Transcript Preparation of teachers

What is a super even number?
 Is 2,097,152 supereven?
How can you decide in an
efficient way? (Roodhardt et al, 1997)
Reflecting on practice: Week 1
 What does it mean to teach? How do you
know?
 What does it mean to productive observe
and discuss teaching?
 What are the benefits of observing and
discussing what goes on inside our
classrooms?
What does it mean to teach?
 To facilitate, plan and guide student
learning
 To have visible evidence of student learning
 To create an environment where students
can learn
 To stimulate student thinking using rigorous
and meaningful tasks
 To express passion and excitement and
encourage students to do the same
You know when students can
 explain their thinking or their work to
their peers
 communicate their thoughts and
reasoning
 apply concepts correctly to novel
situations
 make connections
How do you know students
understand? When they
 Recognize how the concept/idea is related or
connected to other things we know
 Consciously reflect on experiences with the
concept/idea (know what does and does not
make sense
 Communicate with others about the
concept/idea
Hiebert et al, 1999
To facilitate learning teachers can
 Ask what they themselves expect to
see/hear from the students
 Observe the students during a lesson,
measuring understanding, guiding and
adapting as needed
 Structure opportunities for interactions
among teacher-student, student-teacher, and
student-student
Teachers Report (ES/MS)
Rarely or
never
Once/twice Frequently
month
or always
10%/4%
34%/37%
57%/59%
Students
9%/8%
26%/20%
explain to each
other
Open ended
51%/29% 33%/40%
tasks
66&/72%
Make
connections
16%/31%
Hudson et al, 2003
The vehicle for teaching: the
lesson
Reflecting on practice:Weeks 2/3
 In your group: design a lesson related to a
given topic.

First draft of the lesson will be due on
Friday with an electronic copy sent to a
lesson study consultant who will provide
feedback over the weekend on the lesson.
 Revise the lesson Monday and Tuesday.
 Teach a selected portion of the lesson to the
larger group in one of the three rooms on
Wednesday and Thursday.
Typical flow of a class
United States
 Demonstrate a
procedure
 Assign similar
problems to
students as
exercises
 Homework
assignment




