Over the last two decades, students in Singapore consistently scored above students from other nations on the Trends in International Mathematics and Science Study (TIMSS; Provasnik et al., 2012). In contrast, students in the United States have not performed as well on international and national mathematics assessments and students with…

When learning the order of operations, students are instructed to adhere to a directive when determining the numerical value of an arithmetic expression. A more typical approach is the use of a popular mnemonic called PEDMAS (parentheses, exponents, division, multiplication, addition, subtraction). The literature is scant on conceptual approaches…

This article proposes an integrated theory of acquisition of knowledge about whole numbers and fractions. Although whole numbers and fractions differ in many ways that influence their development, an important commonality is the centrality of knowledge of numerical magnitudes in overall understanding. The present findings with 11- and 13-year-olds…

Time perception has not been comprehensively examined in mathematics difficulties (MD). Therefore, verbal time estimation, production, and reproduction were tested in 13 individuals with MD and 16 healthy controls, matched for age, sex, and intellectual skills. Individuals with MD performed comparably to controls in time reproduction, but showed a…

The purpose of this study is to determine how Finnish pupils' problem-solving skills develop from the 3rd to 5th grade. As research data, we use one non-standard problem from pre- and post-test material from a three-year follow-up study, in the area of Helsinki, Finland. The problems in both tests consisted of four questions related to each other.…

Many children in the U.S. initially come to understand the equal sign operationally, as a symbol meaning "add up the numbers" rather than relationally, as an indication that the two sides of an equation share a common value. According to a change-resistance account (McNeil & Alibali, 2005b), children's operational ways of thinking…

Mathematics educators have had a long standing interest in students' understanding of decimal numbers. Most studies of students' understanding of decimals have been conducted within Western cultural settings. Similar research in other countries is important for a number of reasons, perhaps most importantly because it can provide insights that may…

Do your students have the incorrect idea that addition "makes numbers bigger" and subtraction "makes numbers smaller"? Do they believe that subtraction is always "taking away"? What tasks can you offer--what questions can you ask--to determine what your students know or don't know--and move them forward in their…

Of the "big four", division is likely to regarded by many learners as "the odd one out", "the difficult one", "the one that is complicated", or "the scary one". It seems to have been that way "for ever", in the perception of many who have trodden the learning pathways through the world of…

One of the common tasks of inferential statistics is to compare two data sets. Long before formal statistical procedures, however, students can be encouraged to make comparisons between data sets and therefore build up intuitive statistical reasoning. Such tasks also give meaning to the data collection students may do. This study describes the…

The notion of periodicity stands for regular recurrence of phenomena in a particular order in nature or in the actions of man, machine, etc. Many examples can be given from daily life featuring periodicity. Mathematically the meaning of periodicity is that some value recurs with a constant frequency. Students learn about the periodicity of the…

The internal representation of numbers generated during calculation has received little attention. Much of the mathematics learning literature focuses on symbolic retrieval of math facts; in contrast, we critically test the hypothesis that internally generated numbers are represented analogically, using an approximate number system. In an fMRI…

This paper presents a study that investigated the process of change in 5-6-year-old children's successful problem-solving approaches when tackling a multiple-step task in elementary arithmetic. Micro-developmental changes in children's successful problem-solving behavior were analyzed using Karmiloff-Smith's model of representational redescription…

Multichannel EEG traces from healthy subjects are used to investigate the brain's self-organisation tendencies during two different mental arithmetic tasks. By making a comparison with a control-state in the form of a classification problem, we can detect and quantify the changes in coordinated brain activity in terms of functional connectivity.…

While infants' ability to discriminate quantities has been extensively studied, showing that this competence is present even in neonates, the ability to compute ordinal relations between magnitudes has received much less attention. Here we show that the ability to represent ordinal information embedded in size-based sequences is apparent at 4…