Scaffolding 6th graders’ problem solving in technology-enhanced science classrooms: a qualitative case study

In response to the calls to improve and deepen scientific understanding and literacy, considerable effort has been invested in developing sustainable technology-enhanced learning environments to improve science inquiry. Research has provided important guidance for scaffolding learning in mathematics and science. However, these reports have provided relatively little insight into how the different types of scaffolds can (or should) be implemented in dynamic, everyday classroom settings. In this qualitative case study, we examined how students solve scientific problems in technology-enhanced classrooms and how peer-, teacher-, and technology-enhanced scaffolds influenced student inquiry. The results indicated that students manifested distinct inquiry patterns when solving scientific problems and integrated different types of scaffolds to facilitate inquiry activities. These findings suggest that to support scientific inquiry in problem-solving contexts, technology-enhanced scaffolds are effective when supported by clear project goals, relevant evidence, peer- and teacher-assessments, and exemplars of knowledge articulation.     

Primary school students’ strategies in early algebra problem solving supported by an online game

In this study we investigated the role of a dynamic online game on students’ early algebra problem solving. In total 253 students from grades 4, 5, and 6 (10–12 years old) used the game at home to solve a sequence of early algebra problems consisting of contextual problems addressing covarying quantities. Special software monitored the students’ online working when solving the problems. Before and after the intervention a paper-and-pencil test on early algebra was administered. The data analysis revealed that the online working contributed to the students’ early algebra performance. There was a significant gain in performance across all grades. The highest effect was found in grade 6. Out of the three strategy profile clusters that could be distinguished in the whole sample, the cluster dominated by using extreme values and the cluster characterized by the trial-and-error strategy were most influential on the gain in early algebra performance. The students’ level of online working, which was defined as a combination of online involvement and strategy use, appeared to have a marginally significant effect on the gain score for the total sample. Per grade there was no significant effect, yet the levels of online working were significantly related to grade. Free playing was mostly performed in grade 4, looking for answers in grade 5, and exploring relations slightly more in grades 5 and 6. About 17 % of the effect of grade on the gain score was mediated by the level of online working.     

Dutch sixth graders’ use of shortcut strategies in solving multidigit arithmetic problems

Strategy flexibility, adaptivity, and the use of clever shortcut strategies are of major importance in current primary school mathematics education worldwide. However, empirical results show that primary school students use such shortcut strategies rather infrequently. The aims of the present study were to analyze the extent to which Dutch sixth graders (12-year-olds) use shortcut strategies in solving multidigit addition, subtraction, multiplication, and division problems, to what extent student factors and task instructions affected this frequency of shortcut strategy use, and to what extent the strategies differed in performance. A sample of 648 sixth graders from 23 Dutch primary schools completed a paper-and-pencil task of 12 multidigit arithmetic problems, designed to elicit specific shortcut strategies such as compensation. Based on the students’ written work, strategies were classified into whether a shortcut strategy was used or not. Results showed that the frequency of shortcut strategies ranged between 6 and 21% across problem types, and that boys and high mathematics achievers were more inclined to use shortcut strategies. An explicit instruction to look for a shortcut strategy increased the frequency of these strategies in the addition and multiplication problems, but not in the subtraction and division problems. Finally, the use of shortcut strategies did not yield higher performance than using standard strategies. All in all, spontaneous as well as stimulated use of shortcut strategies by Dutch sixth graders was not very common.     

Fourth graders’ adaptive strategy use in solving multidigit subtraction problems

Using the choice/no-choice methodology we investigated Dutch fourth graders’ (N = 124) adaptive use of the indirect addition strategy to solve subtraction problems. Children solved multidigit subtraction problems in one choice condition, in which they were free to choose between direct subtraction and indirect addition, and in two no-choice conditions, in which they had to use either direct subtraction or indirect addition. Furthermore, children were randomly assigned to mental computation, written computation, or free choice between mental and written computation. One third of the children adaptively switched their strategy according to the number characteristics of the problems, whereas the remaining children consistently used the same strategy. The likelihood to adaptively switch strategies decreased when written computation was allowed or required, compared to mandatory mental computation. On average, children were adaptive to their own speed differences but not to the accuracy differences between the strategies.     

