PROPORTIONAL REASONING AMONG 7TH GRADE STUDENTS WITH DIFFERENT CURRICULAR EXPERIENCES
David Ben-Chaim, James T. Fey, William M. Fitzgerald, Catherine Benedetto and Jane Miller
New curricula and teaching strategy have been developed for the topics of middle school mathematics. The Connected Mathematics Project (CMP) is one of the new middle grades curriculum projects to develop a complete mathematics curriculum with teacher support materials for grades six, seven, and eight. This curriculum is structured to develop students’ knowledge and understanding of mathematics that is organized around interesting problem settings. Students solve problems and observe patterns and relationships. The CMP curriculum treats the major areas of mathematics like number, geometry, measurement, algebra, statistics, and probability. In the seventh grade, three units deal with the ideas of similarity, ratio, and proportions and their applications. The approach in those units is to encourage students to construct their own procedures for doing rational number computations, solving proportions, and applying those skills to applied problem solving. Students collaborate in work on the problems, sharing through mathematical reflections, discussion, and journal writing.
In a traditional curriculum, each arithmetic operation is taught with a focus on developing student proficiency in well-defined computation algorithms to ensure speed and accuracy of execution. The text material provides problems and the teacher demonstrate the solutions to sample of problems to the students. After that, students solve problems according to the given format.
There are three focuses in this research comparing between traditional and CMP curriculum:
a. How do the conceptual understandings, computational skills, and problem solving strategies and success of CMP and traditional curriculum students compare?
b. Does the new CMP approach do successfully lead students to construct effective strategies for fraction, decimal, percent, ratio, and proportional computation?
c. Do CMP students develop flexible and/or effective strategies for solving contextual problems involving rational numbers and proportions?
2. PROPORTIONAL REASONING
A mathematician said that a proportion is a statement of the equality of two ratios. The students’ ability to reason proportionally develop through-out grades 5-8. Following ideas of Frudenthal (1978,1983), proportional reasoning problems can be described in three broad categories: a) comparing two part of a single whole; b) comparing magnitudes of different quantities with an interesting connection; and c) comparing magnitudes of two quantities. Three types of tasks are reported in the literature for assessing proportional reasoning: a) Missing value problems; b) Numerical comparison problems; c) Qualitative prediction and comparison problems. Freudenthal (1978,1983) has pointed out that missing value and comparison proportion problems can be solved by three distinguishable approaches related to: a) Internal ratio (within a magnitude); b) external ratio (between two magnitudes); c) refraining from computation until the result has been found formally.
3. PURPOSE OF THE STUDY
The goal of this study is to describe the character and effectiveness of proportional reasoning by students with different curricular experiences. The main purpose was to compare CMP and traditional curriculum students in American middle schools about how they proportional reasoning.
The data was gathered during the first school years (1994-1995) in which CMP sample consisted of students from eight seventh grade classes and seven difference teachers in some areas in America. The control sample consisted of students from six seventh grade and six difference teachers also in America. The equivalence indication of the two samples was given by the standardized test results reported by evaluation team (CMP and Control). These results show that the Control student’s scores were slightly higher than the CMP students at the beginning of the year, and slightly lower at the end of the year. 187 students were in the CMP sample and 128 students in the Control sample.
Both samples were tested on variety of proportional reasoning problems and were distributed randomly in each participating class. Researchers interviewed about 25% students to gain more knowledge about their understanding. There are five rate of density problems that did in both samples about numerical comparison, missing value problem, integer number structural, fraction and structural number, and population density information. There are three major categories that were identified: Correct answer (correct answer, correct answer with correct support work, and correct answer with incorrect support work), Incorrect answer (incorrect answer, incorrect answer with partial understanding, and incorrect thinking), and no response.
5. RESULT AND DISCUSSION
Most of the students from both samples responded to the problems with support work. Analysis of the written support showed that the CMP students demonstrated much more proficiency in writing than Control students. CMP students were more frequently instructed to write and talk about the idea. Incorrect answer with partial understanding often occurred when the problem called for calculations followed by reasoning.
