It has been suggested that calculus is among the greatest achievements in human reasoning ((N.C.T.M., 2000); (, 2017)). The concept of derivatives is one of the most important topics in calculus and has applications in various fields, such as economics ((Feudel, 2019); (Feudel & Biehler, 2021)). Derivatives are presented differently in math curricula across different countries. Derivatives are presented in the new twelfth-grade mathematics textbooks in Iran as well as in the first-year undergraduate curricula of universities. Some studies have shown that understanding the concept of a derivative is very difficult for students because of the complexity of its definition and representations ((Thompson & Thompson, 1996); (Zandieh, 2000); (Auxtero & Callaman, 2020); (Mirin, 2018)). Many studies have investigated students’ thinking about calculus concepts, including derivatives ((Berry & Nyman, 2003); (Oehrtman, 2009); (Rivera-Figueroa & Guevara-Basaldúa, 2019); (Selden et al., 2000)). Historically, derivatives were used for 200 years before being formally defined. The derivative was first used by Fermat and discovered by Newton and Leibniz. It was further developed by Taylor, Euler, and Maclaurin and then named by Lagrange. Subsequently, Cauchy and Weierstrass defined derivatives more precisely ((Desfitri, 2016); (Jaafar & Lin, 2017); (Haghjoo et al., 2020)). Derivatives have since been studied in the contexts of calculus ((Heid, 1988); (Zandieh, 1997), (Zandieh, 2000); (Likwambe & Christiansen, 2008); (Huang, 2011); (Bingolbali et al., 2007); (Weber et al., 2012)), ratios ((Thompson & Thompson, 1996); (Confrey & Smith, 1994); (Byerley & Thompson, 2017)), dynamic simulation ((Roschelle et al., 2012);(Johnson, 2010); (Santos & Thomas, 2003); (Samuels, 2017)), modeling ((Ärlebäck et al., 2013); (D, 2008); (Carli et al., 2020)), and teaching ((Sánchez-Matamoros et al., 2019)).

(Zandieh, 1997) suggested that a basic understanding of derivatives is realized through various representations and tasks in the context of calculus; this is called “derivative conceptual understanding.” (Zandieh, 2000) presented a framework for analyzing students’ understanding of the concept of derivatives. One component of this framework includes multiple representations, such as graphical, verbal, physical, and symbolic; the other includes the contexts and layers of process–object pairs, including ratio, limit, and function ((Zandieh, 1997);(Zandieh, 2000)). Different models have been suggested in the mathematics education literature that aim to describe the understanding of derivative concepts, such as concept image and definition (Tall & Vinner, 1981), action–process–object–schema (APOS) (Asiala et al., 1997), APOS-triad (intra–inter–trance) (Baker et al., 2000), and APOS-ACE (activities, classroom discussion, exercises) (Borji et al., 2018). In the next section, we will describe a model we built for our research. A review of additional models can be found in (Feudel, 2019). In our own experience with teaching differential calculus, we have observed problems with understanding derivatives; we therefore aimed to investigate the cause of this phenomenon. After reviewing studies that focus on understanding the concept of derivatives, we decided to use (Zandieh, 2000) framework in the context of learning the path of the (Hähkiöniemi, 2006) derivative, because it can better reveal the layers and contexts of the concept of derivatives and the nine tasks designed. Based on an analysis of students’ use of different kinds of representations, we considered how these representations could be used in the learning of derivatives (Hähkiöniemi, 2006). (Zandieh, 2000) model is often used in mathematics education research at the college level ((Carlson et al., 2002); (Hähkiöniemi, 2006); (Roorda et al., 2009-01); (Likwambe & Christiansen, 2008); (Roundy et al., 2015); (Jones, 2017); (Abd Hamid et al., 2019)). That model is based on (Tall & Vinner, 1981) construct of concept image, in which understanding a mathematical concept is characterized not only by knowing its definition but also by the so-called concept image: “The concept image represents the total cognitive structure associated with a concept, which includes all the mental pictures and associated properties and processes.”

