The purpose of the reported study was to investigate the social science students’ concept images and concept definitions of antiderivatives. Data were collected through asking students to answer 10 questions related to antiderivatives and also by interviewing them. The theory of concept image and concept definition was used for data analysis. The results of the study show that the students’ definitions of antiderivatives were personal reconstructions of the formal definition. Their concept images were coherent only to a certain extent as there were some conceptions of some ideas that were at variance with those of the mathematical community. These were more evident when students solved problems in the algebraic representation. Some students did not know which integration or differentiation methods they should apply in solving the problems. The significance of such findings is to enable the mathematics educators to pay attention not only to the use of signs and symbols representing mathematical concepts but also to their semantics.
Calculus is an important branch of mathematics in a number of social and natural science disciplines. In the social sciences antiderivatives are required in tackling problems in marginal analysis and optimisation problems. In the natural sciences students need to understand antiderivatives to deal with rates of change for concepts such as velocity, acceleration and rate of flow. However, students fail to understand some of the basic concepts of calculus (Brijlall & Ndlazi,
Like other calculus concepts the learning and understanding of antiderivatives involves significant use of signs and symbols. But these symbols continue to lack meaning for students (Ferrer,
Having highlighted some difficulties that students have in relation to the topic of study, studies related to students’ understanding of this important calculus concept have not yet been conducted in Lesotho, where the current study took place. This could probably be because in Lesotho students encounter calculus for the first time at university or other tertiary institutions, of which there are not many; hence, the idea is not very popular. The findings of this study will not only be useful to the National University of Lesotho where the study was conducted, they will also be useful to researchers elsewhere in terms of either similarities or differences in findings and their implications.
This study addressed the following research questions:
Research question 1 (RQ1): What definitions of antiderivatives do secondyear social sciences students give?
Research question 2 (RQ2): What connections do students make between a function and its antiderivative in algebraic representation?
Research question 3 (RQ3): How do students solve problems involving differentiation and integration?
The foundation of this study involved the characteristics of a specific calculus concept, the antiderivative. The theoretical framework by Tall and Vinner (
The foundation of this study involved characteristics and understanding of a very specific calculus concept. The review of the literature includes the type of difficulties that the students encounter in understanding the concept of antiderivative together with possible causes for such.
In the studies of Hall (
Metaxas’s (
The definition provided in the study of Metaxas (
On the surface the two integrals seem to produce the same function but in essence the
In the study by Kiat (
Giving incorrect answers due to calculation errors, failure to use relevant rules and to link derivatives to their antiderivatives and inability to use appropriate integration techniques in solving indefinite integral problems by students were dominant in the studies by Maharaj (
The theory of concept definition and concept image by Tall and Vinner (
Tall and Vinner (
The term concept image is described as the total cognitive structure that is associated with the concept, which includes all the mental pictures and associated properties and processes (Tall & Vinner,
The definitions of antiderivative provided in some calculus textbooks include a list of connected conditions: ‘An antiderivative of a function
These definitions mean that if we have a function
This study involved two methods of data collection to document students’ understanding of antiderivatives. Questions on antiderivatives and interviews to illuminate the students’ questions responses were used. These methods of data collection were important because they could both enable the researchers to get qualitative data that could answer the research questions by studying students’ responses to get direct answers or make inferences.
The participants in this study were 117 secondyear university social science students whose major subjects were economics and statistics. The sample group was taught algebra and calculus in their first and second years by the first author. The calculus topics covered during their second year included: integrals, techniques of integration and the relationship between integration and differentiation with initial value problems.
Students were also aware that the indefinite integral was used as a general antiderivative. The students had encountered problems in which the terms antiderivative and integral (indefinite) had been used interchangeably in their calculus textbooks. They also had solved application problems such as: marginal cost, marginal revenue, marginal propensity to save, marginal propensity to consume, and initial value problems. Students knew for example that the cost function is the antiderivative of the marginal cost function obtained through the process of integration and to check this they had to differentiate to obtain the marginal cost function. The students had not solved problems in which the integration and differentiation operators were paired. They encountered this for the first time during the study. They were also not familiar with the integration of expressions that cannot be broken down into elementary functions. The learning took place through lectures, class discussions and tutorial sessions. The participants had used mainly the prescribed calculus textbook in the social sciences (Haeussler et al.,
Ten questions (see
The questions, their justification and research questions their data would answer.
