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Exploring pre-university students’ mathematicalconnections when solving Calculus applicationproblems

Javier García-García & Crisólogo Dolores-Flores

To cite this article: Javier García-García & Crisólogo Dolores-Flores (2020): Exploringpre-university students’ mathematical connections when solving Calculus applicationproblems, International Journal of Mathematical Education in Science and Technology, DOI:10.1080/0020739X.2020.1729429

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INTERNATIONAL JOURNAL OF MATHEMATICAL EDUCATION IN SCIENCE AND TECHNOLOGYhttps://doi.org/10.1080/0020739X.2020.1729429

Exploring pre-university students’ mathematical connectionswhen solving Calculus application problems

Javier García-García and Crisólogo Dolores-Flores

Research Centre of Mathematics Education, Autonomous University of Guerrero, Chilpancingo, Mexico

ABSTRACTMathematical connections play an important role in achievingmath-ematical understanding. Therefore, in this article, we report researchwhose objective was to identify mathematical connections that pre-university students make when they solve problems that involve thederivative and the integral. In this research, we consider a math-ematical connection like a true relationship between two or moreconcepts, definitions, theorems, or meanings amongst themselves,with other disciplines or with real life. Task-based interviews thatincluded four application problems were used to collect data from25 students (18 males and 7 females) and Thematic Analysis wasused to analyse them. Our results indicate that mathematical con-nections are dependent on each other, and because of this, theyform systems of mathematical connections around the reversibilityconnection between the derivative and the integral. We found con-nections of five types: different representations, procedural, features,reversibility and meaning as a mathematical connection.

ARTICLE HISTORYReceived 1 October 2018

KEYWORDSMathematical connections;Calculus; derivative andintegral; problem solving;pre-university

1. Introduction

Making mathematical connections is a frequently established goal for mathematics educa-tion (Evitts, 2004), whichmeans to relatemathematics to the real world, to other disciplinesand other mathematical concepts (García-García & Dolores-Flores, 2018, 2019; Özgen,2013). Mathematical connections allow mathematics to be viewed as an integrated fieldand not as a collection of separate parts, which is how students regard it (Evitts, 2004;Jaijan & Loipha, 2012; Mwakapenda, 2008) and how it is often presented in the teaching-learning process. To avoid this, theMexican curriculum proposes that at the pre-universitylevel, students need to:

Construct and interpretmathematicalmodels through the application of arithmetic, algebraic,geometric and variational procedures, for the understanding and analysis of real, hypotheticalor formal situations.

Formulate and solve mathematical problems applying different approaches.

Explain and interpret the results obtained through mathematical procedures and contrastthem with established models or real situations (DGB, 2013a, p. 12, 2013b, p. 13).

CONTACT Javier García-García jagarcia@uagro.mx Research Centre of Mathematics Education, AutonomousUniversity of Guerrero, Av. Lázaro Cárdenas S/N, Chilpancingo, Guerrero 39086, Mexico

© 2020 Informa UK Limited, trading as Taylor & Francis Group

2 J. GARCÍA-GARCÍA AND C. DOLORES-FLORES

Furthermore, teaching ‘must leave aside the memorization of disarticulated topics andthe acquisition of relatively mechanical skills’ (DGB, 2013a, 2013b, p. 6). It aims to pro-mote interdisciplinary work, much like how real facts are presented in everyday life. Inthis sense, countries such as South Africa (Mwakapenda, 2008), Turkey (Özgen, 2013),Israel (Leikin & Levav-Waynberg, 2007), Mexico (DGB, 2013a, 2013b), Australia (Marsh-man, 2014; Sawyer, 2008), and the United States (NCTM, 2014) have as a curricular goal,for students to make mathematical connections. This has led researchers to point out theimportance of connecting school mathematics with the outside world (Gainsburg, 2008)since it offers certain advantages, according to teachers (Karakoç &Alacacı, 2015). If this isachieved in a classroom, studentswill be able to improve theirmathematical understanding(Eli, Mohr-Schroeder, & Lee, 2011; Mhlolo, 2012).

Literature indicates that, while students perceive the process of connectingmathematicswith real life as important, they also recognize that it has not been sufficiently implementedin the classroom (Baki, Çatlıoğlu, Coştu, & Birgin, 2009; Soltani, Mohammad-Hassan,Shahvarani, &Manuchehri, 2013). Nor is there enough information about howmathemati-cal connections (if any) are applied in the classroom nor how often (Gainsburg, 2008). Oneway to make mathematical connections in the classroom could be through problem solv-ing that promotes relationships between different concepts and, as suggested by Leikin andLevav-Waynberg (2007), find different ways to reach the solution. However, some teachersfind it difficult to teach in this manner. Therefore, we agree with Garii and Okumu (2008)when they point out that if teachers do not recognize the various ways math is embeddedin our daily lives, then, regardless of the depth of their mathematical knowledge, they maybe unable to help students make mathematical connections between school mathematicsand the reasons for studying mathematics.

A review of the literature indicates that there is little research that explores mathe-matical connections that emerge when students solve problems. Entirely for reference,we have identified the studies of Lockwood (2011), Özgen (2013), Yoon, Dreyfus, andThomas (2010) and Dolores and García-García (2017), although none of them focuseson the pre-university level in Calculus. Other studies explore mathematical connectionsbetween representations (García-García & Dolores-Flores, 2019; Berry & Nyman, 2003;Dawkins &Mendoza, 2014; Haciomeroglu, Aspinwall, & Presmeg, 2010; Hong & Thomas,2015; Mhlolo, 2012; Mhlolo, Venkat, & Schäfer, 2012; Moon, Brenner, Jacob, & Okamoto,2013) and on the resolution of specific tasks (García-García & Dolores-Flores, 2018; Eliet al., 2011, 2013; Jaijan & Loipha, 2012; Mamolo & Zazkis, 2012). This review has moti-vated us to exploremathematical connections that pre-university studentsmakewhen theysolve problems that involve the use of the derivative and the integral – a part of the Fun-damental Theorem of Calculus (FTC) –, in view of the fact that there is little research withthis focus. Therefore, the research question that is answered in this paper is: What mathe-matical connections do pre-university students make when they solve problems involvingthe derivative and the integral?

