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How Do Widely-Used Calculus Textbooks Introduce the Concepts of Definite Integrals?

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Title: How Do Widely-Used Calculus Textbooks Introduce the Concepts of Definite Integrals?
Language: English
Authors: Dae S. Hong (ORCID 0000-0003-0529-6448), Dennis Kwaka
Source: International Journal of Mathematical Education in Science and Technology. 2025 56(9):1669-1689.
Availability: Taylor & Francis. Available from: Taylor & Francis, Ltd. 530 Walnut Street Suite 850, Philadelphia, PA 19106. Tel: 800-354-1420; Tel: 215-625-8900; Fax: 215-207-0050; Web site: http://www.tandf.co.uk/journals
Peer Reviewed: Y
Page Count: 21
Publication Date: 2025
Document Type: Journal Articles
Reports - Research
Education Level: Higher Education
Postsecondary Education
Descriptors: Mathematics Education, Mathematics Instruction, Mathematics Materials, Textbook Evaluation, Calculus, Mathematical Concepts, College Mathematics
DOI: 10.1080/0020739X.2024.2348145
ISSN: 0020-739X
1464-5211
Abstract: This study explores opportunities to learn definite integrals in three widely-used textbooks in the US, Definitions, worked examples, and exercise problems were coded using research based cognitive resources in definite integrals to examine if widely-used textbooks provide students with opportunities to explore how two quantities are related to show area under a curve or other contexts. The results show that initial introduction, worked examples, and exercise problems all provide limited opportunities for students to explore the multiplicative relationship between two quantities (integrand and differential) and adding small pieces to understand definite integrals. Although students may be able to solve integral problems using anti - derivatives, it is possible that they may not understand the definite integral conceptually (knowing why the definite integral gives the area under a curve) or be able expand their understanding to other contexts. Implications of these results are also discussed.
Abstractor: As Provided
Entry Date: 2025
Accession Number: EJ1483296
Database: ERIC
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  Value: <anid>AN0187890693;imt01sep.25;2025Sep15.05:03;v2.2.500</anid> <title id="AN0187890693-1">How do widely-used calculus textbooks introduce the concepts of definite integrals? </title> <p>This study explores opportunities to learn definite integrals in three widely-used textbooks in the US, Definitions, worked examples, and exercise problems were coded using research based cognitive resources in definite integrals to examine if widely-used textbooks provide students with opportunities to explore how two quantities are related to show area under a curve or other contexts. The results show that initial introduction, worked examples, and exercise problems all provide limited opportunities for students to explore the multiplicative relationship between two quantities (integrand and differential) and adding small pieces to understand definite integrals. Although students may be able to solve integral problems using anti - derivatives, it is possible that they may not understand the definite integral conceptually (knowing why the definite integral gives the area under a curve) or be able expand their understanding to other contexts. Implications of these results are also discussed.</p> <p>Keywords: Cognitive resources; integral; learning opportunities; textbooks</p> <hd id="AN0187890693-2">1. Introduction</hd> <p>In the United States, Calculus 1 is frequently the initial course that mathematics majors need to take. It serves as a foundational and introductory course for students pursuing disciplines in science, technology, engineering, and mathematics (STEM), as well as the social sciences and business. Numerous studies have delved into how students grasp calculus topics such as limits, derivatives, and integrals, revealing the challenges and difficulties they encounter journey (Brijlall & Ndlazi, [<reflink idref="bib3" id="ref1">3</reflink>]; Güçler, [<reflink idref="bib14" id="ref2">14</reflink>]; Jones, [<reflink idref="bib22" id="ref3">22</reflink>]; Sealey, [<reflink idref="bib38" id="ref4">38</reflink>]).