Linking Transformation and Problem Atomization in Algebraic Problem-Solving

Abstract

The transition from arithmetic to algebra requires students to change both their thinking and the way they learn. We often observe students using arithmetic formalism also when solving algebraic problems. This formalism manifests itself primarily in the acquisition of coherent computational procedures. Students must be sufficiently aware that the computation steps are sequential transformations of the problem. This creates a problem for them in solving more complex problems. Our research investigated whether problem transformation coupled with atomization is a suitable alternative for students to learn coherent algorithms. Although atomization is not based on precise rules, it was reported by students to be a comprehensible way of solving problems and providing them with sufficient confidence. If students are motivated to understand a computational method, this understanding represents fulfilling the student’s need for security.

1. Introduction

Since the time of the Arab mathematicians, algebra has been regarded as the part of mathematics whose subject is the solution of equations. Current students begin to study algebra after they have completed their arithmetic studies, usually in high school. Arithmetic represents a formal system of computation where the next step of computation is predicted, leading to a mentality of automatism in students [1]. Individual computational algorithms are mastered by students without the necessary insight. Many types of research have revealed conceptual differences between arithmetic and algebraic thinking [2,3,4,5,6]. According to the findings of [7], students’ success in algebra correlates with how students handle the transition from arithmetic to algebraic thinking. The transition to algebraic thinking is a process that could be clearer for the student. Even in algebra, students work within an arithmetic frame of reference and are focused on counting [8,9]. However, to be successful in algebra, they need to focus on the algebraic relationships, the representation of the problem, and the method of solution, not just the solution to the problem itself [10]. Therefore, teaching algebra also needs to be directed so that students develop algebra-specific skills. Critical activities in algebra include transformation activities, the basic feature of which is to change the form of an expression or equation while maintaining equivalence [11]. In arithmetic, students encountered transformations, which had the same goal as in algebra. For example, transforming $7+8$ to 7 + 3 + 5 allowed the pupil to use a well-known fact—addition to 10 [12,13]. The use of such transformations requires the pupil to solve one extra problem. The number 8 needs to be transformed into the sum of two numbers, one of which must be $10-7$. Once it is done, the original problem, the sum of two numbers, is transformed into a more complex issue, the sum of three numbers. However, solving the transformed problem is less demanding for the pupil and uses his or her well-learned knowledge. The activity that is part of the transformation activity is the atomization of the problem. Task atomization involves breaking the solution of a problem into subproblems and is a special case of preparatory task modification [14,15]. By solving these tasks, we obtain the result of the task at hand.

Students must deal with difficulty in atomization because a given problem does not predict its solution atoms, and the student must discover them. Our research aims to discover how students perceive the atomization method of the problem.

