Private Universe Project in Mathematics: Workshop 2. Are You Convinced?
Proof making is one of the key ideas in mathematics. Looking at teachers and students grappling with the same probability problem, we see how two kinds of proof—proof by cases and proof by induction—naturally grow out of the need to justify and convince others.,Englewood, New Jersey—Teachers Workshop Englewood, a town with unsatisfactory student test scores, is implementing a long-term project to improve math achievement. As part of a professional development workshop designed in part to give
Author(s): Harvard-Smithsonian Center for Astrophysics

Private Universe Project in Mathematics: Workshop 2. Are You Convinced?
Proof making is one of the key ideas in mathematics. Looking at teachers and students grappling with the same probability problem, we see how two kinds of proof—proof by cases and proof by induction—naturally grow out of the need to justify and convince others.,25 min. Englewood, New Jersey—Teachers Workshop Englewood, a town with unsatisfactory student test scores, is implementing a long-term project to improve math achievement. As part of a professional development workshop designed in part
Author(s): Harvard-Smithsonian Center for Astrophysics

Private Universe Project in Mathematics: Workshop 1. Following Children's Ideas in Mathematics
An unprecedented long-term study conducted by researchers at Rutgers University followed the development of mathematical thinking in a randomly selected group of students for 12 years—from first grade through high school—with surprising results. In an overview of the study, we look at some of the conditions that made the students’ math achievement possible.,10 min Building Towers Five-HighThe Kenilworth students in the fourth grade are seen working on the Towers problem (“How many different
Author(s): Harvard-Smithsonian Center for Astrophysics

Private Universe Project in Mathematics: Workshop 5. Building on Useful Ideas
One of the strands of the Rutgers long-term study was to find out how useful ideas spread through a community of learners and evolve over time. Here, the focus is on the teacher’s role in fostering thoughtful mathematics.,Englewood—Second Grade: Probing Student Thinking. How can a teacher know what an individual student is thinking when there are 24 or more students in the room? In Englewood, a second-grade teacher tries to follow her students’ thinking by asking appropriate questions as she
Author(s): Harvard-Smithsonian Center for Astrophysics

Private Universe Project in Mathematics: Workshop 5. Building on Useful Ideas
One of the strands of the Rutgers long-term study was to find out how useful ideas spread through a community of learners and evolve over time. Here, the focus is on the teacher’s role in fostering thoughtful mathematics.,Englewood—Second Grade: Probing Student Thinking. How can a teacher know what an individual student is thinking when there are 24 or more students in the room? In Englewood, a second-grade teacher tries to follow her students’ thinking by asking appropriate questions as she
Author(s): Harvard-Smithsonian Center for Astrophysics

Private Universe Project in Mathematics: Workshop 5. Building on Useful Ideas
One of the strands of the Rutgers long-term study was to find out how useful ideas spread through a community of learners and evolve over time. Here, the focus is on the teacher’s role in fostering thoughtful mathematics.,“Equations" In Colts Neck, New Jersey, fourth-grade teacher and former Rutgers researcher Amy Martino finds out that what started as a 15-minute “warm-up” question evolves into an interesting discussion about equations.
Author(s): Harvard-Smithsonian Center for Astrophysics

Private Universe Project in Mathematics: Workshop 5. Building on Useful Ideas
One of the strands of the Rutgers long-term study was to find out how useful ideas spread through a community of learners and evolve over time. Here, the focus is on the teacher’s role in fostering thoughtful mathematics.,Englewood—Fourth Grade: Towers Fourth-grade teacher Blanche Young attempts the Towers activity for the first time with her students. She feels that their work is valuable, but questions how much time these open-ended activities are taking away from the standard curriculum.
Author(s): Harvard-Smithsonian Center for Astrophysics

Private Universe Project in Mathematics: Workshop 5. Building on Useful Ideas
One of the strands of the Rutgers long-term study was to find out how useful ideas spread through a community of learners and evolve over time. Here, the focus is on the teacher’s role in fostering thoughtful mathematics.,Englewood, NJ - Kindergarten: Stacking Blocks In Englewood, New Jersey, a kindergarten teacher uses blocks as mathematical objects in an addition activity.
Author(s): Harvard-Smithsonian Center for Astrophysics

Private Universe Project in Mathematics: Workshop 4: Thinking Like a Mathematician
What does a mathematician do? What does it mean to think like a mathematician? This program parallels what a mathematician does in real life with the creative thinking of students.,Romina’s Proof - Responding to a problem posed by one of the students, Romina, a 10th-grader, invents a proof solution and shares it with the others.
Author(s): Harvard-Smithsonian Center for Astrophysics

Private Universe Project in Mathematics: Workshop 4: Thinking Like a Mathematician
What does a mathematician do? What does it mean to think like a mathematician? This program parallels what a mathematician does in real life with the creative thinking of students.,Revisiting Problems After Five Years - Kenilworth 10th graders re-examine the same problem they had last seen in the fifth grade—the Pizza problem. One student, Michael, uses the binary number system to his advantage.
Author(s): Harvard-Smithsonian Center for Astrophysics

