IMPLEMENTING THE "PROFESSIONAL STANDARDS FOR TEACHING MATHEMATICS": Questioning in the Mathematics Classroom

IMPLEMENTING THE "PROFESSIONAL STANDARDS FOR TEACHING MATHEMATICS":Questioning in the Mathematics ClassroomAuthor(s): Nancy Nesbitt VaccSource: The Arithmetic Teacher, Vol. 41, No. 2 (OCTOBER 1993), pp. 88-91Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41195922 .

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IMPLEMENTING THE PROFESSIONAL STANDARDS FOR TEACHING MATHEMATICS

Questioning in the Mathematics Classroom many of us, implementing the Pro-

fessional Standards for Teaching Math- ematics (NCTM 1991) in our classrooms makes great sense. It is clearly reasonable that if students are to develop an understanding of and an ability to use mathematical applications in a variety of contexts (NCTM 1989), they should have meaningful and relevant experiences that will actively en- gage them in constructing their own knowledge. Also, that active engagement needs to be accompanied by opportunities for students to talk about what they already know and don't know and what they are doing as they strive to extend or change their current level of understanding. For many teachers, however, offering this type of instruction means changing their beliefs about mathematics instruction. After all, most of us are products of elementary and secondary school classrooms in which the teachers told us what we needed to know or do and we listened to and did what they told us to do. What we were thinking about during this interaction often did not matter, and we were unaware that it should. This same type of discourse existed in many of our methods courses. The instructor spent most of the class telling us what we needed to know so that we could tell our future students what they needed to know. Fortunately, we have come to the realization that this style of teaching is not as effective as once thought, and consequently we need to change what we are doing. However, how we go about

Edited and prepared by Nancy Nesbitt Vacc University of North Carolina at Greensboro Greensboro, NC 27412

The Editorial Panel welcomes readers ' responses to this article or to any aspect of the Professional Stan- dards for Teaching Mathematics for consideration for publication as an article or as a letter in Readers '

Dialogue.

making needed changes in our teaching is unclear.

Cooney (1992) states that the Profes- sional Teaching Standards "califs] for a manner of teaching in which ideas are not packages to be accumulated but terrain to be explored" (p. 64). The same applies to each of us in our role as the leader of a group of explorers. As part of the process of extend-

ing our understanding of what and how mathematics should be taught in the elementary school, we need to explore our classroom terrain carefully to identify what works best as we strive to implement the professional teaching standards. A significant com- ponent of this exploration is how we lead, that is, what we say and do during a lesson. As Green, Weade, and Graham ( 1 988) have found, no matter how scriptlike a lesson may be, it will vary from classroom to classroom because of the teacher's influence on the structure of the lesson, communicative or instructional demands, and interaction with students. Affecting each of these areas is the role of discourse in general and of questioning in particular.

Watson and Young (1986), on the basis of a review of the literature, stated that "[t]eachers commonly ask as many as 50 000 questions a year and their students as few as 10 questions each." In addition, 80 percent of teacher's questions involve memory processes only, with many of the questions to younger students being directed toward correctly labeling phenomena. Civikly (1992) indicated that teacher talk is an expected and extensive part of a student' s environment and that clarity of questions, responses, feedback, directions, and class discussions may be even more critical to students' learning than clarity in structured lectures.

Viewing this need from a perspective of the micropolitics of classroom interaction, Bloome and Willett ( 1 99 1 ) have found that classroom lessons are influenced by the kinds of conversations teachers want to have and believe are appropriate in the classroom. In their study, the elementary school teacher had ultimately set the conversational agenda and had evaluated students within it. When students did not respond appropriately during the teacher's conversational agenda of classroom discussion, she changed the structure to that of lecture, where she had more control of the discourse. Teachers who are uncomfortable with giving students author- ity for interpretations may be unlikely to use open-ended discussions and activities. Yet, if we as teachers are to find out what our students already know so that we can help them use that understanding to construct new knowledge, we need to focus on questions that will assist us in achieving that goal.

Question Types Barnes (1990) identified three categories of teacher questioning that relate directly to classroom instruction: factual, reasoning,

Teachers ask as

many as 50 000

questions a year and students ask as few as 10 each.

88 ARITHMETIC TEACHER



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Directions: Make a list of questions that you might ask your students using this set of figures.

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From the Arithmetic Teacher, October ll)93

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and open. A brief discussion of each of these categories forms a framework for determin- ing the types of questions asked during mathematics lessons so that needed changes, if any, can be identified.

Factual questions As cited earlier, a preponderance of teachers' questions seek factual information, that is, a name or specific information. My expe- rience at various times during the past year with a total of approximately 1 75 mathematics educators (i.e., preservice and in-service elementary school teachers and teacher educators) illustrates this situation well. Each group was given the task presented in figure 1 . With the exception of one teacher who taught in a combined second- and third- grade classroom, all the questions generated by these mathematics educators sought factual information. Students responding to their questions would be required to identify a specific phenomenon but not show insight into its use. Some of their suggested questions are as follows: ♦ What is the name of figure O? ♦ What do we call figures B, D, and S? ♦ How many figures have an acute angle? ♦ What do we call angle X in figure F? ♦ Which figure is congruent with figure G? ♦ Which figure has five sides?

