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30 Manipulatives Rivera, Sinfuego, Tudayan, Limjap Use of Manipulatives to Develop Selected Second Year High School Students’ Understanding of Equality and Linear Equations Melanie Rivera, Mary Grace Sinfuego, Adrian Paul Tudayan, and Auxencia Limjap2 Science Education Department, De La Salle University –Manila Abstract This study investigated how the use of manipulatives can improve the understanding of the concept of equality using the symbol of equal sign, and the procedural skill in solving algebraic equations of second year high school students. The researchers used a triangulation method in the data gathering process through assessment materials, interview of the teacher of the students, and observation of the researchers. The manipulatives served as instrument or pedagogical tool to explain both the concept and the process. Results show that with the help of the manipulatives as pedagogical tool, the students retained and improved their relational understanding of equality and their procedural knowledge as well as conceptual understanding of solving linear equations. Introduction Algebra plays an important role in students’ mathematics education which serves as a foundation for higher mathematics courses. First year high school is the most crucial stage in mathematics education since transition occurs in students’ learning of Mathematics from the numerical skills of Arithmetic to the abstract concepts of Algebra. However, confusion exists in interpreting relationships and procedures given to the students (Kieran, 1981). Students have difficulties in solving algebraic equations. The foundations for learning the fundamental concepts of Algebra might need to be developed. The understanding of symbolic arithmetic maybe so weak, that they are not able to translate those symbols and concepts to Algebra. Thus, students’ knowledge of fundamental concepts is important so they can use those concepts and skills in Algebra. One very important concept is the equality relationship of real numbers. Equality, as defined by Elgueta & Jansana (1999), is an idea of having an “identity relation” between things being compared. It gives the notion that these things being compared are related to each other and that all aspects of the relationship must be balanced. As such, it is very important to find out whether students understand equal sign as a relation or merely an operation (Pratt, &, Jones, 2005). A research (Stephens, Knuth, McNeil, &, Alibali, 2006) was done in the American Midwest which found out that: Students who understand the equal sign as a relational symbol of equivalence are more successful in solving algebraic equations than their peers who do not have such an understanding. (p.310) The researchers also noted that understanding of equal sign matters especially on students’ performance in solving equivalent equations. It is a concept that clarifies to the students the procedures involved in solving equations. This study examines the conceptual and procedural understanding of equal sign of second year high school students when they solve linear equations in Elementary Algebra. According to Pratt et al (2005), students may view the equal sign in different ways. It is treated as a relation by some students, but others treat it as an operation. The conceptual framework is described by Figure 1. 2 Dr. Auxencia A. Limjap is currently a scholar in residence at the Far Eastern University, Manila. Intersection 31 2015 Vol 12 (1) Sec on d Year Hig h Sc hoo l Stud ents’ Un der stand ing o f Equal Si gn -‐ O pera tion al -‐ Re lati onal -‐ O ther Alter nativ e -‐ No kn ow ledge C urr ent S kills in Linear Equatio n co nsis t of: -‐ Pro ced ural Kno wledg e -‐ C onc eptua l Un der stand ing T he Use of Manip ulativ es I mpr ov ed C onc eptua l Un der stand ing of Equ ality I mpr ov ed Pro ced ural Sk ills in So lvin g Linear Equatio ns Figure 1. Process that enhances student’s understanding of equality and linear equation Conceptual Framework Students may have alternative conception or they have no knowledge at all (Knuth, Stephens, Hattikudur, McNeil, &, Alibali, 2008). Students’ skill in solving linear equations consist of their procedural knowledge and their understanding of the concept behind the procedure. Therefore, the kind of understanding that they have on equal sign contributes to the improvement of their conceptual understanding as well as their procedural skill in solving linear equations. The researchers used manipulatives such as counters and rectangles as the intervention in this study in order to enhance what they already know about equal signs and linear equations. These counters and rectangles serve as pedagogical tools to raise the level of understanding of the students from the concrete level using objects to represent balance, to the abstract level using symbols, concepts and relations to solve linear equations. Methodology This research is descriptive in nature using exploratory case study. To investigate students’ understanding of equality and develop their skills in solving linear equations, manipulatives were used. Qualitative data were gathered from the worksheets, pre-‐test and pos-‐test. Interview was conducted to validate their written response and for triangulation. The triangulation method of data collection is used in this study for credibility of the findings in this research. This study is focused on second year high school students, since they already have knowledge of linear equations that was formally introduced to them during