Differentielle Entwicklung arithmetischer Fähigkeiten nach der Grundschule: Manche Schere öffnet und schließt sich wieder
Abstract
Zusammenfassung: Nach der Grundschule differenziert das deutsche Schulsystem zwischen unterschiedlichen Schulformen (Hauptschule, Realschule, Gymnasium oder Gesamtschule). Verschiedene Studien belegen, dass sich der Lernfortschritt zwischen den Schulformen der Sekundarstufe unterscheidet. Die Leistungsunterschiede zwischen den Schulformen scheinen dabei immer größer zu werden (Schereneffekt), jedoch ist die Befundlage dazu nicht immer einheitlich. Speziell für die mathematischen Kompetenzen gibt es divergierende Ergebnisse.In der vorliegenden Studie wurde einer möglichen Ursache der inkonsistenten Ergebnisse bisheriger Studien nachgegangen indem inhaltliche Unterschiede zwischen den verwendeten Aufgaben vermieden wurden. Dazu wurden von 415 SchülerInnen der 5. und 6. Klasse an Haupt- und Realschulen deren arithmetische Basiskompetenzen in den vier Grundrechenarten erhoben. Die Ergebnisse zeigen zum einen konsistent, dass Realschüler in den Aufgaben besser abschnitten als Hauptschüler. Jedoch ergaben sich differentielle Ergebnisse hinsichtlich der Leistungsentwicklung an den beiden Schulformen abhängig von der jeweiligen Grundrechenart. Während für Additionsaufgaben sogar eine Verringerung des Leistungsunterschiedes zwischen Haupt- und Realschule von der 5. zur 6. Klasse beobachtet werden konnte, wurde nur für schwierige Divisionen eine Vergrößerung des Leistungsunterschieds beobachtet. Wir schließen daraus, dass die zur Untersuchung differentieller Leistungsentwicklung an verschiedenen Schulformen verwendeten Aufgaben einen bedeutsamen Einfluss auf die beobachteten Ergebnisse haben können und damit eine mögliche Erklärung für die bisher inkonsistenten Ergebnisse sein könnten. Es wird angeregt die eingesetzten Aufgaben in zukünftigen Studien besser zu kontrollieren.
Basic Arithmetic Competencies in Secondary School: Are there Differential Achievement Gains?
Background: The German school system differentiates between different school types after the end of primary school. Recent research has shown different learning gains for the different school types with the dominant view indicating increasing achievement gaps between school types with increasing grade level. However, there is an ongoing debate whether this holds for mathematical skills as the empirical evidence is inconsistent. On the one hand, there are studies which did not observe differential achievement gains whereas there are also studies in which achievement gains differ between school types. When reviewing the tasks used to evaluate the differential achievement we wondered whether the divergent findings may stem from the fact that different tasks were used to assess mathematics proficiency. These tasks covered a wide range of different mathematical abilities (e. g., word problems, fractions, arithmetic strategies, etc) and differed between as well as within studies.
Aims: To get a more detailed picture of the development of mathematical abilities in secondary school, we took another approach and focused on tasks tapping the four basic arithmetic operations. Unlike more complex tasks such as word problems, for example, basic arithmetic operations allow for identifying basic numerical representations which are to be recruited to solve these operations. Apart from the basic numerical representations as proposed in the Triple Code Model of Dehaene and Cohen [1995; a) visual number form, b) verbal representation of numbers and, c) magnitude representation of numbers], we were interested in two more representations as introduced by Nuerk, Graf and Willmes (2006) strategic, conceptual and procedural components primarily involved in arithmetical tasks as well as a representation of the place-value structure of the Arabic number system.
