embedding innovative assessment in high school mathematics ...€¦ · a teacher’s experience...

Copyright © 2012 Educational Testing Service. All rights reserved.

Embedding Innovative Assessment in High School Mathematics and

Language Arts Classes

Mary Fowles & Liz Marquez Educational Testing Service

Michael Santa Maria & Amanda Perry

North Brunswick High School, New Jersey

CCSSO NCSA Conference June 29, 2012

Copyright © 2012 Educational Testing Service. All rights reserved. Copyright © 2012 Educational Testing Service. All rights reserved.

Overview

• CBAL: Description of the project

• CBAL Mathematics

• CBAL English Language Arts

• Questions and Comments

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Cognitively Based Assessment of,

for, and as Learning (CBAL)

Research initiative to create knowledge and capability, grounded in the learning sciences, that can be used to meet the assessment-innovation needs of the field. CBAL assessment prototypes are designed to

• Document what students have achieved (“of learning”)

• Help identify how to plan instruction (“for learning”)

• Be considered by students and teachers to be a worthwhile educational experience in and of itself

(“as learning”)

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Four Components of CBAL

Conceptual

Framework

Summative

Assessments

Formative

Assessments

Professional

Support

Theory of

Action

Competency

Model

Learning

Progressions

Scenario-

Based Tasks

Scenario-

Based Tasks

Component

Tasks

Communities

of Practice


CBAL Mathematics


Topics • CBAL Competency Model for Middle School

Mathematics

• Link to the Common Core State Standards

• Description and research basis for a learning progression

• CBAL task examples and implications for instruction based on a learning progression

• Examples of teacher support

• A teacher’s experience implementing CBAL tasks

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Competency Model

• Goal of model

• Knowledge of mathematical procedures and mechanics, relative to bigger picture of mathematics

• Relationship to Common Core State Standards

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Competency Model (cont.)

Model informed by

• NCTM Standards

• Achieve Benchmarks

• Learning sciences research

• Other standards documents

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Middle School Math Competency Model: The First Three Levels

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Top Levels of Middle School Math Competency Model

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Learning Progression

In CBAL, a learning progression is defined as a description of qualitative change in a student’s level of sophistication for a key concept, process, strategy, practice, or habit of mind. Change in student standing on such a progression may be due to a variety of factors, including maturation and instruction. Each progression is presumed to be modal--i.e., to hold for most, but not all, students. Finally, it is provisional, subject to empirical verification and theoretical challenge.

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Provisional Learning Progressions (Middle School Grade Levels)

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*Multiple progressions in this content area

CBAL Learning Progressions

Proportional Reasoning

Algebra: Equality/Variable

Algebra: Functions*

Statistics*

There is an iterative refinement process of progressions and how they

are applied to the development of summative and formative

assessment tasks.


Research Basis for the Linear Functions

Learning Progression • Mathematical development involves integration of separate

cognitive systems – within neo-Piagetian frameworks

– Case (1993) : a central conceptual structure

– Kalchman, Moss and Case (2001); Kalchman and Koedinger (2005)

• Mathematical development involves elaboration on concepts from an operational understanding to a structural understanding

– Sfard (1994): reification

• Building on earlier CBAL learning progressions

– Graf (2009): The Concept of Function progression and Linear Functions progression

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Linear Functions Learning Progression

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Level 1 Students understand a variable as represented on a number line Understands One Dimensional Change

Level 2 Students understand two-dimensional representations in terms of

how one variable depends on another. Understands Mutual Change

Level 3

Students understand what constant change looks given one

specific linear function.

Understands Constant

Change

Level 4

Students can compare constant change given more than one

function.

Compares more than one linear function

Level 5

Students can distinguish between linear and nonlinear functions.

Recognizes non-linear

functions

Source: Baxter and Junker, 2001 (cited in Weaver and Junker, 2004)


Common Core State Standards

CONTENT:

• Expressions and Equations

– Understand the connections between proportional relationships, lines, and linear equations.

– Analyze and solve linear equations and pairs of simultaneous linear equations.

• Functions

– Define, evaluate, and compare functions.

– Use functions to model relationships between quantities

PRACTICES:

• Make sense of problems and persevere in solving them

• Construct viable arguments and critique the reasoning of others.

• Model with mathematics.

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Moving

Sidewalks

Moving Sidewalks and the Functions & Linear

Functions Learning Progression

In an airport, two moving sidewalks run parallel to each other in opposite directions at the same speed.

