Transcript Learning Fractions In a Linear Measurement Context

What is Lesson Study and How
Can it Support the Common
Core State Standards?
St. Petersburg, Florida, December 8, 2012
Catherine Lewis
Mills College
www.lessonresearch.net
This material is based
upon work supported by
the National Science
Foundation under Grant No.
0207259. Any opinions,
findings, and conclusions
or recommendations are
those of the author(s) and
do not necessarily reflect
the views of the National
Science Foundation.
This material is based upon
research supported by the
Department of Education
Institute for Education
Sciences, Grant No.
R308A960003. Any
opinions, findings, and
conclusions or
recommendations are those
of the authors and do not
necessarily reflect the views
of the grantors.
Thanks to MSRI for funding substitute teachers for site-based participants.
How I encountered lesson
study….
Good professional
learning is_____________
Lesson Study Cycle
1. STUDY CURRICULUM &
FORMULATE GOALS
Consider long-term goals for student
learning and development
4. REFLECT
Study subject matter, curriculum and
standards, focus the inquiry
Formal lesson colloquium in which
observers:
Select or revise research lesson
•Share data from lesson
Write instructional plan that includes:
•Long-term goals
•Anticipated student thinking
•Data collection plan
•How the lesson fits in the long-term
learning trajectory
•Rationale for chosen approach
•Use the data to illuminate student
learning, disciplinary content, lesson
and unit design, and broader issues
in teaching-learning
•Document the cycle, to consolidate
and carry forward learning and new
questions into next cycle of lesson
study
2. PLAN
3. DO RESEARCH
LESSON
One team member conducts
research lesson, others
observe and collect data
Choosing a Lesson Study Theme
What qualities would you like students to have
when they leave your mathematics program?
What is a gap between those ideal qualities and
their actual qualities that you would really like to
work on as an educator?
“Build children’s mathematical
curiosity, and their ability to notice and
represent mathematical patterns…”
Can patterns help us
find an easy way to
answer the question:
How many seats fit
around any number of
triangles, arranged in a
row as shown?
INPUT
OUTPUT
Number of
Triangle
Tables
Number of
Seats
1
3
2
4
3
4
5
6
Questions About Video
1.Planning: What is similar and
different from planning familiar to
you?
1.Lesson: Practice collecting data
on students.
1.Discussion: What do teachers
learn? What agenda helps them
Recap of “How Many Seats?” Lesson Study
Cycle
Research Lesson 1
• All students filled out chart correctly but
few could verbalize meaning of +2 pattern
Research Lesson 2
• Chart eliminated, students solved
individual problems, shared findings
• Students showed their counting methods
• Many students could verbalize meaning of
+2 pattern
Area of circle
Area of rectangle
Part of a set
Linear measurement
1 meter
US: Harcourt
Volume Measure
Number Line
Grade 5
Area: Rectangle
Set
US: Investigations
Number Line
Area: Rectangle
Money
Japan: GT
Linear Measure: Fraction
Meaning
Linear Measure: Fraction
Meaning
Volume Measure
Volume Measure
Number Line
Number Line
Area: Circle
Fraction bar, strip
Bar Graph
Japan: TS
Set
Clock
Grade 4
Linear Measure: Data Report
3-D Objects
Linear Measure: Data Report
Weight
Fraction track
Area: Rectangle
Area: Rectangle
Volume Measure
Number Line
Volume Measure
Linear Measure: Fraction
Meaning
Linear Measure: Fraction
Meaning
Volume Measure
Volume Measure
Number Line
Number Line
Area: Rectangle
Area: Rectangle
Area: Rectangle
Area: Circle
Area: Other Figure
Set
Fraction bar, strip
Weight
Bar Graph
Linear Measure: Data Report
Music Notes
Money
Number Line
Area: Rectangle
Set
Money
Grade 3
Volume Measure
Number Line
Area: Rectangle
Area: Rectangle
Area: Circle
Area: Circle
Area: Other Figure
Set
Fraction bar, strip
Weight
Grade 2
Money
Area: Other Figure
Set
Money
Volume Measure
Volume Measure
Number Line
Area: Rectangle
Area: Rectangle
Area: Circle
Area: Circle
Area: Other Figure
Area: Other Figure
Set
Set
3-D Object Volume
Fraction bar, strip
Linear Measure: Data Report
Area: Other Figure
Set
Volume Measure
Weight
Linear Measure: Data Report
Area: Circle
Grade 1
Linear Measure: Fraction
Meaning
Number Line
Linear Measurement Context
How Can We Describe the Blue
Mystery Piece in Terms of One
Meter?
