students the freedom to reason through problems in their own ways while still providing some structure for the lesson.
5. Students often worked in small groups or pairs in the lesson episodes. How does collaborative group work or pair work facilitate students' reasoning?
6. Describe effective ways for teachers to assess students' mathematical reasoning and ways to document their progress throughout the year. What are some ways students can assess their own reasoning ability and document their growth and progress?
Extension
Examining Mathematics as Reasoning

Each classroom excerpt profiled in this video is from a featured lesson in TEACHING MATH: A Video Library, K-4. You may want to watch the full version of these lessons to further examine and explore the role of reasoning in learning mathematics. The following list provides information about each full video and the page number of the accompanying print unit. Videos are listed in the order in which their excerpts appear in the Reasoning video.