The Department of Mathematics Education

by

June Jones

University of Georgia

Spring 1997

This paper will examine difficulties in problem-solving among fifty-six college freshmen in precalculus algebra.

Questions to consider:

(1) How do the problem-solving difficulties of these students fit the categories of difficulties as determined by the literature? and

(2) How does use of a graphing calculator affect problem-solving?

Five basic methods were identified: restatement, analysis, analogies, dependencies, and graphic. Studies show that none of the methods stands out as best and that all yield acceptable results. Each method has its own merit and they can be used in conjunction to best satisfy the needs of any variety of problems (Butler & Wren, 1965).

Besides the above stated broad-based information, there has been much research into specific areas within problem-solving. A few of them are related below.

The third NAEP Mathematics Assessment revealed that 13- and 17-year-olds were improving (from 1973 to 1982) in basic skills, but still lacked a deep understanding of mathematics and its applications. It also reported that students had positive attitudes toward mathematics, problem-solving and computers (Carpenter, Lindquist, Matthews & Silver, 1983).

Rich (1990) confirmed that the graphing calculator in precalculus improved students' understanding of graphing and provided an appropriate problem-solving technique.

Carter (1995) found that the graphing calculator seemingly led to improved problem-solving, as less time was consumed with algebraic manipulations. He also reported that the students used the calculators as a monitoring aid while solving word problems. Bitter & Hatfield (1991) also found that students using calculators showed improved problem-solving skills.

Szetela & Super (1987) found a better attitude toward problem-solving when the calculator was used. However, the scores were not significantly higher for those students than their counterparts who did not use a calculator. Schoenfeld (1988) found in a study with geometry students that they could feel confident about their skills and still not make the connection within the topics.

The volume of research has prompted responses from many sources. The National Council of Teachers of Mathematics devoted its 1980 Yearbook to Problem Solving in School Mathematics. The National Council of Supervisors of Mathematics placed great emphasis on problem-solving (Carl, 1989). The NCTM Standards (1989) also emphasized problem-solving at all levels and encouraged extensive use of the calculator. The National Research Council compiled Everybody Counts (1989) and Moving Beyond Myths (1991), which provided a wealth of statistics on various demographics, many of which affect student problem-solving. Madison and Hart reported much of the same in A Challenge of Numbers (1990). The NCTM Position Statement (1996) states:

Research and experience have clearly demonstrated the potential of calculators to enhance students' learning in mathematics. The cognitive gain in number sense, conceptual development, and visualization can empower and motivate students to engage in true mathematical problems solving at a level previously denied to all but the most talented. The calculator is an essential tool for all students in mathematics.

The subjects for this study were from the students enrolled in three freshman pre-calculus algebra classes at Macon College, a two-year institution, during the spring quarter of 1997. The only prerequisite for the course was high school Algebra II or college Developmental Studies Algebra.

The fifty-six students (82% of the total students in the class), ranged in age from 18 to 68, with a mean age of 26 (25 w/o the 68-year-old). 91% of the students had gone through Algebra II in high school, with a mean math grade of 2.5. 77% reported that they had taken a developmental studies class within the last year with a mean grade of 2.9 (38% made an A). 48% reported having used a calculator in high school and 54% used a calculator in their developmental studies class. They reported studying math an average of 6 hours a week. This studying was done alone by 62% and a combination of alone and with others by 38%. 84% thought that mathematics was important to their career plans. The following are the results from the question "Has your attitude toward using the graphing calculator changed since you started using it in this class?" No: 4% still did not like it, 19% liked using it previously, and 7% said no without any explanation. Yes: 66% liked it better, overall, because they knew more about how to use it and what it can do. One person (2%) said she was more frustrated with the calculator than when the class started. One other person (2%) said yes without an explanation.

The apparatus consisted of three parts. Part 1 was a series of personal information questions. Part 2 was an attitude survey (not summarized or analyzed in this paper). Part 3 consisted of six problems complete with instructions.

Part 3 was distributed at the beginning of class. Orally, the students were reminded to follow the general directions at the top of the page for all problems. They were also told that this would in no way adversely affect their grade. It should be noted that the drop date had passed when this was administered, so all students taking part will finish the course. Each concept within the questions had also been mentioned directly or indirectly at some point in the quarter. Each class had 50 minutes to work the six problems. In the interest of time, the students answered Parts 1 and 2 outside of class.

Each of the questions will be evaluated in relation to these difficulties. The students were also asked if they used the calculator on each problem. The use of the calculator will also be addressed.

Comprehension was a problem for 21 of the 56. Eleven people did not attempt to work the problem. One person even stated, "I can't do word problems." Eight people let the 70 meters represent "area," and 2 people used all 4 sides. Structurally, only two people realized that they could orient the plot in two different ways, and one of those students did arrive at both sets of dimensions correctly. Forty-six people did sketch a rectangle and attempt to label it. Thirty-two students managed to set up an equation, and 27 of those solved correctly for the variable. Twenty people gave a correct set of dimensions. Understanding was also lacking in relation to the maximum area inquiry. Nineteen people did not answer the question. Twenty-one said yes because dimensions were involved. Sixteen said no because the "question involved perimeter" or "it didn't ask for area." In explaining why, the one person who found both sets of dimensions, proceeded to find the area of each and identified the one with the largest area. She further found the maximum area when 70 m of fencing was used without the restriction that one side was 5 m longer than the other. The judgment phase was attempted by ten people who checked their answers. Nineteen people said they did not use the calculator mainly because it involved easy arithmetic. Seventeen students did use the calculator to do the arithmetic or to check the answer they obtained by hand.You want to fence a rectangular plot bordering a building. You only need three sides fenced because the building forms the fourth side. If one side is 5 meters longer than an adjacent side, and you have 70 meters of fencing: Find the dimensions of the plot. Does this question have anything to do with the maximum area of the plot? Why?

