Virtual Algebra Tiles: A pedagogical tool to teach and learn algebra through geometry
Abstract
Learning and teaching algebra concepts are frequent problems for both students and teachers alike. Students struggle to learn algebra, and teachers struggle to identify the best way to teach algebra. It is recurrent that teachers use instructional methods that promote memorization, and, consequently, the learning process may prove unsatisfactory. This paper introduces Virtual Algebra Tiles, an open educational resource whose purpose is to enable students to use algebra tiles from a computer to enhance their knowledge of algebra concepts. To evaluate the effectiveness of the system, we designed a case study. The participants were 40 students of engineering degrees who were randomly assigned to experimental or control groups. Results indicate that the system has a large effect size (ES = 0.83, p < 0.001) on learning effectiveness and a promising long‐term knowledge retention (ES = 0.79, p < 0.001). Students concluded that this version of algebra tiles is more motivating than other versions, because it is presented in a format that stands natural to them. Students feel more comfortable in virtual environments, and therefore, the learning process becomes a more pleasant task.
Lay Description
What is already known about this topic:
- Inadequate teaching methodologies can lead to student failure.
- It is necessary to decrease the emphasis on memorization and to favour active learning techniques.
- The use of mathematical manipulatives increases students' learning gains.
- Algebra tiles are an educational resource that helps students understand algebra concepts by using geometric figures.
What this paper adds:
- Virtual Algebra Tiles is a multimedia version of the educational resource algebra tiles.
- Virtual Algebra Tiles has a large effect size on students' learning achievements.
- Virtual Algebra Tiles improves students' long‐term knowledge retention.
Implications for practices and/or policy:
- Teachers should participate in the development of digital pedagogical material to enrich their lectures.
- It is necessary to develop more open educational resources, which can be accessed by students and teachers.
- Digital educational resources must take into account the special needs of particular users.
1 INTRODUCTION
Sometimes, it seems easier to think that the students' failure is their own guilt. We tend to argue that they have to assume the whole responsibility of their performance. We consider that our task as teachers is limited to teaching concepts and the success or failure of the students is beyond our concern. Namely, we teach the subjects, but if our students do not understand, it is just because of them and we have nothing to do about it.
However, reality shows that in many cases, student failure obeys not precisely to lack of talent or academic difficulties. Failure is commonly caused by obsolete teaching methodologies that do not encourage students to achieve the academic goals (Boaler, 2013). The consequences of using inadequate teaching methods are evident in subjects related to mathematics. Different studies (Bowen, Chingos, & McPherson, 2009; Stinebrickner & Stinebrickner, 2014) show that attrition rates in universities attributable to low academic performance are increasing worldwide. One possible cause of this is the old‐fashioned teaching methods that no longer fit the needs of the students.
In contrast, many studies demonstrate that the use of information technologies (IT) in education is effective and helps to improve the academic level of students (Keengwe & Bhargava, 2013; Laurillard, 2002; Sotiriou, Riviou, Cherouvis, Chelioti, & Bogner, 2016). Moreover, the use of IT resources favours academic retention and stimulates students in values such as autonomy, responsibility, and curiosity, among others (Su & Cheng, 2015; Ting, 2015; Zhou & Purushothaman, 2018).
Hence, teachers, researchers, and practitioners from all over the world have developed a large number of pedagogical tools to support teaching processes. These resources provide students with different alternatives for acquiring knowledge, as recommended in the Universal Design for Learning guidelines (2011). All of these are based on the framework of Education 2.0 (Tîrziu & Vrabie, 2015) that is progressively evolving to a new concept: “Education 3.0.” This new concept includes technologies such as augmented reality and virtual reality that allow students to recreate abstract theories in a more significant way (Garzón, Pavón, & Baldiris, 2017; Hung, Chen, & Huang, 2017; Martín‐Gutiérrez, Fabiani, Benesova, Meneses, & Mora, 2015).
This paper has two main objectives. First, we describe general aspects of the development process of a digital version of the educational resource algebra tiles named Virtual Algebra Tiles (VAT). Second, this research proposes to identify the relevance of using the VAT resource, as a tool to improve students' academic performance, addressing the following research questions (RQ):
- RQ1:
What is the effect of the VAT resource on the learning gain of users?
