Strategies in Single-Digit Addition: Patterns and Perspectives

This dissertation focuses on students’ use and development of mental strategies in single-digit addition in the first years of school in Denmark. The overarching aim was to investigate longitudinal and cross-sectional patterns in Danish students’ development and use of strategies for single-digit addition in relation to 1) different teaching practices, 2) later mathematical achievement and 3) teachers’ perspectives on teaching and learning of number and arithmetic.
The work builds on data from two independent studies: A) a longitudinal study of 147 students’ development of strategy use from year one to four, and B) a study of six year one classes (83 students and six teachers) with data on the students development of strategies (assessed twice: October/November – April/May) in year one, teacher interviews on the teaching of number and arithmetic, and classroom observations of the teaching. The latter study was subsequently supplied with data on later achievement in arithmetic, fraction knowledge and word problem solving in year four.
The dissertation consist of a synthesis and five individual papers: Paper I describes the proportional use of four different strategies (‘error’, ‘counting’, ‘derived fact strategy’, ‘direct retrieval’) as a function of school age (year one to four) and sex based on data from Study A, verified on data from Study B; Paper II is a statistical analysis of how much strategy use on student level changes over a half school year (October/November – April/May) in year one and the extent to which development in strategy use varies between classes in Study B; Paper III is an analysis of the extent to which strategy use in year one predicts mathematical achievement in year four (number and arithmetic, fraction knowledge and word problem solving); Paper IV provides a qualitative analysis of teachers’ expectations for their year one students’ additive competence in Study B; and Paper V reports on the same teachers’ perspectives on teaching and learning number and addition in year one.
The key findings can be summarised as follows:
From year one to year four Danish students decreased their use of counting strategies and increased their use of fact-based strategies (Paper I). However, on a time scale of a half learning year (November to April in year one) this longitudinal (developmental) variation in strategy use that reflected increasing number knowledge with age was overrode by considerable individual variation, much of which could be related to sex (Paper I-II). Hence, in year one, girls did on average use counting three times more often than boys, a difference that equalled at least two years’ development (Paper I). Strategy use patterns varied little between classes (Paper I-II), and did not develop differently between classes in year one. From this follows that different teaching practice did not result in traceable differences in the speed by which students’ developed their strategy use during year one (Paper II). The spread in strategy use between students was more or less constant throughout year one with no indications of students with the highest use of counting were decreasing their use of this unsophisticated strategy faster with time than students that used this method less often from the start (Paper II). Measures of strategy use in year one, especially the frequency by which students used counting all, correlated significantly with mathematical achievement in year four. For two of three measurements of mathematical achievement (fraction and word problem solving) strategy use patterns in year one explained variation not explained by measures from a standard achievement test in year one (Paper III). Or in other words, information on how a student did single-digit addition in year one gave a better prediction for later achievement in fraction and word problem solving than how well the student did addition in year one (paper III). The correlations between strategy use in year one and mathematical achievement in year four were generally stronger in boys than in girls (Paper III). It hypothesised that this difference might reflect a higher inclination in girls compared to boys to use counting even when they master retrieval strategies.
Altogether, these results points towards that strategy use habits are deeply rooted within the individual child, as it must have been established either before or at the very beginning of formal schooling, and only change slowly and gradually with time. Overall, these results points to the relevance of students’ early understanding of number and arithmetic, i.e. strategy use not only as indicators of later achievement but also as an important focus for intervention and targeted teaching with the aim of supporting the development of all students. From a learning perspective, the results might be useful in two ways. First, students’ strategy use ‘profiles’ from the assessment interviews could provide information on the student’s risk of developing mathematical difficulties or of becoming low achiever. In that respect, excessive use of counting strategies (counting all in particular) in the early years of school might be of particular value as screening parameter. If teachers are able to detect students in need of special attention early, targeted teaching might be initiated before possible difficulties evolves. Second, if the strong correlation between strategy use patterns in early year one and achievement in year four reflects a causal relation (this remains to be investigated) teaching practice should focus on developing number sense and flexible strategy use for arithmetic.