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.     

When intuitive conceptions overshadow pedagogical content knowledge: Teachers’ conceptions of students’ arithmetic word problem solving strategies

Intuitive conceptions in mathematics guide the interpretation of mathematical concepts. We investigated if they bias teachers’ conceptions of student arithmetic word problem solving strategies, which should be part of their pedagogical content knowledge (PCK). In individual interviews, teachers and non-teaching adults were asked to describe students’ strategies in situational contexts within or outside the scope of the intuitive conception. The results revealed that teachers relied on their PCK and identified student strategies; however, in the presence of the intuitive conception, their PCK was overshadowed and they ceased to differ significantly from non-teachers. This brings the attention to certain biases that can have a strong impact on teachers’ efficient use of PCK.     

Chinese and Singaporean sixth-grade students’ strategies for solving problems about speed

This study examined 361 Chinese and 345 Singaporean sixth-grade students’ performance and problem-solving strategies for solving 14 problems about speed. By focusing on students from two distinct high-performing countries in East Asia, we provide a useful perspective on the differences that exist in the preparation and problem-solving strategies of these groups of students. The strategy analysis indicates that the Chinese sample used algebraic strategies more frequently and more successfully than the Singaporean sample, although the Chinese sample used a limited variety of strategies. The Singaporean sample’s use of model-drawing produced a performance advantage on one problem by converting multiplication/division of fractions into multiplication/division of whole numbers. Several suggestions regarding teaching and learning of mathematical problem solving, algebra, and problems about speed and its related concepts of ratio and proportion are made.     

Children’s use of metacognition in solving everyday problems: An initial study from an Asian context

The aim of this study is to understand the relationship between metacognition and students’ everyday problem solving. Specifically, we were interested to find out whether regulation of cognition and knowledge of cognition are related to everyday problem solving and whether students who perform better in the decision-making problem will better differentiate the various components of metacognition. Two hundred and fifty-four fifth grade students completed a survey. We found evidence to suggest the existence of two major components of metacognition. Our results also suggest that at a higher level of decision-making, knowledge of cognition and regulation of cognition were differentiated in their use by participants.     

Understanding problem solving behavior of 6–8 graders in a debugging game

Debugging is an over-looked component in K-12 computational thinking education. Few K-12 programming environments are designed to teach debugging, and most debugging research were conducted on college-aged students. In this paper, we presented debugging exercises to 6th–8th grade students and analyzed their problem solving behaviors in a programming game – BOTS. Apart from the perspective of prior literature, we identified student behaviors in relation to problem solving stages, and correlated these behaviors with student prior programming experience and performance. We found that in our programming game, debugging required deeper understanding than writing new codes. We also found that problem solving behaviors were significantly correlated with students’ self-explanation quality, number of code edits, and prior programming experience. This study increased our understanding of younger students’ problem solving behavior, and provided actionable suggestions to the future design of debugging exercises in BOTS and similar environments.     

Children’s use of number line estimation strategies

This study tested whether second graders use benchmark-based strategies when solving a number line estimation (NLE) task. Participants were assigned to one of three conditions based on the availability of benchmarks provided on the number line. In the bounded condition, number lines were only bounded at both sides by 0 and 200, while the midpoint condition included an additional benchmark at the midpoint and children in the quartile condition were provided with a benchmark at every quartile. First, the inclusion of a midpoint resulted in more accurate estimates around the middle of the number line in the midpoint condition compared to the bounded and, surprisingly, also the quartile condition. Furthermore, the two additional benchmarks in the quartile condition did not yield better estimations around the first and third quartile, because children frequently relied on an erroneous representation of these benchmarks, leading to systematic estimation errors. Second, verbal strategy reports revealed that children in the midpoint condition relied more frequently on the benchmark at the midpoint of the number line compared to the bounded condition, confirming the accuracy data. Finally, the frequency of use of benchmark-based strategies correlated positively with mathematics achievement and tended to correlate positively also with estimation accuracy. In sum, this study is one of the first to provide systematic evidence for children’s use of benchmark-based estimation strategies in NLE with natural numbers and its relationship with children’s NLE performance.     

