1. Unpacking the relationship between
classroom practice and student learning in
mathematics: Examining the power of
student explanations
Megan Franke,
Mathematics Classroom Practice
Study Group
UCLA

2. Session
Overview
• Understanding
the
rela.onship
between
classroom
prac.ce
and
student
outcomes
• Prior
research
on
students’
explana.ons
and
teachers’
support
of
those
explana.ons
• Engaging
students
in
each
other’s
mathema.cal
ideas
• Findings
related
to
student
par.cipa.on,
teaching
and
student
learning

3. Results
of
a
large-­‐scale
intervention
study
• Recruited
volunteer
teachers
at
19
schools
in
low-­‐performing,
urban
school
district
• On-­‐site
professional
development
focused
on
algebraic
reasoning
• Thinking
Mathema-cally:
Integra-ng
Arithme-c
and
Algebra
in
Elementary
School
• Equal
sign,
Rela.onal
thinking
• Orchestrate
conversa.ons
Jacobs, V., Franke, M.., Carpenter, T., Levi, L. & Battey, D.
(2007). Exploring the impact of large scale professional
development focused on children’s algebraic reasoning. Journal
for Research in Mathematics Education 38 (3), pp. 258-288.

4. The schools and district

5. Overview of Classrooms:
Mean Achievement
80
70
60
50
40
30
20
10
0
Ach
Low (2 classes)
Medium (1
class)
High (3 classes)

6. Collecting
data
around
interactions

7. Student communication
• Explaining
to
other
students
is
posi.vely
related
to
achievement
outcomes,
even
when
controlling
for
prior
achievement
(Brown
&
Palincsar,
1989;
Fuchs,
Fuchs,
HamleG,
Phillips,
Karns,
&
Dutka,
1997;
King,
1992;
NaNv,
1994;
Peterson,
Janicki,
&
Swing,
1981;
Saxe,
Gearhart,
Note,
&
Paduano,
1993;
Slavin,
1987;
Webb,
1991;
Yackel,
Cobb,
Wood,
Wheatley,
&
Merkel,
1990).
• When
describing
their
thinking,
students
must
be
precise
and
explicit
in
their
talk,
especially
providing
enough
detail
and
making
referents
clear
so
that
the
teacher
and
fellow
classmates
can
understand
their
ideas
(Nathan
&
Knuth,
2003;
Sfard
&
Kieran,
2001).

8. Potential Benefits of
Explaining Your Own Thinking
• Transform
what
you
know
into
an
explana.on
that
is
relevant,
coherent,
complete,
and
understandable
to
others
• Bring
concepts/details
together
in
ways
that
you
hadn’t
thought
of
previously
• Recognize
misconcep.ons,
contradic.ons,
incompleteness
in
your
idea
• Develop
a
sense
of
yourself
as
someone
who
can
do
mathema.cs
and
communicate
mathema.cally
8

9. Coding Student Participation
• Accuracy
of
answer
given
• Correct
• Incorrect
• No
answer
• Nature
of
explana.on
given
• Correct
and
complete
• Ambiguous
or
incomplete
• Incorrect
• Further
elabora.on
aPer
teacher’s
ques.ons

10. Types of Student Explaining
• Gives
correct/complete
explana.ons
Five?
‘Cause
10
plus
10
equals
20,
huh?
And
then
it
says
minus
10
equals
5
plus
blank.
So
it
goa
be
10,
so
5
plus
5
equals
10.
And
that’s
how
I
got
it.
• Gives
incorrect
or
incomplete
explana.ons
50
+
50
=
50
+
□
+
25
[50].
It’s
just
like
50
plus
50.
They
are
kind
of
partners
because
they
are
the
same.
8
+
2
=
7
+
3
(True
or
False?)
[True]
because
there’s
a
2
and
a
3
and
a
7
and
an
8.
They’re
like
an
order.
10
10
+
10
–
10
=
5
+
□

11. Table 3. Correlations between Student Participation and Achievement Scores
Highest level of student participation on a
a
problem
Gives explanation
Achievement
b
Score
.69*
Correct and complete
.73*
Ambiguous, incomplete, or incorrect
-.01
Gives no explanation
a
-.69*
Percent of problems in which a student displayed this behavior. Problems discussed
during pairshare and whole-class interaction are included.
b
Percent of problems correct.
Note: Number of students = 35.
*p <.05

12. Profiles of Students’ Contributions
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
Correct
explanations
Incorrect
explanation
Correct answer
only
Incorrect answer
only
Low
Classes
Medium
Class
High
Classes

13. Moving
toward
understanding
the
details
of
practice
in
relation
to
student
outcomes
• Teachers’
support
explaining
(Lampert,
2001),
Revoicing
(Forman
et
al.,
1998;
O’Connor
&
Michaels,
1993,
1996;
Strom
et
al.,
2001)
Press
(Kazemi
&
S-pek,
)
Teachers’
ques.oning
(Wood,
1998)
Filtering
approach
(Sherin,
2002)
• Teachers’
prac.ce
supports
students’
produc.ve
explana.ons
(Gillies,
2004;
Rosja-­‐Drummond
and
Mercer,
2003)
• And
while
evidence
shows
these
prac.ces
are
not
likely
in
many
classrooms,
they
are
even
less
likely
in
classrooms
of
low-­‐income
students
of
color
(Anyon,
1981,
Ladson-­‐Billings,
1997;
Lubienski,
2002;
Means
&
Knapp,
1991).

