Pascal s triangle as assimilation paradigm standard notation l.jpg
This presentation is the property of its rightful owner.
Sponsored Links
1 / 40

PASCAL’S TRIANGLE AS ASSIMILATION PARADIGM: Standard Notation PowerPoint PPT Presentation


  • 94 Views
  • Uploaded on
  • Presentation posted in: General

PASCAL’S TRIANGLE AS ASSIMILATION PARADIGM: Standard Notation. Elizabeth B. Uptegrove Felician College [email protected] Background. Students first investigated combinatorics tasks The towers problem The pizza problem The binomial coefficients

Download Presentation

PASCAL’S TRIANGLE AS ASSIMILATION PARADIGM: Standard Notation

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Pascal s triangle as assimilation paradigm standard notation l.jpg

PASCAL’S TRIANGLE AS ASSIMILATION PARADIGM:Standard Notation

Elizabeth B. Uptegrove

Felician College

[email protected]


Background l.jpg

Background

Students first investigated combinatorics tasks

The towers problem

The pizza problem

The binomial coefficients

Students then learned standard notation


Objectives l.jpg

Objectives

Examine strategies that students used to generalize their understanding of counting problems

Examine strategies that students used to make sense of the standard notation


Theoretical framework l.jpg

Theoretical Framework

  • Students should learn standard notation

  • Having a repertoire of personal representations can help

  • Revisiting problems helps students refine their personal representations


Standard notation l.jpg

Standard Notation

  • A standard notation provides a common language for communicating mathematically

  • Appropriate notation helps students recognize the important features of a mathematical problem


Repertoires of representations l.jpg

Repertoires of Representations

  • Existing representations are used to deal with new mathematical ideas

  • But if existing representations are taxed by new questions, students refine the representations

  • Representations become more symbolic as students revisit problems

  • Representations become tools to deal with reorganizing and expanding understanding


Research questions l.jpg

Research Questions

  • How do students develop an understanding of standard notation?

  • What is the role of personal representations?


Data sources l.jpg

Data Sources

  • From long-term longitudinal study

  • Videotapes of a group of 5 students

    • Ankur, Brian, Jeff, Michael, and Romina

    • After-school problem-solving sessions (high school)

    • Individual task-based interviews (college)

  • Student work

  • Field notes


Methodology l.jpg

Methodology

  • Summarize sessions

  • Code for critical events

    • Representations and notations

    • Sense-making strategies

  • Transcribe and verify


Combinatorics problems l.jpg

Combinatorics Problems

  • Towers -- How many towers n cubes tall is it possible to build when there are two colors of cubes to choose from?

  • Pizzas -- How many pizzas is it possible to make when there are n different toppings to choose from?


Combinatorics notation l.jpg

Combinatorics Notation

  • C(n,r) is the number of combinations of n things taken r at a time

  • C(n,r) gives the number of towers n cubes tall containing exactly r cubes of one color

  • C(n,r) gives the number of pizzas containing exactly r toppings when there are n toppings to choose from

  • C(n,r) gives the coefficient of the rth term of the expansion of (a+b)n

  • These numbers are found in Pascal’s Triangle


Students strategies l.jpg

Students’ Strategies

  • Early elementary: Build towers and draw pictures of pizzas

  • Later elementary: Tree diagrams, letter codes, organized lists

  • High school: Tables and numerical codes; binary coding; organization by cases


Results l.jpg

Results

  • Students used their understanding of the pizza and towers problems to make sense of combinatorics notation and of the numbers in Pascal’s Triangle

  • Students used this understanding to make sense of a related combinatorics problem

  • Students regenerated or extended their work in interviews two or three years later


Generating pascal s identity l.jpg

Generating Pascal’s Identity

  • First explain a particular row of Pascal’s Triangle in terms of pizzas

  • Then explain a general row in terms of pizzas

  • First explain the addition rule in specific cases

    • Towers

    • Pizzas

  • Then explain the addition rule in the general case


Pascal s identity student version l.jpg

Pascal’s Identity(Student Version)