In this research, we seek to validate a framework to study mathematical connectionsthat we have built from the thematic analysis in previous research (García-García &Dolores-Flores, 2018, 2019) as well as extend its validity when we explore the mathemat-ical connections that students make when they solve application problems of Calculus.This is a contribution of this research. On the other hand, the importance of focusing onthe pre-university level is that, at this level of education, students profile their university

INTERNATIONAL JOURNAL OF MATHEMATICAL EDUCATION IN SCIENCE AND TECHNOLOGY 3

education by orienting their professional training towards the sciences, technologies orthe humanities. Therefore, mathematical connections related to Calculus are importantbecause they are part of the pre-university students’ preparatory training and are use-ful for the understanding of ideas within Calculus and of more advanced courses likeMathematical Analysis, Differential Equations and Complex Variable, just to name a few.

2. Mathematical connections andmathematical understanding

We assume that making mathematical connections is a fundamental part of achievingmathematical understanding. In this sense, some authors consider that a student’s math-ematical understanding is related to his or her skill in making mathematical connections(Barmby, Harries, Higgins, & Suggate, 2009; Berry & Nyman, 2003; Boaler, 2002; Busin-skas, 2008; Cai & Ding, 2015; Good, McCasilin, & Reys, 1992; Hiebert & Carpenter, 1992;Noss, Healy, & Hoyles, 1997). For Hiebert and Carpenter (1992), the strongest connectionthat is built is the mathematical understanding that is achieved. Meanwhile, Cai and Ding(2015) point out that some features of mathematical understanding are: it is both a process(or knowing) and a result of the act of understanding (sometimes called knowledge); it isat the same time the process to make mathematical connections and the result of makingthem; it is a dynamic and continuous process; it can have different levels and different types;the aim is to achieve a deeper understanding of mathematics. So, for Cai and Ding (2015)and Hiebert and Carpenter (1992) mathematical connections are the result of mathemat-ical understanding, but also that mathematical understanding can be the act of makingmathematical connections.

We agree with the previous ideas and we accept that making mathematical connectionsbetween mathematical ideas is an important indicator of mathematical understanding(Berry &Nyman, 2003). At the same time, a student who has mathematical understandingwill be able tomakemathematical connections between ideas, concepts, procedures, repre-sentations and meanings (García-García & Dolores-Flores, 2019). Likewise, for Kastberg(2002), mathematical understanding occurs in the mind of the student; she argues that,although this may change, it may become more or less consistent with the standard math-ematical view of the concept; the mediator of understanding is the previous knowledge ofthe student.

On the other hand, for Businskas (2008) mathematical connections are those relation-ships onwhichmathematics is structured and are independent of the student, but are at thesame time, those relationships by which the thought processes build mathematics. How-ever, for Eli et al. (2011), a mathematical connection could be a link in which previousor new knowledge is used to establish or support an understanding of the relationship(s)between two or more mathematical ideas, concepts, strands or representations within amental network. According to Evitts (2004), the connected knowledge can be describedin terms of its personal construction and meaning, the multiplicity of links between con-cepts and procedures, and the power derived from knowing connections. Mathematicalconnections can be made with the everyday world, with previous knowledge, with thefamiliar contexts inside and outside school, with different mathematical concepts, withother disciplines, with the past and the future (NCTM, 2014; Presmeg, 2006). Berry andNyman (2003) point out, furthermore, that students need to be motivated to reflect onmathematical connections in the classroom.

4 J. GARCÍA-GARCÍA AND C. DOLORES-FLORES

In brief, for this research, we assume that there is a strong relationship betweenmathematical connections and mathematical understanding (García, 2018). Likewise, weconsider a mathematical connection as a cognitive process through which a person makesa true relationship between two or more concepts, definitions, theorems, or meanings(García-García &Dolores-Flores, 2018, 2019) to solve application problems.Mathematicalconnections are useful in improvingmathematical understanding (Businskas, 2008) whichis valued as such by the expert (teacher or researcher). We can identify the mathematicalconnections that students make through their written productions and verbal argumentsthat they develop when they solve specific tasks.

In this research, we presented application problems to the students. We assume that,when amathematical model is already present in a given context because it has been intro-duced by others, then we are talking about the application of mathematics (Niss, 2012).That is, we are facing an application problem that may or may not evoke concepts fromother disciplines or real life. In this research,we use problems that evoke concepts of physicsand biology and that can be solved by applying the derivative or the integral.

3. A framework to studymathematical connections in Calculus

There is no consensus on how to study the mathematical connections that students makewhen solvingmathematical tasks. For this reason, we have adopted a framework (Figure 1)

Figure 1. Framework to study mathematical connections (Adopted from García, 2018; García-García &Dolores-Flores, 2018, 2019).

INTERNATIONAL JOURNAL OF MATHEMATICAL EDUCATION IN SCIENCE AND TECHNOLOGY 5

that we have previously constructed from the data collected from pre-university stu-dents when solving algebraic tasks (García-García & Dolores-Flores, 2018) and graphicaltasks (García-García & Dolores-Flores, 2019) and that, in this research we extend its useto analyse mathematical connections that pre-university students make when they solveapplication problems in Calculus.

In Figure 1, we recognize as intra-mathematical connections when a student relatesor associates mathematical concepts between themselves to solve mathematical tasks orapplication problems, this means that mathematical connections emerge inside mathe-matics and among mathematical entities (García-García & Dolores-Flores, 2018, 2019),but that it could involve concepts of other disciplines, as well. In this research, we assumethe following types of mathematical connections (Figure 1) as described below:

(1) Procedural: this mathematical connection occurs when ‘someone uses rules, algo-rithms or formulas to complete any mathematical task’ (García-García & Dolores-Flores, 2019, p. 21). Likewise, include the explanations or arguments that a studentoffers to use those formulas and how he or she applies them to achieve a result. Forexample, a student that uses the power rule to find a derivative of a function is usingthis mathematical connection.

(2) Different representations: this type of mathematical connection could be: (a) alternaterepresentations when a student uses two or more representations to represent a math-ematical idea or concept (verbal–algebraic, algebraic–geometric, etc.). (b) equivalentrepresentations when the same mathematical concept is expressed in two differentways within the same representation. For example, f (x) = (x − 1)2 and f (x) = x2 −2x − 1 are equivalent representations in the algebraic representation.

(3) Features: a student makes this type when he or she identifies some invariant attributeor quality that differentiates it from others, that is, some characteristic that allows himor her to recognize that mathematical concept in other contexts. Likewise, this mathe-matical connection is identifiedwhen a student defines the characteristics or describesthe properties of the mathematical concepts that make them different or similar toothers.

(4) Reversibility: a student makes this type of mathematical connection when he or sherecognizes the bidirectional relationship between derivative and integral as operationsand when he or she uses the FTC as a way of linking both concepts. So, it occurs ‘whensomeone can start froma conceptA to get to a concept B and invert the process startingfrom B to return to A’ (García-García & Dolores-Flores, 2019, p. 21).

