The current researchers, in their work as mathematics lecturers, witnessed poor performance by preservice mathematics teachers in determining limits of trigonometric functions. The students’ poor performance triggered the current study.

The purpose of the study was to get an in-depth understanding of how preservice mathematics student teachers develop mental structures related to limits of trigonometric functions. The study focused particularly on the students’ understanding of the SLI and its application in evaluating limits of other trigonometric functions. The researchers also explored the errors that the participating students made when they attempted to apply SLI in computing limits of trigonometric functions.

The following research questions guided the study.

Q1: What level of understanding do the preservice mathematics student teachers show on the Sine limit identity and its application in solving limits of trigonometric functions?

Q2: What errors do the preservice mathematics student teachers make when computing limits of trigonometric functions involving the Sine limit identity?

Although there are a few pedagogical researches on trigonometry, the reviewed literature revealed a number of challenges faced by students in the area. (Nurmeidina & Rafidiyah, 2019) studied the challenges faced by students when solving equations involving trigonometric functions. In their study, they found that students failed to solve the equations because they could not understand the mathematical statements involved. They attributed the failure by those students to lack of practice.

In a different study, (Siyepu, 2015) explored the errors made by mathematics students in trigonometry. Siyepu classified the errors identified into four categories, which were conceptual errors, procedural errors, extrapolation errors and interpretational errors. According to Siyepu, conceptual errors occur when a student fails to grasp a concept or fails to observe some relationship among concepts. Procedural errors occur when a student fails to apply some formulae, algorithms, rules or theories. Extrapolation errors are errors caused by over-generalization of properties, for instance in the case of limits of functions students assume that all limits of functions can be computed by direct substitution. Errors of interpretation are caused by failure by the students to get the correct meaning of mathematical statements.

(Kamber & Takaci, 2017) proclaim that the challenges faced by students in trigonometry at higher levels of learning start in the secondary school. According to those researchers, lack of understanding of a mathematics concept at secondary school level manifests itself at higher levels. (Kandeel, 2017) and (Elbrink, n.d.) supported the notion by asserting that mathematical knowledge built atop misunderstood concepts is not likely to be successful. If the proclamation by Kamber and Takaci is anything to go by, the challenges faced by students at tertiary level in trigonometry could be a result of their failure to grasp prerequisite concepts taught in secondary school. In Zimbabwe, mathematics students learn trigonometric ratios (sine, cosine and tangent) for the first time in their third year in secondary school (Primary & Education, 2015). At this level, the students learn to calculate trigonometric ratios of acute and obtuse angles. In their fifth year in secondary school, the students

are introduced to trigonometric functions. The assumption is that by the time they complete secondary education, they are able to differentiate, integrate and draw graphs of simple trigonometric functions involving sine, cosine and tangent of angles measured in radians.

In an effort to mitigate the challenges faced by students in computing trigonometric limits, (Sumianto, 2023) suggested the use of the jigsaw-learning model. The jigsaw-learning model is a cooperative learning method where heterogeneous students are put into a group. Each member of the group is tasked to explore and understand a concept. The members then meet as a group. They share and discuss the different areas they explored. Other researchers, (Nordlander, 2021) as well as (Baye et al., 2021), suggested the use of an interactive computer application called Geogebra. Nordlander’s suggestion came after carrying out an experimental study with the application. In the experimental study, Nordlander analyzed how students explored. According to the researcher, the visualizations from Geogebra assisted the students to grasp the SLI.

Figure 1 shows an illustration of how Geogebra shows the changes in the values of θ and sin θ leading to the evaluation of. When using the application, students move the point marked P along the arc. As the point P moves, the values of θ and sin θ change accordingly. At some points, the students calculate the values of the function. The students observe that as the angle θ reduces in size, the function approaches 1.

An analysis of the available literature shows that a number of researchers observed that concepts on trigonometry pose some challenges to many students. Different concepts on trigonometry were studied by different researchers. However, as stated earlier in this report, studies that are specifically targeted on how students learn limits of trigonometric functions are scarce. The knowledge gap found in the reviewed literature and personal experiences with mathematics students prompted the current researchers to carry out the current study.

A constructivist framework known as APOS theory guided the study. APOS theory is an improvement by (Dubinsky, 1984) on Piaget’s genetic epistemology on how mathematics students develop mental structures as they learn mathematics concepts. APOS is an acronym whose letters stand for Action, Process, Object and Schema, which are components of mental structure development ((Dubinsky, 2014); (Maharaj, 2010); (Salado & Trigueros, 2015)).

Figure 2 shows how the components of mental structure development are linked as suggested by the APOS theory. According to (Weller et al., 2011), an action is a reaction to external stimuli. It is synonymous to a situation where a person assembles a gadget by following some guidelines on a manual. Demonstrations, formula, instructions, algorithms are some of the external stimuli. The action conception forms the basis of the concept formation process. A student with an action conception can do nothing more than simply carrying out procedural computations by following a set of given instructions or imitating a demonstration. For instance, in the current study, a student with an action conception could do nothing more than computing limits of the form by simply comparing it to the SLI.