their freshmen year. There were six participants in this study who are considered as average students in their Algebra class from a public high school in Marikina City. Their Algebra teacher picked them based on their class standings in the academic year 2010-‐2011. The researchers used the triangulation method in gathering data for this study. Data were gathered from (1) the students through the series of assessments; (2) the teacher, who gave recommendations and (3) the researchers, through the observed reactions, interactions and interviews that were conducted. The participants were given a five-‐item pre-‐test adapted from a prior research (Stephens et al, 2006) that targets students’ understanding of equality. The first three items are on the students’ interpretation of the equal sign and the next two items are on solving linear equations. Aspart of the pre-‐
assessment, the researchers interviewed the students one at a time regarding their answers in the pre-‐test. This gave the researchers deeper insight and information about the nature of their knowledge and understanding of the concepts. With the data gathered from the students, researchers introduced to them the use of manipulatives – counters and rectangles. There were five one and a half hour sessions wherein the researchers presented the Properties of Equality and the procedures in using the manipulatives. To monitor students’ progress, they answered worksheets for each topic throughout the study. They responded to another seven-‐item post-‐test to check how their understanding and skills have improved. As part of the post-‐assessment, a focused group discussion was conducted by the researchers in relation to their attitude towards the use of manipulatives in solving linear equations. There are four instruments used in this study – the pre-‐test, the worksheets on the use of manipulative counters and rectangles, the post-‐test and the interview protocol. The presentation of the manipulatives was based on the approach introduced by Canete, C., Sta.Maria, M., Soriano, M., &, Limjap, A. (2007). 32 Manipulatives Rivera, Sinfuego, Tudayan, Limjap The five-‐item pre-‐test was adapted from Knuth et al, (2006). It consists of two parts, namely the interpretation of equal sign and the application of their understanding of equal sign. In the interpretation part, students were asked to identify the name of the given symbol, and to interpret its function. In the application, however, the students are asked to solve for the given linear equation. Figure 2. The white rectangle represents a positive variable while the shaded rectangle represents the additive inverse of the variable. + + + Figure 3. Representation of Positive Integer -‐ -‐ -‐ Figure 4. Representation of Negative Integers In the intervention, each participant was given a manipulative kit that includes ten pieces of two-‐
sided rectangles, ten pieces of white chips and ten pieces of colored chips. The two-‐sided rectangles represent the variable or the unknown value while the white chips and the colored chips represent the positive and negative counters respectively. Worksheets about the use of the manipulatives were distributed at the end of each session for students to accomplish inside the classroom. The post-‐test was administered at the end of all the sessions. This test consists of seven items divided into two parts. The first part asks the students to represent given linear equations using counters, rectangles and algebraic symbols. The second part deals with solving linear equations algebraically. The researchers conducted a focused group discussion regarding their experience in using the manipulatives and how they see it as a tool for learning Algebra. The students were asked to explain their answers as they were given probing questions during the interview. Table 1. Summary of the Categories in Interpretation of Equal Sign Relational Views equal sign as a signal of two quantities that are equal or that have the same value. Operational Sees equal sign as “number-‐
operator sequence” (Pratt & Jones, 2005). Interprets it as a signal for the answer or that it separates the given from the answer Other Views equal sign as it is. Alternative Responses that can be categorized under this are the word translation for it (i.e. 4 + 4 = 8, a response could be, four plus four “is equal to” or “equals” eight) No If the students do not have Knowledge response or do not really have a prior knowledge of equal sign In analyzing the data in the pre-‐test stage, there were no frequency counts. Instead, every response was analyzed. The pre-‐test is divided into two parts: the first part is the interpretation of equal sign. The researchers adapted the coding used by Knuth et al (2008) in interpreting students’ responses for items 2 and 3. The interpretation of equal sign is categorized into four namely – operational, relational, other alternative interpretation and no knowledge. The answers of the students were categorized as operational if their response has something to do with “number-‐operator sequence” (Pratt et al, 2005). This means that the students find the equal sign as a signal for “the answer” (Knuth et al, 2008). It is relational when the students express their understanding of equal sign as equivalence or two quantities that are equal (Knuth et al, 2008). While the other alternative category is used if the students’ answers are just the direct translation of “it means is equal to”, “5 + 4 is equal to 9” (Knuth et al, 2008). This means that the student only answers the word translation of the