(5) Meaning: we can identify this type of mathematical connection when a studentattributes a sense to a mathematical concept, that is, what it means to him or her (thatmakes it different from others). It includes those where a student gives a definitionthat he or she has constructed for these concepts. It is different from the featuretype because the properties and qualities of mathematical concepts are not described.According to Kilpatrick, Hoyles, Skovsmose, and Valero (2005), the meaning of amathematical concept may be limited by its definition and the context of its use.

(6) Part-whole: this ‘appears when students establish a logical relationship betweenmath-ematical concepts: generalization or inclusion’ (García-García &Dolores-Flores, 2019,p. 21). In this sense, this mathematical connection occurs when a student identifiesthat A (a general case) contains B (a particular case), or B is contained in A.

6 J. GARCÍA-GARCÍA AND C. DOLORES-FLORES

4. Fundamental theorem of Calculus

FTC is essential in the development of Calculus (Carlson, Persson, & Smith, 2003)and establishes the mathematical connection between the derivative and the integral.Mathematically, this theorem points out that if f is a continuous function on a closed

interval [a, b], then the function g defined by g(x) =x∫af (t)dt where a ≤ x ≤ b is an

antiderivative of f , that is to say, g′(x) = f (x) for a < x < b. Using Leibniz’s notation, wecan write this theorem, whenever f is a continuous function (Stewart, 2010) as

ddx

x∫

a

f (t)dt = f (x)

This is known as the first part of the FTC; whose use is demanded in the applicationproblems that we used in this research. The second part of the FTC establishes that if f isa continuous function on the closed interval [a, b], then:

b∫

a

f (x)dx = F(a) − F(b)

Whenever F is any antiderivative of f , this is F′ = f (Stewart, 2010). Together, the twoparts of the FTC point out that differentiation and integration are inverse processes undercertainmathematical conditions, previously explained.What one operation does, the otherundoes.

Given the theoretical complexity of the FTC, students, when moving to the higher levelin the field of mathematical modelling and applications, present difficulties such as lackof knowledge (mathematical theory and other disciplines), difficulties in formulating pre-cise mathematical problems, interpreting word problems, understanding processes, use ofrepresentations, among others (Thomas et al., 2015).

On the other hand, Radmehr and Drake (2017) explored students’ – first-year univer-sity students and Year 12 and Year 13 students’ –metacognitive experiences, metacognitiveskills, and mathematical problem solving in relation to the questions related to the FTC.Among other findings, Radmehr and Drake (2017) reported: (1) ‘several students madeincorrect pre- and post-judgments which could have affected their success when answer-ing the questions’ (p. 24); (2) ‘students at both levels only rarely used checking strategieson these questions’ (p. 24); (3) the teaching of the FTC could be reconsidered with moreemphasis on understanding it conceptually, and (4) ‘several students had difficulties withsymbols and terminologies involved in the FTC’ (p. 25). Accordingly, they indicated that‘several students had difficulty solving questions associated to the FTC and that students’metacognitive experiences and skills could be further developed’ (Radmehr & Drake,2017, p. 1).

Bajracharya (2014) has explored the students’ understanding of the FTC and its appli-cation to graphically based problems. He reported that students had confusion betweenthe antiderivative difference and the function difference; they used three problem-solvingstrategies: algebraic, graphical and integral; and the students found it difficult to connect

INTERNATIONAL JOURNAL OF MATHEMATICAL EDUCATION IN SCIENCE AND TECHNOLOGY 7

and apply concepts such as rate, integrals, and Riemann sum, among others, to solvephysics problems.

For some authors, the concept of accumulation is central to the idea of integration and,therefore, it is at the centre of the understanding ofmany ideas and applications in Calculus(Jones, 2015; Kouropatov & Dreyfus, 2013, 2014; Thompson, 1994; Thompson & Silver-man, 2007). Thompson (1994) added that the concepts of rate of change and infinitesimalchange are central to understanding the FTC, while Kouropatov and Dreyfus (2013, 2014)argued that accumulation and its rate of change are two sides of the same coin. Accord-ing to Jones (2015), student conceptualizations about the definite integral were related tothree common interpretations: (a) area under a curve, (b) as the values of an antideriva-tive, and (c) as a limit of Riemann sums (the idea of accumulation). In this sense, Jones(2015) inferred that conceptualizations such as the area under a curve and the value of anantiderivative, limited students in their abilities to givemeaning to the contextualized inte-grals, whereas the sum of Riemann based on the idea of accumulation was practical anduseful to givemeaning to a variety of applications, for instance, inmultivariate and physicalcontexts (Jones, 2013). For its part, Rösken and Rolka (2007) reported that students fromgrade 12 have difficulty in distinguishing between concepts such as area and integral, theyconnected the symbol of integration with a specific type of function and, they believe thatcalculation of the area is restricted to a specific graph of a function. However, Kouropa-tov and Dreyfus (2013) recognized that university students successfully work on commonintegrals, such as identifying a primitive function and calculating areas, but usually fail onnon-routine questions that require a reasonable level of conceptual understanding.

Garner and Garner (2001) pointed out that the current reforms of Calculus proposegreater emphasis on conceptual understanding and practical application, above memo-rization and in the procedural. For this reason, solving application problems in Calculusis useful; besides, promoting mathematical understanding, it also allows for identifyingthe mathematical connections that students make when trying to solve those problems.Because of this, we propose to explore the mathematical connections from the resolutionof problems that evoke physical and biological contexts, and, with this, we can also inferabout the understanding that they have about the FTC. The latter allows us to differen-tiate the present study from those who explore the students’ knowledge concerning theFTC, because none of these researches exploremathematical connections in pre-universitystudents. Likewise, this research considered only polynomial functions (which model theproblems of application), since these are continuous, differentiable and integrable. Addi-tionally, because they are the typical functions that are taught in a course of Calculus inpre-university classes in Mexico.

