The purpose of this investigation is to observe the effects of structural variables on the level of difficulty of missing value proportion (MVP) problems. Recently, Harel and Behr (in press) proposed a MVP problem difficulty analysis in terms of structural variables. Based on this analysis, we hypothesized a problem difficulty hierarchy for MVP problems in terms of two of these variables, the location of the missing value and the coordination of measure spaces. Problems which reflect these two variables were given to twenty-five preservice elementary teachers. The results of this study will be reported and the possible implications for instruction will be addressed.

Tournaire and Pulos (1985) listed two categories of variables which have been identified as influencing the difficulty level of proportion problems: student centered (e.g., age and gender) and task centered. The task centered variables are of two types: contextual (e.g., mixture versus rate) and structural. The structural variables include integer ratio, order of the unknown and numerical complexity. Harel and Behr (in press) defined an additional structural variable, the coordination of measure spaces. In their analysis of missing value proportion (MVP) problems. In this paper we will propose a MVP problem difficultly hierarchy, with respect to order and measure spaces coordination.

Order of the Missing Value is a structural variable which refers to the location of the unknown quantity with respect to the three given known quantities in the wording of the problem. It is determined by the order of the two per statements -- a per b, the closed per statement, and c per ?, the open per statement -- in the problem presentation and by the location of the missing value within the open per statement. This variable has four categories: BL,TL,TRand BR, where B (bottom) and T (top) refer to the order of the per statements and L (left), and R (right) refer to the location of the missing value in the open per statement. In the problem: How many candles cost b cents if c candles cost d cents? the location of the missing value is TL (top left).

Coordination of the measure spaces is a structural variable which refers to the syntactical relationship between the units of measures of the four quantities in a MVP problem. The measure spaces in a MVP problem are coordinated when the order of measure spaces referenced in the two per statements is M1 per M2, M1 per M2; they are uncoordinated when the order of the measure spaces mentioned in the wording of the problem is M1 per M2, M2 per M1. For example, 5 candles cost 10 cents, 20 candles cost how many cents? has coordinated measure spaces, but 5 candles cost 10 cents how many cents does 20 candles cost? has uncoordinated measure spaces.

THE STUDY

Careful consideration was taken to hold the other structural variables constant. These include integer ratio (none, one or two), the number of measure spaces (one or two), the number of dimensions of the measure space (one or two) and the partitionability of the measure space (yes or no). For further explanation of these variables see Harel and Behr (in press). The problems used in this study had no integer ratios, two measure spaces, one dimension in each measure space and partitionable measure spaces. The contextual variables were also held constant. The problems used continuous quantities, were of the rate type, were familiar and involved no use of manipulatives.

Problem Types. Eight different problem types resulted from the manipulation of the two structural variables. Four of the problems had coordinated measure spaces and differed in terms of the order of the missing value. The other four problems had uncoordinated measure spaces and differed in terms of the order of the missing value. Since the only variable manipulated in each set of four problems is the order of the missing value, the order of difficulty for each set is the same. There are two factors in this variable, the order of the per statements and the order of the missing value in the open per statement. In terms of the order of the per statements, it is likely easier to solve problems in which the closed per statement appears first because the solver does not have to change the problem presentation. This means that BR and BL problems will be easier then TR and TL problems. Now in terms of the order of the missing value in the open per statement, it is likely easier to solve problems in which the order of the missing value is on the right than when it is on the left. This is because when the missing value is on the right the equation that is formed is of the type c * v = x (v is the computed value from the closed per statement, and * is either multiplication or division), which is a direct computation; but when the missing value is on the left the equation that is formed is of the type x * v = c, which is an indirect computation (Harel & Behr, in press). This suggests that BR and TR problems will be easier than BL and TL problems. By combining these two orderings, the complete order of difficulty for each set of four problem can be hypothesized as BR < BL < TR < TL, where < denotes easier.

In order to hypothesize a difficulty order for all 8 problem types, it is necessary to decide whether the coordination of the measure space or the order of the missing value is more salient. At this time there is no indication which that would be, therefore the order of difficulty will be between pairs of problems only. That is, the comparison will be between the coordinated and the uncoordinated problems which have the some order of the missing value. It is likely that the coordinated problems will be easier than the uncoordinated problems because when solving a coordinated MVP problem, the equation formed by the solver for the closed per statement has the same structure as the equation for the open per statement, this reasoning suggests the following order of difficulty: BL_c < Bl_u, BR_c < BR_u , TL_c < TL_u and
TR_c <TR_u , where c stands for coordinated measure spaces and u stands for uncoordinated measure spaces.

Subjects. The subjects for this study were twenty-five pre-service elementary teachers who were enrolled in a methods of teaching elementary mathematics course at Northern Illinois University. The subjects were given fourteen problems, eight of which were the MVP problem types described above and six were multiplication, division and addition problems which were included to reduce the likelihood of fortuitous correct answers. They were asked to solve each problem and to show all of their work.

