Teacher resources and professional development across the curriculum
Teacher professional development and classroom resources across the curriculum
|Introduction | Developing Mathematical Ideas Through Communication | Analyzing and Evaluating the Thinking of Others | Additional Strategies | Your Journal|
Let's look at several classroom techniques that contribute to students' ability to communicate about mathematics.
Knowing specific vocabulary can make communication easier and more reliable. The tasks below were presented to students after the "Shapes" activities presented in Part A. The teacher wanted to give students an opportunity to extend and consolidate their understanding through the use of tasks. This also allowed the teacher to assess students' depth of understanding of shapes, their use of vocabulary, and their ability to organize and express their thoughts.
Below are some examples of student responses to different writing prompts. Consider how the prompt affects the information the student writes down and how it may affect the thinking the student engages in.
Teacher: What do you know about these shapes? Tell as much as you can.
Jose: One is a triangle. It has three sides of different sizes. One of its corners is a square corner; the other corners have small angles. I know triangles always have three straight sides and three angles, and it doesn't matter which way you turn them. They can be long and skinny or like a tent or upside down. They can have some big and some small angles or three angles the same. The other shape is a rectangle. It has two long matching sides and two short sides. You can call a shape a rectangle as long as it has four square corners and four sides, even if two of the sides are really short. A rectangle can be really long and skinny, or it can be fat like a square.
Teacher: Why do we call these two figures triangles?
Kim: They both have exactly three sides and three corners. That's what triangles have.
Teacher: Are these two shapes congruent? Explain exactly how you would find out. Give more than one method if possible.
Jabari: I think they are congruent because it looks like they both have five sides with matching lengths and in the same order. Both have three right angles, one small angle, and one big angle in the middle. I would cut them out and match them if I had scissors. I could measure the sides with a ruler, and copy and check the angles with a folded paper.
As students use a variety of methods to communicate about worthwhile mathematical problems and concepts, they gradually organize their thoughts about the task at hand and consolidate their knowledge into generalized understanding. A task can challenge a student's less-developed ideas and beliefs and provide a catalyst for embracing more solid or sophisticated concepts.
Writing encourages students to organize, summarize, communicate, and extend their thinking. It provides a visible record that can be reflected on and discussed with others. Also, many people find that the act of writing can increase their retention of concepts.
Prompts (1) and (3) utilize words that encourage students to communicate all they know about the question. It is likely that the students thought about the problem for a longer time as they searched for additional things to write, which allowed the teacher to see the breadth and depth of their understanding. Prompt (2) is more of a factual question. Kim's answer demonstrated that she has a general understanding of the properties of triangles, but it didn't clarify whether she knows that the sides must be straight.
Initially, working with their teachers to co-write responses that are organized, clear, and complete can be helpful for students. Working with others to write clearly, gives students opportunities to clarify their ideas and use their new vocabulary. Students can also write definitions of important terms and directions for procedures in their own words and discuss their work with a partner. Journal writing can provide an opportunity for students to reflect on what they understand and what they still need to learn.
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