- RQ2:
What is the degree of user satisfaction after using the VAT resource?
- RQ3:
What is the long‐term effect of the VAT resource on the learning gain of users?
To answer the research questions, we conducted a case study with 40 engineering students from a private university in Colombia. The study compared the results of students taught using the VAT resource (experimental group) with those of students taught through traditional lecture methods (control group). Traditional lecture method refers to the method in which the teacher imparts knowledge through oral language (Xing‐ju, Lin, & Gui‐feng, 2013). This method includes the teacher, the books, and the classroom as the centre of the process.
To identify the effect of the VAT on users learning gains (RQ1), we evaluated the effect size (ES) of the system on the learning effectiveness of the users. In this context, learning effectiveness is defined as the improvement in a user's learning between the beginning and the end of the intervention through the application. Likewise, to measure the degree of user satisfaction (RQ2), we applied a satisfaction survey that uses a 7‐point Likert scale. Finally, to identify the long‐term effect of the VAT on the learning effectiveness (RQ3), we evaluated the ES of the system on the learning effectiveness of the users in the context of a follow‐up test.
This work can be a guideline for teachers, researchers, and practitioners who want to elaborate their own digital pedagogical material to enrich their lectures. In addition, we want to show the positive results that can be obtained when students are motivated with the right pedagogical tools. The results of this study lead us to consider that using this type of pedagogical aids adds value to the learning process. The usage of IT resources seems to be more attractive for students and motivates them to continue reviewing the topics autonomously in different spaces other than educational institutions.
The remainder of this paper is structured as follows: Section 2 analyses previous related work including a formal definition of the pedagogical resource algebra tiles. Section 3 presents the development process of the VAT through the instructional design model ADDIE and the work done in each of its stages. Section 4 presents the methodology of the case study conducted to validate the efficacy of the VAT. Section 5 presents the main results of the case study. Section 6 discusses the meaning of the results and concludes the paper. Finally, Section 7 presents the limitations of the study and some considerations for future research.
2 RELATED WORK
Algebra tiles are a type of mathematical manipulatives (Kablan, 2016) and represent a pedagogical tool that helps students better understand algebra concepts by using geometric figures (Castro, 2017). Algebra tiles consist of squares and rectangles of different dimensions whose areas represent the monomials (Thornton, 1995). It allows users to simulate, among others, processes of adding and multiplying polynomials, solving linear systems, modelling algebraic expressions, and factoring polynomials, as shown in Figure 1.
These tiles can be acquired commercially or made at home using materials such as plastic, paper, or wood. Every piece is coloured on either side with a different colour to differentiate positive values from negative values (Leitze & Kitt, 2000).
The National Council of Teachers of Mathematics established the Curriculum and Evaluation Standards for School Mathematics (Hirsch et al., 1989). This curriculum encourages teachers to decrease the emphasis on memorizing the rules of algebra and to favour active learning techniques. Algebra tiles provide the students with a means to learn in an experimental way rather than simply studying abstract concepts from a book (Richardson & Bachman, 2017). It allows the teacher to use concrete models to introduce concepts, instead of focusing on instructional methods that encourage memorization (Caglayan, Olive, & Izsak, 2013; Saraswati, Ilma, & Putri, 2016).
Many guidelines aim to instruct on how to construct and how to use algebra tiles. Working With Algebra Tiles (MathBits.com, 2016) and A Concrete Introduction to the Abstract Concepts of Integers and Algebra using Algebra Tiles (MathGains, 2016) are good examples of this. However, after a careful review in web search engines and specialized bibliographic databases, we could not find any digital version of this pedagogical resource. That is why we decided to create a digital version of algebra tiles. Our goal was to generate an interactive, functional, and free tool, which students and teachers can use at any time from a personal computer.
The efficacy of the algebra tiles as a pedagogical tool to teach and learn algebra concepts has been widely proven (Agrawal & Morin, 2016). However, nowadays, students feel more comfortable when they are taught by using new technologies having into account that they were born in a digital era (Venkatesh, Croteau, & Rabah, 2014). Hence, the digital version of this educational resource happens to be interesting for students, who sometimes need extra motivation for learning.
