Preservice Middle School Mathematics Teachers’ Definitions of Algebraic Expression and Equation

Using correct definitions of the mathematical concepts is crucial for learning and teaching of any mathematical content. Being able to make mathematically correct definition of the concepts is an indicator of teachers‟ content knowledge. The purpose of this study is to determine how preservice middle school mathematics teachers define the concept of algebraic expression and equation. The participants of this case study were 35 preservice middle school mathematics teachers. The data were collected through written exam and semi-structured interviews. Written exam includes two questions asking preservice teachers to define equation and algebraic expression and write an example of each. Only 9 of participants correctly defined algebraic expression. Preservice teachers‟ definitions of algebraic expression were classified under three themes which are expressions containing unknown, expressions containing equality, and mathematical expressions. Two themes arose from preservice teachers‟ definition of equation: expressions with unknown, and expressions with equality.


Introduction
Teachers are the most important factors affecting teaching and learning process of mathematics (Ball, 1990;Charalambous, Hill, Chin & McGinn, 2019;Shulman, 1986Shulman, , 1987. Teachers" knowledge about the subject that they will teach is the most important factor affecting student success. Content knowledge is the knowledge of teachers about definitions of concepts, relation between concepts and mathematical notation of subject they teach (Ball, Tames &d Phelps, 2008). For students to learn mathematics topic and concept correctly, teachers should have sufficient content knowledge about the subject (Ball, 1990, Ball, Thames & Phelps, 2008Fennema, Sowder & Carpenter, 1999). Researches show that teachers" content knowledge has effect on their teaching practice and thus students learning (Ball, Lubienski & Mevborn, 2001;Brizuela, 2016;Charalambous et al., 2019;Copur-Gencturk, 2015;Hill & Ball, 2004;Hill, Rowan & Ball, 2005;Tchoshanov, Cruz, Huereca, Shakirova, Shakirova & Ibragimova, 2017). It cannot be expected from a teacher with lack of knowledge on subject he teaches to perform a successful teaching practice (Ball, 1990;Ball et al., 2008;Dreher, Lindmeier, Heinze & Niemand, 2018). To strengthen students" mathematical knowledge, teachers should be strengthened first (Ma, 2010).
Teachers should know the concepts included in content area and different representations of the relationships between these concepts (Ball, 1990). One of the most important dimensions of content knowledge is to be able to make a mathematically correct definition (Ball et al., 2008). Teachers should be able to define the concept correctly first. Making correct definition of concepts is the basis of concept knowledge about the subject that, they will teach (Ball, 1990, Ball et al., 2008. Learning algebra is challenging for students and teachers" content knowledge related to mathematical concepts determine how students make sense of algebraic concepts (Stephens, Ellis, Blanton & Brizuela, 2017). Just focusing on operational properties of algebraic concepts prevents students from correctly structuring their mathematical meaning. Students develop many different misconceptions as they deal with concepts of algebra because they focus on rules and operation (Akkan, Baki & Çakıroğlu, 2012;Dede & Argün, 2003;Weinberg, Dresen & Slater, 2016). Students have difficulty to understand letter symbols used in algebra. They find it difficult to make sense of letter symbols as a numerical value that represents a variable (Brizuela, 2016;Christou & Vosniadou, 2012). Students also learn algebraic concepts incorrectly due to the erroneous meanings they attribute to letter symbols (placeholder, abbreviation of an object, etc.). For example, they define algebraic expression and equations as they include a letter and an operation symbol because of wrong learning related to letter symbol (Stephens et al., 2017).
The most difficult concepts in teaching algebra are the concepts of algebraic expression and equation. The concept of algebraic expression and equation form the basis of teaching advanced algebra (Wasserman, 2016). In order for these concepts to be learned correctly by students, teachers' content knowledge must be strong. Attorps (2003) stated that teachers' understanding of equation concept is incorrect. Teachers explained the meaning of the equation through operational processes. Another study examining conceptions of algebra conducted by Stephens (2008) included 30 preservice elementary teachers. Result of the study indicated that preservice teachers" content knowledge related to algebra is quite limited. Most of participants equated algebra just with manipulation of symbols and implied that solution strategies including traditional symbol manipulation may be more valuable for algebra learning than strategies demonstrating conceptual understanding. Stump and Bishop (2002) found that 30 prospective teachers who participated in their studies defined algebra with a focus on symbol, letter and problem solving, and they did not mention algebraic relations. Zuya (2017) worked with 54 preservice secondary school mathematics teachers to investigate preservice teachers" knowledge and skills related to algebra. Finding of the study showed that the preservice teachers" performances were very low in tasks demanding conceptual knowledge for solution while they performed above average in tasks requiring procedural knowledge. What do teachers view as algebra determines what will worth to teach in the classroom (Stephens, 2008). That is, teachers" definitions of algebra play an important role during organization of teaching process (Stephens, 2008;Stump & Bishop, 2002).
Teachers and pre-service teachers who view algebra not as a way of thinking but as a solution process in which a series of operations are performed do not use algebraic relations in operations involving basic concepts of algebra, such as algebraic expression and equation. They mostly focus on procedural information and cannot make conceptual explanations regarding these procedural processes (Agarwal, 2006;Black, 2007;Hohensee, 2017;Odumosu & Fisayi, 2018;Welder, 2007;Welder & Simonsen, 2011;Zuya, 2017). Therefore, it is important to determine how pre-service teachers define the two basic concepts of algebra; algebraic expression and equation. In this context, this study aims to determine how middle school mathematics teacher candidates define the concept of algebraic expression and equation, which form the basis of algebra learning.

