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

running head from addition to multiplication from addition to multiplication and back the development of students additive and multiplicative reasoning skills wim van dooren 1 dirk de bock1 2 and lieven verschaffel1 1 centre for instructional psychology and technology katholieke universiteit leuven belgium 2 hogeschool universiteit brussel belgium author for correspondence wim van dooren center for instructional psychology and technology katholieke universiteit leuven vesaliusstraat 2 po box 3770 b 3000 leuven belgium wim vandooren ped kuleuven be phone 3216325755 fax 3216326274 abstract this 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 methods 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 sixth 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 indicate 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 proportional problems and proportional methods to additive problems switching between them based on the numbers given in the problem 1from addition to multiplication from addition to multiplication and back the development students additive and multiplicative reasoning skills introduction since several decades 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 nesher 1988 nunes piaget grize szeminska squire davies lin 1991 tourniaire modestou for a review see van dooren de bock janssens spinillo harel hart 1981 1984 kaput karplus pulos lesh post nunes tourniaire vergnaud 1983 1988 among the strategies that students can apply in proportional situations the literature distinguishes correct multiplicative 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 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 sugar 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 students 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 character 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 has 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 multiplication for 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 is 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 additive 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 will be central in the present study is when the numbers given in the problem form non integer ratios hart 1981 kaput karplus et al 1983 lin 1991 tourniaire hinsley hayes reusser verschaffel de corte verschaffel 10from addition to multiplication greer eckenstam fischbein deri nello greer 1987 hart 1981 sowder 1988 as such the detection of predictive correlations between a problem s surface structure and its solution procedure can be very effectual allowing a fluent solution of problems without going through laborious problem solving steps but as argued by ben zeev and star 2001 the detected correlations can be spurious and then problem solving experience can become ineffectual ben zeev and star 2001 gave experienced students several sets of algebraic equations along with algorithms to solve them they showed that these students were susceptible to experimentally induced spurious correlations between irrelevant characteristics of the equations and the algorithm that is used to solve them their results also indicated that students were not necessarily aware that they were responsive to such a spurious correlation it became part of their implicit knowledge when asked directly students indicated that they were choosing randomly between two strategies but actually they were responding systematically to spurious correlations for proportional reasoning the missing value formulation of a word problem is probably the most salient feature for students the majority of the proportional reasoning tasks that students encounter in the upper grades of elementary school and in the lower grades of secondary school are formulated in a missing value format cramer et al 1993 and a lot of attention is paid to the development of fluency in solving such problems at the same time nonproportional problems stated in a missing value format are very rare this may explain why van dooren et al 2005 observed a considerable increase in the number of proportional answers to nonproproportional word problems throughout primary school a similar explanation could be given for the fact that students can associate even the number 11from addition to multiplication characteristics of word problems with a certain solution procedure when students are first introduced in proportionality word problems usually contain numbers that allow calculations with easy multiplicative jumps this way students can focus on recognizing and working through the proportional structure of the situation and applying the taught procedures therefore a spurious correlation association as proposed by ben zeev therefore we can really speak of the overuse of approaches research questions the previous section started with two lines of research that so far developed largely separately relating to the tendency to approach proportional situations additively and the tendency to approach to additive situations proportionally typically these tendencies and more generally even the abilities to reason additively and multiplicatively are not studied simultaneously in the same students nevertheless it is interesting to investigate how both abilities and both types of errors develop over age and more importantly whether it is possible that both seemingly opposite overgeneralisations can occur at the same time in individuals for example in the transition phase from one kind of overgeneralisation to the other as explained before only by examining additive and multiplicative reasoning 12from addition to multiplication simultaneously it is impossible to determine the actual reasoning abilities of students to understand the quantitative relations that distinguish additive from proportional situations based on the available literature reported in the previous section we first of all anticipated a development with age from an overall additive approach to missing value problems consisting of the correct use of additive methods in additive situations and the incorrect use in multiplicative situations towards an overall multiplicative approach involving the correct use of multiplicative methods in multiplicative situations and the incorrect use in additive situations a key question however was how the development from an additive to a multiplicative approach would look like and more specifically how the transition could be characterised would students in the transition tend to apply additive and multiplicative methods appropriately to additive and multiplicative problems if not would they choose randomly for additive or multiplicative methods or would they rely on superficial problem characteristics the literature on additive and proportional reasoning summarised above suggests that if students in the transition rely on irrelevant problem characteristics they will most likely consider the numbers given in word problems and use multiplicative methods when the ratios between given numbers are integer and additive methods when ratios between given numbers are non integer this would imply that students in this intermediate stage at the same time use multiplicative methods in additive situations i e when the ratios between given numbers are integer and use additive methods in multiplicative situations when the ratios between given numbers are non integer method participants students from third to sixth grade from two different