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Mathematics as a didacticaladventureAndrejs Dunkels aa Department of Mathematics & Department of TeacherEducation Lulea University , S‐971 87 LULEÅ, SwedenPublished online: 09 Jul 2006.

To cite this article: Andrejs Dunkels (1995) Mathematics as a didactical adventure,International Journal of Mathematical Education in Science and Technology, 26:3, 417-429,DOI: 10.1080/0020739950260310

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INT. J. MATH. EDUC. SCI. TECHNOL., 1995, VOL. 26, NO. 3, 417-429

Mathematics as a didactical adventure

by ANDREJS DUNKELSDepartment of Mathematics & Department of Teacher Education Lulea University,

S-971 87 LULEÅ, Sweden

Personal views and reflections are presented around some of the mathematicalmilestones I have encountered as a learner, teacher, and teacher educator. Theseencounters are didactical adventures starting with a school boy's ideas aboutdivisibility by 9 and 11, proceeding via a project on multiplication for elementaryschool children to thoughts about algebra, area, integrals, and the derivative ofthe product of two functions. Some aspects of visualization are discussed as wellas procedural and conceptual knowledge.

1. IntroductionThis is meant to be a paper based on views and experiences gathered from my work

as a learner and teacher of mathematics. It is not a research paper but rather anaccount of some of the personal feelings and thoughts I have had over the years.

I work at present as a teacher educator part of the time. My student teachers willteach in elementary school after graduation. Therefore I try to keep in touch withschools, and I teach at least once a week in some elementary class—sometimes justoccasional lessons, sometimes longer projects. Part of the time I teach mathematicsto engineering students. I am also—and have always been—involved in variousin-service course programmes.

Before joining Lulea University in northern Sweden I taught at UmeåUniversity, Uppsala University, schools in the Uppsala area, and 1966-1968 atKenya Science Teachers College in Nairobi, Kenya.

I now invite the readers to accompany me and visit some of the mathematicalmilestones of my life.

2. Learning by discovery versus learning by listeningWhen I first encountered the formula for the derivative of a product

(fg)'=f'g+fg'I was highly surprised. The result was really not what I had, intuitively and naively,expected. The formal proof was not very difficult, I thought. I understood each step,I could reproduce the proof, and I could note that the result was indeed a sum oftwo terms.

But I did not understand the proof as a whole. I did not understand why therehad to be two terms. I had no feel for either the result or the proof. I had no mentalimage of the result, I had no useful mental image of a product, I had no mental imageof what was going on. I accepted the result and I could successfully use the formulain actual cases.

Later I wondered whether it would have been an advantage for me to have hada more appropriate mental image of multiplication. I think it would. How, then, canone think about multiplication?

Before going into this general question allow me to take you to the milestone

0020-739X/95 $10-00 © 1995 Taylor & Francis Ltd.

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Figure 1. A car resembling one of the cars of my childhood days in Uppsala.

where I became truly interested in mathematics—the most important milestone ofthem all. It happened in Uppsala, Sweden, where I grew up.

Most number plates of cars in the 1950s in Sweden had one or two letters—acounty code—followed by a number. In the Uppsala area in those days most carshad four-digit numbers. At school we had learned the nines multiplication table. Iwas fascinated by the number patterns that it contains. We had also learnt the digitalsum test for divisibility by 9. I loved it, and I used to check the numbers of all carsI met.

One day I saw a car with a number like 1386. The digital sum test was positive.Quite often I performed the actual division in my head—for fun, I clearly recall thepleasant feeling of knowing there would be no remainder, and then also being ableto verify that this was indeed true.

This time I noticed that the quotient, 154, was special, the siim of its first andlast digits added up to the middle digit. It also struck me that 154 was divisible by11. Unconsciously, I formulated a rule: Three-digit numbers and divisibility by 11are linked in the way that first digit plus last equals middle. We had not learnt thisrule at school, and so it was a true discovery.