Six people did not attempt to answer the question. One person said "longer" and one person said they were the "same length." Nine people drew a rope and divided it into five pieces. The other 48 answered "shorter," showing class comprehension of only 86%. Those people also followed through showing all phases through judgment with their explanation. Nine people used their calculator citing "division" as the reason. Forty-two people said they did not use the calculator for reasons such as, answer was just common sense, division was basic and could be done easily by hand or mentally.A rope four meters long is cut into five pieces of equal length. Is each piece longer or shorter than one meter? Why?

Eleven people skipped the question leaving the remaining 45 students with some level of comprehension. One person tried to use area instead of perimeter. 41 students sketched a triangle and 31 went on to the structure and operational stages and set up an equation to obtain the numbers 6, 12 and 24. These met the length relationship with a sum of 42. Judgement was incomplete for everyone, as no one realized that a triangle would not be formed by that arrangement of lengths. Eleven people said they used the calculator to do the division or check the answer. 27 said they did not use the calculator because it could be done mentally. Seven people noted you could not use the calculator because this question required a drawing.A triangle has a perimeter of 42 inches. The medium length side is twice as long as the shortest side and half as long as the longest side. How long is each side?

One person did not attempt the question. The remaining 55 people showed comprehension and structural reasoning by dividing. Judgment proved difficult for 21 individuals; seven answered 31_ buses and fourteen rounded down to 31 buses. Forty people used the calculator to do the division, while fifteen did the division by hand. All obtained 31_, then drew their conclusion."An army bus holds 36 soldiers. If 1,128 soldiers are being bused to their training site, how many buses are needed?"(Taken from the National Assessment of Educational Progress (1983))

This problem is similar to the problem Schoenfeld quotes for Wertheimer
who believes the elementary age student should be able to look at the problem
and write as

to illustrate true mastery of the operation (Schoenfeld, 1988).

All 56 students showed basic comprehension. Four people made computational
errors. Three of those errors were due to not entering parentheses in the
calculator. Those students then did not use judgment to realize that 2411.5
could not be the correct answer. Of the 52 students who arrived at the
742 for the solution, only 6 showed the step noted above. 39 people used
the calculator to find the answer saying it made the arithmetic go quicker.
It is of interest to note that two of those students went on to say that
they should have seen the obvious and not used the calculator. The other
17 did not use the calculator because the arithmetic was easy or obvious.

Thirty-one students demonstrated comprehension through judgment here by stating -3 was the answer because it was associated with the function value of 10. Four people did not answer the question. Thirteen people answered 3 because it was biggest. The remaining eight students gave answers that related to the graph such as intercepts. When asked for the associated range, only twelve people showed comprehension by listing the set of values {10, 4, 1, -2, -8}. Six students did not answer the question, and the remaining 40 gave various responses that showed little or no comprehension. Comprehension was getting worse by the third part of the question. When asked to graph, five people left it blank, and 48 drew a solid line. That 48 included 9 of the 12 who answered range with the set of separate points. Forty people used the calculator to graph and thus explains the solid line. This is consistent with Rich (1990) who found that students do not instinctively use and understand graphs just because they have a graphing calculator.Given f(x) = 1 - 3x with Domain = {-3, -1, 0, 1, 3}. Which number in the domain is associated with the highest point of the graph? What is the range of the given function? Graph the given function.

The traditional "army bus" problem seems to indicate that some contextual realization does come with maturity. The 1983 NAEP showed 47% of the 13-year-olds with as erroneous conclusion (Carpenter, et al., 1983). The students in this study only showed 39% erroneous conclusions.

Physical maturity did not help in the problem Wertheimer said should be evident to elementary age children. Only 11% of the college students exhibited this trait.

In many cases, the student was satisfied just to get a number answer. Lack of motivation may be a cause of students not moving on to the judgment phase.

Use of the calculator seems consistent with prior use. 48% of the students used a calculator in high school and 54% used a calculator in their developmental studies class. The calculator was used in this survey in a variety of ways for an average of 54% of solutions.

Lovell (1971) suggested that students develop their concept of function in stages. In early stages, they can carry out an arithmetic relationship in a table or on a line graph. In this stage they have difficulty translating between graphical representations and ordered pairs. They progress to a point where, in the final stage, the student has mastered the function concept and can find domain and range. This may be the case in question six.

Some questions remain. Among them: (1) How are students motivated to want to think through all the steps of problem solving difficulties? (2) Can the graphing calculator to be used in this motivating process, and if so, how?

Bitter, G. G., & Hatfield, M. M. (1991). Here's a math-teaching strategy that's calculated to work.

Butler, C. H., & Wren, F. L. (1965).

Carl, I. (1989). Essential Mathematics for the Twenty-first Century: The Position of the National Council of Supervisors of Mathematics.

Carpenter, T. P., Lindquist, M. M., Matthews, W., & Silver, E. A. (1983). Results of the Third NAEP Mathematics Assessment: Secondary School.

Carter, H. H. (1995).

Lovell, K. (1971). Some aspects of the growth of the concept function.

Krulik, S. (Ed.). (1980).

National Research Council. (1989).

National Research Council. (1990).

National Research Council (1991).

National Council of Teachers of Mathematics. (1989).

National Council of Teachers of Mathematics. (1996). Position Statement.

Rich, B. S. (1990).

Schoenfeld, A. (1988). When Good Teaching Leads to Bad Results: The Disasters of "Well-Taught" Mathematics Courses.

Szetela, W. & Super, D. (1987). Calculators and Instruction in Problem Solving in Grade 7.