North Carolina Math Summit • Growth Mindset in Mathematics
August 10 and 11, 2016 • Cary, NC
Deborah Loewenberg Ball
WHAT ARE WAYS TO SEE AND GROW
STUDENTS’ MATHEMATICAL COMPETENCE?
“Formative Assessment” and the Work of Teaching
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© 2016 Deborah Loewenberg Ball • School of Education • University of Michigan • Ann Arbor, MI 48109 • [email protected]
SEEING AND BUILDING STUDENTS’
MATHEMATICAL LEARNING COMPETENCE
①  Warmup problems and end-of-class checks
②  Open-ended notebooks
③  Presentations at the board
④  Homework
2
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© 2016 Deborah Loewenberg Ball • School of Education • University of Michigan • Ann Arbor, MI 48109 • [email protected]
FOR EACH PRACTICE, THINKING ABOUT THE
WORK OF TEACHING:
!  What are the affordances?
!  What are your questions/concerns?
!  New ideas?
3
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© 2016 Deborah Loewenberg Ball • School of Education • University of Michigan • Ann Arbor, MI 48109 • [email protected]
①  WARMUP PROBLEMS AND
END-OF-CLASS CHECKS
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© 2016 Deborah Loewenberg Ball • School of Education • University of Michigan • Ann Arbor, MI 48109 • [email protected]
USING A WARMUP PROBLEM
!  Problem on small piece of paper, students insert
into notebook
!  Students work silently without conferring for about
5 minutes
!  Sometimes discuss, sometimes teacher circulates
and notices aspects of different students’ work
5
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© 2016 Deborah Loewenberg Ball • School of Education • University of Michigan • Ann Arbor, MI 48109 • [email protected]
6
August 11, 2016
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International License: https://creativecommons.org/licenses/by-nc-nd/4.0/
© 2016 Deborah Loewenberg Ball • School of Education • University of Michigan • Ann Arbor, MI 48109 • [email protected]
USING A WARMUP PROBLEM TO LAUNCH WORK:
IDENTIFYING CONDITIONS OF A PROBLEM
Making sense of and interpreting problems.
What does this involve?
!  Careful reading
!  Identifying conditions
7
Making this explicit
and transparent is a
high-leverage
mathematical
teaching practice.
August 11, 2016
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© 2016 Deborah Loewenberg Ball • School of Education • University of Michigan • Ann Arbor, MI 48109 • [email protected]
WARM-UP PROBLEM
How many different three-digit numbers can you make
using the digits 1, 2, and 3, and using each digit only
once?
Show all the three-digit numbers that you found.
How do you know that you found them all?
8
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© 2016 Deborah Loewenberg Ball • School of Education • University of Michigan • Ann Arbor, MI 48109 • [email protected]
VIDEO: HOW DOES THIS WORK BUILD STUDENTS’
RESOURCES?