Some time later I extended my rule to four-digit numbers after having noticedthat in numbers like 1386 the sum of the first and third digits equals the sum of thesecond and fourth.

Cars were not as common as they are today, of course, and so it was actuallypossible for me to check all cars I saw. One day I had a mathematical shock. I sawa car like the one depicted in Figure 1.

It was obvious without calculation that first + third was not equal to se-cond + fourth and so I concluded that 2816 could not be divisible by 11. There wasno need for checking, I felt, but unconsciously I carried out the division anyway,which led to the unbelievable fact that the remainder was 0. I just could notunderstand it. I added to my rule 'except 2816', and often returned to this number,thought about it but did not find any pattern or explanation.

The years to come brought a few additional exceptions, but the mystery remaineda mystery until I had studied mathematics at university level for a year. I found thefull truth in a number theory book in the library. I had simply to replace 'equal' inmy rule by 'congruent mod 11'.

Later still, when I had become a teacher educator, a teacher at an in-servicecourse requested a visual explanation that could be used with pupils at school level,without highbrow formulas and without concepts like mod. Then I came up withthe explanation based on an observation about remainders when powers of 10 aredivided by 11—one obtains alternately 1 and 10—as shown in Figure 2.

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Mathematics as a didactical adventure 419

A(a)

8 1 6

D

(b)

PO

a P „

(c)

(d)(-2) + 8 + (-1) + 6

Figure 2. (a) First of all we find a suitable visual representation of the number in question,two thousands (triangles), eight hundreds (squares), one ten (circle), and six units(dots). When dividing by 11 we note that 1 leaves the remainder 1, 10 leaves theremainder 10, 100 the remainder 1, 1000 the remainder 10, etc.

(b) We leave the units as they are. We add a unit, 1 (dot), to the ten, making it an eleven, andthen compensating by adding a "hole", ( - 1). Then we make a hole in each hundred,making it a ninety-nine, and compensate by adding a unit, 1, for each hundred. Finallywe add a unit to each thousand, making it divisible by 11, and compensate by addingthe same number of holes. ( — 1)

(c) Now the enclosed number is certainly divisible by 11, because each part of it is, and sothe divisibility of the whole number depends solely on what is outside the enclosure.Thus the original number is divisible by 11 if and only if the sum of all that is outsidethe enclosure is divisible by 11. We can now phrase our findings using the digits of theoriginal number.

(d) This sum must be divisible by 11. We notice that the digits appearing here are exactlythe digits of the original number. The sum is a modified digital sum of our number.Moving from the right we change signs of every second digit, starting with the tens.

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Figure 3. A modification of the old Indian jig-saw puzzle of the Pythagorean theorem. Thepuzzle can be analyzed and transformed into a solid proof. In the setting given here onecould say that a2 + b2 is what one gets in the fourth (right-most) figure of the sequence.The starting figure shows c2.

So one could say that the birth of my interest in mathematics has been closelylinked with discovery and self-initiated investigation—and with multiplication, inparticular the nines table and the elevens table.

3. The teacher's role in promoting discovery: an example using multi-plication mats

There is a widely spread idea that mathematics is logical, strict, sterile, free ofemotions, simply right or wrong. This is partly true and partly a myth. In its polishedform mathematics may give those uninitiated this very impression.

However, no part of mathematics has ever been created the way it is presentedin research papers and most textbooks. Pythagoras, for example did not sit down,right hand at forehead, saying, 'Now I'll think of a theorem...oh, yeah...a2 + b2 = c2,yeah!—OK, and now I have to figure out the proof. Pythagoras, rather, did a jig-sawpuzzle, perhaps like the one described in Figure 3.

The truth is that the formulation of a theorem is something that is done afterhaving discovered that in the bunch of scrap paper something has in fact been proved.From the mess—sketches and notes—one picks out the relevant details and presentsthe result in reverse order to that in which it was produced—without mentioningall the wrong tracks, all the failures. It is important for me to convey this idea to mystudents—in particular if, as is usually the case, the textbook does not discuss thisphenomenon.