3 VIRTUAL ALGEBRA TILES
It is common to follow the guidelines of an instructional design model to develop educational software. These models imply a series of common rules that seek to maximize the efficacy of the process of its creation. To create this open educational resource, we used the instructional design model “ADDIE.” This model represents a guide to facilitate the process of building effective training tools through five stages: analysis, design, development, implementation, and evaluation (Peterson, 2003). Next, we describe the work done in each stage.
3.1 Analysis
This stage represents the basis for creating the educational resource. Together with three mathematics teachers, we defined college‐level introductory algebra students as the target group. Then, we analysed the main characteristics of the target group, the previous knowledge of the students, and their specific needs. In this stage, we established the contents of the VAT. This educational resource has the potential to illustrate a large number of algebraic and geometrical themes. However, the educational resource had to focus on specific topics, at least, for its first version. We agreed to focus on factoring polynomials and assumed that future versions of the resource would cover other topics. The contents were defined according to the standards and competencies required by students in this level of education.
3.2 Design
In accordance with the recommendations of the mathematics teachers, we defined the structure of the resource following a collaborative design approach (Milrad, 2010). In this approach, teachers' knowledge and experience were the driving forces providing ideas and suggestions. Next, we determined that the VAT would contain four main components: first, the metadata that includes the objective, prerequisites, and technical requirements for using the resource; second, a section of examples in which users can find a series of examples on how to factor polynomials through the VAT; third, a section of practice that refers to a free practice workspace where users can practice at will what they have learned; and fourth, a section of activities, in which users are evaluated with eight exercises to assess their learning gains.
3.3 Development
This stage was built on the basis of both analysis and design stages, and its purpose was to elaborate the educational material for the VAT. All the educational contents—examples, activities, and theoretical explanation of the topics—were elaborated with the help of the three mathematics teachers. To structure and present the content on the World Wide Web, we used free software development tools. The software includes the HTML5 canvas element and JavaScript interpreted programming language to draw the graphics and CSS3 to set the visual style of the HTML elements.
An important recommendation at the analysis stage was to include accessibility features to ensure access to people with certain types of disabilities. Therefore, we took into account certain criteria provided by the Web Accessibility Initiative in the mobile accessibility guidelines (Web Accessibility Initiative, 2016). All the text of the application, as well as the explanation of every example, is accompanied with an audio recording. These audio guides improve usability for students with some visual impairments.
3.4 Implementation
This stage refers to the delivery of the resource to the target users. The pedagogical resource VAT was published in June 2016 and can be accessed free of charge. Students from different colleges and secondary schools have used it since then, and we will continue to expand its usage to extend its benefits. To validate the quality of the resource, we conducted a case study that will be explained later in this document.
3.5 Evaluation
The evaluation of the resource was carried out through the entire process: within stages, between stages, and after implementation. Evaluation within stages and between stages is called formative evaluation, and its purpose was to ensure the quality of the resource before implementation. After finalizing the software development and before its publication, we presented it to five mathematics teachers to explain to them its main characteristics. The teachers gave us their opinions about the functionalities of the VAT, its pertinence, the quality of the contents, and possible improvements.
On the other hand, summative evaluation was carried out after the implementation seeking to determine the quality of the pedagogical resource. Summative evaluation was held through the “Assessment Rubric for Open Digital Educational Resources” of the Ministry of Education of Colombia (2014). Four mathematics teachers, five students, and one software engineer evaluated the VAT through the aforementioned rubric in issues related to thematic, education, and technology.
The final average score for this evaluation was 88.35 out of 100. This result corresponds to “good” according to the rubric, confirming that the main goal of the project was achieved. Thematic (91.88) obtained the higher score, followed by education (87.88) and technology (85.38%). The evaluators highlighted the clear and coherent structure of the software, the quality of the contents, and the pertinence of this pedagogical resource. An aspect that should be improved, in accordance with the evaluation, is the number of topics covered in the resource. Teachers signalled the fact that algebra tiles are useful for learning not only factoring polynomials but also learning other algebra topics.
3.6 System functions
As a result of the collaborative work done with the help of the three mathematics teachers, the final version of the VAT presents the structure depicted in Figure 2.