2. Theoretical Basis

Teaching algebra involves three types of algebraic activities, i.e., generational, transformational, and global/meta-level, with transformational activities forming the core of school algebra [11]. Therefore, it is necessary to gradually implement such tasks in the teaching of mathematics that contain different transformational activities in different contexts. Elements of transformational activities are critical to algebraic thinking [16] and should form a significant part of algebra instruction [17]. Transformational algebraic activities are “rule-based” activities that include collecting similar expressions, simplifying expressions, and adding numerical values to expressions [16]. The first transformational activity that students encounter is adding numerical values to algebraic expressions. This activity is significant for developing the concept of equivalence of two algebraic expressions, which is the basis of transformations. Regarding transformations, two expressions are considered transformation equivalents if one can be transformed into the other using appropriate manipulations [18]. Students often solve the problem: “Atomize the expression $4x^2-1$ into the product.” This problem contains transformational activities because it can be formulated as follows: “Replace the expression $4x^2-1$ with an expression that is equivalent to it and is in the product form.” During this transformation activity, students should know that algebraic expressions are equivalent if and only if any substitution of numerical values into the expressions leads to the same results [19]. Students must develop the relational and substitution concepts of the equal sign to successfully use transformations, symbolically expressing the equivalence of two expressions. The relational conception of the equals sign means that the symbol “=” denotes the “sameness” of two objects or expressions [2,20,21]. The relational concept means that we interpret the symbolic notation $4x^2-1=\left(2x-1\right)\left(2x+1\right)$: the expressions $4x^2-1$ and $\left(2x-1\right)\left(2x+1\right)$ have the same value for the permissible values of x. The substitution conception is based on the transitive and symmetric equivalence properties [22], which allow substitution two equivalent expressions. This conception allows the equal sign to be interpreted as the equivalence of unequal—different looking symbolic notations, but with the same value are interchangeable. For example, if $4x^2-1=\left(2x-1\right)\left(2x+1\right)$, the substitution concept of the symbol “=“ means that the expression $4x^2-1$ and vice versa can replace the expression $\left(2x-1\right)\left(2x+1\right)$. Similarly, the substitution concept will allow the number 14 in the expression $14+8$ to be replaced by the expression $12+2$ because 14 = 12 + 2 [23]. However, the necessary development of the relational and substitutional concept of the “=“ symbol does not occur if students manipulate expressions only on the basis of learned rules. Students learn to manipulate expressions based on certain rules. However, mastery of the rules alone does not guarantee that students will realize the connection between the modification of an expression and its transformation [24]. Too much attention by both teachers and learners to maintaining rules when manipulating symbols encourages procedural manipulation of expressions without perceiving the internal structure of the expression [25]. In high school algebra, the notion of structure is associated with activities such as recognizing structure, seeing a part of an expression as a unit; breaking an expression into meaningful subexpressions; recognizing which manipulations are possible and useful to perform; and choosing appropriate manipulations that make the best use of structure [26]. For example, the expression $x^6+3$ can be written in different structural ways: ${(x^2)}^3+3$ or ${(x^2)}^3+3$, or $x^6-1+4$ and then $\left(x^3-1\right)(x^3+1)+4$. According to [27], attention must be paid to the link between internal structure and expression manipulation in transformation. [28] proposed to focus expression manipulation on establishing the concept of equivalence with the use of the internal structure of expressions to justify the execution of transformations. However, transformational algebraic activity need not be limited to following rules and manipulating symbols to simplify an expression. The correct choice of manipulations with the expression can be used to create an expression with desired properties or an expression for a certain purpose [29], which is used to create different structural notations of the expression. It is such manipulations that are part of the solution to many algebraic problems [30], where, for example, substitution is used. Teaching students to recognize the internal structure of expressions is an essential prerequisite for mastering the next transformation activity, namely the transformation problem [31]. Proper identification of object equivalence will allow the solver to transform an unknown problem into one that he/she can solve with acquired skills [32]. The ability to transform a problem, or a part of it, is fundamental to success in higher mathematics [33]. For example, many exponential or logarithmic equations are solved by transformation to solve a quadratic equation. Identifying or creating an expression of the desired properties is often a sub-problem that must be solved, thus opening the way to transform the problem by suitable substitution. Identifying and solving subtasks is called problem atomization and is a special case of preparatory modification [14]. Problem atomization is important in generating alternatives and evaluating their suitability for solving the problem. According to [34], students are uninterested in solving long and complex problems. Therefore, task atomization could be an interesting way of solving tasks for students, because it leads to the solution of often short subtasks with the use of already acquired knowledge and computing skills. The ability to atomize a task also changes pupils’ attitudes toward algorithms. They no longer see them as a procedure to be memorized to solve problems. However, they realize that the procedure to solve a problem combines several short algorithms. Pupils can “embed” these short algorithms into the solution of a problem because of their ability to atomize the problem. The ability to atomize a problem will help to increase pupils’ attainment in algebra and may change pupils’ attitudes toward learning algorithms. While learning arithmetic, pupils often acquire the belief that in mathematics, they are expected to find a result quickly [35], so they prefer to learn coherent algorithms [36]. However, algebra requires students to think and learn differently [37]. Research has shown that purposefully incorporating transformational activities related to task atomization into mathematics instruction positively impacts the development of students’ algebraic thinking [16,38].

