Night Session, Pascal’s Identity, Clip 6 of 7: Examples of Pascal’s Identity in the notation for combinations

Carolyn Alexander Maher, Regina D. Kiczek & Elena Perrone Steencken
This is the sixth of seven clips from the night session. After Jeff draws Pascal’s Triangle in what the students call “choose” notation, the researcher asks the students to express an instance of the addition rule using that “choose” notation (the notation for selecting combinations). Michael gives an example, and then he and the other students discuss the meaning of other specific instances of Pascal’s Identity in “choose” notation, in terms of specific instances of...

Shirts and Pants with Jamie and Michael, Clip 3 of 3: Recording a graphical representation of their solution

Amy Marie Martino & Alice S. Alston
In this clip, Jamie and Michael discuss their solutions to the Shirts and Pants Problem introduced in first clip in this series. They compare their individual solutions and find that they match. Researcher Alice Alston recommends that they depict a graphical representation of their solution on the large sheet in addition to the narrative solution which Jamie has already included. Jamie and Michael then work on drawing their solution of shirts and pants outfits on...

Night session, Pascal's identity, clip 3 of 7: further explorations of factorials and combinations

Carolyn Alexander Maher, Regina D. Kiczek & Elena Perrone Steencken
This is the third of seven clips from the night session. The four students (Ankur, Jeff, Michael, and Romina) investigate the reason for dividing n! by (n-x)! and x! when calculating “n choose x.” In explaining the specific example of “5 choose 2,” they use two analogies: 1) arranging five people on a line when you are concerned about the positions of only two of the people and 2) counting the number of 5-tall towers...

Alan's Infinity

Carolyn Alexander Maher
In this clip from 4th grade classroom study, students discuss the placement of numbers between zero and one on a number line. A debate ensues as to how many numbers exist between zero and one. Andrew suggests that a microscope can show more numbers than might be visible to the eye. Michael and Erik argue that the line would need to be made larger to include more numbers. Alan maintains that in any given interval,...

Equine behavior: behavior test, walking on cardboard test, Genesis

No Name Supplied
Test conducted as part of the Young Horse Teaching & Research Program of the Equine Science Center, Rutgers University, November, 2009. Subject was RU Genesis, a 6-month-old weanling draft cross born May, 2009. Test conducted by Dr. Sarah L. Ralston at the George H. Cook Campus, Rutgers University. Duration of test: 14 seconds. Overall score on a scale of 1-4 (4 is highest compliance): A-plus, according to sound track (4.25).

Early algebra ideas about binomial expansion, Stephanie's interview five of seven, Clip 1 of 10: Combinatorics notation for selecting a particular number of items from a total of 4.

Carolyn Alexander Maher & Elena Perrone Steencken
In the first clip in a series of ten from the fifth of seven interviews in which 8th grader Stephanie explores Early Algebraic Ideas about Binomial Expansion, the researcher, Carolyn Maher, introduces Stephanie to mathematical notation to represent the number of ways to select a specific number of objects from a set. Maher asks Stephanie to recall an earlier problem task, in which she was asked to determine the possible number of unifix "towers", four...

Equine behavior: behavior test, scary aisle test, Firedance

No Name Supplied
Test conducted as part of the Young Horse Teaching & Research Program of the Equine Science Center, Rutgers University. Subject was RU Firedance, a yearling draft cross born May, 2008. Test conducted by Dr. Sarah L. Ralston at the George H. Cook Campus, Rutgers University. Duration of test: 2 minutes. Overall score on a scale of 1-4 (4 is highest compliance): 3.464286.

Equine behavior: behavior test, preliminary test 1 with student (Fall 2010, indoor), Sierra, mustang two-year-old female

No Name Supplied
Test conducted by Sarah Shaw at the George H. Cook Campus, Rutgers University as part of the Young Horse Teaching & Research Program of the Equine Science Center, Rutgers University. One of a series of tests comparing behavior responses of mustangs fed a high starch diet versus behavior responses of mustangs fed a low starch diet. In the preliminary test, all the mustangs were on the same diet consisting of only forage (total mixed ration...

