Assessment and Evaluationin the

Mathematics Classroom

Jane SilvaInstructional Leader K-8

Objectives

> To examine the principles of effective assessment

> To examine how to collect and interpret assessment data

“From their earliest school experience, students draw life-shaping conclusions about themselves as learners on the basis of the information provided to them as a result of classroom assessments.”

- Stiggins, Student-Involved Classroom Assessment, Prentice-Hall, 2001, p.48.

Recall your own assessment experiences as students.

Recall your own assessment experiences as students.

What kind of meaningful classroom assessment information do we want to provide to our students?

Select Expectations

Summative/ Diagnostic culminating Assessment assessment and evaluation Adapt program Select and based on implement formative learning assessments experiences

Ongoing, formative assessments (with feedback to students)

THE TEACHING/ LEARNING

CYCLE

Types of Assessment

Diagnostic Assessment:Assessment FOR Learning

Formative Assessment:Assessment FOR Learning

Summative Assessment:Assessment OF Learning

Assessment AS Learning

Types of Assessment

Diagnostic Assessment:Assessment FOR Learning

Formative Assessment:Assessment FOR Learning

Summative Assessment:Assessment OF Learning

Assessment AS Learning

Knowing what ingredients you already have and those that are still needed to make excellent soup, that’s diagnostic;

When the cook tastes the soup, that’s formative;

When the guests taste the soup, that’s summative.

When they think it’s good or bad that’s evaluation.

Recipe for Assessment

Why Do We Assess?

We Assess

To document student and teacher progress To provide feedback to the student and

family, and the teacher To inform instructional decisions

Wormelli, R. (2006). Fair Isn’t Always Equal

We do not Assess

To motivate students To punish students To sort students

Wormelli, R. (2006). Fair Isn’t Always Equal

Ten Approaches to Avoid in Assessment and Grading

Ten Approaches to Avoid in Assessment and Grading

1. Incorporating nonacademic factors, such as effort, behaviour, and attendance

It is agreed that there is a high correlation between academic success and effort, behaviour and attendance.

However, we don’t have a commonly accepted, legally justifiable, non-subjective method for measuring effort.

Instead, specific feedback on these factors should be communicated to students.

Ten Approaches to Avoid in Assessment and Grading

2. Penalizing students’ multiple attempts at mastery

“Feedback that is given on an assignment that can’t be revised or that is not clearly and specifically related to future work is unlikely to be seen as useful by the student…(s)he can’t hope for a slight improvement in the grade, despite the fact that he now understands how to do the work.” (Nolen & Taylor, 2005, p.60)

Ten Approaches to Avoid in Assessment and Grading

3. Grading Practice (bad homework)

Confabulation

When the mind seeks the big-picture connections of something that is learned, and when it doesn’t find all the pieces of the puzzle, it makes up information or borrows from other memories and inserts false information into holes of missing information. Successful teachers don’t give homework unless their students have already mastered the concepts.” p.116

Ten Approaches to Avoid in Assessment and Grading

4. Withholding assistance with the learning when it’s needed

Accommodations

If providing glasses, for example, allows students to follow and participate in lesson, why not allow this learning tool.

If providing a calculator, for example, allows students to identify and organize salient information that will allow them to be competitive with the best thinkers in the class, why not allow this learning tool.

Ten Approaches to Avoid in Assessment and Grading

5. Assessing students in ways that do not accurately indicate their mastery

Speed

Perseverance

Level of conformity

Mathematical Sophistication

Students Differ in Math via:

Ten Approaches to Avoid in Assessment and Grading

6. Allowing extra credit and bonus points

Assessment “of” or “for” learning?

“Teachers are applying assessment for learning when they plan to revisit a particular big idea later in the program if the concept is not as well understood as it should be.” LNS, P.50 Bonus assignments don’t necessary move students along the continuum.

