Transcript Slide 1

Reading Interactions That
Unnecessarily Hinder Algebra
Learning and Assessment
Carl Lager, PhD
University of California, Santa Barbara
[email protected]
(805) 893-7770
Carl Lager - May 16, 2008 [email protected]
Overview
1) ELs – The numbers
2) ELs – Engaging math items
3) ELs - Uncovering EL engagement
4) Treisman challenge
5) Adding it up (2001) application
6) What you can do to help ELs
Carl Lager - May 16, 2008 [email protected]
English Learners – the numbers
 In the U.S., there are over 5.5 million
English Learners (USDOE, 2004) out
of 48.5 million public school students
(NCES, 2006) – 11.3%
 In California, there are just under 1.6
million ELs (CDE, 2006) out of 6.3
million public school students – 25.1%
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Who are California’s English
learners?
 In 2006 - 2007: 35% K - 2, 27% 3–5,
19% 6 - 8, 19% 9-12
 Over 85% speak Spanish as their
primary language
 Many ELs are U.S.-born children of
immigrants, not immigrants themselves
(Tafoya, 2002)
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Mathematics Achievement of
California’s Middle School
Students
2006-2007 EO/FEP CST mathematics results:
 6th – 21% Below or Far Below basic
 7th – 25% Below or Far Below basic
2006-2007 EL CST mathematics results:
 6th – 54% Below or Far Below basic
 7th – 59% Below or Far Below basic
Carl Lager - May 16, 2008 [email protected]
Mathematics Achievement of
California’s Secondary
Students
2006 - 2007 CAHSEE-M
EOs – 80% passed
ELs - 47% passed
IFEP – 85% passed
RFEP – 86% passed
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Mystery
Quote – When/Who?
Raleigh Schorling,
NCTM, 1926
“…our secondary schools are crowded with
pupils who have little background and
experience and less ability for mathematical
training…Many come from ‘first generation
homes.’ They do not even speak our
language.
Every school subject now has unusual
difficulties with the vocabulary of the
subject…The language difficulties which the
teacher confronts in instructing the children
of recent immigrants,
a
problem
met
in
Carl Lager - May 16, 2008 many high schools,
-is alone very great.”
[email protected]
Equity strand, NCTM, 2000
“Students who are not native speakers of
English, for instance, may need special
assistance to allow them to participate fully
in classroom discussions.
Some of these students may also need
assessment accommodations. If their
understanding is assessed only in English,
their mathematical proficiency may not be
accurately evaluated.”
Carl Lager - May 16, 2008 [email protected]
Large-scale assessment

Because many ELs and (and some nonELs) feel mathematics problems are also
language problems, let’s experience a largescale mathematics problem (or four) like a
English learner.
 You’ll get to work on four problems projected
on the screen. You’ll have 90 seconds to
work on each problem.
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Raising our awareness
 After 90 seconds have elapsed, I’ll say
“time!” You write your answer and the
level of your confidence in the
appropriate box on the blue worksheet.
 You will work silently and independently.
Afterward, you will freewrite to specific
questions and share out lessons learned.
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Problem 1
 El dueño de un huerto de manzanas
manda sus manzanas en cajas. Cada
caja vacía pesa k kilogramos (kg). El
peso medio de una manzana es a kg y
el peso total de una caja llena de
manzanas es b kg. ¿Cuántas
manzanas han sido empacadas en cada
caja?
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Problem 2
 El dueño de un huerto de manzanas
manda sus manzanas en cajas. Cada
caja vacía pesa k kilogramos (kg). El
peso medio de una manzana es a kg y
el peso total de una caja llena de
manzanas es b kg. ¿Cuántas manzanas
han sido empacadas en cada caja?