– Comparing students’ performance on rate problems 1 and 2
The first two rate problems (a numerical comparison and finding a missing value) required the same knowledge of rate, relating to amount and cost. The performance on the missing value problem was better than on the numerical comparison. The missing value problem may be easier because it consisted of two identical parts, each less complex than the numerical comparison problem. The CMP students performed well on the missing value problem because there is possibility of effects o their performance on the comparison problem. One’s evidence is the high percentage of CMP students using unit rate for the comparison. The CMP students performed significantly better than the Control students on the first two rate items.
• A mini case study
There were nine different strategies based on analysis of the problem task and students’ written test papers and individual interviews for solving the numerical comparison rate problem:
1. Comparing the ratio of two different variables using external ratio or a functional method.
2. Comparing ratios of the same variable using internal ratios or a scalar method.
3. Comparing the cost of the some quantity by finding common factor or common multiple quantities such as price per unit.
4. Comparing amounts for the some cost by finding common factor or common multiple costs such as unit per price.
5. Building up strategy.
6. Looking at ratios of differences between the same variables.
7. Responding to the numbers but not the context of a given problem
8. Relating to only one variable by ignoring part of the data in the problem.
9. Affective responses to numerical data and question.
– Comments on rate problem 2
Analysis of students’ responses to the second rate problem also showed use of a variety of strategies to solve the problem. There were still many students who manipulated numbers in ways that did not reflect the structure of the given information or the question. All students from both samples applied the same strategies for both parts of the problem. Very few were successful on one part and not on the other. The students’ thinking is essentially additive when they build up the smaller quantity to the larger one in equal increments.
– Comparing students’ performance on rate problems 3 and 4
Both of this numerical comparison’ problems require the same knowledge of rate relating to distance and time. The second problem was more difficult because it contains both decimal and fraction numbers for the time. Most of the incorrect answers for both samples stem from confusion between distance per unit of time and time per unit of distance. Teachers should be well-advised to use a variety of time and distance units. The CMP students performed better than Control students on both of the distance/time rate items. The CMP students used the unit rate strategies twice as often as the Control students. The superior performance of CMP students is a result of the problem solving approach presented by the CMP curriculum where students construct personal understanding of key mathematical structures. The CMP students are more familiar with this context due to the CMP problem based curriculum.
– Comments on rate problem 5
The study included one other rate problem about density of feral cat populations in two cities. In this problem, a numerical comparison is required, and integers are used in spite of larger density problem. The performance on the density problem for both sample decreased drastically. The causes are not familiar factor and the real activity large integer numbers used in the density problem.
The main purpose of this study was to compare proportional reasoning of seventh grade students with CMP curriculum and Control curriculum. This report has focused on numerical comparison and missing value proportion problem with several different contextual settings and number structure. These two samples were tested with a written exam and interviewed some students randomly. The results, The CMP students clearly outperformed the Control students on the collection of tasks and on each individual rate problem. Reforming curriculum students were capable of providing a good quality of written and oral explanations to their work. Both reform and traditional curriculum students have a long way to attain mastery of basic proportional reasoning strategy and skills.
The CMP students competed very successfully with the Control students and also demonstrated ability to develop a variety of strategies. The CMP students are encouraged to construct their own understanding and procedures for doing rational number computation, solving proportions, and applying those skills to applied authentic problem solving tasks. There are more factors that could affect CMP student performance. First, the CMP curriculum is connectivity between the units, an important factor for acquiring knowledge in the rational number domain. Second factor is the level of CMP teachers’ preparation and background versus the traditional teachers.
7. IMPLICATIONS FOR TEACHING
Researchers identified many demonstrated strengths and many difficulties encountered by students when they were dealing with proportional reasoning concepts and their applications. They also found evidence confirming conjectures about the effects of number structure and context familiarity on task difficulty. Both the complexity of numbers and the contexts should be more manipulated by curriculum materials and instruction. All teachers concern about teaching the complicated subject matter included under the umbrella of proportional reasoning. Freudenthal (1983) has recommended that learning process of ratio and proportionality must be steered in such a way that sources of insight are not clogged during the process. The varieties of strategies and ways students find to solve problems are encouraging. Students are capable of developing their own repertoire of sense-making tools to help them to produce creative solutions and explanations.