Ten Approaches to Avoid in Assessment and Grading

7. Group grades

Learning Skills vs. Content Knowledge

Cooperative learning is an outstanding teaching strategy and technique used to teach students about a topic, not a demonstration of proficiency in that topic itself.

Ten Approaches to Avoid in Assessment and Grading

8. Grade on a curve

How Do We Grade?

We must compare students to the standards and not to each other (grading on a curve).

Let’s use pilots to show why this is a good idea:

In a classroom where all students are above

standard”

Good students

“fail”

Meanwhile, across the hall in a classroom where all

students are “below standard”

Bad students “pass”

Ten Approaches to Avoid in Assessment and Grading

9. Using norm-referenced terms to describe criterion-referenced attributes

Avoid Norm-Referenced Grading

Shift in assessment practice away from comparing students’ performance with that of other students towards comparing students’ performance with established criteria.

Use Criterion-Referenced Grading

Grade students by comparing them to curriculum standards.

Consider the “Mountain Curve” where we try to move all students toward the standards.

Ten Approaches to Avoid in Assessment and Grading

10. Recording zeros for work not done

Why Do We Grade?

Low grades push students farther from our cause, they don’t motivate students.

High grades can have short-term effects on motivation as well, if it affects students’ intrinsic motivation.

Classroom Assessment & Grading that Work

Classroom Assessment & Grading that Work

The “grade” you often observe is a combination of the student’s “true

score” and an “error score”.

A “true score” represents a student’s actual level of achievement.

Classroom Assessment & Grading that Work

How can we find true score?

Classroom Assessment & Grading that Work

How can we find true score?

Consider the Item Response Theory (IRT)

- Uses complex mathematics to translate a students pattern of responses to a trait score on

that distribution- We use a scale that represents performance

along a continuum

For this theory to hold, our assessments must include Type I, II,

and III items.

To relate this to our practices, we may say that our assessments should include level 1, 2, 3, and 4

questions.

Let’s look at Type I, II, and III items.

Type I Items or Tasks

- Address the basic details and processes that are relatively easy for students

- Teacher asks: About this topic, what are the basic details and processes students should understand or be able to do fairly easily if they were paying attention in class?

Type I Items and Tasks for Information

- Vocabulary terms: a common type of basic detail

- Facts: identify characteristics of specific persons, places, living things, nonliving things, events and causes of events

- Time sequences: involve events that occurred between two points in time

Type I Items and Tasks for Mental Procedures

- Single rules: Ex. Capitalization – if the word begins a sentence, then capitalize the word

- Algorithms: procedures that do not vary much in their application once learned

- Tactics: may be used differently from one situation to another

Type II Items or Tasks

- Address more complex ideas and processes and are more difficult for students

- Teacher asks: About this topic, what are the more complex ideas and processes students should understand or be able to do if they were paying attention in class?

Type II Items and Tasks for Information

- Generalizations: statements for which examples can be provided; identify characteristics of classes of the same type of information

- Principals: deals with cause/effect relationships; geared toward predicting what will occur in a given situation

Type II Items and Tasks for Mental Procedures

- Decision making: process of generating and applying criteria to select from among seemingly equal alternatives

- Problem solving: process of overcoming constraints or limiting conditions that are in the way of pursuing goals

- Experimental inquiry: process of generating and texting explanations of observed phenomena

- Investigation: process of identifying and resolving issues that involve confusions or contradiction

- Invention: process of developing unique products or processes that fulfill perceived needs

Type II Items and Tasks

Constructed-Response Items:- Short written response- Essays (addresses a generalization)

Type III Items or Tasks

- Require students to make inferences or applications that go beyond what was taught in class

- Teacher asks: About this topic, what inferences and applications might students be able to make even though they go beyond what was taught in class?