primary schools in flanders participated in the study 88 third graders 78 fourth graders 81 fifth graders and 78 sixth 13from addition to multiplication graders one school was situated in a middle sized city the other in a smaller village both schools were average in size and attracted students from mixed socioeconomic backgrounds mainly from the immediate neighbourhood the sample consisted of approximately equal numbers of boys and girls the educational standards in flanders ministerie van de vlaamse gemeenschap 1997 indicate that by the end of sixth grade students should be able to compare the equality of two ratios and calculate the missing value when confronted with a missing value proportionality problem even though schools in flanders use a variety of textbooks the general instructional approach and the timing for the teaching of proportional missing value problems is very similar in second and third grade the focus is on solving simple multiplication word problems e g 1 pineapple costs 2 euro how much do 3 pineapples cost in fourth grade this focus gradually shifts toward solving proportional missing value problems typically referring to contexts such as unit price weight price and time distance e g 12 eggs cost 2 euro what is the price of 36 eggs these missing value proportional problems are further rehearsed in fifth and sixth grade and some new application contexts are introduced as well e g currency exchanges mixtures in recipes or paints in sixth grade some attention is also paid to tackling word problems with larger and or rational numbers and the use of a pocket calculator to do so and students learn how to solve problems with noninteger ratios usually by means of a unit factor approach mathematics textbooks for primary school and secondary school do not pay attention to contrasting proportional and nonproportional missing value problems materials all students solved four experimental word problems the design of these word problems is explained and illustrated in table 1 14from addition to multiplication insert table 1 here two of the word problems were proportional problems for which proportional calculations i e finding the value of x in b a x c lead to the correct answer the other two were additive word problems for which additive calculations i e finding x in b a x c are required as can be seen in table 1 proportional and additive problems were formulated similarly the crucial difference between proportional and additive situations lies in the second sentence for example for the integer additive problem in table 1 the additive character of the situation lies in the fact that both girls run at the same speed but one started later implying that the difference in laps between both girls remains constant the problem can be easily turned into a proportional one by changing the second sentence into ellen and kim are running around a track they started together but kim runs faster when ellen has run 4 laps kim has run 8 laps when ellen has run 12 laps how many has kim run we also experimentally manipulated the number characteristics of the word problems for two of the problems i e one additive and one proportional the given numbers were chosen so that when doing proportional calculations one has to work with integer ratios i version for the other two problems again one additive and one proportional the numbers formed non integer ratios n version in the latter case care was taken however that the outcome of proportional calculations still would be integer this way we wanted to avoid that students would start to doubt about the correctness of their calculations just because they obtain a non integer outcome eight different test variants were constructed one test variant included the 15from addition to multiplication experimental items as they are shown in table 1 the other variants were created by reformulating the additive problems as proportional problems and vice versa reformulating the non integer variants as integer variants and vice versa or combinations of these this way any uncontrolled variance in our results e g due to the different contexts dealt with in the word problems would be cancelled out procedure in order to be able to detect individual student profiles students should solve at least one variant of each of the four experimental items at the same time it was very important that students would not become aware of the goal of the study and that their response on one experimental item would not influence their behaviour on another one therefore we limited ourselves to offering only four experimental word problems per student the four experimental items were moreover embedded in two larger tests each test contained 15 problems on a wide variety of mathematical topics that were related to the students school curriculum mixed among these 15 buffer items each test had two of the experimental items one proportional word problem i or n version and one additive word problem i or n version the other test then contained the other two experimental items also mixed among 15 buffer items both tests were administered with one week in between students were told that the tests were meant to assess their progress in mathematics in general no further instructions were given as to how to solve the problems except for the fact that a pocket calculator could be used and that we explicitly asked students to record their calculations on the answer sheets results 16from addition to multiplication in a first stage we will look at the responses to the four experimental items separately and consider the extent to which they are affected by students age the additive or proportional character of the experimental word problems and the number characteristics integer non integer of the word problems in a second stage we will look at students solution profiles to the four experimental items together general results responses to the experimental word problems were classified as proportional answer when proportional operations were executed on the given numbers i e calculating x in the expression b a x c additive answer when additive operations were executed i e finding x in b a x c or other answer when the given numbers were combined in another way with arithmetic operations than specified above or when the problem was left unanswered when purely technical calculation errors e g 8 2 14 were committed the answer was not necessarily scored as other if the calculations were clearly proportional or additive we labelled them as such table 2 shows the solutions given by students to the four experimental items taken as a whole the results confirm those of earlier studies by van dooren et al 2005 2009 first of all the proportional word problems elicited significantly more proportional responses 32 3 than the additive word problems 26 2 1 293 535 p 0 0001 and the additive problems elicited significantly more additive responses 56 9 than the proportional problems 44 2 1 42 081 p 0 0001 the differences were not very large however and the results presented in table 2 clearly indicate an overgeneralisation in both directions 17from addition to multiplication students often used proportional strategies on additive problems and additive strategies on proportional problems insert table 2 here second students solutions were clearly affected by the number characteristics of the word problem regardless of the problem type problems in which the numbers form inte