Results from the Second Mathematics Assessment
of the National Assessment of Educational Progress
National Council of Teachers of Mathematics. Edited by Thomas P. Carpenter and Mary K. Corbitt.
The two tables below from this assessment involve addition of fractions. The results have serious implications for teaching when we consider that the students answering these questions have been taught fractions for 5 to 6 years.
Table 1: Estimate 12/13 + 7/8
Only 24 percent of the 13-year-olds correctly estimated the sum of 12/13 + 7/8 to be 2. Twenty-eight percent got an answer of 19 by adding the two numerators, and 27 percent got an answer of 21 by adding the two numerators. The answers to this question show that students do not understand fraction notation and suggest that we should not teach operations with fractions until students can describe fraction amounts and compare them to familiar benchmarks such as 1/2 and 1. For example, students should see and be able to explain that if a cake is cut into 13 equal parts and they get 12 parts, or if it is cut into 8 equal parts and they get 7 parts, that in either case they will have almost the whole cake.
Table 2: Sums of fractions esp. 1/2 + 1/3
It is surprising that only 35 percent of the 13-year-olds were able to correctly determine the sum of 3/4 + 1/2, since 1/2 and 3/4 of a dollar are common examples of fractions. The fact that less than half of the students could compute the sum involving the mixed number is due in part to not knowing that 2 2/5 equals 2 + 2/5. The percentage of correct answers for 1/2 + 1/3 have remained fairly consistent for 13-year-olds from one assessment to the next, with about half of the incorrect answers being given as 2/5. This is another example of where fraction symbols do not make sense to some students, for if something is added to 1/2, the result should be greater than 1/2, and yet 2/5 is less than 1/2. The NCTM Standards say that mathematics should be a "sense-making experience", but this is not the case for many students when it comes to using fractions.