The menu on the home page contains the following functions: the Introduction, which provides users with basic information about the resource; the Objective, which presents the learning objective of the resource; the Methodology, which tells users how to use the software, what the structure of the resource is, and the technical conditions to operate the system; the Prerequisites, which specifies the prior skills that users must have, related to informatics, algebra, and geometry; the Metadata, which gives technical information about the resource; and finally, Access the application. This function is the most important of the resource and contains three sections: Examples, Practice, and Activities.
3.6.1 Examples section
This section includes a series of 24 examples of different factoring methods designed by the mathematics teachers who helped in the process. Each example is explained gradually with the help of a text box and an audio file.
3.6.2 Practice section
In this section, the application divides the screen into two spaces: the “Available figures” panel and the “Workspace.” In the available figures panel, the user disposes six geometric figures that can be dragged to the workspace (Figure 3). This panel contains also a Help button, which offers practical instructions on how to manipulate the tiles and a Home button to return to the main page. Users can select a specific figure and drag it to the workspace. By clicking on the selected figure, it is possible to open a menu to use the functions of “Rotate figure,” “Transform sign,” “Eliminate figure,” and “Change variables” as described in the following:
- Rotate figure: The user can rotate the selected figure from vertical position to horizontal position and vice versa.
- Transform sign: The user can transform a figure from positive to negative and vice versa.
- Eliminate figure: The user can eliminate the selected figure.
- Change variables: By default, the dimensions of the figures are named with the variables x, y, and z. This function allows the users to change the names of the variables that they want to use.
Users can move each figure freely through the workspace in order to join it with other figures to simulate the factorization of any first‐ and second‐degree polynomials.
3.6.3 Activities section
In this section, users can access a test where they are evaluated in order to identify the learning gains. This test consists of eight exercises that the users must solve with the tiles. As users solve each exercise, the application indicates to them if they are doing the exercise well.
4 CASE STUDY
We conducted a case study to identify the relevance of using the VAT resource as a pedagogical tool to improve students' academic performance. The study compared the learning outcomes of the students who learned with the VAT resource and those who learned with traditional lecture methods. The study employed a mixed method approach and took place during a 5‐week period. The topics included in the study were greatest common factor, factoring by grouping, difference of two squares, and factoring quadratic polynomials.
4.1 Participants
The study included 40 first‐year engineering students enrolled in an algebra summer course at a private university in Colombia. Their ages ranged from 17 to 20 (M = 18.05, SD = 1.09). The students (23 males and 17 females) were randomly assigned to the experimental group (n = 20) or the control group (n = 20).
4.2 Treatment
On the first day, all the students gathered to receive basic information about the study. Then, students headed to one of two classrooms (depending on the group to which they belonged) and took the first test (pretest). After that, the students in the experimental group received instructions on how to operate the VAT resource. On the other hand, students in the control group received a list of books they could use as a guide.
On the second, third, and fourth days, the students of both groups received reinforcement of the topics using the respective methodology in 4‐hr sessions. Each session was conducted as follows: the first hour, the teacher explained the theoretical concepts; the second hour, the teacher did some exercises using the respective methodology; and the third and fourth hours, the students completed an activity composed of 20 exercises with the guidance of the teacher.
On the fifth and sixth days, students attended the respective sections. In these days, students were asked to study the subjects on their own following a task‐based approach. On the seventh day, the students took the second test (posttest). After completing the posttest, the students in the experimental group and the two teachers who participated in the study were asked to fill the satisfaction survey. One month later, all the students took the third test (follow‐up test). After completing the follow‐up test, the students in the control group received instructions on how to operate the VAT resource so they could use it afterwards.