Question  Why it was asked  Research question that would be answered by its data 

1. What is an antiderivative of a function?  To find how students would define an antiderivative.  Data collection was useful in responding to Research question 1 (RQ1). 
2. Does a function have only one antiderivative? Explain.  To give students an opportunity to relate a function to its antiderivative.  Data collected was useful in responding to Research question 2 (RQ2). 
3. In order to complete the process of integration a constant must be added to the results. Why is this so?  To give students an opportunity to relate a function to the indefinite integral which is taken to be the general antiderivative of a function.  Data collected was useful in responding to RQ2. 
4. What is the relationship between the processes of differentiation and integration if any?  To find if students could relate the two processes verbally so that they could use this knowledge to solve problems that could not be broken down into simpler functions when integrated.  Data collected was useful in responding to Research question 3 (RQ3). 
5. Find:
Which processes did you use in finding the answer for 5a? 
To find students’ procedural understanding of differentiation.  Data collected was useful in responding to RQ3. 
6. Find
∫ Which process did you use in finding the answer for 6a? 
To find students’ procedural understanding of the process of integration.  Data collected was useful in responding to RQ3. 
7. What is the relationship between the functions 
To provide students an opportunity to relate the general forms of functions where one is the derivative and the other is the integral.  Data collected was useful in responding to RQ2. 
8. Find

To see if students could relate the processes of integration and differentiation in a general algebraic form.  Data collected was useful in responding to RQ3. 
9. Find

To see if students could relate the processes of integration and differentiation in a general algebraic form with functions that could not be broken down into simpler forms.  Data collected was useful in responding to RQ3. 
10. Find

To see if students could relate the processes of integration and differentiation in a general algebraic form with functions that could not be broken down into simpler forms.  Data collected was useful in responding to RQ3. 
Interviews were used to ensure validity and provide richer elaborations of the questionnaire responses. This also provided a form of methodological triangulation (Seale,
The interviews were conducted with a stratified subsample of participants (
Students’ definitions of antiderivative.
Category  Student  Definition 

1. Integral  S41  It is the integral of a function. 
S96  It is the function that is obtained from integrating a function’s orginal function.  
2. Integration  S42  Is the intergration of a function. 
S101  [ 

S12  It is a sign used to change a derived funtion to its original form. Example 

3. Reverse or undoes  S52  It’s a function that undoes the derivates of a function. E.g. marginal cost is a derivative of a cost function and when taking antiderivatives you got cost function. 
S25  It is the reverse of a function.  
4. Back to original function  S66  Is a function when derived gives an original function. 
S67  Antiderivative of a function is a function of which when derived gives the original function. 
Data was analysed using the theory of concept image and concept definition. The theory was complemented by some parts of the literature. For example, in analysing the students’ definition of antiderivatives (see
In analysing data that relates a function to its antiderivative, two possibilities were anticipated by the authors: (1) that a function has one antiderivative or (2) that a function has many antiderivatives. These two categories were anticipated because there is a possibility that students could take the general antiderivative as one function if they are not aware that the
Students were asked if they were willing to take part in the research by the first author who taught them. They were made aware that any information they give will be treated with great confidentiality. This is also evidenced by the fact that their names are not used in the research and they were aware of this. They were also told that they are free to withdraw from the research activity if they felt uncomfortable. But because of the good relationship that existed between the first author and the students, they showed great interest in taking part in the research. Perhaps another factor that contributed to students’ willingness to participate is that they were told that the information that they give will also help their instructor to improve her teaching not only for their benefit but also for the classes that will follow. The permission to conduct the study was also obtained from the Head of Department of Mathematics and Computer Science as the course is offered by the mentioned department.
