The goal of this practice guide is to offer educators specific, evidence-based recommendations that address the challenge of teaching early math to children ages 3 to 6. The guide provides practical, clear information on critical topics related to teaching early math and is based on the best available evidence as judged by the authors. The guide…

This essay argues for the importance of quantitative reasoning skills as part of a liberal education and describes the successful introduction of a mathematics-based quantitative skills course at a small Canadian university. Today's students need quantitative problem-solving skills, to function as adults, professionals, consumers, and citizens in…

In this paper we articulate an approach, termed culturally-mathematically relevant pedagogy (CMRP), for fostering urban English language learners' mathematical progression. CMRP integrates three aspects, the use of (1) adaptive teaching to build on students' funds of knowledge for mathematics, (2) tasks that make sense to students given their…

The combined seventh-grade and eighth-grade class began each day with a mathematical reasoning question as a warm-up activity. One day's question was: Is the product of two odd numbers always an odd number? The students took sides on the issue, and the exercise ended in frustration. Reflecting on the frustration caused by this warm-up activity,…

Two people take turns selecting from an even number of items. Their relative preferences over the items can be described as a permutation, then tools from algebraic combinatorics can be used to answer various questions. We describe each person's optimal selection strategies including how each could make use of knowing the other's preferences. We…

In recent times considerable attention has been devoted to the performance of schools and in particular, of students in literacy and numeracy. As part of a national agenda addressing what is portrayed as a crisis in numeracy, and hence mathematics education, governments have introduced a wide range of reform initiatives to improve performance.…

In this paper, we focus on Finnish pre-service elementary teachers' (N = 269) and upper secondary students' (N = 1,434) understanding of division. In the questionnaire, we used the following non-standard division problem: "We know that 498:6 = 83. How could you conclude from this relationship (without using long-division algorithm) what 491:6…

Numeric magnitudes often bias adults' spatial performance. Partly because the direction of this bias (left-to-right versus right-to-left) is culture-specific, it has been assumed that the orientation of spatial-numeric associations is a late development, tied to reading practice or schooling. Challenging this assumption, we found that preschoolers…

Because number is an abstract quality of a set, the way in which a number is externally represented does not change its quantitative meaning. In this study, we examined the development of the brain regions that support format-independent representation of numerical magnitude. We asked children and adults to perform both symbolic (Hindu-Arabic…

In this article, we use a particular example to illustrate a thematic approach to the teaching and learning of mathematics. Our theme, suitable for undergraduates and able sixth-form students, is the enumeration of mathematical objects associated with the prime factorizations of integers. It is shown in detail how this gives rise to some beautiful…

Speakers are known to make subject-verb agreement errors both when a number-mismatching noun intervenes between the head of the subject phrase and the verb (e.g., "*The key to the cabinets are on the table") and in configurations in which there is a number-mismatching noun that does not intervene (e.g., "*The cabinets that the key open are on the…

Developmental dyscalculia is a learning disability that affects the acquisition of knowledge about numbers and arithmetic. It is widely assumed that numeracy is rooted on the "number sense", a core ability to grasp numerical quantities that humans share with other animals and deploy spontaneously at birth. To probe the links between number sense…

Some researchers have argued that children's earliest symbols are based on their sensorimotor experience and that arbitrary symbol-referent mapping poses a challenge for them. If so, exposure to iconic symbols (such as one-finger-for-one-object manual gestures) might help children in a difficult domain such as number. We assessed 44 preschoolers'…

The main objective of this article is to provide new evidence regarding the impact of bilingualism on the attentional system. We approach this goal by assessing the effects of bilingualism on the executive and orienting networks of attention. In Experiment 1, we compared young bilingual and monolingual adults in a numerical version of the Stroop…

A considerable body of research exists on students' understanding of decimal fractions and the prevalence and persistence of common misconceptions related to this understanding. Results from major studies such as the National Assessment of Educational Progress (NAEP) in the United States and the Concepts in Secondary Mathematics and Science (CSMS)…