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Reading Interactions That Unnecessarily Hinder Algebra Learning and Assessment. Carl Lager, PhD University of California, Santa Barbara [email protected] (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 l.jpg

Reading Interactions That Unnecessarily Hinder Algebra Learning and Assessment

Carl Lager, PhD

University of California, Santa Barbara

[email protected]

(805) 893-7770

Carl Lager - May 16, 2008 - [email protected]


Overview l.jpg
Overview Learning and Assessment

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 - [email protected]


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English Learners – the numbers Learning and Assessment

  • 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 - [email protected]


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Who are California’s English learners? Learning and Assessment

  • 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 - [email protected]


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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 - [email protected]


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Mathematics Achievement of California’s Secondary Students Students

2006 - 2007 CAHSEE-M

EOs – 80% passed

ELs - 47% passed

IFEP – 85% passed

RFEP – 86% passed

Carl Lager - May 16, 2008 - [email protected]


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Mystery Quote – When/Who? Students

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 - [email protected]


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Equity strand, NCTM, 2000 Students

“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 - [email protected]


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Large-scale assessment Students

  • 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 - [email protected]ion.ucsb.edu


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Raising our awareness Students

  • 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 - [email protected]


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Problem 1 Students

  • 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 - [email protected]


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Problem 2 Students

  • 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 - [email protected]


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Problem 3 Students

  • 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 - [email protected]


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Problem 4 Students

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 - [email protected]


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Individual Freewrite (salmon sheet) Students(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 - [email protected]


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Share out Students

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 - [email protected]


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

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 - [email protected]


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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? 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 - [email protected].ucsb.edu


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CAHSEE released item (#88, p. 32, challenges

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 - [email protected]


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

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 - [email protected]


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Reading mathematics texts/items challenges

  • 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 - [email protected]


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Reading Comprehesnion challenges

  • Step 1 in problem solving is understanding the problem (Polya, 1943).

  • Reading comprehension is critical to understanding the problem

Carl Lager - May 16, 2008 - [email protected]


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The RRSG (RAND, 2002) defines reading comprehension as: challenges

…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 - [email protected]


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The RRSG (2002) defines reading comprehension as: challenges

…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 - [email protected]


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The RRSG (2002) heuristic: challenges

Carl Lager - May 16, 2008 - [email protected]


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The Study (Lager, 2006) challenges

Reading Interactions That Unnecessarily Hinder Algebra Learning and Assessment

Carl Lager - May 16, 2008 - [email protected]


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Research Question challenges

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 - [email protected]


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Design challenges

  • 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 - [email protected]


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Design challenges

  • 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 - [email protected]


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Your turn (white sheets) challenges

  • 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 - [email protected]


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Figure 1 challenges

Figure 2

Figure 3

Carl Lager - May 16, 2008 - [email protected]


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Overall Results – Correct Response challenges

Carl Lager - May 16, 2008 - [email protected]


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Most common misconception challenges

  • Figure 4 vs. Figure 5

  • Figure 5 vs. Figure 6

  • Why?

    Lack of empty grid spaces!

Carl Lager - May 16, 2008 - [email protected]


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Example – Task 2 challenges

  • …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 - [email protected]


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Example – Task 2 challenges

  • …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 - [email protected]


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Revisit Task 2 challenges

Figure 1

Figure 2

Figure 3

Carl Lager - May 16, 2008 - [email protected]


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Mathematics Register challenges

  • 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 - [email protected]


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Figure Number (N) – Type II challenges

  • 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 - [email protected]


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Row Fallacy challenges

1

Figure 1

1

2

Figure 2

1

2

3

Figure 3

Carl Lager - May 16, 2008 - [email protected]


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Blue side fallacy challenges

1

1

Figure 1

1

1

2

2

Figure 2

1

1

2

2

3

3

Figure 3

Carl Lager - May 16, 2008 - [email protected]


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Yellow column height fallacy challenges

1

Figure 1

1

2

Figure 2

1

2

3

Figure 3

Carl Lager - May 16, 2008 - [email protected]


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Number of blue squares (B) challenges

0

Figure number (N)

Task 3

Carl Lager - May 16, 2008 - [email protected]


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Number of blue squares (B) challenges

0

Figure number (N)

One possible correct scaling

10

8

6

4

2

1

2

5

3

4

Carl Lager - May 16, 2008 - [email protected]


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Number of blue squares (B) challenges

0

Figure number (N)

Figure number scaling issue #1

10

8

6

4

2

1

2

3

4

5

Carl Lager - May 16, 2008 - [email protected]