3. Methodology

The research participants were 28 first-year high school students aged 15 and 16 years, of which 11 were girls and 17 were boys. Participation in the research was voluntary, outside of school hours, with the consent of the school administration and the parents of the research participants. Before starting the research, the participants covered the curriculum: solving linear, quadratic, and irrational equations and inequalities. When solving linear equations, emphasis was placed on using rules for adjusting algebraic expressions when manipulating the expressions on the left and right sides of the equation. When solving quadratic equations, students used the formula for calculating the roots of a quadratic equation. Emphasis was placed on the term “discriminant”, the value of which determines the number of solutions of the quadratic equation in the set of real numbers. The solution of linear and quadratic inequalities was realized using the method of zero points. The basic method of solving irrational equations was to isolate one square root containing the unknown and then exponentiate the equation. These adjustments were used to show the difference between an equivalent and a consequent adjustment of an equation and when a correctness test should be performed. In two lessons, students solved irrational equations using appropriate substitution. Before starting the research, we conducted individual interviews with two mathematics teachers who taught the students in question the lesson of solving irrational equations. The interview aimed to find out how these teachers taught irrational equations. Therefore, we used open-ended questions in the interview with the teachers. Subsequently, we conducted a group interview with the students. Within the interview, we investigated how the students were taught to solve irrational equations. The interview lasted for 15 min. At the same time, the teachers confirmed to us that the research participants (students) have sufficient knowledge and master the calculation procedures for solving irrational equations. Another part of the research was the presentation of the solution of irrational equations using task atomization. We devoted 45 min to the presentation of task atomization. Within this section, one of the researchers demonstratively solved several more complex irrational ones. When solving, emphasis was placed on making the students aware that the individual adjustments were transformations, i.e., replacing one problem with another based on an equivalence relation. We direct the transformations so that we can use the knowledge already acquired while solving the problem. A discussion with the pupils solved the problems. The used tasks have a longer calculation procedure, but by analyzing the assignment and setting sub-goals—atomization—they can be solved as a sequence of simple sub-tasks with a short calculation. We illustrate the method of solving with the following example (other tasks used are in Appendix A).

Example: Solve the equation on the set R $2\sqrt{x}+\sqrt[8]{x^5}=3\sqrt[4]{x^3}$.

Solution:

Subtask 1: Modify (transform) the equation to contain only powers of the unknown x.

To solve this problem, we will use one of the (transformation) rules for counting with powers, namely

$$\sqrt[m]{a^n}=a^{\frac{n}{m}}.$$

The solution of this subproblem 1 is the exponential equation

$$2x^{\frac{1}{2}}+x^{\frac{5}{8}}=3x^{\frac{3}{4}}.$$

(1)

Subtask 2: Transform equation (1) to the form $a^x=a^y$.

Subproblem 2 has no solution because we do not know the rule by which the expression $2x^{\frac{1}{2}}+x^{\frac{5}{8}}$ transformed into an expression of the form $x^{\frac{n}{m}}$.

Subtask 3: Transform Equation (1) so that a suitable substitution can be used, i.e., a repeating term occurs. To make use of substitution, the equation to be solved needs to be modified so that it contains the so-called repeating term. In our case, this subproblem has two more subproblems. The first sub-problem is the problem from the lesson Modifying fractions to a common denominator, and its solution is the equation:

$$2x^{\frac{4}{8}}+x^{\frac{5}{8}}=3x^{\frac{6}{8}}.$$

We solve the second part of Subtask 3 by reusing the equivalence $\sqrt[m]{a^n}=a^{\frac{n}{m}}.$ The final solution of subtask 3 is an equation of the form

$$2{\left(\sqrt[8]{x}\right)}^4+{\left(\sqrt[8]{x}\right)}^5=3{\left(\sqrt[8]{x}\right)}^6.$$

(2)

Subtask 4: Transform Equation (2) using suitable substitution.

In the previous subtasks, we have successively created a repeating expression $\sqrt[8]{x}$, which we replace with the symbol y, i.e., we use the substitution $\sqrt[8]{x}=y$. After the above substitution, Equation (2) takes the form

$$2y^4+y^5-3y^6=0$$

(3)

and that solves sub-problem 4.

Subtask 5: Modify the expression $2y^4+y^5-3y^6$ into a product.

Solve this subproblem by removing the expression $y^4$ before the parenthesis. The solution to subproblem 5 is the expression

$$y^4\left(2+y-3y^2\right)$$

and Equation (3) takes the form

$$y^4\left(2+y-3y^2\right)=0.$$

Subtask 6: Solve the equation

$$y^4\left(2+y-3y^2\right)=0.$$

The given equation is an equation in product form, so we split its solution into the solution of two partial equations:

$$y^4=0 \vee \left(2+y-3y^2\right)=0.$$

The solution of the first partial equation is $y_1=0$, and the second one $y_2=1;y_3=-\frac{2}{3}$.