For addition and subtraction problems, we were mainly interested whether specific demands on the place-value representation in terms of the carry and borrowing effect differ between 5th and 6th grade and between school types. On the other hand, single-digit multiplication problems are assumed to be solved by fact retrieval (which is supposed to be part of the verbal representation of numbers). Easy division problems are predominantly solved by recasting the division problem as an inverse multiplication problem at this age. Thus, fact retrieval and thereby, the verbal representation of numbers should be most important representation when solving easy division tasks as well. Finally, more complex, for instance two-digit multiplication as well as difficult division problems should primarily rely on procedural and strategic processes. However, scientific knowledge on these operations is still sparse. Taken together, the differentiation between easy and difficult problems for each of the four basic arithmetic operations (i. e., addition/subtraction without vs. with carry/borrowing; single- vs. two-digit multiplication/division should allow us to investigate the influence of basic numerical representations on the question of differential achievement gains between school types.
Methods: Therefore, we assessed arithmetic abilities of 415 5th and sixth 6th graders from 20 classes of two different school types (general secondary school, i. e., “Hauptschule”, and intermediate secondary school, i. e., “Realschule”). Importantly, the tasks employed focused on the four basic arithmetic operations: addition, subtraction, multiplication and division. Furthermore, for each of the basic arithmetic operations we designed easy and difficult tasks. The easy addition and subtractions problems did not require a carry or borrowing operation, respectively, whereas half of the difficult addition and subtractions problems did require a carry or borrowing operation to come to the correct solution. Easy multiplications were composed of two single-digit numbers whereas in difficult multiplication problems one of the multiplicands was a two-digit number and the other one either a single-digit or a two-digit number. The items for the easy division task were created as inverse single-digit multiplication problems (e. g., 3 x 4 = 12 → 12 : 4 = 3). In difficult division problems, the quotient was always a two-digit number. All tasks were administered as speeded tests with a strict time limit to rule out ceiling effects. Furthermore, we assessed cognitive functioning (by the subtest “matrices” of the intelligence test CFT-20-R, Weiß, 2006) and verbal working memory (by students initial memory span for the 15 words of the verbal learning and memory test VLMT, Helmstaedter, Lendt, & Lux, 2001). All of these tasks were completed within one school lesson (i. e., 45 minutes).
Results: Interestingly, initial analyses indicated that verbal working memory as well as non-verbal intelligence had no significant impact on the differences between school types and grades and were thus not considered in the final analyses. In general, we observed differences between school types for all arithmetic operations with children attending intermediate secondary school outperforming children attending general secondary school. Moreover, for all tasks we found that 6th graders solved more problems correctly than did 5th graders. However, differences between school types were not comparable between 5th and 6th grade. While students attending general secondary school were even able to catch up with students attending intermediate secondary school in addition and to a lesser degree for subtraction tasks, the performance difference in the difficult division task indeed increased from 5th to 6th grade. Finally, for multiplication and easy division tasks performance differences between students attending secondary and intermediate general school were comparable in 5th and 6th grade.
Discussion: Taken together, the results of the current study were partly in line with both the notion of increasing achievement gaps (for the difficult division task, cf. Becker et al., 2006) as well as the assumption of no differential achievement gains between different school types (cf. Schneider & Stefanek, 2004) for the multiplication and easy division tasks. Most importantly, however, for addition and subtraction tasks we observed that the difference between students attending general and intermediate secondary school even decreased from 5th to 6th grade. With regard to the underlying basic numerical representations our results were somewhat mixed. On the one hand, manipulating task difficulty by means of the need for a carry/borrowing operation in addition/subtraction, and thus addressing the place-value representation, did not seem to be meaningful with respect to differential achievement gains between school types. For both tasks and both difficulty levels the performance difference between general and intermediate secondary school decreased from grad 5 to grade 6. On the other hand, in all tasks in which the gap between school types did not change between grade levels retrieval of multiplication facts and thus the verbal representation of numbers is assumed to be the predominant way of solving. This was not only true for multiplication tasks, but also for easy division problems, as indicated by a very high correlation between the performance for easy multiplication and division problems. This high correlation suggested that students indeed relied on the strategy of recasting the division problem as a multiplication problem. Interestingly, we found that the difficult division task was the most difficult task of our assessment. Since easy and difficult divisions can be assumed to differ majorly in the involved strategic, conceptual and procedural processes (no more fact retrieval for these problems), this finding indicates that the increasing gap between school types may be a consequence of more advanced solution strategies of students attaining intermediate secondary school.