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http://cbal.ets.org/cbal/toolsAndSimulations/sidewalks2.html


Moving

Sidewalks

a. What is the slope of the line? Show your work.

b. Explain the meaning of the slope in terms of Rider A’s movement from the gate.

Slope = Units:

4

Rider A on Sidewalk

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This question can elicit evidence of student

understanding at levels 1-3 of the learning

progression.


Rider A on Sidewalk

a. What is the slope of the line?

“(30-0)/(15-0)=2/1” (Key: 2)

a. Explain the meaning of the slope in

terms of Rider A’s movement from the

gate. “Rider A is leaving the gate at

2 ft/sec.”

(Key: Rider A is moving away from

the gate at 2 ft/sec.)

Part a: Students will calculate slope by

finding change in y over change in x.

Part b: They will be able to explain slope

in context.

Level 3 Linearity

y=mx and y = mx+b Constant Change

A response consistent with a Level 3 understanding might look like this.


Part a: Students calculate slope by

finding change in y over change in x.

Part b: They will explain slope

procedurally.

Rider A on Sidewalk

a. What is the slope of the line? 2/1 (Key: 2)

b. Explain the meaning of the slope in

terms of Rider A’s movement from

the gate. “(30-0)/(15-0)=2/1” (Key:

Rider A is moving away from the

gate at 2 ft/sec.)

Level 2 Coordinate plane Mutual Change



Implications for Instruction “Next Steps”

level 2 level 3

• Encourage students to look at units on the x and y axis

• Reinforce slope as a “rate of change” to promote the use of units

• Have students explain the title of the graph in context


Part a: Students take a ratio of the

labeled coordinates.

Part b: They explain slope one

dimensionally.

a. What is the slope of the line?

15/30, 0/30, or 15/0 … (Key: 2)

a. Explain the meaning of the slope in

terms of Rider A’s movement from

the gate. “Rider A is moving up

the line.” (Key: Rider A is moving

away from the gate at 2 ft/sec.) Rider A on Sidewalk

Level 1 Number Line One Dimensional Change



Implications for Instruction “Next Steps”

1 2 3 In addition to Level 2 implications,

• Use points on the line to calculate slope other than the endpoints

• Illustrate the movement from one point to the next (vertically and horizontally)

• Reinforce the definition of slope as a “rate of change”


TEACHER SUPPORT—e.g., Handbook

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Online Teacher Support— VIA Features

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• Immediate access to student responses

• Field for feedback to student

• Results can be viewed by class, by individual student, or by item.

• Suggests the level in the learning progression at which a student is performing

• Informs next steps in instruction.


A Teacher’s Experience with CBAL: Math

Advantages

• I am learning how to help students relate mathematical concepts to real world experiences.

• Interactive simulations made topics more enjoyable for the class.

• I receive information about where students may be with respect to a LP and how to help them reach the next level of that progression. This helps get and keep my students on a path to success.


A Teacher’s Experience

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Challenges

• Set-up of computer labs

– Ineffective for teaching and classroom management

• Outdated computers

– Slowed class time down

– Took away from the flow of the task


CBAL English Language Arts

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CBAL ELA Competency Model (Subskills Shared across Reading and Writing)

Skills Foci

Social

Perspective-Taking

Rhetorical Analysis

Reading between the Lines

Stance-Taking

Code-Switching

Conceptual

Research

Hypothesis-Testing

Argument-Building

Discourse

Summarization

Outlining

Comparison

Verbal Vocabulary-Building

Grammatical Analysis

Print

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Summative Assessment Design Strategy

• Embodies a scenario

• Begins with a sequence of lead-in tasks

• Culminates in a sustained writing task

• Focuses on critical learning progressions for the skills focus targeted in that scenario

• Presents grade-appropriate texts for students to read and think about

• Presents critical thinking and writing activities with support (guidelines, tips, planning tools, etc.)

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Ban Ads

Task 2A 1 of 7 15 min

Directions: Read each statement below. Decide whether it gives a reason to ban or allow ads intended for children. Click on the statement to select it. Then click in a box under Ban or Allow to indicate which side of the issue the statement supports.

Statements

Commercials are usually easy to remember and hard to

forget, so an advertisement can get a lot of people to buy

something, even if it is not good for them.

Most healthy adults watched a lot of commercials when they

were young.

