Pedagogical content knowledge (PCK) was introduced by Shulman in 1986 and refers to the knowledge teachers use to translate particular subject matter to students, taking into account possible ...(mis)conceptions. PCK was – and still is – very influential in research on teaching and teacher education, mainly within the natural sciences. The present study aims at a systematic review of the way PCK was conceptualized and (empirically) studied in mathematics education research. Based on a systematic search in the databases Eric, PsycInfo and Web of Science 60 articles were reviewed. We identified different conceptualizations of PCK that in turn had a differential influence on the methods used in the study of PCK.
•PCK is differently conceptualized in empirical mathematics education research.•Large-scale studies often measure PCK through a paper-and-pencil test.•Small-scale studies typically use multiple qualitative data sources.•Half of the PCK-studies investigates the development of (pre-service) teachers' PCK.
Word problems are among the most difficult kinds of problems that mathematics learners encounter. Perhaps as a result, they have been the object of a tremendous amount research over the past ...50 years. This opening article gives an overview of the research literature on word problem solving, by pointing to a number of major topics, questions, and debates that have dominated the field. After a short introduction, we begin with research that has conceived word problems primarily as problems of comprehension, and we describe the various ways in which this complex comprehension process has been conceived theoretically as well as the empirical evidence supporting different theoretical models. Next we review research that has focused on strategies for actually solving the word problem. Strengths and weaknesses of informal and formal solution strategies—at various levels of learners’ mathematical development (i.e., arithmetic, algebra)—are discussed. Fourth, we address research that thinks of word problems as exercises in complex problem solving, requiring the use of cognitive strategies (heuristics) as well as metacognitive (or self-regulatory) strategies. The fifth section concerns the role of graphical representations in word problem solving. The complex and sometimes surprising results of research on representations—both self-made and externally provided ones—are summarized and discussed. As in many other domains of mathematics learning, word problem solving performance has been shown to be significantly associated with a number of general cognitive resources such as working memory capacity and inhibitory skills. Research focusing on the role of these general cognitive resources is reviewed afterwards. The seventh section discusses research that analyzes the complex relationship between (traditional) word problems and (genuine) mathematical modeling tasks. Generally, this research points to the gap between the artificial word problems learners encounter in their mathematics lessons, on the one hand, and the authentic mathematical modeling situations with which they are confronted in real life, on the other hand. Finally, we review research on the impact of three important elements of the teaching/learning environment on the development of learners’ word problem solving competence: textbooks, software, and teachers. It is shown how each of these three environmental elements may support or hinder the development of learners’ word problem solving competence. With this general overview of international research on the various perspectives on this complex and fascinating kind of mathematical problem, we set the scene for the empirical contributions on word problems that appear in this special issue.
The success or failure of education systems in promoting student problem-solving skills depends on attitudinal, political, and pedagogical variables. Among these variables, the design of mathematics ...textbooks is thought to partially explain why students from high-achieving countries show better problem-solving ability in international assessments. In the current study, we delved into this question and compared the frequency and characteristics of arithmetic word problems (AWPs) contained in primary school math textbooks in two countries with different levels of performance in international assessments—Singapore and Spain. In our analyses, we focused on (1) the quantity of arithmetic word problems, (2) the variety of problems in terms of their additive or multiplicative structures and semantic-mathematical substructures, and (3) the quantity and nature of illustrations that were presented together with arithmetic word problems. Although a larger proportion of AWP activities was found in Singaporean textbooks, the results showed a similar variety of AWPs in both Singaporean and Spanish math textbooks. Furthermore, in both countries, math textbooks emphasized the structures classified as (additive) combine 1 and (multiplication) simple rate in AWPs. Notably, the Singaporean textbook contained a larger percentage of illustrations that reflected the semantic-mathematical structures of the problems and helped students learn how to solve AWPs (e.g., bar models). The findings are discussed in light of theories that posit that textbooks constitute a fundamental part of the teaching–learning process in the classroom.
Several studies have shown that children do not only erroneously use additive reasoning in proportional word problems, but also erroneously use proportional reasoning in additive word problems. ...Traditionally, these errors were contributed to a lack of calculation and discrimination skills. Recent research evidence puts forward an additional explanation, namely, children’s relational preference (i.e., in tasks where both, additive and multiplicative reasoning, are appropriate, some children have a preference for additive relations, while others have a preference for multiplicative relations). Children’s relational preference offers a unique explanation for erroneous word-problem solving, after taking into account computation and discrimination skills in 8- to 12-year-olds. However, it is still unclear whether relational preference is also associated with word-problem solving at an earlier age, before the start of formal instruction in word-problem solving. A task measuring children’s relational preference as well as three additive and three proportional word problems was administered to a large group (
n
= 343) of 6- to 7-year-olds. Results show that relational preference is also associated with word-problem solving behavior at this young age: an additive preference is related with better performance on additive word problems but also with more erroneous additive reasoning in proportional word problems. Similarly, a multiplicative preference is related with better performance on proportional word problems but not yet with more erroneous proportional reasoning in additive word problems. The latter is possibly due to the low number of proportional errors that were made in the additive word problems at this young age. The implications of these findings for further research and educational practice are discussed.
