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• Author or Editor: Janet Sharp
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## A Teacher's Journal: Manipulatives for the Metal Chalkboard

Manipulates represent one possible teaching tool for building a child's conceptual foundations. Once mathematical ideas are experienced at the concrete level, they must eventually he matched to a symbolic expression. In this way, conjectures can be explored and verified.

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## A Calculus Activity with Foundations in Geometric Learning

Modified instruction to replace the deductive textbook approach helps calculus students improve their conceptual understanding and precision.

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## Reveal Limitations through Fraction Division Problem Posing

Students notoriously struggle with division of fractions in 5 key areas. Hear what those 5 areas are and how recommendations address the limitations.

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## And Then There Was Luke: The Geometric Thinking of a Young Mathematician

What might students say about the angles of the pentagonal block shown in figure 1? Children might respond in different ways, depending on their abilities and experiences with angles. Some might say that the block “has five angles” after touching each of the corners. Others might observe that “it looks like all the angles are the same size.” Perhaps a few would respond as did Luke, a bright fifth grader who is featured in this article.

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## What Happens to Geometry on a Sphere?

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## Using Moon Phases to Measure Time

A lesson on time incorporates science, mathematics, and literacy while exploring Hopi Native American culture.

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## Using a Pattern Table to Solve Contextualized Proportion Problems

Students in a sixth-grade classroom we visited were celebrating a classmate's birthday and enjoying fun-sized bags of Peanut M&M's candies. We overheard them discussing their curiosity at the small number of blue M&M's each of them had received in their small bags. Because the students were occupied in an informal, party atmosphere, we were pleasantly surprised to hear one student, Rickea, comment on a related mathematical issue. She speculated that the teacher's class-sized bag would have relatively few blue M&M's, as well. What a wonderful teaching opportunity for ratios and proportions Rickea's casual comment posed! In this article, we describe (1) how we built a week-long, problem-based unit around Rickea's original proportion question and (2) the effectiveness of using problem solving to help Rickea and her classmates construct knowledge about ratio and proportional thinking.

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## Implementing the Assessment Standards: The Process of Assessment Applied to Tessellations

A comprehensive process design, which facilitates the analysis of all events that have an impact on students’ mathematical experiences, is outlined in the Assessment Standards for School Mathematics (NCTM 1995). This process of assessment is held to six standards: Mathematics, Learning, Equity, Openness, Inference, and Coherence. These Standards represent those ideas that are valued and by which mathematical assessment should be judged.

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## Using Engineering to Understand Reciprocal Functions

Graphic representations of numerical data can illustrate relationships within the data that students might not otherwise notice. When students investigate common numerical data patterns like 2, 4, 6, 8 or 1, 3, 5, 7, they can easily recognize that differences between terms are constant.

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## The Intersection of Two Unlikely Worlds: Ratios and Drums

When mathematics lessons are linked with personal experiences, typically, the result is that the student gains a stronger understanding of the content than if the lessons are isolated and unconnected. This premise was recently supported in a local fifth-grade classroom. The students learned to play three mathematically disparate rhythms on conga drums as an introduction to an exploration of ratio. Ratios connect naturally with African and Afro-Cuban drumming because the drummer's combination of many rhythms, each with a pattern repetition of different length results in a polyrhythmic song. The pattern repetitions are comprised of a given quantity of one type of beat mixed with a specified quantity of another type of beat, or a ratio of one beat to the other one. Although we completed this lesson with a group of children of whom half were African American, we believe this lesson can be powerful and meaningful for children of all ethnic backgrounds.