14. Teachers’ Supporting of Students
to Share their Thinking
• 98%
of
segments:
Teachers
asked
the
target
students
to
explain
their
thinking
• 91%
of
segments:
Teachers
requested
an
explana.on
at
the
outset
of
the
segment,
or
aPer
an
answer
was
given
• 76%
of
segments:
Teachers
asked
the
student
to
elaborate
further
on
their
explana.on
• Frequent
reminders
about
listening
to
explana.ons:
• “Give
[name]
a
chance
[to
explain]”
• “I
like
the
way
[Student]
is
paying
close
aen.on
to
what
[Students]
are
about
to
share”
• “Let’s
understand
[Student’s]
thinking.”

15. Whether teachers elicited student thinking beyond
initial explanations and how the engagement ended
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
Yes: correct
explanation
Yes: incorrect
explanation
No: correct
explanation
No: incorrect
explanation
No: correct answer
Low
Medium
High
No: incorrect
answer

16. Example: General Question
Problem:
375
=
__
+
(3
x
10)
• Student:
345
• Teacher:
I’m
just
a
lile
unsure
of
how
you
came
up
with
345.
Can
you
show
me
what
you
did?

17. Example:
Speci?ic
Question
• Problem:
100
+
__
=
100
+
50
Student:
The
50
will
go
right
there
because
it
has
to
be
the
same
number.
Teacher:
What
has
to
be
the
same
number?

18. Probing Sequence
• Used
when
teacher
was
unclear
about
a
student’s
explana.on
• Used
to
highlight,
clarify
or
make
explicit
por.ons
of
a
student’s
strategy
• Used
when
teacher
is
trying
to
help
a
student
understand
a
problem

19. Engaging
with
each
others’
ideas
• New
study:
• K-­‐5
teachers
• Mul.-­‐age
school
• School
describes
itself
as
a
learning
environment
that
values
diversity,
encourages
crea.vity
and
innova.on,
supports
disciplined
inquiry,
involves
families
and
their
communi.es,
and
makes
a
commitment
to
mee.ng
the
needs
of
the
whole
child.
• All
teachers
par.cipated
(12
who
taught
mathema.cs)
• 25-­‐35
students
per
classroom
•
Classroom
observa.ons
•
•
•
•
Spent
the
year
in
classrooms
approximately
once
a
week
Video
and
audiotaped
2-­‐3
days
in
March,
April
each
class
Collected
student
work
Researcher
designed
assessment
and
standardized
test

20. Collecting
observation
data
• One
sta.onary
video
camera
with
two
flat
microphones
captured
the
ongoing
flow
of
the
class
and
the
interac.on
of
up
to
two
groups
of
students.
• Four
Flip
video
cameras
captured
the
interac.on
of
the
remaining
students.
• one
Flip
video
camera
was
sta.onary
and
the
other
three
were
operated
by
research
team
members.
• Distributed
six
digital
audio
recorders
to
pick
up
sound
not
captured
by
Flip
• Created
a
single
movie
for
each
classroom
observa.on
by
combining
all
of
the
video
and
sound
sources
• Movie
analyzed
using
Studiocode
so
that
we
could
code
the
details
within
the
context

21. The above figure illustrates the way the codes are applied to a video timeline. Instances of codes are represented by rows. In addition, codes
have labels as can be seen in the table in the upper right hand corner.

25. Relationship between Student Participation
and Achievement
Par%al
correla%on
with
achievement
Provided
fully-­‐detailed
explana.ons
of
how
to
solve
the
problem
.30*
Highest
level
at
which
you
engaged
with
other
students’
ideas
.44*
Highest
level
at
which
other
students
engaged
with
your
ideas
.41*
25

26. 26
• Explain
your
thinking
• Engage
with
others’
ideas
to
a
high
degree
• Have
others
engage
with
your
idea
to
a
high
degree

27. Teacher Support of Students’ Engagement
with Each Other’s Ideas
27

30. Why the invitation was not enough
• Student
had
no
readily
available
response
or
a
response
that
provided
any
detail,
and
so
the
teacher
had
to
find
ways
to
work
with
the
student
to
elaborate
and
extend
their
engagement
with
the
other
student’s
idea
• Student
did
not
discuss
the
mathema.cal
idea
in
what
had
been
shared
or
did
not
address
the
par.cular
mathema.cal
idea
that
the
teacher
wanted
to
address
• Students
did
not
know
how
to
take
up
the
teacher’s
invita.on

