Proposal view
Proposal Type: Individual Paper 
Domain: Assessment and Evaluation 
SIG: Assessment and Evaluation 
Type Submitted Paper 
Equipment Slide projector
Paper Details
Title The free fall of Dutch written arithmetic: what can solution strategies tell us?
Abstract
In the Netherlands, achievement on problems of written arithmetic shows a steady decline over the last two decades. This is reported in the fourth consecutive large-scale national assessment of the level of mathematical competence at the end of primary school.

By investigating the strategies pupils used in solving problems of written division, this study aims at contributing to the explanation of this decline. For the assessments of 1997 and 2004, strategies for solving written division problems were studied and classified as being either traditional long division, or some form of a so-called realistic strategy. Furthermore, it occurred quite often that pupils stated an answer, without writing down any calculations.

Results show that two important changes took place from 1997 to 2004. Firstly, fewer pupils solved the problems by the traditional algorithm in 2004 than in 1997. That could be expected because of the Dutch reform in mathematics education, the Realistic Mathematics Education (RME). However, the amount of pupils using realistic strategies did not increase accordingly, but was quite stable. Instead, more and more pupils stated an answer without working, which turned out to be a quite unsuccessful approach. Secondly, traditional and realistic strategies and stating an answer without working were all significantly less successful in 2004 than in 1997. So, pupils not only shifted to a less successful strategy, they also were less successful in using all strategies.
Summary
Introduction

In the Netherlands, four consecutive large-scale national assessments of achievement in mathematics at the end of primary school have been conducted. Achievement in written arithmetic shows a steep decline since 1987 (Janssen, Van der Schoot, & Hemker, 2005). Pupils are now less successful in solving problems on written addition, subtraction, multiplication, division, and combinations of those procedures than two decades ago.

By investigating the solution strategies pupils used to solve several written division problems, this study aims at contributing to the explanation of this decline in achievement. The focus is on the results for written division of the 1997 and 2004 assessments.

 

Method

In the 1997 assessment, 10 different division problems were administered to a total of 574 pupils (Janssen, Van der Schoot, Hemker, & Verhelst, 1999). In 2004, there were 13 division problems, but each of the 1044 pupils only solved a subset of 3 to 8 of these problems (Janssen et al., 2005). Four problems were part of both assessments, so linking of the results was possible through these common problems.

Looking into the pupils’ written work, their solution strategies were classified, in a similar way as Van Putten, Van den Brom-Snijders, and Beishuizen (2005) did. Four main categories were distinguished. Firstly, pupils solved division problems with a traditional long division algorithm. Secondly, so-called realistic strategies were observed, in which manifolds of the divisor were repeatedly subtracted from the dividend or were repeatedly added.In the appendix, examples of both strategies are shown. Thirdly, it occurred quite often that pupils stated an answer, but did not write down any calculations. Finally, a category remained including unclear strategies, wrong procedures, and skipped problems.

 

Results

Changes in frequency of the used strategies between the two assessments were studied first. From latent class analyses on the strategy categories for the 4 common problems (Vermunt, 1997), a model with four latent classes showed the best fit. Year of assessment was included as covariate, so the differences between 1997 and 2004 can be reported. In 1997, 42% of the pupils belonged to the class with a high tendency to solve problems by the traditional algorithm; in 2004 this was reduced to only 13%. Furthermore, the class with a high tendency to state an answer without working amounted 16% of the pupils in 1997, and increased to 43 % in 2003. The percentage of pupils in the other two classes, one characterized by a high tendency to solve the problems with a realistic strategy, and the other by a combination of a tendency to show no working or to be classified in the remainder category, did not differ much between 1997 and 2004.

So, pupils solved problems with the traditional algorithm less often, as could be expected from the reform in Dutch mathematics education, the Realistic Mathematics Education (RME, Treffers & Beishuizen, 1999). However, the number of pupils solving division problems with a realistic strategy did not increase accordingly. Instead, it seems pupils shifted to not writing down any calculations in solving these problems.

Secondly, the effect of using a particular strategy on the probability to reach a correct answer was assessed by explanatory IRT-analyses (De Boeck & Wilson, 2004). In 1997 traditional strategies led to significantly more success than realistic strategies. This difference disappeared in 2004. Furthermore, both in 1997 and 2004 problems answered without working were less often solved correct than problems solved either by realistic strategies or by traditional strategies. Moreover, traditional and realistic strategies and stating an answer without working were all significantly less successful in 2004 than they were in 1997.

 

Conclusion and discussion

Two changes contributed to the large decline in written division from 1997 to 2004. Pupils not only shifted to more unsuccessful strategies, they were also less successful in using all sorts of strategies. Especially the large group of pupils showing no working in 2004 is worrisome and requires further investigation. Perhaps this is a negative side effect of the implementation of the RME, since traditional strategies appear not to be replaced by realistic ones, but by no working instead. In addition, maybe teachers should emphasize to their pupils the importance of writing down calculations when solving this kind of problems.

Future research is planned into the strategy use in the other domains of written arithmetic, and into the phenomenon of showing an answer without working.

 




References

De Boeck, P., & Wilson, M. (Eds.) (2004). Explanatory Item Response Models: A Generalized Linear and Nonlinear Approach. New York, NY: Springer.

Janssen, J., F. Van der Schoot, F., & Hemker, B. (2005). Balans van het reken- wiskundeonderwijs aan het einde van de basisschool 4 [National assessment of mathematics education at the end of primary school 4]. Arnhem: Cito.

Janssen, J., Van der Schoot, F., Hemker, B., & Verhelst, N.D. (1999). Balans van het reken- wiskundeonderwijs aan het einde van de basisschool 3 [National assessment of mathematics education at the end of primary school 3]. Arnhem: CITO.

Treffers, A. & Beishuizen, M. (1999). Realistic Mathematics Education in the Netherlands. In I. Thompson (Ed.), Issues in teaching numeracy in primary schools (p. 27-38). Buckingham: Open University Press. 

Van Putten, C.M., Van den Brom-Snijders, P.A., & Beishuizen, M. (2005). Progressive Mathematization of Long Division Strategies in Dutch Primary Schools. Journal for Research in Mathematics Education, 36, 44 – 73.

Vermunt, J.K. (1997). LEM 1.0: A general program for the analysis of categorical data. Tilburg: Tilburg University.
Keywords Cognitive skills
Large-scale national assessment projects
Mathematics education
Appendices figure1_EARLI.JPG 
Authors
Name Surname Institution Country e-mail EARLI Number Presenting
Marian Hickendorff Leiden University Netherlands hickendorff@fsw.leidenuniv.nl   *  
Cornelis M. van Putten Leiden University Netherlands putten@fsw.leidenuniv.nl    
Willem J. Heiser Leiden University Netherlands heiser@fsw.leidenuniv.nl    
Norman D. Verhelst CITO, Arnhem Netherlands norman.verhelst@cito.nl    
Visit NQcontent
© European Association for Research on Learning and Instruction, 2012 All rights reserved.