Speaking of a2 + b2 I come to think of an error that many teachers all over theword have encountered, namely

(a + b)2 = a2 + b2

Is this error due to lack of a mental image of multiplication? What does school

a(b + c)

Figure 4. Standard figure meant to help the learner to grasp the distributive law ofmultiplication over addition. Could be from any textbook on introductory algebra.Could be from any country.

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Mathematics as a didactical adventure 421

ooooooooooooo ooo oo

Figure 5. Some rectangular arrays used as models of multiplication. One of the arrays hassquares as objects. This was our choice in the project mentioned in the text. This modelhas its advantages. It shows, for example, the commutative property of multiplicationin a convincing and evident way.

mathematics offer in that respect? Many textbooks have something like Figure 4when algebra is the topic.

In my experience this picture is easy to grasp for all those who have a solidbackground in multiplication as well as the concept of area. It is, however, hard tograsp for those who have not come to grips with areas, and for those who have notdeveloped the region or rectangle model of multiplication. Thus Figure 4 may betrying to explain to the beginner the difficult statement a(b + c) = ab + ac with theaid of the difficult concept of area. And it may be trying to explain something usinga model that is not known by the learner.

All elementary school teachers should think about this and consciously preparetheir pupils for algebra. I have carried out a multiplication project with 8-9-year-oldsaiming at concept development through mental imagery using the rectangle model.I will say a few words about this didactical adventure.

One of the initial questions of the project was whether pupils can learn thenumerical side of multiplication—i.e. a reasonable amount of multiplication facts,e.g. all the products of one-digit numbers—when teaching emphasis is on conceptformation and development.

Multiplication was, of course, introduced as repeated addition, but we did notdo any skip counting at the beginning stage but went straight on to the rectanglemodel, i.e. arranging objects in rows with the same number of objects in each row.See Figure 5.

Fairly early we decided to choose the square as the basic object to use whenvisualizing or modelling multiplication, like the array to the right in the second rowof Figure 5. Each such rectangle of squares was, in Swedish, called a 'matta', whichmeans 'carpet', and which makes it possible to joke about what we were doing bycalling it 'mattmatik'—'carpetmatics'. If I would do this project in English I wouldcall these rectangles 'mats', in which case one could talk about 'matmatics' andperhaps also talk about 'table mats'. In the sequel of this paper I will talk aboutmultiplication mats.

The choice of object of our rectangular array was not a random choice. The idea

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Figure 6. A systematical way of counting squares of a multiplication mat. One starts off withthe first row, marks all the squares with a certain pattern. Then the number of squaresneeded to fill up to 10 is marked with the same pattern as far away from the first rowas possible. The same procedure is carried out for the second row, etc. Here two rowshave been completed giving us 20. When no more full tens can be obtained one stops,counts the number of full tens and the remaining blanks, and gets the answer.

is that later this same model may be used also with fractions. Other objects may behard, or impossible, to split into fractional parts.

In order to know how many squares there are in a particular mat, 7X8, say, onecan of course count squares one by one. It is well known that such a way of countingis unreliable and has no link to our place value system of enumeration. We thereforeavoided counting one by one. Instead we used the method outlined in Figure 6.

The fortunate thing was that not only could we link this systematical way ofcounting to the place value system but the pupils liked it very much too.

There was one thing that worried me a lot. How would we go about larger mats?What would the counting of squares be like with examples like 7 X 19 or 23 X 68?

The resolution came quite unexpectedly, and I consider this one of the milestonesof my professional life. I would like to share it with the reader.