4.3 Measuring tools
The pretest aimed to identify previous knowledge of the students of the topics included in the study. On the other hand, the posttest was designed to evaluate the learning achievement of the students after treatment. Likewise, the follow‐up test sought to identify the long‐term impact of the reinforcement. All three tests consisted of 20 multiple‐choice items with a total score of 5.0. Two experienced mathematics teachers, who had been teaching algebra courses for over 15 years, designed all the tests. The credibility of the tests was calculated using the Kuder–Richardson reliability formula. The values were found to be 0.78, 0.81, and 0.79, which are above the threshold of 0.7, indicating internal consistency reliability. Finally, the satisfaction survey was applied to evaluate the degree of users' satisfaction. It consists of 10 statements that use a 7‐point Likert scale. Each level ranges from strongly disagree (1) to strongly agree (7). This survey was validated and used in other investigations (Joo‐Nagata, Martinez Abad, García‐Bermejo Giner, & García‐Peñalvo, 2017) and modified to be applied in this research.
5 RESULTS
5.1 RQ1: What is the effect of the VAT resource on the learning gain of the users?
In order to guarantee an equivalent prior knowledge of the students in both the experimental and control groups, a t test was performed in terms of their pretest grades. Results in Table 1 show no statistical differences in the students of both groups, t(38) = 0.12, p = 0.91. This indicates that all the students presented similar previous knowledge on the topic “factoring polynomials.”
| Group | N | Pretest | t test | ||
|---|---|---|---|---|---|
| Mean | SD | t value | df | ||
| Experimental | 20 | 1.75 | 0.97 | 0.12 (p > 0.05) | 38 |
| Control | 20 | 1.72 | 0.92 | ||
To identify the outcomes of the students when using different learning methodologies, a t test was performed to compare the posttest grades between the two groups. As shown in Table 2, an important difference, t(38) = 2.74, p = 0.009, was found between the experimental and control groups.
| Group | N | Posttest | t test | ||
|---|---|---|---|---|---|
| Mean | SD | t value | df | ||
| Experimental | 20 | 3.35 | 0.92 | 2.74 (p < 0.05) | 38 |
| Control | 20 | 2.53 | 0.98 | ||
Subsequently, looking to determine the impact of the VAT in learning gains effectiveness, we calculated the ES. We used the means and standard deviations for pretests and posttests using Equations 1 and 2.
(1)
and
are the mean scores of the posttest and pretest of the experimental group, respectively.
and
are the mean scores of the posttest and pretest of the control group, respectively. Finally, SDpost is the pooled standard deviation for the posttest:
(2)| Variable | Value |
|---|---|
| Total sample size (N) | 40 |
| Effect size (ES) | 0.83 |
| p (ES) | <0.001 |
| 95% lower limit | 0.22 |
| 95% higher limit | 1.43 |
5.2 RQ2: What is the degree of user satisfaction after using the VAT resource?
After the posttest, the students in the experimental group and the two teachers were asked to evaluate the learning experience through the satisfaction survey (Table 4). At the bottom of the instrument, they had the opportunity to write personal comments about the experience.
| Question/affirmation | Codification | |
|---|---|---|
| 1 | I am comfortable using the resource. | Item_1 |
| 2 | It is easy to navigate within the resource. | Item_2 |
| 3 | The information displayed in the resource is accurate. | Item_3 |
| 4 | The resource has given me a positive impression about algebra. | Item_4 |
| 5 | The resource has given me important information for my learning. | Item_5 |
| 6 | The graphic design of the resource is visually appealing. | Item_6 |
| 7 | I was given enough information for the use of the resource. | Item_7 |
| 8 | I like the information that shows the resource. | Item_8 |
| 9 | It is easy to use the application. | Item_9 |
| 10 | I prefer the VAT resource to books to study algebra. | Item_10 |
- Note. VAT: Virtual Algebra Tiles.
Results of the satisfaction survey (Figure 4) show an average of 6.38 (out of 7), which indicates that the users felt satisfied with the resource. We can notice that Item_9, Item_1, and Item_2 are the items that have the greater acceptance. These items have to do with the ease of use of the application, ease of navigation within the application, and the convenience of using the application. On the other hand, Item_4 and Item_8 were the items with lower scores. These items have to do with the positive feeling of the user about algebra. However, these scores do not indicate disapproval to the resource, but a sense of indifference regarding algebra.
Students pointed out the benefits of the VAT as a pedagogical tool based on blocks. They argued that learning process becomes more meaningful when they can manipulate real objects. Besides, the students and teachers praised the resource for being highly interactive given that it provides feedback while a user performs an activity. An encouraging phrase used by a student when explaining his or her experience with the VAT was “If I had been taught algebra with these type of tools when I was in secondary school, I would not have had problems learning algebra.”
