*Mathematica in Education and Research
Vol.4 No.4
Copyright 1995 TELOS/Springer-Verlag Publishers
*

- Introduction
- Background
- The Experiment
- The Schedule
- Data Analysis and Results
- Conclusions
- References
- About the Authors
- Download
example Mathematica notebooks

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:

- Declare the variables: ``
*h*represents the cost of the house,*l*represents the cost of the lot.'' - Translate the equations:

Then they would use the computer to solve these equations usingh+l= 89900

h= 1000 + 7l

In[1]:=Solve[h+l == 89000, h == 1000 + 7 l, h, l]

Out[1]=h -> 78000, l -> 11000

Returning to pencil and paper, they would:

- Copy the solution:
*h*--> 78000,*l*--> 11000. - Translate the mathematical answer back into prose: ``The house costs $78000 and the lot costs $11000.''

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*:

- Chapter 3: Solving equations with one variable and solving word problems with only one unknown.
- Quiz 1: Students in all classes took this quiz without using to determine if the students in different classes were significantly different. (They were not.)
- Half of the students studied solving systems of equations while the other half missed algebra class because of proficiency testing.
- Chapter 3: Word problems with more than one unknown.
- Chapter 3 Test (with word problems): half of the classes used while the other half did not.
- Chapter 4: More word problems
- Chapter 4 Test (more word problems): half of the classes used while the other half did not.
- Chapters 5, 6, 7: Factoring, algebraic fractions, applying algebraic fractions to the word problems in the last half of Chapter 7.
- Chapter 7 Quiz: half of the classes used while the other half did not.
- Chapter 8: No word problems.
- Quiz 2: A quiz to look at any differences in problem solving ability between the students using one equation for translation and students using systems of equations for translation. Two of the classes that used one equation and two of the classes that used two equations used , while the other four did not.
- Chapter 9: More word problems.
- Chapter 9 Test: All of the classes now used systems of equations for translation; however, half of the classes continued using while the other half did not.
- Quiz 3: A final quiz on word problems after both groups had learned how to solve word problems using as many variables as unknowns. Half of the students used and the other half did not.

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[2]` 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.

Booth, L. R. * Algebra: Children's strategies and errors. A report of the Strategies and Errors in Secondary Mathematics Project*. Berks NFER-Nelson, Windsor (1984).

Clement, J. Algebra word problem solutions: Thought processes underlying a common misconception. * Journal for Research in Mathematics Education*, 13(1) (1982) 16--30.

Clement, J., Lochhead, J., and Monk, G. S. Translation difficulties in
learning mathematics. * American Mathematical Monthly*, 88 (1981) 286--289.

Davis, R. B. * Learning Mathematics. The Cognitive Science Approach to Mathematics Education*. Croom Helm, London (1984).

Herscovics, N. Cognitive obstacles encountered in the learning of algebra. In S. Wagner and C. Kieran (Eds.), * Research Issues in the Learning and Teaching of Algebra* (pp. 60--86). National Council of Teachers of Mathematics, Reston, VA (1989).

Hinsley, D. A., Hayes, J. R., and Simon, H. A. From words to equations: Meaning and representation in algebra word problems. In, M. A. Just and P. Carpenter (Eds.), * Cognitive Processes in Comprehension* (pp. 89--106). Lawrence Erlbaum, Hillsdale, NJ (1977).

Kaput, J. Towards a theory of symbol use in mathematics. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 159-195). Lawrence Erlbaum, Hillsdale, NJ (1987).

Kirshner, D., Awtry, T., McDonald, J., and Gray, E. The cognitive caricature of mathematical thinking: the case of the students and professors problem. * Proceedings of the Thirteenth Annual Meeting NA Chapter of the International Group for the Psychology of Mathematics Education* (pp. 1--7). Blacksburg, VA (1991).

Kuchemann, D. Algebra. In, K. M. Hart (Ed.), * Children's Understanding of Mathematics*. (pp. 102-119). John Murray, London (1981).

Lochhead, J., and Mestre, J. P. From words to algebra: mending misconceptions. In, A. F. Coxford (Ed.), * The Ideas of Algebra, K--12* (pp. 127-135). National Council of Teachers of Mathematics, Reston, VA (1988).

MacGregor, M., and Stacey, K. Cognitive models underlying students' formulation of simple linear equations. * Journal for Research in Mathematics Education*, 24 (1993) 217--232.

Mathews, S. M. The effect of using as many variables as are needed to solve word problems on the problem-solving skills and attitudes of students in Algebra I. (Doctoral dissertation, The Ohio State University, 1994). * Dissertation Abstracts International* (1994).

Kaput, J. Towards a theory of symbol use in mathematics. In, C. Janvier (Ed.), * Problems of Representation in the Teaching and Learning of Mathematics* (pp. 159--195). Erlbaum, Hillsdale NJ (1987).

Mestre, J. P. The role of language comprehension in mathematics and problem solving. In, R. Cocking and J. Mestre (Eds.), * Linguistic and Cultural Influences on Learning Mathematics* (pp. 201--220). Erlbaum, Hillsdale, NJ (1988).

Reed, S. K. A structure-mapping model for word problems. * Journal of Experimental Psychology: Learning, Memory, and Cognition*, 13 (1987) 124--139.

Susann Mathews ...

Susann Mathews

Department of Teacher Education

Wright State University

Dayton, Ohio 45435

smathews@desire.wright.edu

Latitia Mccallister ...

Latitia McCallister

Wright State University

Dayton, Ohio 45435

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