Assessment in the

Mathematics Classroom

Jane SilvaInstructional Leader K-8

Objectives for the Day

> To examine the principles of effective assessment

> To explore ways of assessing mathematics

>To explore the link between assessment and instruction

>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.

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

Purposes of AssessmentAccording to the Ministry of Education’s curriculum policy documents, the purposes of assessment and evaluation are to:

• improve student learning;• help teachers to determine students’ strengths and weaknesses in

their achievement of the curriculum expectations;• guide teachers in adapting curriculum and instructional

approaches to students’ needs, and;• assess the overall effectiveness of programs and classroom

practices.

The improvement of student learning is the most important focus of assessment.

Key Messages

Assessment is an ongoing awareness of students’ learning and needs, rather than an occasional event in the program.

Key Messages

Of all the assessment strategies, formative assessment is the most valuable strategy for supporting students’ learning and for promoting students’ independence and responsibility as learners.

Key Messages

Observation is the most efficient and effective way for teachers to assess students’ mathematical abilities, and is an integral part of all assessment strategies.

Key Messages

Teachers should use a variety of assessment strategies in order to assess students’ mathematical development as completely as possible.

Key Messages

Teachers should not provide a grade/mark/level on an individual assessment or a collection of assessments unless the grade/mark/level serves a clear purpose: to communicate achievement of curriculum expectations to students, parents, other teachers, and administrators.

Key Messages

SELECT EXPECTATIONS

SUMMATIVE/ DIAGNOSTICCULMINATING ASSESSMENTASSESSMENT ANDEVALUATION

ADAPT PROGRAM SELECT AND IMPLEMENTBASED ON FORMATIVE LEARNING EXPERIENCESASSESSMENTS

ONGOING, FORMATIVE ASSESSMENTS(PROVIDE FEEDBACK TO STUDENTS)

THETEACHING/ LEARNING CYCLEInstruction and

Assessment

Recall your own assessment practices as teachers.

What kinds of assessments do you use?

Types of Assessment

Diagnostic Assessment

Formative Assessment

Summative Assessment

Types of Assessment

Diagnostic AssessmentAssessment FOR Learning

Formative AssessmentAssessment FOR Learning

Summative AssessmentAssessment OF Learning

Assessment AS Learning

Evaluation

Summative (after)

Judging

Assign Grades

Report on Achievement

Assessment

Diagnostic (before)

Formative (during, ongoing)

Feedback

OF LearningFOR Learning

Types of Assessment

Diagnostic AssessmentAssessment FOR Learning

• The collection of information that occurs before a unit or new teaching (e.g., lesson);

• enables teachers to determine student strengths and learning needs;

• enables teachers to build on students’ prior knowledge;

• provides direction for the adjustment and improvement to programs for individual students or for the whole class.

CubeProbability Prompts

IMPOSSIBLE

LIKELY

CERTAIN

Describe probability as a measure of the likelihood that an event will occur, using mathematical language

CubeAddition and Subtraction Prompts

Face 1: I understand…

Face 2: I don’t understand…

Face 3: I find it easy to…

Face 4: I find it difficult to…

Face 5: I learned…

Face 6: I still want to know…

Open QuestionsNumber Sense and Numeration

Work in pairs to decide what this graph might be about.

0102030405060708090

Open QuestionsData Management

What does this conversation tell you about this man’s understanding?

What does this observation tell you about Jane’s understanding?

Mom!Why did Jamie get 4

sandwiches and I only got 2?

Gathering Diagnostic Information• Ask students to explain what they already know about a

concept or skill;

• Have students use manipulative materials to represent and explain a mathematical idea;

• Conduct brief, informal conversations with students about new ideas;

• Observe and talk with students as they solve a problem or complete a task that involves the concept or skill.

Provides ongoing day to day feedback for the teacher regarding the following:

• Identifying where to focus students’ attention and effort and encouraging progress and achievement of goals

• Monitoring of student learning, identifying areas of growth or concern and providing descriptive feedback

• Identifying the effectiveness of instruction and programming in order to modify and adjust it as necessary

Formative Assessment Assessment FOR Learning

Overall

Specific

What do I want my students to learn?

Curriculum Expectations

What evidence will I look for to know that learning has occurred?

Application

Knowledge and

Understanding

Thinking

Mathematical Process

Expectations

Communication

What learning opportunities should I provide to promote learning?

Teaching strategies?

Opportunities?

Experiences?

Resources?

Personal communication?

Paper-and-pencil tasks?

Performance tasks?

Observation?

What are the most appropriate methods of assessing student

learning?

How should I record or document significant assessment information?

What conclusions can be made from assessment?

Addition

Subtraction

What do I want my students to learn?

Big Ideas

What evidence will I look for to know that learning has occurred?

Problem Solving

Representing

Mathematical Process

Expectations

Communication

What learning opportunities should I provide to promote learning?

Modeling problem solving process

Cooperative learning

Manipulatives made available

Pig and Chicken Performance Task

Observation

What are the most appropriate methods of assessing student

learning?

How should I record or document significant assessment information?

Teacher Grade Assessment FOR Learning Observation & Interview

Term Year

Unit of Study Date Mathematics Lesson Task/Problem Learning Goal/Curriculum Expectations Possible Solutions/Strategies

Questions about student learning…

Student Errors

What conclusions can be made from assessment?

Teacher’s Assessment

The student has difficulty understanding the problem (i.e., misread the question, omitted information - 22 legs), or selecting appropriate problem solving strategies.