</p> <p>The integral concept is typically introduced towards the end of the first calculus class in the United States. In comparison to other calculus topics such as limits and derivatives, integral concepts have received relatively less attention from mathematics education researchers (Larsen et al., [<reflink idref="bib26" id="ref5">26</reflink>]). Many experts in teaching calculus consider an understanding of the integral as a necessary central concept or skill for a genuine understanding of first-year calculus (Jones, [<reflink idref="bib22" id="ref6">22</reflink>], [<reflink idref="bib23" id="ref7">23</reflink>]; Sofronas et al., [<reflink idref="bib40" id="ref8">40</reflink>]). Moreover, it is utilised in advanced-level mathematics disciplines such as differential equations, complex analysis, and numerical analysis, serving as a useful tool for numerous real-world applications (Ely & Jones, [<reflink idref="bib12" id="ref9">12</reflink>]; Jones & Ely, [<reflink idref="bib24" id="ref10">24</reflink>]; Özgeldi & Aydın, [<reflink idref="bib33" id="ref11">33</reflink>]). While understanding the integral is necessary, several existing studies demonstrate students' challenges in understanding the integral concept (Jones, [<reflink idref="bib22" id="ref12">22</reflink>]; Mahir, [<reflink idref="bib28" id="ref13">28</reflink>]; Orton, [<reflink idref="bib32" id="ref14">32</reflink>]; Sealey, [<reflink idref="bib38" id="ref15">38</reflink>]), illustrating that, in many cases, students are able to correctly integrate functions but do not know what it means or how to interpret it in context. Researchers also note that being able to interpret integrals in context and model using quantitative structures is critically important in learning calculus (Ely & Jones, [<reflink idref="bib12" id="ref16">12</reflink>]).</p> <p>In more recent years, researchers shifted their attention to describing and identifying different units of knowledge that students need to put together to understand definite integrals (Jones, [<reflink idref="bib22" id="ref17">22</reflink>]; Sealey, [<reflink idref="bib38" id="ref18">38</reflink>]). Identifying various units of knowledge in learning definite integrals is crucial as it directly impacts instructional practices and the curriculum materials provided to students. Understanding which units of knowledge about definite integrals are covered, exposed, and emphasised in classes and textbooks is essential for designing effective teaching strategies. Recent studies have often indicated that when students conceptualise definite integrals as the product of two quantities and the process of adding small pieces, they are more likely to enhance their understanding of definite integrals (Ely & Jones, [<reflink idref="bib12" id="ref19">12</reflink>]; Jones & Ely, [<reflink idref="bib24" id="ref20">24</reflink>]). Examining learning opportunities in widely-used textbooks to understand how they address and cover various units of knowledge regarding definite integrals would be a valuable endeavour. By analysing how textbooks present concepts such as the product of two quantities and the process of adding small pieces, educators and curriculum developers can gain insights into the effectiveness of instructional materials.</p> <p>Researchers recently acknowledged that thinking about curriculum materials is one important area to consider in improving the teaching and learning of calculus (Thompson & Harel, [<reflink idref="bib47" id="ref21">47</reflink>]). Students have several different opportunities to become familiar with mathematical topics including what textbooks offer, what calculus instructors do in their classes, and how they select their implemented tasks. In this study, we examined how widely-used calculus textbooks provide learning opportunities to students. A recent study (Özgeldi & Aydın, [<reflink idref="bib33" id="ref22">33</reflink>]) examined integral lessons in calculus textbooks in terms of competency demands. While we acknowledge the significance of examining textbooks based on competency demands, our study aims to focus specifically on the pieces of knowledge that students need to understand the definite integral. This examination of textbooks will contribute to our understanding of how calculus textbooks provide knowledge about the definite integral, complementing previous studies.</p> <p>These are our research questions that we attempted to answer.</p> <p></p> <ulist> <item> What units of knowledge about definite integral do widely-used calculus textbooks provide to students?</item> <p></p> <item> How do widely-used textbooks provide opportunities for students to become familiar with the concepts of adding up pieces to understand definite integral?</item> </ulist> <hd id="AN0187890693-3">2. Related literature</hd> <p>In this section, we will describe what previous studies noted about students' learning challenges in understanding definite integrals and types of knowledge that students might need to expand their understanding.