  • N choose X represents pizzas with X toppings when there are N toppings to choose from

  • N choose X+1 represents pizzas with X+1 toppings when there are N toppings to choose from

  • N+1 choose X+1 represents pizzas with X+1 toppings when there are N+1 toppings to choose from


Pascal s identity student explanation l.jpg

Pascal’s Identity(Student Explanation)

  • To the pizzas that have X toppings (selecting from N toppings), add the new topping

  • To the pizzas that have X+1 toppings (selecting from N toppings), do not add the new topping

  • This gives all the possible pizzas that have X+1 toppings, when there are N+1 toppings to choose from


Taxicab problem l.jpg

Taxicab Problem

  • Find the number of shortest paths from the origin (at the top left of a rectangular grid) to various points on the grid

  • The only allowed moves are to the right and down

  • C(n,r) gives the number of shortest paths from the origin to a point n segments away, containing exactly r moves to the right


Taxicab problem diagram l.jpg

Taxicab ProblemDiagram


Taxicab problem student strategies l.jpg

Taxicab Problem(Student Strategies)

  • First connect the taxicab problem to the towers problem in specific cases

  • Then form the connection in the general case

  • Finally, connect to the pizza problem


Romina links the taxicab problem to 4 choose 3 and the towers problem l.jpg

Romina Links the Taxicab Problem to “4 choose 3” and the Towers Problem


Brian and romina make some connections l.jpg

Brian and Romina Make Some Connections


Interview with mike 2002 l.jpg

Interview with Mike (2002)


Interview mike l.jpg

Interview (Mike)

  • Recall how to relate Pascal’s Triangle to pizzas and standard notation

  • Call the row r and the position in the row n

  • Write the equation


Pizzas on the triangle l.jpg

Pizzas on The Triangle

  • The nth row is for the n-topping pizza problem

  • The first number in each row is the “no topping” pizza

  • The last number in each row in the “all toppings” pizza

  • The rest of the numbers are for 1, 2, … toppings

  • Michael: “We see like the physical connection”


Pizzas and pascal s identity l.jpg

Pizzas and Pascal’s Identity


Interview romina l.jpg

Interview (Romina)

  • She explained standard notation in terms of towers, pizzas, and binary notation

  • She explained the addition rule in terms of towers, pizzas, and binary notation

  • She explained the taxicab problem in terms of towers


Romina links pizzas towers and binary notation l.jpg

Romina Links Pizzas, Towers, and Binary Notation


Interview ankur l.jpg

Interview (Ankur)

  • He explained standard notation in terms of towers

  • He explained specific instance of the addition rule in terms of towers

  • He explained the general addition rule in terms of towers


Ankur explains pascal s identity in terms of towers l.jpg

Ankur Explains Pascal’s Identity in Terms of Towers


Conclusions l.jpg

Conclusions

  • Students learned new mathematics by building on familiar powerful representations

  • Students built up abstract concepts by working on concrete problems

  • Students recognized the isomorphic relationship among three problems with different surface features

  • Their understanding appears durable


Questions l.jpg

Questions?


The towers solution l.jpg

The Towers Solution

  • For each cube in the tower, there are two choices: blue or red

  • For a tower 4 cubes tall, the number of towers is 2222 = 24 = 16

  • For a tower n cubes tall, the number of towers is 2n.


The pizza solution l.jpg

The Pizza Solution

  • There are two choices for each topping: on or off the pizza. When there are four toppings, there are 2222 = 24 = 16 possible pizzas

  • When there are n possible toppings, there are 2n possible pizzas


Can you explain the addition rule 1998 l.jpg

Can You Explain theAddition Rule? (1998)


Can you explain the addition rule yes l.jpg

Can You Explain theAddition Rule? Yes!


Add a blue to the second 10 l.jpg

Add a Blue to the Second 10


Pascal s triangle l.jpg

Pascal’s Triangle


Taxicab towers isomorphism l.jpg

Taxicab-Towers Isomorphism


  • Login