Subtask 7: Find the result of the original irrational equation.

We get the result of the original entered equation by returning to the original unknown, i.e., by solving three partial equations

$$\sqrt[8]{x}=0 \vee \sqrt[8]{x}=1 \vee \sqrt[8]{x}=-\frac{2}{3}.$$

The solution of the first is $x_1=0$, of the second $x_2=1$, and the third has no solution on the set R. The set of solutions of the given irrational equation is K = {0; 1}.

During problem solving, one of the researchers present recorded students’ reactions to the suggested method of solving irrational equations. In the final stage, we conducted a group interview with the students to find out to what extent the atomization of the problem was an understandable way of solving the problems for them. The above objective is relevant because students can judge the comprehensibility of the explanation [39].

4. Findings

(A)

Interview with teachers

There were mainly two basic procedures used in teaching the solution of irrational equations. One procedure consisted of modifying the irrational equation into the form

$$x^{\frac{n}{m}}=a$$

using the rules for counting with powers. The second procedure was based on the use of suitable substitution. The use of substitution was mainly linked to the case when the given irrational equation had the form

$${\left(V\left(x\right)\right)}^{2n}+b{\left(V\left(x\right)\right)}^n+c=0$$

or it could be modified to that shape. In this case, a substitution of the form $y={\left(V\left(x\right)\right)}^n$ is introduced. In both cases, the teachers emphasized to the pupils that when modifying an irrational equation to the desired form, it is necessary to follow the rules for modifying expressions with powers, which pupils already know from the lower grades.

(B)

First interview with students

The way students are taught to solve irrational equations includes the following elements memorizing the solution procedure as a whole, practicing the solution procedure on similar problems. They considered that the positive thing about solving irrational equations was that there was no need to learn many different ways of solving and they did not need to learn new rules. They found the problems where they needed to use substitution to be the most challenging because they had to remember a longer procedure. A few pupils said that they did not learn the use of substitution for this reason. Pupils said that they had to invest more time in memorizing the procedure. This investment pays off because the rehearsed procedure gives them confidence when solving similar problems.

(C)

Second interview with students

The students expressed that task atomization is a new way of solving problems for them. Their initial expectation was that they would learn a new computational procedure that they could apply when solving more complex irrational equations. One part of the students commented that they gradually realized that this way of solving problems did not require them to memorize lengthy procedures. Some students expressed concern that they had to recall much knowledge from their previous mathematics studies when using the atomization of the problem. The most apt statement of this type was:

“Am I supposed to remember so many things in one task?”

Students responded to task atomization with some degree of surprise that one task contained other tasks. In this context, the problem for them was that they felt that they were solving tasks without an assignment. They said that the assignment instructed them on the activity to perform. One of the students formulated the problem as follows:

“I cannot solve the problem without the assignment.”

An unexpected element of the atomization of the problem for the students was that even subproblems that did not have a solution were set during the solution. Students considered this type of subtask to be a wrong step in the solution. This caused that for some of them, task atomization needed to be a more reliable method. One student expressed this by saying:

“I prefer to learn something that surely leads to the result.”

As part of this interview, and during presenting the atomization of the task, several (23) questions were asked by the students. The most frequently represented question was:

“How do I know I am supposed to atomize a task?”

Other questions were of the type:

“Is problem atomization applicable to other types of equations?”

“Is the solution of inequalities also atomized?”

“Is there a tutorial on how to atomize a task?”

After the researchers answered these, the students rated task atomization as an appropriate method of problem solving. They also expressed that they had to take a different approach to learning mathematics. The main difference they saw was the need to consider what steps of the calculation they would perform. At the same time, they were aware of the strong interconnectedness of individual mathematics knowledge. One student commented:

“Teachers always told us that math knowledge is related, but only now I really see that it is.”

