Marian Hickendorff

Solution Strategies and Achievement in Dutch Complex Arithmetic: Latent Variable Modeling of Change

In the Netherlands, national assessments at the end of primary school (Grade 6) show a decline of achievement on problems of complex or written arithmetic over the last two decades. The present study aims at contributing to an explanation of the large achievement decrease on complex division, by investigating the strategies students used in solving the division problems in the two most recent assessments carried out in 1997 and in 2004. The students’ strategies were classified into four categories. A data set resulted with two types of repeated observations within students: the nominal strategies and the dichotomous achievement scores (correct/incorrect) on the items administered.It is argued that latent variable modeling methodology is appropriate to analyze these data. First, latent class analyses with year of assessment as a covariate were carried out on the multivariate nominal strategy variables. Results showed a shift from application of the traditional long division algorithm in 1997, to the less accurate strategy of stating an answer without writing down any notes or calculations in 2004, especially for boys. Second, explanatory IRT analyses showed that the three main strategies were significantly less accurate in 2004 than they were in 1997.

Multilevel Latent Class Analysis for Large-Scale Educational Assessment Data: Exploring the Relation between the Curriculum and Students' Mathematical Strategies.

ABSTRACT A first application of multilevel latent class analysis (MLCA) to educational large-scale assessment data is demonstrated. This statistical technique addresses several of the challenges that assessment data offers. Importantly, MLCA allows modeling of the often ignored teacher effects and of the joint influence of teacher and student variables. Using data from the 2011 assessment of Dutch primary schools’ mathematics, this study explores the relation between the curriculum as reported by 107 teachers and the strategy choices of their 1,619 students, while controlling for student characteristics. Considerable teacher effects are demonstrated, as well as significant relations between the intended as well as enacted curriculum and students’ strategy use. Implications of these results for both more theoretical and practical educational research are discussed, as are several issues in applying MLCA and possibilities for applying MLCA to different types of educational data.

Grade-related differences in strategy use in multidigit division in two instructional settings

We aimed to investigate upper elementary childrens strategy use in the domain of multidigit division in two instructional settings: the Netherlands and Flanders (Belgium). A cross‐sectional sample of 119 Dutch and 122 Flemish fourth to sixth graders solved a varied set of multidigit division problems. With latent class analysis, three distinct strategy profiles were identified: children consistently using number‐based strategies, children combining the use of column‐based and number‐based strategies, and children combining the use of digit‐based and number‐based strategies. The relation between childrens strategy profiles and their instructional setting (country) and grade were generally in line with instructional differences, but large individual differences remained. Furthermore, Dutch children more frequently made adaptive strategy choices and realistic solutions than their Flemish peers. These results complement and refine previous findings on childrens strategy use in relation to mathematics instruction. Statement of contribution What is already known? Mathematics education reform emphasizes variety, adaptivity, and insight in arithmetic strategies. Countries have different instructional trajectories for multidigit division. Mixed results on the impact of instruction on childrens strategy use in multidigit division. What does this study add? Latent class analysis identified three meaningful strategy profiles in children from grades 4–6. These strategy profiles substantially differed between children. Dutch and Flemish childrens strategy use is related to their instructional trajectory.

Training can increase students’ choices for written solution strategies and performance in solving multi-digit division problems.

Making adaptive choices between solution strategies is a central element of contemporary mathematics education. However, previous studies signal that students make suboptimal choices between mental and written strategies to solve division problems. In particular, some students of a lower math ability level appear inclined to use mental strategies that lead to lower performance. The current study uses a pretest-training-posttest design to investigate the extent to which these students’ choices for written strategies and performance may be increased. Sixth graders of below-average mathematics level (n = 147) participated in one of two training conditions: an explicit-scaffolding training designed to promote writing down calculations or a practice-only training where strategy use was not explicitly targeted. Written strategy choices and performance increased considerably from pretest to posttest for students in both training conditions, but not in different amounts. Exploratory results suggest that students’ strategy choices may also be affected by their attitudes and beliefs and the sociocultural context regarding strategy use.

How to Measure and Explain Achievement Change in Large-Scale Assessments: A Rejoinder.

In this rejoinder, we discuss substantive and methodological validity issues of large-scale assessments of trends in student achievement, commenting on the discussion paper by Van den Heuvel-Panhuizen, Robitzsch, Treffers, and Köller (2009). We focus on methodological challenges in deciding what to measure, how to measure it, and how to foster stability. Next, we discuss what to do with trends that are found. Finally, we reflect on how the research findings were received.