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OTICING
UMERACY
OW!
RESEARCH FUNDED BY THE NATIONAL SCIENCE FOUNDATION:
Transforming Undergraduate Education in STEM (TUES) Award # 1043667, 1043656, 1043831
About Us
Preservice Teacher Preparation Collaborative
Jonathan Thomas
Northern Kentucky University [email protected]
KY Center for Mathematics
Edna O. Schack
Morehead State University
Sara Eisenhardt
Northern Kentucky University [email protected]
Molly H. Fisher
University of Kentucky
[email protected]
Margaret Yoder
Eastern Kentucky University
[email protected]
Janet Tassell
Western Kentucky University
[email protected]
Cindy Jong*
University of Kentucky
[email protected]
Todd Brown*
University of Louisville
[email protected]
Greg Gierhart*
Murray State University
[email protected]
* Comparison Implementers
[email protected]
Instructional Module
Professional Noticing
Attending to the children’s work
Interpreting children’s work in
context of mathematics
Deciding appropriate next steps
Jacobs, V. A., Lamb, L. L. C., & Philipp, R. A. (2010). Professional Noticing of Children’s
Mathematical Thinking. Journal for Research in Mathematics Education, 41, 169-202.
Pedagogies of Practice
Decomposition
of professional noticing
Representations
video of early number sense diagnostic
events
Approximations
PSETs conduct diagnostic interview with child
Grossman, P. (2011). Framework for teaching practice: A brief history of an idea. Teachers
College Record. 113, 12, 2836-2843.
Early Numeracy
Stages of Early Arithmetic Learning
•
•
•
Learning Progression
Early Quantitative Understanding
Examination of Counting Schemes
Olive, J. (2001). Children's number sequences: An explanation of Steffe's constructs and an extrapolation to
rational numbers of arithmetic. The Mathematics Educator, 11, 4-9.
Steffe, L. (1992). Learning stages in the construction of the number sequence. In J. Bideaud, C. Meljac, & J.
Fischer (Eds.), Pathways to number: Children’s developing numerical abilities (pp. 83–88). Hillsdale:
Lawrence Erlbaum.
Wright, R. J., Martland, J., & Stafford, A. (2000). Early numeracy: Assessment for teaching and intervention.
London: Paul Chapman Publications/Sage.
Early Numeracy
Stages of Early Arithmetic Learning
Stage 0:
Stage 1:
Stage 2:
Stage 3:
Stage 4:
Stage 5:
Emergent Counting Scheme
Perceptual Counting Scheme
Figurative Counting Scheme
Initial Number Sequence
Intermediate Number Sequence
Facile Number Sequence
PRIMARY RESEARCH QUESTION
To what extent can teacher
educators facilitate the development
of Preservice Elementary Teacher
(PSET) professional noticing
(attending, interpreting, and
deciding) of children’s mathematics?
Professional Noticing Assessment
“I have seven little bears . . . But now I have too
many shells. I have eleven shells. (Jon shows the
eleven shells then covers them with his hand.) How
many shells am I going to have left over?”
Professional Noticing Prompts
1. Please describe in detail what this child did in
response to this problem. (Attending)
2. Please explain what you learned about this
child’s understanding of mathematics.
(Interpreting)
3. Pretend that you are the teacher of this child.
What problem or problems might you pose
next? Provide a rationale for your choice.
(Deciding)
Jacobs, V. A., Lamb, L. L. C., & Philipp, R. A. (2010)
Assessment Score Levels
Level
Attending
Interpreting
Deciding
4
Elaborate
3
Salient
Accurate
2
Limited
Limited
Adequate,
Disconnected
1
Inaccurate
Inaccurate
Appropriate
& Connected
Inappropriate,
No Rationale
Growth of PSET PN: Attending
“He knew that since the teacher
said he had too many shells he
had to do subtraction. He also
knew that because the teacher
said left over he had to do
subtraction, or see what the
difference was. The child
understood key words and
phrases and understood how to
take away to get the right
answer. He used the bigger
number and took away using
the smaller number and
realized that 11-7=4.”
“In response to this
problem this child first
counted the bears and
found that there were
seven. From there he
used his fingers and
counted up from seven
until he got to the
number eleven. He had
four fingers up so he
said that that was his
answer.”
Growth of PSET PN: Interpreting
“It seemed that
instead of subtracting
seven from eleven he
used the problem
7+?=11, and came
up with four by
counting from seven
to eleven instead of
from eleven to seven.”
“This child does not count
on; he needed to count
the bears from one in
order to count the
remainder of the shells.
He uses his fingers to
count when materials are
unavailable to him. He
understands associating
one object with a number
and adding a value with
each corresponding
object added.”
Growth of PSET PN: Deciding
“I would pose more
bears than shells. Or
only have shells
exposed, so he couldn't
count the bears. How
many shells must I take
away to get 7
bears? Other ways of
getting answer and
using subtraction.”
“I would screen both of
the counters. This
requires the student to
use a different type of
counters (fingers) but he
might run into trouble
because he will be
counting past 10. I[t]
would be interesting to
see how he got the
answer.”
