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Questioning in the Elementary Mathematics Classroom

Linda Proudfit Division of EducationGovernors State UniversityUniversity Park, Illinois 60466

Thetypesofquestions that areaskedby teachers in elementaryclassrooms determine to a large extent how children viewmathematicsandwhatchildren decide is important. Forexample,if the only questions that are asked during mathematicsinstruction concern correct answers, children can onlyconclude that the most important things in mathematics arecorrect answers; however, ifchildren are to become successfulproblem solvers and intelligent users of mathematics, theyneed to focus on more than just correct answers. In addition,they need to focus on the decisions that they make while tryingto obtain correctanswers. Solvingreal-world problems requiresthe ability to make decisions about which methods to use anddecisions about the reasonableness ofresults, to namejust two.

TheNational Council ofTeachers ofMathematics (NCTM),in its Professional Standards for Teaching Mathematics(NCTM, 1991), stresses the importance of teachers "posingquestions and tasks that elicit, engage, and challenge eachstudent’s thinking" (p. 35). One way of doing this is to askquestions which focus on the various methods of solvingproblems and on thereasons forchoosing these methods. "Howdo you think we might solve this problem? Why did you decideto use that method to solve this problem? Why does the answerseem reasonable?" Questions such as these may be used in avariety of situations to help students understand and applymathematics. The teaching situations which will be discussedin this article are: (a) developing operation concepts, (b)developing computational algorithms, (c) solving routineworldproblems, and (d) solving non-routine problems.

Developing Operation Concepts

When introducing an operation such as addition, a real-world problem which presents an addition situation should beused. For example, the following situation mightbe presented:

Three children are working with clay at the art table.Five other children join (hem to paint at the art table.How many children are now at the art table?

This problem situation provides an opportunity to ask manyquestions which will focus the children’s attention on theimportance aspects of addition. Questions which might beasked are:

1. Will the answer be more than three children? (Yes,some children join the three children.)

2. Will the answer be more than five children? (Yes,five children join some other children.)

3. How could we find out how many children are at the arttable? (Children from the class could act out the problemsituation, and the total number of children could be counted.Children might also suggest that they could use counters ordraw pictures to represent the children in the problem.)

Ifconcrete objects such as chips are used to solve the aboveproblem, children might use this model and be asked thefollowing questions:

1. How many chips should we use to show how manychildren are working with clay at the art table? (3)

2. How many chips should we use to show how manychildren join them to paint? (5)

3. How will the chips help you to answer the question? (Thetotal number of chips shows how many children are working atthe art table in all.)

After solving this problem, the following problem could bepresented:

The next day, five children are working with clay at theart table. Three other cfiildrenjoin them to paint at theart table. How many children are now at the art table?

If children do not see the relationship between this problemand the previous one, the following questions might be asked:

1. How could we solve this problem? (Children mightsuggest acting out the problem or using counters.)

2. Havewesolvedaproblemlikethis before? (Aftersolvingthis problem, the similarity between this problem and theprevious one will be apparent to most children.)

3. How are the two problems alike? (The numbers are thesame; they are just in a different order. The answers must bethe same.)

Developing Computational Algorithms

Mathematics educators have long suggested thatcomputational algorithms be developed in a manner which ismeaningful to students; however, most instruction regardingalgorithms focuses on the mechanical procedures, rather thanon the conceptual framework which might make thesealgorithms seem reasonable.

One ofthe greatfallaciesof the elementary curriculum

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is to classify arithmetic as a skill or a drill or a toolsubject.... The teaching process, according to thetool conception of arithmetic, undertakes to tellchildren what to do (but not why they do it) and thenby ceaseless drill to have them do it until they candemonstrate some degree of mastery. (Brownell,1945, p. 494)

While practice is necessary, ceaseless drill is not ifmeanings are taught. As learning a sensible sentence is easierthan memorizing a string of nonsense syllables, children find iteasier to perform an algorithm which makes sense to them thanto follow a list of unrelated steps. Also, children will be morelikely to apply these procedures to unfamiliar situations if thecomputational algorithms are more than mechanical skills.

An instructional approach which emphasizes questioningcan assist students to develop an understanding ofcomputationalalgorithms in addition to proficiency with them. This can bedone by developing algorithms using real-world problems andconcrete models.

To develop the division algorithm, for example, thefollowing problem could be used:

There will be 142 fourth-grade students attendingJefferson Elementary School nextyear. There will be5 fourth-grade classes. If each class should haveabout the same number ofstudents, howmany studentsshould be assigned to each class?

Students who are being introduced to the division algorithmcan solve this problem by using concrete materials such asbundling sticks. Ifstudents begin with 14 groupsof 10 sticks and2 single sticks, they can distribute the sticks equally among fivegroups to solve the problem. To guide this work, the followingquestions could be asked:

1. Howcould we use the bundling sticks to help us solve thisproblem? (The bundling sticks could stand for students.)

2. How many bundling sticks do you need? (142)3. How could we arrange the sticks to help answer this

question? (Put them into 5 equal groups.)After giving the students an opportunity to separate the

bundling sticks into five groups, ask:1. How many tens did you put in each group? (2 tens)2. Did you use all of the tens when you did this? (No)3. What did you do with the remaining tens? (The groups of

ten were unbundled.)4. How many ones did you put in each group? (8 ones)5. How many sticks are in each group? (28 with 2 left over)6. What did you do with the extra sticks? (Put 1 in each of

two groups.)7. How many students should be in each class? (28 in three

classes and 29 in two classes)The algorithm can be viewed as a way of recording what

was done with the bundling sticks.

