Mathematics Assessment Project (MAP)
Mary Bouck / Hugh Burkhardt-October 30 2012
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• Research and development project out of
University of California, Berkeley
• Purpose—develop assessments that support
the Common Core State Standards for
Mathematics:
--Content
--Eight Practices
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Well-engineered tools to support those
facing new challenges with CCSSM
Tools for Formative Assessment
oLessons, “Classroom Challenges”
oProfessional Development Modules
 Test Tasks
 Prototype Summative Tests
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 Assess and develop student’s integrated
understanding
 Provide diagnosis and treatment
• Use rich tasks that assess reasoning, linking
content and practice standards
• Help teachers and students move reasoning
forward,
• A “protein supplement” to any curriculum
diet
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Concept development lessons
To reveal and develop students’ interpretations of
significant mathematical ideas and how these connect to
their other knowledge.
“Proficient students expect mathematics to make sense”
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Problem solving lessons
To assess and develop students’ capacity to apply their
math flexibly to non-routine, unstructured problems,
both from pure math and from the real world.
“They take an active stance in solving mathematical
problems”
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MAPs FAL are Lessons, they are
opportunities for students to learn as well
as opportunities of teachers and students
to note what students can do on a topic
and with the practices and what they are
struggling with.
MAPs FAL’s—are examples/models of how
one can do formative assessment and
instruction at the same time in a lesson.
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Classroom Challenge: a formative
assessment lesson for
“concept development”
FAL Task: Repeating Decimals
Read through the task and answer
the questions the way you think your
“average” student would do so.
Directions to students:
•Read through the opening text
‘How to write repeating decimals’ on
the handout,.
•Answer as much as you can. Show
your work so I (the teacher) can
understand your reasoning. Don’t
worry if you can’t do everything.
There will be a lesson on this
material that will help you improve
your work.
FAL Task: Repeating Decimals
Collect students’
responses to the task.
Make some notes on
what their work reveals
about their current levels
of understanding and
their problem solving
strategies.
We suggest that teachers
do not score students’
work. The research
shows that this will be
counterproductive.
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On your own, take
about 5 minutes and
change these
fractions into
decimals.
Do as many as you
can.
You likely will be
able to do some
mentally and others
may need to do some
working out on
paper.
You might see some
patterns that will
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help you!
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Use a
calculator and
try to finish the
sheet.
Note any
patterns you
see in the table
and write them
down on the
back of your
paper.
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Look at the pattern in the thirds-- 0.3333333... Do
the threes in this pattern go on forever? How can
you be sure?
What does the pattern for two thirds and three
thirds look like?
Look at the pattern in the sixths-- 0.16666... (or
0.16666....7, if the calculator rounds the number)
Is a sixth a repeating decimal? Why? How can we
write this using “a line” in the notation?
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Look at sevenths. Do you see any patterns?
0.142857142...; 0.285714285...;
0.428571428...; ...
How can we write these using our notation?
Look at the ninths: 0.111111...; 0.222222...;
0.333333...; 0.444444…
How does this pattern continue?
What happens with nine ninths?
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If needed, teach the students a method for
figuring out the fractional value of a repeating
decimal (shown on the first individual task).
x = 0.17
100x = 17.7
100x - 10x = 17.7 – 0.17
90x = 16
x = 16/90
x = 8/45
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Collaborative Activity: Continuing with
the same work and Ideas.
•Please get into groups of 2 or 3.
•Each group will be given three sets of
cards: Set A—Decimals, Set B—
Equations, and Set C—Fractions
(along with large sheet of paper)
•Take turns matching cards. the
expressions and the words. Place the
matches next to each other on the paper.
Each time you make a match explain to
your partner(s) why.
•If you think there is no suitable card
that matches, write one of your own.
There are some blank cards for doing
this.
Collaborative Activity: Teacher’s role
Please get into groups of 2 or 3 (self selected).
•Each group will be given three sets of cards: Set A—Decimals, Set B—Equations, and Set C—Fractions
•Take turns matching cards. the expressions and the words. Place the matches next to each other on the
paper. Each time you make a match explain to your partner(s) why.
•If you think there is no suitable card that matches, write one of your own. There are some blank cards for
doing this.
Teachers are to:
•Listen and watch students carefully.
•Note different student approaches to the task.
•Notice any difficulties that students encounter, and the
ways they justify and explain to each other.