32. Teacher
support
for
engaging
in
other’s
ideas
• Student
did
not
have
much
of
a
detailed
response

33. Jack ate 6 peanut butter sandwiches.
He ate 1/6 of a sandwich and decided he didn't want more.
How much does Jack have left?
Ms.
A:
Okay.
Who
can
explain
what
Yadira
did?
Who
can
explain,
Cole,
who
can
explain
what
Yadira
did
here?
Cole,
can
you
come
and
explain?
invita.on
Cole:
She
took
these
things
(poin.ng
to
6
rectangles)
and
then
she
did
this
(mo.oning
over
the
lines
dividing
one
of
the
rectangle
into
sixths)
so
that
she
can
throw
away
this
(poin.ng
to
the
shaded
part).
Ms.
A:
But
what
did
she…
what
are
these?
(poin.ng
to
the
5
wholes
in
Yadira’s
picture)
probe
Cole:
Wholes.
Sandwiches.
Ms.
A:
Those
are
the
sandwiches.
Ok.
Those
are
the
whole
sandwiches.
Yes.
And?
probe
Cole:
And
then
she
did
that
(mo.oning
to
the
lines
dividing
one
whole
into
sixths
again)
so
you
can
see
that
she
colored
in
one,
and
that's
the
one
he
ate.
Ms.
A:
So
how
many
does
he
have
leP?
Cole:
Umm.
5
wholes
and
5
sixths
sandwiches.
Ms.
A:
Do
you
agree
with
Yadira
and
Cole?
(to
class)

34. Teacher
support
for
engaging
in
other’s
ideas
• Student
needed
support
to
get
to
the
mathema.cal
ideas

35. The students were in the middle of a conversation about 0/3. The question
arose as the students were counting backwards by 1/3 from 4. Ben stated that
he thought 0/3 was a whole and should follow 1/3 in counting backwards.
Ms. J:
Sara has her hand raised. Sara do you want to add something? Invitation
Sara:
I don’t agree with, I wanted to actually kind of come up and, Ben said that
0/3 is a whole. Ben it is not. It is not a whole.
Ms. J:
Can you give him any evidence of that? Probe
Sara:
It is not a whole because none of the, like like, for 3/3 [she is drawing a
picture and all of it is shaded in and walks over to Ben’s picture and shows
that none of it is shaded in] None of it is shaded in, in this one.
Ms. J:
Well Sara what is this, 3 parts of what [pointing to her picture] Lets make
it a real thing, it is easier to talk about scaffold
Sara:
3 part of, pieces of chocolate, a brownie [students in the class are also calling
out different things it could be]
Ms. J:
So this is chocolate, a, brownie, a tray of brownies, alright brownies. So
you are saying that these 3 pieces, this is a whole, a whole brownie that
has been what probe
cut
into?
3 Pieces
Ben do you have something to say? invitation
yeah
come on up.
Sara:
Ms. J:
Sara:
Ms. J:
Ben:
Ms. J:

36. Teacher
support
for
engaging
in
other’s
ideas
• Student
did
not
know
how
to
take
up
the
teacher’s
invita.on

37. If Seily has five-thirds liters of soda, what would that look like?
Draw and label all parts. Two students have written their solutions on the board
Ms J:
Carlos. come on up and explain Daniel’s because you said that yours was
more like Daniel’s (she had asked earlier for students to point to the strategy
on the board that was like theirs). You said yours was a little more like
Daniels. (As Carlos is walking to the board with his paper) Can you explain
that one? invitation
Carlos:
(walking slowly and pauses) No.
Ms J:
You can’t explain this picture (pointing to Daniel’s picture)? Yours is very
much like it. positioning
Carlos:
(looks at the drawing for about 6 sec) I understand that (points to one part of Daniel’s
picture) but not the lines.
Ms J:
Oh, Can you ignore the lines and explain the picture? scaffolding
Carlos:
Yes
Ms J:
Okay
Carlos:
What Daniel did, right here (points to his picture) is 5/3 which is one liter
(pointing to Daniel’s picture) and 2/3 of a liter (pointing to the picture).

38. Supporting teachers to engage students in
mathematics
• A
number
of
researchers
have
not
only
engaged
in
significant
research
in
this
area
but
they
have
also
begun
to
support
teachers
• Mercer
and
colleagues:
rules
for
par.cipa.on
• Gillies
and
colleagues:
communica.on
skills
• O’Connor
and
Michaels:
Talk
moves
• Consistent
with
• Seymour
&
Lehrer
• Hueris.c
moves
vs.
specialized
version
of
the
move
• Kazemi
&
S.pek
• Norms
for
press
• Cengiz,
Kline
&
Grant
(2011)
• Combina.on
of
instruc.onal
ac.ons
• Lampert

39. Teacher
Practice
0.198
(0.082)
Student
Achievement

40. 0.323
(0.104)
Teacher
Practice
Student
Participation
0.297
(0.092)
Student
Achievement