Transcript Mathematical notation - Literacy Research Association

Exploring Written Math
Explanations as a Tool to
Inform Math and Writing
Instruction
Mary A. Avalos ([email protected] )
Mileidis Gort ([email protected])
Jennifer Langer-Osuna ([email protected])
December 4, 2013
Literacy Research Association
This work is possible due to a grant funded by
The Institute of Educational Sciences, Grant Award
Number: R305A100862: The content of the presentation
does not necessarily reflect the views or policies of IES or
the U.S. Department of Education nor does mention of
trade names, commercial products, or organizations imply
endorsement by the U.S. Government.
Writing Instruction
• Dearth of Research (Applebee & Langer, 2011; Ball, 2006; Gilbert &
Graham, 2010; Graham, 2013; Kiuhara, Graham, & Hawken, 2009)
• Common Core State Standards
– High expectations concerning writing across the
curriculum (Calkins, Ehrenworth, & Lehman, 2012; Graham, Gillespie, &
McKeown, 2012)
• Disparity across groups for writing achievement (NAEP;
National Center for Education Statistics, 2012)
– Race/Ethnicity
– Gender
– Suburban vs. Urban and Rural
Writing Across the Curriculum
• Writing in content areas may assist in
students’ understandings of the content
(Alvermann & Phelps, 2002; Daniels & Zemelman, 2004; Klein
& Rose, 2010; Manzo, Manzo & Estes, 2001; Vacca & Vacca,
2002)
• Students need to produce knowledge (i.e.,
using writing) to be competent in the domain
(Moje, 2007)
Language in Math Intervention
Project (2010-2013)
• Focus on English learners (ELs) and teachers of
ELs in 4th-8th grade;
• Reform oriented approach to math instruction
(NCTM 1991; 2000)
• Explicit language teaching during math
instruction to raise student achievement and
knowledge of academic language
– Genre of word problems and prominent language
features (syntax/semantics) to better comprehend
problems
– Discussion and justification of solution processes
– Explanation writing of problem solutions
LiM Intervention
• Focus on conceptual understanding
• Task set-up
– Structure of word problems (genre)
– Academic language
• Technical vocabulary and prominent language
features (noun and verb groups--chunking,
referents, connectives, language confounds)
• Written explanations (genre and
math register)
• Students discuss problem solutions
Genre Features of Explanations
• Topic statement about the phenomenon in
question
• Followed by explanation of how/why
something occurs.
• Sometimes there is a concluding or
summarizing statement.
• Explanations have a process focus rather
than a thing focus- they are often concerned
with a logical sequence
• Solution and answer is justified based on
knowledge of math concept or procedure.
(Knapp & Watkins, 2005)
Math Explanations
• Introduction → Description
of the problem to be
solved
– Includes a model and plan
• Explanatory Sequence
– Procedural steps taken to
solve the problem
• Conclusion → Justification
– Defend the reasonableness
behind the solution
Adapted from Knapp & Watkins, 2005
Research Question
• How can written explanations of
mathematics problem-solving be
used as a tool to inform
instructional needs related to
language of word problems?
Genre of Word Problems
Stephanie’s mom is planning a Halloween
party. She wants to make cookies that are
shaped like jack-o-lanterns, hissing black cats,
and ghosts. The cookie recipe calls for 3 ¼ cups
of flour, ¼ cup of butter, and 2/3 cup of sugar.
If Stephanie’s mom is going to triple the
recipe, how much flour, butter, and sugar
does she need?
Linguistic & Grammatical Features Commonly
Found in Mathematic Explanations
• Generalized non-human participants—classes of things rather than
specific participants (the problem; the garden; markers, etc.)
• Connectives to note time relationships (first, then, next, following,
finally)
• Connectives to note cause-and-effect relationships (if/then, so, as a
consequence, since)
• Mainly action processes (multiplied, added, timed)
• Mental processes (I know; We thought)
• Some passives (is multiplied, is solved, is found)
• Diagrams, labels, pictures, and drawings to assist with explanations
• Subject specific vocabulary (multiply, sum, product, area, base, height)
• Mathematical notation (30 – 24 = 6)
• Existential Processes (There was; There is; It is; etc.)
Methods
• Urban elementary school in Southeastern U.S.
– 78% F/R lunch
– Diverse population (Latino/a American—48%;
African/Caribbean American—25%; European American—
17%; Asian American—6%; Multi-racial—3%)
– 22% English learners
• 4th grade classroom
– Teacher participated in LiM project
– Assigned explanation writing in math
– 7 target students
• English learners
• High Achievers/Gifted to Struggling
• Data Sources
– Collected 21 explanation writing samples in 2012-13
(September, March, May)
Analyses
• Mathematics:
– Scored explanation writing samples using rubric
(Lane, Stone, Ankenmann, & Liu, 1994) for students’
• Mathematical Knowledge
• Strategic Knowledge
• Communication
• Language
– Systemic Functional Linguistics (Halliday, 1978)
– Connectives to note time relationships (first, then, next,
following, finally)
– Connectives to note cause-and-effect relationships (if/then, so,
as a consequence, since)
– Mental processes (I know; We thought)
– Diagrams, labels, pictures, and drawings to assist with
explanations
– Subject specific vocabulary (multiply, sum, product, area, base,
height)
– Mathematical notation (30 – 24 = 6)
Math Rubric Scores
Math
Knowledge
Strategy
Communication
TP
1
TP
2
TP
3
TP
1
TP
2
TP
3
TP
1
TP
2
TP
3
Student 1
3
4
2
2
4
2
3
4
3
Student 3
0
1
1
1
1
1
0
0
0
Framework to Answer Research
Question:
• What do we see in the written
explanations concerning mathematical
understandings?