Figure 7 has a small selection of children's work at a beginning stage.The method worked quite well, and the pupils learned their tables with the aid

of the mats and without traditional drill.We worked with the model without thinking of separate multiplication tables to

begin with. After about 3 months work with all kinds of problems withmultiplications we looked at the possibility of arranging all one-digit multiplicationsin a table. The pupils drew their mats on squared paper, at first with squares withside length 1 cm, later with 0-5 cm squares.

One day one of my classes had a substitute teacher. The week before we hadstarted going beyond 10 with everybody in the class in a serious and systematicalway by allowing one of the factors to be a two-digit number. The pupils had workedwith examples like 8X17. The main idea was to make use of the place value systemand split 17 into 10 and 7. The method is shown in Figure 8.

With this approach there is no need for exercises of the following type:

laying the ground for errors like

62X 3

rors

51X 8

like

74X2

91X6

68X5

43X2

83X3

3040

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Mathematics as a didactical adventure 423

, „ _..., —I—/L— — - ^ — j * -

Figure 7. Examples of beginners' work with multiplication mats.

With the table mats approach there is no need to start with examples withoutcarry digits, since this problem does not arise at all. Therefore the pupils could, andindeed did, make up their own exercises.

I thought that it would be nice for the substitute teacher to see what we had done,and so I asked one of the pupils to go to the chalk board and show an example ofhis own choosing. He chose something like 7 X 24 and his presentation looked likethe one given in Figure 9.

The chalk board did not have any squares, and so it was pretty hard for this pupilto draw parallels. He could not show each separate square in the mat the way he wasused to. The mat he produced was not rectangular. But I was pleased. This pupilhad showed me how we would proceed. From that moment on we would use blankpaper without lines or squares. Our mats would become symbolic. We wouldremember all the time that what we do is count squares, but the actual squares would

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424 A. Dunkels

10 7

808

8x17 = 80 + 56= 136

Figure 8. At this stage the pupils could write down partial results without having to findthe number of squares in our systematical way. The addition is easily performedmentally. The carry digit causes no trouble in this setting—it does not manifest itselfas a carry—and there is no need for designing exercises in particular ways to avoid carrydigits. Instead the pupils can make up their own exercises.

from now on exist in our imagination only. They would not need to be drawnexplicitly.

I realized that some children had taken a huge step towards the situation in Figure4. We had indeed hit the true track from arithmetic to algebra. This was exactly theintermediate step that I had been searching for and not found in the literature.

The work with the symbolic mats went excellently. Some of the pupils choseexamples with larger numbers. I had not demonstrated anything beyond one-digittimes two-digit, and so I was pleased about the way the pupils could manage thisextension by themselves, often in cooperation; Figure 10 shows one example of apupil's work.

With a clear conscience I could decide that there was no need for the pupils ofthe project to learn the traditional algorithm with carry digits. Those readers whohave not tried this model before may perform 23 X 68,165 X 37, 59 X 638, say, usingsymbolic multiplication mats. Gradually the size of the mats can be decreasedmaking them still more symbolic, more like tables rather than actual pictures ofmultiplications.

Later I used mats to make clear the fact that 3a + 5b cannot in general besimplified. A well-known error, revealing week concept attainment, is to write Sab

MJL

Figure 9. A pupil's pencil-and-paper work resembling what a pupil drew on the chalk boardto show what we had done with a one-digit number times a two-digit number. Sincethe board had no grid, he could not follow lines and could not actually show each squarethe way he was used to with squared paper.

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Mathematics as a didactical adventure 425

/6

/OO +?0 +2.0 -hW» 0/

IC

/to

1L4'It

Figure 10. A pupil's way of using a mat to find 12 X 17. The extension to two-digit numberswas found by this pupil on his own. He had seen and used mats with a one-digit numbertimes a two-digit number.

for this expression. Once one understands multiplication mats then the situation,described in Figure 11, is obvious.

