This article was downloaded by: [University of Chicago Library]On: 13 November 2014, At: 04:48Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

International Journal ofMathematical Education in Scienceand TechnologyPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/tmes20

Designing questions that developunderstandingRuth Hubbard aa School of Mathematics , QUT , GPO Box 2434, Brisbane4001, AustraliaPublished online: 09 Jul 2006.

To cite this article: Ruth Hubbard (1997) Designing questions that develop understanding,International Journal of Mathematical Education in Science and Technology, 28:6, 793-801,DOI: 10.1080/0020739970280602

To link to this article: http://dx.doi.org/10.1080/0020739970280602

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information(the “Content”) contained in the publications on our platform. However, Taylor& Francis, our agents, and our licensors make no representations or warrantieswhatsoever as to the accuracy, completeness, or suitability for any purposeof the Content. Any opinions and views expressed in this publication are theopinions and views of the authors, and are not the views of or endorsed by Taylor& Francis. The accuracy of the Content should not be relied upon and should beindependently verified with primary sources of information. Taylor and Francisshall not be liable for any losses, actions, claims, proceedings, demands, costs,expenses, damages, and other liabilities whatsoever or howsoever caused arisingdirectly or indirectly in connection with, in relation to or arising out of the use ofthe Content.

This article may be used for research, teaching, and private study purposes.Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expresslyforbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

INT. J. MATH. EDUC. SCI. TECHNOL., 1997, VOL. 28, NO. 6, 793-801

Designing questions that develop understanding

by RUTH HUBBARDSchool of Mathematics, QUT, GPO Box 2434, Brisbane 4001, Australia

(Received 20 November 1995)

Although the development of an understanding of mathematical concepts isusually one of the stated goals of instruction, the types of exercises mostfrequently given to students do not appear to contribute to this goal. Standardexercises ask students to carry out mathematical procedures, not to think aboutthem. A model of what it means to understand mathematics is presented,followed by examples of exercises which are designed to develop understandingin terms of the model in specific ways. It is assumed that understanding is morelikely to develop, if it is the direct focus of exercises rather than if thedevelopment is left to chance.

1. IntroductionAn important goal of much mathematics instruction is that students should

understand mathematics. But what exactly do we mean by understanding mathe-matics? Cognitive scientists and mathematics educators have developed a model ofwhat it means to understand mathematics and although there is still much to belearned about the details of the model, it is sufficiently well defined to serve as abasis for designing instruction. In their summary of research into mathematicalunderstanding, Hiebert and Carpenter [1] describe the model as follows:

We begin by defining understanding in terms of the way information isrepresented and structured. A mathematical idea or procedure or fact isunderstood if it is part of an internal network. More specifically, themathematics is understood if its mental representation is part of a networkof representations. The degree of understanding is determined by thenumber and strength of the connections. A mathematical idea or procedureor fact is understood thoroughly if it is linked to existing networks withstronger and more numerous connections.

By way of contrast, Mitchelmore and White [2] have proposed a model ofmathematical knowledge in which understanding is lacking. Their model consistsof isolated mathematical ideas which are connected neither to each other nor to anyphysical contexts. Unfortunately, we have all encountered large numbers ofstudents whose mathematical knowledge appears to be structured according tothe Mitchelmore and White model rather than the ideal one. Why do so many ofour students fail to construct links between their mathematical ideas and thecontexts that gave rise to them?

Mitchelmore [3] explains the problem by observing that in exemplary primaryand secondary mathematics teaching the dominant paradigm is 'development-result-practice' but the teacher does all the developing and the students just do the

0020-739X/97 $12.00 © 1997 Taylor & Francis Ltd.

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 0

4:48

13

Nov

embe

r 20

14

794 R. Hubbard

practice. Frequently the practice consists of very restricted types of exercises andas a result, the students construct very restricted mental representations of theconcept or procedure and fail to construct links to other knowledge.

