24 SES 02, Teaching Professional Development Part 1
Paper Session to be continued in 24 SES 08 A
It is a truth universally acknowledged that problem solving forms the basis for successful mathematics education. Solving of carefully selected problems helps to develop, refine and cultivate creativity (Kopka, 2010, Foreword; Silver, 1997). (Eysenck, 1993) states that problem solving skills develop fast if the solver gets new experience with it.
Word problems have been identified as an area of concern in many countries. Teachers report that word problems are among those in which pupils show the poorest performance (Rendl, Vondrová et al., 2013) and studies with pupils confirm this assertion (e.g. Hembree 1992; Zohar and Gershikov 2008; Van Dooren et al., 2010; for Czech pupils, Hejný, 2003; Novotná, 2000, etc.)
The contribution will present a part of a large longitudinal research focusing on deeper understanding of the relationships between formal characteristics of word problems and the procedures pupils use for their solution and mistakes they make. Vondrová, Novotná and Havlíčková (submitted) studied the influence of the order of numerical data, context, position of the unknown transformation and the length of the assignment in an additive word problem on the performance and reasoning of primary pupils from grades 4 and 5.
The proposed contribution focuses on the influence of the order of numerical data in the assignment on lower secondary pupils’ performance in solving word problems depending on familiarity/unfamiliarity of the context. The findings are illustrated on two quartets of word problems in which the order of numerical data in the assignments are varied in two different contexts.
The research question: How does the order of numerical data in contexts familiar and unfamiliar to pupils influence achievement of lower secondary pupils when solving word problems?
To answer this research question several quartets of word problems were posed. Each quartet had the same structure: Problem 1 and 2 had a context familiar to pupils, problems 3 and 4 less familiar. Problems 1 and 2 and problems 3 and 4 differed in the order of presentation of numerical data in the assignment. In each pair the change in the order was identical. “Classical” school problems on division of a whole into unequal parts were used. The problems were based on division of the whole into three unequal parts. Their difficulty was adapted to the pupils’ age. The variants were solved by pupils in Grade 6 (N = 182) and 9 (N = 282) in four Prague schools. The schools involved in the experiment fulfilled the following criteria: they were schools with no specialisation, of medium size, for children from their immediate neighbourhood with a varied socio-economic background, with a percentage of children from abroad not exceeding the average for the whole Czech Republic, not established exclusively for children with special needs and located in the area of outer Prague. All the participating classes were from the same grade. The pupils were assigned three tests. The Initial Test in mathematics and the Czech language (assigned at the beginning of the school year in October 2016) aiming to divide pupils into four equally able groups. In the other two tests, each group was assigned one variant of the problems. Item Response Theory (Lord, 1980; Van der Linden, Hambleton, 1997) was used both for the division of the pupils into equally able groups, each solving a different variant of the word problem, and for the quantitative interpretation of data. To analyse the parameters of problems, we used Item Response Theory (or IRT) in IRTpro 3 software. A two-parameter logistic model was used. To find the latent ability of pupils, a scale was put on a z-score with Bayes estimation EAP, in an iterative way using results in the Initial Test (for which a model graded for the total result in the test was used) and Test 1 and Test 2. This enabled us to compensate for any inconsistencies in terms of ability grouping which we may have made at the beginning.
The analyses conducted using IRT brought the following results: In the 6th grade, the difficulty of the quartet was relatively low (the difficulty parameter in the two-parameter logistic model was on the scale from -2.1 to -1.3). The easiest was problem 1 (pupils proceeded in the order of the assignment). The difficulty of problems 2, 3, 4 was comparable (-1.5 to -1.3). The order of information played its role in case of a familiar context. In a less familiar context its impact was negligible. Discrimination was best in problem 1 (2.5), which can be explained by the fact that the pupils did not come across any “intricacy” and could use school procedures. The change in the order of information and context had more impact on discrimination of problems (it decreased to 0.9 in problem 2). There was no difference in discrimination in problems 3 and 4 (1.3 and 1.4), which can be explained by the fact that work in a less familiar context made the pupils think harder about the assignment. The difficulty of the quartet in the 9th grade was average (the difficulty parameter was on the scale from -0.1 to 0.3). Problems 1, 2, 3 were comparably difficult (the same parameter -0.1), Problem 4 combining a less familiar context and a change in the order of information was slightly more difficult for the pupils (0.3). As in the 6th grade, familiarity/unfamiliarity of the context had impact on discrimination of problems. Discrimination of problems 3 and 4 was the same and lower than in problems 1 and 2 (1.4 and 1.5 in contrast to 3.2 and 4.5). The best discrimination 4.5 was again in problem 1 that had “no intricacy” in its assignment. In the presentation, some selected results and examples of pupils’ errors will be presented.
Eysenck, M.W. (1993). Principles of Cognitive Psychology. Hove: Lawrence Erlbaum Associates Ltd. Hejný, M. (2003). Anatómia slovnej úlohy o veku. Disputationes scientificae Universitatis Catholicae in Ružomberok. Ružomberok: Katolícka univerzita, 3(3), 21–32. Hembree, R. (1992). Experiments and relational studies in problem solving: A meta-analysis. Journal for Research in Mathematics Education, 23, 242–273. Kopka, J. (2010). Ako riešiť matematické problémy.Ružomberok: Katolická univerzita v Ružomberku. Lord, F. M. (1980). Applications of item response theory to practical testing problems. Hillsdale, N.J.: Lawrence Erlbaum Associates. Novotná, J. (2000). Analýza řešení slovních úloh. Prague: UK-PedF. Rendl, M., Vondrová, N. et al. (2013). Kritická místa matematiky na základní škole očima učitelů. Prague: UK-PedF. Silver, E. (1997). Fostering creativity through instruction rich in mathematical problem solving and problem posing, ZDM, 3, 75–80. Van der Linden, W.J., Hambleton, R.K. (Eds.) (1997). Handbook of Modern Item Response Theory. New York: Springer Verlag. Van Dooren, W., De Bock, D., Vleugels, K., & Verschaffel, L. (2010). Just answering … or thinking? Contrasting pupils' solutions and classifications of missing-value word problems. Mathematical Thinking and Learning, 12(1), 20–35. Vondrová, N., Novotná, J., & Havlíčková, R. (submitted for ZDM 2019(1)). The influence of solving and situational information on pupils’ achievement in additive word problems with several states and transformations. ZDM, 2019. Zohar, A., & Gershikov, A. (2008). Gender and Performance in Mathematical Tasks: Does the Context Make a Difference? International Journal of Science and Mathematics Education, 6, 677–693.
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