

RATIONAL
NUMBER KNOWLEDGE OF PRESERVICE ELEMENTARY Kathleen
CramerUniversity of Wisconsin at River Falls ABSTRACT This study assessed the rational number knowledge of 48 elementary education majors. The test used, in part, a subset of items from a testing instrument developed by the Mathematics Leadership Project at Northern Illinois University and University of Minnesota. The paper discusses results of the test and presents sample test items. The data suggest that some 20% of the preservice teachers who took this test do not understand rational number ideas well enough to be expected to teach fractions meaningfully. 

Introduction A subset of the teacher profile developed as part of the Mathematics Leadership Project (see paper in these proceedings by LaCampagne, Post, Harel, and Behr) was administered to preservice teachers in elementary education enrolled in an elementary mathematics methods class at the University of WisconsinRiver Falls. The purpose of the testing was to assess students' conceptual understanding of rational number ideas. Many of the test items reflect content taught to fourth graders participating in the National Science Foundation sponsored Rational Number Project (Behr, Lesh, Post & Silver, 1983). The test assessed content that these preservice teachers will shortly be leaching upon completion of their degree. Participants Fortyeight junior and senior level students were given a 45 item fraction test covering the categories listed in Table 1. All but one student had completed two, 3 quarter credit mathematics classes designed for elementary education majors. These two courses met the mathematics requirements for the University Fiftyfour percent of the students had taken no other college level mathematics class; 13 percent had taken one or two remedial classes (below college algebra); 29 percent had taken classes ranging from college algebra to calculus; 4 percent " had a mathematics minor. Test Results Table 1 Results of Fraction Test Given to Preservice Elementary Education Teachers 





Test scores ranged from 10 items correct to 44. The overall mean correct was 63 percent. While 10 students (20%) answered more than 84% of the items correctly, another ten students answered less than 42% of the items correctly. Upon completion of the mathematics techniques class these twenty students will have completed all the required mathematics preparation needed to teach elementary school. But, unfortunately, there will be ten students who do not know (at least with respect to fractions) ideas that they will be certified to teach. It is unlikely that the mathematics techniques class they were enrolled in would have been sufficient to remediate their mathematics deficiencies. Rational Number Test The items selected for the test emphasized conceptual understanding of rational number ideas that should be part of the elementary school curriculum. Even though many of the items could have been solved procedurally, if the preservice teacher forgot the particular procedure, he/she could have relied on intuitive, less formal reasoning to solve the problem. For example, to determine the larger of this fraction pair, 16/31 or 9/19, changing both fractions to equivalent fractions with the same denominator is an unnecessary and tedious step. A decision can be reached by comparing to 1/2; 16/31 is more than 1/2 while 9/19 is less than 1/2. Results from the Rational Number Project (Post & Cramer, 1987) showed that fourth graders who participated in the rational number teaching experiment which emphasized the development of conceptual understanding over procedural skill did order fractions pairs as the one above by comparing the fractions to onehalf. The fraction division problems were also solvable by relying on conceptual understanding of division. For example, 5/4 divided by 1/4 could be solved by thinking: How many 1/4's in 5/4? The least successful division problem (40% correct) was 1/2 divided by 4. While students struggled to find a procedure to solve the problem, they could have solved it by drawing a picture: 

. If they divided the region from zero to 1/2 into 4 equal parts, each part would be 1/8. Students also could have thought of a number to multiply 4 by to obtain 1/2.  
Examples of items from
the different categories are described below. Percent correct for the particular
sample item is also noted in the parentheses after the test question.
PartWhole Concepts




Here are 7 number lines. All have a length of 2 units. The top line is labeled A, E C, D, and E. In the space below write the numbers represented by each letter.  
Ordering Choose the larger fraction: 



Name a proper fraction between 7/8 and 1 (54%)  
This last problem shows students' reliance on procedures. The most common incorrect answer was the fraction 14/16 which, of course, was generated by multiplying the numerator and denominator by two. Other incorrect answers included, 8/8, 5/6, 4/8, 7/9.  
Fraction
Equivalence