An investigation of standard school texts reveals long lists of very similarexercises and many texts for tertiary students follow the same model. The exercisesmainly involve substituting into formulae and carrying out standard procedures.At the end of each chapter there are usually a few 'problems' but many experiencedteachers omit these because they know that their students would not be able tomanage them. Even if the teacher does ask the students to attempt the problems,many students will choose to omit them because they know from past experiencethat 'I can't do problems'. The exercises rarely ask students to reflect on the resultsthey obtain, generalize from them, make comparisons, observe patterns or relatethe new procedures to previous knowledge. Yet these are the very activities whichhave the potential for creating links in the student's cognitive structure. A furtherdefect of standard exercises is that they never contain any justification as towhether and why they are important or necessary. Exercises are presented withoutany context and apart from the obvious fact that practising the exercises issupposed to enable the student to pass examinations, no mathematical or pedago-gical reasons are given for the selection of particular exercises. If students hadsome answers to questions such as, 'Why am I doing this page of exercises?','Where in mathematics is this procedure used?', their work might become moremeaningful.

On the other hand, the mathematics education literature, for example Vinner[4], Tall [5] contains many ingenious types of questions designed specifically toinvestigate students' representations of mathematical ideas. Such questions have tobe designed very carefully, because students' representations of mathematical ideasare internal and cannot be observed directly. In order to obtain information aboutthese internal representations the questions have to probe for the existence of ideasand the connections between them in the students' minds. Similar types ofquestions can be used, Hubbard [6], to help students to increase the number ofconnections between their internal representations of mathematical ideas. Specificexamples of such questions will be used to illustrate some strategies for improvingunderstanding.

2. Improving understanding2.1. Finding out about students' understanding

At every level, a teacher must decide where to begin a course of instruction.The starting point is usually determined by the syllabus rather than by aninvestigation of the understanding that students bring to the course. Even whentests are used to determine students' 'readiness', the types of questions used in thetests often reveal only whether or not the students have learned to carry out certainprocedures, and provide very little information about the robustness of theirnetworks of mathematical ideas. But students are ready to learn new material onlyif they have built internal representations to which the new information can beconnected. In the words of Ausubel [7]:

If I had to reduce all of educational psychology to just one principle, I wouldsay this: The most important single factor influencing learning is what thelearner already knows.

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 0

4:48

13

Nov

embe

r 20

14

Questions that develop understanding 795

•f s

+- •

Figure 1.

But how do we address this question of readiness to learn? Most of the detailedresearch on learning mathematics has been done with young children learning tocount and to understand the basic operations of arithmetic. Even at this level, thediversity in the mental networks that children create is astonishing. Now considerhow this diversity expands during 12 years of schooling! The task of determiningwhat our students understand is a daunting one but this does not mean it should beignored. Well thought out questions can reveal a great deal about students'understanding and misunderstanding, particularly if the students are asked togive reasons for their responses. Here is an example of such a question.

Figure 1 shows the position versus time graph for a certain object moving on astraight line.

(a) Is the object moving faster at time t\ or at time <2? How do you know?(b) What is the initial velocity (i.e. velocity when t = 0) of the object? How do

you know?(c) During the time interval \ti, £3], is the object speeding up or slowing down?

How do you know?

This question was included in readiness tests for first-year science and engineeringstudents. The response was scored as correct only if the answer and the reasongiven were both correct. The percentages of correct responses are shown inTable 1.

The differences in scores for the two groups are probably explained by the factthat the engineering students were forewarned about the test because passing itgained them exemption from a preliminary subject. The science students were notforewarned and knew that the test had no assessment implications.

Science (w = 218) Engineering (n = 263)

(a) 52(b) 33(c)70

635985

Table 1. Percentages of correct responses.

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 0

4:48

13

Nov

embe

r 20

14

796 R. Hubbard

The main problems that emerged from the explanations were:

• A failure to distinguish between average and instantaneous rate of change.• Confusion between distance travelled and speed.• The inability to proceed without a formula or scales on the axes.• The fact that the graph did not go through the origin.• Application of an inappropriate formula.• Division by zero.

All of these problems were noted in responses to part (b) and many students clearlyhad difficulty with several of the above. The widespread problem of confusionbetween average and instantaneous velocity was probably exacerbated by thestudents' reluctance to abandon a formula, v = s/t, which had produced correctanswers in the past and clearly had some connection with the subject matter butwas inadequate for answering these questions.

