ACCESS FOR ALL: READING COMPREHENSION STRATEGIES FOR UNDERSTANDING AND SOLVING MATHEMATICS TASKS

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ACCESS FOR ALL READING COMPREHENSION STRATEGIES
FOR UNDERSTANDING AND SOLVING MATHEMATICS TASKS

• Carl Lager, PhD
• University of California, Santa Barbara

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Overview

• Three problem situations will be used to
  identify, explicate, and model specific reading
  comprehension strategies in different contexts.
• You will experience them as problem solvers
  and learn how to facilitate them with your
  students.

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Large-scale assessment

• Because for many students (ELs and non-ELs)
  mathematics problems are also language problems,
  lets experience a large-scale mathematics item
  (or four) like a English learner.
• Youll get to work on four items projected on
  the screen. Youll have 90 seconds to work on
  each problem.

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Raising our awareness

• After 90 seconds have elapsed, Ill say time!
  You write your answer and the level of your
  confidence in the appropriate box on the
  worksheet.
• Work silently and independently.
• At the end, we will do a freewrite and share out.

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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 C) b / a
• B) (b - k) / 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 C) 48
• B) 40 D) 56

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Individual Freewrite (4 minutes)

• 1) What specific meaning-making strategies did
  you employ and when?
• 2) How effective were your strategies?
• 3) How confident were you in your
  strategies/answers?
• 4) What mental movies were you generating?
  What were you seeing?
• 5) How did you feel?

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Whole group share out

• 1) What specific meaning-making strategies did
  you employ?
• 2) How effective were your strategies?
• 3) How confident were you in your
  strategies/answers?
• 4) What mental movies were you generating?
  What were you seeing?
• 5) How did you feel?

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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? How many boxes are we talking about here?
  Is 2 kg the total weight of all the empty boxes?
  If so, dont I need to know how many boxes so I
  can divide that quantity by 2 kg to find the
  weight of one box?
• Translation The owner of an apple orchard ships
  apples in boxes. Each empty box weighs 2 kg.

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CAHSEE released item (88, p. 32, CDE, 2006)

• How many apples are packed in each box?
• Whos packing the apples? This passive voice
  (PV) construction is very common in mathematics
  texts because it purposefully focuses our
  attention on the subject of the action, the
  apples. However, by doing so, PV obfuscates who
  is doing the action, making the construction of a
  mental movie of the problem more difficult.

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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)
• lt90 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
• Think about what you just experienced with the
  Spanish CAHSEE math item

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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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RC Classroom Strategies - I

• 1) Activate related prior knowledge and
  experience.
• 2) Break down task into smaller chunks.
• 3) Use manipulatives or real-world objects.
• 4) Predict the problem.

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RC Strategies - I

• This problem will be presented, bit-by-bit, in 5
  stages.
• At each stage, you will be given 1 minute to
  predict the problem. Generate as many possible
  problems as you can that fit the information
  provided.
• At stage 5, you will receive the actual problem.
  Solve it.
• When you finish, write down how and why your
  answer based on the paper version of the problem
  might differ from the answer for the real-world
  version.

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RC Strategies I (10 minutes)

• And away we go

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Tennis Ball small group discussion(5 minutes)

• What were some of your questions?
• How, why, and when did they change?
• Was there a progression or schema that organized
  the changes?
• How and why would your answer based on the paper
  version of the problem differ from the answer for
  the real-world version?

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Tennis Ball share out

• What were some of your questions?
• How, why, and when did they change?
• Was there a progression or schema that organized
  the changes?
• How and why would your answer based on the paper
  version of the problem differ from the answer for
  the real-world version?

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Treismans 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
  (Martinez-Cruz and Delaney 2005)

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(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 answers.
• 3) Then, apply the KWC and CI strategies to the
  problem. Another words, answer in writing the
  questions Ill provide.
• 4) You have 10 minutes. Go.

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(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?

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(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).
• NCTMs (2000) Grades 9 12 Reasoning and Proof
  standard (p. 342, p. 344)

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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? ???

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(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)

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(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

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(Polya, 1943 and Hyde, 2006)

• Step 1 in problem solving is understanding the
  problem (Polya, 1943)
• Hyde infuses Pólyas (1943) work with reading
  comprehension strategies to create his five-phase
  Braid Model of Problem Solving Situation,
  Representations, Patterns, Connections, and
  Extensions.

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(Polya, 1943 and Hyde, 2006)

• Two such strategies the KWC strategy and the
  Checking Inferences strategy were applied to this
  problem.
• 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?

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Checking Inferences

• What inferences did you make? Are the inferences
  accurate?
• What information is implied by the problem
  writer?
• What are the significances of these
  inference-implication interactions for problem
  solvers?

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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).

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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)

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Metalinguistic awareness

• Metalinguistic awareness - the reflection upon
  and analysis of oral or written language in
  mathematics (MacGregor and Price 1999 Herriman
  1991).

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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)

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Wordwalking

• Wordwalking - the changing of the original
  questions wording to convey similar meaning but
  actually change the problems mathematical
  structure.

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In conclusion

• In conclusion

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The overriding challenge

• 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.

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NCTMs PSSM (2000)

• Though NCTM (2000) expects secondary mathematics
  teachers to help studentslearn to read
  increasingly technical text (p. 351), many
  teachers not adequately prepared to do so (RAND,
  2002).

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NCTMs PSSM (2000)

• Telling students, and language learners in
  particular, to read mathematics problems
  carefully (NCTM 2000, p. 54) without teaching
  them how is innately unjust and unfair.

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The RRSG (RAND, 2002)

• Reading instruction is seldom effectively
  integrated with content-area instruction.
• Teaching in the content areas relies on texts as
  a major source of instructional content.

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The RRSG (RAND, 2002)

• These texts are not designed as a context for
  comprehension instruction, but comprehension
  instruction that uses these texts may be crucial
  if students are to understand or learn from them.
  

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www.huertodemanzanas.com

• For this sessions materials, powerpoints, and
  papers, and more related work on ELs, go to my
  website and click on these tabs
• 1) Huerto de Manzanas
• 2) Tennis Ball
• 3) Stamp and Change

Thank you.
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http//www.todos-math.org

• Specific reading comprehension tasks and
  strategies were modeled today.
• Lets help our students by meeting these
  challenges together.