Here are possible explanations for how the students got the wrong answers in A through F: A. The student subtracted the numbers incorrectly as integers instead of decimals: 6 - 4 = 2 instead of 6.25 - 4 = 2.25. B. The student added the fractions as integers instead of converting to a common denominator first. C. The student subtracted the decimals incorrectly, likely doing 42 - 6.7 instead of 42.13 - 6.7. D. The student likely just wrote down the original first number instead of performing the subtraction. E. The student likely subtracted the second number from the first instead of vice versa. F. The student likely divided the first number by the

1. First example:
Teacher: Is 25% of 15 greater, less than, or equal to 15?
Student: It is less than 15.
Teacher: Would you explain how you decided on that answer?
Student: You subtract. 25%-15=10, and 10 is less than 15.
Second example:
1986, National Assessment of Educational Progress
1. Which of the following is true about 87% of 10?
A. It is greater than 10
B. It is smaller than 10
C. It is equal to 10
Correct answer: 45% of the students (7th and 8th grade students)
2. Explain how you decided on your answer
Of the 45% who had the correct answer, about half of them wrote an
explanation of an appropriate solution process.
About ¼ did not write any explanation.
About ¼ gave inappropriate explanations.
Examples of answers:
A. “100 % is all of 10 and 87% is smaller than 100%, so 87% is
not all of 10 so it’s smaller.”
B. “10 is a whole number; 87% = .87; .87 is smaller than 10.”
C. “50% of 10 would be 5. 100% of 10 would be 10. 87% of 10
would be between 5 and 10 so the answer is less than 10”
D. “I thought about how much 87% would be of 10. Almost to the
nine on a scale from one tot ten. It was less than 10”
E. “ There’s 8 as you count up to 10, so I think it’s less.”
F. “Because 10% of 87 is 8.7, and 8.7 is less than 10”
G. “You take the numbers and see how many times the smaller
number goes into the larger number”

2. A. 6.25 - 4 = 6.21
2 1 3
B. + =
5 4 9
C. 42.13 – 6.7 = 36.6
D. 42.13 – 6.7 = 44.63
E. 47.1 – 0.65 = 22.1
F. 47.1 – 0.65 = 4.06
For A. to F., figure out how the student got to the wrong answer.
It might help to write then numbers under one another, just the way
you would do to calculate the answer on scratch paper.