Early algebra ideas involving one variable, Clip 4 of 11: Working on equations two and three

Davis, Robert B. (Robert Benjamin), Carolyn Alexander Maher, Alice S. Alston, Amy Marie Martino & Thomas LaMonde Purdy
In the fourth of eleven clips from the first day of the Early Algebra Ideas 6th grade class sessions, the students work in small groups to find and record solutions for the first two of a series of open sentences that have been presented as a worksheet by the researcher, Robert B. Davis. As the students test different values in the equations, Davis keeps track on the white board of answers that the students propose...

Shirts and Pants Group Sharing, Clip 2 of 3: Michael, Gerardo, Jeff, Brian, and Matt share their solutions

Alice S. Alston
In this clip, students continue to share their solutions to the Shirts and Pants Problem, which they have worked on in the Shirts and Pants series. Michael and Gerardo share their solution in which they have drawn lines between the shirts and pants, saying that their solution is more like Stephanie and Dana’s than Jamie and Michael’s. Jeff and Brian, who have drawn six outfits of shirts and pants without using lines, share their solution...

A86, Discovering equivalent fractions and introducing fraction notation (classroom view), Grade 4, October 4, 1993, raw footage

Carolyn Alexander Maher
Researcher Maher began the session by discussing the relative size of fractions within different models. Using the analogy of two different sized dioramas of children in a fishing boat, the class agreed that the proportions within each model need to be the same, but that the parts of the models cannot be interchanged. Then, the class worked in partners on the problem: Which is larger, one half or two thirds, and by how much? Meredith,...

Continuing to explore fraction comparisons, Clip 2 of 7: Kimberly and Audra compare one half and three fourths

Amy Marie Martino
In the second clip, Kimberly and Audra worked together on the task: Which is larger one half or three fourths. Kimberly built a model using an orange and red train, two dark green rods, and four light green rods , and reasoned directly that three fourths was larger than one half by one fourth. She then set out to find another model, and built a model that was the same length using a different train...

Early algebra ideas involving one variable, Clip 8 of 11: Losing the "secret"

Davis, Robert B. (Robert Benjamin), Carolyn Alexander Maher, Alice S. Alston, Amy Marie Martino & Thomas LaMonde Purdy
In the eighth of eleven clips from the first day of the Early Algebra Ideas 6th grade class sessions, the researcher, Robert B. Davis, challenges the students to find values for the equation printed below. Students suggest a number of the factors of 60. However, when each factor is tested by substituting it into the equation and evaluating the result, none of the factors makes a true statement. Various students suggest that the "secret idea"...

Equine behavior: Marley has his temperature taken, 2009 or 2010

No Name Supplied
Produced as part of the Young Horse Teaching & Research Program of the Equine Science Center, Rutgers University, under the oversight of Dr. Sarah L. Ralston. Recorded at George H. Cook Campus, Rutgers University. Subject was RU Marley, a yearling mustang born ca. May, 2008. Student Elyse Conway holds Marley as Dr. Daniel P. Keenan takes his temperature.

Early algebra ideas involving two variables, Clip 1 of 18: Open sentences that can be made true or false with legal or illegal substitutions

Davis, Robert B. (Robert Benjamin), Carolyn Alexander Maher, Alice S. Alston & Amy Martino
This first of 18 clips focusing on Early Algebra Ideas Involving Two Variables occurred toward the end of the first of two consecutive 6th grade class sessions. Researcher, Robert B. Davis, builds on earlier exploration with a single variable to introduce algebraic equations with two variables. He uses the symbols of square (referred to as a “box”) and a “triangle” as to represent the variables. Using the equation printed below, the class discusses ways to...

The twentieth anniversary New Jersey Book Arts Symposium. Afternoon session

Karen Guancione, Lynn S. Mullins, Rachel Hadas & Michael Joseph
The New Jersey Book Arts Symposium is a daylong event held on the first Friday in November to explore and celebrate all aspects of the book arts, from time-honored arts and crafts used in traditional book-making practices, such as calligraphy, printing, illustration, binding, typography and paper-making, to avant-garde explorations of the ontology of the book in its adaptation to artistic expression in artists' books. It customarily focuses its activities on a specific theme, but this...