In addition there were responses such as:

Initial velocity of the object is 0 by v = s/t = s/0 = 0. But, physically (inreality) if an experiment starts at 0 time the object may be moving.

In this student's mental network, mathematical formulae and the real world arenot necessarily connected.

The purpose of this example was to show that it is possible to obtain valuableinformation about students' mental representations of concepts to which newmaterial will need to be connected. Such information is not usually obtainablefrom standard questions which ask the student to carry out a procedure which isreadily memorized. For example, 77% of the engineering students referred to inTable 1 were able to correctly identify the derivative of y = 3/x.

3. Addressing students' misconceptionsJust re-teaching a topic such as the interpretation of distance—time graphs will

not necessarily solve the students' problems because it could fail to alter theirmathematical representations of the relevant concepts. Students do not abandontheir representations of mathematical ideas or formulae such as v = s/t withoutsome kind of cognitive conflict. We have to devise activities which will force themto confront the inconsistencies or inadequacies in their representations. One way todo this is to make a list of statements about a similar graphical problem some ofwhich contain the actual misconceptions that have been observed. Students canwork in pairs to try to convince each other which statements are false and whichare true. When they have made their decisions, they should justify them withwritten explanations. Some examples of such statements could be:

The object is moving faster at time ti because the speed at that point ($2/^2) i s

greater than the speed at t\, (s\/t\).

The velocity at t = 0 must be greater than zero because although there isno scale, it does have some velocity because it doesn't go through theorigin.

A less confrontational way to generate discussion of the same misconceptions is tocreate a conversation between two or more students which again contains the

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 0

4:48

13

Nov

embe

r 20

14

Questions that develop understanding 797

misconceptions that need to be addressed. Then the students can try to pinpointthe errors in the arguments of 'other people'. For example:

James: I think you have to use the formula v = s/t to solve problems aboutvelocity, that's what we always did at school.

Maria: But there aren't any numbers to substitute for s and t. You need numbersto work it out.

Anna: Well in part (b) we do have a number, t = 0, so why don't we put that intothe equation?

James: Well if you do that you get v = s/0 and I remember that s/0 is undefinedso that means the velocity must be zero.

etc. etc.

Of course one exercise will not be sufficient to remove misconceptions from astudent's mental network, particularly if they have become entrenched. We are allvery reluctant to abandon habitual ways of thinking. Students will need severalquestions involving graphs relating different pairs of variables over a period oftime, before they will be able to build strong connections between all the ideasinvolved and thereby develop an understanding of rate of change.

4. Developing a new topicWhen a good teacher develops a new topic they try to relate it to ideas that they

think are familiar to their students and then assume that the students understandthe development and are ready to practise its results. Unfortunately many studentsare directed to practise before they have connected the new idea to anything intheir network of mathematical ideas and as a result the new knowledge is isolatedand not linked to related ideas. The danger in practising what is not understoodwas discussed by Comenius [8] in The Great Didactic:

That nothing should be learned by heart that has not been thoroughlygrasped by the understanding.

written in the seventeenth century, so it is hardly a new idea. Yet it is clear that thepredominant paradigm in mathematics instruction is quite the opposite ofComenius' advice.

In order to connect new ideas to existing ones the student needs to be involvedin activities which assist in the connection building process. The teacher cansuggest useful connections but the teacher cannot create the connections in thestudents' mind.

In developing the concept of the derivative at a point x, we make use of aformula such as

f(x + h)-f(x)h

and we relate the formula to the gradient of a secant. But do we ask the student todo exercises in which the formula has to be reconstructed and attached to thegraph? The following question from Hughes-Hallett and Gleason [9] does exactlythis.

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 0

4:48

13

Nov

embe

r 20

14

798 R. Hubbard

+->JC

Show how you can represent the following on Figure 2:

(a) /(4)(b)

( c )V ; 4-2A number of questions along these lines would be helpful in linking the concept ofthe derivative to concepts of functions and their graphs.