Selecting a large and diverse sample of 5-6-year-old preschool children (179 boys and 174 girls;
= 70.03 months,
= 3.43), we aimed to extend previous findings on variability in children's home math ...environment (i.e., home math activities, parental expectations, and attitudes) and its association with children's mathematical skills. We operationalized mathematics in a broader way than in previous studies, by considering not only children's numerical skills but also their patterning skills as integral components of early mathematical development. We investigated the effects of children's gender and socioeconomic status (SES) on their home math environment, examined the associations between children's home math environment and their mathematical skills, and verified whether these associations were moderated by children's gender and/or SES. Parents of 353 children completed a home math environment questionnaire and all children completed measures of their numerical (e.g., object counting) and patterning skills (e.g., extending repeating patterns). Results indicated no effect of children's gender on their home math environment. There was no effect of SES on the performed home math activities, but small SES differences existed in parents' math-related expectations and their attitudes. We found no evidence for associations between children's home math environment and their mathematical skills. Furthermore, there were no moderating effects of gender or SES on these associations. One explanation for these findings might relate to the characteristics of the general preschool system in the country of the present study (Belgium). Future studies should consider the effect of the preschool learning environment because it might explain differences between studies and countries with regard to the home math environment and its association with mathematical skills.
This study is the first to examine the associations between the occurrence, frequency, and adaptivity of children’s subtraction by addition strategy use (SBA; e.g., 712 − 346 = ?; 346 + 54 = 400, 400 ...+ 300 = 700, 700 + 12 = 712, and 54 + 300 + 12 = 366) and their underlying conceptual knowledge. Specifically, we focused on two rarely studied components of conceptual knowledge: children’s knowledge of the addition/subtraction complement principle (i.e., if
a
+
b
=
c
, then
c
−
b
=
a
and
c
−
a
=
b
) and their knowledge of different conceptual subtraction models (i.e., understanding that subtraction can be conceived not only as “taking away” but also as “determining the difference”). SBA occurrence was examined using a variability on demand task, in which children had to use multiple strategies to solve a subtraction. SBA frequency and strategy adaptivity were investigated with a task in which children could freely choose between SBA and direct subtraction (e.g., 712 − 346 = ?; 712 − 300 = 412, 412 − 40 = 372, and 372 − 6 = 366) to solve 15 subtractions. We measured both children’s knowledge of the addition/subtraction complement principle, and whether they understood subtraction also as “determining the difference.” SBA occurrence and frequency were not related to conceptual knowledge. However, strategy adaptivity was related to children’s knowledge of the addition/subtraction complement principle. Our findings highlight the importance of attention to conceptual knowledge when teaching multi-digit subtraction and expand the literature about the relation between procedural and conceptual knowledge.
Research on rational numbers suggests that adults experience more difficulties in understanding the numerical magnitude of rational than natural numbers. Within rational numbers, the numerical ...magnitude of fractions has been found to be more difficult to understand than that of decimals. Using a number line estimation (NLE) task, the current study investigated two sources of difficulty in adults’ numerical magnitude understanding: number type (natural vs rational) and structure of the notation system (place-value-based vs non-place-value-based). This within-subjects design led to four conditions: natural numbers (natural/place-value-based), decimals (rational/place-value-based), fractions (rational/non-place-value-based), and separated fractions (natural/non-place-value-based). In addition to percentage absolute error (PAE) and response times, we collected eye-tracking data. Results showed that participants estimated natural and place-value-based notations more accurately than rational and non-place-value-based notations, respectively. Participants were also slower to respond to fractions compared with the three other notations. Consistent with the response time data, eye-tracking data showed that participants spent more time encoding fractions and re-visited them more often than the other notations. Moreover, in general, participants spent more time positioning non-place-value-based than place-value-based notations on the number line. Overall, the present study contends that when both sources of difficulty are present in a notation (i.e., both rational and non-place-value-based), adults understand its numerical magnitude less well than when there is only one source of difficulty present (i.e., either rational or non-place-value-based). When no sources of difficulty are present in a notation (i.e., both natural and place-value-based), adults have the strongest understanding of its numerical magnitude.
The current paper presents an introduction to a special issue focusing on mathematical flexibility, which is an important aspect of mathematical thinking and a cherished, but capricious, outcome of ...mathematics education. Mathematical flexibility involves the flexible, creative, meaningful, and innovative use of mathematical concepts, relations, representations, and strategies. In this introduction we discuss the most relevant theoretical, methodological, and educational considerations related to mathematical flexibility, which form the background of the empirical studies presented in the special issue. Collectively, these studies provide a broader understanding of the mathematical flexibility, its subcomponents, influences, and malleability.
Some authors argue that age-related improvements in number line estimation (NLE) performance result from changes in strategy use. More specifically, children's strategy use develops from only using ...the origin of the number line, to using the origin and the endpoint, to eventually also relying on the midpoint of the number line. Recently, Peeters et al. (unpublished) investigated whether the provision of additional unlabeled benchmarks at 25, 50, and 75% of the number line, positively affects third and fifth graders' NLE performance and benchmark-based strategy use. It was found that only the older children benefitted from the presence of these benchmarks at the quartiles of the number line (i.e., 25 and 75%), as they made more use of these benchmarks, leading to more accurate estimates. A possible explanation for this lack of improvement in third graders might be their inability to correctly link the presented benchmarks with their corresponding numerical values. In the present study, we investigated whether labeling these benchmarks with their corresponding numerical values, would have a positive effect on younger children's NLE performance and quartile-based strategy use as well. Third and sixth graders were assigned to one of three conditions: (a) a
condition with an empty number line bounded by 0 at the origin and 1,000 at the endpoint, (b) an
condition with three additional external benchmarks without numerical labels at 25, 50, and 75% of the number line, and (c) a
condition in which these benchmarks were labeled with 250, 500, and 750, respectively. Results indicated that labeling the benchmarks has a positive effect on third graders' NLE performance and quartile-based strategy use, whereas sixth graders already benefited from the mere provision of unlabeled benchmarks. These findings imply that children's benchmark-based strategy use can be stimulated by adding additional externally provided benchmarks on the number line, but that, depending on children's age and familiarity with the number range, these additional external benchmarks might need to be labeled.
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