9
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© 2016 Deborah Loewenberg Ball • School of Education • University of Michigan • Ann Arbor, MI 48109 • [email protected]
THE ROLE OF CONDITIONS OF A PROBLEM
!  Identifying conditions can help in making sense of and
interpreting a problem
!  Using the conditions can help in persevering in solving
a difficult problem
!  Conditions are useful in constructing a mathematical
argument
!  Referring to the conditions is useful in critiquing an
argument
10
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© 2016 Deborah Loewenberg Ball • School of Education • University of Michigan • Ann Arbor, MI 48109 • [email protected]
11
August 11, 2016
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International License: https://creativecommons.org/licenses/by-nc-nd/4.0/
© 2016 Deborah Loewenberg Ball • School of Education • University of Michigan • Ann Arbor, MI 48109 • [email protected]
FOREGROUNDING AND USING “ERRORS”
!  Deliberately inviting or featuring “stuck” or “wrong”
solutions
!  Asking students to share and analyze “wrong”
interpretations, answers, methods
!  Posing incorrectly solved problems and asking students to
explain what is wrong and why someone might do this
!  Designing problems that entail appraisal of a solution
12
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© 2016 Deborah Loewenberg Ball • School of Education • University of Michigan • Ann Arbor, MI 48109 • [email protected]
END-OF-CLASS CHECKS (AKA “EXIT TICKETS”)
!  Questions can include
!  An example of something done in class today
!  A review question
!  A process, learning, or reflection question
!  Should take no more than 3 minutes
!  Useful to be something the teacher can eyeball and
approve before each student leaves
13
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© 2016 Deborah Loewenberg Ball • School of Education • University of Michigan • Ann Arbor, MI 48109 • [email protected]
14
August 11, 2016
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International License: https://creativecommons.org/licenses/by-nc-nd/4.0/
© 2016 Deborah Loewenberg Ball • School of Education • University of Michigan • Ann Arbor, MI 48109 • [email protected]
15
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International License: https://creativecommons.org/licenses/by-nc-nd/4.0/
© 2016 Deborah Loewenberg Ball • School of Education • University of Michigan • Ann Arbor, MI 48109 • [email protected]
16
August 11, 2016
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© 2016 Deborah Loewenberg Ball • School of Education • University of Michigan • Ann Arbor, MI 48109 • [email protected]
①  WARMUP PROBLEMS AND
END-OF-CLASS CHECKS
!  What are the affordances?
!  What are your questions/concerns?
!  New ideas?
17
August 11, 2016
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International License: https://creativecommons.org/licenses/by-nc-nd/4.0/
© 2016 Deborah Loewenberg Ball • School of Education • University of Michigan • Ann Arbor, MI 48109 • [email protected]
②  OPEN-ENDED NOTEBOOKS
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International License: https://creativecommons.org/licenses/by-nc-nd/4.0/
© 2016 Deborah Loewenberg Ball • School of Education • University of Michigan • Ann Arbor, MI 48109 • [email protected]
19
August 11, 2016
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International License: https://creativecommons.org/licenses/by-nc-nd/4.0/
© 2016 Deborah Loewenberg Ball • School of Education • University of Michigan • Ann Arbor, MI 48109 • [email protected]
TYPES OF WRITING IN NOTEBOOKS
! 
! 
! 
! 
! 
20
Warmup problems
In-class problems
Notetaking on others’ ideas as well as one’s own
“Note to self”
End-of-class checks
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© 2016 Deborah Loewenberg Ball • School of Education • University of Michigan • Ann Arbor, MI 48109 • [email protected]
21
August 11, 2016
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International License: https://creativecommons.org/licenses/by-nc-nd/4.0/
© 2016 Deborah Loewenberg Ball • School of Education • University of Michigan • Ann Arbor, MI 48109 • [email protected]
22
August 11, 2016
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International License: https://creativecommons.org/licenses/by-nc-nd/4.0/
© 2016 Deborah Loewenberg Ball • School of Education • University of Michigan • Ann Arbor, MI 48109 • [email protected]
23
August 11, 2016
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International License: https://creativecommons.org/licenses/by-nc-nd/4.0/
© 2016 Deborah Loewenberg Ball • School of Education • University of Michigan • Ann Arbor, MI 48109 • [email protected]
24
August 11, 2016
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International License: https://creativecommons.org/licenses/by-nc-nd/4.0/
© 2016 Deborah Loewenberg Ball • School of Education • University of Michigan • Ann Arbor, MI 48109 • [email protected]
25
August 11, 2016
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International License: https://creativecommons.org/licenses/by-nc-nd/4.0/
© 2016 Deborah Loewenberg Ball • School of Education • University of Michigan • Ann Arbor, MI 48109 • [email protected]
DAILY STUDENT-ENGAGED FEEDBACK
26
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© 2016 Deborah Loewenberg Ball • School of Education • University of Michigan • Ann Arbor, MI 48109 • [email protected]
HOW WE TALK ABOUT “ERROR” MATTERS
!  Making the environment “safe” is not all there is
!  “Errors” are a necessary part of mathematical work
!  Being “meta” about mathematical work is an
important mathematical competence
!  Dwelling on things that go wrong or make you stuck
!  Analyzing solutions or methods that do not work, are
not right
27
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© 2016 Deborah Loewenberg Ball • School of Education • University of Michigan • Ann Arbor, MI 48109 • [email protected]
②  OPEN-ENDED NOTEBOOKS
!  What are the affordances?