14 7 chickens have 14 legs

16

Inference From Assessment

The student may have difficulty identifying important information when reading problems or lacks a strategy for selecting an appropriate problem solving strategy.

Next Instructional StepThe teacher models the problem solving process. The teacher reads and rereads the problem aloud, underlining important information and crossing out extraneous information. The teacher prompts the student restate the problem in his/her own words. The teacher models how to develop a plan to solve the problem, suggesting appropriate tools and strategies. The teacher uses questions and cue to prompt the student.The teachers asks the student to explain his/her thinking to determine whether he/she understand the problem.

Teacher’s Assessment

A grade 5 student has difficulty remembering the order of the steps when using the standard long division algorithm.

Inference from the Assessment

Based on this observation, what inference can you make about this student?

Inference From AssessmentThe student may not understand, at the conceptual level, the procedure for long division.

Feedback• Telling whether response is correct or incorrect

• Informing the learner of the correct answer

• Explaining what is accurate and inaccurate

• Asking the learner to keep working on a task until he/she succeeds

• Timing of feedback

Feedback• Telling whether response is correct or incorrect

– negative effect• Informing the learner of the correct answer

• Explaining what is accurate and inaccurate

• Asking the learner to keep working on a task until he/she succeeds

• Timing of feedback

Feedback• Telling whether response is correct or incorrect

– negative effect• Informing the learner of the correct answer

– positive effect• Explaining what is accurate and inaccurate

• Asking the learner to keep working on a task until he/she succeeds

• Timing of feedback

Feedback• Telling whether response is correct or incorrect

– negative effect• Informing the learner of the correct answer

– positive effect• Explaining what is accurate and inaccurate

– most positive effect• Asking the learner to keep working on a task until he/she succeeds

• Timing of feedback

Feedback• Telling whether response is correct or incorrect

– negative effect• Informing the learner of the correct answer

– positive effect• Explaining what is accurate and inaccurate

– most positive effect• Asking the learner to keep working on a task until he/she succeeds

– positive effect• Timing of feedback

Feedback• Telling whether response is correct or incorrect

– negative effect• Informing the learner of the correct answer

– positive effect• Explaining what is accurate and inaccurate

– most positive effect• Asking the learner to keep working on a task until he/she succeeds

– positive effect• Timing of feedback

– the more delay, the less improvement

Feedback• Telling whether response is correct or incorrect

– negative effect• Informing the learner of the correct answer

– positive effect• Explaining what is accurate and inaccurate

– most positive effect• Asking the learner to keep working on a task until he/she succeeds

– positive effect• Timing of feedback

– the more delay, the less improvement

Students who are given comments only – rather than marks or marks and comments – make more gains in achievement and feel more positive about the experience (Butler, 1998).

Feedback

>Numerical score

>Right/wrong

>No grade necessary for formative tasks

>Anecdotal feedback

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>General(“need more practice”)

Identify what is done well,

what needs improvement,

and how to get there.

Specific,Focused

Feedback

What does this work tell you about the student’s understanding of addition?Identify what is done well, what needs improvement, and how to get there.

What does this work tell you about the student’s understanding of addition?Identify what is done well, what needs improvement, and how to get there.

What does this work tell you about the student’s understanding of addition?Identify what is done well, what needs improvement, and how to get there.

Let’s Try a Question

Using the Rubric: Code the Student Exemplars

What Code is This?

Code 20Annotation:

Student demonstrates some understanding of the relationships between important elements of the problem; shows 11 quarters equals $2.75 and draws quarters to continue adding towards the target of $5.25. 13 more quarters does not give a total of $5.25.

Code 10Annotation:

Student demonstrates a minimal evidence of a solution process; draws 14 quarters with no justification and subtracts the 11 given to conclude 3 more.

What Code is This?

What Code is This?

Code 30Annotation:

What Code is This?

Code 40Annotation:

Student demonstrates a thorough understanding of the relationships between all of the important elements of the problem; shows groups of 4 quarters to make $1.00; uses diagram to show an additional 10 quarters are needed to make $5.25.

• Analyse students’ written work

• Observe students as they work on learning tasks

• Conduct conversations with students about their learning

Gathering Formative Information

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Planning for InstructionPlanning for Assessment

• Usually carried out at the end of a unit, term, semester or year;

• informs teachers’ formal judgments about students’ achievement to be used in reporting;

• helps confirm what students know and can do and whether they have achieved the curriculum expectations.

Summative Assessment Assessment OF Learning

Summative Assessment…

Examples:

• Interviews • Problem solving situations• Performance tasks• Paper-Pencil Tests

Evaluation is the summative process of:

• judging the quality of student work on the basis of previously established criteria

• assigning a value (letter or percentage grade) to represent that quality.

Evaluation

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

Evaluation Decide early which assessments will form the basis for

grades and which will be used in a formative way

Compare student performance against criteria and standards

No single measure will give you all the information you need

Communicate your explicit expectations about both quantity and quality to students

Involve students in the selection process

Assessment AS Learning

• Focuses on the active involvement of students in the assessment of their own learning;

• Students personally monitor their own learning and use the feedback to adjust or adapt their understanding;

• Emphasizes students as critical assessors who can use metacognition in the self-assessment process.

Let’s Review:

• Diagnostic (Assessment for learning): is assessment helps to determine entry points.

• Formative (Assessment for learning): is assessment helps to promote learning and progress.

• Summative (Assessment of learning): is assessment reflects student achievement at a given time in a specific context e.g. at the end of year to determine placement or programs.

Knowing what ingredients you already have and may require 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