From Addition to Multiplication and Back The Development of Students' Additive and Multiplicative Reasoning Skills

1、running head: from addition to multiplicationfrom addition to multiplication and backthe development of students additive and multiplicative reasoning skillswim van dooren 1, dirk de bock1 2, and lieven verschaffel11 centre for instructional psychology and technology, katholieke universiteit leuven,

2、 belgium2 hogeschool-universiteit brussel, belgiumauthor for correspondencewim van dooren, center for instructional psychology and technology, katholieke universiteit leuven, vesaliusstraat 2, po box 3770 b-3000 leuven, belgium, wim.vandoorenped.kuleuven.be, phone +3216325755, fax +3216326274.abstra

3、ctthis study builds on two lines of research that so far developed largely separately: the use of additive methods to solve proportional word problems and the use of proportional methods to solve additive word problems. we investigated the development with age of both kinds of erroneous solution met

4、hods. they key question is whether and how an overall additive approach to word problems develops into an overall multiplicative approach, and how the transition from the first kind of errors to the second occurs. we gave a test containing missing-value problems to 325 third, fourth, fifth, and sixt

5、h graders. half of the problems had an additive structure and half had a proportional structure. moreover, in half of the problems the internal and external ratios between the given numbers were integer while in the other cases numbers were chosen so that these ratios were noninteger. the results in

6、dicate a development from applying additive methods “anywhere” in the early years of primary school to applying proportional methods “anywhere” in the later years. between these two stages many students went through an intermediate stage where they simultaneously used additive methods to proportiona

7、l problems and proportional methods to additive problems, switching between them based on the numbers given in the problem. 1from addition to multiplicationfrom addition to multiplication and backthe development students additive and multiplicative reasoning skillsintroductionsince several decades,

8、a lot of research has focused on the development of multiplicative reasoning, and more particularly, on the transition from an additive to a multiplicative way of thinking (clark & kamii, 1996). as repeatedly argued by mathematics education researchers, multiplication and division are more than just

9、 a different set of arithmetic operations that are taught after addition and subtraction, and multiplicative thinking ranges much further than merely a faster way of doing repeated addition. although repeated addition often remains an “implicit, unconscious, and primitive model” for multiplication (

10、fischbein, deri, nello, & marino, 1985, p. 4), scholars have stressed that the repeated addition model for multiplication is incomplete, and that a significant qualitative change is required to get from additive to multiplicative thinking (e.g., greer, 1994; nesher, 1988; nunes & bryant, 1996; piage

11、t, grize, szeminska, & bangh, 1977). studies (e.g., nunes & bryant, 1996; squire, davies, & bryant, 2004) have confirmed that children at 8-9 years old perform well on multiplicative tasks about one-to-many correspondence (“every car has 4 wheels, so how many wheels do 6 cars have?”) that can be con

12、ceived as and solved via repeated addition but fail on tasks where multiplication needs to be conceived and handled differently, for example as a cartesian product (“with 6 shorts and 4 t-shirts, how many outfits can i make?”). we will describe the results of a study on how primary school students a

13、pproach multiplicative situations described in word problems, and contrast these with the same students approach of situations that are not multiplicative but additive. our study focuses on a specific subset of additive and multiplicative problems, namely those referring to situations 2from addition

14、 to multiplicationof co-variation. we will clarify these situations, and the difference between additive and multiplicative ones using the following examples: tom and his sister ana have the same birthday. tom is 15 years old when ana is 5 years old. they are wondering how old ana will be when tom i

15、s 75 (= additive situation)rick is at the fish store to buy tuna. the customer before him bought a piece of 250 grams of tuna and had to pay 10 euro. rick needs 750 grams of tuna, and he wonders what he will have to pay (= multiplicative situation)both situations in a certain way deal with co-variat

16、ion: the older ana gets, the older tom gets, and the more fish rick buys, the more he has to pay. but there is an important difference between the two situations. mathematically speaking, the first situation can be described by a function of the form f(x) = x + a, and the second by a function of the

17、 form f(x) = bx, and the reasoning that is required in both situations is very different. the first situation about the age of tom and ana can be called additive, in the sense that the given numbers are linked by the operations of addition and subtraction. there are two variables in the situation (t

18、he age of ana and the age of tom), and the difference between these variables is invariant: adding 10 years to anas age will always provide toms age. the second situation about rick buying tuna is a multiplicative one, and, more specifically, a proportional situation (nunes & bryant, 1996). the two

19、variables in this problem (weight and cost) are linked by multiplication and division with an invariant: knowing that the tuna costs 40 euro per kilogram, multiplying the kilos of tuna with 40 yields the price, and dividing the price by 40 yields the kilos of tuna bought. to summarise, we will focus

20、 on students thinking in additive and multiplicative situations of co-variation. approaches where the difference between two values is considered and added to a third value are called additive, whereas approaches where the ratio between 3from addition to multiplicationtwo values is considered and mu