•Try not to make suggestions that move students
towards a particular approach to the task. Instead, ask
questions that help students to clarify their thinking.
Post completed posters and compare solutions
It is suggested teachers:
 Select a set of cards that most groups matched
correctly as this may encourage good
explanations
 Then select one or two matches that most
groups found difficult. Ask students to share
their thinking with the whole
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FAL Task: Repeating Decimals: revisited
Directions to students:
• Look at your original
responses and the
questions (on the
board/written on your
script.)
• Think about what you
have learned.
• Look at the new task
sheet, Repeating
Decimals (revisited).
• Use what you have
learned to answer these
questions.
Turn to a couple of people near you and talk about:
•What
mathematics could your students learn or deepen their
understanding of with this lesson?
•What
would they struggle with in this lesson?
•How
is this lesson like what goes on in your district’s
mathematics classrooms?
•How
•What
this?
•What
is it different?
do you think might be the benefits of using a lesson like
could teachers learn about their students from using
lessons like this? What could students learn about themselves
from lessons like this?
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Items: The tasks used with the students look like classwork
and homework implied by standards (content and practices in
the CCSS).
•
Feedback: The task provides information to both the teacher
and the students as to his/her mathematical
understanding/skills. The feedback has content (mathematical)
not just value judgment (grade)
•
Revisions: There is an opportunity for students to revise their
work.
•
Timing: The assessment is available and used by teachers
while instruction is still going on for the topic, while there is still
time to adapt the teaching to meet the needs of the students.
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Assessment Task: Design
• Recommend these be given to
students one or more days before
lesson
• Suggest students answer the
questions without assistance from
the teacher
• Collect work, but do not give a score,
as the purpose is to help teachers
see what sense students are making,
so as to prepare for the lesson
• Suggest written comments be given;
“help students to make further
progress by summarizing their
difficulties as a list of questions.”
Constructive Feedback Recomendations
Recommendation:
Research shows that students benefit most from feedback that:
• Focuses on the task, not on grades or scores.
• Is detailed rather than general.
• Explains why something is right or wrong.
• Makes clear what has been achieved and what has not.
• Suggests what the student may do next
• Offers specific strategies for improvement
Lesson Structure
Before the lesson
Students work on an assessment task.
The teacher reviews the work, noting mistakes and misconceptions.
Whole-class introduction
Teacher asks questions or provides students with guidance on the context
of the lesson.
Collaborative activity
Students then work in small groups, producing a poster of their work.
Whole-class discussion
Students engage in an interactive discussion where they justify their
solutions.
Review individual solutions to the assessment task
Students use comments from the teacher regarding their work and what
they have learned in the lesson to try to improve their solution.
Collaborative Activity: Teacher’s role
Please get into groups of 2 or 3 (self selected).
•Each group will be given three sets of cards: Set A—Decimals, Set B—Equations, and Set C—Fractions
•Take turns matching cards. the expressions and the words. Place the matches next to each other on the
paper. Each time you make a match explain to your partner(s) why.
•If you think there is no suitable card that matches, write one of your own. There are some blank cards for
doing this.
Teachers are to:
•Listen and watch students carefully.
•Note different student approaches to the task.
•Notice any difficulties that students encounter, and the
ways they justify and explain to each other.
•Try not to make suggestions that move students
towards a particular approach to the task. Instead, ask
questions that help students to clarify their thinking.
Reviewing Work
• Students revisit their answers to the assessment
task and revise their work.
• They either use a different colored pen when
reviewing their work or complete a fresh copy or
slightly different version of the task sheet.
Lesson Structure
Before the lesson
Students work on an assessment task.
The teacher reviews the work, noting mistakes and misconceptions.
Whole-class introduction
Teacher asks questions or provides students with guidance on the context
of the lesson.
Collaborative activity
Students then work in small groups, producing a poster of their work.
Whole-class discussion
Students engage in an interactive discussion where they justify their
solutions.
Review individual solutions to the assessment task
Students use comments from the teacher regarding their work and what
they have learned in the lesson to try to improve their solution.
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Good tasks—CCSS content and opportunities to
engage in one or more practices
Collect information (intro task), inform teacher
and lesson plan
Given written feedback and not scores
During “collaborative” work, teacher is to work
hard at listening to what students can do and
what they struggle with
Opportunity for revising work (understanding,
thinking)
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Math goals
Standards addressed: content and practice
Introduction/overview
Materials needed
Time needed
Lesson outline
--Before lesson (individual task and analysis)
--Launch, Explore, Summarize
Solutions
Student sheets
Mathematical Content
Number
8.NS:Know that there are numbers that are not rational,
and approximate them by rational numbers.
