Shaping the River

Shaping the river:
Contexts of mathematical development and their
implications for assessment and standardsetting.
.
Christopher Correa
Kevin F. Miller
University of Michigan
Overview
Quality control inspectors?
Getting from standards and assessment to learning
Historical perspectives
Poor mathematical performance in the U.S. is a comparatively
recent phenomena
Sources of differences in mathematical performance between
children in China and the U.S.
Symbol structure
Educational practices
The case of fractions
Implications for assessment & standards setting
The uses of assessment
“Quality control
inspectors”?
How might assessment
information lead to
increased learning?
Interaction between
• Kinds of information
provided
• Models of learning
• Instructional practices
Overview
Quality control inspectors?
Getting from standards and assessment to learning
Historical perspectives
Poor mathematical performance in the U.S. is a comparatively
recent phenomena
Sources of differences in mathematical performance between
children in China and the U.S.
Symbol structure
Educational practices
The case of fractions
Implications for assessment & standards setting
Historical perspectives:
“A calculating people”
“When a country has a national sport as part of its culture—
that is, contests that almost all members find interesting to
participate in or watch—the team representing that country
tends to be strong, even if its population is small…I think that
school mathematics is a national (in the sense used above)
intellectual pursuit in Asian countries, but not in the United
States. Whereas students cannot escape from it in the former,
in the latter they are told (at least implicitly) that, if they are
poor at or dislike mathematics, they are free to seek
achievement in some other area.
Hatano, G. (1990). Toward the cultural psychology of mathematical
cognition. Comment on Stevenson, H. W., Lee, S-Y. (1990). Contexts of
achievement. Monographs of the Society for Research in Child Development,
55 (1-2, Serial No. 221), 108-115.
These things can change…
America in the 1830s
“We are a traveling and a calculating people,” said Ohio
booster James Hall in his book Statistics of the West, with
evident satisfaction. “Arithmetic I presume comes by instinct
among this guessing, reckoning, expecting and calculating
people,” said English traveler Thomas Hamilton, with
evident distaste. Each man attached a different value to the
idea, but both agreed that Americans in the 1830s had some
sort of innate reckoning skill that set them apart from
Europeans.
Cohen, P. C. (1982). A calculating people: The spread of numeracy
in early America. Chicago: U. of Chicago. (pp. 3-4)
Geary, D. C., Salthouse, T. A., Chen, G-P, & Fan, L. (1996).
Are East Asian Versus American Differences in
Arithmetical Ability a Recent Phenomenon?
Developmental Psychology, 32, 254-262
Younger (20 year old) and Older (late 65-75 year old)
adults
Matched for health, education
Given Arithmetic and basic cognitive tasks
Addition performance
Age & Addition Performance in the U.S. & China
50
# problems in 2 minute
Addition problems
like 19 + 8 + 27 = ?
# correctly solved in 2
minutes
Interaction between
age and culture
40
30
20
U.S.
China
10
0
Younger
Older
Subtraction performance
Age & Addition Performance in the U.S. & China
70
60
# problems in 2 minute
Subtraction problems
like 35-9 = ?
# correctly solved in 2
minutes
Interaction between
age and culture
50
40
30
20
U.S.
China
10
0
Younger
Older
Perceptual speed – No culture effect
60
Age & Perceptual Speed Performance in the U.S. &
China
50
# correct in 3 minutes
Perceptual speed task
Identifying whether number
strings are the same or not
• 179359821
• 178539821
# correctly solved in 3
minutes
ETS kit of factor-referenced
tests (Ekstrom et al., 1976)
40
30
20
U.S.
China
10
0
Younger
Older
Mental rotation – no culture effect
Rotation of figures in 3dimensional space
# solved in 3 minutes
# problems in 3 minutes
From ETS kit of factor-referenced
tests (Ekstrom et al., 1976)
20
Age & Mental Rotation Performance in the U.S. &
China
10
U.S.
China
0
Younger
Older
Conclusions
Differences are
Large
Specific
Probably recent in origin
Where might they come from?
Differences in what children learn
Differences in educational environment
Differences in teaching processes
Overview
Quality control inspectors?
Getting from standards and assessment to learning
Historical perspectives
Poor mathematical performance in the U.S. is a comparatively
recent phenomena
Sources of differences in mathematical performance between
children in China and the U.S.
Symbol structure
Educational practices
The case of fractions
Implications for assessment & standards setting
Language and Learning to Count
Children need to learn a system of number
names as they learn to count
Not a trivial task
Number names in Chinese & English - Part I
Counting to Ten
Numeral
Chinese
(written)
Chinese
(spoken)
English

1 2
一 二
3
三
4
四
5
五
6 7
六 七
8
八
9
九
10
十
yi
san
si
wu liu qi
ba
jiu
shi
er
one two three four five six seven eight nine ten
Both languages share an unpredictable list
– No way to induce “five” from “one, two, three, four”

