Exploring Written Math Explanations as a Tool to Inform Math and Writing

InstructionMary A. Avalos (mavalos@miami.edu )

Mileidis Gort (gort.4@osu.edu)

Jennifer Langer-Osuna (jlangerosuna@miami.edu)

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 solutionAdapted 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

TP1

TP2

TP3

TP1

TP2

TP3

TP1

TP2

TP3

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 1Gary 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) – 30B. 30 – (12 x2)C. (30 – 12) x 2D. (12 – 2) x 30

Time Point 1: Student 1

Worked from answer choices

Connectives-Time Relationships

Use of diagram to model problem

Mathematical notation

Logical and correct procedure; clearly written; selected A as correct answer

Mental Process

Connective-Cause/Effect

Time Point 1: Student 3

Subject-specific vocabulary

Mental process

Mathematical notation

Connective-time relationship

Connective-cause/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

Mathematical notation

Subject-specific vocabulary

Connective-Cause/Effect

Notes formula for area

Time Point 2: Student 1

Subject-specific vocabulary

Worked from answer choices

Notes formula for area

Mathematical notation

Mental process

Time Point 2: Student 3

Attempts to write formula for area

Attempts diagram to model problem

Subject-specific 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

Subject-specific vocabulary

Connective-Cause/Effect

Time Point 3: Student 1 Connectives-Time Relationships

Subject-specific vocabulary

Mathematical notation

Time Point 3: Student 1 Connective-Cause/Effect

Mental Process

Subject-specific vocabulary

Connective- Time 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 ProblemGary 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