Method
Case study design was adopted for this study. Case study is a qualitative inquiry which aims to explore a program, an activity, process or one or more cases in depth through multiple data sources (Creswell, 2007). Written exams and interviews were used as data sources. Case studies help researcher to investigate a problem in detail. Therefore, this study was conducted as a case study in order to investigate preservice middle school mathematics teachers" knowledge related to basic algebra concepts.

Participants
Participants of this study were 35 second grade preservice middle school mathematics teachers studying in a state university in Turkey. For selection of participants, convenience sampling and criterion-sampling methods were used. A convenience sample is a group of people who are chosen for a study because they are available (Fraenkel, Wallen & Hyun, 2012). , In convenience sampling, participants are selected from a group of people easy to reach in order to prevent loss of time, money and labour. (Büyüköztürk, Çakmak, Akgün, Karadeniz & Demirel, 2013). The criteria about determining the participant was to take "Introduction to Algebra" and "Linear Algebra" courses, which are important for improvement of preservice teachers" content knowledge of algebra. Taking and succeeding these course were taken as criteria because contents of these courses includes basic algebra concepts and proof of algebraic theorems. Preservice teacher learn background theorem of basic and simple algebra rules. So, at the end of these courses, preservice teachers should have deep understanding of algebraic concepts, theorems and logical explanation of operation with symbols. Also, preservice teachers" algebraic thinking skills are expected to develop after taking these courses.

Data Collection
Research data were collected through written exam and semi-structured interviews. In this context, the research was carried out in two stages. First, written exam that includes two open-ended questions was administered to preservice teachers to determine the definitions of the equation and algebraic expression. To ensure intelligibility of questions, written exam questions were checked and revised in terms of content and language by two experts and one in-service mathematics teacher. Preservice teachers were asked to define both equation and algebraic expression and write an example of each. To ensure reliability and validity, two experts checked content and language of questions. The written exam lasted 15-20 minutes. Second, clarification interviews were conducted with 7 preservice teachers. Interviewing is an effective way of checking accuracy of inferences that the researcher makes through written exam or observations (Fraenkel, Wallen & Hyun, 2012). Interviews help the researcher to understand what participants think. Interviewees were selected based on diversity of definitions. The interviews took on average 5 to 7 minutes. Interviews were recorded with an audio recorder.

Data Analysis
Content and discourse analysis were utilised to discover hidden meaning in preservice teachers" written and spoken expression. For content analysis, the processes of coding data, creating codes and themes, defining and interpreting the findings were followed (Patton, 2002). First, written exam papers of preservice teachers were examined and a summary of their answers to all questions were written up. Then, codes are established based on definition of preservice teachers. Lastly, codes were reviewed and general themes were created. Also, interviews were transcribed and data were analysed. To ensure reliability, all data were coded separately by two researchers. Then, codes and themes were discussed with the participation of third researcher and final version of codes and themes were created. The reliability rate between two coding was determined as 92%. Preservice teachers" definitions of algebraic expression led to creation of three major themes, which are expressions containing unknown, expressions containing equality, and mathematical expressions Also, preservice teachers" definitions of equation collected under two themes: expressions with equality and expressions with unknown.