Pre-service teachers’ flexibility with referent units in solving a fraction division problem

This study investigated 111 pre-service teachers’ (PSTs’) flexibility with referent units in solving a fraction division problem using a length model. Participants’ written solutions to a measurement fraction division problem were analyzed in terms of strategies and types of errors, using an inductive content analysis approach. Findings suggest that most PSTs could calculate fraction division and make equivalent fractions procedurally but did not have the quantitative meanings of measurement division with fraction quantities or of making equivalent fractions. Implications are discussed for the improvement of PSTs’ specialized knowledge for teaching fraction division.     

Exploring the Development of Conceptual Understanding through Structured Problem‐solving in Physics

A study on the effect of a structured problem‐solving strategy on problem‐solving skills and conceptual understanding of physics was undertaken with 189 students in 16 disadvantaged South African schools. This paper focuses on the development of conceptual understanding. New instruments, namely a solutions map and a conceptual index, are introduced to assess conceptual understanding demonstrated in students’ written solutions to examination problems. The process of the development of conceptual understanding is then explored within the framework of Greeno’s model of scientific problem‐solving and reasoning. It was found that students who had been exposed to the structured problem‐solving strategy demonstrated better conceptual understanding of physics and tended to adopt a conceptual approach to problem‐solving.     

Understanding students’ game experiences throughout the developmental process of the number navigation game

Educational technology research and development - Serious games for learning have received increased attention in recent years. However, empirical studies on students’ gaming experiences...     

Assessing kindergarteners’ mathematics problem solving: The development of a cognitive diagnostic test

This study aimed to develop an instrument for assessing kindergarteners’ mathematics problem solving (MPS) by using cognitive diagnostic assessment (CDA). A total of 747 children were recruited to examine the psychometric properties of the cognitive diagnostic test. The results showed that the classification accuracy of 11 cognitive attributes ranged from .68 to .99, with the average being .84. Both the cognitive diagnostic test score and the average mastery probabilities of the 11 cognitive attributes had moderate correlations with the Applied Problem subtest and the Calculation subtest of the Woodcock–Johnson IV Tests of Achievement. Moreover, the correlation between the cognitive diagnostic test and the Applied Problems subtest was higher than that between the cognitive diagnostic test and the Calculation subtest. The results indicated that the formal cognitive diagnostic test was a reliable instrument for assessing kindergarteners’ MPS in the domain of number and operations.     

Reflections on the type of question as a determinant of the form of interaction in decision‐making and problem‐solving discussions

This essay advances the thesis that scholars interested in communication in decision‐making and problem‐solving groups have focused in their research on questions of policy at considerable neglect of questions of fact, conjecture, and value. One should not presume that the process involved in discussions of questions of policy is similar to those involved in discussions of the other three types. In fact, there is good reason to believe that because the decision frame presumably is different for each type of question, the pattern of interaction characteristic of discussions involving each type of question will be distinct. Such distinctiveness is suggested and better explained by viewing each type of question from a unique theoretical perspective. Accordingly, this essay explores discussions of questions of fact from a narrative perspective, questions of conjecture from a cognitive perspective, questions of value from a deontological perspective, and questions of policy from a social‐influence perspective in terms of expected patterns of interaction and the possibilities each presents in accounting for variation in the consensual outcomes of group decision‐making and problem‐solving discussions, as well as the appropriateness of the final choices the participants make.     

Initial judgment of solvability in non-verbal problems – a predictor of solving processes

Metacognition and Learning - Meta-reasoning refers to processes by which people monitor problem-solving activities and regulate effort investment. Solving is hypothesized to begin with an initial...     