Teachers asking questions in this category will find out whether their students know specific mathematical facts, but they will gain little, if any, information about whether their students actually understand the given concept. Likewise, students responding to these questions will be chal- lenged only to recall previously learned data. They will not be encouraged to make comparisons between or among acquired facts and new observations, nor will they have much opportunity to question previously learned facts. Also, students responding to factual questions are usually unable to agree or disagree with another student or the teacher as a means of confirming or questioning what they already know; they have no opportunity to reconstruct what they have previously learned. To give students this opportunity, non-fact-seeking questions need to be a major part of classroom discourse. These include a variety of reasoning questions, as well as open questions that do not call for reasoning.

Reasoning questions Reasoning questions require students to

90

construct, or reconstruct from memory, logically organized information. They are divided into four main types: closed reasoning - recalled sequences, closed reasoning - not recalled, open reasoning, and observation (Barnes 1971). A brief overview and representative questions for each, based on figure 1 , are presented subsequently.

Closed reasoning - recalled sequences. Questions in this category require respon- dents to develop one acceptable, logically organized response based on previously acquired knowledge. Sample questions of this type include the following: • Using figure G as a unit of measure, how

much carpet would you need to cover the floor in a room the size of figure U?

• How many of these figures can be divided into three equal parts?

Closed reasoning - not recalled. Teach- ers asking questions in this category will expect one correct answer only, but it is not based on previously learned knowledge. The following, using figure 1, illustrate this type of questioning: • If you put a fence around a field shaped

like figure J, and the distance between each fence post is the same size as the shortest side of figure L, how many fence posts will you need to buy?

• Which five figures, when put together so only one side of one figure is touching one side of another figure, will form a new figure that is the same shape and size as figure U?

Open reasoning. Questions in this category have more than one acceptable answer. Because answers will vary, these ques-

Learning is affected by opportunities

to relate incoming information to what is

already known.

tions afford valuable opportunities for students to listen to, and question and learn from, each other. The following are representative of this type of reasoning questions: • In what ways are figure К and figure S

similar? • Why aren't figures C, J, and К called

triangles? • Which figures can be divided into two

equal parts? • How many different ways can you make

figure U using other figures on this page?

Observation. Reasoning questions in this category require students to interpret what they perceive concerning given data. Ex- amples of questions that could be used, based on figure 1 , include the following. • Which is larger, figure Ρ or Q, and how do

you know? • Of figures L, T, and U, which two are

most alike? • In what ways are figures С and J

different?

Open questions not calling for reasoning The second-third grade teacher cited earlier indicated that she would ask the students the following questions about figure 1 :

• What do you notice about these shapes?

Although this question could be consid- ered factual because it elicits previously learned knowledge, it is an open question, since a wide range of acceptable answers exists. Most important, a question of this nature presents an opportunity for students to describe observed phenomena for which they have not yet learned a name. Additional questions in this category that could be asked, using figure 1, include these:

• What can you tell me about figures B, D, andS?

• What is the difference between figures Ε and M?

• Which figure is a polygon? • How is figure Ε like figure O?

Teachers who ask questions in this category gain specific information about their students' cognitions that can be used in introducing new concepts and in planning instruction that specifically meets students' individual needs.

ARITHMETIC TEACHER



Summary Learning is affected by the opportunities students have to relate incoming information to what they already know and then restructure their existing knowledge or construct new ideas when appropriate. As the Professional Teaching Standards indicates, classroom discourse, or "the ways of repre- senting, thinking, talking, agreeing, and dis- agreeing" (NCTM 1991, 34), is central to helping students develop mathematical understanding and skills. This development, however, cannot be achieved without teachers' asking a variety of questions that chal- lenge students' thinking - questions that require much more than factual recall. There- fore, a place for us to begin as we strive to implement the standards is with a self- evaluation of the types of questions we ask our students. Comparing the types of questions we ask with those presented in the foregoing is a way to initiate this analysis. The results should form a basis for modify- ing or changing our day-to-day practices so that our students talk about what they know and think; we listen to what they are telling us; and we plan instruction based on what we are hearing from, and learning about, our students.

References

Barnes, Douglas. "Language in the Secondary Class- room: A Study of Language Interaction in Twelve Lessons in the First Term of Secondary Education." In Language, the Learner and the School, edited by Douglas Barnes, James Britton, and Harold Rosen, 1 1-77. Baltimore, Md.: Penguin Books, 1990.

Bloome, David, and Jerri Willett. "Toward the Micropolitics of Classroom Interaction." In The Politics of Life in Schools. Power, Conflict, and Cooperation, edited by John Blase, 207-36. Newbury Park, Calif.: Sage Publications, 1991.

Civikly, Jean M. "Clarity: Teachers and Students Making Sense of Instruction." Communication Education 41 (April 1992): 1 38-52.

Cooney , Thomas J. "Evaluating the Teaching of Math- ematics: The Road to Progress and Reform." A rith- metic Teacher 39 (February 1992):62-64.

Green, Judith L., Regina Weade, and Kathy Graham. "Lesson Construction and Student Participation: A Sociolinguistic Analysis." In Multiple Perspective Analysis of Classroom Discourse, edited by Judith Green and Judith Harker, 1-11. Norwood, N.J.: Ablex Publishing Corp., 1988.

National Council of Teachers of Mathematics. Cur- riculum and Evaluation Standards for School Math- ematics. Reston, Va.: The Council, 1989.

. Professional Standards for Teaching Math- ematics. Reston, Va.: The Council, 1991.

Watson, Ken, and Bob Young. "Discourse for Learn- ing in the Classroom." Language Arts 63 (February 1986): 126-33. W

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IMPLEMENTING THE "PROFESSIONAL STANDARDS FOR TEACHING MATHEMATICS": Questioning in the Mathematics Classroom