A) b + k
B) (b - k) / a
C) b / a
D) (b + k) / a
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Problem 3
 El dueño de un huerto de manzanas
manda sus manzanas en cajas. Cada
caja vacía pesa 2 kilogramos (kg). El
peso medio de una manzana es 0.25 kg
y el peso total de una caja llena de
manzanas es 12 kg. ¿Cuántas
manzanas han sido empacadas en cada
caja?
 A) 14
C) 48
 B) 40
D) 56
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Problem 4
The owner of an apple orchard ships
apples in boxes that weigh 2 kilograms
(kg) when empty. The average apple
weighs 0.25 kg, and the total weight of a
box filled with apples is 12 kg. How many
apples are packed in each box?
A) 14
B) 40
C) 48
D) 56
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Individual Freewrite (salmon sheet)
(3 minutes)
1) What specific meaning-making strategies did
you employ?
2) How effective were your strategies?
3) How confident were in your answers?
4) What “mental movies” were you generating?
What were you seeing?
5) How did you feel?
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Share out
1) What specific meaning-making strategies did
you employ?
2) How effective were your strategies?
3) How confident were in your answers?
4) What “mental movies” were you generating?
What were you seeing?
5) How did you feel?
Carl Lager - May 16, 2008 [email protected]
CAHSEE released algebra item
(#88, p. 32, CDE, 2006)
The owner of an apple orchard ships
apples in boxes that weigh 2 kilograms
(kg) when empty. The average apple
weighs 0.25 kg, and the total weight of a
box filled with apples is 12 kg. How many
apples are packed in each box?
A) 14
B) 40
C) 48
D) 56
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CAHSEE released item (#88, p. 32,
CDE, 2006) – possible challenges
The owner of an apple orchard ships apples in
boxes that weigh 2 kilograms (kg) when empty.
 Boxes? Singular and plural meanings at the same
time! How many boxes are we talking about here?
Is 2 kg the total weight of all the empty boxes?
 Translation: The owner of an apple orchard ships
apples in boxes. Each empty box weighs 2 kg.
Carl Lager - May 16, 2008 [email protected]
CAHSEE released item (#88, p. 32,
The average apple….
When have you ever heard this phrase? Average as
an adjective? An EL would expect red, juicy, ripe,
etc. to describe the physical characteristics of the
apple, not average.
The owner of an apple orchard ships apples…
Ships as a verb? What
about ships on the water?
Carl Lager - May 16, 2008 [email protected]
CAHSEE released item (#88, p. 32,
CDE, 2006)
How many apples are packed in each box?
 Who’s packing the apples?
 Passive voice (PV)
 Focusing on subject of action, the apples,
obfuscates who is doing the action
 More difficult to generate a “mental movie”
of the problem.
Carl Lager - May 16, 2008 [email protected]
Reading mathematics texts/items
 Vocab, syntax, symbols, multiple meanings of
words make math reading difficult (Gullatt 1986;
Harris & Devander, 1990)
 Math texts require different reading demands
than other texts (Bye, 1975)
 <90% meaningful words = frustration (Betts,
1946)
 Second language learning is more difficult when
textbook English is the first English – discourse
very different from ordinary talk (Fillmore, 1982)
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Reading Comprehesnion
 Step 1 in problem solving is understanding
the problem (Polya, 1943).
 Reading comprehension is critical to
understanding the problem
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The RRSG (RAND, 2002) defines
reading comprehension as:
…the process of simultaneously
extracting and constructing
meaning through interaction and
involvement with written language.
It consists of three elements: the
reader, the text, and the activity or
purpose for reading.
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The RRSG (2002) defines reading
comprehension as:
…these elements interrelate in reading
comprehension, an interrelationship
that occurs within a larger sociocultural
context that shapes and is shaped by
the reader and that interacts with each
of the elements iteratively throughout
the process of reading.
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The RRSG (2002) heuristic:
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The Study (Lager, 2006)
Reading Interactions That
Unnecessarily Hinder Algebra
Learning and Assessment
Carl Lager - May 16, 2008 [email protected]
Research Question
1) What are the specific language
difficulties that hinder Spanishspeaking ELs in grades 6 and 8, from
understanding one set of visual-based
linear function activities?
