Mathematica in Education and Research
Copyright 1995 TELOS/Springer-Verlag Publishers
High school math teachers may not even consider using Mathematica because of its great complexity. Some feel that because they cannot master all of Mathematica, they cannot use it in their teaching. Indeed, it would be easy to overdo the use of Mathematica. Nevertheless, even a small subset of can be a powerful aid to learning. Therefore, for our project, we concentrated on only the part of Mathematica that eighth- and ninth-grade students would need to help them solve word problems: the Solve command for one or two algebraic equations. Neither Ms. McCallister nor Mr. Elsass had used Mathematica before. However, they were neither intimidated nor overwhelmed because they concentrated their efforts on learning and then helping their students learn just one task: how to solve equations using Mathematica. For example, consider one of the Quiz 3 word problems.
A house and lot together cost $89,000. The house costs $1000 more than seven times the cost of the lot. How much does the house and the lot each cost?
(In spite of the efforts toward mathematics education reform, the reality is that Algebra I students mostly still learn---or attempt to learn---to solve the problems in their textbooks, and these kinds of problems still populate the Algebra I texts.) Students in the classes using Mathematica would do the following, using pencil and paper:
h + l = 89900Then they would use the computer to solve these equations using Mathematica:
h = 1000 + 7l
In:= Solve[h+l == 89000, h == 1000 + 7 l, h, l]
Out= h -> 78000, l -> 11000
Returning to pencil and paper, they would:
This may seem like such a small usage of that one may ask, ``Why bother with Mathematica'' We need to provide high school students with real tools to help them in understanding mathematics. Furthermore, we need to show them when and how to apply those tools appropriately. Mathematica is a wonderful tool to help students solve word problems. Perhaps more to the point, it helps them, as we discuss below.
A primary goal in teaching algebra is to help students learn to solve word problems. However, translating from the prose in algebraic word problems to algebraic equations is particularly difficult ([Clement, Lochhead, and Monk, 1981] and [Lochhead and Mestre, 1988]). Many studies have been done concerning the sources of translational errors and different schemas students use for translating ([Booth 1984], [Clement 1982], [Davis 1984], [Herscovics 1989], [Hinsley, Hayes, and Simon 1977], [Kaput 1987], [Kirshner, Awtry, McDonald, and Gray 1991], [Kuchemann 1981], [MacGregor and Stacey 1993], and [Reed 1987]). I have looked at the number of variables used in translating word problems from prose to algebraic equations [Mathews 1994]. I found that, when Algebra I students are taught to declare a variable for each unknown in a word problem and, hence, write an equation for each relationship, they can translate the word problems significantly better than when they use only one variable in problems with 2 or more unknowns. However, even though the students do not solve the systems of equations any worse than when trying to solve only one equation, they do not solve them any more correctly. (There was no statistically significance difference between the solutions of the equations of word problems translated into one equation and solutions from systems of equations.) After defining the unknowns and setting up equations to solve a problem, students are often unsure if they are moving in the right direction, which leads to stumbling through the solving of equations. By taking away that unsure feeling of how to solve the equations, Mathematica could allow students to try their equations and then critically examine the answer the computer gives them. If the answer seems to be a wrong one, they no longer have to wonder, ``Did I set it up wrong or did I solve it wrong?'' The students can immediately go back to their translation of the problem and try to determine where a mistake was made. This vision of the learning process lead us to select our research questions: ``If taught how to use a computer algebra system to solve their equations, can students concentrate more on the translations and solve word problems correctly? Is Mathematica an appropriate tool for this?''
My father, a secondary school principal for many years, once explained how he judged a teachers effectiveness. He would ask the teachers students two questions: ``In this teachers class, did you learn anything? If you learned, did the teacher help or get in the way?'' This seems to be the bottom line about the effectiveness of any instruction we provide our students. In asking if Mathematica is an appropriate tool as a computer algebra system for eighth and ninth-grade algebra students, we need to look for the answers to these questions: ``Did students learn how to solve word problems? Did Mathematica help or get in the way?''