5.3 RQ3: What is the long‐term effect of the VAT resource on the learning gain of the users?
Finally, we performed a t test to determine the long‐term impact of the course according to the learning methodology using the scores of the follow‐up test. As shown in Table 5, a significant difference, t(38) = 3.43, p = 0.001, was found between the experimental and control groups, indicating that the VAT favours the long‐term knowledge retention of students. Likewise, the calculated ES for the follow‐up test was 0.79 with a 95% confidence interval of [0.33, 1.27] (Table 6), which corresponds to a medium ES, as indicated by Cohen (1992). Figure 5 shows the evolution of the scores of the students in the experimental and control groups during the whole process.
| Group | N | Follow‐up test | t test | ||
|---|---|---|---|---|---|
| Mean | SD | t value | df | ||
| Experimental | 20 | 3.00 | 0.76 | 3.43 (p < 0.01) | 38 |
| Control | 20 | 2.21 | 0.70 | ||
| Variable | Value |
|---|---|
| Total sample size (N) | 40 |
| Effect size (ES) | 0.79 |
| p (ES) | <0.001 |
| 95% lower limit | 0.33 |
| 95% higher limit | 1.27 |
6 DISCUSSION AND CONCLUSIONS
This study presented and evaluated VAT, a digital educational resource to learn basic concepts of algebra. This resource corresponds to an interactive web‐based visual representation of the mathematical manipulative “algebra tiles.” It enables students to engage and control the physical actions of each figure, allowing the users to construct mathematical knowledge.
The study compared this educational resource with traditional teaching methods (lectures) to identify its learning effectiveness. Experimental results and feedback from students and teachers suggest that the use of this resource is pertinent. Therefore, it should be included as an alternative pedagogical tool to teach and learn algebra in both secondary school and college‐level introductory algebra courses.
The results of the pretest revealed a similar academic level in each group. In the posttest, students in the experimental group showed a significant increase in their scores, outperforming students in the control group. The students in the control group improved their performance notably, albeit with a lower average score compared with that of the students in the experimental group. In the follow‐up test, the grades of the experimental and control groups decreased with respect to the posttest. However, the average score of the experimental group remained higher than the average score of the control group and was higher than the average scores in the pretest.
On the basis of the results, we conclude that learners who received instruction with the VAT achieved better results than did those who received instruction using traditional methodologies. In addition, results of the follow‐up test indicate that the long‐term knowledge retention was better in students in the experimental group.
These findings support the outcomes of previous studies (Agrawal & Morin, 2016; Saraswati et al., 2016; Richardson & Bachman, 2017). These studies have shown that the use of these mathematical manipulatives helps to increase student learning because they encourage relational thinking and promote algebraic reasoning.
The VAT resource allows users to represent models through the manipulation of geometric figures. As stated by the National Council of Teachers of Mathematics of the United States, representations are necessary for students to understand mathematical concepts. Representations allow students to recognize connections among related concepts and apply mathematics to problems solution (Joan & Martin, 2000).
7 LIMITATIONS OF THE STUDY AND CONSIDERATIONS FOR FUTURE RESEARCH
This study has some limitations that have to be kept in mind for further research. The benefits of the usage of the software VAT with respect to traditional methodologies were evaluated. However, the study did not include a comparison between the virtual version and the physical version of this educational resource. Besides, further research should include secondary education students in order to verify the effectiveness of the resource with this target group.
This study included two different teachers to teach each target group. This fact must be taken into account when interpreting the findings of this study, because it could bias the final results. Therefore, we recommend that future research include only one teacher to teach the two groups. In addition, to eliminate possible “novelty effect,” it would be interesting to conduct longitudinal case studies. These kinds of studies could help to identify more deeply the real benefits that this technology can contribute to education.
Given the good results found in this study, we will continue to spread the use of the VAT through secondary education institutions. The purpose is to extend the benefits of using this resource as an alternative to teach and learn algebra concepts. Further development includes the use of augmented reality technologies to enhance the pedagogical resource with the purpose of covering solution of polynomials of degree 3.