5. Methodology

This research is qualitative and task-based interviews were used to collect data. Accordingto Goldin (2000), this method involves an interviewer (in our case the researcher) and aninterviewee (in this case a student) interacting in relation to one ormore tasks (in this study,we proposed four application problems and one open question) presented to the intervie-wee in a pre-planned way. In analysing verbal and non-verbal behaviour or interactions,the researcher hopes to make inferences about mathematical thinking, learning and prob-lem solving of the student. Likewise, Assad (2015) recognizes that task-based interviews

8 J. GARCÍA-GARCÍA AND C. DOLORES-FLORES

provide opportunities to assess students’ conceptual knowledge, but also to extend thatunderstanding. According to Assad, the interview protocol can be semi-structured, allow-ing the interviewer to judge the appropriate answer of students’ mathematical reasoning.Through the questions, the interviewer can motivate students to self-correct when theymake mistakes or to extend or generalize a problem (Assad, 2015). Task-based interviewsallow researchers to observe, record, and interpret complex behaviours and patterns inbehaviour, including words spoken by students, interjections, movements, writing, draw-ings, actions in and with external materials, gestures, facial expressions, etc. (Goldin,2000).

According to Goldin (2000) some principles that we can consider important whendesigning the tasks are: choose tasks that incorporate rich representational structures thatare attainable for students, encourage free problem solving, decide what could be recordedand record as much as possible, prepare clinical interviews and pilot tests, as well as antic-ipate new or unforeseen possibilities. These points were considered in the design of thisresearch.

6. Task-based interview design

For the interview, we used a semi-structured protocol which incorporated four applicationproblems of Calculus whose solution could be found using the derivative or the integraland, finally, we asked an open question when students finished solving those problems.

In order to design the problems, we are guided, first to a review of the curriculaof Differential Calculus (DGB, 2013a) and Integral Calculus (DGB, 2013b) of Mexicanpre-university courses, and second, to an exploratory review of four books (Contreras,Martínez, Lugo, & Montes, 2009; Larson, Edwards, & Hostetler, 2002; Mora & Del Río,2009; Stewart, 2010) of Differential and Integral Calculus proposed as basic bibliographyin those programmes. In these programmes, we have found that it is suggested as a goalin teaching, the solving of application problems that evoke physical, chemical, economic-administrative, biological, and other contexts (DGB, 2013a, 2013b). Likewise, the fourbooks reviewed, present, at different levels and frequencies, problems of this type bothas illustrative examples and as tasks to be solved by the students. Finally, considering boththe revision of the curricula of Differential Calculus and Integral Calculus as to the prob-lems proposed in the reviewed books, we designed two application problems that evokedphysical contexts (position and velocity of an object), whose data (position or velocity)were provided by a graphical representation. Also, we consider it appropriate to designtwo problems that evoke biological concepts (population and speed of growth of certainspecies of animals), whose data (the function that models the total population and thefunction that models the growth rate) were presented using algebraic representations. Inall problems, we used models associated with polynomial functions of grade 1, 2 and 3.

Participants were given sheets with the following problems to be solved by them:

(1) The population of certain species of animals is calculated by the expression

p(t) = 2t3–t2 + 100

where t (time) is measured in years.

INTERNATIONAL JOURNAL OF MATHEMATICAL EDUCATION IN SCIENCE AND TECHNOLOGY 9

Figure 2. Position of a stone thrown vertically for four seconds.

(a) Find an expression that allows you to calculate the speed of growth of thepopulation for any year.

(b) What is the population growth rate at t = 2?

(2) The position of a stone thrown vertically is represented in the following graph(Figure 2), where s(t) is measured in metres and t, the time, in seconds.(a) How can you calculate the speed of the stone at any given second?(b) What is the speed of the stone when 2 s have elapsed, that is to say, at t = 2?

(3) A population of animals is growing at annual rate given by r(t) = 6t2–2t.(a) Find the function that describes the total population at a given time.(b) If the initial population is 100 animals, what is the function that describes the

total population?

(4) The speed v(t) of a body thrown vertically upwardswith an initial velocity of 20metresper second is represented by the following graph (Figure 3):(a) What is the formula of the function that gives the position that governs that

movement?(b) What is the formula of the position function that governs that movement if, after

a second, the body is 4 metres away, that is, s(1) = 4m?

While each student solved each problem, the researcher asked auxiliary questions to iden-tify whatmotivated him or her to solve the task in that way, his or her procedures and his orher justification, as well as the meaning of his or her results. In this way, the student playedan active role in solving the proposed problems and the researcher also had relevant par-ticipation by questioning every action that the student performed. That is, the researcher

10 J. GARCÍA-GARCÍA AND C. DOLORES-FLORES

Figure 3. Speed of a body thrown vertically upwards.

was interested in what reasoning the student had and what knowledge he used, regardlessof whether he or she was consistent from the point of view of mathematics.

For instance, some questions that researchers asked were: Do you understand theproblem in general, that is, what is asked of you? What do you think you should do tosolve the problem proposed? Why do you think the problem should be solved in thatway? Why do you use the derivative (or integral, if the students indicate the impor-tance of using these mathematical concepts) to solve the proposed problem? Do youknow any other way to solve this problem? What does your result mean? And so on.At the end, when the student finished solving the four problems, the researcher askedthe following question: From the problems solved by you or from your previous knowl-edge, do you think there is any relationship between the derivative and the integral? (if youranswer is no) why do you think that? or (if your answer is yes) what is that relationshipbetween the derivative and the integral? Also, some personal data of the students werecollected.

Based on the results of a pilot application of the proposed problems, we consider it con-venient to provide students the algebraic representation associated with the graphs given in

INTERNATIONAL JOURNAL OF MATHEMATICAL EDUCATION IN SCIENCE AND TECHNOLOGY 11

problems 2 and 4 if the students could not deduce such representation and if they asked forthis information to solve the proposed tasks. In the second problem, this is s(t) = 20t–5t2and in the fourth problem the algebraic representation is v(t) = 20–10t.

7. Research context and participants

The research was carried out in a high school (pre-university level) located inChilpancingo, Guerrero, Mexico. The curriculum of this school follows the competen-cies approach, which promotes the know (knowledge), the know-how (the application ofknowledge) and the know how to be (behaviour, attitudes and values).

The participants were 25 students (18 males and 7 females), aged from 17 to 18. All ofthemhad just completed and approvedDifferential and Integral Calculus with high scores1– a grade higher than 8 –, as well as physics and biology, so we assumed that the studentswere familiar with the tasks that this research considered and that it was confirmed duringthe interviews of each student. In this research, we use the convenience sampling; the req-uisites were: students that passed with high scores in Calculus and, those that were willingto participate in an interview solving mathematical problems, that is, volunteer students.Hereafter we will refer to them as S1, S2, S3, . . . , S25.

The interviews were conducted over four working days by the first author of thisresearch and by a doctoral student with previous experience in interviewing, who knewin detail its aim. The activity was recorded in audio and video formats, for later analysis.The interviews were transcribed in their entirety to have the narratives of the students andwere analysed along with the students’ written productions.

