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…

In mathematics education, there has been tension between deep learning and repetitive learning. Western educators often emphasize the need for students to construct a conceptual understanding of mathematical symbols and rules before they practise the rules (Li, 2006). On the other hand, Chinese learners tend to be oriented towards rote learning…

The article reports aspects of the evolution of a teaching approach over repeated trials for beginning symbolic algebra. The teaching approach emphasized the structural similarity between arithmetic and algebraic expressions and aimed at supporting students in making a transition from arithmetic to beginning algebra. The study was conducted with…

Four experiments examined the effects of training a "drawing" response to each of three stimuli in a 5-member equivalence class. In Experiment 1 the stimuli were an arbitrary word, a shape, or a mathematical symbol. Subjects then were trained to draw a separate component of a stickman at each of the 3 stimuli. Subsequent tests for function…

After students have constructed a conceptual understanding of a mathematical idea, applying mathematical symbolism to represent and define the idea concisely becomes an easier task. In fact, Moore (1994) found that students "often needed to develop their concept images through examples, diagrams, graphs, and other means before they could…

This study investigated how the use of virtual manipulatives in integer instruction impacts student achievement for integer addition and subtraction. Of particular interest was the influence of using virtual manipulatives on students' ability to create and translate among representations for integer computation. The research employed a…

Prior neuroimaging work on visual perceptual expertise has focused on changes in the visual system, ignoring possible effects of acquiring expert visual skills in nonvisual areas. We investigated expertise for reading musical notation, a skill likely to be associated with multimodal abilities. We compared brain activity in music-reading experts…

Various measures have been used to investigate number processing in children, including a number comparison or a number line estimation task. The present study aimed to examine whether and to which extent these different measures of number representation are related to performance on a curriculum-based standardized mathematics achievement test in…

As students progress through primary school, they are introduced to a growing collection of notation and symbols. From a teaching perspective, before introducing new mathematical symbols it is important to consider the role they play. If students cannot see a meaningful relationship between these signs and symbols and mathematical concepts, coming…

The study's goal was to examine the metacognitive practices and problem solving by preservice secondary mathematics teachers within secondary core content areas. The effort to analyze and describe the processes led to qualitative methods, a multiple case study. Transcripts of problem solving sessions and interviews were coded for the presence of…

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…

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'…

Chance, randomness and probability constitute statistical notions that are interrelated and characterize the logicomathematical thinking of children. Traditional theories support that probabilistic thinking evolves after the age of 7. However, recent research has underlined that children, as young as 4, may possess and develop basic notions,…

When teachers listen to, and learn from, students, they begin to see the world from a child's perspective, and this helps make what they teach more accessible to students. Thus it is important for teachers and educators to understand mathematics from the children's perspectives to help them improve their classroom instructions, by anticipating…

"Two and Two Make Zero" considers children's acquisition of numerical concepts from a wide range of perspectives including topics that are often overlooked, most notably: the principal properties of the counting numbers in and of themselves; the role that numerical symbols play in number acquisition; the underlying conceptual structure of number…