5. Discussion

The teachers we interviewed aimed to teach their students to solve basic types of irrational equations by passing on ready-made computational procedures—algorithms. The teaching method they chose was justified because students expect them to have a reliable “recipe” for finding the solution to the given equation. The goal and teaching method of the teachers in our research corresponds to the findings of several studies [40,41,42,43], in which it was found that many teachers teach mathematics intending to impart to students a reliable computational algorithm that will provide them with the certainty of successfully solving a particular set of problems. This approach to teaching mathematics leads students to adopt computational templates to solve problems [44]. Students adopt these computational templates as formal procedures without understanding why they lead to the correct solution of the problem [45]. Handing out and practicing ready-made algorithms leads students to believe that learning many rules and computational procedures is the essence of mathematics [46]. A few successfully solved (similar) problems using a memorized algorithm (template) during a mathematics class further fixates students on learning the procedures because they associate memorizing them with success in mathematics [47]. The above attitudes of students toward learning mathematics, which are caused by being handed ready-made algorithms, were also observed during the first interview. In their statements, students confirmed that they were willing to invest time in memorizing the computational algorithm without questioning why the problem was computed in that way. They consider solving problems of this type as practice and checking whether they have memorized the steps of the algorithm in the correct order. Although students are not interested in solving long problems [48], most students are willing to learn a more extended solution procedure if an irrational equation requires it. In particular, they identified solving an irrational equation using substitution as too long a computational method. This evaluation of the procedure with substitution by the students can be taken as an indication that the students did not sufficiently realize that substitution means moving on to solving a new problem whose solution they already know. Based on the first interview, we confirmed that many students acquire the computational procedure largely without understanding and do not analyze its steps [49]. Students only need to know the rule (instructions) to find solutions to a given problem [50]. Based on analyzing students’ responses in the first interview, we find that students’ instrumental understanding of solving irrational equations is predominant. Instrumental understanding is mastering and using rules without knowing the reasons [51].

At the beginning of the second interview, students expressed that the atomization of the task was a new way for them to problem-solving and initially had trouble accepting the change in the way equations were solved. Their biggest concern was whether they could recall the different knowledge from previous mathematics lessons. This confirms that teaching and learning ready-made algorithms cause students to rely on memory to solve problems [42,52,53,54]. A notable finding is that students positively evaluated task atomization because this method does not require them to learn long computational procedures. According to [55], adherence to learned problem-solving procedures does not promote students’ reasoning development and discourages logical thinking. Related to this research-demonstrated consequence of learning coherent problem-solving procedures is a significant concern of the students in our research. They were reluctant to learn to atomize the problem and expected to be presented with rules to atomize the problem. In particular, they needed to know what types of tasks are atomized and on what basis the subtasks are formulated. We consider their concerns manifestations of their desire to obtain some rules that would give them the same sense of certainty as the learned procedures provided [56,57]. Instead of rules, they explained that each step of solving a task transforms the task, i.e., replacing the given task with a new one. Thus, a sequence of different elementary tasks emerges, from which “thematically” coherent subtasks are formed by atomization. Different solvers of the same task may atomize the task differently. Based on this explanation, students believe that the use of task atomization is not based on precise rules; instead, it requires a different way of learning mathematics based on relational understanding, which may cause uncertainty in students in the short term [58]. Relational understanding means that the student knows what calculation method to use and why it is appropriate to use that particular method [51]. Relational understanding is a prerequisite for finding one’s solution to a problem and allows the student to relate the computational procedure to the problem or adapt the computational procedure to a new problem [59]. We consider a significant finding from the second interview with students that they identified atomization as a way of solving understandable problems, even though it disrupted their habitual approach to learning mathematics. An equally important finding is that our experiment made them aware of the strong connections between different pieces of mathematics knowledge. At the end of the interview, the prevailing view among the students was that the confidence provided by the algorithms they had learned could be increased by learning why a given algorithm was used for a given set of tasks. They realized that the reason for using algorithms is precisely the tool for identifying subtasks. The identification of the subtask is a fundamental activity for the successful use of task atomization.

6. Conclusions

Based on our research, the fact that students already knew the method of solving irrational equations proved to be advantageous. Students already knew the rules they needed to feel their mathematical security. Therefore, while solving more complex irrational equations, they were able to focus more on finding a solution procedure by atomizing the problem. Moreover, during the second interview with the students, we confirmed that students need rules and expect new rules when they encounter a new problem. They expect to use the same rules for more complex problems in the same area of mathematics (in our case, solving irrational equations). Although atomization is not based on precise rules, it has been reported by students as an understandable way of solving problems and provides them with sufficient confidence. If students are motivated to understand the computational method, this understanding represents the fulfillment of students’ need for security. At the same time, task atomization linked to transformations can contribute significantly to the development of students’ creative approach to solving mathematical problems.