Many children have significant mathematical learning disabilities (MLD, or dyscalculia) despite adequate schooling. The current study hypothesizes that MLD partly results from a deficiency in the Approximate Number System (ANS) that supports nonverbal numerical representations across species and throughout development. In this study of 71 ninth…

Proficiency with fractions serves as a foundation for later mathematics and is critical for learning algebra, which plays a role in college success and lifetime earnings. Yet children often struggle to learn fractions. Educators have argued that a conceptual understanding of fractions involves learning that a fraction represents a magnitude…

We present a research report on addition and subtraction conducted with Down syndrome students between the ages of 12 and 31. We interviewed a group of students with Down syndrome who executed algorithms and solved problems using specific materials and paper and pencil. The results show that students with Down syndrome progress through the same…

When young children attempt to locate the positions of numerals on a number line, the positions are often logarithmically rather than linearly distributed. This finding has been taken as evidence that the children represent numbers on a mental number line that is logarithmically calibrated. This article reports a statistical simulation showing…

The study sought out to extend our knowledge regarding the origin of mathematical learning disabilities (MLD) in children by testing different hypotheses in the same samples of children. Different aspects of cognitive functions and number processing were assessed in fifth- and sixth-graders (11-13 years old) with MLD and compared to controls. The…

We examined the role of executive attention, which encompasses the common aspects of executive function and executive working memory, in children's acquisition of two aspects of mathematical skill: (a) knowledge of the number system (e.g., place value) and of arithmetic procedures (e.g., multi-digit addition) and (b) arithmetic fluency (i.e.,…

A dedicated, non-symbolic, system yielding imprecise representations of large quantities (approximate number system, or ANS) has been shown to support arithmetic calculations of addition and subtraction. In the present study, 5-7-year-old children without formal schooling in multiplication and division were given a task requiring a scalar…

Chinese pre-school children perform well in learning mathematics compared with English-speaking children. This study investigates the scenes behind Chinese preschool children's mathematics performance using teacher questionnaires and interviews. Results indicated that the Chinese number system appeared to afford advantages to Chinese children in…

We examined relations between symbolic and non-symbolic numerical magnitude representations, between whole number and fraction representations, and between these representations and overall mathematics achievement in fifth graders. Fraction and whole number symbolic and non-symbolic numerical magnitude understandings were measured using both…

Place value underpins much of what people do in number. In this article, the author describes some simple tasks that may be used to assess students' understanding of place value. This set of tasks, the Six Tasks of Place Value (SToPV), takes five minutes to administer and can give direct insight into a student's understanding of the number system…

Student performance in mathematics has been linked to the mathematical knowledge of the teacher. Based on this finding, a 5-day professional development module was created to improve teachers' mathematical knowledge and their understanding of number sense. We found no difference prior to the professional development in mathematical content…

As all physicists know, all units are arbitrary. The numbering system is anthropocentric; for example, the Celsius scale of temperature has 100 degrees between the boiling point of water at STP and the freezing point of water. The number 100 is chosen because human beings have 10 fingers. The best units might be based on physical constants, for…

Developing an understanding of place value and the base-ten number system is considered a fundamental goal of the early primary grades. For years, teachers have anecdotally reported that students struggle with place-value concepts. Among the common errors cited are misreading such numbers as 26 and 62 by seeing them as identical in meaning,…

The view of infinity as a metaphor, a basic premise of modern cognitive theory of embodied knowledge, suggests in particular that there may be alternative ways in which one could formalize mathematical ideas about infinity. We discuss the key ideas about infinitesimals via a proceptual analysis of the meaning of the ellipsis "..." in the real…

Results from a project conducted in Mexico are discussed, in which a group of 17 indigenous teachers analyzed the numeration systems of their first language. The main goal of the project is to develop resources that help teachers in supporting students' understanding of the systems. In the first phase of the project, the central organizing ideas…