Findings Preservice Teachers' Definitions of Algebraic Expression
Preservice teachers" definitions of algebraic equation were collected under 3 different themes. Preservice teachers defined algebraic expressions as expressions containing unknown, expressions containing equality, and mathematical expressions (Table 1) "It is expressions that we establish equality between numbers." (3x = 7) (PT22) "They are numerical expressions containing equality." (2 + 3 = 5) (PT33) The preservice teachers" examples for algebraic expression are also incorrect. Two preservice teachers gave an equation as an example of algebraic expression, and other two of them gave number sentence example. In the semi-structured interview conducted with Participant 6, the preservice teacher made the following explanation.
Researcher: You gave 2x + 5 = 11 as an example for algebraic expression. Is 2x + 5 algebraic expression in this expression, or is this entire expression (2x+5=11) an algebraic expression? PT6: Yes, whole of it is an algebraic expression. Mathematical expression: 3 of preservice teachers who participated in the study defined algebraic expression as "mathematical expressions" in written exams. 2 pre-service teachers defined algebraic expressions as they are "mathematical expression", while 1 preservice teacher defined algebraic expression as they are "mathematical expression, something like rational and root numbers". For algebraic expression, they gave 7/3, and sin30 = 1/2 as examples. It was determined that algebraic expression definitions and examples of these pre-service teachers were wrong.

Preservice Teachers' Definitions of Equation
Equation definitions of the preservice teachers who participated in the study were collected under two different themes. In this context, preservice teachers defined the equation as expressions with equality and expressions with unknown. Name of themes and distribution of number of preservice teachers are given in Table 2. PT: Yes. Expressions with unknown: 8 preservice teachers defined equation as "expressions that contain unknown". Although these preservice teachers did not use the concept of equality in their definitions, examples they gave contain equality and they gave the correct examples for the equation. In interviews conducted with 2 preservice teachers, they stated that the equation contains equality and the unknown is found thanks to equality. Examples of preservice teachers" definitions are presented below.