Type II Items and Tasks for Information

Sometimes involves:- Comparing: identifying similarities and

differences- Classifying: grouping into categories based

on their like characteristics- Creating metaphors: identifying a general

pattern that connects information that is not related at the literal level

- Creating analogies: identifying the relationship between two sets of items

- Analyzing errors

Type III Items and Tasks for Mental Procedures

- Address the extent to which students can apply the procedure in a context not addressed in class

Type III Items and Tasks

Constructed-Response Items:- Short written response- Essays (go beyond expectations)

Sample Assessment

Section I:

1. Which company has the highest daily rate?2. Which company has the most free mileage?3. If each company had the same daily rate and the same amount

of free mileage, which would be cheapest?4. If each company had the same amount of free mileage and the

same cost per mile, which company would be the most expensive?

5. Once you’ve used up your free mileage, which company would cost the least amount of money to travel 100 miles in a single day?

Red Rental Easy Rental Reliable Rental

J&S Rental

Daily Rate $43.00 $27.50 $40.00 $35.25

Free Mileage

1 200 500 900 800

Cost per Mile

$0.22/mile $0.32/mile $0.25/mile $0.20/mile

Sample Assessment

Section II:

6. If you travel 100 miles per day, which company is the least expensive for5 days:10 days:15 days:20 days:Create a table or graph that shows how expensive each company is for each of the four options above (5 days, 10 days, 15 days, 20 days), and explain how you calculated your answers.

Red Rental Easy Rental Reliable Rental

J&S Rental

Daily Rate $43.00 $27.50 $40.00 $35.25

Free Mileage

1 200 500 900 800

Cost per Mile

$0.22/mile $0.32/mile $0.25/mile $0.20/mile

Sample Assessment

Section III:

7. Each of the four companies could be considered the “best deal” under certain conditions. For each company, describe the situation under which it would be the best selection. In your answer and explanation, use the daily rate, free mileage, and the rate per mile after free mileage.

Red Rental Easy Rental Reliable Rental

J&S Rental

Daily Rate $43.00 $27.50 $40.00 $35.25

Free Mileage

1 200 500 900 800

Cost per Mile

$0.22/mile $0.32/mile $0.25/mile $0.20/mile

How do you Grade this type of Assessment?

Use the scoring scale. A “+” indicates that the student was

successful with this type of question.

Quick Reference Guide for the Simplified Scoring Scale

Student Pattern of Responses

Type I Items

+ + + Some understanding with help

0 with help

Type II Items

+ + 0 Some understanding with help

0 with help

Type III Items

+ 0 0 0 with help

0 with help

Score 4.0 3.0 2.0 1.0 0.0

Quick Reference Guide for the Complete Scoring Scale

Student Pattern of ResponsesType I Items

+ + + + + part part with help

part with help

0with help

Type II Items

+ + + part 0 0 part with help

0with help

0with help

Type III Items

+ part 0 0 0 0 0with help

0with help

0with help

Score 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0

Quick Reference Guide for the Complete Scoring Scale

Score on Scale

Description of Place on Scale

4.0 In addition to Score 3.0 performance, in-depth inferences and applications that go beyond what was taught.

3.5 In addition to Score 3.0 performance, partial success at inferences and applications that go beyond what was taught.

3.0 No major errors or omissions regarding any of the information and/or processes (simple or complex) that were explicitly taught.

2.5 No major errors or omissions regarding the simpler details and processes and partial knowledge of the more complex ideas and processes.

2.0 No major errors or omissions regarding the simpler details and processes but major errors or omissions regarding the more complex ideas and processes.

1.5 Partial knowledge of the simpler details and processes but major errors or omissions regarding more complex ideas and processes.

1.0 With help, a partial understanding of some of the simpler details and processes and some of the more complex ideas and processes.

0.5 With help, a partial understanding of some of the simpler details and processes but not the more complex ideas and processes.

0.0 Even with help, no understanding or skill demonstrated

What to do if Students Responses Don’t Follow a

Logical Pattern- Realize that no assessment can ever provide perfectly accurate data about a student

- Drop some items because they are deemed to be invalid

- Rethink the classification of specific items

- Meeting individually with students