Increasing Student Engagement During Questioning Strategy SessionsAuthor(s): JAMES S. CANGELOSISource: The Mathematics Teacher, Vol. 77, No. 6 (September 1984), pp. 470-472, 469Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27964137 .

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Increasing Student

Engagement During

Questioning

Strategy Sessions

By JAMES S. CANGELOSI, Utah State University, Logan, UT 84322

AN AGENDA FOR ACTION

40A4

4AAA

4AQC

1984

An

alarming proportion of secondary .school students are unable to apply

mathematical principles that they have memorized to the solutions of realistic prob lems (Florida Department of Education

1978). (An individual displays behavior at

the application level whenever he or she de cides which one of a number of possible principles or processes is appropriate in for

mulating a solution to a problem.) Conse

quently, in its published recommendations for the 1980s, the National Council of Teachers of Mathematics (1980) argued that schools should focus on developing stu dents' abilities to apply principles. But ac

cording to numerous published schemes for

categorizing cognitive levels of learning (Bloom et al. 1956; Guilford 1959; Bruner, Goodnow, and Austin 1967; Skemp 1971;

Cangelosi 1982), a student cannot learn to

apply a principle without having first con

ceptualized that principle. (An individual

conceptualizes a set when she or he dis

tinguishes between attributes that define set membership and psychological noise that distinguishes set members from one an

other; an individual conceptualizes a prin ciple or a process when she or he under stands why that principle is true or why that concept works.) One commonly ad vanced argument is that students fail to

apply a principle correctly because they lack the degree of concept attainment nec

essary to learn that principle at the appli cation level (Cangelosi 1980).

For at least the past 150 years, the literature in mathematics education has

consistently indicated that to conceptual ize, students must become involved in in ductive reasoning (Hendrix 1973). The same

literature suggests that mathematics teach ers should ask their students questions (i.e., use Socratic methods) to motivate students to reason (Cooney 1981). The questioning

model developed by Taba (1966) for social studies lessons has been popularized by general methods textbooks (Joyce and Weil

1980) and education courses as one mecha nism for eliciting inductive reasoning in students for all subject areas, including

mathematics. However, teachers attempting to apply such questioning-strategy models with typical classroom-sized groups com

monly have students answer questions aloud as soon as those students volunteer to do so. It appears that only the few more

outspoken and quick-to-respond students become highly engaged in this type of lesson (Duval County Schools 1977). For ex

ample, I recently observed the following large-group questioning strategy session

(Cangelosi in press) :

Ms. Ling uses an overhead projector to display six

sequences to twenty-nine mathematics students. She

asks, "What do you see?" Willie: "Some numbers." Ms. Ling: "Anything special about all six sets of num bers?" Anna Mae: "There's an order." Ms. Ling: "What's an ordered set called?" Anna Mae: "A se

quence!" Nettie: "Or a vector." Ms. Ling: "So we

have six sequences or vectors_What else do you notice?... Okay, Willie?" Willie: "Three of them are

470 Mathematics Teacher

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written in blue and the rest in red. Why is that?" Woodrow: "Because she used different pens, you ...". Ms. Ling (interrupting Woodrow): "The sequences in red are special. They belong together for a reason other than I used the same pen to write them out." "I know!" shouts Ory, raising his hand. Ms. Ling: "Okay?" Ory: "The blue numbers are all perfect squares!" Nettie: "No, ninety isn't a perfect square!"

Ms. Ling: "Anna Mae, thanks for raising your hand. What do you think?" Anna Mae: "All the members of the red sequences have a common factor." ...

Ms. Ling's inductive questioning strategy session

leading to the discovery of geometric sequences con tinues.

