Transcript Are Deaf Students` Answers to Mathematics Word Problems Really

Are Deaf Students’ Answers
to Mathematics Word
Problems Really Illogical?
Fourth Biennial TELA Conference
Ohio School for the Deaf
June 27, 1998
Judy MacDonald, NTID Mathematics Department
Kathleen Eilers crandall, NTID English Department
Can English teachers
contribute to Mathematics
learning?
The students can do the mathematics, at
least in some situations, but have unexpected
difficulties with word problems.
What effect does English have on the
students’ success?
Could this happen in your
class?
Symbols and words have
unique meanings.
Today, we will discuss:
 Explanations
answers
 Strategies
for seemingly outlandish
for understanding six
categories of confusions in order to
maximize student success
1. What is the goal?
 Problem
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?
1. What is the goal?
 Student
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?
1. What is the goal?
 Explanation
The student needs to distinguish -What is currently happening?
What is finished?
and
What is the goal?
Hint
Notice
clues in
verbs.
1. Strategies for promoting
understanding
 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
sense to check answers.
1. Strategies for promoting
understanding
 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
meaning?
 Problem
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:
2. What words show the
meaning?
 Student
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 + 23¼
Key Sequence: 40 23 1 4
Write your answer in a complete sentence: He
need 63¼ to work.
2. What words show the
meaning?
 Problem
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:
2. What words show the
meaning?
 Student
solutions II
• Mary earns $2 an hour more than Andy.
Mary earns $8 an hour. How much does
Andy earn?
Represent: A: how much Andy earns
Translate: 8 + 2 = A
Solve: 10 = A
Write your answer in a sentence: Andy earns $10 an hour.
2. What words show the
meaning?
 Problem
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:
2. What words show the
meaning?
 Student
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 + 10
Solution: 1. 55 + 10 = 1 1. 55
Answer in words: Maria earns $ 1 1. 55.
2. What words show the
meaning?
 Problem
IV
• Translate: two and a quarter
2. What words show the
meaning?
 Student
solutions IV
• Translate: two and a quarter
2. What words show the
meaning?
 Problem
V
• Locate a point on the number line that is
twice as far from -1 as it is from 5.
2. What words show the
meaning?
 Student
solutions V
• Locate a point on the number line that is
twice as far from -1 as it is from 5.
2. What words show the
meaning?
 Explanation:
Students may always translate
some words the same way.
more, raise (“ADD”)
quarter (“$0.25”)
as (“SAME”)
Hint
Words may
have inflexible
meanings.
2. Strategies for promoting
understanding
 Display
data on a number line.
• Mary earns $2 an hour more than Andy.
Mary earns $8 an hour. How much does
Andy earn?
2. Strategies for promoting
understanding
 Practice
using context clues.
 Use clue words (more, less, higher,
lower, greater, smaller) that
correspond to the mathematical
process.
 Foster the use of common sense.
3. The parts do not add up to
the whole!
 Problem
I
• Four times a number is added to its
square. The sum is -1. Find the
number(s).
3. The parts do not add up to
the whole!
 Student
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.)
3. The parts do not add up to
the whole!
 Problem
II
• Shade in the areas represented by the
given fractions:
a) 5/8
b) 2¾
3. The parts do not add up to
the whole!
 Student
solutions II
• Shade in the areas represented by the
given fractions:
a) 5/8
b) 2¾
3. The parts do not add up to
the whole!
 Problem
III
 Construct line segment MN that is three
times as long as line segment PQ below.
3. The parts do not add up to
the whole!
 Student
solutions III
 Construct line segment MN that is three
times as long as line segment PQ below.
3. The parts do not add up to
the whole!
 Explanation
“is added to”
equals |
add
“2¾”
2
| ¾
“three times as long as”
3 times | same length
Hint
Parts remain
individual
entities.
3. Strategies for promoting
understanding
 Teach
students how to analyze
problems.
 Ask students to compare
information in a problem.
 Encourage students to use
common sense.
 Be aware of possible interlanguage
confusions.
4. An end hath no means.
 Problem
I
• Translate and solve: one hundredth
divided by fifty.
4. An end hath no means.
 Student
solutions I
• Translate and solve: one hundredth
divided by fifty.
100
50

4. An end hath no means.
 Problem
II
• Select the fourth largest number: 50, 20,
75, 15, 32, 64, 4, 84.
4. An end hath no means.
 Student
solutions II
• Select the fourth largest number: 50, 20,
75, 15, 32, 64, 4, 84.
50 75 64 84
4. An end hath no means.
 Explanation
• Word endings may not be
meaningful for students.
Hint
Endings
change
numbers.
4. Strategies for promoting
understanding
 Emphasize
word endings that
change number values.
 Provide additional practice to
recognize endings such as -th, -st,
-est, -er.
5. Just plain bad
 Problem
• Find the product of six and seven eighths
and twelve.
Translate:
Solve:
Key sequence:
5. Just plain bad
 Student
solutions
• Find the product of six and seven eighths
and twelve.
Translate: 6 12
Solve: = 63
Key sequence: 6 7 8 12
5. Just plain bad
 Explanation
• The problem is ambiguous.
Hint
Ambiguous
problems
are unfair.
5. Strategies for promoting
understanding
 Read
and re-read problems.
 Cluster information in problems.
Example: Find the product of two
numbers: twelve, and six and seveneighths.
 Avoid
ambiguous problems.
6. What did you order?
 Problem
• 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:.
6. What did you order?
 Student
solutions II
• Mary earns $2 an hour more than Andy.
Mary earns $8 an hour. How much does
Andy earn?
Represent: A: how much Andy earns
Translate: 2 + 8 = A
Solve: 10 = A
Write your answer in a sentence: Andy earns $10 an hour.
6. What did you order?
 Explanation
• Students prefer the order the
numbers have in the problems.
• Students may overlook
words that signal
order.
Hint
Order
is critical.
6. Strategies for promoting
understanding
 Practice
ordering information on a
number line.
 At first, restate with left to right
order. Example: Mary earns $8 an hour. Andy earns
$2 less than Mary. How much does Andy earn?
 Then,
practice orders that violate
the left to right principle.
 Pay particular attention to words
such as: after, before, now, then, until.
Summary






Words may have inflexible meanings.
Endings change numbers.
Ambiguous problems are unfair.
Parts remain individual entities.
Order is critical.
Notice clues in verbs.
Presenters:
Judy MacDonald
Kathleen Eilers crandall
NTID Mathematics Department
NTID English Department
Rochester Institute of Technology
Rochester Institute of Technology
Rochester, NY 14623
Rochester, NY 14623
Phone: (716) 475-6028
Phone: (716) 475-5111
Fax: (716) 475-6500
Fax: (716) 475-6500
Email: [email protected]
Email: [email protected]
Web: http://www.rit.edu/~kecncp