Equations and Expressions
8.EE: Analyze and solve linear equations.
Mathematical Practices
1.Make sense of problems and persevere in solving
them.
2. Reason abstractly and quantitatively.
6. Attend to precision.
Classroom Challenge: a formative
assessment lesson for
“problem solving”
FAL Task: Gold Rush
Read through the task and
think about how a “typical”
middle school student in your
district would answer the
questions.
Directions to students: Read
through the questions carefully
and try to answer them as well
as you can. Show all your work
so that I (the teacher) can
understand your reasoning. Try
to show your findings in an
organized way. Don’t worry if
you can’t do everything. There
will be a lesson on this material
that will help you improve your
FAL Task: Sample Teacher Feedback
• What does the length of the
rope given to a prospector
measure?
• How could you measure the
amount of land enclosed by
the rope?
• Now investigate if combining
ropes affects how much land
each prospector gets.
• How can you now organize
your work?
• How do you know for sure
your answer is the best
option?
Collaborative Activity: Producing a Joint solution
Please get into groups of 2 or 3 (self selected).
1.Take turns to explain your method and how you think
your work could be improved.
1.Listen carefully to each other. –
don’t understand.
Ask questions if you
1.Once everyone in the group has explained their method,
plan a joint method that is better than each of your
separate ideas.
1.Make sure that everyone in the group can explain the
reasons for your chosen method.
1.On poster paper, write a brief outline of your group’s
method. State any assumptions your group has made.
Collaborative Activity: Teacher’s role
Please get into groups of 2 or 3 (self selected).
1.Take turns to explain your method and how you think your work could be
improved.
2.Listen carefully to each other. – Ask questions if you don’t understand.
3.Once everyone in the group has explained their method, plan a joint method that is
better than each of your separate ideas.
4.Make sure that everyone in the group can explain the reasons for your chosen
method.
5.On poster paper, write a brief outline of your group’s method. State any
assumptions your group has made.
Teachers are to:
•Listen and watch students carefully.
•Note different approaches to the task and what
assumptions students make.
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Display posters
Hold a class discussion that has groups
explaining different approaches and/or
different assumptions.
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Imagine you are the teacher and have to assess
the student work. Individually review each piece of
work.
Note the mathematics the student has used and if
they have made any errors.
Explain your findings to the rest of the group.
Listen carefully to explanations by others in your
group. Ask questions if you don't understand.
Once everyone agrees on what the student has
done, complete the questions below the students’
work. (Make sure the student who writes the
answers is not the student who explained them.)
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P-38
P-39
P-40
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Hold a class discussion that has groups
explaining the different approaches
assumptions in the samples.
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Which method did you like best? Why?
Which method did you find most difficult
to understand? Why?
How could the student (Ann, Mark, . . . )
improve his/her answer?
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Lesson Structure
Before the lesson
Students work on an assessment task
The teacher reviews the work, noting mistakes and misconceptions, and
writes questions that will push students to improve their solutions.
Whole-class introduction
Provides students with feedback and they work to improve solution
Collaborative activity
Students work in small groups to provide a joint solution.
Analyzing Student Work
Students discuss sample work and give written analysis.
Whole-Class Discussion
Organize a discussion about what has been learned and some of the
different approaches.
Review or Reflect on work and learning
Students reflect on their individual and group solutions and work on an
improved solution or they reflect on their work and learning.
Mathematical Content: Geometry:
6.G. Find the area of right triangles, other triangles, special
quadrilaterals, and polygons by composing into
rectangles or decomposing into triangles and other
shapes; apply these techniques in the context of solving
real-world and mathematical problems.
7.G. Draw, construct, and describe geometrical figures and
describe the relationships between them. Solve real-life
and mathematical problems involving angle measure,
area, surface area, and volume.
Mathematical Practices
1. Make sense of problems and persevere in solving
them.
3. Construct viable arguments and critique the reasoning
of others.
4. Model with mathematics.
After looking at two lessons,
what questions, comments or
concerns do you have :
 About the formative
assessment lessons and or
MAP Lessons?
 Implementation of those
lessons in classrooms??
Mathematics Assessment Project (MAP)
Open source:
http://map.mathshell.org.uk/materials/
Mary Bouck
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