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

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### Janet M. Sharp

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.

### Janet Sharp and Rachael M. Welder

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

### Janet Sharp, Tracie Lutz and Donna E. LaLonde

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

### Janet Sharp and Karen Hoiberg

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.

### Janet M. Sharp and Corrine Heimer

WE HAVE TO SHARE THIS WITH OUR students! They will love it!” This statement was all we could think about after a professional development session dealing with geometry. Spherical geometry challenged our capabilities in geometry but greatly interested us. Before we could teach our students about spherical geometry, we needed to learn more about this strange new world ourselves. In this article, we describe our discoveries and some of the activities we developed for our sixth-grade students.

### Janet M. Sharp and Barbara Adams

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.

### Janet Sharp, Loren Zachary and Greg Luttenegger

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.

### Janet M. Sharp and Karen Bush Hoiberg

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.

### Karen Bush Hoiberg, Janet Sharp, Ted Hodgson and Jim Colbert

Why Do Some Plants Have Larger leaves than other plants? Why do the overall shapes of different kinds of leaves vary? How does one determine the area of a peculiarly shaped leaf? Plant biologists are extremely interested in these questions. A biologist might wonder, for instance, which of two plant species carries out the most photosynthesis. Since the amount of light a plant can absorb for use in photosynthesis is related to its area, the biologist might investigate the photosynthetic capacity of two species by comparing the amount of chlorophyll in leaf pieces of the same area.