21、ltiplied with a third value are called multiplicative. two remarks are important here. first, when classifying students approaches to problems, also repeated-addition approaches will be considered multiplicative, because repeated addition also adequately considers the multiplicative relations of a s

22、ituation. second, given our exclusive focus on situations of co-variation, the only multiplicative situations that we consider are proportional situations, and various other multiplicative situations (e.g. cartesian product situations) are left aside. in the rest of the article, we will, therefore,

23、use the terms “proportional” and “multiplicative” as synonyms. we investigated the way students approach situations that are described in a word problem and conceive them as additive or multiplicative. it may happen that students solve word problems that refer to a multiplicative situation erroneous

24、ly in an additive way, and, inversely, solve additive word problems multiplicatively (de bock, 2008). students tendency to approach proportional situations additively instead of multiplicatively has been amply documented in studies of students development of proportional reasoning (e.g., hart, 1981;

25、 lin, 1991; tourniaire & pulos, 1985). at the same time, research has shown that students often tend to use multiplicative approaches beyond their applicability range (e.g., fernndez, llinares, & valls, 2008; modestou & gagatsis, 2007; for a review, see van dooren, de bock, janssens, & verschaffel,

26、2008), including on situations with an additive structure (e.g., van dooren, de bock, hessels, janssens, & verschaffel, 2005). the main goal of the present study is to characterise the development with age of the use of additive and multiplicative models simultaneously. we agree with verschaffel, gr

27、eer, and de corte (2007) that “although separate analyses of the conceptual fields of additive and multiplicative structures will doubtless continue, there is a strong need for a comparative analysis between or a synthesis of these two hitherto rather separate bodies of research” (p. 588). only by 4

28、from addition to multiplicationinvestigating the simultaneous development of students use of additive and multiplicative models rather than studying them in separate studies as mostly happened in the past it is possible to determine whether and how students develop an understanding of the quantitati

29、ve relations underlying additive and multiplicative situations. before going into detail on this study, we provide some theoretical and empirical background concerning each of the overgeneralisation phenomena. we also frame these phenomena in the broader literature on problem solving. theoretical an

30、d empirical backgroundmultiplicative reasoning and the over-reliance on additive methodsbecause of its wide applicability, proportionality takes a pivotal role in primary and secondary mathematics education. full mastery of the multiplicative reasoning that is required in proportional situations is

31、not achieved easily, but several studies have shown that children already at a young age can successfully handle simple proportional situations (e.g., lamon, 1994; spinillo & bryant, 1999), for instance by relying on repeated addition strategies: “if 2 pineapples cost 4 euro, 6 pineapples cost 4+4+4

32、 = 12 euro”. the actual teaching of proportionality generally only starts in the upper elementary (or lower secondary) grades, where students intensively practice proportional reasoning skills with missing-value proportionality problems where three values are given and a fourth is unknown (kaput & w

33、est, 1994), and are confronted with various typical contexts in which proportional reasoning is required (mixtures, costs, currency conversion, ). for example: “grandma adds 2 spoonfuls of sugar to juice of 10 lemons to make lemonade. how many lemons are needed if 6 spoonfuls of sugar are used?” giv

34、en the pivotal role of proportional reasoning in mathematics education, a lot of research has focused on how students acquire proportional reasoning skills, which difficulties 5from addition to multiplicationthey experience, and how it can be enhanced by instruction (e.g., freudenthal, 1973, 1983; h

35、arel & behr, 1989; hart, 1981, 1984; kaput & west, 1994; karplus, pulos, & stage, 1983; lesh, post, & behr, 1988; nunes & bryant, 1996; tourniaire & pulos, 1985; vergnaud, 1983, 1988). among the strategies that students can apply in proportional situations, the literature distinguishes correct multi

36、plicative approaches and erroneous additive approaches. before going into the latter, let us first briefly explain the various correct multiplicative approaches. students approaching the above lemonade problem multiplicatively will most often use a scalar approach (vergnaud, 1983, 1988), focusing on

37、 the internal ratio of sugar to sugar (6 spoonfuls / 2 spoonfuls), and apply this to the number of lemons (3 10 = 30 lemons for 6 spoonfuls of sugar). the alternative is a functional approach (vergnaud, 1983, 1988), focusing on the external ratio of sugar to lemon juice (10 lemons / 2 spoonfuls of s

38、ugar 6 5 = 30 lemons are needed). a variant of the functional approach is the unit factor approach (vergnaud, 1983, 1988), which goes first to the unit value of one of the quantities (e.g., 10 lemons for 2 spoonfuls of sugar 5 lemons for 1 spoonful of sugar 5 6 = 30 lemons are needed). finally, stud

39、ents can approach the situation by a more elementary approach, that could be called building up or replication: for 2 + 2 + 2 spoonfuls of sugar, 10 + 10 + 10 lemons are needed. it is clear that this approach for solving missing-value proportionality problems is based on the repeated-addition charac

