Are Deaf Students’ Answers to Mathematics Word Problems Really Illogical? Fourth Biennial TELA Conference Ohio School for the Deaf June 27, 1998 Judy

Are Deaf Students’ Are Deaf Students’ Answers to Mathematics Answers to Mathematics Word Problems Really Word Problems Really Illogical?Illogical?

Fourth Biennial TELA ConferenceFourth Biennial TELA Conference

Ohio School for the DeafOhio School for the Deaf

June 27, 1998June 27, 1998

Judy MacDonald, NTID Mathematics Judy MacDonald, NTID Mathematics DepartmentDepartment

Kathleen Eilers crandall, NTID English Kathleen Eilers crandall, NTID English DepartmentDepartment

Can English teachers Can English teachers contribute to contribute to Mathematics learning?Mathematics learning?

The students can do the mathematics, The students can do the mathematics, at least in some situations, but have at least in some situations, but have unexpected difficulties with word unexpected difficulties with word problems. problems.

What effect does English have on the What effect does English have on the students’ success?students’ success?

Could this happen in your Could this happen in your classclass??

Symbols and words have Symbols and words have unique meaningsunique meanings. .

Today, we will discuss:Today, we will discuss:

Explanations for seemingly Explanations for seemingly outlandish answersoutlandish answers

Strategies for understanding six Strategies for understanding six categories of confusions in order to categories of confusions in order to maximize student successmaximize student success

1. What is the goal?1. What is the goal?

ProblemProblem I am digging a hole 7/8 meters deep. I dug 1/2

meter this morning, 1/6 meter this afternoon, and I plan to dig 1/16 meter tomorrow morning. • After tomorrow morning, how much will I have dug?

• How much will I have left to dig?

Student solutionsStudent solutions I am digging a hole 7/8 meters deep. I dug 1/2

meter this morning, 1/6 meter this afternoon, and I plan to dig 1/16 meter tomorrow morning. • After tomorrow morning, how much will I have dug?

• How much will I have left to dig?



ExplanationExplanation

The student needs to distinguish --The student needs to distinguish --

What is currently happening? What is currently happening?

What is finished? What is finished? andand

What is the goal?What is the goal?

HintHint Notice Notice

clues in clues in

verbs.verbs.

1. Strategies for 1. Strategies for promoting understandingpromoting understanding

Add redundancy for verbs.Add redundancy for verbs.Original: I am digging a hole 7/8 meters deep. I dug 1/2

meter this morning, 1/16 meter this afternoon, and plan to dig 1/16 meter tomorrow morning.

New: My job is to dig a hole that is 7/8 meters deep. Today, I finished digging 1/2 meter in the morning and 1/16 meter in the afternoon. Tomorrow, I plan to dig 1/16 meter.

Teach students to use common Teach students to use common sense to check answers.sense to check answers.


Make use of illustrations.Make use of illustrations. My job is to dig a hole that is 7/8 meters deep. Today, I finished

digging 1/2 meter in the morning and 1/6 meter in the afternoon. Tomorrow, I plan to dig 1/16 meter.

• After tomorrow morning,

how much will I have dug?

• How much will I have left

to dig?

2. What words show the 2. What words show the meaning?meaning?

Problem IProblem I • John must work 40 hours this week. So

far he has worked 23¼ hours. How many more hours does he need to work? Translate:

Key Sequence:

Write your answer in a complete sentence:


Student solutions IStudent solutions I • John must work 40 hours this week. So

far he has worked 23¼ hours. How many more hours does he need to work? Translate: 40 + 23¼Key Sequence: 40 23 1 4 Write your answer in a complete sentence: He need 63¼ to work.


Problem IIProblem II• Mary earns $2 an hour more than Andy.

Mary earns $8 an hour. How much does Andy earn?

Represent:

Translate:

Solve:

Write your answer in a sentence:


Student solutions IIStudent solutions II• Mary earns $2 an hour more than Andy.


Represent: A: how much Andy earnsTranslate: 8 + 2 = A Solve: 10 = A Write your answer in a sentence: Andy earns $10

an hour.


Problem IIIProblem III• If Maria gets a raise of $1.55 per hour,

she will be earning $10 an hour. How much does Maria earn now?

Translation:

Solution:

Answer in words:


Student solutions IIIStudent solutions III• If Maria gets a raise of $1.55 per hour,

she will be earning $10 an hour. How much does Maria earn now?

Translation: 1. 55 + 10Solution: 1. 55 + 10 = 1 1. 55Answer in words: Maria earns $ 1 1. 55.


Problem IVProblem IV

• Translate: two and a quarter


Student solutions IVStudent solutions IV

• Translate: two and a quarter


Problem VProblem V• Locate a point on the number line that is

twice as far from -1 as it is from 5.


Student solutions VStudent solutions V• Locate a point on the number line that is

twice as far from -1 as it is from 5.


Explanation:Explanation: Students may always translate Students may always translate

some words the same way. some words the same way.

more, raisemore, raise (“ADD”) (“ADD”)

quarter quarter (“$0.25”)(“$0.25”)

as as (“SAME”)(“SAME”)

HintHint Words mayWords may

have inflexiblehave inflexible

meanings.meanings.


Display data on a number line.Display data on a number line.• Mary earns $2 an hour more than Andy.