The present study aimed to investigate the understanding of the concept of derivatives among undergraduate students at universities in Tehran based on (Zandieh, 2000) framework. The “Zandieh framework” has been used in several studies. One advantage of this framework is its two-dimensional table of layers and representations. For this study, we added a numerical representation to Zandieh’s framework. Our research questions are as follows:

According to (Zandieh, 1997), the fundamental understanding that leads to the concept of derivatives is realized through different representations and tasks in the context of calculus. The process object is a basic concept in this framework and is derived from (Sfard, 2008) perspective in the previous section. The two main components of this framework include multiple representations (contexts) and layers of process–object pairs, each of which is briefly described below. Representations of the derivative concept include the following: 1) graphical, as the slope of the tangent to the curve at a point; 2) verbal, as the instantaneous rate of change; 3) physical, as velocity (acceleration and the general state of motion); and 4) symbolic, as the limit of the difference quotients. The derivative layers that can act as processes and objects are as follows: The ratio is obtained by dividing the numerator by a fraction. An object is considered a pair of integers or the output of the division process. This limit refers to the process of approaching a certain value. The objective is the limit. A function is regarded as the process of correspondence between two nonempty sets. Finally, the object is a set of ordered pairs.

Layers of Zandieh’s framework for connecting limits and derivatives

Figure 1 illustrates the layers and processes between the limit and derivative and their relationship, following Zandieh’s framework. If a student’s conceptual structure is not developed in one of the layers, Zandieh uses a pseudostructural concept (Zandieh, 1997). Figure 1 illustrates the process for the ratio, which should be perceived first to understand the concept of derivatives; as its result, the difference quotient is turned into an object. Then, the limiting process is formed, the result of which is an object derivative at one point. In the third layer, the derivative is calculated at several points, the result of which is the derivative as a function. It should be noted that verbal and pictorial representations are equally significant in making meaning (Cuoco & Curcio, 2001).

Expanding Zandieh’s framework

(Roundy et al., 2015) expanded (Zandieh, 1997). The highlights of this expanded framework include the following: 1) expanding physical representations to those interested in physics, 2) introducing the thickness idea for the approximate value of the derivative (in physical and engineering problems, it is necessary for values to be very close to zero for calculation, but can also be considered), 3) adding a numerical context, and 4) adding an instrumental understanding row to the layers, but in a completely separate table, for those who merely know the derivative formula to reflect its weak relationships with aspects of the derivative concept.

Table 1 presents students’ understanding of the (Roundy et al., 2015) derivative. The numerical calculation column and the focus on its physical aspects are among the parameters of this framework. The instrumental comprehension row is also considered

Rules to "take a derivative" separately, thus, the layers of instrumental comprehension should be considered separately, according to (Roundy et al., 2015).

As shown in Table 1, the example given for physical representation is a derivative, where is the volume of an air-filled piston relative to the pressure on the piston, which is controlled by a set of weights on the piston (in this example, a weight of one unit is used for the concept of thickness). The derivatives include compressibility, velocity, and thermal conductivity (Gundlach & Jones, 2015).

(Hähkiöniemi, 2006) suggested that there are two paths for understanding the perceptual derivative, which is based on intuitive representations such as the steep slope of a function or the slip of a pencil between the function and the tangent line, as well as the symbolic derivative, which is shown with difference quotients. Based on Hähkiöniemi, students usually learn based on a motion context and learn the derivative through tasks covering the visual and symbolic worlds. In addition to Zandieh’s expanded framework, this study adopted Hähkiöniemi’s hypothesis. Hähkiöniemi derived his idea from Tall’s three mathematical worlds of the visual, symbolic, and formal (Tall, 2008).