Parents who are concerned about protecting their children

rarely worry about the advertisements their children watch.

Junk food advertisers like advertising to children because they

know that children are very easy to influence and that the

habits people form as children will often last the rest of their

lives.

When a child sees an advertisement and keeps asking the

parents to buy an advertised product, the parents usually give

in, no matter what the child is asking for.

Most people ignore most of the advertisements they see.

Children may not understand that advertisements are trying to

trick them into buying something.

If an advertisement makes a child want a product, parents can

use that as an opportunity to teach the child how to make

smart choices.

Parents don’t usually take the time to check what their children

are watching on TV, so they don’t know what advertisements

their children are seeing.

Some of the people who argue the hardest in favor of ads for

children earn big profits from the advertising industry.

Ban Allow

If you change your mind, click on the undo arrow.

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Formative Tasks:

Literary Interpretation

Literary Classics

Developed in collaboration with NBTHS Teachers

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Romeo & Juliet Intro

Romeo and Juliet

Pre-Reading Task

1. Paraphrasing Seventeenth Century Language About Love

Prologue 2. Predicting Themes Based on the Prologue

Act I 3. Understanding Act I: Characters, Events, Important Quotations

II 4. Comparing and Contrasting Versions of the Balcony Scene

I–V 5. Understanding Key Literary Terms

II 6. Exploring Cause and Effect in the Fight Scene

III 7. The Role of Geography

I–IV 8. Perspectives on Marriage

I–V 9. Essay: Characterizing Romeo or Juliet

I–V 10. Grand Jury Trial: Who Is Responsible?

Click NEXT to continue. 37


Romeo & Juliet Intro Task 1

TASK 1

Paraphrasing Seventeenth Century Language About Love

Romeo and Juliet was written over 400 years ago, but people then were just as interested in love and romance as

we are today.

People in the seventeenth century even had self-help books on the subject. A handbook called The Mysteries of

Love and Eloquence offered advice for romantic “wooing and complimenting.” The language may seem strange

and old-fashioned, but the ideas can be paraphrased in modern-day terms.

Your goals:

• Show that you understand seventeenth century language

• Relate a big idea from Romeo and Juliet to today’s world

© ssstep/iStockphoto # 10241281



Romeo & Juliet

Click NEXT to continue.

Task 1 1

Love in the Seventeenth Century

On the left is an example of what a man might say to a

woman to persuade her to fall in love with him. Select the

most accurate modern paraphrase for the quotation.

The bright light from the stars makes your eyes even

brighter.

Your eyes are so bright they lend extra light to the stars.

I can’t tell if the stars or your eyes are brighter.

Quotation from

The Mysteries of Love and Eloquence “The stars borrow new light from your more radiant eyes.”

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Romeo & Juliet Task 1 5

More quotations appear below. In the right-hand column, type a paraphrase of each quotation in your own

words. You can use your answers from the previous questions as models.To view tips on paraphrasing, click on

the above tab.

Quotations from

The Mysteries of Love and

Eloquence

Your Translation

“Your perfections have so far

incaptived my senses and affection,

that I have resolved neither to love

nor serve any other but yourself.”

“Among all the days of my life I must

account this the happiest above all

the rest, wherein I had the honor first

to know you.”

Translation


Tips for Paraphrasing

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Romeo & Juliet

Task 2 Predicting Themes Based on the Prologue

Task 2 Intro

A prologue is an introduction to a play. Prologues typically announce and explain the most important ideas in the

play.

Your goal:

Predict the major themes of Romeo and Juliet based on evidence from the prologue

© 221A/iStockphoto # 15600362



Romeo & Juliet

Paraphrase your excerpt here:

Task 2 2

Prologue to Romeo and Juliet

Two households, both alike in dignity,

In fair Verona, where we lay our scene,

From ancient grudge break to new mutiny,

Where civil blood makes civil hands unclean.

From forth the fatal loins of these two foes

A pair of star-crossed lovers take their life;

Whose misadventured piteous overthrows

Doth with their death bury their parents’ strife.

The fearful passage of their death-marked love,

And the continuance of their parents’ rage,

Which, but their children’s end, naught could

remove,

Is now the two hours’ traffic of our stage;

The which if you with patient ears attend,

What here shall miss, our toil shall strive to mend.

Working with one or more partners, paraphrase the

excerpt assigned by your teacher. Rewrite it in modern

language in the box below.