RESULTS AND DISCUSSION

Due to the small sample size, the results indicate the trends identified in the percentage of students who correctly answered the problems. For the coordinated measure space problem types, the results are as follows: TR_c < BR_c < BL_c < TL_c. The results for the uncoordinated measure space problem types are as follows: TL_u < BL_u < TR_u < BR_u. When comparing the problems which have the same order of the missing value but differ in terms of the coordination variable, the results are as follows: TR_c < TR_u , TL_u < TL_c, BR_c < BR_u, and
BL_u < BL_c.

Since the results do not match the proposed order of difficulty, it is necessary to identify the error in the assumptions made. Harel and Behr (in press) stated that when solving MVP problems in which the open per statement appears first, it is necessary to change the problem presentation so that the closed per statement appears first. This process is accomplished through problem transformations and results in a problem representation, the solver's conceptualization of the problem. Since the percentage of correct answers for the problems were relatively high, we can assume that the students were able to change the order of the per statements with little difficulty. Therefore, the order of the per statements may be eliminated when considering problem difficulty in MVP problem presentation. When this is taken into consideration we can combine the results of TR with BR and TL with BL which gives the following results: R_c < L_u < R_u < L_c.

This indicates that if the measure spaces are coordinated the problems are easier when the order of the missing value is on the right than when it is on the left, Rc < Lc. This agrees with the hypothesis. Though, when the measure spaces are uncoordinated the hypothesis appears not to be supported by the data when the missing value was on the left the problems are easier than when it was on the right, L_u < R_u. It will be argued that the hypothesis is supported for the problems which have uncoordinated measure spaces.

When solving MVP problems which have uncoordinated measure spaces, it may be necessary as a first step to use a transformation on the problem in order to change the uncoordinated problem presentation into a coordinated problem representation. There are a couple of ways this could be accomplished. For instance, if the student changes the problem so that the unknown is always on the right then TL_u -> TR_c', BL_u -> BR_c', TR_u -> TR_c' and BR_u -> BR_c' where the -> indicates a transformation and the prime,', indicates the problem structure under the transformation, the problem representation. When these new problem representations are substituted in place of the corresponding problem presentations in the results from the uncoordinated set of four problem types, the following order of difficulty occurs: TRc' < BRc' < TR_c', < BR_c'. This result and the assumption that one problem type has consistent difficulty are contradictory, therefore the assumption that the students change the problem so that the unknown is on the right is probably incorrect. Another possible transformation that could be used when solving an uncoordinated measure space MVP problem would change the order of the quantities in the open per statement. the resulting structure changes are as follows:
TL_u -> TR_c', BL_u -> BR_c', TR_u -> TL_c', and BR_u -> BL_c'. When these new problem representations are substituted in place of the corresponding problem presentations in the results from the uncoordinated set of four problem types, the following order of difficulty occurs: TR_c' < BR_c' < TL_c' < BL_c'. When the order of the per statements is eliminated from this hierarchy, the resulting hierarchy is R_c'< L_c'. That is, when the unknown is on the right the problem is easier than when the unknown is on the left, which agrees with the results from the coordinated MVP problems. When combining the results for R_c with R_c' and L_c with L_c', the results are as follows: R_c < L_c. Thus the assumption that students change the order of the quantities in the open per statement may possibly be correct.

The analysis of the results reveals that the most salient variable, with respect to percentage correct on MVP problems, is the order of the missing value within the open per statement. This variable not only influences the percentage correct, but also the type of transformation that the student uses on MVP problems with uncoordinated measure spaces. It is also true that once students are able to solve MVP problems, the order of the per statements and the coordination of the measure spaces has little or no affect on problem difficulty.

CONCLUSIONS

The results indicate that the order of the missing value and the coordination of the measure spaces may effect problem difficulty for pre-service teachers. A similar study indicates that the coordination of the measure spaces do effect problem difficulty for seventh grade students (Bezuk, 1986). It is doubtful that these issues are addressed in the classroom. It seems that making students aware of these slight variations In MVP problems would be an easy task for the teacher and may improve performance on MVP problems. It will also help students to realize that the MVP problems have many representations and therefore give them more flexibility in their solution strategies. Teachers can easily manipulate the problem presentation with respect to the structural variables and teach the students the possible transformations that can be applied to the problem presentation to make the problem more easy to solve. There are eight types of transformations, which form a mathematical group under the composition operation. For more information refer to Harel and Behr (in press).

In general, since the percentage correct for all these MVP problems was not impressively high (the highest being 66%) and the subjects were preservice elementary teachers, this signals a need to educate these preservice teachers about MVP problems, their representations, the variables which affect problem difficulty and the various solution strategies which are appropriate. A larger study of this design should be pursued in order to validate the findings of this study. Further research should systematically consider all of the variables which have been described in Harel and Behr (in press).

REFERENCES

Bezuk, N. B. (1986). Task variables affecting seventh grade students' performance and solution strategies on proportional reasoning word problems. Unpublished doctoral dissertation, University of Minnesota.

Harel, G. & Behr., M. (In press). Structure and Hierarchy of Missing Value Proportion Problems and Their Representations. Journal of Mathematical Behavior.

Tournaire, F., & Pulos, S. (1985). Proportional reasoning: A review of the literature. Educational Studies in Mathematics, 16, 181-204.