8. Data analysis

To analyse data, we used thematic analysis (Braun&Clarke, 2006, 2012), whose objective isidentifying patterns of meanings (themes) through a set of data provided by the answers toresearch questions raised. According to Braun and Clarke (2006), ‘a theme captures some-thing important about the data in relation to the research question and represents somelevel of patterned response or meaning within the data set’ (p. 82). Thematic analysis canbe used for a wide range of frameworks and for different research questions, allows forwork with large or small data sets, and can be applied to produce theory-driven analysis(Braun & Clarke, 2012) as we did in this research.

To identify mathematical connections in this research we followed the six phases of thethematic analysis: (1) familiarizing yourself with the data, (2) generating initial codes, (3)searching for themes (type of mathematical connections) and subthemes (specific math-ematical connections), (4) reviewing potential themes and subthemes, (5) defining andnaming themes and subthemes and, (6) producing the report. In this research, we con-sidered a theme as a type of mathematical connection (provided in the framework) and asubtheme, a specific mathematical connection within Calculus (built from the data).

Also, from phase 3, we found that the mathematical connections made by the stu-dents were strongly related to each other forming a system that is described in the resultssection. This relationship, as well as the mathematical connections found, were discussedin repeatedwork sessions between the authors and an external researcher, to gain reliabilityand validity in the data analysis.

12 J. GARCÍA-GARCÍA AND C. DOLORES-FLORES

9. Results

We found a variety of mathematical connections associated with the first part of FTC inthe students’ productions (Table 1). We believe that the variability in the frequency of eachidentified mathematical connection indicates that the level of understanding that studentscan achieve is different in each case. Next, we describe these results.

9.1. Mathematical connection of procedural type

Three subthemes were grouped in this type of mathematical connection. The participantsin this study used formulas such as d

dx (xn) = nxn−1 (to solve problems involving the cal-

culation of speed or the rate of growth, n = 20) and ∫ xndx = xn+1

n+1 + C (to solve problemsinvolving the calculation of position or the total population, n = 16). For example, in prob-lem 1 students used the rule of power to derive polynomial functions after they reflect onthe problem and its demand (see Figure 4).

On the other hand, when the students solved part b of problem 4 (physical problem),they indicated that ‘finding a primitive function of the position of an object given an initialcondition implies, first, calculating an integral and then solving a linear equation (n = 7.)’.In this sense, participants who made this mathematical connection recognized that ifv(t) models an object’s speed and s(1) = 4 is an initial position, then to find the prim-itive function that models the position at any instant they must first solve the integral∫ v(t)dt = s(t) + C and then solve the equation 4 = s(1) + C. With this last one, they will

Table 1. Intra-mathematical connections made by the students.

Type of mathematical connection Subtheme Frequency

Procedural 1. The formula ddx (x

n) = nxn−1 is used to solve problems involvingthe calculation of speed or the rate of growth.

20

2. The formula ∫ xndx = xn+1

n+1 + C is used to solve problemsinvolving the calculation of position or the total population.

16

3. Finding a primitive function of the position of an object given aninitial condition implies, first, calculating an integral and thensolving a linear equation.

8

Different representations 4. The speed of an object at its maximum height is 0. 135. s′(a) = 0 means that the object is motionless at t = a. 6

Feature 6. The integral of a polynomial function is an increase of its degreeby one and the derivative of a polynomial function is a reductionof its degree by one.

2

Meaning 7. In ∫ r(t)dt = p(t) + C the integration constant C means theinitial population of animalsa.

11

8. p′(2) = k means that just in the second year the populationincreased k species.

7

Reversibility 9. The derivative and integral are inverse operations. 2210. The integral of the acceleration function of an object is itsspeed and the integral of the speed function is its position.

15

11. Calculating the speed of growth means finding the derivative. 1412. The derivative of the position function of an object is its speedand the derivative of its speed is its acceleration.

14

13. The integral of the growth rate function r(t) of a population isthe total population function.

12

Total 160ar(t) is the function that models the rate of growth and the p(t)models the total population of certain species of animals.

INTERNATIONAL JOURNAL OF MATHEMATICAL EDUCATION IN SCIENCE AND TECHNOLOGY 13

Figure 4. Answer for task 1 provided by S8.

Figure 5. Calculations made by S3 to answer problem 4.

find the value of C and finally, they will have the particular antiderivative function that weasked them for (see the extract from S3 and Figure 5).

Interviewer: Do you understand the second question [of problem 4]?S3: Yes. It is replaced. (Then, she wrote the equation 4 = −5(1)2 + 20(1) + C.

and solved it. Her solution is C = −11. Finally, she wrotehat the requestedfunction is s(t) = −5t2 + 20t − 11).

Interviewer: Can you tell me what you did?S3: If after a second the body is four meters away (points to the equation she

wrote 4 = −5(1)2 + 20(1) + C) and here what we would be looking for isthe constant [of integration], that is the unknown. By developing all this wearrive at the result that the constant is equal to minus eleven.

9.2. Mathematical connection of different representations type

For this type of mathematical connection, we constructed two subthemes: the speed reach-ing an object at its maximum height is 0 (n = 13) and s′(a) = 0 means that the object ismotionless at t = a (n = 6, see Table 1). In the first one, for example, S4 associated his ver-bal and numeral result s′(a) (where s(t) models the position of a stone) with its graphicalrepresentation. This student identifies this characteristic from the derivative at a point andvisualizes the behaviour of the stone in a graph (which was provided as a mathematicalmodel for the phenomenon described in the problem).

14 J. GARCÍA-GARCÍA AND C. DOLORES-FLORES

Interviewer: Could you solve the following problem (points to part b of problem 2)?S4: Yes (he derives the position function and calculates the speed for t = 2 in the

derivative obtained previously). The speed would be zero meters per second.Interviewer: How does that look on the graph or what does that result mean?

S4: For example, here are the two seconds (points to the axis of the abscissathe value of t = 2) and here it stops (it indicates the maximum point ofthe parabola), that is, here it does not have any speed. When it reaches itsmaximum point is when it comes to a stop.

The second mathematical connection which relates the meaning of s′(a) is like the firstone because students say that the object reaching the maximum height (has zero velocity)there exists a moment when it stops. So, students indicated that if s(t) is a function thatmodels the position of an object, then s′(a) = 0 means that the object is motionless at theinstant t = a (see the excerpt from S8).When the object is released there is an instant whenits speed is 0, between the time it is thrown and when it reaches the ground. Next, becauseof gravity, the object begins to descend to the ground.