The mats 3a and 56 cannot be put together so as to produce a genuine mat, unlesscertain of the numbers 3, a, 5, and b are equal, in which case the two mats 3a and56 would fit along at least one side, for example, with a = 6 we would get 8a. Thisis easily seen by modifying the mat containing 6 in Figure 11. Of course, if we altersome of the sides we may be able to fit the two mats together to make one single mat,e.g. by thinking of the sum as

3

5

a

b

3a

5b

3a

3a

5b

5b

3a

5b

Figure 11. A way of using mats to show that the expression 3a + 5b cannot, in general, besimplified.

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> • 4

' • / \

Figure 12. Two geoboard figures used when introducing the concept of area. The area ofa figure is given by the number of squares that cover the figure exactly. Gradually thepossibility of a fractional number of squares is encountered. The triangle in this figurecan be thought of as half of a 3-square rectangle.

-b=3\a + ^-

but that is not the aim here. Such important transformations would be treated later.What we wish to investigate at this stage is whether the mats 3a and 5b can be puttogether unaltered.

4. AreaThe concepts of multiplication and area are linked through the idea that the area

of a figure equals the number of squares of a particular kind that can be fitted ontothe figure with the squares covering the figure completely. This is particularlytransparent with multiplication seen as a mat. In practice, the squares are, forexample, mm2, cm2, acres, hectares or km2.

In a learning situation with beginners the squares may be the smallest squaresof the geoboard (see Figure 12). Gradually figures that cannot be covered by a wholenumber of squares will be encountered as the concept of area is developed. Laterthe learner reaches the most general situation where areas of figures bounded bycurvilinear sides may be found using more sophisticated methods, for example,integrals.

Of course it is easier for learners with a background in multiplication mats toacquire the ideas behind the integral. In order to define the integral one does nothingbut find the sums of two sequences of mats between which the value of the integrallies, and then uses the concepts of upper bound and lower bound and limits.

In the typical calculus course there is quite a lot of area calculation. The learnereasily gets the impression that areas are indeed one of the most important topics ofmathematics.

To me a milestone in this respect was the study of physics where I seriouslyrealized that areas were not of interest per se, but that the concept of area can be usedas a model of very many situations. We have

distance = speed X timemass = density X volumeforce = pressure X areamomentum = force X lengthwork = force X distance

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Mathematics as a didactical adventure 427

0

Figure 13. The effect as a function of time. When we want to find the energy consumptionduring a tiny time interval then we may consider the effect as constant throughout thatinterval. But in order to get the energy consumption from / = 0 through (= 1 then wemust use an integral.

impulse = force X timevoltage = current X resistanceeffect = voltage X currentweight = mass X gravitational accelerationfriction force = friction coefficient X normal forcearea = height X base

Each of these situations may be given an interpretation as an area. Each of theserelations can be visualized using a multiplication mat, provided that the factors areconstants, of course.

If we have a situation with

f(t) = the effect in watts at time t (hours)

as in Figure 13, and we want to calculate the energy consumption between time(measured in hours) t = 0 and t=\, then we are led to the use of integrals.

This time there is no immediate interpretation in terms of multiplication matsbut in terms of area, a concept that from a didactical viewpoint may build on mats.It sounds now as if I thought that all integrals are areas. Of course I realize that thereis yet another step of abstraction, the learner needs to digest the idea of a Riemannsum. The terms of such sums are products where one factor may be negative, andthey resemble multiplication mats.

Generally we have a line of development that goes straight from multiplicationwith table mats to integrals; from counting the number of squares of a mat via areasand Riemann sums to thinking of infinitesimal mats like dA =/(x) da:.

One of the reasons for learning mathematics is that the subject is general—a singleresult may often be possible to apply in many, as it may seem, different situations.Its results are expressible in short ways. It is international. It is created by humansto serve humans. It is a way of communicating thoughts about phenomena in natureand in our minds.

5. EpilogueFor several years I have asked my engineering students when they start their

university studies what their immediate thought is when they encounter the wordderivative.

