Arithmetic is a major mathematical focus in elementary school curriculum, and researchers such as Mason (2008) claim that "algebraic thinking is required in order to make sense of arithmetic" (p. 58). When adding, subtracting, multiplying, and dividing, learners must rely on a small set of fundamental properties also important for the…

Understanding the decimal system is challenging, requiring coordination of place-value concepts with features of whole-number and fraction knowledge (Moloney and Stacey 1997). Moreover, the learner must discern if and how previously learned concepts and procedures apply. The process is complex, and misconceptions will naturally arise. In a…

Many of us embrace the order and beauty in mathematics. The order of operations is an iconic mathematics topic that seems untouchable by time, reform, or mathematical discoveries. Yet, think for a moment about a commonly heard statement in teaching the order of operations: "You work from left to right." At another point in the…

Students' success with fourth-grade content standards builds on mathematical knowledge learned in third grade and creates a conceptual foundation for division standards in subsequent grades that focus on the division algorithm. The division standards in fourth and fifth grade are similar; but in fourth grade, division problem divisors are only one…

When a second-grade class struggled to make sense of the algorithms for multidigit addition and subtraction, attitude and morale were down, and student discourse was at an all-time low. When daily mathematics class started, the teacher was hearing an overall class moan. The instructed needed to turn it around, to find a way to help excite and…

Several detours prompted me to find time in an overcrowded school day to incorporate important, powerful, daily, whole-class application of fact strategies. A few years ago, I embarked on a journey with my second graders, a journey through the strategies for mental computation of addition and subtraction facts. The focus of that journey was to…

This article reports on a study in which researchers asked children to "make up as story and a picture about marbles for this number sentence: 3 x 5 = 15." Students in this study came from pre - dominantly low- to average-income families living in three distinct geographical areas within the United States. A similar division task was…

Counting all and counting on are distinct counting strategies that can be used to compute such quantities as the total number of objects in two sets (Wright, Martland, and Stafford 2010). Given five objects and three more objects, for example, children who use counting all to determine quantity will count both collections; that is, they count…

As students progress from working with whole numbers to working with integers, they must wrestle with the big ideas of number values and order. Using objects to show positive quantities is easy, but no physical negative quantities exist. Therefore, when talking about integers, the author refers to number values instead of number quantities. The…

The learning of "basic facts"--single-digit combinations for addition, subtraction, multiplication, and division--has long been a focus of elementary school mathematics. Many people remember completing endless worksheets, timed tests, and flash card drills as they attempted to "master" their basic facts as children. However,…

The Common Core State Standards for Mathematics (CCSSI 2010) expect second grade students to "fluently add and subtract within 20 using mental strategies" (2.OA.B.2). Most children begin with number word sequences and counting approximations and then develop greater skill with counting. But do all teachers really understand how this…

Moving beyond physical interactions with materials is a significant mathematical step for students that is often difficult to take. Persistent tally-mark use, for example, among older children is a testament to this challenge. For many students, shifting away from tangible tools begins a precarious journey; teachers should support it with…

The Standards for Mathematical Practice (SMP) in the Common Core State Standards for Mathematics (CCSSM) (CCSSI 2010) highlight the mathematical habits that educators should be fostering in mathematics classrooms throughout K-grade 12 education. That argumentation and discourse are important components of developing mathematically proficient…

Incorporating the use of children's literature when teaching mathematics to young children is a developmentally appropriate practice: "Literature … provides a means for children to encounter mathematical concepts and vocabulary in the context of something familiar, a story" (Fogelberg et al. 2008). Moreover, introducing culturally…

Quite a bit of the arithmetic in elementary school contains elements of algebraic reasoning. After researching and testing a number of instructional strategies with Irish third graders, these authors found effective methods for cultivating a relational concept of equality in third-grade students. Understanding equality is fundamental to algebraic…