MATHEMATICAL COMMUNICATION THROUGH STUDENT-CONSTRUCTED QUESTIONS




MATHEMATICAL COMMUNICATION THROUGH STUDENT-CONSTRUCTED QUESTIONS
Mathematical communication among students should receive increased emphasis in the classroom according to NCTM's Curriculum and Evaluation Standards for School Mathematics (1989). Two modes of student communication are evident in the mathematics classroom, the expressive mode and the receptive mode, as identified by Del Campo and Clements 1987). Discussing, creative writing, drawing, and acting out are examples of use of the expressive mode, whereas completing standard worksheets and exercises prepared by the teacher are examples of use of the receptive mode. An instance of use of the receptive mode would be a child's attempt to identify which of a given set of shaded rectangles represents a fourth, as seen in figure 1. The task in figure 1 involves simple recognition rather than creation and production of a result using the student's own words and pictures, such as when the student either draws a diagram to represent a fourth or finds a fourth of a given length of string. According to Del Campo and Clements, the receptive mode of communication has dominated mathematics classrooms, and they urge more expressive modes of communication so as to engender student ownership and interest. One way to tap the expressive mode is for children to construct their own questions (e.g., Silverman, Winograd, and Strohauer [1992]; Walter [1988]). Three types of student-constructed responses that can be used to promote discussion and creative writing in mathematics are the following:
1. What is a question to ask? (i.e., given some information, constructing a question)
2. What is a problem to solve? (i.e., constructing a word problem to be answered by peers)
3. What related questions can be asked? (i.e., asking questions related to a given problem) WHAT IS A QUESTION TO ASK? Give students an incomplete word problem with all the information needed to solve some problem but without an actual question. The students' job is to complete a problem by preparing a question based on the information. EXAMPLE 1 A collector wants to buy a doll costing $20 but only has $12. Possible questions:
* Does the collector have enough money to buy the doll?
* How much more money is needed to be able to buy the doll? EXAMPLE 2 Three children, Ann, Bert, and Amy, share a chocolate bar. Ann takes a fourth of the chocolate bar and Amy takes half. Possible questions:
* Who ate the most?
* What fraction of the chocolate bar was left for Bert?
* How much did Ann and Amy eat altogether? EXAMPLE 3 The whole length of a 36-cm piece of string is used to form different-sized rectangles. Possible questions:
* What are the dimensions of the rectangles?
* How many rectangles can be made?
* What are some values for the areas of the rectangles? Comments: Creating such questions may allow students to feel a sense of ownership and help them to communicate their mathematical understandings to teachers and peers. Children's questions will vary according to their previous experiences and understanding of the given information. WHAT IS A PROBLEM TO SOLVE? The teacher selects a topic and asks the class to work in groups to prepare a word problem, together with a method of solution, on that topic. Later the problems may be solved by members of other groups. The word problem is then handed in to the teacher, who edits it for readability, compiles the students' word problems, and distributes them to the class to be used as exercises. The names of those who prepared the word problems may be shown on the "exercise sheet," but the solutions are not. EXAMPLE OF STUDENTS' PROBLEMS
1. There were 600 balloons at the circus. 1/4 popped. 1/3 were red. 1/4 were green. 1/4 were blue. How many blue balloons were at the circus? (Marissa)
2. Stepheny wants to watch her favorite show at 5:30 p.m. It is a 1/2 hour show. In the morning she gets up at 10 a.m. She is at the ice arena an hour later. She gets off the ice at 2/3 of regular skating time which is 3 hours. She eats lunch for 1/3 of an hour. Then she goes shopping for twice as long as she went skating. It takes her 1/2 an hour to walk home. How much of her show will she get to watch? (Tanya)
3. The Winnipeg Jets were on a 4-game road trip. Through all games, the Jets scored 24 goals. One-third were against the Vancouver Canucks, 1/2 were on the Sharks, 1/8 were on the Nordiques. How many did they score against the Canadiens? (Tommy) Comments: If students prepare the problems to be turned in on Friday, then the teacher has time to edit and compile the word problems to use as class exercises on Monday. If five word problems are prepared, students may take two to three periods to solve and present solutions for the four word problems prepared by other groups. Even if the word problems were ambiguous or had insufficient or superfluous information, the discussions generated by such questions, both in their preparation and during the intergroup attempts at solutions, would be valuable communicative and learning experiences. The word problems might also reflect a range and variety of individual interests. Encouraging students to present their solutions on overhead transparencies enhances interest and discussion. As an extension, students could be asked to prepare word problems individually. WHAT RELATED QUESTION CAN BE ASKED? Students are given a problem and are required to list a number of questions related to the problem. A number of questions are then selected from this list for further investigation. EXAMPLE 1 The average of four numbers is 10. What are some possible values for the numbers? Possible related questions
* What is an average?
