This chapter presents the answers to the four research questions proposed for this study in chapter one, based on the analyses carried out in chapter four.  Conclusions, limitations of the study and recommendations follow in order to complete this chapter.


            The following research questions were investigated in this study:

1.      How did the students in the two groups, experimental and control, compare in their prior achievement and attitudes, and their experiences with technology?


2.      What relationships appear to exist between attitudes and achievement in the learning of linear functions activities?


3.      At which level and in what ways, can the use of multiple representations be supported by spreadsheets learning activities promote to better promote understanding of linear functions in students at college level algebra?


4.      How well does the medium of a powerful spreadsheet like Excel, lend itself to promoting instruction through multiple representations?


The following sections will answer each one of these questions separately.


Answering the First Research Question

1.      How did the students in the two groups, experimental and control, compare in the prior achievement and attitudes, and their experiences with technology?


Prior achievement on mathematics based on grades reported by students from both groups was examined.  The students’ achievement in mathematics is represented by grades referred to a general knowledge in this field, not necessarily in the topic of linear functions.  It is important to point out here that students in both groups had completely different prior experiences, as well as, performances in mathematics.  This was attributed to the fact of receiving the course from different schools programs (public and private).  The research question was formulated, as the first one, since the prior achievement in mathematics was the only variable that students exhibited before treatment for the study started.

The results dealing with this variable, reported on chapter four revealed no significant difference (p > .05) between the control and experimental groups on prior achievement in mathematics.  Based on these findings, prior achievement in mathematics based on grades seemed not to be determinant factor in helping student attain a broad understanding about linear functions.  These results show that a good prior achievement in mathematics did not necessarily imply a better achievement in linear functions.  Different mathematics topics were taught in a college level algebra course.  In this study, linear functions were strongly emphasized.  According to the results, a good performance in prior mathematics classes did not guarantee a good performance in linear functions.

            In terms of technology use in prior mathematics courses, the students’ profiles revealed that the use of computers, calculators, spreadsheets, Internet, among other tools, were almost nonexistent.  Further, the profiles indicated that for the majority of the students in the study, it was their first experience using technology, particularly, spreadsheets, in their mathematics courses. The findings indicate that in the experimental group, previous mathematics achievement , particularly those supported by technology were not a decisive factor in promoting better understanding in linear functions using spreadsheets and multiple representations.


Answering the Second Research Question

2.      What relationships appear to exist between attitudes and achievement in the learning of linear functions activities?


            Attitudes toward mathematics were explored in this study by collecting data through a scale that was administered in the beginning and at the end of the treatment, for control and experimental groups.  The results on this variable, summarized on the previous chapter, indicate that no significant (p > .05) was found on the comparisons made. 

The items of the attitudes scale toward mathematics were organized in two clusters: attitudes toward technology and its uses, and opinion or feelings toward mathematics as a subject.  In terms of technology uses in mathematics and by inspecting students’ responses on items 1 and 2 of the attitudes scale, the experimental group had not significant (p > .05) more positive experiences than the control group with the use of calculators to perform routine calculations in their mathematics courses prior the study.  In terms of the attitudes toward technology, the control group exhibited a very slight (1 point of difference) positive attitude at the beginning of the study, according to the inspection on item 3 of the scale.  Once the treatment concluded, the experimental group seemed to have a gain in positive attitudes toward technology.  These results agreed with the treatments that each group received, as described on chapter three.

The treatment received by the experimental group as part of the study was based on intensive use of spreadsheets while emphasizing multiple representations of linear functions.  The improvement in attitudes toward mathematics from this group is noticeable.  It could be inferred that the traditional approach, based strictly in the use of textbook and not technology allowed to enhance learning, did not promote an improvement in the attitudes of the control group at the end of the study.

            These results on attitudes toward mathematics agreed with the findings on achievement discussed in a previous section of this chapter.  Based on this information, it would be sensible to conclude in this study, that attitudes toward mathematics seemed to have limited effects on mathematical achievement. 