Carl Lager - May 16, 2008 [email protected]
Design
 Theory-driven, standards-based, studentcentered algebra activity was adapted from a
Navigating Through Algebra activity in Grades 3
- 5 (NCTM, 2001)
 9 generative tasks, including problem solving
 “Concrete”
Representations
Abstraction
 Based on linguistic and mathematical
frameworks
Carl Lager - May 16, 2008 [email protected]
Design
 221 middle school students (grades 6 & 8)
 2 low-performing urban SoCal middle schools
60/40 split for 6th & 8th and EL/Non-EL
 56/44 split female to male
 82% Lat, 10% AA, 2% As, P, An, 1% In
 Almost all ELs were Spanish-speakers
Students worked silently, independently, and
without notes
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Your turn (white sheets)
 To give you a taste of the tasks, four of
them have been shared, in a modified
form, for today.
 Take 10 minutes to look over and do the 5
tasks silently and independently. If you
finish early, try to predict EL strengths and
challenges on the task.
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Figure 1
Figure 2
Figure 3
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Overall Results – Correct
Response
80%
70%
60%
50%
T1
T2
40%
30%
20%
10%
0%
T1
T2
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Most common misconception
 Figure 4 vs. Figure 5
 Figure 5 vs. Figure 6
 Why?
Lack of empty grid spaces!
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Example – Task 2
 …For each figure, record the figure number
(N) and the corresponding number of blue
squares (B) in your table. For Figure 1, N=1,
so B=3.
Figure number (N)
Number of blue squares (B)
1
2
3
5
3
7
Carl Lager - May 16, 2008 [email protected]
Example – Task 2
 …For each figure, record the figure number
(N) and the corresponding number of blue
squares (B) in your table. For Figure 1, N=1,
so B=3.
Figure number (N)
Number of blue squares (B)
1
42
3
5
93
7
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Revisit Task 2
Figure 1
Figure 2
Figure 3
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Mathematics Register






Shifts of Application (Durkin & Shire)
Polysemy (Durkin & Shire)
Form of label
Form of square vs. form of figure
Parentheses
Unknown lang., unknown concept (Garrison
& Mora)
 Semantic – Complex words or phrases
(Spanos et al.)
Carl Lager - May 16, 2008 [email protected]
Figure Number (N) – Type II
 Writing a correct response for the wrong
reason!
 1) Row fallacy
 2) Blue side fallacy
 3) Yellow column height fallacy
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Row Fallacy
1
Figure 1
1
2
2
Figure 2
1
3
Figure 3
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Blue side fallacy
1
1 Figure 1
1
1
2
2 Figure 2
1
2
1
2
3
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3 Figure 3
Yellow column height fallacy
1
Figure 1
1
2
Figure 2
1
2
3
Figure 3
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Task 3
Number of
blue squares
(B)
0
Figure number (N)
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One possible correct scaling
Number of
blue squares
(B)
10
8
6
4
2
0
1
2
3
4
5
Carl Lager - May 16, 2008 [email protected]
Figure number (N)
Figure number scaling issue #1
Number of
blue squares
(B)
10
8
6
4
2
0
+1
1
2
+2
3
4
5
Carl Lager - May 16, 2008 [email protected]
Figure number (N)
Figure number scaling issue #2
Number of
blue squares
(B)
10
8
6
4
2
0
1
4
9
16
25
Carl Lager - May 16, 2008 [email protected]
Figure number (N)
Number of blue squares scaling
issue
Number of
blue squares
(B)
11
9
7
+2
5
3
+3
0
1
2
3
4
5
Carl Lager - May 16, 2008 [email protected]
Figure number (N)
Research Questions
2) What are the specific language
difficulties that hinder Spanishspeaking ELs in grades 6 and 8, from
communicating their mathematical
understandings of one set of visualbased linear function activities?