• What is the writer on the verge of
understanding as evident by use of
language features in their writing?
• What are the implications for teaching
and learning of word problem genre
and/or language features?
Adapted from Seidel (1998) and Reddy-Butkovich (2008)
Time Point 1
Gary bought 12 markers at $2 each. He gave the
cashier $30. Which expression can be used to find
how much change Gary should receive?
A. (12 x 2) – 30
B. 30 – (12 x2)
C. (30 – 12) x 2
D. (12 – 2) x 30
Use of diagram
to model
problem
Mathematical
notation
Mental
Process
Time Point 1: Student 1
Connectives-Time
Relationships
Worked
from answer
choices
ConnectiveCause/Effect
Logical and correct procedure; clearly written; selected A as correct
answer
Time Point 1: Student 3
Mathematical
notation
Mental process
Connective-time
relationship
Subjectspecific
vocabulary
Connectivecause/effect
relationship
No diagram or introductory statement; use of vague referents
throughout; assumes answer choice D (12 – 2) x 30 means equal
to 30
Time Point 2
Reggie wants to plant a garden in his
backyard. He plans to plant the garden
in the shape of a rectangle and draws a
diagram to determine its size. The
garden has an area of 128 feet with a
length of 16 feet. Which of the following
equations can be used to find w, the
width of the side of Reggie's garden?
A) w x 4 ==128
B) w ÷ 16 = 128
C)
w x 16 = 128
D) 128 ÷ 4 = w
Time Point 2: Student 1
Use of
diagram to
model
problem
Subjectspecific
vocabulary
ConnectiveCause/Effect
Mathematical
notation
Notes formula
for area
Time Point 2: Student 1
Worked from
answer
choices
Subjectspecific
vocabulary
Mental process
Notes formula
for area
Mathematical
notation
Time Point 2: Student 3
Attempts
diagram to
model
problem
Attempts to write
formula for area
Subjectspecific
vocabulary
Time Point 2: Student 3
3. Explanatory Sequence (how you solved the
problem):
I know my answer is correct because it wants you
to multiply not divide. So, B and D are wrong.
Now, look at A. Then look at the problem. It does
not say it anywhere. So, C is the remaining answer.
That means C is the correct answer.
Time Point 3
The baker made a total of 2,687 Italian
cookies over the weekend. They sold
1,263 more cookies on Saturday than
on Sunday. How many cookies did they
sell on Saturday? How many cookies did
they sell on Sunday? Please include a
model.
Time Point 3: Student 1
Use of graphic to organize
data
ConnectiveCause/Effect
Subjectspecific
vocabulary
Time Point 3: Student 1
Connectives-Time Relationships
Subject-specific
vocabulary
Mathematical notation
Time Point 3: Student 1
ConnectiveCause/Effect
Subjectspecific
vocabulary
Mental
Process
ConnectiveTime
Relationship
Time Point 3: Student 3
Mathematical notation
Findings
• Difficulties when language of word
problem incongruent with order of
mathematical expression or equation
• Regularly working from given multiple
choice answers (test-taking strategy)
does not encourage reasoning through
the problem
• Vague referents ≠ precise language
choices
• Depth of conceptual and procedural
understandings may impact writing
proficiency, particularly for struggling
students
• Mathematical components are key to
effective explanations.
Implications
• Language of Word Problem
– As a scaffold, teachers should consider modifying the
word problem when first introducing a concept so that
the word order of the problem is congruent with the
order of the mathematical expression or equation →
Change the order of declarative sentences in the
problem.
– Eliminating multiple choice answer possibilities will
create the necessity for students to reason through
the problem rather than work from the possible
answer choices → Change the request of the problem.
• Explanation Writing Instruction
– Vocabulary is important, but not enough for effective
explanations when precise language is valued 
instruct students regarding use of referents and when
to explicitly refer to the participant or process.
– Depth of conceptual and procedural understandings
may impact writing proficiency
–
Modification of Language in Word Problems
Original Problem
Modified Problem
Gary bought 12 markers at $2 each. He gave the
cashier $30. Which expression can be used to find
how much change Gary should receive?
A) (12 X 2) - 30
B) 30 - (12X2)
C) (30-12) X2
D) (12 X 2) X 30
Gary gave the cashier $30. He bought 12 markers
at $2 each. Write an expression to find how much
change Gary should receive?
Reggie wants to plant a garden in his backyard. He
plans to plant the garden in the shape of a
rectangle and draws a diagram to determine its
size. The garden has an area of 128 feet with a
length of 16 feet. Which of the following
equations can be used to find w, the width of the
side of Reggie's garden?
A)w x 4 ==128 B) w ÷ 16 = 128
C) w x 16 = 128
D) 128 ÷ 4 = w
Reggie wants to plant a garden in his backyard. He
plans to plant the garden in the shape of a
rectangle and draws a diagram to determine its
size. The garden has an area of 128 feet with a
length of 16 feet. What is the equation needed to
find w for Reggie's garden?
The baker made a total of 2,687 Italian cookies
over the weekend. They sold 1,263 more cookies on
Saturday than on Sunday. How many cookies did
they sell on Saturday? How many cookies did they
sell on Sunday? Please include a model.
The baker sold some cookies on Saturday. On
Sunday, the baker sold the same amount plus 1,263
more cookies. If the baker sold a total of 2,687
cookies over the weekend, how many cookies did
the baker sell each day?
Conclusion
• Students’ explanation writing can
serve as a tool to assist teachers in
determining:
– Mathematical understandings related
to genre and language of word
problems
– Instructional decision-making (content
and writing)
• Explicit instruction of effective math
explanations, which may lead to increased
writing and math proficiency
– Academic language development