Student Engagement

Ling's Socratic strategies were probably highly successful in helping Willie, Anna

Mae, Ory, and Nettie to achieve an under

standing of geometric sequences. But what

happened to the other twenty-five students

during the lesson? What were they learn

ing? Ling seems to know how to use ef fective questioning strategies, but only a

small proportion of her students are benefit

ing from her attempts. For Ling's strategies to be effective, a

student must attempt to answer her

questions. It is not necessary for a student to be recognized and tell his or her re

sponses to Ling to obtain full benefit from the lesson. But it is necessary for each stu dent at least to attempt to formulate an answer to each question in her or his own mind. Because Ling allowed Anna Mae to answer immediately after asking the second

question, most students did not have

enough time to formulate their own ?n swers to that question. They quit thinking about how they would answer the question and instead listened to Anna Mae's answer and to the ensuing discussion. But simply listening to the answers provided by other students is insufficient. Only the quick-to respond, outspoken students (e.g., Anna

Mae and Willie) really became engaged in the lesson.

Students' engagement rates have been estimated to be lower when a teacher inter acts with only a few in a group than when he or she interacts with the majority ofthat

group (Rosenshine 1980). Borg and Ascione

(1982) found that teachers need to use spe cific techniques to encourage students to

attempt to answer questions that are posed

in classroom-sized groups. Studies of the time intervals between teachers' questions and students' answers suggest that teachers do not generally use techniques that allow for the majority of students to formulate an swers to teacher-initiated questions during classroom-sized group sessions (Arnold, Atwood, and Rogers 1974). One must keep in mind that with a questioning strategy that is designed to effect inductive student

reasoning, it is critical that students at

tempt to answer questions and not just hear answers given by others.

Improving Student Engagement

How, then, could Ling have conducted her lesson so that while retaining the advan

tages of her Socratic teaching technique, all or almost all her students formulate an swers to each question? Here are three pos sibilities :

1. Ling might preface her questions with directions for all students to answer each question in their minds without

answering aloud or volunteering until she asks them to do so. Had Ling taken this

optioii, part of her lesson might have gone as follows :

Ms. Ling uses an overhead projector to

display six sequences to twenty-nine stu dents. She says, "I am going to ask a

question. Each of you is to answer the

question in your mind. Don't tell us your answer or volunteer to do so until I call on you. Just silently hold your answer in

your mind. Okay, how do the sequences of numbers written in red differ from those written in blue?" Anna Mae and

Willie eagerly raise their hands and say, "Oh, Ms. Ling!" Ms. Ling is tempted to call on them and encourage their en

thusiasm, but she resists and quiets them down with a stern look and a silent

gesture. She waits, watching students'

faces, and then says, "Have you thought of an answer yet, Eddie?" Eddie: "Yes, ma'am." Ms. Ling: "Fine, hang on to it. Give us your answer, Judy." Judy re

sponds. Ms. Ling: "How about yours, Willie?" Willie responds. Ms. Ling:

September 1984 471

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"Compare Willie's answer to Judy's, everyone_Nettie, how would you com

pare the two?"

2. Another possibility is for Ling to re

quire each student to write answers to

questions on a sheet of paper as she circu lates around the room, quietly reading an swers while looking over students' shoul ders. This technique might have resulted in

part of the session going as follows :

Ms. Ling uses an overhead projector to

display six sequences to twenty-nine stu dents. She directs her class, "On a sheet of paper, use two or three sentences to describe how the number sequences writ ten in red differ from those written in blue." As the students begin writing, she walks among them ahd silently reads a

sample of their responses. Some students write nothing until Ms. Ling prods them into putting something down. After notic

ing that all have written something, she

asks, "Would you please read your answer to the class, Ory?" Ory reads aloud. Ms. Ling: "Okay, now read yours, Jamal." Jamal reads his, and Ms. Ling says, "Draw a comparison between what Jamal and Ory read, Anna Mae."

Because she has seen their responses when moving about the class, Ms. Ling can select responses that she believes will stimulate a productive discussion. The

procedure is followed for subsequent questions.

3. Ling might also consider having stu dents discuss answers to her questions in small subgroups. Each subgroup could then

report its conclusions to the class as a whole.

The idea behind these suggestions is to increase the number of students who become engaged in developing their own answers to the questions. Teachers need to

keep in mind that in these types of

questioning-strategy sessions, the process of trying to answer each question is far more important to aiding a student's con

ceptualization than knowing the "right" answer to each question.