Linguistically, learning to count to ten should be
of equal difficulty in both languages
Number names in Chinese & English - Part II
From Ten to Twenty
Numeral
11
12
Chinese 十一 十二
(written)
13
14
15
16
十三
十四
十五 十六
17
18
19
20
十七
十八
十九
二十
shi wu shi liu shi qi
shi ba
shi jiu
er shi
Chinese shi yi shi er shi san shi si
(written)
English eleven twelve thirteen fourteen fifteen sixteen seventeen eighteen nineteen twenty

Chinese has a clear base-ten structure
– similar to Arabic numerals: 11 = “10…1”

English lacks clear evidence of base-ten structure
– Names for 11 and 12 not marked as compounds with 10.
– Larger teens names follow German system of unit+digits name,
unlike larger two-digit number names
 compare “fourteen” and “twenty-four”
Number names in Chinese & English - Part III
Above Twenty
Language
Rule
Example
三十七
Decade unit (two,three,four,five,six,seven,eight,nine) + ten
(shi) + unit
san shi qi
Decade names (twen,thir,for,fif,six,seven,eight,nine) + ty + unit
thirty-seven
Chinese
(written)
Chinese
(written)
English

Both languages share a similar structure
– similar to Arabic numerals: 37 = “3x10 + 7”


For Chinese, this extends previous system
For English, it represents a new way of naming
numbers
A longitudinal view
110
110
2-year-olds
100
100
90
US