Early algebra, investigating linear functions, series 5 of 7, ladder problem, Clip 7 of 7: Combining a six step ladder with a two step ladder

John Francisco & Prashant V. Baldev
In the final clip of the series of seven from an after-school enrichment session in an urban middle school, Ariel, a 7th grade boy, revisits his method for solving a specific example of the Ladder Problem. Researcher John Francisco asks Ariel explain how he had predicted the number of rods for a ladder with 125 steps. The procedure that Ariel had articulated and followed for smaller numbers had the following steps: (1) Find the closest...

Comparing fractions, a whole class debate, Clip 3 of 5: Brian challenges the girls' argument

Carolyn Alexander Maher
In the third clip from this session, the class reviewed the problem: Which is bigger, one half and one third, and by how much? Laura and Jessica built a model to show that one half was larger than one third using an orange and red train to represent one, dark green rods to represent one half, and purple rods to represent one third. Audra joined Laura and Jessica to corroborate their findings. When researcher Carolyn...

Early algebra ideas involving two variables, Clip 16 of 18: Working on problems 8 and 9

Davis, Robert B. (Robert Benjamin), Carolyn Alexander Maher, Alice S. Alston & Amy Marie Martino
In the sixteenth of 18 clips from Early Algebra Ideas Involving Two Variables on the second of two consecutive classroom sessions with the class of 6th grade students, the students are looking for relationships among the values in the tables that will help them to find equations for problems 8 and 9. Researcher Robert B. Davis points out to the group that they are to construct equations that do not depend on knowing the value...

Topological frustration of artificial spin ice supplemental movies

John Cumings, Thomas Marsh & Jasper Drisko
Supplemental movies for the manuscript entitled "Topological frustration of artificial spin ice."

The twentieth anniversary New Jersey Book Arts Symposium. Morning session

Michael Joseph, Gaunt, Marianne (Marianne Irene), Anna M. Aschkenes, Béatrice Coron, Karen Guancione, Asha Ganpat, Celeste Regal & Diane Savona
The New Jersey Book Arts Symposium is a daylong event held on the first Friday in November to explore and celebrate all aspects of the book arts, from time-honored arts and crafts used in traditional book-making practices, such as calligraphy, printing, illustration, binding, typography and paper-making, to avant-garde explorations of the ontology of the book in its adaptation to artistic expression in artists' books. It customarily focuses its activities on a specific theme, but this...

Building large models to show equivalence, a generalization, Clip 1 of 3: Finding the smallest model

Carolyn Alexander Maher
At the start of the session, researcher Carolyn Maher asked the students if they remember working on comparing two thirds and three quarters. She mentioned that she had seen students build more than one model, and asked the students how many models they thought were possible to build. Michael conjectured that if one knew the “secret” behind the models, he could build models using all the rods. Amy commented that they had found six models,...

A96, Building large models to show equivalence, a generalization (student view), Grade 4, Oct. 11, 1993, raw footage

Carolyn Alexander Maher
During this session, conducted as a whole class discussion, Researcher Carolyn Maher revisited the problem that the students had worked on during the previous three sessions: Which is larger, two thirds or three fourths, and by how much? First, she asked students to show her the smallest model that could be built to show the difference between the two fractions. After it was built, she asked the students to convince her that it was indeed...

A95, Building large models to show equivalence, a generalization (presentation view), Grade 4, Oct. 11, 1993, raw footage

Carolyn Alexander Maher
During this session, conducted as a whole class discussion, Researcher Carolyn Maher revisited the problem that the students had worked on during the previous three sessions: Which is larger, two thirds or three fourths, and by how much? First, she asked students to show her the smallest model that could be built to show the difference between the two fractions. After it was built, she asked the students to convince her that it was indeed...

Shirts and Pants with Jamie and Michael, Clip 3 of 3: Recording a graphical representation of their solution

Amy Marie Martino
In this clip, Jamie and Michael discuss their solutions to the Shirts and Pants Problem introduced in first clip in this series. They compare their individual solutions and find that they match. Researcher Alice Alston recommends that they depict a graphical representation of their solution on the large sheet in addition to the narrative solution which Jamie has already included. Jamie and Michael then work on drawing their solution of shirts and pants outfits on...

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