Skills Needed for Mathematical Problem Solving (4)

Problem 1: It takes Carla 1 hour longer to mow the lawn than it takes Sharon to mow the lawn. If they can mow the lawn in 5 hours working together, then how long would it take each girl by herself? [13]

Step 2: Students were then asked to solve the following problem as a home work, where the same concept as problem 1 is needed.

Problem 2: John takes 3 hours longer than Andrew to peel 500 pounds (lb) of apples. If together they can peel 500 lb of apples in 8 hours, then how long would it take each one working alone? [13]

Although the above problem was discussed in class before they attempted to solve it, to make sure that students understood it, they had difficulties in solving it. I decided that the concept of rate of work has to be discussed again. Few days later few students managed to solve the given problem and the solution to problem 2 was discussed with the whole class.

Step 3: In order to assess students’ understanding of the concept of the rate of work and the process of problem solving, I assigned the following problem as a home work.

Problem 3: It takes pump A 2 hours less time than pump B to empty a certain swimming pool. Pump A is started at 8:00 A.M., and pump B is started at 11:00 A.M. If the pool is still half full at 5:00 P.M., then how long would it take pump A working alone? [13]

Problem 3 also needs a deep understanding of the concept of the rate of work. Except for a few students, most found the problem very challenging and could not solve it. I carefully examined the solutions generated by students and I understood that the concept of rate of work was their main difficulty. I decided to give a full hour lesson on the rate of work with many examples and I made sure that not only they understood the concept but also how to use it to formulate problems. I requested that they look again at the problem. Few days later, more than half the class solved problem 3 correctly.

Step 4: I then assigned the following problem in a quiz.

Problem 4: It takes pump B 2 hours more time than pump A to fill a swimming pool. Both pumps are started at 7 am. At 10 am pump A breaks down. It took 1 hour to repair it and then was restarted again. At 3 pm 80 % of the swimming pool was filled with water. How long would it take each pump working alone to fill the swimming pool?

About half the class solved the problem correctly and a quarter of the class had solution with minor mistakes. I carefully examined the solutions generated by students and it was clear that the students had a better understanding of the concept of the rate of work and more importantly they knew how to apply it to solve problems.

Students fail to solve problems involving concepts that are not thoroughly understood. Moreover, mathematical problems may be used as teaching methodologies not only to introduce concepts but also to help students gain a deeper understanding of these concepts [9]. In fact some concepts cannot be thoroughly understood unless they are used in problem solving or any other activity where critical thinking and reasoning are involved.

Mathematical problem solving may also be used to introduce a new concept. An example of a mathematical problem that may be used to introduce a new concept is now presented.

Problem 5: The present population of the UAE is 4.5 million. If we assume that the population grows at an annual rate r = 3% for the next 15 years, what will be the population P of the UAE in t years?(assume t is smaller than 15)

Students can easily be guided to use percentages and come up with the following result.
P(t)=4.5(1+3%)^t

At this point the concept of exponential functions can easily be introduced using the result obtained. Students will have understood that exponential functions may be used to solve population problems and make a connection between the concept and its possible application. Experience shows that students are more motivated when solving problems related to their daily life [12].

1, 2, 3, 4, 5, 6, 7,