</p> <hd id="AN0187890693-4">2.1. How students learn integral concepts</hd> <p>According to experts in teaching calculus, there are three important goals of teaching and learning definite integrals: (<reflink idref="bib1" id="ref23">1</reflink>) the integral as net change or accumulated total change, (<reflink idref="bib2" id="ref24">2</reflink>) the integral as area, and (<reflink idref="bib3" id="ref25">3</reflink>) techniques of integration (Sofronas et al., [<reflink idref="bib40" id="ref26">40</reflink>]). Other studies note that students interpret definite integral notation in similar ways: (<reflink idref="bib1" id="ref27">1</reflink>) as the area under a curve, (<reflink idref="bib2" id="ref28">2</reflink>) as an anti-derivative, and (<reflink idref="bib3" id="ref29">3</reflink>) as a sum-based interpretation (Ely, [<reflink idref="bib10" id="ref30">10</reflink>]). With these different ways to learn and interpret definite integrals, previous studies show that students are able to calculate integrals without being able to explain why they are doing it (Orton, [<reflink idref="bib32" id="ref31">32</reflink>]) and fail to provide a clear definition of the definite integral (Rasslan & Tall, [<reflink idref="bib35" id="ref32">35</reflink>]). Another study observes that students can apply techniques of integration but struggle to grasp the underlying meaning (Brijlall & Ndlazi, [<reflink idref="bib3" id="ref33">3</reflink>]). Jones et al. ([<reflink idref="bib25" id="ref34">25</reflink>]) reported that nearly every student in their study used either 'area under a curve' or 'antiderivative' when describing definite integrals.</p> <p>What we learn from these studies is that, while students are often familiar with finding the area under a curve and using anti-derivatives (or techniques of integration), they frequently find it challenging to understand why the definite integral represents the area under a curve or to apply this concept in other contexts, such as physics and engineering. These studies suggest that the ability to use the concept of 'area under a curve' and 'antiderivative' is less productive when applying integrals to contextualised problems or quantitative situations (Ely & Jones, [<reflink idref="bib12" id="ref35">12</reflink>]; Jones, [<reflink idref="bib22" id="ref36">22</reflink>]; Jones & Ely, [<reflink idref="bib24" id="ref37">24</reflink>]; Sealey, [<reflink idref="bib38" id="ref38">38</reflink>]). Studies show that, while students are familiar with concepts like 'area under a curve' and 'antiderivative', using quantities and quantitative relationships is more productive for understanding the meaning of definite integrals and for interpreting integrals in various contexts (Jones et al., [<reflink idref="bib25" id="ref39">25</reflink>]; Oehrtman & Chhetri, [<reflink idref="bib30" id="ref40">30</reflink>]). Since being familiar with finding the area under a curve and finding the anti-derivative can limit students' understanding of why the definite integral represents the area under a curve or definite integrals in other contexts, researchers have been seeking ways to understand the meaning of integrals based on quantities and quantitative relationships (Ely, [<reflink idref="bib10" id="ref41">10</reflink>]; Jones & Ely, [<reflink idref="bib24" id="ref42">24</reflink>]).</p> <hd id="AN0187890693-5">2.2. What students need to expand their understanding of definite integrals</hd> <p>If understanding definite integrals based on quantities and quantitative relationships is more productive, how can we present definite integrals using quantitative relationships? By identifying units of knowledge that students can effectively use to understand integrals based on quantitative relationships, we can redesign our lessons and textbooks to offer those units of knowledge to students. In recent years, researchers have paid more attention to identifying the units of knowledge that students need to understand quantitative relationships, in addition to demonstrating students' tendency towards procedural understanding or the area conception (Larsen et al., [<reflink idref="bib26" id="ref43">26</reflink>]). This idea prompted the consideration of various ideas that students may use to understand the meaning of definite integrals through quantitative relationships because it is overly simplistic to assume that a single idea is sufficient (e.g. area under a curve or anti-derivative) to understand integral or any mathematical topic (Harlow & Bianchini, [<reflink idref="bib15" id="ref44">15</reflink>]; Jones, [<reflink idref="bib22" id="ref45">22</reflink>]). By examining additional units of knowledge, we can look at the more fine-grained elements of knowledge students need to understand the meaning of definite integrals.</p> <p>Sealey ([<reflink idref="bib38" id="ref46">38</reflink>]) described four different layers – limit, summation, function, and product – in understanding definite integrals, and reported that it is most challenging for students to understand the product layer, specifically when contexts of integral tasks are not necessarily about getting the area under a curve (Sealey, [<reflink idref="bib38" id="ref47">38</reflink>]). When the context is the area under a curve, the product layer describes the product of the height (function value) and the base of rectangles. If students are able to understand the product layer, they are able to use quantitative relationship as they assign meanings to quantities in the given situations and contexts (Ely, [<reflink idref="bib10" id="ref48">10</reflink>]). Then, what this means is understanding how the product is formed and what each factor and