The present survey study included one test to evaluate undergraduate basic sciences and engineering students' understanding of the concept of derivatives at nine universities in Tehran. The population included students at Tehran universities studying fields in engineering and basic sciences (Farhangian University was added as well) who had passed Calculus I. According to the Ministry of Science, Research, and Technology, there were 174,000 undergraduate students in Tehran at the time of this research. Basic science fields include mathematics and its applications, computer science, chemistry, and physics. The students in this research were studying during the 2017-2018 academic year.

We selected participants through multistage random cluster sampling. The minimum required sample size based on the Morgan table was 383 students; 604 students were selected for this study. Test power was obtained at 80.3%. Participants were selected from

among undergraduates at universities in Tehran; the students came from all over Iran. The sample students studied during different semesters (most were in their first year), and we targeted those who had passed Calculus I (simple questions were posed based on the concept of derivatives). Table 2 lists the number of students at each university.

For this study, we designed a test based on (Hähkiöniemi, 2006) hypothesis for the derivative learning path. The framework was designed based on Table 1 because all representations were considered in the tasks, and in the physical layer, velocity was considered.

Data collection

A researcher-made test was used to evaluate students' understanding of the concept of derivatives. The face and content validities of the tests were confirmed. The reliability coefficient of the test was obtained using Cronbach's alpha (r=0.88), indicating the appropriate reliability of the test. To design the test questions, we first examined the literature related to the concept of derivatives as well as calculus books and selected 102 questions. After reviewing the questions based on Hähkiöniemi's hypothetical learning path for the derivative (Hähkiöniemi, 2006), we reduced them to 16 questions; finally, nine questions were approved for the experimental stage. The testing time was 25-35 minutes (multiple-choice questions without description). In the experimental stage, the questions were descriptive and required one hour to complete. At this stage, students were asked about the reasons for their choices. After analyzing the questions and students' answers (correct and incorrect), the questions were designed as multiple-choice questions so the students could demonstrate their maximum ability and minimize mistakes. The students were asked to present their solutions simultaneously.

Data analysis

Students' answers were coded on a five-point Likert scale. Scores of 0-4 were assigned to each task choice. Thus, the total test score was considered a maximum of 32 (without task 1). Task 1 was analyzed only qualitatively; other tasks were analyzed both quantitatively and qualitatively. Data analysis was performed using SPSS version 24. After collecting the answers, the layers of ratio, limit, function, and various task contexts were analyzed quantitatively and qualitatively following the Zandieh framework.

First, nine tasks were analyzed quantitatively and qualitatively; the results were then summarized. Task 1 was descriptive, whereas the remaining tasks involved multiple-choice questions; thus, it is not listed in Table 3. Task 1 is explained below. These findings may hold relevance for university professors, teachers, textbook authors, and researchers with regard to teaching and learning the concept of derivatives.

Task 1 (the term “derivative”): “What does ‘derivative’ mean to you? Explain it as you can. If possible, provide a tangible example from your surroundings.”

Table 4 shows students correct and incorrect answers in Task 1. In Task 1, we considered what students generally understood about the word “derivative” and wanted them to express it using simple language. Almost 65% of the students described derivatives as the tangent line slope; that is, they were at the layer of the limit and graphical context in terms of Zandieh’s framework. In addition, 15% of the students were in the verbal context; that is, the rate of change (e.g., velocity, acceleration, rate) and function layers; 10% were in the symbolic context and function layers; and another 10% misunderstood and had instrumental or nonlayer understanding. One student stated, “In my opinion, a derivative means a shift from a difficult theory into an easy one. For example, , and sometimes the opposite occurs, but the derivatives of difficult formulas are very difficult.” Most engineering students described derivatives based on their perspectives and fields. For example, one said that the derivative means “creating changes in a function to achieve some information about it—changing the function into another one to use it in signals.” Some conceptual mistakes were observed in students’ interpretations, such as, “Derivative means , or derivative means putting the power of an expression as its coefficient and subtracting one from its power. For example, .”