China

US
80
80
70
70
70
60
60
50
50
40
40

30

60



40

0




1 2




















10
5 6 7 8 9 10 11 12
Month





China

US















30














20

0
3 4



20
10


50

20

90
80
30

4-year-olds
100
China

90
Median Abstract Counting
110
3-year-olds
10
0
1 2 3 4
5 6 7 8 9 10 11 12
Month
1 2
3 4 5 6 7 8 9 10 11 12
Month
Learning difficulties reflect language
structure
..and they don’t stop here!
The Panda’s snack
No language difference
for counting-principle
errors such as doublecounting
• Mastering number list
and understanding
numerosity not the same
•
•
Producing sets of n items
No language difference
% Correct
•
100
US
China
80
60
40
20
0
3
4
5
Age
Producing Sets of 12
100
% Correct
• Language affects only
some aspects of early
number knowledge
Producing Sets of 4
80
60
US
China
40
20
0
3
4
Age
5
Teens numbers
Continuing effects
% Correct
Learning Arabic
numerals involves a
mapping from verbal
number names
100
80
US
China
60
40
20
0
K
1
2
Grade
Larger 2-digit
numbers
Teens continue to
cause problems
% Correct
100
80
US
China
60
40
20
0
K
1
Grade
*
2
Summary
Early mathematical development is a mix of
language-dependent and universal factors
Sensitivity to symbol structure begins very
early
Base-ten concepts and “teens” are
problematic for speakers of English
Foundation for later mathematics
Other sources
Overview
Quality control inspectors?
Getting from standards and assessment to learning
Historical perspectives
Poor mathematical performance in the U.S. is a comparatively
recent phenomena
Sources of differences in mathematical performance between
children in China and the U.S.
Symbol structure
The case of fractions
Educational practices
Implications for assessment & standards setting
Interaction between symbol structure and
classroom processes – Rational numbers
English terms
somewhat opaque
Numerator
Denominator
Chinese terms more
transparent
分子 (“fraction child”)
分母 (“fraction mother”)
*****
Transparent fraction terms – does it make a difference?
Not entirely clear
Miura et al. (1999)
vs.
1/3 =
?
Paik & Mix (2003)
Children in both countries
likely to pick “numerator + denominator”
foil
Language support can only take you so far…
return
Student thinking – components of knowledge
In addition to content domains, the most recent
TIMSS student assessment also considers different
types of thinking:
Knowing facts, procedures, and concepts
• What students need to know
Applying knowledge and conceptual understanding
• Applying what the student knows to solve routine problems or
answer questions
Reasoning
• Solving for unfamiliar situations and multi-step problems
Thinking about Fractions
Knowing
Applying
Reasoning
Thinking about Fractions
Knowing
Applying
Reasoning
Thinking about Fractions
Knowing
Applying
Reasoning
Fourth-grade students in
many high-achieving
countries are proficient
in problem solving.
Fourth-grade student achievement in Japan, TIMSS 2003
Fourth-grade student achievement in Netherlands, TIMSS 2003
U.S. fourth-grade
students are more
proficient at knowing
procedures and
knowledge than
applying knowledge to
solve problems.
Fourth-grade student achievement, TIMSS 2003
…though this changes
over time.
Eighth-grade student achievement, TIMSS 2003
Teacher Knowledge
Teachers may be more comfortable
facilitating complex problem solving if they
have a strong understanding of
mathematical concepts
Teacher Knowledge in the U.S.
Zhou et al. (2006)
compare pedagogical
content knowledge of
U.S. and Chinese
teachers
Why?
K-12 experiences
Teacher Training
Few content specialists
Changes in workforce
Few opportunities for learning
Cultural Beliefs and Scripts
Beliefs about learning
Can you get smarter?
• Americans more likely to believe that intelligence is
fixed and that success is due to ability rather than
effort
• Can be maladaptive
• Teaching & Assessment practices can reinforce this
idea
Self-Theories of Intelligence
Dweck, 1999
Entity Theorists
Intelligence is fixed
Trait largely determined by nature
Incremental Theorists
Intelligence is malleable
Quality that can be increased through
nurture
Desire similar outcome
• achieving good scores, doing “well”
Different motivation for pursuing this outcome
• Performance goals
seeking
to validate
abilityitasis
“When –I take
a course
in school,
good relative
to others
very important
for me to
validate that I
am smarter than other students.”
• Learning goals
– seeking
to develop
ability
“In school
I am always
seeking
opportunities to develop new skills
and acquire new knowledge.”
Beliefs about Teaching
We used multiple correspondence analysis to represent individual
teachers and their responses to the interview question, “How do
students best learn mathematics?”
Beliefs about Teaching
…and find within-group similarities
Practice
“I think that drill is very important… I think that
there is just no substitute for going over it and
over it and over it until it’s very firmly entrenched
in their minds.” (Teacher U4-2)
With some students, there can be something said
for a lot of rote practice at the very beginning of a
concept… there are some students who need 40
problems of it just to get the pattern of what
they're doing. (Teacher U4-3)
Remembering and Understanding
US Students
Chinese Teachers
5
US Teachers
Mean frequency
Attributions for
student errors by
Chinese & American
teachers and college
students
Remembering
vs.
Understanding
Chinese Students
6
4
3
2
1
0
learn
understand
remember
forget
Scripts for teaching
Customary teaching practices that are
informed by a culture’s underlying values
and assumptions of what teaching should be
like.
National Scripts
National Scripts
National Scripts
Identifying a Script for Chinese Lesson
Introductions
Data
Participants include 19 fourth- and fifth- grade teachers from six
schools in Beijing, China
Videotaped Classroom Observations
Lesson introductions are defined as “what happens between the
beginning of class and the final solution of the first novel
problem.”
Interviews
A Proposed Script for Lesson
Introductions in China
I. Teacher Reviews Old Knowledge
(whole-
class)
II. Students Develop Methods and Solutions
(small groups)
III.Students Share Results (whole-class)
I. Teacher Reviews Old Knowledge
(whole-class)
II. Students Develop Methods and
Solutions (small groups)
III.Students Share Results (whole-class)
I. Teacher reviews old knowledge
Video example
I. Reviewing Old Knowledge
Review Definitions of Key Concepts
• Students, not the teacher, provided definitions
• More than one student shares definition
Quick Computation
• Teacher uses visual aid to present between five and
ten review problems to whole class
• Teacher elicits whole-class or individual student
responses
• The types of problems chosen are directly related to
new content taught later in the lesson.
I. Reviewing Old Knowledge
“First, I begin to study the textbook and
prepare the lesson carefully. Then, I try to
find the connector between the original
knowledge and the new knowledge so as to
lead them from the original one to the new
one. After that, students can get a network
of knowledge.”
I. Teacher Reviews Old Knowledge
(whole-class)
II. Students Develop Methods and
Solutions (small groups)
III.Students Share Results (whole-class)
II. Students Develop Methods and
Solutions
Students ask questions about novel problem
II. Students Develop Methods and
Solutions
Group Discussion
I. Teacher Reviews Old Knowledge
(whole-class)
II. Students Develop Methods and
Solutions (small groups)
III.Students Share Results (whole-class)
III. Students Share Their Ideas
“The teacher is just a guide in the class and should leave the
initiative to the students. Teacher should give more
opportunities for the students to talk. It doesn’t matter if
they say something wrong if only they can express
themselves. I can just correct them in time or let other
students correct them. Anyway, we should get the students
actively participate in the learning activities in class.”
III. Students Share Their ideas
After group discussions, the teacher calls on
students to explain their methods
III. Students Share Their Ideas
Individual students demonstrate their solution
in front of class
• Teacher asks for volunteers
• Students use visual aid, such as overhead projector,
to demonstrate their (or their group’s) solution.
• Usually, more than one student shows work
• Teacher stands to the side and probes students to
provide more complete explanations or more precise
definitions.
Summary
Teacher Reviews Old Knowledge (wholeclass)
Students Develop Methods and Solutions
(small groups)
Students Share Results (whole-class)
Scripts and Beliefs are hard to change
Cohen & Hill (2001) found nearly all teachers
endorsed reform beliefs generally, but they
agreed with specific ideas that were
consistent with their existing models of
mathematics instruction.
• 89.6% agree that mathematics should “study
mathematics in the context of everyday situations”
• However, 52.1% also agreed “teachers should make
sure that students are not confused a the end of a
mathematics period.”
Overview
Quality control inspectors?
Getting from standards and assessment to learning
Historical perspectives
Poor mathematical performance in the U.S. is a comparatively
recent phenomena
Sources of differences in mathematical performance between
children in China and the U.S.
Symbol structure
The case of fractions
Educational practices
Implications for assessment & standards setting
Implications
What are the qualities of mathematics
education in the U.S?
What is the role of standards and
assessment in influencing student thinking?
What is the role of standards and
assessment in influencing teaching?