quantity means within the product. To conceptualise adding 'small pieces' infinitely many times, it is important to have unit of knowledge that allows them to understand the meaning of the product of two quantities,</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mrow><mrow><msub><mi>x</mi><mi>i</mi></msub></mrow></mrow><mo stretchy="false">)</mo></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Δ</mi><mi>x</mi></math> </ephtml> , the product layer (Sealey, [<reflink idref="bib38" id="ref49">38</reflink>]). Being able to understand the meaning of the product of two quantities is challenging but it is an important unit of knowledge to understand the underlying concepts of definite integrals as conceptualising the product layer can help students to understand definite integral in different contexts such as also the distance travelled, force, and energy (Jones & Ely, [<reflink idref="bib24" id="ref50">24</reflink>]; Stevens & Jones, [<reflink idref="bib42" id="ref51">42</reflink>]). However, a recent study shows even advanced mathematics students struggled to understand the product layer (Soto & Oehrtman, [<reflink idref="bib41" id="ref52">41</reflink>]).</p> <p>Jones ([<reflink idref="bib22" id="ref53">22</reflink>]) introduced several cognitive resources (referred to as 'fine-grained' elements of knowledge in a person's cognition) that students utilise in understanding integrals, asserting that it is challenging for students to grasp integrals because they struggle to activate productive cognitive resources. Students can activate a variety of cognitive resources through symbolic forms, such as the perimeter and area symbolic form, the function matching symbolic form, and the <emph>adding up pieces</emph> symbolic form (Jones, [<reflink idref="bib22" id="ref54">22</reflink>]). They can interpret definite integrals using these cognitive resources, but the <emph>adding up pieces</emph> perspective – understanding that infinitely many tiny rectangles will be added to compute the area – is most productive in understanding definite integral. In the perimeter and area symbolic form, students interpret</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> </ephtml> as a graph in the <emph>x</emph>–<emph>y</emph> plane and</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo largeop="false">∫</mo></math> </ephtml> as a measure of the area under the graph (as one undivided piece instead of infinitely many small pieces) while in the function matching, the integral symbol is interpreted as an instruction to finding the original pre-derivative function (Jones et al., [<reflink idref="bib25" id="ref55">25</reflink>]). With adding up pieces, students can conceptualise several ideas, partitions of given intervals (length or time), target quantities (area, distance and volume etc.) and the sums of these (Stevens & Jones, [<reflink idref="bib42" id="ref56">42</reflink>]). With consideration of these, students who use adding up pieces often indicate that there is a multiplicative relationship between differential and integrand to understand target quantities and see small pieces being added instead of considering one undivided piece of area and domain (Jones, [<reflink idref="bib22" id="ref57">22</reflink>]). As previously mentioned, students are often familiar with interpreting definite integral as the area under a curve or anti-derivative. When they are able to use adding up pieces, they know what each symbol in definite integral means as it is important for students to carefully interpret quantities in Riemann sum and assign meaning to those symbols to conceptually understand how Riemann sum and definite integral are related (Tallman et al., [<reflink idref="bib45" id="ref58">45</reflink>]). Since recognising the multiplicative relationship in <emph>adding up pieces</emph> is useful in applied contexts in physics and engineering (Ely, [<reflink idref="bib11" id="ref59">11</reflink>]; Jones, [<reflink idref="bib22" id="ref60">22</reflink>]; Sealey, [<reflink idref="bib38" id="ref61">38</reflink>]), using adding up pieces will provide learning opportunities to students where interpretations of area or anti-derivative (function matching) alone may not. For example, without understanding the product layer, it is challenging for students to see why the product of two quantities produces distance travelled or energy (Ely, [<reflink idref="bib10" id="ref62">10</reflink>]). Moreover, anti-derivative provides a technique of evaluating integral rather than something about the context (Ely, [<reflink idref="bib10" id="ref63">10</reflink>]). As described earlier, these different resources would provide different units of knowledge that allow students to interpret and understand definite integral differently (e. g. undivided piece vs. small pieces and the whole length of the bottom side vs. infinitesimally small piece of the domain).