+1

+2


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Number of blue squares (B) challenges

0

Figure number (N)

Figure number scaling issue #2

10

8

6

4

2

1

4

9

16

25

Carl Lager - May 16, 2008 - [email protected]


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Number of blue squares (B) challenges

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 - [email protected]


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Research Questions challenges

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 - [email protected]


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Did the students successfully express what they understood? challenges

  • 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 - [email protected]


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Precise challenges

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 - [email protected]


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“By 1” challenges

  • 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 - [email protected]


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“By 1” challenges

  • Meant goes up by 1 on each “side”

  • – student unsuccessfully communicated his understanding

+1

+1

+1

+1

+1

+1

Carl Lager - May 16, 2008 - [email protected]


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“By 1” challenges

  • Meant goes up by 1 “side” only

  • – student successfully communicated his misunderstanding

1

1

1

2

2

3

Carl Lager - May 16, 2008 - [email protected]


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“Two spaces” challenges

  • One student wrote: It changes for two spaces.

  • The response is appropriate, but vague and does not accurately describe two blue squares

  • – student unsuccessfully communicated her understanding

Carl Lager - May 16, 2008 - [email protected]


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Treisman’s Challenge (May, 2007) challenges

  • 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)

Carl Lager - May 16, 2008 - [email protected]


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(Posamentier and Salkind, 1996) challenges

  • 1) Solve the following 3-part problem singly or with a partner. Write all work on the problem itself.

  • 2) When finished, circle your solutions.

  • 3) Grab the green sheet from your folder and answer the 5 questions

Carl Lager - May 16, 2008 - [email protected]


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(Posamentier and Salkind, 1996) challenges

  • 4) Do your answers change? Why or why not?

    5) We will share out immediately afterward.

    6) ARE YOU READY…..?!

Carl Lager - May 16, 2008 - [email protected]


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(Posamentier and Salkind, 1996) challenges

  • 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?

Carl Lager - May 16, 2008 - [email protected]


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(Posamentier and Salkind, 1996) challenges

  • 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).

  • NCTM’s (2000) Grades 9 – 12 Reasoning and Proof standard (p. 342, p. 344)

Carl Lager - May 16, 2008 - [email protected]


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Would 76 cents be correct? challenges

  • 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? ???

Carl Lager - May 16, 2008 - [email protected]


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(Posamentier and Salkind, 1996) challenges

  • 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)

Carl Lager - May 16, 2008 - [email protected]


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(Hyde, 2006) challenges

  • 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?

Carl Lager - May 16, 2008 - [email protected]


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Checking Inferences (Hyde, 2006) challenges

  • What inferences did you make?

  • Are the inferences accurate?

  • What information is implied by the problem writer? (Lager)

Carl Lager - May 16, 2008 - [email protected]


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(Hyde, 2006) challenges

  • K: What do I know for sure?

  • • A man buys 120 stamps.

  • • Each purchased stamp is either a 3-cent stamp or a 6-cent stamp.

  • • He buys at least two 3-cent stamps and at least two 6-cent stamps.

  • • He pays for the 120 stamps with a $5.00 bill.

  • • The man receives $0.75 in change.

Carl Lager - May 16, 2008 - [email protected]


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(Hyde, 2006) challenges

W: What do I want to figure out, find out, or do?

Carl Lager - May 16, 2008 - [email protected]


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Checking Inferences (Hyde, 2006) challenges

  • C: Are there any special conditions, rules, or tricks I have to watch out for?

  • What inferences did you make?

  • Are the inferences accurate?

Carl Lager - May 16, 2008 - [email protected]


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Unknown! challenges

Find the inferences

  • x + y = 120

  • 3x + 6y = 425

  • x + y = 120

  • 3x + 6y = 424

  • x + y = 120

  • 3x + 6y = 426

Carl Lager - May 16, 2008 - [email protected]


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Find the inferences challenges

Amount of money paid – Amount of change received = Total cost of stamps purchased

  • If the man received the correct change, the equation must be true. Applying the known data in this problem generates:

  • $5.00 – $0.75 = $4.25

  • $5.00 – ? = unknown

Carl Lager - May 16, 2008 - [email protected]


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Find the inferences challenges

x + y = 120

  • 3x + 6y = ?

  • 3 unknowns and only 2 equations!

  • Therefore, there is no way to determine with certainty if anychange amount is correct!

