Middle school students' understanding of number sense related to percent

School Science and Mathematics, Jan 1997 by Gay, A Susan, Aichele, Douglas B

A different question showed six circles with the first five shaded. For one seventh-grade student, the number of shaded objects was the same as the percent. In this case, since five circles were shaded, 5% was shaded. When asked what percent would be shaded if all six circles were shaded, he stated that 6% would be shaded. The interviewer asked him to think about the six circles with only three shaded. In that case, he stated that one-half of the circles would be shaded, and that would be 3%.

The last question in Part 1 pictured six circles with three shaded and asked the student to compare the shaded part with 87%. Two seventh-grade students chose "equal to 87%" from the multiple-choice responses. Their identical explanations noted that onehalf or three circles were shaded while the other half or three of the circles were not shaded, and so it was equal. Neither student, in this explanation, mentioned 87%.

Interpreting a quantity expressed as a percent of number. Some students showed a solid understanding of the meaning of a quantity expressed as a percent of a number, while others revealed a state of confusion, as well as a number of inventive reasoning strategies. Many students with a good concept of 100% as the whole often used this fact to explain why " 110% of 145 is greater than 145." Several students explained that 110% of 145 is 10% more than 145. When asked to estimate how much 110% of 145 would be, a common response was ten more or 155. Others responded that it would be a little bit more than 145, or gave a range of 10 to 20 more than 145.

While some students showed an understanding of 100% and 110%, others demonstrated a lack of understanding. Stating that 110% of 145 was less than 145, one seventh-grade student drew a rectangle to use as a model, labeled the middle of the rectangle as 150%, and noted that 110% was less than 150%. Some students who claimed that "110% of 145 was less than 145," were asked what percent of 145 would be equal to 145. Two students responded that 145% of 145 would be 145.

Fractional relationships were used by some students who noted that 50% was 1/2 and 25% was 1/4. In determining that 25% of 15 was less than 15, one seventh grader said "there's four 25 percents to be 100, so take four into 15." This same student decided that 33 1/3% of 30 was less than 30 because "you could get about three of those into 30."

For students who were confused or had difficulty answering the questions on this part of the test, the use of a visual model was often helpful. Some students selected their own models, while the interviewer offered a possibility for others. A seventh-grade student used his ten fingers as a model for 100% and illustrated 33 1/3% by showing three fingers and part of a fourth finger. For other students, a suggested model was something familiar and real to them, such as a number of soccerballs, football passes, or test questions. Using these models, the students could often correctly answer questions to which they had previously given no answer.