</p> <p>Students struggle most to understand the product layer, which is critical in <emph>adding up pieces</emph> (Jones, [<reflink idref="bib23" id="ref64">23</reflink>]; Sealey, [<reflink idref="bib38" id="ref65">38</reflink>]). A recent study also stated that advanced mathematics students also struggled to understand the product layer and recommended more careful attention to the product layer (Soto & Oehrtman, [<reflink idref="bib41" id="ref66">41</reflink>]). However, since <emph>adding up pieces</emph> is more productive in understanding definite integral, being able to use <emph>adding up pieces</emph> along with the product layer may be a key additional unit of knowledge students need to possess as this was recommended by other researchers as well (Jones, [<reflink idref="bib23" id="ref67">23</reflink>]; Oehrtman & Simmons, [<reflink idref="bib31" id="ref68">31</reflink>]). However, students are unable to use the idea or only few use the idea even if they are able to correctly answer integral tasks with anti-derivative (Jones et al., [<reflink idref="bib25" id="ref69">25</reflink>]; Wagner, [<reflink idref="bib49" id="ref70">49</reflink>]) and some students struggle to use this idea (Stevens & Jones, [<reflink idref="bib42" id="ref71">42</reflink>]). Identifying adding up pieces as a key unit of knowledge does not mean that other cognitive resources (the perimeter and area symbolic form and the function matching symbolic form) are not important in calculus. Including adding up pieces along with what students are familiar with (area under a curve and anti-derivative) provides learning opportunities that allow students to interpret definite integral in various contexts where activating other symbolic forms alone may not (Jones, [<reflink idref="bib22" id="ref72">22</reflink>], [<reflink idref="bib23" id="ref73">23</reflink>]).</p> <hd id="AN0187890693-6">2.3. Exploring calculus students' learning opportunities</hd> <p>When considering opportunity to learn (OTL), we can consider content coverage (list of topics and sub-topics covered), content exposure (amount of time spent to instruction), and content emphasis (which topics are selected for emphasis) to understand opportunities to learn (OTL) (Reeves et al., [<reflink idref="bib36" id="ref74">36</reflink>]). The term 'coverage' is used to denote the topics and sub-topics taught (Stigler & Hiebert, [<reflink idref="bib43" id="ref75">43</reflink>]). On the other hand, 'content exposure' refers to the overall amount of time devoted to a given subject (Floden, [<reflink idref="bib13" id="ref76">13</reflink>]; Lee, [<reflink idref="bib27" id="ref77">27</reflink>]; Wang, [<reflink idref="bib50" id="ref78">50</reflink>]). Lastly, 'content emphasis' indicates the relative amount of time students spend on various topics (Wang, [<reflink idref="bib50" id="ref79">50</reflink>]). With this idea, we can ask 'To what extent are the topics emphasized in the national curriculum, textbooks and teachers' lessons?' (Floden, [<reflink idref="bib13" id="ref80">13</reflink>]). When particular procedures, algorithms, and problems are presented in textbooks (content coverage), teachers and students have potential learning opportunities to become familiar with those procedures and problems. In seeking more meaningful ways to conceptualise OTL, researchers also investigate the allocation of class time to specific topics or the level of student engagement (Floden, [<reflink idref="bib13" id="ref81">13</reflink>]). One significant aspect of OTL is that it is unreasonable to expect students to grasp certain mathematical topics if they have not been provided with adequate opportunities to learn them (Floden, [<reflink idref="bib13" id="ref82">13</reflink>]).</p> <p>When teachers prepare and plan their lessons, they select (and modify) tasks and activities from textbooks and other resources in planning their lessons (Remillard & Heck, [<reflink idref="bib37" id="ref83">37</reflink>]). While we acknowledge that not every task and activity in textbooks may be included or implemented in lessons, textbooks play a crucial role in providing opportunities to learn (OTL) to students, particularly through selected tasks and activities to be included in the lessons (content coverage and exposure) Additionally, when some tasks are assigned as homework or when students read textbooks to study on their own, these can be transformed into learning opportunities for students. With these different layers of opportunities, we think of calculus students' opportunities to become familiar with the concepts of definite integrals and what calculus textbooks can offer to students. As other studies have examined opportunities to learn in various mathematics textbooks, it becomes evident that textbooks serve as significant resources for both teachers and students in the teaching and learning process as textbooks offer content coverage, encompassing a list of topics and sub-topics covered, content exposure, representing the amount of time potentially devoted to instruction, and content emphasis, highlighting