said equation. Answers that are just the definition of the symbol such as “it means equals” or “it means is equal to” as stated in Knuth et al (2008) coding, fall under the other alternative. If the student said that he or she does not know the answer or if there is no response for the item, it is considered under no knowledge category. The second part of the pre-‐test focuses on Intersection 33 2015 Vol 12 (1) students’ skills in solving linear equations. These are categorized into two – procedural knowledge and conceptual understanding. Students’ skills in solving linear equation were categorized as either procedural knowledge or conceptual understanding. According to Schneider et al (2005), procedural knowledge is when students are able to solve linear equations by following methods to solve problems using operations. Procedural knowledge is where students understand the rules and procedures for solving linear equations. There are students who solve fast resorting to “automated” or “unconscious thinking” by “combining two rules without knowing why they work” (Haapasalo & Kadijevich, 2003). This can be improved through drill practices in solving problems. These students may know the procedures but not necessarily the concept behind the process. On the other hand, conceptual understanding is when students are able to solve problems by “connecting rules, algorithms and concepts given various representation forms” (Haapasalo & Kadijevich, 2003). Students with conceptual understanding are able to understand very well the concepts of the rules, algorithms and procedures. In contrast with the procedural knowledge, a student who has the conceptual understanding not only recognizes how to solve the problem but also comprehends what happens when rules are combined. In the intervention, the researchers taught the students how to use the manipulatives (counters and rectangles) in solving equations for them to enhance their conceptual understanding of the equal sign. With this, the researchers discussed the representation of rectangles, positive and negative counters. Rectangles represent the unknown value. Positive and negative counters are used in representing positive and negative numbers respectively. Worksheets were given at the end of each session to assess the students’ understanding and learning in the discussion for each day and whether they have misconceptions. Each worksheet was assessed by item. Researchers analyzed the common items where the participants had difficulties. The post-‐test deals with the learning that the students gained and their ability to apply it in solving linear equations. The first part summarizes their learning during the intervention. This is assessed, analyzed and compared with their performance in their outputs. The second part determines if the participants are able to apply the relationship found in equality. Responses to every item were analyzed by comparing these with the common items in the worksheets where the group had difficulties. To probe deeper into their understanding of equality the researchers interviewed the students, as a group, regarding their answers to the post-‐test. Evidences of “relational” understanding of equality were sought during the focused group discussion. The purpose is not only to report the result of the intervention but to help the students revisit the lessons given during the intervention. Results and Discussion A pretest preceded the seven-‐day implementation of the instruction and the performance of Student A, Student B, Student C, Student D, Student E, and Student F, to assess students’ prior understanding of equal sign, the following questions were asked: 5𝑥 – 9 = −4𝑥 (1) The arrow above points to a symbol. What is the name of the symbol? (2) What does the symbol mean? (3) Can the symbol mean anything else? If yes, please explain. According to Student C, she sees equal sign as “equality or if the number is equal to the other number or they have the same the value”. Thus it was classified as relational understanding. On the other hand, Student D exhibits an operational understanding when he wrote the following: Figure 5. Student D’s answer to the second item Student F exhibits alternative conception when he gave the following response to the written inquiry: Figure 6. Student F’s response to the third item. 34 Manipulatives Rivera, Sinfuego, Tudayan, Limjap When asked the following probing questions, it became clear that he has a different understanding of equality and equation. Reseacher: What do you mean by this? *pointing to his answer* Student F: Pinaghihiwalay po ‘yung equation sa sagot. (It separates the equation from the answer). Researcher: So, 5x – 9 is an equation while –4x is the answer? Student F: Yes. For him, equality is not part of the equation and these are two different entities. The Students’ understanding of the symbol of equality can be summarized as follows: Table 2. Summary of Students’ Understanding of Equality Prior Understanding Understanding of of Equality Equality after the (Interpretation of Intervention Equal Sign) Student A Relational Relational Student B Relational Relational Student C Relational Relational Student D Operational Relational Student E Operational Relational Student F Other Alternative Relational -‐ Misconception Half of the participants in this study have relational understanding of equality. These students (Student A, Student B and Student C) see equal sign as denoting equality or as both sides having same value. Student D and Student E exhibited operational understanding of equal sign. They see it as a signal or a means to separate the answer from the operations or the solutions. With this kind of understanding, they do not see the relationship that exists between the two quantities being separated by this symbol. Student F was classified as having other alternative understanding of equality. Probing his answer shows that this understanding is a misconception. After the pretest, there were six worksheets given to the students in the intervention program. At the start, the researchers reviewed the students on the fundamentals of equations such as the definition, examples of equations and distinguishing equations from the non-‐equations. Each participant was given a manipulative kit that includes ten pieces of two-‐