To carry out the research to answer these questions, I worked with two Algebra I teachers, Tish McCallister and Tom Elsass, who each taught four sections of Algebra I during the 1994-1995 school year. They teach at St. Mary's Memorial High School, a public secondary school with about 900 students in a small town in rural Ohio. Ms. McCallister had four sections of ninth-grade algebra students, and Mr. Elsass had three sections of ninth graders and one section of eighth graders. This was a two-factor, two-level experiment: (1) use of to solve the algebraic equations that the students wrote to solve the word problems vs. solving them by hand and (2) translating directly to one equation in one unknown vs. translating to two equations in two unknowns. To control for teacher as a possible confounding variable, the teachers both taught word problems to their four sections as follows: one section using one variable without Mathematica, one section with one variable with Mathematica, one section with two variables without Mathematica, and one section with two variables with Mathematica.
The experiment was carried out during the entire school year so that word problems were integrated with their other algebra topics as naturally as possible. This is a continuation of the research of the use of one variable versus the use of two variables while adding another dimension---exploring the use of Mathematica. In this article, we focus on the use of Mathematica.
One hundred sixty-nine students participated in this study. Students were assigned to these classes as their schedules permitted. Although they were not assigned randomly, they were not self selected, and the classes were assigned their treatments based solely on when the teachers could have access to all fifteen computers in the two teachers' rooms, which were directly across the hall from one another. Ms. McCallister scheduled her classes that would use during the times when Mr. Elsass had a preparatory period or lunch, and Mr. Elsass scheduled his classes that would use during the times when Ms. McCallister had a preparatory period or lunch. Thus, each teacher could use the computers in both rooms.
All of the algebra classes were treated as similarly as possible. They studied the same material at the same time, except during a transitional period in the fall when half of the class times were used for taking the Ohio Ninth-Grade Proficiency Tests. While half of the algebra classes were not meeting because of the proficiency testing, the other half was taught to solve systems of equations both by substitution and by linear combinations. (The other classes were taught this late in the school year.) The classes had the same assignments, the same notes, and the same quizzes and tests. All sections were taught how to solve equations both by hand and using Mathematica, in the interest of helping all of the students learn something that might be helpful. No students regularly solved their homework word problems using Mathematica. (They did their assignments with paper and pencil.) However, all of the students were given the opportunity to use Mathematica to check their homework in assignments containing word problems throughout the year. Students checked their homework in groups of size three. They would check their solutions with each other, and then were allowed to send one member of their group to the computer to confirm their solutions. This step was not required; students in classes that were allowed to use Mathematica during quizzes or tests tended to check their homework with Mathematica whenever they were allowed to do so, but in the non- classes only students who liked computers used them regularly for checking.
Ms. McCallister and Mr. Elsass spent only a short time in class teaching their students how to use the features of Mathematica that they would need to solve algebraic equations. They taught the students the Solve command (including its syntax, the uses of brackets and braces, and the use of the double equals symbol). The students learned how to use with little difficulty, but the teachers were not surprised that some students had some difficulty in getting used to the syntax of Mathematica. This was attributed to the students inexperience with computers. They felt that it was no more difficult to introduce students to Mathematica than to any other new computer language. The two teachers helped the students correct their syntax errors until the students were able to do that for themselves. Most students responded to the use of Mathematica with enthusiasm. A few students did not like using the syntax, but Ms. McCallister felt that their lack of enthusiasm was due simply to their inexperience at the computer.
Only the four sections in the experimental groups (two classes for each teacher) were allowed to use to solve their equations during tests containing word problems. Furthermore, during tests students had to have their word problems translated into one or two equations before they were allowed to use Mathematica to solve their equations. During the Mathematica parts of their tests, the students were, of course, allowed to change their equations if they discovered that they had translated the word problems incorrectly (such as when they got disconcerting solutions to their equations). All of the students in all of the groups were required to write out five parts for each solution to word problems and received points for each part: writing down the knowns and unknowns, 1 point; declaring the variable(s), 2 points; translating the prose into algebraic equations, 3 points; solving the equations correctly, 2 points; and translating the Mathematica results back into prose, 2 points. Thus, any solution to a word problem had ten total points possible, with only two allocated for the correct solution to the equation(s).