Type the class’s full paraphrase here:

Click NEXT to

continue.

Prologue Tips for Paraphrasing

42


Romeo & Juliet

Now that you’ve read the prologue in its original form and in

present-day English, what do you think Romeo and Juliet

will be about?

Write your prediction (two to three sentences) in the box

below.

Task 2 3


Prologue to Romeo and Juliet

Two households, both alike in dignity,

In fair Verona, where we lay our scene,

From ancient grudge break to new mutiny,

Where civil blood makes civil hands unclean.

From forth the fatal loins of these two foes

A pair of star-crossed lovers take their life;

Whose misadventured piteous overthrows

Doth with their death bury their parents’ strife.

The fearful passage of their death-marked love,

And the continuance of their parents’ rage,

Which, but their children’s end, naught could

remove,

Is now the two hours’ traffic of our stage;

The which if you with patient ears attend,

What here shall miss, our toil shall strive to mend.

Prologue Tips for Paraphrasing

43


Romeo & Juliet


Task 10 Intro

Task 10 Grand Jury Trial: Who is Responsible?

At the end of Romeo and Juliet, the Prince of Verona tells the crowd, “All are punished.”

Nearly everyone in the play suffers, but are some more guilty of causing that suffering than others?

You are part of a grand jury that has been set up to decide exactly who should be held responsible for the

tragic events in Romeo and Juliet. First, you will review the tragic events that applied to other characters

besides Romeo and Juliet.

© Andrey Kiselev/fotolia #31342422 © msdnv/fotolia #22142938

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Romeo & Juliet


Task 10 1

If you change your mind, click on the undo arrow.

Tragic Events

Son commits suicide

Dies after learning of her son’s exile

Commits suicide

Daughter commits suicide

Killed in a fight

Two relatives are killed in a fight

Kills someone in a fight

Nephew is killed in a fight

Characters Tragic Events

Paris

Montague

Lady Montague

Tybalt

The Prince

Capulet

Lady Capulet

Mercutio

Match the characters on the right with the tragic events that

apply to them. Click on an event to select it. Then click next

to the name of the character to whom that phrase applies.

You may use the same phrase more than once. You

may also use more than one phrase for some

characters.


Romeo & Juliet


Task 10 2

Choose one:

Montague

Capulet

Friar Lawrence

The Prince

The Nurse

Lady Montague

Tybalt

Lady Capulet

Mercutio

Benvolio

The Apothecary

Paris

Choose one:

Romeo

Juliet

Now that you have revisited the tragic events of the play and the way different characters are affected by

those events, you should be ready to consider who might be held responsible.

From the following lists, choose two characters to bring to trial for the tragedies in Romeo and Juliet. You

must choose either Romeo or Juliet (not both), but the second character is up to you.

Click on the two characters that you hold most responsible for the tragic events in the play.

46


Romeo & Juliet


The name of the first character you have chosen is in the box below. Add at least two reasons

why you think this character is responsible. What does he or she do or say to cause the events

that happen in the play? You do not need to use complete sentences.

Character’s name:

Why s/he is responsible:

The name of the second character you have chosen is in the box below. Add at least two

reasons why you feel this character is responsible. What does he or she do or say to cause the

events that happen in the play? You do not need to use complete sentences.

Character’s name:

Why s/he is responsible:

Task 10 3

47


Romeo & Juliet

Your Opening Statement for the Trial

In this exercise, you will write an opening statement that explains why you think that the two characters you’ve

chosen are guilty and should be brought to trial. An opening statement is a speech a lawyer makes to assert an

accused person’s guilt or innocence.

First, restate the names of the characters you hold responsible, and explain why you think each one is guilty. Use

complete sentences. For each character, also include one counterargument and explain why you think it is not

valid.

First character’s name:

Reasons for guilt:

Second character’s name:

Reasons for guilt:

Task 10 5

Opening Statement Previous Answers



Summary

• CBAL’s tasks assess the CCSS and suggest levels at which students are performing with respect to relevant learning progressions

• CBAL’s learning progressions are intended to serve as a guide to effective instruction.

– Assessments are intended to discriminate between levels of learning progressions.

– Evidence on suggested level of performance potentially informs more focused and effective interventions.

49


Questions

http://www.ets.org/research/topics/cbal/initiative

50

http://www.ets.org/research/topics/cbal/initiative

embedding innovative assessment in high school mathematics ...€¦ · a teacher’s experience...

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