!  What are your questions/concerns?
!  New ideas?
28
August 11, 2016
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International License: https://creativecommons.org/licenses/by-nc-nd/4.0/
© 2016 Deborah Loewenberg Ball • School of Education • University of Michigan • Ann Arbor, MI 48109 • [email protected]
③  PRESENTATIONS AT THE BOARD
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© 2016 Deborah Loewenberg Ball • School of Education • University of Michigan • Ann Arbor, MI 48109 • [email protected]
VIDEO: FIRST VIEWING
SEEING/HEARING STUDENTS
30
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© 2016 Deborah Loewenberg Ball • School of Education • University of Michigan • Ann Arbor, MI 48109 • [email protected]
QUIET WRITING TIME (QWT)
!  Choose a blank row in the Google doc
!  Jot a few notes
!  What did you see/hear
with respect to
particular students?
0:01
0:02
0:03
0:04
0:05
0:06
0:07
0:08
0:09
0:10
0:11
0:12
0:13
0:14
0:15
0:16
0:17
0:18
0:19
0:20
0:21
0:22
0:23
0:24
0:25
0:26
0:27
0:28
0:29
0:30
0:31
0:32
0:33
0:34
0:35
0:36
0:37
0:38
0:39
0:40
0:41
0:42
0:43
0:44
0:45
0:46
0:47
0:48
0:49
0:50
0:51
0:52
0:53
0:54
0:55
0:56
0:57
0:58
0:59
2:00
1:00
1:01
1:02
1:03
1:04
1:05
1:06
1:07
1:08
1:09
1:10
1:11
1:12
1:13
1:14
1:15
1:16
1:17
1:18
1:19
1:20
1:21
1:22
1:23
1:24
1:25
1:26
1:27
1:28
1:29
1:30
1:31
1:32
1:33
1:34
1:35
1:36
1:37
1:38
1:39
1:40
1:41
1:42
1:43
1:44
1:45
1:46
1:47
1:48
1:49
1:50
1:51
1:52
1:53
1:54
1:55
1:56
1:57
1:58
1:59
End
31
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© 2016 Deborah Loewenberg Ball • School of Education • University of Michigan • Ann Arbor, MI 48109 • [email protected]
VIDEO: WATCHING THE WORK OF TEACHING TO
SEE AND BUILD STUDENTS’ STRENGTHS
32
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© 2016 Deborah Loewenberg Ball • School of Education • University of Michigan • Ann Arbor, MI 48109 • [email protected]
QUIET WRITING TIME (QWT)
!  Choose a blank row in the Google doc
!  Jot a few notes
!  What did you see/hear
with respect to
the work of teaching?
0:01
0:02
0:03
0:04
0:05
0:06
0:07
0:08
0:09
0:10
0:11
0:12
0:13
0:14
0:15
0:16
0:17
0:18
0:19
0:20
0:21
0:22
0:23
0:24
0:25
0:26
0:27
0:28
0:29
0:30
0:31
0:32
0:33
0:34
0:35
0:36
0:37
0:38
0:39
0:40
0:41
0:42
0:43
0:44
0:45
0:46
0:47
0:48
0:49
0:50
0:51
0:52
0:53
0:54
0:55
0:56
0:57
0:58
0:59
2:00
1:00
1:01
1:02
1:03
1:04
1:05
1:06
1:07
1:08
1:09
1:10
1:11
1:12
1:13
1:14
1:15
1:16
1:17
1:18
1:19
1:20
1:21
1:22
1:23
1:24
1:25
1:26
1:27
1:28
1:29
1:30
1:31
1:32
1:33
1:34
1:35
1:36
1:37
1:38
1:39
1:40
1:41
1:42
1:43
1:44
1:45
1:46
1:47
1:48
1:49
1:50
1:51
1:52
1:53
1:54
1:55
1:56
1:57
1:58
1:59
End
33
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© 2016 Deborah Loewenberg Ball • School of Education • University of Michigan • Ann Arbor, MI 48109 • [email protected]
PARTNER READING TIME
Read what people wrote; discuss with the person next
to you
!  What did people see, hear, and feel?