Answering the Third Research Question

3.      At which level and in what ways, can the use of multiple representations be supported by the spreadsheets learning activities to better promote understanding of linear functions in students at college level algebra?


Since the capabilities of spreadsheets to illustrate the multiple representations (symbolic, graphical, tabular, and verbal) of a linear function, this technology was intensively used in this study, particularly with the experimental group, to teach the mathematics concepts in a college level algebra course.  The use of spreadsheets allowed students to work with multiple representations of linear functions.  As Kaput (1992) affirms, linking one representation with the others together in the same computer screen was essential to understanding the advantages of technology in learning.  This capability also provided student with the understanding of how all the representations referred to the same concept (Keller and Hirsch, 1998).  Through the use of spreadsheets on the mathematics lessons, students interacted with all representations and they examined all the effects caused on representations when the x variable from a linear functions assumed different values.

            Mathematics achievement, specifically on linear functions, was one of the variables explored in this study.  This variable was measured through an achievement test, emphasizing multiple representations administered at the beginning and at the end of the treatment.  The results, reported on chapter four, indicate that at the beginning of the study, the control group exhibited a significant (p < .05) higher achievement than the experimental group.  It is assumed that the experimental group’s intensive use of spreadsheets and multiple representations in the mathematical lessons through the treatment reflected a significant (p < .05) improvement in achievement at the end of the study. 

Moreover, in order to explore students’ performance on achievement in specific content topics related to linear functions, the research instrument on achievement was divided in a cluster dealing with these topics.  A significant (p < .05) higher achievement was observed in favor of the control group in all of these areas at the beginning of the study.  Once the treatment concluded, no significant (p > .05) was found between groups. It was observed in the experimental group a significant gain (p < .05) in achievement in mathematics in the areas of graphs and slope.  Both groups exhibited significant (p < .05) improvements in the content of Cartesian coordinates once the teaching experiment concluded.  Interestingly, the two-way ANOVA reported significant (p < .05) interactions between effects (pre-post administrations and groups) in the topic of slope.

In terms of the representations of the linear function, the results revealed that in three of these representations (symbolic, tabular, and verbal) significant differences  (p < .05) were found at the end of the study in the experimental group.  That is, students in the experimental group performed higher at the end of this study in achievement on linear functions through symbolic, tabular, and verbal representations.  In terms of the graphical representations, both groups exhibited a significant improvement (p < .05) on achievement at the post administration of the test.  The analysis of variance (ANOVA) carried out between effects (groups and administrations of the test) reported significant interactions (p < .05) in graphical and verbal representations.

            The results of this study on achievement in mathematics, through the emphasis placed on multiple representations supported by the use of spreadsheets, suggest that the approach presented served to promote a better understanding of linear functions on students in a college algebra course.  It was observed that students, who used multiple representations and spreadsheets as part of their mathematics course, performed higher in achievement in mathematics (linear functions) at the end of the study that did the students in the control group.  The data from this study suggest that the multiple representations approach supported by technology was more successful in promote achievement gain than the traditional approach.  The findings of this research are supported by previous research in the field of representations done by Porzio (1994).  His studies revealed that the emphasis placed in multiple representations and technology was more adequate to promote understanding and connections between representations.

Answering the Fourth Research Question

4.      How well does the medium of a powerful spreadsheets like Excel, lend itself to promoting instruction through multiple representations?


            Spreadsheets were selected as the technology medium to be used in this research project because their capabilities to promote and show the multiple representations of the linear functions.  The software provided an engaging environment in where students explored all the representations separately first, and then, all together in the same worksheet.  The linking process between representations strongly recommended by Kaput (1992), Keller and Hirsch (1998), and Dufour-Janvier, et al. (1987) was particularly shown on the spreadsheets. 