The role of number words in preschoolers’ addition concepts and problem‐solving procedures

Preschoolers’ conceptual understanding and procedural skills were examined so as to explore the role of number‐words and concept–procedure interactions in their additional knowledge. Eighteen three‐ to four‐year‐olds and 24 four‐ to five‐year‐olds judged commutativity and associativity principles and solved two‐term problems involving number words and unknown numbers. The older preschoolers outperformed younger preschoolers in judging concepts involving unknown numbers and children made more accurate commutativity than associativity judgements. Children with conceptual profiles indicating a strong understanding of concepts applied to unknown numbers were more accurate at solving number‐word problems than those with a poor conceptual understanding. The findings suggest that an important mathematical development during the preschool years may be learning to appreciate addition concepts as general principles that apply when exact numbers are unknown.     

Procedural and conceptual changes in young children’s problem solving

This study analysed the different types of arithmetic knowledge that young children utilise when solving a multiple-step addition task. The focus of the research was on the procedural and conceptual changes that occur as children develop their overall problem solving approach. Combining qualitative case study with a micro-genetic approach, clinical interviews were conducted with ten 5–6-year-old children. The aim was to document how children combine knowledge of addition facts, calculation procedures and arithmetic concepts when solving a multiple-step task and how children’s application of different types of knowledge and overall solving approach changes and develops when children engage with solving the task in a series of problem solving sessions. The study documents children’s pathways towards developing a more effective and systematic approach to multiple-step tasks through different phases of their problem solving behaviour. The analysis of changes in children’s overt behaviour reveals a dynamic interplay between children’s developing representation of the task, their improved procedures and gradually their more explicit grasp of the conceptual aspects of their strategy. The findings provide new evidence that supports aspects of the “iterative model” hypothesis of the interaction between procedural and conceptual knowledge and highlight the need for educational approaches and tasks that encourage and trigger the interplay of different types of knowledge in young children’s arithmetic problem solving.     

Social and socio‐mathematical norms in collaborative problem‐solving

Based on the notions of social and socio‐mathematical norms we investigate how these are established during the interactions of pre‐service teachers who solve mathematical problems. Norms identified in relevant studies are found in our case too; moreover, we have found norms related to particular aspects of the problems posed. Our results show that most of these norms, once established, enhance the problem‐solving process. However, exceptions do exist, but they have a local orientation and a relatively small influence.

En s'appuyant sur les concepts des normes sociales et ‘socio‐mathématiques’, nous avons étudié comment ces normes se sont établies au cours des interactions entre les enseignants et les étudiants en activité de résolution des problèmes mathématiques. Aux résultats de la recherche apparaissent d'une part les mêmes normes qui ont été déjà remarquées à d'autres recherches relatives et d'autre part des normes liées plus particulièrement aux problèmes posés. Les résultats de la recherche montrent que dans la majorité des cas les normes aident le processus de la resolution des problèmes. Il existe bien sûr des exceptions, mais elles ont une influence et une orientation locale.

Basierend auf den Begriffen der sozialen und sozio‐mathematischen Normen untersuchen wir, wie diese in die Interaktionen von angehenden Lehrern beim Lösen von mathematischen Problemen einfliessen. Normen, welche in relevanten Studien identifiziert werden, wurden in unserem Fall ebenfalls gefunden. Wir haben ausserdem Normen gefunden, welche sich auf bestimmte Aspekte der Fragestellungen beziehen. Unsere Resultate zeigen, dass die meisten Normen, sind sie einmal etabliert, die Problemlösungsprozess verbessern. Es bestehen zwar Ausnahmen, doch diese haben eine lokale Orientierung und einen relativ kleinen Einfluss.

Basados en las nociones de la norma social y sociomatemática, estamos investigando cómo se establecen dichas normas durante las interacciones de los profesores en pre‐servicio, que resuelven problemas matemáticos. Las normas identificadas en estudios relevantes también se encuentran en nuestro caso; de hecho, hemos hallado normas relacionadas con aspectos particulares de los problemas expuestos. Nuestros resultados demuestran que la mayor parte de estas normas, una vez establecidas, mejoran el proceso de solución de problemas. Sin embargo, existen también excepciones pero éstas tienen una orientación local y una relativamente menor influencia.