1,720,959 research outputs found
Pascal’s Triangle and Pascal’s Identity: Contextualizing and Decontextualizing
No full textAccording to the Common Core State Standards, the ability to contextualize and the ability to decontextualize are important mathematical skills for students to develop. Learners should have "the ability to decontextualize--to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents--and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved" (CCSS, 2012).
This analytic focuses on a problem-solving session involving four high school students named Ankur, Jeff, Michael, and Romina, participants in a long-term study of students’ mathematical learning (Maher, 2002). (A fifth student, Brian, joins the group briefly, but does not participate in the discussions highlighted in this analytic.) The students make sense of Pascal’s Triangle and Pascal’s Identity (the addition rule for Pascal’s Triangle) by referring to combinatorics problems they know well, including the pizza problem. * The analysis of this session, called the Night Session,** includes seven events. In the first two, students demonstrate the ability to contextualize the numbers in Pascal’s Triangle by referring to the pizza problem and to binary numbers. In the third and fourth events, they demonstrate the ability to decontextualize the pizza problem in order to generate Pascal’s Identity in general form using standard mathematical notation. The last three events provide further examples of decontextualization: working with symbolic notation only, the students convert Pascal’s Identity to factorial notation, correct an arithmetic error, and simplify the equation.
*The generalized pizza problem is: How many pizzas is it possible to make when there are N toppings to choose from? The answer is: there are 2^N possible pizzas because there are two choices for each of the N toppings: on or off the pizza.
**The Night Session took place on the evening of May 12, 1999, during the students’ junior year in high school. The five students had participated in the longitudinal study since elementary school. At the time of the Night Session, they had worked together on combinatorics problems involving Pascal’s Triangle on seven occasions between December 1997 and January 1999 (Uptegrove, 2005).
References
Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Retrieved February 11, 2014, from www.corestandards.org/standards/mathematics
Maher, C. (2002). How students structure their own investigations and educate us: What we have learned from a fourteen year study. In A. Cockburn & E. Nardi (Eds.), Proceedings of the Twenty-sixth Annual Meeting of the International Group for the Psychology of Mathematics Education (PME26), Vol I, pp. 31-46. Norwich, England: School of Education and Professional Development, University of East Anglia.
Uptegrove, E. (2005). To symbols from meaning: Students’ investigations in counting. (Unpublished doctoral dissertation) Rutgers University, NJ
David, Erik, and Meredith Use Reasoning to Resolve Conjectures
No full textMeredith rebuilds the previous day’s models for comparing 2/3 and 3/4. The longer model is 24 cm long; a train of a blue rod, a black rod, and a brown rod, representing 1. In this model, a dark green rod represents 1/4, a brown rod represents 1/3, a red rod represents 1/12, and a white rod represents 1/24. Meredith shows that the difference between three greens and two browns is one red (1/12) or two whites (2/24). The other model is 12 cm long; it consists of an orange rod plus a red rod. In this model, a light green rod represents 1/4, a purple rod represents 1/3, and a white rod represents 1/12. Meredith shows that in this model, the difference (1/12) is represented by one white rod.
Researcher Maher challenges Meredith, David, and other students to predict what a larger model would look like without building it. David and Meredith conjecture that there could be a larger model in which the white represents 1/48.
Students are asked to work on a model to verify David’s conjecture. Erik builds another model of length 36 cm (3 blue rods plus 9 white rods), which can be used to show thirds and fourths, but which does not model David’s conjecture.