Private Universe Project in Mathematics: Workshop 4: Thinking Like a Mathematician
What does a mathematician do? What does it mean to think like a mathematician? This program parallels what a mathematician does in real life with the creative thinking of students.,How a Mathematician Approaches Problems Fern Hunt, a mathematician at the National Institute for Standards and Technology, is seen as she collaborates with colleagues to solve difficult technical problems. Using the metaphor of the children’s game Towers of Hanoi, she explains her approach to solving problems. 15 min
Author(s): Harvard-Smithsonian Center for Astrophysics

Private Universe Project in Mathematics: Workshop 4: Thinking Like a Mathematician
What does a mathematician do? What does it mean to think like a mathematician? This program parallels what a mathematician does in real life with the creative thinking of students.,How a Mathematician Approaches Problems. Fern Hunt, a mathematician at the National Institute for Standards and Technology, is seen as she collaborates with colleagues to solve difficult technical problems. Using the metaphor of the children’s game Towers of Hanoi, she explains her approach to solving problems. In a
Author(s): Harvard-Smithsonian Center for Astrophysics

Private Universe Project in Mathematics: Workshop 3. Inventing Notations
We learn how to foster and appreciate students’ notations for their richness and creativity. We also look at some of the possibilities that early work in creating notation systems might open up for students as they move on toward algebra.,Kenilworth Study: Pizzas In the fourth grade, the students encounter counting problems where the solutions cannot be built using standard manipulatives. As he invents his own notation systems, one student, Matt, builds on previous work to arrive at a solution
Author(s): Harvard-Smithsonian Center for Astrophysics

Private Universe Project in Mathematics: Workshop 3. Inventing Notations
We learn how to foster and appreciate students’ notations for their richness and creativity. We also look at some of the possibilities that early work in creating notation systems might open up for students as they move on toward algebra.,15 min. Pizzas in the Classroom In Englewood, New Jersey, Blanche Young, who attended the summer workshop, tries out one of the problems with her fourth-grade students. Later, she meets with Arthur Powell to discuss the lesson. 5 min. New Brunswick, New Jersey
Author(s): Harvard-Smithsonian Center for Astrophysics

Private Universe Project in Mathematics: Workshop 2. Are You Convinced?
Proof making is one of the key ideas in mathematics. Looking at teachers and students grappling with the same probability problem, we see how two kinds of proof—proof by cases and proof by induction—naturally grow out of the need to justify and convince others.,8 min. Working With Towers- In the third grade, students in the Kenilworth study build towers four-high, and hypothesize about towers three-high. In the fourth grade, they build towers five-high. 20 min. “Gang of Four” In the fourth g
Author(s): Harvard-Smithsonian Center for Astrophysics

Private Universe Project in Mathematics: Workshop 2. Are You Convinced?
Proof making is one of the key ideas in mathematics. Looking at teachers and students grappling with the same probability problem, we see how two kinds of proof—proof by cases and proof by induction—naturally grow out of the need to justify and convince others.,Englewood, New Jersey—Teachers Workshop Englewood, a town with unsatisfactory student test scores, is implementing a long-term project to improve math achievement. As part of a professional development workshop designed in part to give
Author(s): Harvard-Smithsonian Center for Astrophysics

Private Universe Project in Mathematics: Workshop 1. Following Children's Ideas in Mathematics
An unprecedented long-term study conducted by researchers at Rutgers University followed the development of mathematical thinking in a randomly selected group of students for 12 years—from first grade through high school—with surprising results. In an overview of the study, we look at some of the conditions that made the students’ math achievement possible.,10 min Building Towers Five-High The Kenilworth students in the fourth grade are seen working on the Towers problem (“How many different
Author(s): Harvard-Smithsonian Center for Astrophysics

Private Universe Project in Mathematics: Workshop 1. Following Children's Ideas in Mathematics
An unprecedented long-term study conducted by researchers at Rutgers University followed the development of mathematical thinking in a randomly selected group of students for 12 years—from first grade through high school—with surprising results. In an overview of the study, we look at some of the conditions that made the students’ math achievement possible.,5 min. Mathematics in Free Play? Prof. Herbert Ginsburg, a psychologist at Columbia University Teachers College, finds that when you exam
Author(s): Harvard-Smithsonian Center for Astrophysics

Science In Focus: Energy
Interview with Dr. Sallie Baliunas about forms of energy including springs and magnets.,Dr. Sallie Baliunas explains the energy transfers that occur when she pushes down on a spring and then releases it. She explains that she adds potential energy to the spring when she pushes it down, energy that is bound up in the coils. When she releases the spring, the potential energy becomes energy of motion. Some potential energy is used to move air molecules, producing sound, and some is lost to heat
Author(s): Harvard-Smithsonian Center for Astrophysics

Private Universe Project in Mathematics: Workshop 5. Building on Useful Ideas
One of the strands of the Rutgers long-term study was to find out how useful ideas spread through a community of learners and evolve over time. Here, the focus is on the teacher’s role in fostering thoughtful mathematics.,Jersey City: Ice Cream Problem Algebra II teacher Gina Kiczek introduces a problem that helps her students learn the difference between permutations and combinations. What Is Pascal’s Triangle? An overview of the “Arithmetic Triangle”: what it is, its history, and how it is
Author(s): Harvard-Smithsonian Center for Astrophysics