Practice using context clues.Practice using context clues. Use clue words (Use clue words (more, less, more, less,

higher, lower, greater, smallerhigher, lower, greater, smaller) ) that correspond to the that correspond to the mathematical process.mathematical process.

Foster the use of common Foster the use of common sense.sense.

3. The parts do not add up 3. The parts do not add up to the whole!to the whole!

Problem IProblem I• Four times a number is added to its

square. The sum is -1. Find the number(s).


Student solutions IStudent solutions I• Four times a number is added to its

square. The sum is -1. Find the number(s).

4n = n2 (Then, the student is unable

to progress to the next sentence.)


Problem IIProblem II• Shade in the areas represented by the

given fractions:a) 5/8 b) 2¾


Student solutions IIStudent solutions II• Shade in the areas represented by the

given fractions:a) 5/8 b) 2¾


Problem III Problem III Construct line segment MN that is three

times as long as line segment PQ below.


Student solutions III Student solutions III Construct line segment MN that is three

times as long as line segment PQ below.


ExplanationExplanation “ “is added to” is added to”

equals | add equals | add

“ “2¾” 2¾” 2 | ¾2 | ¾

“ “three times as long as”three times as long as”3 times | same length3 times | same length

HintHint Parts remainParts remain

individualindividual

entities.entities.


Teach students how to analyze Teach students how to analyze problems.problems.

Ask students to compare Ask students to compare information in a problem.information in a problem.

Encourage students to use Encourage students to use common sense.common sense.

Be aware of possible Be aware of possible interlanguage confusions.interlanguage confusions.

Problem IProblem I• Translate and solve: one hundredth

divided by fifty.

4. An end hath no means. 4. An end hath no means.

Student solutions IStudent solutions I• Translate and solve: one hundredth

divided by fifty.

100 50


Problem IIProblem II• Select the fourth largest number: 50, 20,

75, 15, 32, 64, 4, 84.


Student solutions IIStudent solutions II• Select the fourth largest number: 50, 20,

75, 15, 32, 64, 4, 84.

50 75 64 84


ExplanationExplanation• Word endings may not be Word endings may not be

meaningful for students.meaningful for students.


HintHint Endings Endings

changechange

numbers.numbers.

Emphasize word endings that Emphasize word endings that change number values.change number values.

Provide additional practice to Provide additional practice to recognize endings such as recognize endings such as -th, -th, -st, -est,-st, -est, -er.-er.


Problem Problem • Find the product of six and seven eighths

and twelve.Translate:

Solve:

Key sequence:

5. Just plain bad5. Just plain bad

Student solutions Student solutions • Find the product of six and seven eighths

and twelve.Translate: 6 12Solve: = 63Key sequence: 6 7 8 12


ExplanationExplanation• The problem is ambiguous.The problem is ambiguous.


HintHint AmbiguousAmbiguous

problemsproblems

are unfair.are unfair.

Read and re-read problems.Read and re-read problems. Cluster information in Cluster information in

problems.problems. Example: Example: Find the product of two Find the product of two

numbers: twelve, and six and seven-numbers: twelve, and six and seven-eighths.eighths.

Avoid ambiguous problems.Avoid ambiguous problems.


Problem Problem • Mary earns $2 an hour more than Andy.


Represent:

Translate:

Solve:

Write your answer in a sentence:.

6. What did you order? 6. What did you order?

Student solutions IIStudent solutions II• Mary earns $2 an hour more than Andy.


Represent: A: how much Andy earnsTranslate: 2 + 8 = A Solve: 10 = A Write your answer in a sentence: Andy earns $10

an hour.


ExplanationExplanation• Students prefer the order the Students prefer the order the

numbers have in the problems.numbers have in the problems.• Students may overlook Students may overlook

words that signal words that signal order.order.


HintHint Order Order

is critical.is critical.

Practice ordering information on Practice ordering information on a number line.a number line.

At first, restate with left to right At first, restate with left to right order. order. Example: Example: Mary earns $8 an hour. Andy earns

$2 less than Mary. How much does Andy earn? Then, practice orders that violate Then, practice orders that violate

the left to right principle.the left to right principle. Pay particular attention to words Pay particular attention to words

such as: such as: after, before, now, then, until.after, before, now, then, until.

6. Strategies for 6. Strategies for promoting understanding promoting understanding

WWords may have inflexible ords may have inflexible meanings.meanings.

EEndings change numbers.ndings change numbers. AAmbiguous problems are unfair.mbiguous problems are unfair. PParts remain individual entities.arts remain individual entities. OOrder is critical.rder is critical. NNotice clues in verbs. otice clues in verbs.

SummarySummary

Presenters:Presenters:

Judy MacDonald

NTID Mathematics Department

Rochester Institute of Technology

Rochester, NY 14623

Phone: (716) 475-6028

Fax: (716) 475-6500

Email: [email protected]

Kathleen Eilers crandall

NTID English Department

Rochester Institute of Technology

Rochester, NY 14623

Phone: (716) 475-5111

Fax: (716) 475-6500

Email: [email protected]

Web: http://www.rit.edu/~kecncp

Documents

Are Deaf Students’ Answers to Mathematics Word Problems Really Illogical? Fourth Biennial TELA Conference Ohio School for the Deaf June 27, 1998 Judy