the emphasised topics emphasis (Ding, [<reflink idref="bib8" id="ref84">8</reflink>]; Thompson et al., [<reflink idref="bib46" id="ref85">46</reflink>]; van Zanten & van den Heuvel-Panhuizen, [<reflink idref="bib48" id="ref86">48</reflink>]), One interesting finding from a previous study is that calculus students often do not see the need to learn the structure of Riemann sum or the adding up pieces idea because area under a curve or anti-derivatives provide correct answers to many integral problems (Wagner, [<reflink idref="bib49" id="ref87">49</reflink>]). If widely-used textbooks give more attention to the perimeter and area symbolic form or the function matching symbolic form than adding up pieces (content coverage and emphasis), students' learning opportunities may be limited to area symbolic form or the function matching symbolic form than adding up pieces (less coverage, exposure, and emphasis). This will lead them likely to use certain interpretations of the integral (area under a curve or anti-derivative) and less likely to interpret definite integrals in other contexts or not feel the need to learn and interpret integral in other contexts. For example, when Riemann sums are covered as a simple calculation procedure instead of a conceptual idea, only a few students interpreted definite integral by describing adding up pieces and the multiplicative relationship (Jones et al., [<reflink idref="bib25" id="ref88">25</reflink>]). However, including more tasks that require the idea of accumulation, the product layer (or the multiplicative relationship between two quantities) provides and increases learning opportunities for students to interpret definite integral not just as an area of one piece or anti-derivative, but also as adding small pieces. By examining learning opportunities in textbooks, we can understand the types of learning opportunities that students have when textbooks are used in and outside of classes.</p> <hd id="AN0187890693-7">3. Method</hd> <p></p> <hd id="AN0187890693-8">3.1. Data sources</hd> <p>According to Mathematical Association of America's (MAA) Calculus study (Bressoud, [<reflink idref="bib2" id="ref89">2</reflink>]), Stewart's Calculus, Calculus Single & Multivariable, and Thomas' Calculus are three widely-used Calculus 1 textbooks in the United States. We used the latest editions of these textbooks; Calculus 9E 9th Edition (Stewart, 2021, abbreviated to SC), Calculus Single & Multivariable 8th Edition (Hughes-Hallett et al., 2021, abbreviated to HC), and Thomas' Calculus 14th Edition (Thomas et al., 2018, abbreviated to TC). In addition to MAA's report, these three textbooks were also analysed in other recent studies by international scholars (Chang et al., [<reflink idref="bib4" id="ref90">4</reflink>]; Chen, [<reflink idref="bib6" id="ref91">6</reflink>]; Mkhatshwa, [<reflink idref="bib29" id="ref92">29</reflink>]; Özgeldi & Aydın, [<reflink idref="bib33" id="ref93">33</reflink>]; Pogorelova, [<reflink idref="bib34" id="ref94">34</reflink>]), indicating the widespread use of these textbooks. However, these studies are focused on derivatives (Chen, [<reflink idref="bib6" id="ref95">6</reflink>]), competency demands (Özgeldi & Aydın, [<reflink idref="bib33" id="ref96">33</reflink>]), multiple representations (Chang et al., [<reflink idref="bib4" id="ref97">4</reflink>]), and optimisation (Mkhatshwa, [<reflink idref="bib29" id="ref98">29</reflink>]). Due to the importance of definite integrals, we feel the need to examine integral more deeply, especially focusing on specific pieces of knowledge needed to understand definite integrals. To examine how three widely-used textbooks introduce concepts of definite integrals and opportunities to learn definite integrals, we analysed the first two sections of the integral chapter from SC and HC, and the first and the third sections from TC (the second section was about sigma notation). We chose to examine these sections to investigate how widely-used textbooks introduce and define definite integrals. Our primary interest lay in understanding how the definite integral was conceptualised in each textbook. Specifically, we aimed to determine whether it focused more on the area under a curve or on the concept of adding up pieces. We also sought to identify the types of knowledge covered when the definite integral was initially introduced. In total, 132 integral tasks (both worked examples and exercise problems) from SC, 157 tasks from TC, and 106 tasks from HC were analysed.