Carl Lager - May 16, 2008 [email protected]
Did the students successfully
express what they understood?
 Task 4 - Your friend, José, asks you:
“Each time the figure number goes up by
one, the number of blue squares changes
by how many?” Help José by answering
his question.
 Answer: By 2
 Appropriate/Inappropriate
 Vague/Precise
 Lenses for examining
incorrect
responses
Carl Lager - May 16,
2008 [email protected]
Appropriate/Precision matrix
Precise
Vague
Appro
priate
“the blue squares
change by 1”
“You add one square
on each side.”
Inappr
opriate
“The blue squares
can’t be added by
1 becouse then the
figure would be
different from the
other figures.”
“Each time you add
one blue square it’s
going to make the
triangle bigger
because the numbers
are mostly odd.”
Carl Lager - May 16, 2008 [email protected]
“By 1”
 Most popular incorrect answer was 1 (by
far)
 Variety of reasons for this response
 Confused the “skip (1)” with the “jump (2)”
(e.g. 3, 5, 7, 9, 11)
– student successfully communicated his
misunderstanding of the term change by
Carl Lager - May 16, 2008 [email protected]
“By 1”
 Meant goes up by 1 on each “side”
+1
+1
+1
+1
+1
 – student unsuccessfully communicated
his understanding
Carl Lager - May 16, 2008 [email protected]
+1
“By 1”
 Meant goes up by 1 “side” only
1
1
1
2
2
3
 – student successfully communicated his
misunderstanding
Carl Lager - May 16, 2008 [email protected]
“Two spaces”
 One student wrote: It changes for two
spaces.
 The response is appropriate, but vague
and does not accurately describe two
blue squares
 – student unsuccessfully
communicated her understanding
Carl Lager - May 16, 2008 [email protected]
Treisman’s Challenge (May, 2007)
 We need to systematically and coherently
go beyond generic instructional strategies
to address EL needs through on-target
long-term professional development
 Borrow from Using the Language to
Increase Deeper Mathematical Meaning
and Understanding session (MartinezCruz and Delaney 2005)
Carl Lager - May 16, 2008 [email protected]
(Posamentier and Salkind, 1996)
 1) Solve the following 3-part problem singly
or with a partner. Write all work on the
problem itself.
 2) When finished, circle your solutions.
 3) Grab the green sheet from your folder and
answer the 5 questions
Carl Lager - May 16, 2008 [email protected]
(Posamentier and Salkind, 1996)
 4) Do your answers change? Why or why
not?
5) We will share out immediately afterward.
6) ARE YOU READY…..?!
Carl Lager - May 16, 2008 [email protected]
(Posamentier and Salkind, 1996)
 A man buys 3-cent stamps and 6-cent
stamps, 120 in all. He pays for them with a
$5.00 bill and receives 75 cents in change.
 Does he receive the correct change?
 Would 76 cents change be correct?
 Would 74 cents be correct?
Carl Lager - May 16, 2008 [email protected]
(Posamentier and Salkind, 1996)
 A man buys 3-cent stamps and 6-[cent] stamps,
120 in all. He pays for them with a $5.00 bill and
receives 75 cents in change.
 Does he receive the correct change? No
 Would 76 cents change be correct? No
 These apparent problems to find (Pólya, 1943) are
really problems to prove (Pólya 1943).
 NCTM’s (2000) Grades 9 – 12 Reasoning and
Proof standard (p. 342, p. 344)
Carl Lager - May 16, 2008 [email protected]
Would 76 cents be correct?
 A man buys 3-cent stamps and 6-cent
stamps, 120 in all. He pays for them with
 a $5.00 bill and receives 75 cents in change.
 Would 74 cents be correct? ???
Carl Lager - May 16, 2008 [email protected]
(Posamentier and Salkind, 1996)
 A man buys 3-cent stamps and 6-cent
stamps, 120 in all. He pays for them with
 a $5.00 bill and receives 75 cents in change.
