Math Courses Online

The students did not seem to know how to paraphrase or translate qualitative information provided in the task into simple proportionality statements with mathematical symbols. Students ‘ difficulties with problem restatement were common and frequently encountered. This influenced students’ positioning of the proportional variables in their solutions using their adopted proportional reasoning patterns.

Mathematics Test – UMB PTN 2009 Code 120

Occasionally, the positions of the variables were switched. In most cases, this caused students to retrace their steps to the beginning of their solutions, thereby wasting valuable time on the task. However, the students did not have any difficulty restating Questions 5 and 6. They reasoned that this was because the content of Questions 5 and 6 were practical and familiar:
Student E: The questions are simple and straight forward. Question 5 is easy because no formula is needed….We buy nuts everyday.
Student D: Question 6 has names of places that we know. “Lagos” and “Ilorin” are familiar words, which makes it easier to understand the question.
Hence, the degree of familiarity and relevance of the tasks appeared to influence students’ understanding and task representation. Questions 5 and 6 were, however, not regular physics problems but mathematical proportional reasoning tasks that were introduced for a specific purpose.
In Questions 14 and 16, the students did not know how to mathematically represent the “ratio of the mass of hot water to that of cold water” and the “ratio of the number of turns of the primary coil to the secondary coil in the transformer.” They did not know what to write down and how to go about solving the problem. In addition, 3 students had difficulty with the long division operation involving complex figures and numbers with decimal points. Two students had some difficulty with multiplication. These difficulties with long division and multiplication affected students’ time management during problem solving. The students reckoned that their difficulty with multiplication and division was due to careless mistakes on their part and distractions posed by the tasks.
Students’ Difficulty in Recognizing Proportional Variables in Physics Tasks
The students could not recognize the proportional variables in some tasks and had some difficulties identifying the multiplicative relationship existing among the proportional quantities. All the students had this problem with Questions 2, 7, and 10. In some cases, students could not generate proportional variables from the given tasks. In Question 2, for example, the students could not recognize the proportionality relationship existing between the 50 divisions on the thimble of the screw gauge and the smallest unit on the sleeve of the screw gauge. Students needed to know this relationship before they could apply their multiplicative strategy to solve the task. Although the students had seen the micrometer screw gauge previously during one of their physics lessons, they claimed that they did not pay any attention to the parts called “thimble”and “sleeve.”
Student B summarized his difficulty thus:
Without my basic knowledge of this relationship, there is no way I will be able to do that. I will just say that, okay, in our practical class this is what we did and that will be all.
Students were able to solve the problem on the micrometer screw gauge only after they had learned how to use it. This indicated that the students lacked meaningful practical experience with the micrometer screw gauge. Similarly, in Question 7 students could not identify the set of variables needed to employ the multiplicative proportional reasoning strategy. Students were unable to solve the problem using proportional reasoning because they felt that they were only provided with two variables (one value of the density and one value of the volume), and no third variable was given. The students believed that there had to be three known values for two variables in any given task before they could use proportional reasoning strategies to solve for a fourth value. Hence, they could only see the density of 2.70 g/cm^sup 3^ in the task as being one variable, rather than seeing it in terms of two variables, which is evident in the representation 2.70g:1cm^sup 3^ or 2.70g to lcm^sup 3^. Student B solved the task correctly using an algorithm. The following dialogue took place between the researcher and Student B when he was asked to solve the task using proportional reasoning:
Researcher: Can you use the ratio method for this question?
Student B: [Attempts the problem]. I can see from the question that it won’t be easy to use ratio.
Researcher: Why not?
Student B: Because of the way the question is. Researcher: What is it about the question that makes it difficult for you to use the ratio method?
Student B: In the other [questions] there is a specific pattern which they follow. Like the way they frame the question, you can easily use ratio. Like this [Question 4]. But from this you cannot easily guess. The students needed to represent the density in its broken down form (that is, mass and volume) to be able to recognize the variables and use their multiplicative strategies. Another comment from Student D provides further insight into the problem:
It didn’t occur to me that I could look at it and break up the density. I know it’s mass over volume. But when it is already density, it seems like one thing to me….Okay, mass over volume gives you density but not density to give you mass over volume.