In this article, the authors discuss a special type of multiplication-and-division-of-fractions problem that elementary school teachers can use to promote children's understanding of fractional quantities and their relationships. These problems are accessible to students working at different levels of fraction understanding, and they can be solved…

The Smarties-Box Challenge encourages students to apply several different mathematical capabilities and concepts--such as, estimation, multiplication, and the notion of being systematic--to solve a complex, multistep problem. To effectively engage in the Smarties-Box Challenge, students are required to demonstrate aspects of all four proficiency…

As a new physics teacher, I was explaining how to find the weight of an object sitting on a table near the surface of the Earth. It bothered me when a student asked, "The object is not accelerating so why do you multiply the mass of the object by the acceleration due to gravity?" I answered something like, "That's true, but if the table were not…

Algorithms and representations have been an important aspect of the work of mathematics, especially for understanding concepts and communicating ideas about concepts and mathematical relationships. They have played a key role in various mathematics standards documents, including the Common Core State Standards for Mathematics. However, there have…

One of the challenges facing teacher educators is providing our pre-service teachers (PSTs) with authentic experiences that cross the boundaries between Initial Teacher Education (ITE) and the classroom. An additional challenge facing the mathematics teacher educator, is addressing and deepening PSTs' mathematical content knowledge (MCK), which…

Individual learning is out of sync with the elements of a curricula and the daily program of a teacher. At a time when multidigit multiplication methods are taught, many children are not perfectly performing the basic multiplication table. Teachers organize settings for learning and they usually have no time to give an individual feedback to every…

Mathematics interventions for struggling students supplement regular instruction and are frequently implemented through computer programs and activity sheet practice (Swanson et al. 2012). However, students with foundational misunderstandings do not benefit from additional drill and practice alone. Instead, interventions that balance concepts with…

With the adoption of the Common Core State Standards for Mathematics (CCSSM) (CCSSI 2010), many concepts related to area are covered in third grade: (1) Recognizing area as an attribute of a plane figure; (2) Understanding that a square with a side length of one is a unit square; (3) Measuring area by tiling figures and counting the squares it…

We show that students rearranging the terms of a mathematical equation in order to separate variables prior to integration use gestures and speech to manipulate the mathematical terms on the page. They treat the terms of the equation as physical objects in a landscape, capable of being moved around. We analyze our results within the tradition of…

Of the four mathematical operators, division seems to not sit easily for many learners. Division is often described as "the odd one out". Pupils develop coping strategies that enable them to "get away with it". So, problems, misunderstandings, and misconceptions go unresolved perhaps for a lifetime. Why is this? Is it a case of "out of sight out…

Dot arrays provide opportunities for students to notice structures like commutativity and distributivity, giving these properties an image that can be manipulated and explored. These images also connect to ways that we organize discrete objects in everyday life. This article describes how the authors developed an array of dot tasks that have been…

This article presents a sequence of learning activities that lead to using the area model of multiplication to understand the distributive property (DP). The connection between area and multiplication is an important one, both for algebraic thinking and for geometry, as indicated in two of the critical areas for the third grade in the Common Core…

As students progress through elementary school, they encounter mathematics concepts that shift from additive to multiplicative situations (NCTM 2000). When they encounter fraction problems that require multiplicative thinking, they tend to incorrectly extend additive properties from whole numbers (Post et al. 1985). As a result, topics such as …

Many rules taught in mathematics classrooms "expire" when students develop knowledge that is more sophisticated, such as using new number systems. For example, in elementary grades, students are sometimes taught that "addition makes bigger" or "subtraction makes smaller" when learning to compute with whole numbers,…

For centuries, the basic operations of school mathematics have been identified as addition, subtraction, multiplication, and division. Notably, these operations are "basic," not because they are foundational to mathematics knowledge, but because they were vital to a newly industrialized and market-driven economy several hundred years…