Humans count to indefinitely large numbers by recycling words from a finite list, and combining them using rules--for example, combining sixty with unit labels to generate sixty-one, sixty-two, and so on. Past experimental research has focused on children learning base-10 systems, and has reported that this rule learning process is highly…

The solution and verification of single-digit multiplication problems vary in speed and accuracy. The current study examines whether the number of different digits in a problem accounts for this variance. In Experiment 1, 41 participants solved all 2-9 multiplication problems. In Experiment 2, 43 participants verified these problems. In Experiment…

Students entering sixth grade operate with three different multiplicative concepts that influence their reasoning in many domains important for middle school. For example, students who are operating with the second multiplicative concept (MC2 students) can begin to construct fractions as lengths but do not construct improper fractions as numbers.…

This study used units coordination as a theoretical lens to investigate how whole number and fraction reasoning may be related for preservice teachers at the conclusion of a math methods class. The study contributes quantitative evidence that units coordination provides a common foundation for both mathematical knowledge for teaching whole number…

Urban and rural children have different levels of performance in arithmetic processing. This study investigated whether such a residence difference can be explained by phonological processing. A total of 1,501 Chinese primary school students from urban and rural areas were recruited to complete nine cognitive tasks: two in arithmetic performance…

Specialised Content Knowledge (SCK) is defined by Ball, Hoover-Thames, and Phelps (2008) as mathematical knowledge essential for effective teaching. It is knowledge of mathematics that is beyond knowledge which would be required outside of teaching; for instance, the capacity to determine what misconception(s) may lie behind an error in…

Background: Adaptive expertise is a highly valued outcome of mathematics curricula. One aspect of adaptive expertise with rational numbers is adaptive rational number knowledge, which refers to the ability to integrate knowledge of numerical characteristics and relations in solving novel tasks. Even among students with strong conceptual and…

In three experiments, we used a masked prime in a verification task to investigate the processing stages occurring during multiplication fact retrieval. We aimed to investigate the retrieval process by overlapping its execution with the processing of a masked prime consisting of a number. Participants evaluated the correctness of multiplication…

This study investigates the teaching and learning of single-digit whole number multiplication in China. Analysis of data from documents, classroom teaching, and semi-structured interviews revealed three salient characteristics of emphasizing "oral calculation," "calculation speed," and "understanding" across…

The number one plays a special role in mathematics because it is the identity element in multiplication and division. The present findings, however, indicate that many middle school students do not demonstrate mathematical flexibility representing one as a fraction. Despite possessing explicit knowledge of fraction forms of one (e.g., 95% of…

Skills and understanding of operations with negative numbers, which are typically taught in middle school, are crucial aspects of numerical competence necessary for all subsequent mathematics. To more swiftly and coherently develop the field's understanding of how to foster this critical competence, we need shared measures that allow us to compare…

The aim of this study is to investigate Danish pre-service teachers' (PSTs) mathematical and didactical knowledge of operations with rational numbers. This knowledge is studied through their collaborative activities to certain tasks related to the teaching of operations with rational numbers. An explicit model of the teachers' mathematical and…

This research paper reports strategies used by Grade 6 learners in multiplying whole numbers in five selected primary schools in Kavango East and West regions. A total of 200 learners' mathematics exercise books were analysed in order to identify the commonly used strategies by learners in multiplying whole numbers. A total of ten teachers…

This article focuses on how two mathematics teachers (Amy and Beth -- pseudonyms) cope with the changes of meanings in multiplication due to the changes of contexts. It highlights the qualitative similarities and differences between these two teachers in the sense-making process of multiplication. A potentially useful framework of supportive and…

This study aims to investigate the obstacles in eighth-grade students' understanding of integer exponents using a mixed method research design. A total of 165 eighth-grade students were given a paper-pencil task and clinical interviews were conducted with 12 students. The findings indicated that achievement of the participants was low, especially…