Interviewer: Can you answer the following question (interviewer points to the secondquestion of problem 2)?

S8: Yes. (Substitute for t = 2 in the function s′(t) = 20 − 10t) it gives zero.Interviewer: What does that mean?

S8: That when two seconds have elapsed, the stone that is thrown goes withoutspeed.

Interviewer: Interviewer: Graphically, what does it look like?S8: It is assumed that in two seconds it must have a speed of zero (while locating

a point in (2, 0) and in (2, 20)). It is here when it begins to descend and by theeffect of gravity runs out of speed.

9.3. Mathematical connection of feature type

We found one mathematical connection associated with a characteristic of the derivativeand the integral. From students’ productions, we constructed the subtheme: ‘the integral ofa polynomial function is an increase of its degree by one and the derivative of a polynomialfunction is a reduction of its degree by one’ (n = 2). This was possible because the studentsused the formula of the derivative ( d

dx (xn) = nxn−1) and the integral (∫ xndx = xn+1

n+1 + C)that gives information about what happens to the degree of a polynomial function that isderived or integrated; that is, the participants focus their attention on the variation of theexponent. Participants whomade it, explained the relationship that they identified betweenthe derivative and the integral (see the excerpt from S19); they argue that since the opera-tions are opposites then the process to obtain the derivative has to do with decreasing itsexponent by one because in the integral it is adding one. From the answer that these stu-dents offer, it is clear that they are thinking of the mathematical relationship between thederivative and the integral for polynomial functions that they havemost often worked within the classroom.

INTERNATIONAL JOURNAL OF MATHEMATICAL EDUCATION IN SCIENCE AND TECHNOLOGY 15

Interviewer: from the exercises you did or from your previous knowledge do you thinkthere is any relationship between the derivative and the integral?

S19: Integrate [a function] is adding one to its exponent and derive [a function]implies a reduction of its degree by one.

9.4. Mathematical connection ofmeaning type

When a student finds a solution to a problem it is important to know if he or she knowswhatthe meaning is of his or her result. In this sense, in the students’ productions we found twomeanings for them: p′(2) = k means that just in the second year the population increasedk species (n = 7) and in ∫ r(t)dt = p(t) + C the integration constant C means the initialpopulation of animals (n = 11). The first one was possible because in part b of problem 1we asked the students to calculate the growth rate at a specific time, at t = 2 (i.e. p′(2)),and the auxiliary question during the interview was what the result means. In this respect,students indicated that just in the second year the population increased 20 species (see theexcerpt from S24).

Interviewer: This 20 (interviewer points to the student’s answer to part b of the problem1), what does it mean?

S24: It’s the rate of population growth.Interviewer: Interviewer: And, in other words, what does it mean?

S24: Growth . . . that is, according to me, if t is supposed to be time, then it couldbe said that just in the second year the population grew by 20. It is what I canobserve.

In the second subtheme, r(t) is a functionwhichmodels the growth rate of a certain speciesof animals, p(t) models the total population and,t is measured in years. Here, studentsassociate the meaning of the constant of integration with the initial population of animals,that is a biological concept (see excerpt from S13).

Interviewer: Interviewer: Let’s move on to the next question of the problem raised (part bof problem 3).

S13: (read the question and wrote p(t) = 2t3 − t2 + 100). This would be it.Interviewer: Interviewer: why?

S13: Because, if it speaks of initial population, then at t the value would be 0.They are no longer counting anything, they are not counting howmany yearspassed or anything and, if it says that the initial population is 100, I mustsubstitute 100 in C.

9.5. Mathematical connection of reversibility type

For this type ofmathematical connection, we found five subthemes in the students’ produc-tions, that have as a main characteristic the bidirectional relation between the derivativeand the integral. In this sense, students considered that ‘the derivative and the integral areinverse operations’ (n = 22),mathematical connections that recognize the first part of FTC.This appeared through a question in which students were asked for a retrospective view on

16 J. GARCÍA-GARCÍA AND C. DOLORES-FLORES

solving the problems raised. The question focused on the relationship that they identifiedbetween the derivative and the integral (see the excerpt from S2).

Interviewer: From the problems you solved or from your previous knowledge do you thinkthere is any relationship between the derivative and the integral?

S2: Yes, yes, I do.Interviewer: What is that relationship?

S2: the derivative and the integral are related. Since from one we can find theother, it´s an inverse process.

This mathematical connection has been validated performing algebraic tasks that proveits validity according to S2. On the other hand, this allowed students to make other mathe-matical connectionswith physical and biological concepts. So, the relation that the studentsidentified between the derivative and the integral was from two perspectives: in themathe-matical context (the derivative of the integral of a polynomial function is equal to the samefunction and vice versa, see García-García and Dolores-Flores, 2018) and, in the physicaland biological context (as amathematical relationship between the position and velocity ofan object and, between the rate of growth of certain species of animals and the total popu-lation; see mathematical connections 10–13 in Table 1). In the mathematical context, thismathematical connection allows students tomake the fourth one as well (seemathematicalconnection number 6 in Table 1), through the first and second (see mathematical connec-tions number 1 and 2 in Table 1), because they said, if the derivative of a polynomial func-tion implies a reduction of its degree by one, and the derivative and the integral are inverseoperations such as addition and subtraction, then the integral of a polynomial functionmeans an increase of its degree by one. Therefore, this mathematical connection allowedstudents to identify the relationship between problems 1 and 3, and 2 and 4, respectively.In this way, this mathematical connection affected how the students solved the proposedtasks.

For instance, when the students solved the fourth problem, they said ‘the integral ofthe acceleration function of an object is its speed and the integral of the speed function is itsposition’ (n = 15) which relates three physical concepts: acceleration, speed and positionfrom the mathematical point of view, where the mediator is the integral. These studentsindicated that if a(t) is a function that models the acceleration of an object and v(t)modelsthe speed, then ∫ a(t)dt is the speed and ∫ v(t)dt is the position of that object at a giventime (see the excerpt from S3).

Interviewer: There you can see other problemspresented. Read themand try to solve them.S3: (he read problem 4, then wrote vertically x, v, a, then he draws an arrow from

v to variable x, this is accompanied by the integral symbol. Then he wrotev(t) = −10t + 20. Finally, he integrated this function and he obtained s(t) =−5t2 + 20t + C).