Mathematical lessons based on spreadsheets developed in this study, allowed students in the experimental group to explore the effects of different values on the representations. This technology fulfilled the objectives set for this study and supported the instructional activities.  Not only multiple representations were supported by the use of spreadsheets,  the mathematical concepts (Cartesian coordinates, graphs, and slope) taught during the course, were introduced through this technology.  Recognized scholars such as Fey (1989), Goldenberg (1987), Kaput (1992), and Porzio (1994) have noticed that technology has contributed to increase the access of multiple representations of  mathematical concepts.

            In terms of promoting instruction, the results on achievement for this study showed that the spreadsheets approach using multiple representations was more adequate than the traditional approach.  Furthermore, this approach based on spreadsheets seemed to serve to enhance higher attitudes toward mathematics and technology.

Comparing the Two Approaches Used in this Study: The Multiple Representations and the Traditional


            The purpose of this section is to present how the same mathematical concept was taught using two different approaches.  The multiple representations, supported with spreadsheets approach used with the experimental group and the traditional used with the control group.  The topic discussed in this class was Cartesian coordinates.


Multiple Representation Approach

            In the class dealing with Cartesian coordinates, students in the experimental group used a worksheet.  In the first part of this activity, six different coordinates were given and students had to fill in the corresponding blanks the location of each one of these points.  In the second part of the activity, students offered their own coordinates, different from the presented in part one, and satisfying certain given locations. Then, using spreadsheets they were asked to locate the coordinates and explore the effects of points locations when the values of x and y in (x, y) changed.  To end the activity, students were encouraged to get printouts in order to show their work.  Finally, they  find an equation related to these coordinates and told a story about the application of Cartesian coordinates in their fields of study.  Figures 19, 20, and 21 show the first and the last parts of this activity done by Student 22.  The answers to the questions on the worksheet appear in italic font and printouts from spreadsheets appear in the following set of figures.

Figure 19. Worksheet sample Cartesian coordinates.

Locate in the Cartesian plane the following coordinates. Also, determine the quadrant where the points are located.


A (2, 4)


First Quadrant


B (-5, 3)


Second Quadrant


C (-9, -7)


Third Quadrant


D (6, -1)


Fourth Quadrant


E (0, 2)


Positive y-axis


F (-5, 0)


Negative x-axis



Figure 20. Spreadsheet sample on Cartesian coordinates.



Figure 21. Sample worksheet on Cartesian coordinates.


Give an example about an equation that can relate one of the coordinates stated above.


The equation y =  is satisfied by the point (2,4). Because if you substitute the values on the equation: 4 =  you get a true statement: 4 = 4.


Tell a story about the applications that Cartesian coordinates may have in daily situations or another fields:


I think that in medicine they serve to imagine two cut points in a surgery.  Also, to make a map of the head where it is divided in sections.  Finally, coordinates may be used to measure distances between different bones of the body.


Traditional Approach

            This approach was based on instructor lectures.  The only resource used in this lesson was the course textbook.  Figures 22 and 23 show samples of a student’s notes from the control group.  All notes are in Spanish.

Figure 22. Sample of student’s notes on Cartesian plane.


Figure 23. Sample of student’s notes on Cartesian coordinates.


Lessons Learned by the Researcher

            Mathematics Reasoning (MRSG 1010) is a college mathematics core course, intended primarily for freshmen students, where technology has been slightly used throughout the semesters that it has been taught.  Although this course has been reviewed in multiple times in terms of its objectives and how they have been fulfilled or not, the use of more technology has been limited and unfortunately, out of the realm of discussion. Nevertheless, regular instructors have supported mainly the use of calculators as a supplement to instruction.  The researcher was convinced that another kind of technology (beyond calculators) could be used in this course to enhance achievement in mathematics.  Therefore, the main challenge for the investigator of this research was to incorporate the intensive use of computers, particularly the use of spreadsheets to the instructional activities dealing with linear functions of this course.                                                

The incorporation of this technology did not constitute an easy activity in this project.  In the beginning of the study, the students exhibited surprise and sometimes lack of belief about how through the use of spreadsheets they could learn the same mathematical content knowledge as did the traditional students.  Furthermore, the researcher had to deal with the lack of expertise on students in the use of spreadsheets.  As stated earlier, this experience using technology represented to the majority of the students their first time using computers in a mathematics course.  In order to correct this deficiency, the instructor of this study spent some class periods teaching the basic features of spreadsheets and how they can be used in mathematics.