When a student repeats an action a number of times, there is a possibility that the student can interiorize the action into a mental process ((Asiala et al., 2015); (Dubinsky & Mcdonald, 1991)). The student becomes aware of the action and can now perform it without an external stimuli or guidance((Dubinsky & Mcdonald, 2001); (Soku et al., 2025)). Unlike an action, a process takes place in the mind of the student and it takes place under the student’s control((Dubinsky, 2014); (Makonye, 2017); (Van & Tong, 2022)) . A student with a process conception is able to reflect, describe, reverse or combine processes. In our study, we expected a student with a process conception to have interiorised the action to evaluate limits of the form to an extent of applying the knowledge to compute limits of the form .

According to (Asiala, 1996), when a student understands the entire processes and actions involving a mathematical concept then a mental object is said to have been formed. The process of developing a mental object from a mental process is called encapsulation((Dubinsky, 1984); (Mukavhi et al., 2021)). For instance, once the process and actions involved in applying SLI in computing limits of the form is encapsulated in a student’s mental structure, the student is considered to have developed the mental object which helps the student to apply the SLI in solving more complex trigonometric limits. The student can carry out suitable algebraic manipulations to change the form of the given functions in order to apply the SLI. At this stage, the student perceives the SLI concept as a mental object upon which actions and processes can act. The existence of the mental object in the student’s mental structure enables the student to solve problems by applying a combination of processes. The mental object can be de-capsulated back to processes and actions in the student’s mental structure whenever it is necessary.

When the student interconnects objects, processes and actions to do with a particular mathematics concept, then a schema for the concept is formed ((Dubinsky & Lewin, 1986); (Maharaj, 2010); (Tsafe, 2024)). In our case, it is the schema for the SLI and its application in computing limits involving trigonometric functions. The schema comprises linked and related mental objects, processes and actions. Lower order schemata in the student’s mental structure are thematized into objects for the purposes of learning higher-order concepts. In other words, lower-order schema forms a basis for higher-order schema. For example, the ‘limit’ schema forms the basis for the derivative schema.

In the APOS context, the students’ level of conceptual understanding is analyzed by means of a tool called a genetic decomposition(GD)((Dubinsky & Lewin, 1986); (Jimenez & Aguillar, 2024)).The GD spells out the observable characteristics that indicate the level of mental structure development exhibited by the students. It is constructed using ideas from the researcher’s own understanding of the concept, information obtained from reviewed literature and the nature of the mathematical concept (Cetin, 2019).

For the current study, the students’ level of understanding was analysed using a genetic decomposition (GD) adopted from (Maharaj, 2010). The adopted GD was adjusted in order to suit the context of the study. According to the constructed GD, a student with an action conception was expected to at least compute limits of the form correctly.

A student with a process conception of the SLI concept and its application could show that

(1)

A student who encapsulated the actions and mental processes involved in computing limits using SLI was expected to apply the SLI in solving more complex trigonometric limits. The student was deemed to have attained the object conception of the SLI. Such a student could change the form of the given indeterminate limits by multiplying or dividing with suitable functions and then apply the SLI to compute the given limits.

When the SLI mental object develops fully in the student’s mental structure, the mental object together with the actions and the mental processes involved form the SLI schema. A student with the SLI schema could compute the limits of the given trigonometric functions by means of carrying out the necessary algebraic operations and then apply the SLI.

The current study followed a qualitative paradigm. It was a case study of sixty-eight preservice mathematics teachers studying for a Bachelor of Education degree at a university in Zimbabwe. This constituted the entire class of first year pre-service mathematics student teachers for the 2024 intake at the university. Calculus was a compulsory course for them. One of the concepts in their Calculus course was limits of functions, which included limits of trigonometric functions.

For purposes of anonymity, the field researcher assigned the students some numbers from one to sixty-eight. The numbers replaced the students’ names. The students were informed of the study and its purpose. They were assured of the confidentiality of the information shared by them.

The students attended two-hour Calculus tutorials five times a week for four weeks. In two of the tutorials, the students learnt the SLI, and its applications in computing limits of other trigonometric functions. In one of the two tutorials, the students, with the assistance of their tutor, explored the different ways of evaluating . In the other tutorial, they learned how to apply the SLI in evaluating limits of other trigonometric functions.

The field researcher, who was also the tutor, administered a short test to the students as a way of assessing their understanding of the concept learnt. The test instructed the students to evaluate the following limits:

1)

2)

3)

4)

5)

After administering the test, the field researcher interviewed the students who got the test items wrong in order to get a deep understanding of their level of understanding.