The presentation of results in this section is categorised into the assertions and evidence aligned with the research questions. The assertions are accompanied by supporting evidence composed of summaries of the questions’ responses and specific indepth quotes from the student interviews from different performance groups and discussion.
From the 117 students’ responses about the definition of antiderivative, 111 yielded four categories: an antiderivative is the integral of a function (67 students – 60% of students including interviewees S41, S59, S96 and S100); an antiderivative is the integration of a function (34 students – 31% of students including interviewees S12, S42, S48 and S101); an antiderivative is the function that undoes (change, reverse) the derivative (7 students – 6% of students including interviewees S25 and S52); an antiderivative is the function that when derived gives the original function (3 students – 3% of students including interviewees S66 and S67). One student defined an antiderivative as a function, there were no responses from two students and three students’ responses were unclassified.
The given definitions are students’ own personal reconstruction of the formal concept definition; similar results were obtained in the studies of Metaxas (
What do you mean when you say that an antiderivative is an integral of a function? What is an integral?
The integral is the inverse of the derivative.
Can you show me what you mean by this.
She then writes [
[
Is it what you mean by inverse?
Yes.
S41 associates the term antiderivative with two terms, integral and inverse (of the derivative). We believe that this emanates from the use of the prefix ‘anti’ and also from the fact that differentiation and integration are referred to as reverse processes. In her working she starts by talking about the derivative, then shows how one can get the indefinite integral from the derivative. This shows her concept image of inverse which matches her personal concept definition. Her working shows that the process of integration was performed twice. Although the relationship between 2
In the second category, S42 and S101 use the term integration while S12 refers to the sign, an integration operator, which has the same fundamental meaning as integration. These showed that the students’ concept images of antiderivative had the process of integration as part of their cognitive structure. In the third category the words
In comparison with the findings of Hall (
The two questions that required students to explicitly relate a function to its antiderivative are Question 2 and Question 7. Question 2 read: ‘Does a function have one antiderivative? Explain’ while Question 7 read: ‘What is the relationship between the functions
In responding to Question 2, 79 students (69%), including S12, S25, S52, S59, S96 and S101, said that a function has many antiderivatives. Of these respondents, 51 were aware that a function has many antiderivatives that differ by a constant while 22 students, including S12 and S96, said that a function will have many antiderivatives depending on the order of differentiation of the function. Six students did not explain their answers. Another 32 students (28%), including interviewees, S42, S48, S66, S67 and S100, said that the function has only one antiderivative, by which they meant a general antiderivative. Four students (3%) including interviewee S41 gave unique responses and two students did not respond. The response that a function has many antiderivatives does not mean that all the students had the same interpretation, as interview excerpts from S12 and S25 illustrate. Their questionnaire responses are first presented:
It depends on how many times it has been derived. If it has been derived twice, then it will be antiderived twice.
No. It is because any antiderivative has the constant C which represents any number so function has many antiderivatives.
S12’s interview:
When asked if a function has only one antiderivative, you say that it depends on how many times it has been derived. Can you explain what you mean?
I mean if a function has been derived twice as in
Can you explain what you mean.
[
[
Can you show me on what you have written which functions are the antiderivatives of the others?
This integral is the derivative of
Which one?
This (
So the antiderivative of
[
Does this
It stands for any particular number if we are not told how to find the
Can it be
Yes madam.
So is
Yes it is one.
S12’s concept image shows that a function has many antiderivatives but these antiderivatives are counted based on the number of times in which a function is integrated to its original form. The example she gives is that of a function which has two antiderivatives because it has been integrated twice to get back to
Can you explain what you mean when you say that a function has many antiderivatives?
Taking
If you were to differentiate each of these antiderivatives what would you get?
They will be the same.
What will they be?
They will be 2
Is this why you say a function has many antiderivatives?
Yes madam.
The excerpt shows that S25 gave the interpretation of antiderivative as taught. Which also matches his personal concept definition of a reverse process. He takes an indefinite integral to be a set of antiderivatives that differ by a constant as reflected by his questionnaire response. Thus his concept image of antiderivative was coherent. The multiple interpretations of antiderivatives given by students, though different, were the same for each student. This means that there were no conflicts that were experienced between the individual students’ evoked concept images.