<� When you distribute the remaining sticks.how many sticks are in each group?

2 <� How many tens can you put in each group?5)"l42

100 <- How many sticks have been used?42 <� How many sticks are left to distribute?

285)T42

10042402 <� How many sticks are left?

When students areconfronted with division problems in thefuture, the images of groups of bundling sticks may helpstudents to ask themselves questions similar to the ones listedabove. These questions can help students follow through theprocess of this algorithm and develop an understanding of it.

Students who are directed to follow a list of steps must relysolely on their memory. If they forget, they are lost. The manyerrors that students make indicate that they forget frequently.On the other hand, students who are asked questions whichhelp them to understand the process may be able to reconstructthis algorithm and apply it to real-world situations.

Solving Routine Word Problems

Most word problems found in textbooks are intended toprovide practice with basic number combinations and withcomputational algorithms and to provide opportunities forchildren to translate real-world situations into mathematicalsentences. As such, the major decisions required are todetermine what information is to be found, what information touse, and which operation or operations to perform.

Questions which focus on the actions occurring in theproblems or the relationships among the quantities involvedmay help students to make these decisions. Consider thefollowing problem:

To celebrate her birthday, Sara is going to bring anapple for each of the 32 students in her class. Hermother bought 3 bags of apples, and each bag has 12apples in it. Does Sara have enoughapples to take toschool?

Questions which teachers might ask to help students solvethis problem follow:

1. About how many apples did Sara’s mother buy? (30)2. How did you determine this? (3 bags of 10 would be 30

apples.)3. How could we find an exact answer? (Add 12+12+12

or multiply 3 X 12.)4. Why do you think that you should add (or multiply) to

help solve this problem?

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Questioning in Mathematics135

If students can think about the process with smaller oreasier numbers, this might suggest an appropriate procedure touse to solve the original problem.

For some children, determining the appropriate operation tosolve routine problems is a difficult task. For these children, thequestions suggested in the following section should also behelpful.

Solving Non-Routine Word Problems

Much has been written about the importance of teachingproblem solving. Somegroups have stated thatproblem solvingshould be the major emphasis of school mathematics (NCSM,1977; NCTM, 1980). At the same time. it is acknowledged thatproblem-solving ability is difficult to develop. Some believethat asking the right questions is a key factor in successfulproblem solving. Krulik and Rudnick (1984) recommend "thatthe students develop an organized set of questions to askthemselves and that they constantly refer to them when they areconfronted by a problem situation" (p. 37). Teachers can helpstudents become better question askers by modeling goodquestioning techniques.

The wide variety ofquestions might be organized accordingto thecomponents oftheproblem-solving process. Thisproccssis frequently discussed in terms of a model proposed by Polya(1957). Polya outlined four aspects of the problem-solvingprocess: (a) understanding the problem, (b) devising a plan, (c)carrying out the plan, and (d) looking back.

These aspects will be used to help organize the questionsdiscussed. Consider the following problem:

Some children are seated at a large round table. Theypass around a box of candy containing 25 pieces. Tedtook thefirst piece. Eachchild tookone piece ofcandyas the box was passed around. The box was passedaround the table until there was no candy left in thebox. Ted got the last piece. How many children couldthere have been seated around the table?

Teachers might ask some or all of the following questionsto facilitate a large-group problem-solving activity or to helpstudents as they work individually orin small groups. Questionsrelated to the first and fourth aspects of the problem-solvingprocess are listed first.

Understanding the ProblemHow many pieces of candy were in the box?How was the candy passed out to the children?Which pieces did Ted get?How many pieces of candy did each child get?

Looking BackDoes the solution give Ted the first and last pieces?Is there a relationship among the solutions?

Questions related to Devising a Plan and Carrying Out thePlan are sometimes difficul t to create since the questions shouldnot suggest a strategy. Rather, they should focus on theinformation given in this problem, relationships among theobjects or quantities involved, and critical decisions thatwould help the student decide on a strategy and carry it out. Itmay be useful to group these questions into several categories.When studentsarehaving difficulties with determining a strategyto use or implementing the strategy, teachers might ask aquestion which:

Asks for an estimate or a range for the solution.Could there have been more than 24 children?Could there have been fewer than 24 children?

Asks about a word or condition which might not beunderstood.Does the problem tell you that each child except Ted getsone piece of candy?

Asks about an important decision.Could the box of candy be passed around the table morethan once?

Suggests an incorrect answer.Could there have been 7 children seated around the table?

Suggests simplifying the problem.What if there were only 9 pieces of candy in the box?

These categories may be helpful in creating questions tohelp students solve other problems, as well.

Conclusion

Teachers who use effective questioning techniques mayhelp students ask themselves questions to help them through theproblem-solving process. Students who acquire the habit ofasking themselves questions maynot need toask many questionsof others. This should help students develop confidence indetermining the correctness of their solutions and in becomingindependent thinkers.

References

Brownell. W. A. (1945). When is arithmetic meaningful.Journal of Education Research, 38,481-498.

Krulik, S., & Rudnick, J. (1984). A sourcebookfor teachingproblem solving. Boston: Allyn & Bacon.

National Council of Supervisors of Mathematics. (1977).Position paper on basic skills. Arithmetic Teacher, 25(2),19-27.

National Council of Teachers of Mathematics. (1980). Anagenda for action. Rcston, VA: Author.

National Council of Teachers of Mathematics. (1991).Professional standards for teaching mathematics. Reston,VA: Author.

Polya, G. (1957). How to solve it (2nd ed.). Princeton. NJ:Princeton University Press.

Volume 92(3), March 3992