Discussion and Conclusion
Mathematical definitions have an important role in mathematics teaching process. Defining mathematical concepts correctly is important for development of conceptual understanding (Edwards & Ward, 2004;Morgan, 2005;Mosvold & Fauskanger, 2013). Conceptual understanding is important for students" learning as well as for defining teachers' content knowledge competence (Morgan, 2005). Teachers' content knowledge is a determining factor of their teaching practices (Ball et al.,, 2008). Thus, it is very important to define basic algebra concepts correctly during algebra teaching process. In this context, this research focuses on how middle school mathematics teacher candidates define the concept of algebraic expression and equation as a dimension of content knowledge.
In the middle school mathematics curriculum (MEB, 2018), algebraic expression is defined as "expressions that contain at least one unknown and one operation". The algebraic expression definitions of the participating 35 preservice teachers classified under three themes: expression containing unknown (28 preservice teachers), expression containing equality (4 preservice teachers) and mathematical expression (3 preservice teachers). It is seen that definitions of preservice teachers (28 preservice teachers) who define the algebraic expression as expressions that contain unknown are insufficient. Most of these preservice teachers (12 preservice teachers) used only expressions containing unknown statements in their definitions. Chalouh and Herscovics (1988) stated that algebraic expression is often defined as expressions that involve variables. It was emphasized that such formal definition is not sufficient and meaningful for students to understand the concept. Defining algebraic expression only to a limited extent (defining algebraic expressions limited to variables) can make it difficult for students to understand other mathematical concepts (coefficient, term, constant term, and similar term) (MEB, 2018) contained in algebraic expression. Failure to understand these concepts correctly by students will make it difficult for them to learn advanced algebra topics (Kieran, Pang, Schifter & Ng, 2016;Stephens, Ellis, Blanton & Brizuela, 2017;Tirosh, Even & Robinson, 1988). For example, it is necessary to understand terms, coefficients and similar term concepts in order to learn the process of operations with algebraic expressions. For this reason, it is important to correctly express these concepts when defining algebraic expression. On the other hand, defining algebraic expression only as expressions containing unknown may cause students to develop different misconceptions that other mathematical concepts, which contain unknown, are also algebraic expressions.
Participants who defined algebraic expression as expressions containing unknown stated that algebraic expression also contains numbers and operations in their definitions (9 preservice teachers). This definition coincides with the definition given in the middle school mathematics curriculum (MEB, 2018). These preservice teachers gave correct examples for algebraic expression. 7 preservice teachers stated that algebraic expression does not involve equality. The reason why the preservice teachers emphasized that algebraic expression does not involve equality is due to their understanding that relation between algebraic expression and equation is operation-oriented. Interview data also support this situation. Two preservice teachers stated that there was no equality in algebraic expression, but it turns out to an equation when equality was added to algebraic expression. These explanations showed that preservice teachers explained the difference between algebraic expression and equation as operation-oriented (there is no equality in algebraic expression). They did not make any conceptual explanations regarding the mathematical meaning of why algebraic expression does not involve equality.
Researches also show that preservice teachers explain basic algebra concepts with a focus on letters and symbols and focus on the operational process (Attorps, 2005;Black, 2007;Welder & Simonsen, 2011). Understanding the underlying mathematical meaning of operations is important for meaningful learning (Van de Walle, Karp, Bay-Williams, 2013). 4 pre-service teachers incorrectly defined algebraic expression as expressions with equality. These preservice teachers mentioned only the concept of equality in their definitions. 2 teacher candidates wrote down equation (e.g. 3x = 7) and two teacher candidates wrote down number sentence (eg 2 + 3 = 5) as an example of algebraic expression. 3 preservice teachers defined algebraic expression as mathematical expressions. These definitions are not an explanatory definition and examples given by the preservice teachers for algebraic expression (7/3, and sin30 = 1/2) are incorrect. Based on these findings, it is understood that preservice teachers have wrong knowledge about algebraic expression. In the middle school mathematics curriculum (MEB, 2018), the equation is defined as "equality that involve unknown and true for some values of the unknown." The majority of participants (28 preservice teachers) defined the equation as expressions with equality. 17 of these pre-service teachers defined equation as equality, which contain unknown. Preservice teachers gave correct examples for the equation. On the other hand, the teacher candidates' definitions do not exactly match up to the definition given in the middle school mathematics curriculum (MEB, 2018). In addition, since not every expression (eg identity) containing equality and unknown is an equation, preservice teachers" definitions are incorrect. These findings show that pre-service teachers define the concept of equation with a focus on letters and symbols. In the interviews, teacher candidates could not elaborate their definitions. They could not make conceptual explanations about the meaning of the equation concept. The examples given by preservice teachers for the equation also support these findings. Preservice teachers think that the right side of equal sign is result while right side is the problem (Van de Walle, Karp, Bay-Williams, 2013). For this reason, they gave examples of equations where the unknown is on the left side of the equal sign. This shows that pre-service teachers consider the equation as computing the amounts and do not have knowledge about the relational meaning of equality. 4 pre-service teachers defined equation as the equality of algebraic expression to a number and 2 pre-service teachers defined as the equality of two algebraic expressions. The definitions of teacher candidates do not match the definition of the equation (MEB, 2018). These definitions do not meet the mathematical meaning of the equation concept. This shows that pre-service teachers think that the difference between equation and algebraic expression is existence of equality. The statements of prospective teachers that there is no equality in algebraic expression support this situation. 4 pre-service teachers defined the equation, using only the concept of equality, as equality of expressions. 8 of preservice teachers defined the equation incorrectly as expressions with unknown. These findings show that pre-service teachers focus on operational processes while defining the equation. Similar findings were obtained in other studies in the literature (Attorps, 2003;Tanisli & Köse, 2013;Yıldız, 2016). In the study carried out by Yıldız (2016) with three secondary school mathematics teachers, teachers defined the concepts of algebraic expression and equation with a focus on letters, operations and equality. They stated that the difference of the equation from the algebraic expression is the equation contains equal sign.

Recommendations
The results obtained from this research show that majority of participating preservice middle school mathematics teachers defines algebraic expression and equation wrongly. An important result of the study is that preservice teachers define algebraic expression and equation based on letters, operations and symbols. Similarly, teacher candidates did not make any explanations about the mathematical meaning of the concepts in the interviews and they focused on the operational processes. Similar results were obtained in the studies conducted in the literature (Attorps, 2003;Stephens, 2008;Strumps & Bishops, 2002;Yıldız, 2016). Ball (1990) pointed out that teachers should know the underlying meanings of operational processes, and that concept knowledge is an important part of teachers' content knowledge competencies. These results indicate that preservice education program should include applications that will enable preservice teachers to learn the definition and meaning of mathematical concepts. In the mathematics teaching courses, a teaching environment should be created in which preservice teachers can examine and discuss mathematical concepts not only theoretically but also practically. In this study, the data were obtained from the written answers given by the preservice teachers to the questionnaire. One-to-one interviews were held with only some of the teacher candidates. In future studies, more in-depth findings can be obtained by conducting extensive interviews with fewer participants and using different data collection tools (focus-group interviews, observation of practice and internship courses, etc.).