In this research project, the investigator dealt mainly with two important aspects: (a) the mathematics topics included in the course syllabus taught during the length of the study, and (b) students attitudes dealing particularly with the concern if spreadsheets will work or not in order to learn linear functions and related themes.  The first aspect was fulfilled when the instructional topics were taught parallel with the traditional group and in accordance with the syllabus.  With the purpose to fulfill the second aspect, the investigator had to motivate, encourage and over all, show and introduce each content topic, in each class session using spreadsheets.  In this way, students realized, once the treatment concluded, that this technology constituted a really invaluable tool to learn mathematics.

Some Comments from Students

            This section presents some sample of comments from students in the experimental group about their experiences using spreadsheets to learn linear functions in mathematics.  These data was collected through a weekly electronic journal that students sent to the instructor of the course. The electronic journal contained two questions: a) What is the big idea that you learned in the class? and b) What topic was difficult or unclear for you?.

            During the first week of instruction.  Here are some comments from students.

            Student 27 said:

The most important thing was how do the graphs in Excel.


Student 25 said:


The most important idea in this class was how work with the computer and learn something new with the computer.


Student 22 said:


In my opinion, the most important thing in our class discussion was the explanation of the spreadsheets as tool for the course.


            During the second week of instruction. Here are some additional comments from students in the experimental group.           

Student 22 said:

The most important idea for me was the different representations of the linear equation and how works with them based in paper and pencil and using spreadsheets.


            Student 13 said:

The Cartesian plane has been explained perfectly.  Although I realize that I found it a little difficult to understand.  It is not that it was not well explained, but I am not skillful with computers.


Student 26 said:


The class with the computer becomes more easy for me that the class just explaining on the board.


            Other colleagues who were teaching the course during the semester that the study took place, or who had taught the course before, expressed interest in knowing how students were performing in the lessons activities based on spreadsheets.  For the majority of the regular instructors of MRSG 1010, the use of spreadsheets and multiple representations to teach linear functions constituted something new and innovative.

            As a result of the intensive use of technology in a college level mathematics course, the researcher’s approach to teaching changed rather dramatically and was reinforced in terms of promoting technology use to enhance achievement in mathematics.  As the report Shaping the Future from the National Science Foundation (George, et al., 1996) affirms, technology should be available to all students, and they need the opportunity to work with it, and get expertise using it as a tool of their learning.  It is reasonable to interpret here that technology in mathematics is intended to first; to provide an environment to promote understanding and achievement, to get help, offer alternatives, and sometimes solutions to certain problems.

            This report discusses some barriers found in educational settings.  The ineffective use of instructional technology is one of them, pertinent to this study.  This problem consists in “a specific lack of knowledge about the hardware and technology that has been spreading into increasing use, and to which many students are already attracted” (p. 44).  Discussing on this issue, the report cited the Director of the Science and Technology Resource Center at the Prince Georges County Community College.  She said:

I see the following as serious problems… the challenge of teaching faculty and students how to access, utilize, and incorporate the vast amounts of information available in print and electronically, and learning how to utilize technology in making education more attractive to students who might otherwise lack motivation or interest in science, mathematics, engineering, and technology courses. (p. 44)


These problems identified in the report were also found in this research experience.  The first situation was discussed at the beginning of this section.