Interviewer: Can you tell me what you did?S3: This is what I told you a while ago (he points to x, v, a, that he wrote on his

answer sheet) that it is as if we have x represent the position, v is speed andhere a would be acceleration. If you want to go from speed to position youmust integrate, that is, if you go here (indicates the arrow from v to x), it is

INTERNATIONAL JOURNAL OF MATHEMATICAL EDUCATION IN SCIENCE AND TECHNOLOGY 17

integrated. If they go to this side (points out the arrows with a different andinverse direction) you need to derive. And here what they are givingme is thespeed graph. If I find the [function associated with the] graph I would havewhat is the speed, and the speed would have to move to the position.

In the S3 answer, we identify that he recognized the reversibility between position, veloc-ity and acceleration from the mathematical point of view, through the derivative and theintegral. This response allowed this student tomake othermathematical connections whenhe solved the other problems. Likewise, other students related the position, the speed andthe acceleration in an inverse sense, that is, they identified that ‘the derivative of the positionfunction of an object is its speed and the derivative of its speed is its acceleration’ (n = 14); aninversemathematical connection to the previous one. Similar to themathematical connec-tion identified by number 9 in Table 1, this also occurred when students were questionedabout the relationship that they identified between the derivative and the integral (see theexcerpt from S12).

Interviewer: Interviewer: From the activities you did or from your previous knowledge, doyou think there is any relationship between the derivative and the integral?

S12: Yes, to begin with, they both evaluate any function. The derivative can beobtained fromposition to speed and starting from speed to acceleration.Withthe integral it’s the other way around, starting from acceleration I can get tospeed and from speed to position. They are only inverse operations that if theygo together are cancelled and that one evaluates the slope of the function andthe other the area that it has underneath.

From S12’s answer, we can argue that he understands the mathematical connection9, both in the physical and mathematical context, especially when he recognized in hisdiscourse the mathematical relation between the slope of a curve and the area under acurve accepting and understanding as inverse processes the derivative and the integral,even in the graphical contexts. Therefore, we can argue that S12 has a more completeunderstanding concerning the mathematical connection between the derivative and theintegral.

Other mathematical connections that were possible because students identified thatthe derivative and the integral are inverse operations were in the context of biology. For

Figure 6. Calculations performed by S5.

18 J. GARCÍA-GARCÍA AND C. DOLORES-FLORES

instance, students mentioned that if a problem gives a function that models a certain phe-nomenon and the student is asked to calculate the speed, then themathematical calculationrequested is the derivative. That is, they use the term ‘speed’ as a connector to decide theuse of the derivative (see the excerpt from S5 and Figure 6) and, this allows them to makethe mathematical connection ‘calculating the speed of growth means finding the derivative’(n = 10).

Interviewer: Interviewer: Some problems arise immediately, read them calmly and try tosolve them.

S5: (reads the first problem and derives the function that models the total popu-lation of animals. Then substitutes the value of t = 2 in the derived functionto answer the second question).

Interviewer: Can you explain what you did?S5: To calculate the speed of some expression, it would be the first derivative,

that is what I did. As they ask me the speed of population growth and theyare giving me the value of t, I told you that the speed was the first derivative,so I only replaced the 2 where the t was.

10. Mathematical connections system

From phase 3 on the thematic analysis, we identified that the mathematical connectionswere strongly related forming a system of mathematical connections. This system wasstrengthened in phase 4 of the thematic analysis through discussion in repeated work-ing sessions. Our data indicate that there is a central mathematical connection (Figure 7)related to a level of hierarchy and abstraction superior to other mathematical concepts thatare included in it. In our case, this was the first part of the FTC because it relates to thederivative and the integral as inverse operations; also, this can be used in the physics andbiology fields when someone is solving application problems that involve the use of bothmathematical concepts. For this reason, the mathematical connection ‘the derivative andthe integral are inverse operations’ is the central one and it appeared in the retrospective viewthat the students manifested once they solved the proposed problems, but it also appearedduring the interview at different moments to help them solve the proposed problems.

Other mathematical connections that appeared as the students solved the proposedproblems were organized around this central mathematical connection. The data indicatedthat from this one, there are others that are derived immediately that we call first orderderivative connections. The characteristic of these is that they are mathematically associ-ated with the two central concepts of Calculus: the derivative or the integral which, at ahierarchical and abstraction level, correspond to a lower level of understanding comparedto the FTC. Among these we recognize, for example, the mathematical connections: ‘theformula d

dx (xn) = nxn−1 is used to solve problems involving the calculation of speed or the

rate of growth’ and ‘the formula ∫ xndx = xn+1

n+1 + C is used to solve problems involving thecalculation of position or the total population’ (first-order derivative connections). Thesemathematical connections allow one to derive the following: ‘the integral of a polynomialfunction is an increase of its degree by one and the derivative of a polynomial function is

INTERNATIONAL JOURNAL OF MATHEMATICAL EDUCATION IN SCIENCE AND TECHNOLOGY 19

Figure 7. Mathematical connections system constructed from the data.

a reduction of its degree by one’ (second-order derivative connection). The latter connec-tion was possible because the students worked with polynomial functions, in addition, thesupport offered them through the use of the formula of the derivative and the integral.

Therefore, the data indicate that there is a system of mathematical connections that arederived almost immediately from each other and are related to a higher and abstract level ofunderstanding. There aremathematical connections that explain themeaning of a result oran algebraic symbol, or those that define some particular characteristics of a mathematicalconcept (such as the variability of the exponent for the derivative and the integral), whichare of second order. Some are associated withmore general concepts (such as the derivativeand the integral) or that involve the use of these to solve problems that are of first-order.Finally, there is a central mathematical connection that is related to a superior concept ofunderstanding that is more abstract (such as the first part of the FTC in this research). Astudent has this system of mathematical connections, but that does not mean that all reachthe same level, as shown in Figure 7.We also found logical relationships between the identi-fiedmathematical connections (Figure 7), in particular, the relationship of implication anddouble implication. In the first one, the development of amathematical connection justifiesthe existence of another of the form: if A is true then B is also true. The logical relation ofimplication allows us to differentiable between first and second-order derivative connec-tions. On the other hand, the relation of double implication indicates the bidirectionality ofthe mathematical connections, that is to say, the development of one justifies the presenceof the other and vice versa. In other words, they are of the form if A is true then B is alsotrue, and if B is true then A is true as well.

20 J. GARCÍA-GARCÍA AND C. DOLORES-FLORES

Figure 7 indicates that the students’ activities were organized around the central math-ematical connection, through which they were able to recognize the bidirectionality ofconcepts such as position-speed, speed-acceleration, total population-growth speed.Whenstudents can create a greater number of interrelated mathematical connections that makeup the mathematical connections system of Figure 7, then we can say that they have abetter understanding of Calculus and the first part of the FTC. This results in improvedstudent performance in solving proposed tasks and helps them to provide consistent argu-ments from the point of view ofmathematics about why they solve the problems in the waythey do. For instance, students who made 9 (S4, S13), 10 (S3, S8, S12, S23) and 11 (S14,S15) mathematical connections respectively, were more successful in solving the proposedproblems.