This is the second assertion made in trying to answer the second research question on how students related the function to its antiderivative. S42 was among the students who said that a function has only one antiderivative, the general, consistently. S12 also consistently related a function to its antiderivative by saying that it depends on the number of times the function had been differentiated. S42’s interview about her Question 2 response illustrates this point. S12’s response to Question 2 has already been discussed and reference will be made to it when discussing both of their responses to Question 7.
Yes. But it has different constants.
S42’s interview:
[
No!
If I have the integral
We can differentiate.
What about the derivative of
It is 3
Is
No! The antiderivative is
S42’s conception of antiderivative is that of a general antiderivative. She said that
Some responses to Question 7 had one dimension while others had two dimensions. Onedimensional responses included:
Questionnaire responses of S12 and S42 show consistency of interpretation.
S12 was asked to explain her response in the interview:
Here you say that
What I meant is that we got this one [
In this case she says that there is one antiderivative because the integration or differentiation had been performed once. This is consistent with her understanding that the number of antiderivatives of a function depends on the number of times a function had been differentiated.
S42 had earlier described an antiderivative as the general antiderivative only. Her response reflects the same earlier interpretation. She had also referred to antiderivative as integration of a function and she emphasises this conception by enclosing the word integration in brackets after writing antiderivative. Since in Question 5 the students obtained
The previous sections have shown that students’ descriptions associated antiderivatives with integration and differentiation. When relating functions to their antiderivatives, the two processes also played a major role. This is in alignment with the literature review about antiderivatives. This section presents findings of how students’ conceptions may have influenced the way the students solved problems involving integration and differentiation. Before we present the findings of how such problems were solved we begin by giving the findings of how the students related the two processes explicitly in Question 4. The question required the students to show how differentiation and integration are related.
In responding to Question 4, four categories of correct explanations were identified as: reverse (39 students; 33%), opposite (16 students; 14%), inverse (12 students; 10%) and back to original form (8 students; 7%). Of the remaining students, 14 (12%) gave individual responses, such as opposite as reverse, opposite as negative and viceversa, 23 (20%) gave wrong responses such as ‘differentiation is the gradient while integration is the area’ (S24) and 5 (4%) did not respond to the question. Some errors made include confusing the processes of integration and differentiation, the incorrect use of integration and integration operators, and giving the descriptions of how some techniques (e.g. power rule) of the two processes are individually performed without relating the processes. Most of students’ descriptions used words such as inverse, reverse and back to original form which resonated with the categories of descriptions of antiderivative in
Students’ responses to Question 9 follow. This question required students to find (a)
Summary of responses to Question 9.
Question  Correct  Incorrect (interviewees)  No response 

a.  111  6  0 
b.  71  43 (S12, S41, S101)  3 
c.  29  86 (S12, S41, S101)  2 
Responses to Questions 9(a) and 9(b) were reasonable, but responses to Question 9(c) were problematic because students wanted to break the expressions down into elementary functions that did not exist. This may be because for Questions 9(a) and (b) the differentiation operator was to be applied before the integration operator and the majority of students managed to find the derivative of
How did you get
I integrated.
What did you get as you integrated?
I got
But the question does not require you to find
Yes but when you differentiate it
When S42 finds that the 2
In responding to Question 4 you said that differentiation and integration are the opposites of each other, what do you mean by this?
If we take the function
Can you tell me what the result would be here without working,
Now with that knowledge what do you think the result here,
[
What encouraged you to work out
Because I knew how to work it out.
What about here,
I just wrote the answer because I did not know how to work it out.
S42 had made connections between the processes of integration and integration. She managed to solve the problems successfully when differentiation was carried out first before integration. But, when integration precedes differentiation she makes the connection between the two processes beyond the algebraic manipulation. This appears to mean that the correct answer was obtained with the understanding that integration and differentiation are opposites of each other but could not give reasons beyond saying she could not work the problem out. In addition, the integrals were written without a constant of integration, which might also be a sign that the order in which the integration and differentiation operators are presented has no significance in terms of the concept they signify or it could be that the student at this moment just forgot to write the constant of integration.