 Would 74 cents be correct?
 A) yes
 B) Likely, but not certain (P & S)
 C) Possible, but not likely (me)
Carl Lager - May 16, 2008 [email protected]
(Hyde, 2006)
 K: What do I know for sure?
 W: What do I want to figure out, find
out, or do?
 C: Are there any special conditions,
rules, or tricks I have to watch out for?
Carl Lager - May 16, 2008 [email protected]
Checking Inferences (Hyde, 2006)
 What inferences did you make?
 Are the inferences accurate?
 What information is implied by the
problem writer? (Lager)
Carl Lager - May 16, 2008 [email protected]
(Hyde, 2006)
 K: What do I know for sure?
 • A man buys 120 stamps.
 • Each purchased stamp is either a 3-cent
stamp or a 6-cent stamp.
 • He buys at least two 3-cent stamps and at
least two 6-cent stamps.
 • He pays for the 120 stamps with a $5.00
bill.
 • The man receives $0.75 in change.
Carl Lager - May 16, 2008 [email protected]
(Hyde, 2006)
W: What do I want to figure out, find out,
or do?
Carl Lager - May 16, 2008 [email protected]
Checking Inferences (Hyde, 2006)
 C: Are there any special conditions,
rules, or tricks I have to watch out for?
 What inferences did you make?
 Are the inferences accurate?
Carl Lager - May 16, 2008 [email protected]
Find the inferences


x + y = 120
3x + 6y = 425


x + y = 120
3x + 6y = 424


x + y = 120
3x + 6y = 426
Unknown!
Carl Lager - May 16, 2008 [email protected]
Find the inferences
Amount of money paid – Amount of change
received = Total cost of stamps purchased
 If the man received the correct change, the
equation must be true. Applying the known
data in this problem generates:
 $5.00 – $0.75 = $4.25
 $5.00 – ? = unknown
Carl Lager - May 16, 2008 [email protected]
Find the inferences
x + y = 120
 3x + 6y = ?
 3 unknowns and only 2 equations!
 Therefore, there is no way to determine with
certainty if any change amount is correct!
Carl Lager - May 16, 2008 [email protected]
(Posamentier and Salkind, 1996)
 A man buys 3-cent stamps and 6-[cent]
stamps, 120 in all. He pays for them with
 a $5.00 bill and receives 75 cents in change.
 Does vs could? No vs. No
 Would vs. could? (76) No vs. No
 Would vs. could? (74) ??? vs. Yes
Carl Lager - May 16, 2008 [email protected]
Wordwalking
 Interpreting Does he receive the correct
change? (explicitly asking about one
transaction) as Is it possible he received the
correct change? (explicitly asking about all
possible transactions) is wordwalking (Mitchell
2001)
 Wordwalking - the changing of the original
question’s wording to convey similar meaning
but actually change the problem’s mathematical
structure
Carl Lager - May 16, 2008 [email protected]
Awareness of potential ambiguity
 MacGregor and Price (1999) define
awareness of potential ambiguity as “…the
recognition that an expression may have
more than one interpretation, depending on
how structural relationships or referential
terms are interpreted…” (p. 457).
Carl Lager - May 16, 2008 [email protected]
Awareness of potential ambiguity
 The Northwest Regional Education
Laboratory (NRWEL) mathematics problem
solving scoring guide states that, with regard
to insight, “an exemplary solution should
document possible sources of error or
ambiguity in the problem itself (NRWEL
2000)”
Carl Lager - May 16, 2008 [email protected]
Metalinguistic awareness
 Metalinguistic awareness - the
reflection upon and analysis of oral or
written language in mathematics
(MacGregor and Price 1999; Herriman
1991).
Carl Lager - May 16, 2008 [email protected]
Solutions II - RC Strategies
1) Activate related prior knowledge and experience.
2) Break down task into smaller chunks.
