Preservice elementary teachers (PSTs) often enter their teacher preparation programs with procedural and underdeveloped understandings of area measurement and its applications. This is problematic given that area and the area model are used throughout K–Grade 12 to develop flexibility in students’ mathematical understanding and to provide them with a visual interpretation of numerical ideas. This study describes an intervention aimed at bolstering PSTs’ understanding of area and area units with respect to measurement and number and operations. Following the intervention, results indicate that PSTs had both an improved ability to solve area tiling tasks as well as increased flexibility in the strategies they implemented. The results indicate that PSTs, similar to elementary students, develop a conceptual understanding of area from the use of tangible tools and are able to leverage visualizations to make sense of multiplicative structure across different strategies.

# Browse

Excerpts from discussion threads on the online MyNCTM community

### Karl W. Kosko

Use Cuisenaire™ Rods to emphasize the column-and-row structure in arrays for meaningful multiplication.

### Krista Francis and Michael Poscente

Lego Mindstorms™ robotics quickly draws children in and provides ample opportunities for engaging them in robust mathematical learning. Two introductory programming tasks empower children to use their creativity and problem-solving skills to build number sense in a fun, engaging learning environment. Contributors to the iSTEM (Integrating Science, Technology, Engineering, and Mathematics) department share ideas and activities that stimulate student interest in the integrated fields of science, technology, engineering, and mathematics (STEM) in K–grade 6 classrooms. Send submissions of no more than 1500 words to this department by accessing http://tcm.msubmit.net. See detailed submission guidelines for all departments at http://www.nctm.org/WriteForTCM.

### James Metz

A t a party that I attended, the hosts gave their guests the Tower of Hanoi puzzle with alternating dark and light discs and a challenge to move the 7 discs to a new post. (I disqualified myself because I knew how to solve the challenge.) However, the hosts' son and daughter-in-law misunderstood the directions and moved the dark discs to one side post and the light discs to the other side post. I immediately wondered, “How many moves did they take, assuming that they made the most efficient moves? How can their interpretation of the problem be generalized to n discs?”

### Jennifer Throndsen

This simple dice game supports students' development of flexibility with numbers, the properties of the four operations (+, −, ×, ÷), and the order of operations. It requires only dice and a game board for each player.

### Joe Champion and Ann Wheeler

A classic manipulative, used since the 1960s, continues to offer opportunities for intriguing problem solving involving proportions.

### Rachel Levy

The mathematical concept of slope can be made real through a set of simple, inexpensive, and safe experiments that can be conducted in the classroom or at home. The experiments help connect the idea of slope with physical phenomena related to surface tension. In the experiments, changes in surface tension across the surface of the water, which correspond to greater slopes on the graph, lead to increased motion of the fluid. The mathematical content, targeted to middle school and high school students, can be used in a classroom or workshop setting and can be tailored to a single session of thirty to ninety minutes.

### Kyle T. Schultz and Stephen F. Bismarck

A geometric approach using exact square manipulatives can promote an understanding of the algorithm to dismantle radical expressions.

### Jessica F. Shumway

Three days of using building blocks significantly enriched second graders' thinking about multiple dimensions.