Special education students’ use of indirect addition in solving subtraction problems up to 100—A proof of the didactical potential of an ignored procedure

In this study, we examined special education students’ use of indirect addition (subtraction by adding on) for solving two-digit subtraction problems. Fifty-six students (8- to 12-year-olds), with a mathematical level of end grade 2, participated in the study. They were given a computer-based test on subtraction with different types of problems. Although most students had not been taught indirect addition for solving subtraction problems, they frequently applied this procedure spontaneously. The item characteristics were the main prompt for using indirect addition. Context problems that reflect an adding-on situation and problems that have a small difference between the minuend and subtrahend most strongly elicited the use of the indirect addition procedure. Moreover, indirect addition was identified as a highly successful procedure for special education students, and the best predictor of a correct answer was found in combination with a stringing strategy.     

Teachers’ use of rational questioning strategies to promote student participation in collective argumentation

Educational Studies in Mathematics - Teachers’ questioning plays an essential role in shaping collective argumentative discourse. This paper demonstrated that rationality dimensions in...     

Flowery math: a case for heterodiscoursia in mathematics problems solving in recognition of students’ authorial agency

In an innovative, progressive school, students were asked to solve a fairly routine mathematical problem using real money in a “real-world” scenario. Even though the school values students’ ideas, the reaction of the teacher to one student’s alternative modelling of the problem suggests that he was expecting a particular answer to be provided using routine mathematical models and thinking while not being interested in exploring the student’s unexpected alternative. We place his reasoning for doing so within broad pedagogical discourses that we think define the “allowable” responses of teachers and students in ways that inhibit meaning-making for both. These broad discourses are defined as the progressive constructivist approach, the scaffolding discursive approach, the situation modelling approach and the dialogic approach. We consider the advantages and the potential consequences each might bring to the case. We suggest that extensive consideration of pedagogical discourses in mathematics classes must be reconsidered both for how we understand students’ mathematical meaning-making and how we construct student agency in relationship to culture, whether as apprentices or authors.     

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...     

Reasoning strategies of students in solving chemistry problems as a function of developmental level,functional M‐capacity and disembedding ability

Achievement in science depends among other factors on hypothetico‐deductive reasoning ability, that is, developmental level of the students. Recent research indicates that the developmental level of students should be studied along with individual difference variables, such as Pascual‐Leone's M‐capacity (information processing) and Witkin's Cognitive Style (disembedding ability). The purpose of this study is to investigate reasoning strategies of students in solving chemistry problems as a function of developmental level, functional M‐capacity and disembedding ability. A sample of 109 freshman students were administered tests of formal operational reasoning, functional M‐capacity, disembedding ability and chemistry problems (limiting reagent, mole, gas laws). Results obtained show that students who scored higher on cognitive predictor variables not only have a better chance of solving chemistry problems, but also demonstrated greater understanding and used reasoning strategies indicative of explicit problem‐solving procedures based on the hypothetico‐deductive method, manipulation of essential information and sensitivity to misleading information. It was also observed that students who score higher on cognitive predictor variables tend to anticipate important aspects of the problem situation by constructing general figurative and operative models, leading to a greater understanding. Students scoring low on cognitive predictor variables tended to circumvent cognitively more demanding strategies and adopt others that helped them to overcome the constraints of formal reasoning, information processing and disembedding ability.     

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.     

Promising intervention strategies to reduce parents’ use of physical punishment

The strong and ever-growing evidence base demonstrating that physical punishment places children at risk for a range of negative outcomes, coupled with global recognition of children’s inherent rights to protection and dignity, has led to the emergence of programs specifically designed to prevent physical punishment by parents. This paper describes promising programs and strategies designed for each of three levels of intervention − indicated, selective, and universal − and summarizes the existing evidence base of each. Areas for further program development and evaluation are identified.     

Pizzas or no pizzas: An advantage of word problems in fraction arithmetic?