The data obtained was analysed using content analysis. In order to get the underlying themes and patterns related to the students’ level of understanding, the researchers analysed the students’ written responses to the given test items and the explanations they shared during interview sessions.

In order to ensure validity and reliability of the results, the researchers used instrument triangulation. Data obtained from the students’ written responses to the given test items was triangulated with data obtained through interviews. Extracts from the students’ written responses and verbatim statements shared by the students during interview sessions were used to support the results.

The first question required the students to compute . The correct answer was -1/2. In order to compute the limit, one could apply the L’ Hospital rule once and then apply the SLI. The working could proceed as shown in equation 2.

According to the GD, a student was expected to have attained at least a process conception of the SLI and its application in order to be able to identify the relationship between the given question and the SLI. Three quarters of the students got the question correct. Student 21 was one of the students who failed to provide a correct response to this question. Figure 3 shows the working provided by student 21.

Student 21 realised the need to apply the L’ Hospital’s rule. However, the student made a procedural error by leaving a negative sign after differentiating cos x. The student went on to make another error by replacing sin x with 2 sin 2x. The field researcher asked the student to explain how he got 2sin 2x from sin x. The student gave the following response.

“I introduced to sin x so that I get in the numerator,” (Student 21, pers.com).

(2)

The last part of the working shows that the object had developed in the student’s mental structure; however, the student had not attained the action conception of the application of the limit.

Although Student 22 got a different answer to Student 21, the two students made similar errors and the same analysis applies to both students. Student 22 failed to differentiate Cos x. During the interview session, the student failed to explain how he got Sin 2x from Cos x. Figure 4 shows the working shown by Student 22.

The correct answer to question was . A possible way of computing the limit could be

About 37% of the students did not manage to provide correct responses to the question. One of those students was Student 47 who provided the working in Figure 5. Although Student 47 got the final answer correct, the student provided incorrect working. It was by coincidence that the answer was correct. The student had a misconception that Sin(4x-x) was equal to sin 4x - sinx. In support of his wrong working the student gave the following explanation.

“ I noticed that there was 4x in the denominator. To get 4x in the numerator, I replaced 3x with 4x - x. The next step was to expand. I got 4x - sin x,”(Student 47,pers.com).

There was evidence in Student 47’s working to show that the object had developed in the student’s mental structure, however the student lacked the required action conception to manipulate in order to apply the SLI.

Student 17 made a serious procedural error. In an attempt to create the SLI, the student failed to factorise the given function. Like Student 47, he lacked the action and process conceptions required to manipulate the given function before applying the SLI . Figure 6 shows the working provided by Student 17.

Student 17 shared the following expalanation.

“ In the first stage, I had to factor out 3x over 4 so that I have the same number inside the brackets, as you see in my working. By applying rules of limits, I got multiplied by . The product was 0 since the other part resulted in a zero,” (Student 17, pers. com).

About 96% of the students managed to give correct responses to question . The question requested the students to compute . A possible working could be

(4)

According to the GD, the question demanded at least an action conception. Only three students failed to provide correct responses to this question. One of those students was Students 31 who gave the response in Figure 7. Student 31 had not attained the action conception of the SLI and its

application in solving other limits. The student failed to explain how she got from . She had the following to say.

“ Isn’t it that I take out 3. Aaah I don’t know,”(Student 31, pers.com).

Question appeared to be the most difficult question for more than half of the students. The question required at least an object conception of the SLI and its application. The correct answer to the question was 1/2. A possible working could be:

(5)

Student 54 was one of the students who failed to provide a correct response to the question. The student provided the working shown in Figure 8. Student 54 attempted in vain to express as partial fractions. The student made a serious conceptual error in the process. He did not show evidence of having attained the action conception required to solve problems of the nature given. The following excerpt was obtained from Student 54.

“………mmmmmm I am not sure but I separated the function. It then resulted in something undefined, “(Student 54, pers.com).

The fifth question of the test requested the students to compute . The correct answer to the question was . The following is a possible way of computing the limit by applying the SLI.

(6)

Student 34 got the final answer to the question correct. The student applied the L’ Hospital’s rule correctly. However, he did not show all the parts of his working. Figure 9 shows how Student 34 proceeded with his working. When the field researcher asked Student 34 to explain the last two stages of his working, he had the following to say.

“cos 3x over cos 4x gives 0 over 0 when we substitute x for 0, that’s where the 1 came from,”(Student 34, pers.com).

The explanation given by Student 34 revealed that although the student got the answer correct, he had not attained the process conception of the SLI application.

Student 19 got the final answer to question correct using incorrect working. She simply cancelled and to remain with 3/4. She produced the working shown in Figure 10. During the interview session, Student 19 simply said that she cancelled common factors. The student’s response indicated that she had not attained the action conception of the SLI.