Preliminary Analysis of
Three Research Sites
Attending
University 0
University 1
University 2
All Participants
Interpreting
N
M
SD
M
SD
M
SD
Pre-Test
37
2.14
.79
1.59
.797
1.54
.61
Post-Test
37
2.43
.87
2.05
.84
2.22
.79
Pre-Test
23
2.39
.99
1.82
.89
2.04
.56
Post-Test
23
3.09
1.04
2.43
.73
2.70
.56
Pre-Test
34
2.38
1.10
1.76
.78
1.97
.67
Post-Test
34
3.00
1.10
2.15
.89
2.47
.75
Pre-Test
94
.96
1.82
2.43
.66
94
1.71
2.18
.81
Post-Test
2.29
2.80
1.03
Descriptive statistics of professional noticing measures by university
Scale
1-4
N
F (3,91)
P
94
16.800
< .001
Partial Eta
Squared
.156
Interpreting
1-3
94
15.617
< .001
.146
Deciding
1-3
94
40.130
< .001
.306
Attending
Deciding
Results of ANOVA comparing pre and post assessments of all universities
.84
.74
Professional Noticing Measure
Descriptive Statistics – All Sites
Attending Interpreting
Deciding
M
1-4
SD
M
1-3
SD
M
1-3
SD
Pre94
assessment
2.29
.96
1.71
.81
1.82
.66
Post94
assessment
2.80
1.03
2.18
.84
2.43
.74
N
Scale
ANOVA comparing pre and post
All
Universities
Attending
Interpreting
Deciding
Scale N
1-4 94
1-3 94
1-3 94
Partial
Eta
F(3,91)
P
Squared
16.800 <.001
.156
15.617 <.001
.146
40.130 <.001
.306
Questions?
tinyurl.com/noticingnumeracynow
Attending Benchmarks
“He counted from one up
when counting all of the
bears. He then counted
the remaining shells on his
fingers to get the answer
4.”
“Counted the bears
individually then used his
fingers to count up to 11.”
“Instead of subtracting
11-7, he counted to seven
and then used his fingers
to see how many more it
took to get to 11.”
“The child subtracted in
response to this question
using his fingers as a
manipulative. Starting with
11 & working backwards.”
Attending Benchmarks
“He counted from one up
when counting all of the
bears. He then counted
the remaining shells on his
fingers to get the answer
4.”
“Counted the bears
individually then used his
fingers to count up to 11.”
“Instead of subtracting
11-7, he counted to seven
and then used his fingers
to see how many more it
took to get to 11.”
“The child subtracted in
response to this question
using his fingers as a
manipulative. Starting with
11 & working backwards.”
Interpreting Benchmarks
“This child understands a one-to-one
correspondence with objects, he needs to
touch the objects and he still uses his
fingers to count on.”
“I learned that this child can add
easier than subtract because
instead of 7-11 he did
7+__=11. I also learned that he
needs a representation of the
numbers to solve the problem (the
bears, his fingers, and shells).”
“I learned that the child
is able to count on from a
given number. He didn't
have to go back and
start at 1.”
Interpreting Benchmarks
“This child understands a one-to-one
correspondence with objects, he needs to
touch the objects and he still uses his
fingers to count on.”
“I learned that this child can add
easier than subtract because
instead of 7-11 he did
7+__=11. I also learned that he
needs a representation of the
numbers to solve the problem (the
bears, his fingers, and shells).”
“I learned that the child
is able to count on from a
given number. He didn't
have to go back and
start at 1.”
Deciding Benchmarks
Appropriate Decision with…
“I believe that the next task should be a really small
number subtracted by a very large number. Ex. 20-6. This
problem would be harder to count on your hands and you
could get a better understanding of his conceptual
knowledge of the problem and addition itself.”
Adequate Decision with…
“I might say "How did
you get this answer" to
see how they explained
their logic.”
Inappropriate Decision with…
“I would ask the child to tell me why
there were four shells leftover. This
would tell us whether or not the child
had an understanding of remainders.
This will tell us if he has the concept
of sharing equally, rather than giving
the four shells to select bears.”
Deciding Benchmarks
“I believe that the next task should be a really small
number subtracted by a very large number. Ex. 20-6. This
problem would be harder to count on your hands and you
could get a better understanding of his conceptual
knowledge of the problem and addition itself.”
“I might say "How did
you get this answer" to
see how they explained
their logic.”
“I would ask the child to tell me why
there were four shells leftover. This
would tell us whether or not the child
had an understanding of remainders.
This will tell us if he has the concept
of sharing equally, rather than giving
the four shells to select bears.”
Deciding Benchmarks
“I believe that the next task should be a really small
number subtracted by a very large number. Ex. 20-6. This
problem would be harder to count on your hands and you
could get a better understanding of his conceptual
knowledge of the problem and addition itself.”
“I might say "How did
you get this answer" to
see how they explained
their logic.”
“I would ask the child to tell me why
there were four shells leftover. This
would tell us whether or not the child
had an understanding of remainders.
This will tell us if he has the concept
of sharing equally, rather than giving
the four shells to select bears.”
PSET Professional Noticing in Clinical Context
Instructional Module