Erien and her students examine soil profiles of the wetlands near their school. The focus of their investigation is on the soil's profile, what organisms live in the soil, and at what level the students encountered water when they were digging.
State Standard NSES . Soil consists of weathered rocks and decomposed organic material from dead plants, animals, and bacteria. Soils are often found in layers, with each having a different chemical composition and texture.
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Students make models of the life cycle of amphibians and go on a field trip.
State Standard NSES . Plants and animals have life cycles that include being born, developing into adults, reproducing, and eventually dying. The details of this life cycle are different for different organisms.
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Gina Kiczek introduces a problem that helps her students learn the difference between permutations and combinations. Students in the Kenilworth study examine "Pascal's Triangle:" what it is, its history, and how it is linked to the Towers and Pizza problems. In the 11th grade, the Kenilworth students build on their earlier thinking as young children to tackle a complex—and realistic—probability problem, the World Series Problem. In an after-school research session, Mike derives the formula for the general combinatorics problem “compute n choose r.”
State Standard PSSM . Understand the meaning of operations and how they relate to one other (specifically, develop an understanding of permutations and combinations as counting techniques) (PSSM Number Standard for grades 9-12)
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In a voluntary two-week summer workshop, high school seniors from Kenilworth and New Brunswick work on a real-life problem ("The Catwalk") based on Eduard Muybridge's sequence of 24 photographs of a cat in motion. The question, “How fast is the cat moving in frame 10 and frame 20?,” deals with some of the fundamental ideas leading toward an understanding of calculus. Students find several ways to represent their growing understanding: visual, symbolic, and kinesthetic.
State Standard PSSM . Use representations to model and interpret physical, social, and mathematical phenomena (PSSM Representation Standard for Grades 9-12)
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In a voluntary two-week summer workshop, high school seniors from Kenilworth and New Brunswick work on a real-life problem ("The Catwalk") based on Eduard Muybridge's sequence of 24 photographs of a cat in motion. The question, “How fast is the cat moving in frame 10 and frame 20?,” deals with some of the fundamental ideas leading toward an understanding of calculus. Students find several ways to represent their growing understanding: visual, symbolic, and kinesthetic.
State Standard PSSM . Develop and evaluate mathematical arguments and proofs (PSSM Reasoning and Proof Standard for Grades 9-12)
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A high school teacher describes an activity where students receive bags of M&M's and make predictions about the number of each color in one bag, a group of bags, etc. They consider how the data changes as the sample size increases. The teacher provides an assessment list called a "Did I?" sheet and makes his expectations clear by giving his assessment rubric to students. Also there is some discussion of a bilingual classroom where students share the same experience but discuss it in groups using different languages.
Topic Probability and Statistics
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This clip shows an elementary school teacher explaining an activity she uses where students begin with a picture of several hexagons and then have to add other shapes to make up the same area.
Topic Ratios and Proportionality
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Inquiry and Habits of Mind are two learning goals which may be particularly amenable to embedded assessment. There is a discussion about a science project that students work on over an extended period as a means of assessment.
Topic Habits of Mind
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This video clip shows groups of students using calculators to graph functions. The students are discussing whether the function xsin(x) goes through the origin. The teacher realizes that the students are having difficulty with composing the graphs of the functions x and sin x and makes a modification in her planned instruction by going back and making a "T-table" showing the values of xsin(x) for particular values of x.
Topic Graphs of functions.
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This clip shows a discussion of an activity which makes a connection between math and science: students launch metal spheres from different heights on a ramp. The students plot the relationship between the height and the distance the spheres travel. The students are given a new height and asked to predict how far the ball will travel.
Topic Graphing
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