3) Use visual representations of real-world objects.
4) Predict the problem.
Let’s experience them as learners…use back of blue
paper
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Predict the problem
Carl Lager - May 16, 2008 [email protected]
Predict the problem
Carl Lager - May 16, 2008 [email protected]
Predict the problem
 A cycle shop has a total of 36 bicycles and
tricycles in stock.
 Collectively, there are 80 wheels.
 The bicycles come in 5 colors and the
tricycles in 4.
 How many bikes and how many tricycles are
there?
Carl Lager - May 16, 2008 [email protected]
Bicycles and Tricycles (Adapted
from Adding It Up, 2001; p. 126)
 A cycle shop has a total of 36 bicycles and
tricycles in stock.
 Collectively, there are 80 wheels.
 The bicycles come in 5 colors and the
tricycles in 4.
 How many bikes and how many tricycles are
there?
Carl Lager - May 16, 2008 [email protected]
(Hyde, 2006)
 K: What do I know for sure?
 W: What do I want to figure out, find
out, or do?
 C: Are there any special conditions,
rules, or tricks I have to watch out for?
Carl Lager - May 16, 2008 [email protected]
Checking Inferences (Hyde, 2006)
 What inferences did you make?
 Are the inferences accurate?
 What information is implied by the
problem writer? (Lager)
Carl Lager - May 16, 2008 [email protected]
What does this system of equations
tell us about this situation?
b + t = 36
2b + 3t = 80
Carl Lager - May 16, 2008 [email protected]
What does this system of equations
tell us about this situation?
What does b + t = 36 represent?
Carl Lager - May 16, 2008 [email protected]
What does this system of equations
tell us about this situation?
What does b + t = 36 represent?
Carl Lager - May 16, 2008 [email protected]
What does this system of equations
tell us about this situation?
What does b + t = 36 represent?
b + t = 36
Number of
bikes and
tricycles
Number of bicycles
and tricycles
Carl Lager - May 16, 2008 -
[email protected]
Number of bicycles
Number of bicycles
=
and tricycles
and tricycles
Number of bicycles
=
and tricycles
Number of bicycles
(b) + number of
=
tricycles (t)
b
+
t
=
Carl Lager - May 16, 2008 [email protected]
36
36
36
What does this system of equations
tell us about this situation?
What does 2b + 3t = 80 represent?
Carl Lager - May 16, 2008 [email protected]
What does this system of equations
tell us about this situation?
What does 2b + 3t = 80 represent?
Number of bicycle
wheels and tricycle
wheels
Number of wheels
Carl Lager - May 16, 2008 [email protected]
What does this system of equations
tell us about this situation?
What does 2b + 3t = 80 represent?
Number of bicycle
wheels and tricycle
wheels
Number of bicycle
wheels and tricycle
wheels
Carl Lager - May 16, 2008 [email protected]
Cognitive organizer
Number of
wheels
Bicycle
wheels
Tricycle
wheels
Carl Lager - May 16, 2008 [email protected]
In conclusion…
 In conclusion…
Carl Lager - May 16, 2008 [email protected]
The overriding meaning-making
and meaning-sharing challenges
 For ELs (and some non-ELs), the ongoing
triple challenge of handling “everyday” and
mathematical English, unfamiliar contexts
and cultural norms, and mathematics content,
all at the same time during an on-demand
assessment and classroom setting can be
quite daunting.
Carl Lager - May 16, 2008 [email protected]
What you can do
 1) Don’t let teachers do the reading
comprehension for the students!
 2) Teach students to become active readers
of their own mathematics tasks
 3) Demand and support long-term,
systematic professional development to
teach mathematics teachers how to do #2
Carl Lager - May 16, 2008 [email protected]
http://www.todos-math.org
 Let’s help our students by meeting these
challenges together. Join TODOS!
Thank you.
clager@education
.ucsb.edu
Carl Lager - May 16, 2008 [email protected]