Ms. McCallister and Mr. Elsass conducted the experiment with half of their classes using Mathematica and half of the classes not using Mathematica for solving equations from word problems that they had translated during the entire 1994--1995 school year. The experimental design provides the ordered sequence of topics taught, including only tests which contained word problems and on which half of the students were allowed to use Mathematica:
To analyze the data, we examined descriptive statistics with a table and with box plots, plotted the empirical cumulative distributions for comparison, and performed appropriate nonparametric hypothesis tests. The quiz data are from research quizzes that were given as special word-problem quizzes. The students took Quiz 1 in September, Quiz 2 in March, and Quiz 3 at the end of the year. The test results shown here are from tests corresponding with chapters (or parts of chapters) that contained word problems. The scores from all of the quizzes and all of the tests were used in determining the students' grades, and, therefore, were taken seriously by the students. All data are given as percentages to make the comparisons meaningful.
Descriptive statistics of the research quizzes are displayed in Table 1. Although the groups on the first quiz are designated as No Mathematica and Mathematica, neither group used Mathematica on Quiz 1 so that we could determine whether or not the groups were significantly different.
Table 1: Descriptive Statistics of the Scores of the Without-Mathematica and With Mathematica Groups on Quizzes 1--3
Table 1 provides an organized way to display the descriptive statistics, but it does not provide the reader with a sense of how the groups compared overall. A good way to visually compare the quiz and test results of the groups not using with the groups using is to study the box plots of the quiz and test data.
Figure 1: Box plots of quiz and test scores by classes with vs. without Mathematica.
In Figure 1, the box plots are displayed with the time line reading from the bottom of the figure to the top. The ends of the whiskers are at the 10th percentiles on the left and the 90th percentiles on the right. The left ends of the boxes mark the lower quartiles; the narrow vertical bars in the middle of the boxes show the medians while the wide bars mark the means; and the right ends of the boxes are at the upper quartiles. It appears that there is no significant difference in the two groups on the first research quiz because the lower quartiles, the medians, and the upper quartiles are within a few points of one another for both groups. The ninetieth percentiles are both at 100 the boxes of the groups using Mathematica are to the right of the boxes of the groups not using Mathematica. In the groups using Mathematica, the lower quartiles, medians, means, and upper quartiles are to the right of (at higher scores than) in the groups not using Mathematica.
Looking at the box plots, we can tell the overall trends of each of the groups and the overall comparisons of the two groups; however, we cannot distinguish comparisons of individual data. To study the data more closely, we need more detail of the distributions of the scores from the group not using Mathematica and the group using Mathematica. For that, a graph of the empirical cumulative distributions is just what we need. Although I have not found a way to make this type of graph in any statistical software that I have used, it is fairly straight forward to do in Mathematica. To get the data into Mathematica, we copied the data from Minitab into text files that were read into Mathematica lists. Smirnov.ma (included in the electronic distribution) is an example notebook we used to make these plots and to perform the Smirnov tests.
Figure 2: Empirical cumulative distributions of scores on Quiz 1, without (line) and with (dashed) Mathematica.
Figure 3: Empirical cumulative distributions of scores on Quiz 2, without (line) and with (dashed) Mathematica.
Figure 4: Empirical cumulative distributions of scores on Quiz 3, without (line) and with (dashed) Mathematica.
Figure 2 shows the empirical cumulative distributions of the scores on the first research quiz. The solid line represents the results of the group who were to use paper and pencil for the remainder of the experiment and the dashed line represents the group who were to use Mathematica; however, all students used paper and pencil for the first research quiz. The plot shows that there is no important difference between the two groups when they started the experiment. (The statistical significance of the differences is addressed by the hypothesis testing described below.)
In Figures 3 and 4, the solid lines represent the group that used paper and pencil for Quizzes 2 and 3, and the dashed lines represent the group that used Mathematica. Note that if the cumulative distribution representing the paper and pencil groups scores is higher, then it is to the left as well; this happens if the paper and pencil groups scores are lower. In both the second and third plots, the solid line is higher and to the left. These two plots show the extent of the improvement attributable to Mathematica. These are important differences in problem solving performance.