!  What differences do you notice?
!  Are there patterns of commonality?
34
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© 2016 Deborah Loewenberg Ball • School of Education • University of Michigan • Ann Arbor, MI 48109 • [email protected]
③  PRESENTATIONS AT THE BOARD
!  What are the affordances?
!  What are your questions/concerns?
!  New ideas?
35
August 11, 2016
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© 2016 Deborah Loewenberg Ball • School of Education • University of Michigan • Ann Arbor, MI 48109 • [email protected]
④  HOMEWORK
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© 2016 Deborah Loewenberg Ball • School of Education • University of Michigan • Ann Arbor, MI 48109 • [email protected]
HOMEWORK: THREE TYPES OF TASKS
1.  Review today’s work
2.  Looking ahead
3.  You Be the Teacher
37
!  You should know how to
do this, and if not, see
what you need to work on
!  Try things that you were
not yet taught
!  Teach someone else and
assess whether they
understand
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© 2016 Deborah Loewenberg Ball • School of Education • University of Michigan • Ann Arbor, MI 48109 • [email protected]
1. REVIEWING TODAY’S WORK
38
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© 2016 Deborah Loewenberg Ball • School of Education • University of Michigan • Ann Arbor, MI 48109 • [email protected]
2. LOOKING AHEAD
39
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© 2016 Deborah Loewenberg Ball • School of Education • University of Michigan • Ann Arbor, MI 48109 • [email protected]
3. YOU BE THE TEACHER
THE TASK
!  Pick someone at your house
to be your “student.”
!  Show this rectangle to the
person and ask him/her to tell
you what fraction is shaded.
!  Explain why some people
might answer ¼.
!  Explain the correct answer.
40
TO NOTE:
!  Because some students
speak Spanish at home, the
assignment can be done in
English or Spanish.
!  This assignment both bridges
the gap with the student’s
family AND foregrounds the
student’s competence.
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© 2016 Deborah Loewenberg Ball • School of Education • University of Michigan • Ann Arbor, MI 48109 • [email protected]
41
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© 2016 Deborah Loewenberg Ball • School of Education • University of Michigan • Ann Arbor, MI 48109 • [email protected]
JAIWAN
42
EFRAIN
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© 2016 Deborah Loewenberg Ball • School of Education • University of Michigan • Ann Arbor, MI 48109 • [email protected]
EMELINE
43
ANAYJAH
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© 2016 Deborah Loewenberg Ball • School of Education • University of Michigan • Ann Arbor, MI 48109 • [email protected]
④  HOMEWORK
!  What are the affordances?
!  What are your questions/concerns?
!  New ideas?
44
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© 2016 Deborah Loewenberg Ball • School of Education • University of Michigan • Ann Arbor, MI 48109 • [email protected]
WRAP UP
!  Structures that support students to engage in a
broader terrain of mathematical work
!  Leaving different things open, structuring others
!  Persistently opening up a wider space of what
counts as “mathematics” and “being smart” or
“good at math”
45
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© 2016 Deborah Loewenberg Ball • School of Education • University of Michigan • Ann Arbor, MI 48109 • [email protected]
THANK YOU!
[email protected]
Slides will be available on my website
(deborahloewenbergball.com)
N
A
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A
V
A
!
W
E
!
6
1
/
2
1
/
8
B LE
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© 2016 Deborah Loewenberg Ball • School of Education • University of Michigan • Ann Arbor, MI 48109 • [email protected]
CREDITS
Graphic on slides 11, 15, and 38:
Papy, F. (1970). Papy’s mincomputer. Mathematics Teaching, 50,
40–45.
47
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© 2016 Deborah Loewenberg Ball • School of Education • University of Michigan • Ann Arbor, MI 48109 • [email protected]