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Figure 14. To visualize the derivative of a product of two functions when the first is constant.The change of g(x) causes a change of the rectangle that represents the product functionp(x), and the rate of change of the whole rectangle must be C times the rate of changeof the non-constant side.

The answers I have received over the years indicate that the first thoughts of themajority are procedural—'the exponent comes down and you get one less upthere'—rather than conceptual. Rarely I get answers that involve 'rate of change' or'velocity'. Thus I have to help my students cultivate their conceptual awareness andknowledge. Otherwise they will not be able to use their mathematics appropriatelyin their studies of science and technology. I am not saying that procedural knowledgeis bad or unnecessary. What I mean is that it is not sufficient for a learner to acquireprocedural knowledge alone, a view that is generally held by most people today. Itis absolutely necessary that such knowledge is based on conceptual knowledge andunderstanding. Also with tools such as Mathematica, Derive, etc., available,procedural skills can be done whereas the importance of conceptual knowledgeremains high.

The interplay between procedure and concept intrigues me and makes thedidactics of mathematics an exciting adventure.

It is important, as I see it, that my students develop their intuition about theconcept of derivative and can think oif'(x) in several different ways, one of whichis as the instantaneous velocity of/(x) at time x. Then, among other things, they cangain understanding of the derivative of a product.

Let us look at the product function p defined by p(x) =f(x)g(x), where / and gare given differentiable functions. To simplify matters at first let us take/(*) to beconstant. Then p(x) may be visualized as a multiplication mat as in Figure 14.

Then p'(x) is the instantaneous rate of change of the area of the rectangle. Therate of change of the side g{x) is g'(x), and so for p'(x) we must have Cg'(x).

Turning next to the general case we think again of p(x) as a mat as in Figure 14but with C substituted by the function/(x) as in Figure IS.

The change of p(x) consists of two parts, one due to the change off(x), one dueto the change of g(x). Therefore the rate of change of the whole area, at the instantx, will be the sum of f(x)g'(x) and g(x)f'(x), or, rearranging factors,

(fgy(x)=f(x)g(x)+f(x)g'(x)

This reasoning is of course not a proof but it makes transparent the fact that theexpression for the derivative of a product consists of two terms.

The situation here is like the jig-saw puzzle of the Pythagorean theorem. Thereasoning, the sketches and jig-saws, provide insight, and they can be transformedinto complete proofs.

This is the case for the situation of Figure 15 too. All the ideas necessary for astringent proof are there. The actual process of transforming the diagrams of Figure

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Mathematics as a didactical adventure 429

g'M

Figure 15. To visualize the derivative of a product of two functions when both factors vary.The change of each factor causes a change of the rectangle that represents the productfunction p(x), and the rate of change of the whole rectangle must therefore consist oftwo contributions to be added.

15 into a complete proof is a true didactical challenge and adventure for the learner.Then one must first consider an interval Ax and draw a mat representing/>(# + Ax).I leave the details of this to the reader. I also recommend the reader to try theadventure of using a multiplication mat for the reciprocal function r defined by

Mathematics has many adventures of different kinds to offer. There is no endingto the problems—already posed and not yet posed—one can attempt to solve anddevelop, there are challenging applications to think about, and there is so much aboutthe learning of mathematics to consider. Personally I am fascinated by everythingthat has to do with mathematics, particularly didactical matters, issues of how welearn the subject and how we communicate our thoughts about mathematics to oneanother. I am pleased with the privilege of having found a profession with neverending adventures. I do hope that this paper has given the reader some insight inthe way I conceive mathematics as a didactical adventure.

Ackno wle dgmentsSincere thanks to my friends and colleagues, in alphabetical order, Christer

Bergsten, Gunnar Jakobsson-Ahl, Ruth Maurer, Peter Mogensen, Barbara Reys,Robert Reys, Lennart Rade, and Kerstin Vannman for their constructive and mostvaluable comments.

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