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

1. Exploring Written Math Explanations as a Tool to Inform Math and Writing Instruction 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. Mary A. Avalos (mavalos@miami.edu ) Mileidis Gort (gort.4@osu.edu) Jennifer Langer-Osuna (jlangerosuna@miami.edu)

2. 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

3. 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)

4. 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

5. 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

6. 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)

7. 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

8. 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?

9. 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?

10. 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.)

11. 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)

12. 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)

13. Math Rubric Scores

14. 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)

15. 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? • (12 x 2) – 30 • 30 – (12 x2) • (30 – 12) x 2 • (12 – 2) x 30

16. Use of diagram to model problem Connectives-Time Relationships Time Point 1: Student 1 Mathematical notation Worked from answer choices Mental Process Connective-Cause/Effect Logical and correct procedure; clearly written; selected A as correct answer

17. Time Point 1: Student 3 Mathematicalnotation Connective-time relationship Subject-specific vocabulary Connective-cause/effect relationship Mental process No diagram or introductory statement; use of vague referents throughout; assumes answer choice D (12 – 2) x 30 means equal to 30

18. 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

19. Time Point 2: Student 1 Use of diagram to model problem Connective-Cause/Effect Subject-specific vocabulary Mathematical notation Notes formula for area

20. Time Point 2: Student 1 Worked from answer choices Subject-specific vocabulary Mental process Notes formula for area Mathematical notation

21. Time Point 2: Student 3 Attempts diagram to model problem Attempts to write formula for area Subject-specific vocabulary

22. Time Point 2: Student 3 3. Explanatory Sequence (how you solved the problem): I know my answer is correct becauseit wants you to multiplynot 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.

23. 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.

24. Time Point 3: Student 1 Use of graphic to organize data Connective-Cause/Effect Subject-specific vocabulary

25. Time Point 3: Student 1 Connectives-Time Relationships Subject-specific vocabulary Mathematical notation

26. Time Point 3: Student 1 Connective-Cause/Effect Subject-specific vocabulary Mental Process Connective- Time Relationship

27. Time Point 3: Student 3 Mathematical notation

28. 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.

29. 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

30. Modification of Language in Word Problems

31. 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