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Children's estimation skills on a bounded and unbounded number line task were assessed in the light of their familiarity with numbers. Kindergartners, first graders, and second graders (N = 120) estimated the position of numbers on a 1--100 number line, marked with either two reference points (i.e., 1 and 10: unbounded condition) or three reference points (i.e., 1, 10, 100: bounded condition). Estimations were more accurate and less variable in older compared to younger children and, beyond age, in children who were familiar with larger numbers. The number of reference points yielded neither a main effect nor an interaction effect with familiarity. Our results underline that children's familiarity with numbers is a pivotal factor for the quality of number line estimations that might even obliterate potential effects of additional reference points.

External number representations are commonly used throughout the first years of instruction. The twenty-frame is a grid that contains two rows of 10 dots each, and within each row, dots are organized in two groups of five. The assumption is that children can make use of these structures for enumerating the dots, rather than relying on one-by-one counting. We compared first-grade children's performance on two types of computerized enumeration tasks, in which between one and 20 dots were presented in random arrangements or on a twenty-frame. The number of dots was a strong predictor of response times and accuracy rates in the enumeration task with random arrangements but not in the twenty-frame task. Performance on the twenty-frame task was correlated with performance on a number and arithmetic test, even when other cognitive variables were statistically controlled. We discuss these findings in the light of theories on utilizing external representations to support numerical learning.

People often have difficulties in understanding situations that involve non-linear processes. Also, the topic of non-linear functions is introduced relatively late in the curriculum. Previous research has nevertheless shown that already children aged 6 years and older are able to discriminate non-linear from linear processes. Within the present article, we examined in more detail the cognitive foundations of this ability in kindergartners. In three experiments, 5-year-olds were presented with the first steps of linear and non-linear (i.e., polynomial and exponential) growth processes and had to forecast future growth. Their forecasts indicated that they distinguished between linear and non-linear growth by taking into account (1) the initial magnitudes of the first steps of the growth processes, (2) the slopes exhibited by these first growth steps, and sometimes also (3) the constant or increasing rates of change. However 5-year-olds' performances appeared somewhat unstable across the experiments, which will be discussed in terms of strategy acquisition and task presentation. Nevertheless, children's sensitivity toward these three characteristics indicates that 5-year-olds possess the cognitive foundations that allow them in principle to develop knowledge about both linear and non-linear processes. (Contains 3 figures, 1 table, and 2 footnotes.)

Many learners have difficulties with rational number tasks because they persistently rely on their natural number knowledge, which is not always applicable. Studies show that such a natural number bias can mislead not only children but also educated adults. It is still unclear whether and under what conditions mathematical expertise enables people to be completely unaffected by such a bias on tasks in which people with less expertise are clearly biased. We compared the performance of eighth-grade students and expert mathematicians on the same set of algebraic expression problems that addressed the effect of arithmetic operations (multiplication and division). Using accuracy and response time measures, we found clear evidence for a natural number bias in students but no traces of a bias in experts. The data suggested that whereas students based their answers on their intuitions about natural numbers, expert mathematicians relied on their skilled intuitions about algebraic expressions. We conclude that it is possible for experts to be unaffected by the natural number bias on rational number tasks when they use strategies that do not involve natural numbers. [ABSTRACT FROM AUTHOR]