Carl Lager - May 16, 2008 - [email protected]


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(Posamentier and Salkind, 1996) challenges

  • 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

Carl Lager - May 16, 2008 - [email protected]


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Wordwalking challenges

  • 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)

  • Wordwalking - the changing of the original question’s wording to convey similar meaning but actually change the problem’s mathematical structure

Carl Lager - May 16, 2008 - [email protected]


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Awareness of potential ambiguity challenges

  • 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).

Carl Lager - May 16, 2008 - [email protected]


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Awareness of potential ambiguity challenges

  • 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)”

Carl Lager - May 16, 2008 - [email protected]


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Metalinguistic awareness challenges

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

Carl Lager - May 16, 2008 - [email protected]


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Solutions II - RC Strategies challenges

1) Activate related prior knowledge and experience.

2) Break down task into smaller chunks.

3) Use visual representations of real-world objects.

4) Predict the problem.

Let’s experience them as learners…use back of blue paper

Carl Lager - May 16, 2008 - [email protected]


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Predict the problem challenges

Carl Lager - May 16, 2008 - [email protected]


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Predict the problem challenges

Carl Lager - May 16, 2008 - [email protected]


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Predict the problem challenges

  • A cycle shop has a total of 36 bicycles and tricycles in stock.

  • Collectively, there are 80 wheels.

  • The bicycles come in 5 colors and the tricycles in 4.

  • How many bikes and how many tricycles are there?

Carl Lager - May 16, 2008 - [email protected]


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Bicycles and Tricycles (Adapted from Adding It Up, 2001; p. 126)

  • A cycle shop has a total of 36 bicycles and tricycles in stock.

  • Collectively, there are 80 wheels.

  • The bicycles come in 5 colors and the tricycles in 4.

  • How many bikes and how many tricycles are there?

Carl Lager - May 16, 2008 - [email protected]


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(Hyde, 2006) 126)

  • 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?

Carl Lager - May 16, 2008 - [email protected]


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Checking Inferences (Hyde, 2006) 126)

  • What inferences did you make?

  • Are the inferences accurate?

  • What information is implied by the problem writer? (Lager)

Carl Lager - May 16, 2008 - [email protected]


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What does this system of equations tell us about this situation?

b + t = 36

2b + 3t = 80

Carl Lager - May 16, 2008 - [email protected]


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What does this system of equations tell us about this situation?

What does b + t = 36 represent?

Carl Lager - May 16, 2008 - [email protected]


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What does this system of equations tell us about this situation?

What does b + t = 36 represent?

Carl Lager - May 16, 2008 - [email protected]


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What does this system of equations tell us about this situation?

b + t =36

What does b + t = 36 represent?

Number of bikes and tricycles

Number of bicycles and tricycles

Carl Lager - May 16, 2008 - [email protected]


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Number of bicycles and tricycles situation?

Number of bicycles and tricycles

Number of bicycles and tricycles

=

36

36

36

=

=

=

Number of bicycles (b) + number of tricycles (t)

b + t

Carl Lager - May 16, 2008 - [email protected]


What does this system of equations tell us about this situation86 l.jpg
What does this system of equations tell us about this situation?

What does 2b + 3t = 80 represent?

Carl Lager - May 16, 2008 - [email protected]


What does this system of equations tell us about this situation87 l.jpg
What does this system of equations tell us about this situation?

What does 2b + 3t = 80 represent?

Number of bicycle wheels and tricycle wheels

Number of wheels

Carl Lager - May 16, 2008 - [email protected]


What does this system of equations tell us about this situation88 l.jpg
What does this system of equations tell us about this situation?

What does 2b + 3t = 80 represent?

Number of bicycle wheels and tricycle wheels

Number of bicycle wheels and tricycle wheels

Carl Lager - May 16, 2008 - [email protected]


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Bicycle wheels situation?

Tricycle wheels

Cognitive organizer

Number of wheels

Carl Lager - May 16, 2008 - [email protected]


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In conclusion… situation?

  • In conclusion…

Carl Lager - May 16, 2008 - [email protected]


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The overriding meaning-making and meaning-sharing challenges situation?

  • 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.

Carl Lager - May 16, 2008 - [email protected]


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What you can do situation?

  • 1) Don’t let teachers do the reading comprehension for the students!

  • 2) Teach students to become active readers of their own mathematics tasks

  • 3) Demand and support long-term, systematic professional development to teach mathematics teachers how to do #2

Carl Lager - May 16, 2008 - [email protected]


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http://www.todos-math.org situation?

  • Let’s help our students by meeting these challenges together. Join TODOS!

Thank you.

[email protected]

Carl Lager - May 16, 2008 - [email protected]


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