Takahashi, 2005
Japan
Present a problem to the
students without first
demonstrating how to
solve the problem
Individual or group
problem solving
Compare and discuss
multiple solution
methods
Summary, exercises and
homework assignment
The Lesson
Introduction: Hatsumon
Thought provoking question
Key question – shu hatsumon
Individual or small group work
Walking among the desks – kikanshido
Anticipated student solutions
Student solutions - Noriage
Massaging students’ ideas
Summing up- Matome
Bass et al, 2002
The Teaching Principle
Effective teaching requires
understanding what students know
and need to learn and challenging
and supporting them to learn it well.
(NCTM, 2000)
The Teaching Principle
Effective teaching requires
understanding what ALL students
know and need to learn and
challenging and supporting them to
learn it well.
(NCTM, 2000)
Research suggests that students
learn when they
•are doing something they consider
worthwhile
•are actively involved in choosing
strategies
•get feedback on what they are thinking
before it is too late
•struggle with a concept before they are
given a lecture
NRC, 1999
The lesson
 Context
 Explicit mathematical goals
 The investigation
 Launch
 Tasks
 How students will work
 Managing discussion
 Questions
 Student work
 Evidence of learning- begin
with the end in mind
Our work
Be explicit about the grade level and
context of the lesson
Include all of the components of the
lesson
Mathematical objectives
“Explore the Pythagorean Theorem”
“Deepen their understanding of proof”
“Learn about lines”
“Section 2.4”
“Learn to work in groups and share”
How will you gain and maintain
student attention: A launch
 Invites students into the mathematics
 Engages students
 Connects to the mathematics in the
lesson
 Short
How might you launch a first
discussion of solving systems of
equations?
Choosing Tasks - a purpose
 Master routine procedures
 Develop conceptual understanding
 Explore new mathematical terrain
 Secure understanding
 Do mathematics
Opportunities for discussion
Tasks have to be justified in terms of the
learning aims they serve and can work
well only if opportunities for pupils to
communicate their evolving
understanding are built into the
planning. (Black & Wiliam, 1998)
In the figure below, what fraction of
the rectangle ABCD is shaded?
B
a) 1/6
A
b) 1/5
c) 1/4
d) 1/3
e) 1/2
D
C
D
NCES, 1996
Dekker & Querelle, 2002
Characterizing Tasks & Knowledge
Cognitive Demand Knowledge Needed
Doing Mathematics
Procedures with
Connections
Procedures without
Connections
Memorization
(Stein, et.al., 2001)
Strategic
Choosing, formulating
strategies
Schematic
explaining, justifying,
predicting
Procedural
performing procedures
Declarative
defining, giving
examples
(Shavelson et al, 1999)
Which shape will hold the same
amount of spaghetti and be the most
economical?
Area Surface Volume Ratio of
of
area
surface
base
area to
volume
Cylinder
Rectangular
prism
Shape 3
MDoE, 2003
The questions
 “… the only point of asking questions is to raise issues
about which a teacher needs information or about which
students need to think.”
 The responses a task might generate and the ways of
following up have to be anticipated
 Questions are a significant part of teaching where attention
is paid to how they are constructed, used to explore, then
develop learning
(Black et al, 2004)
Using Inquiry
 Engaging in mathematical thinking
 Justifying, thinking procedurally and
reflecting
 Making and explaining connections
 Doing procedures
 Knowing facts
Inquiry Questions
 Compare and contrast: How are they alike? How different?
 Predict forward: “What would happen if . . ?”
 Predict backward: “How can I make . . happen?” “Is it
possible to ... ?”
 Analyze a connection/relationship: “When will . . . be
(larger,equal to, exactly twice, …) compared to . . .?” “When
will . . be as big as possible?”
 Generalize/make conjectures: “When does . . . work?”
“Under what conditions does … behave this way?” “Describe
how to find . . .?” “Is this always true?”
 Justify/prove mathematically: “Why does . . . work?”
 Consider assumptions inherent in the problem and what
would happen if they were changed
 Interpret information, make/ justify conclusions: “The data
support… ; “This… will make ….happen because…”
Scaffolding
 Framing of tasks
 Implementing the tasks
- the questions posed
- the way questions are answered
Types of math problems
Stigler & Hiebert, 2004
How teachers implemented math
problems
Stigler & Hiebert, 2004
Managing learning- about the
details
 How will students work? How will they be
arranged and for what reason?
 What tools will be useful and how should
they be made available?
 How will the work be recorded?
 When will you break for a discussion and
why?
 How will they share their work?
Managing Student Responses
“…when teachers learn to see and hear
students’ work during a lesson and to
use that information to shape their
instruction, instruction becomes clearer,
more focused, and more effective.” (NRC,
2001, p.350 )
What and how the work is
recorded matters
2x 15 + 25x3 = 150
15
25
x2
x3
30
+
105
2x.15 + 3x.25 = 1.50
75
What and how the work is
recorded matters
2x15 + 3x25 = 30+ 75 = 105
3x15 + 2x25 = 45 + 50 = 95
4x15 + 2x25 = 60 + 50 = 110
Goal: Ax+By = C
Strategies for learning
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Brainstorming discussion
Drawing/artwork
Field trips
Games
Humor
Graphic Organizers, maps,
Manipulatives/labs/
experiments
Music
Movement
Metaphor/analogy/simile
Mnemonic devices
•Projects/problem based
instruction
•Reciprocal teaching
(peer teaching)
•Role play
•Storytelling
•Technology
•Visualization
•Visuals
•Apprenticeships
•Writing/journals
(M. Tate, 2005)
Our work
Area of a quadrilateral
Slope
Similarity
Our work
Tuesday: Specific mathematical goal and
context selected; tasks chosen
Wednesday: Initial framing of lesson,
anticipated student solutions
Thursday: Description of how the class will
be organized and managed, key questions
identified
Friday: First draft produced
Suggestions
Tuesday: Brainstorm with whole group but
come to consensus
Everyone does the task(s) and shares
solutions - thinking about their own
approach and how their students might
work
Divide the work on Thursday so a subset is
responsible for each segment; share the
last 20 minutes of the hour
Working together
 Gain insights
 Better able to understand students
 Accountability
 Troubleshoot
 Stimulate our own teaching
 Gives you perspectives you may not have
thought of
References
 Bass, H., Usiskin, Z, & Burrill, G. (Eds.) (2002). Classroom
Practice as a Medium for Professional Development.
Washington, DC: National Academy Press.
 Black, P. & Wiliam, D. (1998). “Inside the Black Box:
Raising Standards Through Classroom Assessment”. Phi
Delta Kappan. Oct. pp. 139-148.
 Black, P., Harris, C., Lee, C., Marshall, B. & Wiliam, D.
(2004). Working inside the black box: Assessment for
learning in the classroom.. Phi Delta Kappan. (86,1) p8
 Hiebert, J., Carpenter, T., Fennema, E., Fuson, K., Wearne,
D., & Murray, H. (1992). Introducing the critical features of
classrooms. in Making sense: Teaching and learning
mathematics with understanding. Portsmouth, NH:
Heinemann
 Dekker,T. & Querelle, N. (2002). Great assessment
problems (and how to solve them). CATCH project
www.fi.uu.nl/catch
 Hudson, S., McMahon, K., & Overstreet, C. (2002). The
2000 National Survey of Science and Mathematics
Educators: Compendium of Tables. Chapel Hill NC:
Horizon Research, Inc. Michigan Department of
Education. (2003). MMLA Lesson Study Project. Burrill,
G., Ferry, D., & Verhey R. (Eds). Lansing, MI..
 National Center for Educational Statistics. (1996).
National Assessment for Educational Progress Released
Item.
 National Council of Teachers of Mathematics. (2000).
Principles and Standards for School Mathematics. Reston
VA: Author
 National Research Council. (1999). How People Learn:
Bain, mind, experience,and school. Bransford, J. D.,
Brown, A. L., & Cocking, R. R. (Eds.). Washington, DC:
National Academy Press.
 Roodhardt, A., Kindt, M., Burrill, G., & Spence, M. (1997).
Patterns and Symbols. From Mathematics in Context.
Directed by Romberg, T. & de Lange, J. Chicago IL:
Encyclopedia Britannica
 Shavelson, R., Ruiz-Primo, M.A., Li, M., and Ayala, C.C.
(August 2003). Evaluating New Approaches to
Assessing Learning, CSE Report 604, National Center
for Research on Evaluation, Standards, and Student
Testing (CRESST), Center for the Study of Evaluation
(CSE), and the UCLA Graduate School of Education &
Information Studies
 Stigler, J. and Hiebert, J., Improving Math Teaching. In
Improving Achievement in Math and Science. 2004 (pp.
12-17).
 Stein, M., Smith, M., Henningsen, M., & Silver, E.
(2000). Implementing standards-based mathematics
instruction: A casebook for professional development.
New York NY: Teachers College Press.
 Takashi, A. (2005). Presentation at Annual Meeting of
Association of Mathematics Teacher Educators.
 Tate,M. (2006). Designing Lessons for Learning.
Presentation at TI International Technology Conference.
Denver CO.