sided rectangles, ten pieces of white chips and ten pieces of colored chips. The two-‐sided rectangles represent the variables or the unknown values while the white chips and the colored chips represent the positive and negative counters respectively. The students were taught how to represent an equation using chips and rectangles. The first worksheet measured the basic knowledge of the students about the definition of equation. It asks if the given mathematical statement is an equation or not an equation. All of the students got the correct answer. Students knew that a mathematical statement is an equation, if it has an “equal sign.” The students retained and improved their understanding of equality to the relational level. In the post-‐test, the conceptual understanding of all students were classified under relational. The participants already have a background of solving linear equations since they learned it in algebra in the previous year. Inspite of that, two-‐
thirds of the participants have procedural knowledge in solving linear equations. They exhibited misconceptions when they solved linear equations. An example of the lack of conceptual and procedural understanding in solving linear equations is shown in the following solution of Student C. Second Fourth & Fifth lines Figure 1. Student C’s solution for the 4th item of the Pre-‐test Student C puts together an algebraic expression and a constant thus 2x-‐3=-‐1x and -‐7x+3=-‐4x. She writes a meaningless expression in the fifth line “ = -‐
6x,” where she seems to think that “24x ÷ -‐4x = -‐6x.” While the computation is procedural, she lacks understanding of the procedures. Even with manipulatives, participants struggle with the concepts. Take the work of Student A as an example. Intersection 35 2015 Vol 12 (1) Figure 8. Student A’s work for the third item in Worksheet 3 However, it was easy to refer to the concrete representations of unknown or variable x and constants and help her realize that we can put together only the figures with the same shape. Consequently, she raised her knowledge of the procedure to the conceptual level in the post test. Table 3. Summary of Students’ Skills in Solving Linear Equations Prior Skills in Improved Skills in Solving Linear Solving Linear Equations Equations Student A Conceptual Conceptual Student B Conceptual Conceptual Student C Procedural Conceptual Student D Procedural Conceptual Student E Procedural Conceptual Student F Procedural Procedural The second part of the first worksheet and all other worksheets were about the representation of algebraic equations using counters and rectangles. These worksheets include application problems. Their math difficulty is evident in the work of Student E who failed to translate this problem situation algebraically. Figure 9. Student E’s work for Application in Worksheet 3 The intervention remedied this situation by using concrete objects to represent the problem situation. See the work of Student D. Figure 10. Student D’s work in Worksheet 4 After the intervention, it was seen that five out of six participants exhibited conceptual understanding. Student F remained to have misconceptions and thus kept his skill at the procedural level. He had difficulty explaining how he came up with his answers. However, since 5 of the 6 participants improved their understanding of equality and linear equations, this study shows the value of manipulatives as a visual pedagogical tool in teaching abstract algebraic concepts. References Canete, C., Sta.Maria, M., Soriano, M., &, Limjap, A. (2007). Essential Mathematics: Elementary Algebra. Phoenix Publishing House. Elgueta, R. & Jansana, R. (1999). Definability of Leibniz Equality. Retrieved on November 9, 2009 from http://www.jstor.org/stable/20016085. Haapasalo, L and Kadijevich, D. ( 2003). Simultaneously activation of conceptual and procedural mathematical knowledge by means of ClassPad. Retrieved on August 13, 2010 from http://hal.inria.fr/docs/00/05/45/38/PDF/de52t
h1.pdf. Kieran, C. (1981). Concepts associated with equality symbol. Retrieved on August 18, 2009 from http://www.jstor.org/stable/3482333. Knuth, E., Alibali, M., Hattikudur, S., McNeil, N., & Stephens, A. (2008). The Importance of Equal Sign Understanding in the Middle Grades. Retrieved on December 7, 2009 from http://itp.wceruw.org/Fall%2009%20seminar/K
nuthetal08.pdf. Pratt, D., &, Jones, I. (2005). Three utilities of equal sign. Retrieved on August 15, 2009 from http://www.eric.ed.gov/ERICDocs/data/ericdocs
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derStern2005.pdf. Stephens, A., Knuth, E., McNeil, N., &, Alibali, M. (2006). Does understanding the equal sign matter? Evidence from solving equations. Retrieved on August 7, 2009 from http://psych.wisc.edu/alibali/files/Knuth_Stephe
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