Another approach is to prepare students in advance for a difficult new conceptwhich will be the subject of a forthcoming lecture by setting an exercise that leadsin to the new concept. Then during the lecture, the new concept can be developedfrom the example that students have already encountered. To make sure thatstudents do the exercise and reflect on what they are doing, it may be necessary toflag the exercise as important or better still, explain why it is important. In otherwords it may not be sufficient for the teacher to have a purpose in setting theexercise, the purpose needs to be communicated to the student as well. Twoquestions which illustrate this approach follow.

Purpose. One purpose of this question is to draw your attention to the specialproperties of the first and second derivatives of some standard functions. Asecond purpose is to start you thinking about the kinds of functions y =f(x),which could provide solutions to the equations below. These are examples ofdifferential equations and we will begin to study them in detail next week.Using what you know about the derivatives of standard functions like ex, \n(x)and sin(x), find functions which satisfy the following equations.

(a) y'-y =(c)y"+y =

(b) y'+y(d)y"-y

Purpose. The purpose of the next question is to start you thinking how manythings you have already learned in statistics, such as the standard error and thedistribution of sample means, can be used to make important decisions. You

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 0

4:48

13

Nov

embe

r 20

14

Questions that develop understanding 799

will learn how to make these decisions formally next week, but you will findthat easier to understand if you think carefully about this question.A physiology book states that the mean resting pulse rate for young women(18—25 years) is 70 beats per minute. The 56 women members of the physiologyclass decide to see if they conform to this standard. After a bit of practice, theyall succeed in measuring their pulse rate for one minute. Then they calculatethe mean 73 and standard deviation 12 of their pulse rates.

(a) How much does their mean pulse rate differ from that in the physiologybook?

(b) Does this suggest that they are freaks or could the difference havehappened by chance?

Suppose they calculate the standard error of the mean and that they know thatsample means have a normal distribution.

(c) How many standard errors away from 70 is the students' mean pulse rate?What does this suggest?

(d) How would you reassure the students that they are not freaks?

5. Making connections between ideasAlthough good teachers explain the connections between mathematical ideas,

there is no guarantee that those connections are internalized by students. Accord-ing to Heibert and Carpenter [1] the potential danger in teaching connectionsexplicitly is that the information required to make the connections explicit will beinternalized as one more piece of isolated knowledge rather than supporting theconstruction of useful connections. Students who believe that mathematics con-sists of many unrelated and perhaps meaningless procedures are very likely tomisuse their teacher's best efforts. In the two following questions students arerequired to connect specific ideas. These are not 'problems' because to solveproblems students often need to connect unspecified ideas and this is much moredifficult.

In order to test a hypothesis about a population mean, the following Minitabprintout was obtained.

MTB > Z T E S T 2 5 3-4 Cl;SUBC> A L T E R N A T I V E - 1 .

TEST OF MU = 25-0 VS MU L.T.25-0THE ASSUMED SIGMA = 3-4

N40

MEAN23-63

STDEV3-95

SEMEAN0-54

Z- 2 - 55

P0.

VALUE0054

If you had used a T T E S T instead of a ZTEST on the same data, whichnumbers in the last line of the output would remain the same and which wouldchange. Do you have enough information to decide whether each of thenumbers that would change, would increase or decrease? Justify your answer.

Notice that it is not possible to answer the above question by rote learning the rulesfor deciding whether to use the t or normal distributions. The rules have to be

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 0

4:48

13

Nov

embe

r 20

14

800 R. Hubbard

manipulated and used in new ways and this should encourage the students to makenew connections between these rules.

(a) Solve the system of equations:

Sx + 5y - 2z = 5

5x — z = —5

(b) Explain your solution geometrically.(c) Does the matrix of this system have an inverse? How do you know?

This question specifically asks the student to connect three aspects of the solutionof a system of equations which could easily remain unconnected.

6. Strengthening the linksThe standard method for reinforcing understanding is drill and practice

exercises. One problem with such exercises is that they reinforce skills in isolationand may in fact contribute to a fragmentation of knowledge. A different way tostrengthen the links between concepts is to ask questions which require thestudents to reverse their thinking. Apart from strengthening the links betweenideas, thinking in reverse questions help to ensure that existing links can betraversed in both directions.

A very simple and effective way to do this, in any context, is to ask the studentto make up the question instead of answering it. The instructions for making upthe question may be very specific as in the following example.