From these differential findings for the four basic arithmetic operations we conclude that differences in the tasks employed between and within recent studies may be a possible reason accounting for the so far inconsistent results on differential achievement gains in German secondary schools. It is suggested that future studies should pay particular attention to the controlling of assessment tasks.
Literatur
(2004). Einblicke Mathematik 1. Stuttgart: Klett Verlag.
(1996). “The TIMSS Test Design”, in M.O. Martin and D.L. Kelly (Eds.), Third International Mathematics and Science Study (TIMSS) Technical Report, Volume I: Design and Development.. Chestnut Hill, MA: Boston College.
(1986). Working memory. Oxford: Oxford University Press.
(2005). Sozialer Hintergrund, Bildungsbeteiligung und Bildungsverläufe im differenzierten Sekundarschulsystem. In V. Frederking, H. Heller & A. Scheunpflug (Hrsg.), Nach PISA. Konsequenzen für Schule und Lehrerbildung nach zwei Studien (S. 9– 21). Wiesbaden: VS Verlag für Sozialwissenschaften.
(2006). Schulstruktur und die Entstehung differenzieller Lern- und Entwicklungsmilieus. In J. Baumert, P. Stanat & R. Watermann (Hrsg.). Herkunftsbedingte Disparitäten im Bildungssystem, (S. 95– 186). Wiesbaden: VS Verlag für Sozialwissenschaften.
(2003). Schulumwelten: Institutionelle Bedingungen des Lehrens und Lernens. In J. Baumert, C. Artelt, E. Klieme, M. Neubrand, M. Prenzel, U. Schiefele, W. Schneider, K.-J. Tillmann & M. Weiß (Hrsg.), PISA 2000: Ein differenzierter Blick auf die Länder der Bundesrepublik Deutschland (S. 259– 330). Opladen: Leske + Budrich.
(2006). Leistungszuwachs in Mathematik. Evidenz für einen Schereneffekt im mehrgliedrigen Schulsystem?. Zeitschrift für Pädagogische Psychologie, 20, 233– 242.
(2008). Low working memory capacity impedes both efficiency and learning of number transcoding in children. Journal of Experimental Child Psychology, 99, 37– 57.
(2008). Das Bildungswesen in der Bundesrepublik Deutschland. Strukturen und Entwicklungen im Überblick. Reinbeck: Rowohlt.
(1995). Towards an anatomical and functional model of number processing. Mathematical Cognition, 1, 83– 120.
(2003). Three parietal circuits for number processing. Cognitive Neuropsychology, 20, 487– 506.
(2004). The role of working memory in mental arithmetic. European Journal of Cognitive Psychology, 16, 353– 386.
(2006). Der Übergang von der Grundschule in die Sekundarstufe I. Zeitschrift für Erziehungswissenschaft, 9, 348– 372.
(2005). Some assumptions and facts about arithmetic facts. Psychology Science, 47, 96– 111.
(2002). Counting on working memory in simple arithmetic when counting is used for problem solving. Memory and Cognition, 30, 447– 455.
(2001). Verbaler Lern-und Merkfähigkeitstest. Göttingen: Beltz Test GmbH.
(2007). The role of working memory in the carry operation of mental arithmetic: Number and value of the carry. The Quaterly Journal of Experimental Psychology, 60, 708– 731.
(2007). The role of working memory in carrying and borrowing. Psychological Research, 60, 708– 731.
(2006). How specifically do we learn? Imaging the learning of multiplication and subtraction. NeuroImage, 30, 1365– 1375.