1 meter
Common Challenges in Understanding
Fractions
• Seeing fraction as number (“I can’t put 2/3 on
number line because it’s two different numbers”)
• Understanding the magnitude of the
denominator (that 1/6 is smaller than 1/5)
• Knowing what is the whole (construct whole
from a fraction)
• Seeing that fractions can be greater than one
How Linear Measurement
Context Might Help
Length helps students attend to
magnitude of fractions (how much)
rather than just count pieces (how
many)
1 meter
Understanding Meaning of Denominator
Only 1 dimension (length) varies, making it easier to
see that ½ is bigger than ¼
1 meter
1 meter
Understanding the Whole
Standard measurement unit gives clear,
stable image of the “whole”
1 meter
Understanding 4/3 as 4
1/3’s
Length may support multiplicative
image that 3 times 1/3 meter is 1
meter and x times 1/n meter is x/n
meter
meter
Lesson Study Resource Kit
1. Mathematics tasks to solve and discuss
(& related student work to analyze)
2. Curriculum inquiry: Japanese textbook,
Takahashi lesson video, teachers’ materials
3. Lesson study materials (template for lesson plan,
protocol for discussion, etc.)
4. Suggested teacher-led inquiry process to explore
and use resource kit
Teachers Solve and Discuss Student Tasks
Problem 1
Estimate the answer to
You will not have
time to solve the problem using paper and
pencil.
Discussion Questions:
How did you solve the problem, and how might
students solve the problem?
Student responses to this task are provided at the
end of this section (p. 14). Discuss why students
chose each of the responses shown.
Teachers try a problem: Find the length
of the mystery strip
Sample
• 11 States and 27 districts
• 13 Groups per condition (4-9 teachers per
group, locally formed)
• 213 Teachers
41% New to Lesson Study
78% Elementary Teachers
• 1,059 Students (Grades 2-5)
Teachers’ Fractions Knowledge (Z-score)
Teachers’ Reflections
“The question of linear versus a “pie”
understanding was really compelling for
me. It’s a distinction in the concept of
fractions that I hadn’t considered and I
wonder what my own understanding of
fractions would be like if I had been first
introduced that way.”
Change in Students’ Fractions
Knowledge (Absolute Score, N=1059)
7
6
5
4
3
2
1
0
Resource
Kit
Control
Gr 2/3
Resource
Kit
Control
Gr 4
Resource
Kit
Control
Gr 5
Teachers’ Reflections
I've always understood how to work with
fractions but didn't understand all of the
"why's" behind the procedures…My students
made many awesome discoveries because I
learned how important a true conceptual
understanding of unit fractions is to students’
overall understanding of other fraction
concepts. [39-680]
Survey Item Examples
Expectations for Student Achievement
“By trying a different teaching method, I can
significantly affect a student’s achievement” (7
items; α=.64 )
Using and Promoting Student Thinking
“I have some good strategies for making students’
mathematical thinking visible” (4 items; α=.68)
When Solar Energy Was Added to the
Japanese National Curriculum….
• Hundreds of elementary schools applied
for small grants as “designated research
schools” on how to teach solar energy
• After about a year of experimentation,
often in collaboration with universitybased colleagues, schools brought to life
their thinking in large public research
lessons
Solar energy, cont’d
• Thousands of educators saw these research
lessons and questioned teachers about why
they chose these approaches, what had
worked and hadn’t
• Knowledge quickly spread about the science
content itself, good teaching materials (what
toys work and don’t to illuminate the
principles), and student thinking.
Solar energy, cont’d
A teacher observing a public research lesson asked
about three student strategies she saw:
•moving a solar cell closer to a light source
•adding a second light source
•using a magnifying class to “concentrate” light
“I want to know whether the three conditions the
children described – ‘to put the solar cell closer
to the light source,’ ‘to make the light stronger’
and to ‘gather the light – would all be considered
the same thing by scientists. They don’t seem the
same to me. But I want to ask the teachers who
know science whether scientists would regard
them as the same thing.”
Using Lesson Study, Japanese Teachers:
•Develop and share their practice-based
knowledge about how to teach new standards
•Shifts policy from “pushed in” by
administrators to “pulled in” by teachers, who
actively experiment with ideas, bring them to life
in the classroom
•Allow policymakers to do “formative research”
on policy
Using Lesson Study, Japanese Teachers:
•Changes the national conversation as well, so
that policy is talked about in the context of
actual, shared classroom practice seen by
policymakers
•Allow policymakers to do “formative research”
on policy
Silicon Valley Mathematics Initiative’s
Lesson Study Project
www.svmimac.org
Lesson Study has
grown and prospered
since 2000
Teams report to schools
and instruction improves.
SVMI provides a
mini-grant to a LS Many teams attend a 5team at a school day summer institute on
or district.
