Assessment
in the
Mathematics Classroom
Jane Silva
Instructional 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 Assessment
According 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.
Key Messages
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.
SELECT EXPECTATIONS
SUMMATIVE/
CULMINATING
ASSESSMENT AND
EVALUATION
DIAGNOSTIC
ASSESSMENT
THE
Instruction
and
TEACHING/LEARNING
CYCLE
Assessment
ADAPT PROGRAM
BASED ON FORMATIVE
ASSESSMENTS
SELECT AND IMPLEMENT
LEARNING EXPERIENCES
ONGOING, FORMATIVE ASSESSMENTS
(PROVIDE FEEDBACK TO STUDENTS)
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 Assessment
Assessment FOR Learning
Formative Assessment
Assessment FOR Learning
Summative Assessment
Assessment OF Learning
Assessment AS Learning
Types of Assessment
FOR Learning
Assessment
Diagnostic (before)
Formative (during, ongoing)
Feedback
OF Learning
Evaluation
Summative (after)
Judging
Assign Grades
Report on Achievement
Diagnostic Assessment
Assessment 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.
Cube
Probability Prompts
Describe probability as a measure of the likelihood that an
event will occur, using mathematical language
IMPOSSIBLE
CERTAIN
Cube
Addition 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 Questions
Number Sense and Numeration
Open Questions
Data Management
Work in pairs to decide what this graph might be about.
90
80
70
60
50
40
30
20
10
0
What does this conversation tell you about this man’s
understanding?
Mom!
Why did Jamie get 4
sandwiches and I only
got 2?
What does this observation tell you about Jane’s
understanding?
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.
Formative Assessment
Assessment FOR Learning
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
What do I want my students
to learn?
Curriculum
Expectations
Overall
Specific
What evidence will I look for to
know that learning has occurred?
Mathematical
Process
Expectations
Knowledge and
Understanding
Application
Thinking
Communication
What learning opportunities should
I provide to promote learning?
Teaching
strategies?
Opportunities?
Experiences?
Resources?
What are the most appropriate
methods of assessing student
learning?
Personal
communication?
Paper-and-pencil
tasks?
Performance
tasks?
Observation?
How should I record or document
significant assessment
information?
What conclusions can be made
from assessment?
What do I want my students
to learn?
Big Ideas
Addition
Subtraction
What evidence will I look for to
know that learning has occurred?
Mathematical
Process
Expectations
Problem Solving
Representing
Communication
What learning opportunities should
I provide to promote learning?
Modeling problem
solving process
Cooperative
learning
Manipulatives
made available
What are the most appropriate
methods of assessing student
learning?
Pig and Chicken
Performance Task
Observation
How should I record or document
significant assessment
information?
Teacher
Grade
Assessment FOR Learning
Observation & Interview
Unit of Study
Mathematics Lesson Task/Problem
Term
Learning Goal/Curriculum Expectations
Possible Solutions/Strategies
Student Errors
Questions about student learning…
Year
Date
What conclusions can be made
from assessment?
Teacher’s Assessment
14
7 chickens have
14 legs
16
The student has difficulty understanding the problem (i.e.,
misread the question, omitted information - 22 legs), or
selecting appropriate problem solving strategies.
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 Step
The 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 Assessment
The 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
Feedback
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).
>Numerical score
>No grade necessary for
formative tasks
>Right/wrong
>Anecdotal feedback
Feedback
>General
(“need more practice”)
Identify what is done
well,
Specific,
Focused
what needs
improvement,
and how to get there.
57
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 20
Annotation:
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.
What Code is This?
Code 10
Annotation:
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?
Code 30
Annotation:
What Code is This?
Code 40
Annotation:
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.
Gathering Formative Information
•
Analyse students’ written work
•
Observe students as they work on learning tasks
•
Conduct conversations with students about their learning
Planning for Instruction
Planning for Assessment
69
Summative Assessment
Assessment OF Learning
•
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…
Examples:
•
•
•
•
Interviews
Problem solving situations
Performance tasks
Paper-Pencil Tests
Evaluation
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.
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.
Recipe for Assessment
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.