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Reading Interactions That Unnecessarily Hinder Algebra Learning and Assessment

Reading Interactions That Unnecessarily Hinder Algebra Learning and Assessment. Carl Lager, PhD University of California, Santa Barbara clager@education.ucsb.edu (805) 893-7770. Overview. 1) ELs – The numbers 2) ELs – Engaging math items 3) ELs - Uncovering EL engagement

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Reading Interactions That Unnecessarily Hinder Algebra Learning and Assessment

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  1. Reading Interactions That Unnecessarily Hinder Algebra Learning and Assessment Carl Lager, PhD University of California, Santa Barbara clager@education.ucsb.edu (805) 893-7770 Carl Lager - May 16, 2008 - clager@education.ucsb.edu

  2. 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 - clager@education.ucsb.edu

  3. 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% Carl Lager - May 16, 2008 - clager@education.ucsb.edu

  4. 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) Carl Lager - May 16, 2008 - clager@education.ucsb.edu

  5. 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 - clager@education.ucsb.edu

  6. Mathematics Achievement of California’s Secondary Students 2006 - 2007 CAHSEE-M EOs – 80% passed ELs - 47% passed IFEP – 85% passed RFEP – 86% passed Carl Lager - May 16, 2008 - clager@education.ucsb.edu

  7. 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 many high schools, -is alone very great.” Carl Lager - May 16, 2008 - clager@education.ucsb.edu

  8. 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 - clager@education.ucsb.edu

  9. Large-scale assessment • Because many ELs and (and some non-ELs) feel mathematics problems are also language problems, let’s experience a large-scale 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. Carl Lager - May 16, 2008 - clager@education.ucsb.edu

  10. 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. Carl Lager - May 16, 2008 - clager@education.ucsb.edu

  11. 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? Carl Lager - May 16, 2008 - clager@education.ucsb.edu

  12. 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 + kC) b / a • B) (b - k) / a D) (b + k) / a Carl Lager - May 16, 2008 - clager@education.ucsb.edu

  13. 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) 14C) 48 • B) 40D) 56 Carl Lager - May 16, 2008 - clager@education.ucsb.edu

  14. 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 C) 48 B) 40 D) 56 Carl Lager - May 16, 2008 - clager@education.ucsb.edu

  15. 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? Carl Lager - May 16, 2008 - clager@education.ucsb.edu

  16. 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 - clager@education.ucsb.edu

  17. 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 Carl Lager - May 16, 2008 - clager@education.ucsb.edu

  18. 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 - clager@education.ucsb.edu

  19. 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 - clager@education.ucsb.edu

  20. 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 - clager@education.ucsb.edu

  21. 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) Carl Lager - May 16, 2008 - clager@education.ucsb.edu

  22. Reading Comprehesnion • Step 1 in problem solving is understanding the problem (Polya, 1943). • Reading comprehension is critical to understanding the problem Carl Lager - May 16, 2008 - clager@education.ucsb.edu

  23. 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. Carl Lager - May 16, 2008 - clager@education.ucsb.edu

  24. 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. Carl Lager - May 16, 2008 - clager@education.ucsb.edu

  25. The RRSG (2002) heuristic: Carl Lager - May 16, 2008 - clager@education.ucsb.edu

  26. The Study (Lager, 2006) Reading Interactions That Unnecessarily Hinder Algebra Learning and Assessment Carl Lager - May 16, 2008 - clager@education.ucsb.edu

  27. Research Question 1) What are the specific language difficulties that hinder Spanish-speaking ELs in grades 6 and 8, from understanding one set of visual-based linear function activities? Carl Lager - May 16, 2008 - clager@education.ucsb.edu

  28. Design • Theory-driven, standards-based, student-centered 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 - clager@education.ucsb.edu

  29. 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 Carl Lager - May 16, 2008 - clager@education.ucsb.edu

  30. 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. Carl Lager - May 16, 2008 - clager@education.ucsb.edu

  31. Figure 1 Figure 2 Figure 3 Carl Lager - May 16, 2008 - clager@education.ucsb.edu

  32. Overall Results – Correct Response Carl Lager - May 16, 2008 - clager@education.ucsb.edu

  33. Most common misconception • Figure 4 vs. Figure 5 • Figure 5 vs. Figure 6 • Why? Lack of empty grid spaces! Carl Lager - May 16, 2008 - clager@education.ucsb.edu

  34. 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 3 2 5 3 7 Carl Lager - May 16, 2008 - clager@education.ucsb.edu

  35. 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 3 2 5 4 3 9 7 Carl Lager - May 16, 2008 - clager@education.ucsb.edu

  36. Revisit Task 2 Figure 1 Figure 2 Figure 3 Carl Lager - May 16, 2008 - clager@education.ucsb.edu

  37. Mathematics Register • Shifts of Application (Durkin & Shire) • Polysemy (Durkin & Shire) • Form of label • Form of squarevs. form of figure • Parentheses • Unknown lang., unknown concept (Garrison & Mora) • Semantic – Complex words or phrases (Spanos et al.) Carl Lager - May 16, 2008 - clager@education.ucsb.edu

  38. 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 Carl Lager - May 16, 2008 - clager@education.ucsb.edu

  39. Row Fallacy 1 Figure 1 1 2 Figure 2 1 2 3 Figure 3 Carl Lager - May 16, 2008 - clager@education.ucsb.edu

  40. Blue side fallacy 1 1 Figure 1 1 1 2 2 Figure 2 1 1 2 2 3 3 Figure 3 Carl Lager - May 16, 2008 - clager@education.ucsb.edu

  41. Yellow column height fallacy 1 Figure 1 1 2 Figure 2 1 2 3 Figure 3 Carl Lager - May 16, 2008 - clager@education.ucsb.edu

  42. Number of blue squares (B) 0 Figure number (N) Task 3 Carl Lager - May 16, 2008 - clager@education.ucsb.edu

  43. Number of blue squares (B) 0 Figure number (N) One possible correct scaling 10 8 6 4 2 1 2 5 3 4 Carl Lager - May 16, 2008 - clager@education.ucsb.edu

  44. Number of blue squares (B) 0 Figure number (N) Figure number scaling issue #1 10 8 6 4 2 1 2 3 4 5 Carl Lager - May 16, 2008 - clager@education.ucsb.edu +1 +2

  45. Number of blue squares (B) 0 Figure number (N) Figure number scaling issue #2 10 8 6 4 2 1 4 9 16 25 Carl Lager - May 16, 2008 - clager@education.ucsb.edu

  46. Number of blue squares (B) 0 Figure number (N) Number of blue squares scaling issue 11 9 7 5 +2 3 +3 1 2 5 3 4 Carl Lager - May 16, 2008 - clager@education.ucsb.edu

  47. Research Questions 2) What are the specific language difficulties that hinder Spanish-speaking ELs in grades 6 and 8, from communicating their mathematical understandings of one set of visual-based linear function activities? Carl Lager - May 16, 2008 - clager@education.ucsb.edu

  48. 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 - clager@education.ucsb.edu

  49. Precise Vague Appropriate “the blue squares change by 1” “You add one square on each side.” Inappropriate “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.” Appropriate/Precision matrix Carl Lager - May 16, 2008 - clager@education.ucsb.edu

  50. “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 successfullycommunicated his misunderstandingof the term change by Carl Lager - May 16, 2008 - clager@education.ucsb.edu

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