After examining the distributions of the data, we proceeded to formal hypothesis testing, first using the Mann-Whitney U-test to test for a difference between medians of the quiz and test scores of the group not using and the group using Mathematica. The U-test assumes that the distributions differ only in location, but not in shape; thus, the median difference is a measure of that difference in location. To compare the entire cumulative distributions without this assumption, we applied the Smirnov test for difference between distributions. We used Minitab to compute the descriptive statistics and perform the Mann-Whitney U-tests. We used Mathematica to perform the Smirnov tests, as shown in appendix I.
The results of the Mann-Whitney U-tests are listed in Table 2. The quizzes and tests are in chronological order. For the statistical test applied to Quiz 1, the alternate hypothesis is ``There is a difference between the medians of the two groups.'' For the remaining quizzes and tests, the alternate hypothesis is ``The median of the scores of the groups using Mathematica is higher than the median of the scores of the groups not using Mathematica.''
Table 2: Mann-Whitney U-Test Statistics of the Scores of the with-Mathematica vs. without-Mathematica Groups
The results show that we cannot reject the hypothesis that the median scores from the two groups are the same for Quiz 1, in which neither group used Mathematica. On all other quizzes and tests, the median scores from the groups using are significantly higher than those from the groups not using Mathematica. As the school year progressed, the difference between the medians of the groups using and of the groups not using it increased nearly monotonically.
Table 3: Smirnov Test Statistics of the Scores of the Scores of the with-Mathematica vs. without-Mathematica Groups
The results of the Smirnov tests are listed in Table 3. The quizzes are in chronological order. For each Smirnov test, the alternate hypothesis is ``The Mathematica group scored higher, in general, than the paper and pencil group. The critical level of significance on Quiz 1, p = 21.4%, indicates that the students who were to use Mathematica were not significantly better prepared than the students who were to use paper and pencil. The critical levels of significance, p = 2.4% and p = 0.8% on Quiz 2 and Quiz 3, respectively, indicate strongly that the use of Mathematica is statistically significant in improving these high school students test scores in algebra word problems.
The data support the hypothesis that teaching Algebra I students to use to solve their equations once they have translated the prose in a word problem to mathematical equations helps them solve word problems better. There was no statistically significant difference between the two groups on Quiz 1, before one group began using Mathematica to solve their translated equations. But there was a statistically significant difference in the problem-solving performance between the groups that did not use Mathematica and those that did, on every quiz and test.
This research provides the answers to our initial questions. If taught how to use a computer algebra system to solve their equations, students can concentrate more on the translations and solve word problems correctly.
The teachers observed that when students used Mathematica to solve their equations, if they got a weird answer, such as an answer in the form 5 + Sqrt when trying to find somebody's age, they would realize that something was wrong. This phenomenon does not usually happen when students use calculators. They think nothing of getting 6.141 for someone's age, because their calculators often give them decimal results. Furthermore, the students using Mathematica knew that if they had any kind of weird answer that they were not expecting, then they had written the equations wrong. They relied on Mathematica to solve their equations correctly, but they could not usually rely on themselves to do so.
I believe that Mathematica allowed the students to concentrate more on their translations because when students are learning something difficult in mathematics, they often temporarily lose skills that are not yet firmly in place and that are only tentatively owned. Thus Mathematica could solve the students' equations correctly without their having to depend on their own equation solving skills when they are first learning to solve algebraic word problems.
Did students in the experiment learn how to solve word problems? Yes. With the mean score of 83% and a median of 90% on the final research quiz, the group did learn to solve the word problems of Algebra I.
Mathematica is an appropriate tool for these students. Did it help or get in the way? Mathematica helped. The data support the conclusions that eighth and ninth graders can learn to use Mathematica and can use it to help them solve word problems.
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Susann Mathews ...
Department of Teacher Education
Wright State University
Dayton, Ohio 45435
Latitia Mccallister ...
Wright State University
Dayton, Ohio 45435