MTB > ZTEST 25 3-4 Cl ;SUBC> ALTERNATIVE - 1 .

Cl

Make up a problem which someone could solve using the above printout. Makesure that your problem contains sufficient detail so that the person solving itwould have enough information to type the commands in the first two lines.

Alternatively, the instructions for making up the question can be quite general.

Without referring to a textbook or to your notes, try to construct a system ofthree linear equations in three variables so that your system has an infinitenumber of solutions.

To answer the above question a student must have a clear understanding of theconditions which will produce an infinite number of solutions. It is not sufficient tomemorize the conditions or to recognize them; the conditions have to be used tocreate an object. Students who only have a process conception [10] of solving asystem of equations, i.e. students who can carry out row operations and perhapswrite down the solution may have great difficulty reversing the procedure.

Thinking in reverse by making up the question, can also be used to helpstudents distinguish between different models they have studied over a period oftime. Because students usually learn about one type of model and then workthrough exercises on that model before they meet a different model, they may

N40

MEAN23-63

STDEV3-95

SE MEAN0-538

Z-2-55

P0-

VALUE0054

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 0

4:48

13

Nov

embe

r 20

14

Questions that develop understanding 801

never focus on the essential properties of each model. For example, students mayrespond to the question below by describing an experiment for comparing twomeans.

A few weeks ago you studied simple regression. Describe a problem whichcould occur in one of the other subjects you are studying or in the context ofone of your hobbies, which could be solved using the simple regressionmodel. Pretend that you have collected some data to help solve your problem.Write the data down and label it but do not do any calculations.

7. ConclusionIf the description of understanding given in the introduction is accepted, then

it must also be accepted that the stereotyped exercises that are a feature of mostmathematics texts do not actively encourage the development of understanding. Ofcourse some students do develop a thorough understanding of mathematics fromthese exercises. These are the mathematically gifted students who have learnedhow to think mathematically unaided. For the vast majority of students mathe-matical knowledge consists of isolated facts and procedures with only a few weaklinks between them.

The main objective of this paper was to show that it is possible to constructquestions which aim to improve understanding in quite specific ways. Theparticular aspects of developing understanding which have been addressed arejust examples from a wide range of possible approaches. Likewise, the questiontypes used as illustrations, are not intended to be comprehensive but rather tostimulate others into constructing questions of their own.

Whether responding to such questions on a regular basis actually improvesstudents' understanding of mathematics is very difficult to measure. However, itseems logical to accept that if measures are taken to improve understanding, theoutcome will be better than if no measures are taken and the development ofunderstanding is left to chance.

References[1] HIEBERT, J., and CARPENTER, T. P., 1992, Handbook of Research on Mathematics

Teaching and Learning, edited by D. A. Grouws (New York: Macmillan).[2] MITCHELMORE, M. C., and WHITE, P., 1995, Math. Educ. Res. J., 7, 50-68.[3] MITCHELMORE, M. C., 1992, Abstracts of Short Presentations at the 7th International

Congress on Mathematical Education, edited by M. Meilleur (Quebec: LavalUniversity).

[4] VINNER, S., 1991, Advanced Mathematical Thinking, edited by D. Tall (Dordrecht:Kluwer).

[5] TALL, D., 1992, Handbook of Research on Mathematics Teaching and Learning, editedby D. A. Grouws (New York: Macmillan).

[6] HUBBARD, R., 1995, 53 Ways to Ask Questions in Mathematics and Statistics (Bristol:Technical and Educational Services).

[7] AUSUBEL, D. P., 1968, Educational Psychology: A Cognitive View (New York: Holt,Rinehart & Winston).

[8] COMENIUS, J. A., 1967, The Great Didactic (New York: Russell & Russell).[9] HUGHES-HALLETT, D., and GLEASON, A. M., 1992, Calculus (New York: Wiley).

[10] SFARD, A., 1992, The Concept of Function: Aspects of Epistemology and Pedagogy, editedby E. Dubinsky and G. Harel (MAA Notes, Washington: Mathematical Associationof America).

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 0

4:48

13

Nov

embe

r 20

14