The suggested techniques are more time

consuming than the more commonly used procedure for conducting questioning strategy sessions. Certainly the teacher can cover more content with the more conven tional procedure. However, what the teach er can cover is not the concern, The con cern should be with what students learn. And what students learn is dependent on how well they become engaged in lessons

(Berlinger 1975). From that perspective, one can argue that the more time-consuming procedures suggested here use time more ef

fectively than the less time-consuming con ventional procedure.

REFERENCES

Arnold, Daniel S., Ronald . Atwood, and Virginia M.

Rogers. "Question Response Levels and Lapse Time Intervals." Journal of Experimental Education 43

(Fall 1974):11-15.

Berlinger, David C. The Beginning Teacher Evaluation

Study: Overview and Selected Findings, 1974-75. San Francisco: Far West Regional Laboratory for Edu cational Research and Development, 1975.

Bloom, Benjamin, Max Englehart, Walker Hill, Edward F?rst, and David Krathwohl. Taxonomy of Educational Objectives, the Classification of Educa tional Goals, Handbook I: Cognitive Domain. New

York: DavidMcKay Co., 1956.

Borg, Walter R., and Frank R. Ascione. "Classroom

Management in Elementary Mainstreaming Class rooms." Journal of Educational Psychology 74 (Feb ruary 1982) :85-95.

Bruner, Jerome, Jacqueline Goodnow, and George Austin. A Study of Thinking. New York: Science Editions, 1967.

Cangelosi, James S. "Four Steps to Teaching for Mathematical Application." Mathematics and Com puter Education 14 (Winter 1980):54-59.

-. Measurement and Evaluation: An Inductive Ap proach for Teachers. Dubuque, Iowa: William C. Brown Co., Publishers, 1982.

-. Cooperation in the Classroom: Teachers and Students Together. Washington, D.C. : National Edu cation Association, in press.

Cooney, Thomas J. "Teachers' Decision Making." In Mathematics Education Research: Implications for the 80's, edited by Elizabeth Fennema. Alexandria,

Va.: Association for Supervision and Curriculum

Development, 1981.

Duval County Schools. Accountability in Citizenship Training Report. Jacksonville, Fla.: Duval County Schools, 1977.

Florida Department of Education. Results of the Flo rida Functional and Basic Skills Tests. Tallahassee: The Department, 1978.

Guilford, Joy P. Personality. New York: McGraw-Hill Book Co., 1959.

(Continued on page 469)

472 ? ?

Mathematics Teacher

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Fig. 8. TAoD2-D2oTV2A

interpreted F as TA <> DA, so we have shown that TA o D2

= #2,which is illustrated

in figure 9. The preceding examples illustrate the

power of visual imagery in making the im

portant concept of composition of functions

meaningful to students. The pictures help the students actively to participate in the definition of new functions by enabling them to compose simpler known functions.

O \

-A

Fig. 9. TAoD2=D2oT_A

Conversely, functions can be factored into the composition of simpler functions, and the factorization can be vividly illustrated

with an appropriate picture, m

Increasing Student Engagement During Questioning Strategy Sessions

(Continued from page 472)

Hendrix, Gertrude. "Learning by Discovery." In Teach ing Mathematics: Psychological Foundations, edited by F. Joe Croeswhite, Jon L. Higgins, Alan R. Os borne, and Richard Shumway. Worthington, Ohio: C. A. Jones Publishing Co., 1973.

Joyce, Bruce, and Marsha Weil. Models of Teaching. 2d ed. Englewood Cliffs, N.J.: Prentice-Hall, 1980.

National Council of Teachers of Mathematics. An Agenda for Action: Recommendations for School Mathematics of the 1980s. Reston, Va. : The Council, 1980.

Rosenshine, Barak. "How Time Is Spent in Ele mentary School Classrooms." In Time to Learn, edited by Carolyn Denham and Ann Lieberman.

Washington, D.C.: U.S. Department of Education, 1980.

Skemp, Richard. The Psychology of Learning Math ematics. Middlesex, England: Penguin Books, 1971.

Taba, Hilda. Teaching Strategies and Cognitive Func tioning in Elementary School Children. Cooperative

Research Project No. 2402. Washington, D.C.: U.S. Office of Education, 1966. W

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September 1984-?-?-4?9

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