INFORMative Assessment of
Student Understanding;
Ratio and Rate of Change
August, 2012
Thinking About Your Practice
• Why do you assess students?
• What purposes do your assessments
serve?
Turn to your neighbor and share your list
INFORMative Assessment
• To determine what students know and
understand
• To determine potential misunderstandings
• To facilitate student recognition of what the
student knows or needs to know
• To make instructional decisions for/with a
student
• To make instructional decisions for/with a
class
Not for evaluation and grading
Teaching-Learning Cycle
Clear
Learning
Targets
Decisions
About
Next Steps
Student SelfAssessment &
Responsibility
Collaboration
Around
Assessment
Questioning &
Instructional
Tasks
Making
Inferences
& Giving
Feedback
Standards for Math Practice
• In your table groups highlight the verbs that
you see within the Standards of
Mathematical Practice
• Talk in your groups and be ready to share...
– Who will be doing these actions?
– What actions are already present in your
classroom?
– Which actions will you be working to include?
Ratio, Rate, Proportional
Reasoning
• It is estimated that less than half of the
adult population can be viewed as
proportional thinkers
Teaching Fractions and Ratios for Understanding, Lamon (1999)
• This implies that proportional reasoning is
not a natural development, but rather
requires some special experience or
learning- perhaps from the classroom
Ratio and Rate for Proportional
Reasoning
In order to reason proportionally students
need to be able to• Select equivalent ratios
• Compare ratios using a variety of
strategies
• Reason multiplicatively
• Reason multiplicatively as the quantities
change, but the relationship remains the
same
Rates and Rates of Change
To reason multiplicatively as the quantities
change, but the relationship remains the
same
• A rate represents a single relationship that
is shared by an infinite number of ratios.
• The rate shows how one quantity in the
pair will change with respect to how the
other quantity changes.
Seeing the Forest...
• Each table group will be creating a poster
that shows their solution strategy,
explaining how that strategy works, and
employs the strategy to count the trees
• With your table groups read through the
“Counting Trees Task”
• When your group is ready have 1 person
come get the poster paper and markers
your group will need
Seeing the Forest and the Trees
• Be ready to share your solution strategy
with the group
• If your group finishes before the whole
group is done, discuss how your students
might approach this problem and then list
the criteria that an “exceptional student
response” would have to include
Teaching-Learning Cycle
Clear
Learning
Targets
Decisions
About
Next Steps
Student SelfAssessment &
Responsibility
Collaboration
Around
Assessment
Questioning &
Instructional
Tasks
Making
Inferences
& Giving
Feedback
What Standards for
Mathematical Practices
could be potential
Learning Targets
corresponding to a
lesson involving this
task?
Or in other words, what
Practices could this task
be used to assess?
Seeing the Forest in Action
The following video is from MARSMathematics Assessment Resource
Service
Before watching this video let’s assume
• There is a lot we don’t know
• That the teacher has good intent and
teaching expertise
Seeing the Forest in Action
Be ready to share and discuss...
• What do you notice most about this video
and
– The task in which students are working
– What the students are doing
– What the teachers are doing
– The understandings about ratios that the
students utilize
– What aspects of the Teaching-Learning Cycle
are shown
Providing Actionable Feedback
• When working with your groups you listed the
criteria that an “exceptional student response”
would have to include
• Use your list and work with your group to provide
feedback to each of the students in the student
work samples
• Ideally, your feedback will
provide the student with an
action they can perform to
improve their thinking.
Clear
Learning
Targets
Decisions
About
Next Steps
Student SelfAssessment &
Responsibility
Collaboration
Around
Assessment
Questioning &
Instructional
Tasks
Making
Inferences
& Giving
Feedback
Impede or Empower
“Involving students in their learning is a key
characteristic of formative assessment.”
(Black and Wiliam, 1998)
“ When children continually participate in the
assessment process, they learn to recognize
their own expertise. As active assessors, they
necessarily exercise a more autonomous and
decision-making role in their learning.
Consequently, instead of being used to gain
power over a child, assessment empowers
the child.”
(Anderson, 1993)
Impede or Empower
• Read the classroom scenarios given in your
handouts
• With a partner determine whether each
scenario impedes or empowers students in
developing their responsibility for themselves
as learners of mathematics
Are there other examples you could add to this
set of scenarios? What practices do you
employ that empower your students to have
an active role in their learning?
Teaching-Learning Cycle
Clear
Learning
Targets
Decisions
About
Next Steps
Student SelfAssessment &
Responsibility
Collaboration
Around
Assessment
Questioning &
Instructional
Tasks
Making
Inferences
& Giving
Feedback
Summing It Up…
• We can look to BOTH the CCSS content standards
and the mathematical practices to help us define
clear learning targets.
• By including our students in the assessment
process, we share the assessment responsibility
with them and empower them to distinguish “right”
from “wrong”.
• By implementing INFORMative assessment in a
more deliberate manner, we could help students
make important connections between “big”
mathematical ideas.