Number relationships, which go far beyond counting skills, refer to the ability to represent a quantity in multiple, flexible ways. It is arguably among the most important mathematics concepts in number and quantity. The current study examined the effectiveness of number relationships instruction in preschool classrooms. Participants included 73…

With careful consideration given to task selection, students can construct their own solution strategies to solve complex proportional reasoning tasks while the teacher's instructional goals are still met. Several aspects of the tasks should be considered including their numerical structure, context, difficulty level, and the strategies they are…

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 goal of this study was to understand errors that student make when simplifying exponential expressions. College students enrolled in four college mathematics courses were asked to simplify and compare such expressions. Quantitative analysis identified three persistent errors: interpreting negative bases, negative exponents, and parentheses.…

Having decided to spend some time looking at one's understanding of numbers, the author was inspired by "Alex's Adventures in Numberland," by Alex Bellos to look at one's innate appreciation of number. Bellos quotes research studies suggesting that an individual's natural appreciation of numbers is more likely to be exponential rather…

The purpose of this study is to provide an account of preservice elementary mathematics teachers' understandings about irrational numbers. Three dimensions of preservice mathematics teachers' understandings are examined: defining rational and irrational numbers, placing rational and irrational numbers on the number line, and operations with…

It is assumed that basic numerical competencies are important building blocks for more complex arithmetic skills. The current study aimed at evaluating this interrelation in a longitudinal approach. It was investigated whether first graders' performance in basic numerical tasks in general as well as specific processes involved (e.g., place-value…

In the third century, Diophantus, the "Father of Algebra" no less, described equations of the form x + 20 = 4 as "absurd." The absurdity stemmed from the fact that the result of four is obviously less than the addend of twenty. And more than 1300 years later, Pascal argued that subtracting four from zero leaves zero because of the impossibility of…

Mathematics has a level of structure that transcends untutored intuition. What is the cognitive representation of abstract mathematical concepts that makes them meaningful? We consider this question in the context of the integers, which extend the natural numbers with zero and negative numbers. Participants made greater and lesser judgments of…

We used both student interviews and diagnostic testing to reveal students' misconceptions about number representations in computing systems. This article reveals that students who have passed an undergraduate level computer organization course still possess surprising misconceptions about positional notations, two's complement representation, and…

Tasks that ask students to label rational number points on a number line are common not only in curricula in the upper elementary school grades but also on state assessments. Such tasks target foundational rational number concepts: A fraction (or a decimal) is more than a shaded part of an area, a part of a pizza, or a representation using…

Number sense is critical to the development of higher order mathematic abilities. However, some children have difficulty acquiring these fundamental skills and the knowledge base of effective interventions/remediation is relatively limited. Based on emerging neuro-scientific research which has identified the association between finger…

This paper reports on one phase of a long-term project investigating mathematical content knowledge of pre-service teachers. A cohort of second year PSTs conducted a diagnostic assessment and a series of associated tutoring sessions with a primary aged child. The focus here is on the PSTs' ability to make appropriate task choices following the…

In this important book for pre- and in-service teachers, early math experts Douglas Clements and Julie Sarama show how "learning trajectories" help diagnose a child's level of mathematical understanding and provide guidance for teaching. By focusing on the inherent delight and curiosity behind young children's mathematical reasoning,…

This paper presents the results of a pre-test, instruction, post-test study that investigated students' integer mental models and how their models changed based on instruction. Sixty-one first graders' responses to questions about the values and order of negative numbers were categorized according to a series of mental models. The models reveal…