Fractions are an important but notoriously difficult domain in mathematics education. Situating fraction arithmetic problems in a realistic setting might help students overcome their difficulties by making fraction arithmetic less abstract. The current study therefore investigated to what extent students (106 sixth graders, 187 seventh graders, and 192 eighth graders) perform better on fraction arithmetic problems presented as word problems compared to these problems presented symbolically. Results showed that in multiplication of a fraction with a whole number and in all types of fraction division, word problems were easier than their symbolic counterparts. However, in addition, subtraction, and multiplication of two fractions, symbolic problems were easier. There were no performance differences by students’ grade, but higher conceptual fraction knowledge was associated with higher fraction arithmetic performance. Taken together this study showed that situating fraction arithmetic in a realistic setting may support or hinder performance, dependent on the problem demands.     

Are first graders' arithmetic skills related to the quality of mathematics textbooks? A study on students’ use of arithmetic principles

A major goal of the first years of schooling is students' development from using counting strategies to using calculation strategies – or even recall – to solve addition or subtraction problems. It requires the perception and usage of arithmetic principles. Although the continued use of counting strategies is problematic for further learning progress, they are frequently used by many students beyond Grade 1. While learning resources possibly play a role in students' development in this field, our knowledge about their impact is limited. In this study we investigated the presented learning opportunities of four different textbooks regarding arithmetic principles in Grade 1, and subsequently the relation to first graders' strategy use by analyzing a one-year dataset of 1614 students from 86 classes. The analyses show differences in the textbooks' quality concerning arithmetic principles as well as a considerable connection to students' strategy use at the end of Grade 1.     

An Investigation of the Potential of Interactive Simulations for Developing System Thinking Skills in Elementary School: A case study with fifth‐graders and sixth‐graders

The purpose of this study was to investigate the impact of a simulation‐based learning environment on elementary school students’ (11–12 years old) development of system thinking skills. The learning environment included interactive simulations using the Stagecast Creator software to simulate the ecosystem of a marsh. Simulations are an important tool in any effort to develop system thinking, because they have the potential to highlight the dynamic nature of systems. Before the implementation of the learning environment (over a period of five 90‐min lessons) two written tests were administered to the students, investigating the development of seven aspects of system thinking. The same tests were administered after the implementation. Specifically, four of the tasks included in each test were associated with skills concerning the structure and the elements of a system and three were associated with the processes and interactions taking place within a system. The findings indicated that elementary school students have the potential to develop system thinking skills. The proposed learning environment provoked considerable improvements in some system thinking skills during a relatively brief learning process. However, the learning environment was not successful in promoting feedback thinking. We interpret these results in view of the difficulties encountered by the students. We also discuss the implications of our findings for the design of learning environments.     

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.     

Instructional approaches to foster third graders’ adaptive use of strategies: an experimental study on the effects of two learning environments on multi-digit addition and subtraction

The adaptive use of strategies, that is selecting a strategy which allows an efficient solution for a given problem, can be considered as an important individual ability relevant in various domains. Based on models of subjects’ skills of adaptive use of strategies, two idealized instructional approaches are suggested to foster students in their strategy development. The explicit approach aims at reducing cognitive load by demonstrating and practicing strategies combined with an explicit identification of criteria for strategy efficiency by contrasting problem solutions. The implicit approach capitalizes on the generation effect and stimulates students to generate their own strategies and efficiency criteria based on the analysis of task characteristics and the comparison of problem solutions. In a 1-week experimental study (16 lessons) with 73 third-graders, we examined the effectiveness of these instructional approaches in the domain of multi-digit addition and subtraction. Results from post- and two follow-up tests after 3 and 8?months did not yield different effects of the two approaches on students’ skills in adaptive use of strategies. A comparison of strategies used by the students showed that the students of the explicit approach more frequently applied complex strategies whereas the students from the implicit approach showed a more sustainable use of self-generated strategies. Hence, for the adaptive use of those strategies students are able to generate, the implicit approach turned out to be more effective than the explicit approach. However, this generation effect does not hold for strategies which are too complex to be generated by students.     

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

Metacognition and Learning - The original version of this article unfortunately contained a mistake in page 7, particularly in the second paragraph of “Procedure” section of Experiment...