Coding was performed based on Zandieh’s framework, as shown in Figure 1. The arrows in Table 5 indicate that some students established connections between two or more contexts in the concept of the derivative. Table 5 shows that the students reached all three layers of the Zandieh framework in terms of slope, but there were very few layers in the numerical context. In the case of the connection between contexts, slope and symbol had the strongest connection.

Task 2 (relative rate of change): "Every second, Ali rides a bicycle meters, and Reza walks on foot, so that j>s. How can the distance traveled by Reza and Ali be compared at a given time? (Byerley & Thompson, 2017)

a) Ali will j-s travel meters more than Reza.

b) Ali will j. s travel meters more than Reza.

c) Ali will j/s travel meters more than Reza.

d) Ali will j. s travel equal to Reza.

e) Ali will j/s travel equal to Reza. Give a reason for the selected option.”

One of our goals in assessing students’ understanding of derivatives was to put one physical position in the task and understand the relative rate of change. Table 3shows that 548 subjects completed Task 2, whereas 56 subjects did not. Further, 240 subjects (43.8%) chose option “a” while 206 subjects (37.6%) chose option “e” (correct option).

Putting a physical position in the problem and understanding the relative rate with a simple question were the main objectives of this study in terms of evaluating students' understanding of derivatives. Only 37.6% of students answered correctly. In addition, 43.8% of the students selected option "a," representing the prevailing collective multiplicative thinking among them. We normally act as follows to compare two expressions of : i) and ii) . For t=1 , equations i and ii can be used for comparison. However, the time parameter should be eliminated by dividing it such that it does not depend on the given task and options. Therefore, formula ii is used. However, most students attempted to use formula i, unaware that this relationship could be established for , t=1 which is the same prevailing thinking among the students.

As shown in Table 3, the maximum number of answers is related to options a and e, although a is greater and cannot be a good sign. The students who chose option a argued that “option a is correct because Ali, who traveled a longer distance, is subtracted from the distance traveled by Reza to obtain the distance Ali has traveled more.” Alternatively, “Option a is correct because when we want to calculate the distance between two points, we use subtraction.” Most students who answered incorrectly were in the ratio layer or had an instrumental understanding of derivatives in the verbal context (see Table 6).

Task 3 (calculating the graph slope without uniting (approximate)): “What is the approximate slope of the following line?

a) Number between 2 and 3

b) Number less than 2

c) Number greater than 3

d) It cannot be calculated approximately

e) Give a reason for your choice”

(Byerley & Thompson, 2017) with modifications.

The aim of Task 3 was to investigate the numerical context of the Zandieh framework using a graph without a unit. Table 3 shows that 42.3% of the students chose the correct answer of option a. Many students could not calculate the slope approximately; they looked for a specific angle or uniting of axes and constantly questioned whether the question had a problem. As shown in Table 7, most students were situated in the layer of the ratio or instrumental understanding of the framework in the numerical context of calculating the slope.

Task 3 was used to determine the relative size of the changes in relative to the changes in . The purpose of this problem is to calculate the numerical value of the slope.

Task 3 was used to determine the relative size of the changes in ^y relative to the changes in ^x. The purpose of this problem is to calculate the numerical value of the slope. Students calculated and approximated in terms of unit .

Table 7 displays the variety of correct and incorrect answers. As shown, 42.3% of the students answered correctly and obtained an approximate score of 2.5. The multiple strategies and conceptual errors in this problem are among the points to be noted.Table 3 shows that 18.7% of students obtained a slope from , meaning they understood the slope.

Task 4 (finding the function criterion): “The values in the following table are true for

Task 4 (finding the function criterion): “The values in the following table are true for

i) Find a relation for .

ii) How many different answers can be found for ?

a) 0 b)1 c)2 d) 4 e) Infinite”

(Pino-Fan et al., 2018) with modifications.

The aim of this task is to determine how students obtained the antiderivative function from the numerical values of the derivative function. Table 3 shows that 65.8% of students chose the correct answer.

