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.     

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.     

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.     

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.     

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.     

Fourth graders’ dyadic learning on multi-touch interfaces—versatile effects of verbalization prompts

Educational technology research and development - Multi-touch interfaces allow for direct and simultaneous input by several co-present learners and afford hands-on learning experiences. Additional...     

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.     

Factors influencing Filipino children’s solutions to addition and subtraction word problems

Young Filipino children are expected to solve mathematical word problems in English, which is not their mother tongue. Because of this, it is often assumed that Filipino children have difficulties in solving problems because they cannot read or comprehend what they have read. This study tested this assumption by determining whether presenting word problems in Filipino or reading them aloud to children in either language facilitated solution accuracy. Contrary to the initial hypothesis, reading word problems aloud did not seem to improve student performance (p > 0.10). In contrast, presenting word problems in Filipino significantly improved solution accuracy (p < 0.0001) and led to differences in error patterns – children were less likely to use an inappropriate arithmetic operation when problems were presented in Filipino. However, the language of the problem had minimal effects on the more difficult Compare problem type. Finally, the benefits of using Filipino were more pronounced for low‐achieving students who may have lower proficiency in English than their high‐achieving peers (p < 0.01).     

Children’s use of subtraction by addition on large single-digit subtractions

Subtractions of the type M − S = ? can be solved by various strategies, including subtraction by addition. In this study, we investigated children’s use of subtraction by addition by means of reaction time analyses. We presented 106 third to sixth graders with 32 large non-tie single-digit problems in both subtraction (12 − 9 = .) and addition format (9 + . = 12). We examined the fit of three regression models, which represented the consistent use of direct subtraction, of subtraction by addition and of flexibly switching between both strategies based on the relative size of the subtrahend. Findings revealed that children did not switch flexibly between the two strategies, as adults do, but that they rely on direct subtraction for problems presented in subtraction format and on subtraction by addition for problems in addition format. We end with the major theoretical, methodological and educational implications of these results.     

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.     

Chinese business students’ changes in beliefs and strategy use in a constructively aligned PBL course

This study adopted a longitudinal retrospective case study approach to investigate Chinese business students’ transitional learning experience in a problem-based learning (PBL) course with innovative assessment practices. The study focused on students’ beliefs and strategy use in a constructively aligned PBL course for business communication. Eight students who had made notable progress were chosen for retrospective analysis. The data included 48 journal entries, interviews, and writing samples collected at different stages of the course. This study identified taxonomies of participants’ beliefs about learning and writing, their perceptions of assessment, and their strategy use for learning. It also examined changes in beliefs, perceptions, and strategy use to determine the nature of the students’ learning experience in this PBL course. Findings suggest a recognised need to design PBL courses that align social constructivist learning principles with students’ beliefs and strategies. The results also highlight the importance of developing appropriate assessment rubrics to enhance student engagement with PBL learning for improved outcomes.     

Educators’ use of questioning as a pedagogical strategy in long day care nurseries

Educators’ questions can encourage children to engage in extended conversations, facilitate comprehension and stimulate thinking. Many studies of educators’ questioning have focused on children aged 3 years and older. Little is known about the manner in which educators of infants in non-parental group care settings use questioning as a pedagogical strategy. Six university-qualified early childhood educators in Sydney, Australia, were videorecorded for 40 min each in their long day care centres. Informed by systemic functional linguistic theory, the educators’ questions addressed to infants (defined as children under 2 years of age) were coded according to whether they served a pedagogical or regulatory function. Pedagogical questions were defined as questions intended to promote children’s learning, whereas regulatory questions functioned as commands and offers which were intended to manage behaviour. The educators’ pedagogical questions were then further analysed according to the response required of the infant. Confirm questions sought a yes or no response, specify questions (who, when, where, what) sought the name of a person, time, place or entity and explain questions (why or how) sought an explanation or justification. The findings demonstrated that educators differed greatly in terms of the frequency and manner in which they used questioning as a pedagogical strategy when interacting with infants. So-called open questioning, which has been shown to stimulate conversation with older children, may not be appropriate for infants whose distinctive developmental trajectories differ from those of preschool-aged children. The findings suggest the need for a more nuanced understanding of the role of questioning in promoting infants’ language, literacy and intellectual development.     

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.     

Students’ meaning making in science: solving energy resource problems in virtual worlds combined with spreadsheets to develop graphs

The aim of this study is to scrutinize the characteristics of conceptual meaning making when students engage with virtual worlds in combination with a spreadsheet with the aim to develop graphs. We study how these tools and the representations they contain or enable students to construct serve to influence their understanding of energy resource consumption. The data were gathered in 1st grade upper-secondary science classes and they constitute the basis for the interaction analysis of students?? meaning making with representations. Our analyses demonstrate the difficulties involved in developing students?? orientation toward more conceptual orientations to representations of the knowledge domain. Virtual worlds do not in themselves represent a solution to this problem.     

Exploring Postgraduate students’ challenges and strategy use while writing a master’s thesis in an English-medium University in Kazakhstan

Tertiary Education and Management - With the growth of English medium instruction (EMI) on a global level, the number of multilingual students writing their master’s theses in English is...     

An international comparison using Cognitive Diagnostic Assessment: Fourth graders’ diagnostic profile of reading skills on PIRLS 2016

Cognitive diagnostic assessment (CDA) approach has been increasingly applied to non-diagnostic large-scale assessments to extract fine-grained diagnostic feedback about students’ ability in a given domain and meet accountability demands for student achievement. This study aimed to diagnose the reading abilities of 4324 students from 19 European Union (EU) member countries that participated in the 2016 Progress in International Reading Literacy Study (PIRLS), one of the most comprehensive international studies that investigate students’ reading achievement. The PIRLS data were analyzed by using the Log-linear Cognitive Diagnosis Modeling (LCDM), a type of cognitive diagnostic classification model (DCM). Students’ weaknesses and strengths were identified based on a four-skill reading ability model. The results revealed that the methodology could provide more fine-grained diagnostic information about students’ reading skills than traditional aggregated-test scoring could. Such information could be utilized by teachers, school administrators, decision-makers, and students for maximizing the learning outcomes of reading programs and instruction.     

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...     

Curriculum strategy framework: investigating patterns in teachers’ use of a reform‐based elementary mathematics curriculum

The goal of this paper is to introduce the curriculum strategy framework as a way to characterize teachers’ interactions with curriculum materials. The framework focuses on three key interpretive activities: reading, evaluating, and adapting curriculum materials. Describing an individual teacher’s curriculum strategy involves identifying the manner in which a teacher engages with each of these activities before, during, and after instruction. This paper presents the results of a study in which the framework was used to identify patterns in the curriculum strategies of 10 elementary‐school teachers who were using a reform‐based mathematics curriculum for the first time. It concludes with directions for further research using the curriculum strategy framework, and implications of this work for curriculum designers.