The lack of success
with these items reflect the students discomfort with having fractions
within a fraction. One student's comment that he thought you could not
have fractions in the numerator most likely characterizes the students'
misunderstanding. The students were more successful with the second problem
because the fraction 8/14 could be reduced to 4/7. Now the problem 4/7 a
/21 can be solved by multiplying numerator by the whole number 3 to obtain
a whole number answer of 12.
Concept of Unit




Below is 3/4 of some unit. Show 2/3. (56%) 



The concept of unit
questions test a more indepth understanding of fraction concepts. To answer
the first problem, for example, the student needs to reflect on what 3/7
means (1/7 + 1/7 + 1/7). By partitioning the picture into 3 equal groups
the student sees that 1/7 equals 3 circles. Since 7/7 makes up the whole
unit, 21 circles is the correct answer.
The third problem was more difficult because of the arrangement of the circles into two groups of three. Behr and Post (1981) called these problem types perceptual distracters and suggested that the ability to solve these problem types was one index of the stability of the learner's rational number concepts. If the student disregarded the way that the circles were clustered, the problem is essentially the same as the first problem (3/2 = 1/2 + 1/2 + 1/2 so 2 circles equals 1/2; now 2/2=4 circles). The two most common incorrect answers were 3 and 6. Fraction Division 



Though the fraction division problems were presented as complex fractions, most students knew to interpret them as division problems. Students struggled to find some procedure to solve the problems, but were inconsistent with the procedure they used. For example, some students who successfully answered the problems when the denominator was a fraction made an error on the problem where the denominator was a whole number. The division problem, (1/2)/ 4 , was changed to [1/2 x 4/1 = 4/2 = 2]. If the students reflected on the answer they would have seen the unreasonableness of it. A proper fraction divided by a whole number must result in an answer less than the fraction. The incorrect strategies applied to these questions yielded mostly unreasonable answers. The students did not attempt to assess the validity of their procedure based on the answers it generated. Fraction Story Problems Mary has 1/2 yard of ribbon. She cuts it into 4 equal lengths. How long is each length? (73%) The same numbers used in fraction division were embedded in fraction story problems. Interestingly, the most successful category was the fraction story problems (78% correct), while the second least successful category was fraction division (51%). Students had more understanding of fraction division than would have been attributed to them given the fraction division results. These preservice teachers had not thought to use their informal understanding of division to solve a problem presented in symbolic form but did so within a story context. No student actually wrote a division number sentence to solve this story problems. When given a problem in symbolic form, they only thought to solve it procedurally. Qualitative Questions What will happen to the fraction 3/15 if the numerator is increased while the denominator slays the same? (83%)
Students were relatively successful on three of the four items in this category. When the question staled that the numerator and denominator were both increased, success level dropped to 40%. Since the answer depends on knowing the increase is proportional or not, the correct answer was choice d. there is not enough information to tell. Summary Students were not as successful on this lest as educators wouldn't expect them t be. As with the data from the Mathematics Leadership Project, there is a group of potential teachers (20% in this study) who do not have the mathematics background sufficient enough to teach fractions with understanding. As mathematics educators continue to suggest changes in the elementary curriculum, and to continue to emphasize teaching for conceptual understanding, we in the position of training teachers need to be sure that new teachers have the needed mathematics understandings themselves to teach the mathematics we want them to teach in the way we want them to teach. REFERENCES Behr, M., & Post, T. (1981). The effect of visual perceptual distractors on children's logicalmathematical thinking in rational numberssituations. In T. Post & M. P. Roberts (Eds.), Proceedings of the Third Annual Meeting of the North American chapter of the International Group for the Psychology of Mathematics Education 816. Behr, M., Lesh, R.. Post. T. & Silver, E. (1983). Rational number concepts. In R. Lesh & M. Landau (Eds.). Acquisition of mathematics concepts and processes. New York: Academic Press, 961. Post, T. & Cramer, K. (1987). Children's strategies in ordering rational numbers. Arithmetic Teacher. 28 (9), 26  31. *This research was supported in part by the National Science Foundation under grant TEI8652341. Any opinions, findings and conclusions expressed are those of the authors and do not necessarily reflect the views of the NSF. 
