The sandwich problem described in this article would likely be considered more appropriate for third grade or above. However, as members of the research team for a lesson study involving teachers from kindergarten through fourth grade, the authors saw kindergarten students rise to the challenge. The authors invite readers to review two scenarios,…

In this article, Lindsey Fowler describes how an animal rhyme--that she shared with her fifth-grade students with the intention of assisting them with the concept of metric conversions--made fostering the Standards for Mathematical Practice (SMP) more difficult. The addition of animals, which added an extraneous element that students mistakenly…

Paper folding is an easy and inexpensive way to engage students artistically, culturally, and mathematically. The pinwheel base is the source for many origami playthings, from a pinwheel to a boat to a butterfly. This article explores a fourth grade activity that repurposes the pinwheel base into a rich and inexpensive manipulative for the review…

Using Cuisenaire Rods, metric measurement, and mapping, students worked collaboratively to calculate, keep records, build, and problem solve with use of decimal fractions as a key element.

This article describes a process called "Backing Up" which is a way to preassess student understanding of a topic and gauge student readiness to move forward in the learning process. This process of backing up begins with using responses to a word problem to identify categories of students' understandings in relation to the expectations…

Without a doubt, fractions prove to be a stumbling block for many children. Researchers have suggested a variety of explanations for why this is the case. The introduction of symbolization and operations before the development of conceptual understanding of fractions, a lack of understanding of the role of the numerator and denominator, and an…

Young children spend a much greater amount of time on practicing multiplication facts compared to understanding the concept of multiplication. When students have long-term, foundational concepts rather than a series of fragmented algorithms or facts, they are more likely to understand and generalize the mathematics. Using generalized models that…

In this article the authors describe a project during which they unpacked fraction standards, created rigorous tasks and lesson plans, and developed formative and summative assessments to analyze students' thinking about fraction multiplication. The purpose of this article is to (1) illustrate a process that can be replicated by educators…

The Common Core State Standards for Mathematics (CCSSM) (CCSSI 2010) outlines ambitious goals for fraction learning, starting in third grade, that include the use of the number line model. Understanding and constructing fractions on a number line are particularly complex tasks. The current work of the authors centers on ways to successfully…

The authors recently conducted a randomized controlled trial that showed a significant impact of teachers' lesson study, supported by mathematical resources, on both teachers' and students' understanding of fractions. The research and mathematical resources are described in the second part of this article. First the authors examine some of the…

One of the challenges in learning fractions is understanding how and why a fraction can have multiple interpretations. As presented in one textbook, a fraction is "a symbol, such as 2/3, 5/1, or 8/5, used to name a part of a whole, a part of a set, a location on a number line, or a division of whole numbers" (Charles et al. 2012, p.…

This article introduces the formative assessment probe--a powerful tool for collecting focused, actionable information about student thinking and potential misconceptions--along with a process for targeting instruction in response to probe results. Drawing on research about common student mathematical misconceptions as well as the former work of…

The authors of this article collaborated with fourth-grade teachers from two schools to support implementation of a research-based fraction and decimal curriculum (Rational Number Project: Fraction Operations and Initial Decimal Ideas). Through this study, they identified five indicators of rich conceptual understanding of decimals, which are…

Investigates why decimal understanding lags behind fractional understanding as demonstrated by students performance on the Third International Mathematics and Science Study (TIMSS). (Author)

Contrasts nominal, cardinal, and ordinal uses of numbers with linguistic and historical examples. (MKR)