* How do we compute an average?
*Are averages used in real life?
* Could the average of two numbers be 10?
* Could the average of more than four numbers be 10?
* Do the numbers have to be whole numbers? EXAMPLE 2 After a 20 percent discount, the price of a shirt was $16. What was the original price of the shirt? Possible related questions
* What is the meaning of 20 percent?
* What is the meaning of discount?
* What is the meaning of 20 percent discount?
* Is the original price more or less than $16?
* What is 20 percent of $16?
* As customers, would we be interested in the original price of the shirt? EXAMPLE 3 The numerals 1, 2, 3, 4, 5, 6, 7, 8, and 9 are written on cards, with one numeral on each card. List all the possible combinations of any three numeral cards that give a sum of 15. Possible related questions
* What is the meaning of numeral?
* What is the meaning of sum?
* What is the meaning of all the possible combinations?
* Would 5, 5, 5 be a possible solution?
* Is the combination 4, 5, 6 the same as 5, 6, 4?
* Why should the sum be 15? Comments: The related questions need not be independent of each other. Indeed, some of the questions may be equivalent but worded differently. The questions need not be in any particular order or be couched in "how to" terms: "what" and "why" questions are also encouraged. Because students are expected to ask questions, problem posing is a less threatening activity than problem solving. In fact, some "poor" problem solvers might be "good" problem posers. Listing the related questions on the chalkboard would benefit students by evoking more questions. Furthermore, such a listing of related questions would help students in "understanding the problem," a crucial aspect of Pólya's (1957) problem-solving heuristics. Once the questions have been listed, groups of students could investigate questions of their choice and see whether such investigations could lead to solving the original problem. All three activities--listing, selecting, and investigating related questions--are enhanced by discussion and, hence, by mathematical communication. CONCLUSION The three types of student-constructed responses discussed here indicate how to communicate mathematics in a mathematics classroom by using the expressive mode of communication. In "What is a question to ask?" students analyze and synthesize the information given and decide on an appropriate question to fit the information. In "What is a problem to solve?" students use their own interests, experience, and mathematical knowledge to construct as well as to solve the problem. In "What related questions can be asked?" students ask questions to clarify their understanding of the problem, investigate related problems, and possibly find solutions to the original problem. Hence all three types of responses enhance communication in the mathematics classroom through students' active involvement in, and empowerment by, mathematics learning. REFERENCES
Del Campo, Gina, and Ken A. Clements. A Manual for the Professional Development of Teachers of Beginning Mathematicians. Victoria, Melbourne: Catholic Education Office of Victoria and Association of Independent Schools of Victoria, 1987. National Council of Teachers of Mathematics. Curriculum and Evaluation Standards for School Mathematics. Reston, Va.: The Council, 1989. P lya, George. How to Solve It. New York: Anchor-Doubleday, 1957. Silverman, Fredrick L., Ken Winograd, and Donna Strohauer. "Student-Generated Story Problems." Arithmetic Teacher 39 (April 1992):6-12. Walter, Marion. "Some Roles of Problem Posing in the Learning of Mathematics." In Mathematics, Teachers and Children, edited by David Pimm, 190-200. Sydney, Australia: Hodder & Stoughton, 1988.
WBN: 9612200445003 AUTHOR: Ramakrishnan Menon Ramakrishnan Menon teaches at the National Institute of Education, Singapore 1025. He is interested in language and mathematics and collaborates with teachers to promote numeracy and meaningful learning of mathematics among elementary school children. SOURCE: Teaching Children Mathematics v2 p530-2 My '96 Reproduced with permission from Teaching Children Mathematics, copyright 1996 by the National Council of Teachers of Mathematics. All rights reserved.