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

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PASCAL’S TRIANGLE AS ASSIMILATION PARADIGM: Standard Notation. Elizabeth B. Uptegrove Felician College uptegrovee@felician.edu. Background. Students first investigated combinatorics tasks The towers problem The pizza problem The binomial coefficients

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PASCAL’S TRIANGLE AS ASSIMILATION PARADIGM: Standard Notation

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## PASCAL’S TRIANGLE AS ASSIMILATION PARADIGM:Standard Notation

Elizabeth B. Uptegrove

Felician College

uptegrovee@felician.edu

### Background

The towers problem

The pizza problem

The binomial coefficients

Students then learned standard notation

### 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

• Students should learn standard notation

• Having a repertoire of personal representations can help

• Revisiting problems helps students refine their personal representations

### 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

• 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

• How do students develop an understanding of standard notation?

• What is the role of personal representations?

### 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)

• Student work

• Field notes

### Methodology

• Summarize sessions

• Code for critical events

• Representations and notations

• Sense-making strategies

• Transcribe and verify

### 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

• 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

• 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

• 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

• 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)

• 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)

• 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

• 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(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

### 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

• 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”

### 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

### 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

### 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

### 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

• 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