Technology has been erroneously used if it promotes misconceptions, confusion, and unclear solutions.  It is important to point out here, that certain students who participated in this study developed an excessive dependency to the calculators or computers use to perform simple mathematics computations and they were unable to perform them without these technological equipments.  This represents an example about the inappropriate use and promotion of technology, where it is believed that computers and calculators are magic boxes to solve all the problems in mathematics.  As cited in the NSF report, Noam (1995), said: “Technology would augment, not substitute” (p. 32).



            This study through the design of mathematics lessons on linear functions, examined first, three major variables: prior achievement in mathematics (based in reported grades), achievement in mathematics, and attitudes toward mathematics.  Second, it was compared two teaching approaches: the multiple representation supported by technology (spreadsheets) and the traditional.

            Based on the analyses of the data provided by the various instruments, this researcher draws the following conclusions regarding the variables of concern.

            With respect to the prior achievement in mathematics based on reported grades, it is concluded that this variable did not play a significant role in determining how much mathematics was learned in the two groups, experimental and control.

            Regarding students attitudes toward mathematics, it is concluded that if positive and higher attitudes are observed, they are a possible factor to enhance achievement in mathematics.  Students in the experimental group had somewhat positive more attitudes toward mathematics and performed higher in achievement.

            In regard to the achievement in mathematics, this researcher concludes that mathematics lessons emphasizing multiple representations supported by the use of spreadsheets constitute an appropriate teaching approach to enhance a broad achievement on linear functions.  Students taught with the multiple representations approach achieved higher on linear functions than students taught with the traditional approach. 

            Finally, regarding the comparison of the two teaching approaches, it was found in this study that the approach based on multiple representations and spreadsheets is more effective than the traditional, in promoting and enhancing achievement in linear functions as well as more positive attitudes toward mathematics.

Limitations of the Study

            The following section presents the limitations of this study.  The short period of time (just four weeks of instruction) devoted to the teaching experiment, instead the whole academic semester, was the first limitation of this research project.  Other findings could be expected if more time was added to the treatment.

            The numbers of topics taught during this study was other limitation of this project.  Content topics strongly related with linear functions were only emphasized throughout the study.  Future studies could explore the use of representations with other content topics discussed in a college level algebra course.

            The researcher was the instructor of the two groups of this study: control and experimental.  Additional research in this field can determine the effects, and other results of using more than one teacher in similar conditions.

            The “Hawthorne effect” constituted the another limitation of the study.  The changes observed at the end of the experiment on attitudes and achievement in linear functions between control and experimental groups could be as a result of other factors involved during the length of the treatment, rather than the emphasis placed on multiple representations of linear functions and the intensive use of spreadsheets.  Examples of these factors could be: teacher style, class environment, and students’ interests.



            The results of this study suggest the following recommendations.  Some of these recommendations agree with the included in the NSF report Shaping the Future (George, et. al, 1996).

            To the university and administrators:

1.      To re-focus the mission of the General Education Program, particularly the section dealing with reasoning skills, where MRSG 1010 is part, taking into consideration students’ needs and interests.


2.      To review this mathematics core course establishing an appropriate description in accordance to technological advances.


To the department:


3.      To provide an attractive curriculum in undergraduate mathematics that students can feel that mathematics is useful in their fields of specialization.


4.      To encourage faculty to use in the teaching of this course other technological tools, such as, spreadsheets.


5.      To provide and promote curricular innovations in the teaching of this course.


6.      To encourage faculty members to do research in all mathematics courses and the publication of their findings.


To the mathematics faculty:


7.      To believe that all students can learn mathematics in different ways and to create and harmonious and attractive environment that can engage students in their learning.


8.      To suggest a curricular review, specifying the mathematical topics that should be taught in MRSG 1010.  To recommend what content should receive more emphasis and what should receive less.


9.      To believe that technology has changed the education in mathematics and the use of it should not be omitted.


10.  To promote and encourage the use of technology (such as spreadsheets) in all mathematics courses, not only used as supplement of instruction.


11.  To explore the use of other teaching approaches, such as multiple representations.  Using this approach, each representation of the same concept is taught and emphasized letting students effectively manage these representations.


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