The authors asked fifth-grade and eighth-grade students to pose stories for number sentences involving the addition and subtraction of integers. In this article, the authors look at eight stories from students. Which of these stories works for the given number sentence? What do they reveal about student thinking? When the authors examined these…

This study examines students' procedural and conceptual achievement in fraction addition in England and Taiwan. A total of 1209 participants (561 British students and 648 Taiwanese students) at ages 12 and 13 were recruited from England and Taiwan to take part in the study. A quantitative design by means of a self-designed written test is adopted…

This study investigated the promotion of children's understanding and acquisition of arithmetic concepts and the effects of inhibitory skills. Children in Grades 3, 4, and 5 solved two sets of three-term addition and subtraction problems (e.g., 3 + 24 - 24, 3 + 24 - 22) and completed an inhibition task. Half of the participants received a…

Although fraction operations are procedurally straightforward, they are complex, because they require learners to conceptualize different units and view quantities in multiple ways. Prospective secondary school teachers sometimes provide an algebraic explanation for inverting and multiplying when dividing fractions. That authors of this article…

Students often misinterpret the equal sign (=) as operational instead of relational. Research indicates misinterpretation of the equal sign occurs because students receive relatively little exposure to equations that promote relational understanding of the equal sign. No study, however, has examined effects of nonstandard equations on the equation…

When solving mathematical problems, many students know the procedure to get to the answer but cannot explain why they are doing it in that way. According to Skemp (1976) these students have instrumental understanding but not relational understanding of the problem. They have accepted the rules to arriving at the answer without questioning or…

This paper argues for teaching pre-service teachers about remediation strategies for learners who encounter problems in mathematics in the early grades. The premise is that all teachers should be equipped with theory-based practical knowledge to support learning. A few teaching sessions to develop the concepts that underlie the mathematical…

Both additive and proportional reasoning are types of quantitative analogical (QA) reasoning. We investigated the development and nature of primary school children's QA reasoning by offering two missing-value word problems to 3rd to 6th graders. In one problem, ratios between given numbers were integer, in the other ratios were non-integer. These…

This single-subject multiple-baseline across participants study was designed to investigate the effects of explicit instruction with manipulatives on the conceptual and procedural knowledge of addition and subtraction of like and unlike fractions of elementary school students with autism. This study included six 8- to 12-year-old students with…

Do your students have the incorrect idea that addition "makes numbers bigger" and subtraction "makes numbers smaller"? Do they believe that subtraction is always "taking away"? What tasks can you offer--what questions can you ask--to determine what your students know or don't know--and move them forward in their…

The Fibonacci series has been studied since it was first described by Leonardo of Pisa--Fibonacci--in 1202. It begins with the sequence 1, 1, 2, 3, 5, 8... Each succeeding number is the sum of the previous two. In number theory courses, students are introduced to the concept of modulo arithmetic, sometimes called "clock" arithmetic. In modulo…

The transition from whole numbers to integers involves challenges for both students and teachers. Leadership in mathematics education calls for an ability to translate depth of understanding into effective teaching methods, and this landscape includes alternative treatments of familiar topics. Noting the multiple meanings associated with the…

During the winter of 2011, the author faced an instructional dilemma: Should she teach the concept of adding and subtracting fractions with unlike denominators using a "quick trick" for the sake of staying on schedule, or should she try something different in hopes that her students would gain understanding through their own methods? The latter…

This study explores the impact that co-designing a virtual manipulative, Fractions Lab, had on teachers' professional development. Tapping into an existing community of practice of mathematics specialist teachers, the study identifies how a cooperative enquiry approach utilising workshops and school-based visits challenged 23 competent primary…

The longitudinal progress of 42 Grade 1 students who participated in a 10-20 week Extending Mathematical Understanding (EMU) intervention program was examined to evaluate the effectiveness of the program for enhancing and accelerating mathematics learning. Overall the students made accelerated progress during Grade 1 and their learning was…