Lesson Study.
All teams engage in the
Annual Public Lesson
Open House
All teams participate
in Fall orientation.
SVMI’s Lesson
Study Project
Model
Teams research, design and
plan a lesson (often building
off existing lessons).
When research lesson is
refined the team conducts
exchange lessons with
another team in the project
(often from a different
district).
Teams use the
student data to revise
and polish the lesson.
Teams try out
initial lesson
designs in
classrooms.
Team members
observe student
thinking.
Which is a 4 X 5 rectangle? What was
each student thinking? (During
lesson by Akihiko Takahashi, 2002)
Lesson Inspired Teachers to Develop and
Spread “Re-engagement”
“Re-engagement” strategy to revisit and reconsider
student thinking. Spread across:
•
•
•
•
At least 7 districts
Elementary and secondary classrooms
Subject areas (math, language arts, etc.)
School-foundation boundary
How Does Lesson Study Fit With
Your Ideas About Good
Professional Learning?
Professional Development
TRADITIONAL
LESSON STUDY
• Begins with answer
• Begins with question
• Driven by expert
• Driven by participants
• Communication
-> teachers
trainer
• Communication among
teachers
• Relationships hierarchical
• Relationship reciprocal
• Research informs practice
• Practice is research
By Lynn Liptak, Paterson School #2, New Jersey.
40
Teachers’ Activities to Improve
Instruction
Choose curriculum,
write curriculum,
align curriculum,
write local standards
Plan lessons individually
Plan lessons collaboratively
Watch and discuss each other’s
classroom lessons
U.S.
copyright Catherine C. Lewis 2005
JAPAN
41
How does lesson study improve instruction?
Visible
Features of
Lesson Study
•Planning
•Curriculum Study
•Research Lesson
•Data Collection
•Discussion
•Revision
•Etc.
?
Instructional
Improvement
42
Figure 1
A Common (Mis)Conception of Lesson Study
Visible Features
of Lesson Study
 Plan
 Teach
 Observe
 Discuss
 Etc.
Key Pathway
Lesson Plans
Improve
Instructional
Improvement
43
How Does Lesson Study Improve Instruction?
Visible
Features of
Lesson Study
•Planning
•Curriculum Study
•Research Lesson
•Data Collection
•Discussion
•Revision
•Etc.
Pathways
Teachers’ Knowledge
Teachers’ Beliefs
Teachers’
Community
Instructional
Improvement
Teaching-Learning
Resources
Policies, Routines
that Support
Learning
44
Building Knowledge Through Lesson Study
“10 years ago, I probably knew only
one way of deriving the area formula
for circles….”
Tad Watanabe
–Professor, Department of Mathematics,
Kennesaw State University, Georgia)
Other ways learned from lesson study
activities…
From Hironaka et al., 5B, p.91
Other ways….
School-wide Lesson Study
California Standards Test in Mathematics: Mean Scale Scores,
Grades 2-5, School and District
3-year net increase for school more than triple that for district (F=.309,
845df p<.001) (Lewis, Perry, Hurd, O’Connell, Phi Delta Kappan, 2006, 88:4)48
Percent Statements Focused on Student
Thinking
60
50
40
Planning
Debrief
30
20
10
0
1
Year 1
2
Year 3
Schoolwide Lesson Study School in U.S.
49
Percent Statements
on Evaluation/Ability
9
8
7
6
5
Planning
Debrief
4
3
2
1
0
1
2
Year 1
Year 3
50
Successful Lesson Study
• Allows teachers to develop knowledge and
beliefs to support improvement– allows
teachers to develop their minds and
hearts
• Changes relationships among teachers.
“Changes the talk around the water
cooler” at a school so that teachers
become more willing to share difficulties
Teachers’ Reflections
The opportunity to focus on two to four
students’ learning was incredible…You feel
like you are in a true research mode.
Elementary Teacher, California
52
Until lesson study we never discussed the value of
the content being taught. We discussed the
different ways students learn (multiple
intelligences), how the brain works, how to
differentiate an inclusion class.
Never had those discussions involved a
discussion of how to develop problemsolving techniques, how to develop a
particular concept …what to expect for
outcomes, and how to adjust the lesson
to meet student needs.
Secondary Teacher, Massachusetts
Teachers’ Reflections
The lesson study has taught me: We must
never assume that all students understand.
It was observed several times that even
our "good" students did not have full
understanding….Lesson study is staff
development in its purest form. Rich
discussion occurs. Team members are
allowed to be creative, curious, selfmotivated participants. The team building
was incredible.”
Elementary Teacher, NY #562
Thank you!
Catherine Lewis
[email protected]
Videos, articles, more information at:
www.lessonresearch.net