40、ter of multiplication, and therefore has characteristics of additive reasoning. nevertheless, we categorise it as multiplicative, as it appropriately handles the multiplicative character of the problem situation. besides these correct multiplicative approaches, there is one erroneous approach that h

41、as received a lot of attention in the literature: the additive one, whereby the relationship between given values is computed by subtracting one value from another, and applying the difference to the third one. for example, in the lemonade problem above, students reason that 6from addition to multip

42、licationfor the second mixture there are 6 2 = 4 spoonfuls of sugar more, so 10 + 4 = 14 lemons are needed. research has identified both subject- and task-related factors that influence the occurrence of such additive errors on proportional problems. as an example of the former, this kind of error i

43、s more typical for younger children with limited instructional experience with the multiplicative relations in proportional situations. but also after instruction, additive errors still occur, particularly on more difficult proportional problems. an important task-related factor in preventing additi

44、ve errors is when the rates (external ratios) in the problem have a dimension that is familiar to students, e.g., speed in kilometres per hour, cost in price per unit (karplus et al., 1983; vergnaud, 1983). another task-related factor strongly related to the occurrence of additive errors and that wi

45、ll be central in the present study is when the numbers given in the problem form non-integer ratios (hart, 1981; kaput & west, 1994; karplus et al., 1983; lin, 1991; tourniaire & pulos, 1985). for instance, when the lemonade problem mentioned before is transformed into “grandma adds 2 spoonfuls of s

46、ugar to juice of 5 lemons to make lemonade. how many lemons are needed if 3 spoonfuls of sugar are used?” it becomes more difficult to execute the multiplicative operations because the ratios are not integer (5/2 and 3/2), and students will therefore more often fall back to erroneous additive reason

47、ing (2+1 spoonfuls of sugar in the lemonade, so 5+1 lemons). research on the overuse of proportionalitybesides the extensive body of evidence of students reasoning additively in multiplicative situations, other lines of research have indicated that students are also inclined to apply multiplicative

48、methods outside their applicability range. especially for nonproportional problems presented in a missing-value format, students tend to erroneously apply multiplicative methods. this has been shown in various domains of mathematics, including elementary arithmetic, geometry, probability, or algebra

49、ic generalisation (for a 7from addition to multiplicationreview, see van dooren et al., 2008). for instance, many students answer “2/6” to the probabilistic problem “the chance of getting a six when rolling a fair die is 1/6. what is the chance of getting at least one six when you roll the die twice

50、?” (van dooren, de bock, depaepe, janssens, & verschaffel, 2003). particularly relevant for the present study is the erroneous application of proportional methods to problems with an additive structure. van dooren et al. (2005) gave a test containing both proportional and various kinds of nonproport

51、ional word problems to large groups of third to eighth graders. among the nonproportional problems, there were additive problems including the following: ellen and kim are running around a track. they run equally fast but ellen started later. when ellen has run 4 laps, kim has run 8 laps. when ellen

52、 has run 12 laps, how many has kim run?generally speaking, and in line with the research on proportional reasoning summarised above, their study showed that students in the early years of primary school already could provide correct answers to proportional word problems, but performance on the propo

53、rtional problems further improved until eighth grade, with most learning gains being made between third and fifth grade, i.e., the years in which this is instructed in classrooms. however, already before the start of formal instruction in proportionality students also gave proportional answers to th

54、e nonproportional word problems: in third grade, 30% of all nonproportional word problems were answered proportionally, and this increased considerably until sixth grade. for the additive problem mentioned above, the percentage of wrong proportional responses (“24 laps”) increased from 10% in third

55、grade to more than 50% in sixth grade whereas correct additive answers (“16 laps”) decreased from 60% in third grade to 30% in sixth grade (and then increased again to 45% in eighth grade).8from addition to multiplicationbuilding further on this observation, a recent study by van dooren, de bock, ev

56、ers, and verschaffel (2009) with fourth to sixth graders showed that just like the tendency to reason additively in multiplicative situations is affected by the fact that given numbers do not form integer ratios (e.g., kaput & west, 1994) the nature of the given numbers also affects students tendenc

57、y to use multiplicative strategies in situations that are not multiplicative. solutions to additive word problems where the numbers formed integer ratios, such as the runner problem mentioned above, were compared with problems where the numbers formed non-integer ratios. a variant of the above-menti

58、oned runner problem above with non-integer ratios (6/4 and 10/4) would be ellen and kim are running around a track. they run equally fast but ellen started later. when ellen has run 4 laps, kim has run 6 laps. when ellen has run 10 laps, how many has kim run? it was found that for the latter non-int

59、eger variants students were less inclined to use proportional strategies, and therefore performed better than on the versions with integer ratios between numbers. this was particularly the case in fourth graders. apparently, the fifth, and especially sixth graders had already become so skilled in do

60、ing proportional calculations involving non-integer ratios that they therefore also overused these skills. combining the two research lines described so far, we endorse the claim by cramer, post, and currier (1993) that “we cannot define a proportional reasoner simply as one who knows how to set up