After realizing that they have not built a model for David’s conjecture, students help David build a model in which the white rod represents 1/48. When asked, David says that he is not surprised at this result; that’s what he thought it would be
James Finds the Difference Between 1/4 and 1/9
No full textThe class is investigating the difference between 1/4 and 1/9. In initial discussions, some students agree with Meredith, who says that the difference is 1/5 because 9 - 4 = 5. Researcher Maher indicates to the class that if they apply Meredith’s rule to 1/2 and 1/4, they would get an answer of 1/2, challenging their earlier reasoning that produced an answer of 1/4. James offers a model consisting of 3 orange rods and a dark green rod (36 white rods) that can be partitioned into ninths and fourths. He presents the model to Researcher Davis. First, James claims that the difference is 1/5, illustrating this difference by 5 white rods. Researcher Davis responds by asking James to clarify what number name is assigned to one white rod, and then of two white rods. James responds 1/36 and 2/36, respectively. James realizes that if the difference between 1/4 and 1/9 is represented by 5 white rods, then that difference must be 5/36. James then presents this finding to the class
Students Compare 1/4 and 1/9
No full textThe class is investigating the difference between 1/4 and 1/9. In initial discussions, some students agree with Meredith, who says that the difference is 1/5 because 9 – 4 = 5. Researcher Maher indicates to the class that if they apply Meredith’s rule to 1/2 and 1/4, they would get an answer of 1/2, challenging their earlier reasoning that produced an answer of 1/4.
Some students build one model to show ninths and a second model (of a different length) to show fifths. For example, Alan builds one model where the blue rod represents 1 and so the white rod would have the number name of 1/9 and a second model where the purple rod represents 1and so the white rod would have the number name of 1/4. Alan concludes that 1/9 is smaller than 1/4, but he is unable to determine the difference using these models. After he is unsuccessful in his attempt to construct a model that can be divided into ninths and fourths, Alan conjectures that the reason is that it is not possible to build a model when one denominator is odd and the other is even. Researcher Maher reminds him that in previous sessions, he built a model for comparing 1/2 and 2/3; in another previous session, he built a model to compare 3/4 and 2/3.
Meanwhile, James builds a model consisting of 3 orange rods and a dark green rod (36 cm.) that can be partitioned into ninths and fourths. He presents the model to Researcher Davis. First, James claims that the difference is 1/5, illustrating this difference by 5 white rods. Researcher Davis responds by asking James to clarify what number name is assigned to one white rod, and then of two white rods. James responds 1/36 and 2/36, respectively. James realizes that if the difference between 1/4 and 1/9 is represented by 5 white rods, then that difference must be 5/36. James then presents this finding to the class
Extending the Doubling Conjecture
No full textIn the previous day’s session, students conjectured that, once they had a model for comparing the fractions 2/3 and 3/4, it was possible to build another model twice the size of that original model.
In this session, Alan uses the doubling conjecture to build three models to compare 1/2 and 2/5. The first model uses one orange rod to represent 1; the second uses two orange rods to represent 1; the third model uses four orange rods to represent 1. Alan conjectures that adding four orange rods to his largest model would result in a very large model that will also work.
This analytic is from raw video A73, October 8, 1993
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Variations on the Author
Full text link“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
Appropriate Similarity Measures for Author Cocitation Analysis
Full text linkWe provide a number of new insights into the methodological discussion about author cocitation analysis. We first argue that the use of the Pearson correlation for measuring the similarity between authors’ cocitation profiles is not very satisfactory. We then discuss what kind of similarity measures may be used as an alternative to the Pearson correlation. We consider three similarity measures in particular. One is the well-known cosine. The other two similarity measures have not been used before in the bibliometric literature. Finally, we show by means of an example that our findings have a high practical relevance.information science;Pearson correlation;cosine;similarity measure;author cocitation analysis
Dispelling the Myths Behind First-author Citation Counts
Full text linkWe conducted a full-scale evaluative citation analysis study of scholars in the XML research field to explore just how different from each other author rankings resulting from different citation counting methods actually are, and to demonstrate the capability of emerging data and tools on the Web in supporting more realistic citation counting methods. Our results contest some common arguments for the continued
use of first-author citation counts in the evaluation of scholars, such as high correlations between author rankings by first-author citation counts and other citation
counting methods, and high costs of using more realistic citation counting methods that are not well-supported by the ISI databases. It is argued that increasingly available digital full text research papers make it possible for citation analysis studies to go beyond what the ISI databases have directly supported and to employ more
sophisticated methods
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