11. Discussion and conclusion

The results found in this research indicate that students made five types of mathematicalconnections: procedural, different representations, feature, meaning and reversibility as weexpected in the framework. In each type, we identified specific mathematical connectionsthat students made when they solved the proposed problems. All of them were stronglyrelated, forming a mathematical connections system (as was previously presented).

The mathematical connections of procedural and reversibility type occurred with highfrequency. Procedural mathematical connection has been identified in university and pre-university students when they solve Calculus tasks (García-García &Dolores-Flores, 2018,2019; Dolores & García-García, 2017), but it is also used by math teachers as reported byEvitts (2004), Businskas (2008) and Eli et al. (2011, 2013). Considering the frequency withwhichmathematical connections 1 and 2 emerged (Table 1), our results are consistent withHaciomeroglu (2007), Dawkins and Mendoza (2014) and, Hong and Thomas (2015) whopoint out that there is persistence in the use of the algebraic procedural, even when wepropose graphs as a procedure to find the solution.

On the other hand, we recognize that in Calculus, developing reversible thinkingin the students is important as indicated by Haciomeroglu, Aspinwall, and Presmeg(2009), so, it was significant that the students, at least the participants in this research,recognized during the interview, some inverse operations such as addition-subtraction,multiplication-division and derivative-integral; in addition, concepts such as position-velocity and velocity-acceleration whenever there is a mathematical model that helps topredict them (position, velocity or acceleration). Due to the reversibility of thought mak-ing available a compensatory thought that will restore the original condition after anyinterruption (Sparks, Brown, & Bassler, 1970), the reversibility mathematical connectionis significant. This mathematical connection has previously been identified when pre-university students solve algebraic and graphical tasks (García-García & Dolores-Flores,2018, 2019). As a result, its occurrence was important when students solved applicationproblems in this research.

Due to the latter, in this research, we found the mathematical connection of differ-ent representations type. This result is consistent with those found by Mhlolo (2012) andMhlolo et al. (2012) in their research with teachers; however, they reported that this math-ematical connection appeared on a weak and superficial level. Our results indicated thatin problem 2, not one student was able to find the algebraic representation associated with

INTERNATIONAL JOURNAL OF MATHEMATICAL EDUCATION IN SCIENCE AND TECHNOLOGY 21

the given parabola, whereas in problem 4 only eight students were able to deduce the alge-braic representation from the given graph. The limitations of the students are aggravatedin the context of problem-solving, a result consistent with that reported by Özgen (2013);however, once we provided the algebraic representation associated with the graphs thenstudents proceeded to derive or to integrate by establishing a variety of mathematical con-nections just as it happened when the students solved graphical tasks (García-García &Dolores-Flores, 2019). On the other hand, we believe that this mathematical connectionwould occur possibly with major frequency if we would have proposed algebraic or graph-ical tasks as we found in previous research (García-García & Dolores-Flores, 2018, 2019).This type has been recognized in research like Businskas (2008) and Evitts (2004) thatreveal the role of the teacher to help students to use it in the classroom.

As we expected in the framework, we found the mathematical connection of mean-ing type. This result is consistent with those we found in previous research when studentssolved graphical tasks (García-García & Dolores-Flores, 2019). This is important to high-light because research that studies mathematical connections has not yet reported its use.For its part, themathematical connection of feature type appeared when the students iden-tified the variation of the degree of the exponent of a function when they derived andintegrated it, that is, it appeared once the students made a procedural mathematical con-nection. This result is consistentwith those reported by Eli et al. (2011, 2013)who indicatedthat the teachers used this mathematical connection when they were solving geometrictasks. Likewise, this is important to achieve other mathematical connections like ‘commonfeatures’ presented by Businskas (2008) in her results. We also believe that this mathemat-ical connection plays an important role in achieving generalization and abstraction forsome mathematical concepts.

As we presented previously, some of the mathematical connections identified in thisresearch are used by math teachers as well (Businskas, 2008; Eli et al., 2011, 2013; Evitts,2004). For this reason, we assume that one of the sources of those mathematical con-nections that students make, is their teachers, but also, textbooks and independent workthat the students do when they solve mathematical tasks outside the classroom, as theparticipants in this research recognized during the interview.

The results found in this research are also consistent with those obtained by Lockwood(2011), in the sense that students also made unexpected mathematical connections. Forexample, during the interview, we could verify that the students used, very naturally, thereversibility connection between the position-speed concepts associated with the ideas of aderivative and an integral. This mathematical connection allowed them to solve problemsthat evoked concepts of biology. In particular, it enabled connections to emerge: ‘calculatingthe speed of growth means finding the derivative’. In its turn, this allowed for the emergenceof ‘the integral of the growth rate function r(t) of a population is the total population func-tion’. Therefore, these are unexpected mathematical connections that were made possiblebecause the students extrapolated their knowledge of Calculus and Physics to the field ofBiology, where the term ‘speed’ was a mediator to achieve those results.

Finally, in this research, we were able to verify the validity and effectiveness of the the-oretical framework – which we built in previous research – to study the mathematicalconnections that appear when students solve application problems. We consider that itis important to continue exploring both the intra-mathematical and extra-mathematicalconnections that students and teachers use when they are solving problems from other

22 J. GARCÍA-GARCÍA AND C. DOLORES-FLORES

disciplines or real-life that involve the use ofmathematical concepts such as derivatives andintegrals. These results could be used to design learning sequences centred on the math-ematical connections between different mathematical concepts, including these and withthose of other disciplines or in the modelling of real-life problems. We also believe that itwill be important for future research to explore the mathematical connections proposedin the curriculum and those that math teachers make in the classroom. Likewise, it is alsoimportant to design an analytical tool that allows the study ofmathematical understandingbased on the quality of the mathematical connections that students or teachers make whenthey are solving or working with mathematical tasks.

Note

1. In Mexico, the numerical scale of grades is from 0 to 10. In some pre-university schools (as inthe school where this research was done), the minimum passing grade is 6.

Disclosure statement

No potential conflict of interest was reported by the author(s).

ORCID

Javier García-García http://orcid.org/0000-0003-4487-5303Crisólogo Dolores-Flores http://orcid.org/0000-0002-2748-6042

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