The knowledge required to solve Task 4 is integral to the calculus theorems. Table 8 categorizes the correct and incorrect answers for Task 4. Multiple solutions and conceptual errors were observed during the task. Based on the results, 88 students reached number 1 while answering Section (ii), indicating that they had no appropriate understanding of the concept of initial function, different from what they called “derivative” as contrary to integral in the derivative phrase section. The percentage of students in the function and limit layers was more than 50%.

Task 5 (numerical representation of average rate of change): “A ball is thrown into the air from a bridge at an 11-meter height. ^f(t) represents the distance between the ball and the ground at time ^t . Some values ^f(t) of are shown in the following table:

Based on Table 9 , what is the approximate velocity of the ball per seconds?

Justify your answer.

a) , b) , c) , d) , e) Something else” (Pino-Fan et al., 2018).

The purpose of this task was to calculate the derivative at a point using a table of numerical values. Table 3 indicates that only 29.3% of the students answered correctly, whereas 70.7% answered incorrectly

Most students looked for physical formulas to answer this question, and they seemed highly confused ( Table 10). Approximating velocity using the table was difficult, and a connection between the table data and the secant lines on the graph could not be established. The percentage of students who answered the questions incorrectly and were in the numerical context in the ratio or instrumental understanding layer was greater than 70%.

Task 5 (deriving the function with several variables): “The gravitational force of the Earth F , which depends on the distance between the object and the center of the Earth ^(r) , is given by the formula ( M= mass of the Earth, m= mass of the object, and G= gravitational constant). Which is dr or F'(r) , and what does this answer tell you?

a).

b).

c).

e).

d).

(Jones, 2017)

The purpose of this task was to calculate the derivative of a particular variable from several variables. As shown in Table 3, 66.8% of the students provided correct answers. Since this task had a procedural aspect, a higher percentage of students were expected to answer correctly. The percentage of students in the functional layer was greater than 60% ( Table 11).

Task 7 (tangent line function): “For function , we have in the following figure. What is the point of based on its width?

a) b) c) d)

(, 2017)

The purpose of this task was to identify and apply a tangential line at a specific point. Table 3 shows that 57.1% of the students answered correctly. Table 13 shows multiple solutions and misconceptions.

Table 3 indicates that more than 50% of the students were in the functional layer in the slope context.

Task 8 (understanding the slope of points and comparing them): “Consider the points A,B,C,D,E,F on the following curve: Which expression is correct for a curved slope at these points?

b.

a.

c.

d.

e.

(, 2017)

The purpose of this task was to assess students’ ability to compare slopes on a graph. As Table 3 shows, 69.1% of the students gave correct answers for this task. As shown in Table 12, the students used tangent lines as a strategy to solve Task 8.

Table 13 indicates the prevailing strategy of the students in Task 8, in which 69.1% gave correct answers. More than 80% of students were in the function or limit layer of the slope context.

Task 9 (magnification of graph and derivative): “The graphs of the two functions, and , as well as their magnifications, are drawn in the intervals and . Which one of the functions is differentiable at point ? Explain.

a) Both functions b) Just function

c) Just function d) None e) Information is not adequate”

(Giraldo & Carvalho, 2003).

The purpose of this task was to identify the linearization and derivative approximation at a point on the graph. As shown in Table 3, 67.4% of students answered this task correctly.

Table 14 indicates that more than 60% of students were in the limit and function layers in terms of graphical analysis.

This study aimed to investigate undergraduate engineering and basic science students' understanding of the concept of derivatives based on Zandieh's framework (Zandieh, 2000). A review of the literature on derivatives shows that many students and teachers have difficulty connecting multiple representations and partial meanings associated with derivatives (Borji et al., 2018)(Pino-Fan et al., 2018)(Fuentealba et al., 2019)(Feudel & Biehler, 2021)(Rodríguez-Nieto et al., 2020)(Biza, 2021). Basic science students performed better than engineering students in terms of understanding slopes (Tasks 3, 7, and 8). These results are consistent with (Bingolbali et al., 2007), (Maull & Berry, 2000), and (Jaafar & Lin, 2017). Although the mean scores for the engineering and basic science groups were not significantly different, the engineering students performed better than the basic science students in the layers of ratio, limit, and function, as well as in graphical, physical, symbolic, and numerical contexts.

