Assessing Students
In assessing students, it is important to consider a problem and find the best strategy to solve it. Consider the following problem: Jane and Judy were running at the same speed around the school track. Jane was the first to start the race. Judy started the race and when she had completed 3 laps, Jane had completed 9 laps. When Jane has finished 15 laps, how many laps had Judy completed? Thirty –two out of 33 pre-service basic educators in a mathematics methods class worked this problem by setting up and evaluating a proportion: 9/3=x/15=135; x=45 .While these learners were conversant with the procedure for evaluating a proportion ,they failed to recognize the fact that the question did not represent a proportional problem. Consequently, the customary proportion algorithm was no suitable strategy to apply. The problem at hand was one of the two situations given to the learners as part of introduction to the teaching of facts on ratios and proportion. The other situation entailed currency exchange problem: 3 U.S dollars can be exchanged for 2 U.K pounds .How many U.K pounds can be exchanged for 21 U.S dollars? This kind of a problem is a proportional one, and all the learners solved the problem correctly using the traditional method of algorithm .In the situation, no single student was able to give an explanation why this problem revealed a proportional problem whereas the racing lap’s situation did not.
Apparently the two problems are similar; each of the problems comprised of three pieces of facts each with one unknown variable. The system does provide itself to a proportion: a/b=c/x where a, b, and c are the known variables and x is the unknown variable. The greater variability between the two problems shows what is unique concerning proportional problems. At this point it is easier to conclude that it not easy to define proportional reasoned as one who understands how to set and resolve a proportion. Research on proportions enables one expand the scope of understanding of proportional reasoning which in turn persuades classroom instructions and strategies.
The crucial element of proportion problems is the multiplicative relationship that subsists among the variables that represent the problem. In the problem of laps, the link between the number of laps Jane ran and the one Judy ran can be represented using addition or subtraction. Jane’s laps=Judy’s laps+6; Judy’s laps =Jane’s laps-6.In the problem concerning money exchange, can be worked out through multiplication: pounds=2/3 U.S dollars or U.S dollars =3/2 pounds. The running problem is not a proportional problem in nature whereas the currency exchange problem is proportional. According French psychologist, Gerard Vergnaud, applied the concept of measure of space to enable one to figure out the nature of multiplicative link that exists in proportional problem. A measure space can be connected to physical degree which is quantified as with 2cm,5 dollars or 3kg .A proportion can be then be seen as a multiplicative connection between the two variables in two measure spaces. Vergaud’s notation specifies that :
M1 and M2 shows any two measure spaces while a, b, c and d are the variables that form rates in a proportion problem. For instance the currency problem can be represented using the above concept of measure space. If 2 pounds can be exchanged for 3U.S dollars, then it follows that 14 pounds can be exchanged for 21 U.S dollars. This can represent in measure space as follows.
Dollars Pounds
The concept of proportional situations, the variables across or between measures spaces are always linked by multiplication:
- Dollars
- Pounds
In each scenario, the quantity of pounds can be derived by multiplying the number of dollars by2/3 where this becomes a constant for the expression for this proportional problem y=2/3 x where y represents pounds and x dollars .The constant factor in this case represents the number of pounds for every dollar: 2/3 pound for every 1 dollar which is known as unit rate. The characteristic of proportional problem graphs is that it passes through the point of origin and is a straight line which always leans toward the right. The other characteristic of proportional problems can be represented by use of fractions. In the example which deals with currency exchange, the relationship of pounds to U.S dollars can be represented as 2/3,4/6,6/9,8/12 .The fractions are equivalent an always decrease to the same number :2/3, a special multiplicative connection among the aspects within each measure space. Beginning with 3 U.S dollars for 2 pounds, if the number of dollars is multiplied by two and the equivalently the number of pounds by 2 then this leads to another rate pair in the table: 6:4. Proportional link is just a kind of relationship which can exist within two sets of variables.
Percentages of Correct Solutions by Strategies for and Numerical Comparison (NC) and Missing Value (MV) Word Problems.
In dealing with the proportional problems, unit rate strategy became the most popular which resulted into many learners getting the answers correct (15). This method used the multiplicative approach between measure spaces. The unit rate was got through division. The unit rate is a constant factor that relates two variables. The approach was common with the seventh-grade learners who had no prior instruction in the normal cross-multiply and divide algorithm .The approach was popular was friendly to learners because it touched the usual experiences. A few numbers of learners applied the fraction strategy but was used by larger number of eighth –grade students. The fraction strategy approach is applied without considering the context of the problem whereby the student using this strategy would calculate the answer, using the multiplication rule to get corresponding fractions as follows:
If 6/120=6/? Then 6/120x2/2=12/240
The percentage of the right answers for single quantities that did not apply integral multiples was relatively minimal than situations with integer connection. The existence of non-integer in the problem has two impacts: first it considerably lowers the level of student attainment, and second it changes the thinking of the student in approaching a problem. This is clearly indicated by the lower number of students who used the unit rate and factor methods in solving the problems. It is evident form the last row of the above table that large number of students in the seventh-grade and eight-grade were not in a position to solve these problems right. The missing data in the table shows that scaling problems were considerably trickier than density, speed, or buying questions.