S12 was among the students who said that integration is the opposite of integration; however, her explanation is still clouded by language issues. She wrote:
Because when we differentiate we subtract the constant to complete the differentiation. So since integration is the opposite of differentiation, we add instead of subtract.
S12 persistently uses the words subtract and add to show that the two processes are opposites of each other. Everyday language gets in the way of her explanation. What she means is that when differentiating the constant of integration the result is zero, thus the constant disappears so she uses the everyday word subtract because by subtraction we mean take away and when integrating we add thus we see the constant coming back. Thus she shows a coherent concept image about the way the two processes are related. The only problem is how to explain her actions using proper mathematical language. These findings on language issues are similar to those of Hall (
Her (S12) written work to Question 10(c) which has the same external structure as Question 9(c) follows:
How did you get your answer here?
Maybe I should have differentiated.
But the integration operator comes before the differentiation one.
But I think I prefer differentiation first before integration.
Why do you prefer differentiation first?
Because I think it is going to be a little bit simpler.
S12 divides by the derivative of the power
The first step shows that S12 seems to be applying the power rule
This seems to resonate with the view that symbols in themselves have no meaning until the learner attaches meaning to them (Sengul & Katranci,
The study has shown that students gave personal definitions of antiderivatives that overlapped with the formal definition as reflected in their interviews. These findings are different from those of Hall (
In some cases the students interpreted the word antiderivative the same. Students persistently showed that a function has one antiderivative or many in the interviews. Some students failed to use their conceptions of how integration and differentiation are related in solving problems where differentiation and integration operators were paired. The main problem was that students could not go beyond the manipulation of symbols to relating them to the concepts they represent by taking them as reverse processes and sometimes there was no need to carry out any algebraic manipulations but just to write the correct answers. While some of students’ conceptions were unique to the context of the study (
The purpose of teaching is to promote understanding; questionable understanding means questionable teaching. This may be true as in the case of the way Question 5, Question 6, and Question 7 in the reported study are written. The restriction,
It is evident from the study that students do sometimes make their own reconstruction of mathematical ideas taught. The mental structures that students have about the concept antiderivative is not fully coherent as a result. There are a number of instances where students failed to solve problems on integration and differentiation even though they had related the two concepts correctly verbally. This shows that making connections between symbols and the concepts they represent is still a challenge that needs further attention.
Although there has been a claim about data triangulation, when dealing with a very large group of students it is not possible to interview each and every one of them because of the enormity of the exercise. Because of this the choice of 10 students for interviews cannot be claimed to be a fair representation of the whole class. If a different group was chosen for interviews the result would perhaps have been slightly different. This is one of the limitations of the reported study.
The second limitation is that of the time that elapsed between administering of questions to students and the conducting of the interviews. The researchers had to go through responses to questions given in order to choose the interviewees. This on its own would also affect the results as through curiosity students might have discussed their responses with their peers and might also have checked their textbooks to see if the answers they had given were correct. While this has been a limitation the interview excerpts do show that students still had problems with solving problems given regardless of time that had elapsed. Thus it could be that some students did not bother to check if they were correct based on the fact that they were not going to be given any marks for the responses they gave for the posed questions.
We would like to wholeheartedly thank Larry Yore and Anthony Essien for their powerful and thoughtful comments in the writing of this article.
The authors declare that they have no financial or personal relationships that may have inappropriately influenced them in writing this article.
E.K.M. contributed in writing the article, reviewed the literature, collected data and analysed data. All these were done by holding a series of discussions with M.Q. where necessary.
This research received no specific grant from any funding agency in the public, commercial, or notforprofit sectors.
Data sharing is not applicable to this article as no new data were created or analysed in this study.
The views and opinions expressed in this article are those of the authors and do not necessarily reflect the official policy or position of any affiliated agency of the authors.