(2009). TEDI-MATH 4 – 8: Test zur Erfassung numerisch-rechnerischer Fertigkeiten für 4 – 8 Jährige. Bern: Huber.
(2001). Leistungsgruppierungen in der Sekundarstufe 1. Ihre Konsequenzen für die Mathematikleistung und das mathematische Selbstkonzept der Begabung. Zeitschrift für Pädagogische Psychologie, 15, 99– 110.
(2002). Entwicklung schulischer Leistungen. In R. Oerter & L. Montada (Hrsg.). Entwicklungspsychologie: Ein Lehrbuch (S. 756– 786. Weinheim: Psychologie Verlags Union.
(2004). Mathematik konkret 1. Realschule Baden-Württemberg.. Berlin: Cornelsen.
(2004). LAU 11. Aspekte der Lernausgangslage und Lernentwicklung, Klassenstufe 11. Ergebnisse einer Längsschnittstudie. Hamburg: Behörde für Bildung und Sport.
(im Druck). Diagnostics and intervention in dyscalculia: Current issues and novel perspectives. In D. Molfese, Z. Breznitz, & O. Rubinsten (Eds.). Reading, writing, mathematics and the developing brain: Listening to many voices. Dordrecht, Netherlands: Springer.
(2011). Three processes underlying the carry effect in addition – Evidence from eye tracking. British Journal of Psychology, 102, 623– 645.
(2009). Zahlenverarbeitung ist nicht gleich Rechnen – eine Beschreibung basisnumerischer Repräsentationen und spezifischer Interventionsansätze. Prävention und Rehabilitation, 21, 121– 136.
(2011). Early place-value understanding as a precursor for later arithmetic performance – a longitudinal study on numerical development. Research in Developmental Disabilities, 32, 1837– 1851.
(2007). Schulformen als differenzielle Lernmilieus. Zeitschrift für Erziehungswissenschaft, 10, 399– 420.
(2006). Grundlagen der Zahlenverarbeitung und des Rechnens. Sprache, Stimme, Gehör, 30, 147– 153.
(2011). Extending the Mental Number Line – A Review of Multi-Digit Number Processing. Journal of Psychology, 219, 3– 22.
(2010). Schereneffekte im ein- und mehrgliedrigen Schulsystem. Zeitschrift für Pädagogische Psychologie, 24, 259– 272.
(2010). Working memory and mathematics: A review of developmental, individual difference, and cognitive approaches. Learning and Individual Differences, 20, 110– 122.
(2008). Entwicklungen von Lesekompetenz und Lesemotivation. Schereneffekte in der Sekundarstufe?. Zeitschrift für Entwicklungspsychologie und Pädagogische Psychologie, 40, 179– 188.
(2006). Stability and change in children’s division strategies. Journal of Experimental Child Psychology, 93, 224– 238.
(2004). Entwicklungsveränderungen allgemeiner kognitiver Fähigkeiten und schulbezogener Fertigkeiten im Kindes- und Jugendalter. Evidenz für einen Schereneffekt?. Zeitschrift für Entwicklungspsychologie und Pädagogische Psychologie, 36, 147– 159.
(2002). Phonological loop and central executive processes inmental addition and multiplication. Psychologische Beiträge, 44, 275– 302.
(2009). Zur Veränderung der Mathematikleistung von Klasse 4 bis 6. Welchen Einfluss haben Kompositions- und Unterrichtsmerkmale. Zeitschrift für Erziehungswissenschaft, Sonderheft 12, 302– 327.
(2006). Grundintelligenztest Skala 2 – Revision. CFT 20-R. Göttingen: Hogrefe.
(2008). Grundintelligenztest Skala 2 – Revision (CFT 20-R) mit Wortschatztest und Zahlenfolgentest – Revision (WS/ZF-R). Göttingen: Hogrefe.
(2009). On the language-specificity of basic number processing: Transcoding in a language with inversion and its relation to working memory capacity. Journal of Experimental Child Psychology, 102, 60– 77.