In general, students' understanding of the concept of derivatives was weak. The students understood the concept of slope better than the other concepts, although the performance of the basic sciences students in the slope context was better than that of engineering students. These results are consistent with (Zandieh, 2000) and (Byerley & Thompson, 2017).

Students performed the weakest in terms of rate of change. In particular, numerically calculating the average rate of change had the lowest mean score and was answered by the lowest number of students. While students referred to physical formulas to give answers, they often chose the wrong approach. Byerley and Thomson (2017) and (Mirin, 2018) confirm this finding. Although the additive meaning of slope and rate results in certain states, it can lead to invalid models of physical situations. The relative rate of change revealed an increase in the rate of change. (Byerley & Thompson, 2017), (Weber et al., 2012), and (Auxtero & Callaman, 2020) discussed this issue. Similar results were obtained in this study, with most students selecting the wrong option, a. In general, engineering students performed better in the context of rate of change than basic science students. Regarding the second question, undergraduate students passed more than 50% of the ratio layer in all tasks, except for Tasks 2 and 5. In Tasks 1, 6, 8, and 9, more than half of the students covered the limit layer, and vice versa for other tasks. In Tasks 2 and 5, less than half of the students were in the function layer, and vice versa in the other tasks.

In tasks 3, 7, and 8, the basic sciences students performed better at all levels of ratio, limit, and function (except for the limit layer of Task 7) than the engineering students. In other words, basic science students understood the slope of the tangent line better than the engineering students. In the case of other tasks, the engineering students performed better than the basic science students in the layers, that is, in the concept of rate of change and the intuitive understanding of graphs. These results are consistent with those reported by (Bingolbali et al., 2007), (Maull & Berry, 2000), and (Carli et al., 2020). Finally, in Tasks 3, 7, and 8 (the slope of a graph without uniting, the use of tangent lines, and understanding the slope of points and comparing them), the basic science students performed better than the engineering students, and vice versa in other cases. In general, engineering students performed better in the graphical, verbal, numerical, and physical contexts. In the intuitive part of the derivative and the magnification of the graph, the students did not perform well, which requires consideration. These results are consistent with those reported by (Bingolbali et al., 2007) and (Desfitri, 2016).

This study's findings suggest that students lack an appropriate understanding of the basic concepts of derivatives in numerical, physical, verbal, and graphical contexts. Engineering students had significantly higher relative frequency percentages and mean scores than basic sciences students for understanding derivatives in terms of the layers of ratio, limit, and function. Basic science students, meanwhile, performed better than engineering students in terms of understanding tangent line slope, while engineering students performed better in terms of rate of change. In mathematics, derivatives have different meanings in areas such as slope, difference quotient limit, rate of change, velocity, and acceleration. Certain meanings are typically seen more in the undergraduate curricula of specific fields. In engineering, for example, rate of change is more often considered, whereas tangent line slope arises more in the basic sciences. Inconsistencies between the interpretation of slopes in mathematics classes and of rate of change in engineering classes may cause problems for students. Therefore, the concepts of slope and rate of change should be considered simultaneously in the curriculum. Students' instrumental understanding of the concept of derivatives in their arguments was evident in this study; they knew the derivative formulas but could not express their meanings. Even the students who passed the differential equations test had weak arguments for interpreting the concepts of derivatives. Perhaps paying significant attention to the instrumental understanding of derivatives led to these results. Students could not establish relationships between the different representations of derivatives, or the relationships were poor. This means they had poor conceptual understanding. This issue should be considered when teaching derivatives. Exploring the concept of derivatives with other frameworks is also suggested for researchers.

This study is part of a dissertation completed at Shahid Rajaee Teacher Training University.