</p> <hd id="AN0187890693-9">3.2. Textbook analysis</hd> <p>This document analysis study involves a careful reading and interpretation of the content of textbooks (Bowen, [<reflink idref="bib1" id="ref99">1</reflink>]). Within document analysis, we utilised thematic analysis to read and examine each textbook, aiming to identify emerging patterns or themes (Bowen, [<reflink idref="bib1" id="ref100">1</reflink>]). To conduct this analysis, we examined how each part of the textbook lessons introduces and develops mathematics topics, as in other textbook analysis studies (Charalambous et al., [<reflink idref="bib5" id="ref101">5</reflink>]; Smith et al., [<reflink idref="bib39" id="ref102">39</reflink>]; Hong, [<reflink idref="bib16" id="ref103">16</reflink>]; Hong & Choi, [<reflink idref="bib18" id="ref104">18</reflink>]; Hong et al., [<reflink idref="bib20" id="ref105">20</reflink>]). In this study, we focused on units of knowledge about the definite integral in widely-used calculus textbooks. Initially, we investigated how three textbooks define and introduce Riemann sum and definite integral concepts. Furthermore, we explored how these textbooks address the concepts of adding up pieces and the product layer. Secondly, each worked example and exercise problem was coded using our codes (refer to Table 1) to examine how these examples and problems provide the concepts of adding up pieces and the product layer. By scrutinising each part of the textbooks to see how widely-used textbooks cover and address adding up pieces, the product layer, and conceptualise the definite integral, we identified learning opportunities provided by each textbook, which emerged as themes or patterns for each textbook. In our analysis, each worked problem and exercise problem serves as one unit of analysis. In total, we analysed 395 problems across the 3 textbooks.</p> <p>Table 1. Analytic framework.</p> <p> <ephtml> <table><thead valign="bottom"><tr><td>Code</td><td>Unit of knowledge</td></tr></thead><tbody><tr><td>Adding up pieces</td><td><list list-type="Bullet"><list-item><p>Adding rectangles under a curve as the number of rectangles increases</p></list-item><list-item><p>Multiplication between two quantities (integrand and differential)</p></list-item><list-item><p>Partitions of interval into small pieces</p></list-item><list-item><p>Providing meaning of the product of two quantities means</p></list-item></list></td></tr><tr><td>Area under a curve</td><td><list list-type="Bullet"><list-item><p>Finding area under a curve as one undivided piece</p></list-item></list></td></tr><tr><td>Anti-derivative (function matching)/area under a curve</td><td><list list-type="Bullet"><list-item><p>Finding anti-derivative to evaluate integral</p></list-item></list></td></tr><tr><td>Expressing mathematical symbols to integral</td><td><list list-type="Bullet"><list-item><p>Expressing each mathematical symbol in Riemann sum to integral</p></list-item></list></td></tr><tr><td>Evaluating Riemann sum or integral with finite number of rectangles</td><td><list list-type="Bullet"><list-item><p>Evaluating Riemann sum or integral with several rectangles</p></list-item></list></td></tr><tr><td>Using integral properties</td><td><list list-type="Bullet"><list-item><p>Verifying various properties of integrals</p></list-item></list></td></tr></tbody></table> </ephtml> </p> <hd id="AN0187890693-10">3.3. Examples of coding</hd> <p>We describe examples of how we examined each task from the three widely-used textbooks. When examining textbook tasks, we considered different units of knowledge that each task presents to students. Table 1 shows the codes we used and units of knowledge we identified. These codes were developed as we reviewed previous studies and examined each task in textbooks (Jones, [<reflink idref="bib22" id="ref106">22</reflink>]; Sealey, [<reflink idref="bib38" id="ref107">38</reflink>]). Examples of these codes are provided in this section. These examples provide different units of knowledge about definite integral to students. We describe units of knowledge that students are exposed to when they try to solve each example.</p> <p>Table 2 shows examples of each code that we used for integral tasks from the three textbooks.</p> <p>Table 2. Coding examples.</p> <p> <ephtml> <table><thead valign="bottom"><tr><td>Item</td><td>Code</td></tr></thead><tbody><tr><td><list list-type="Bullet"><list-item><p>Partition the interval into <italic>n</italic> = 100, 200, and 1000 subintervals of equal length, and evaluate the function at the midpoint of each subinterval. <graphic href="tmes_a_2348145_ilm0005.gif" content-type="Graph" /><math xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">f</mi><mo stretchy="false" xmlns="">(</mo><mi xmlns="">x</mi><mo stretchy="false" xmlns="">)</mo><mo xmlns="">=</mo><mi xmlns="">x</mi><mrow xmlns=""><mtext>sin</mtext></mrow><mfrac xmlns=""><mn>1</mn><mi>x</mi></mfrac></math> on <graphic href="tmes_a_2348145_ilm0006.gif" content-type="Graph" /><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mo>[</mo><mrow><mfrac><mi>π</mi><mn>4</mn></mfrac><mo>,</mo><mi>π</mi></mrow><mo>]</mo></mrow></math> (TC, p. 277)</p></list-item></list></td><td>Adding up pieces</td></tr><tr><td><list list-type="Bullet"><list-item><p>What is the area between the graph of <italic>f</italic>(<italic>x</italic>) in Figure 5.41 and the <italic>x</italic>-axis, between <italic>x</italic> = 0 and <italic>x</italic> = 5? (HC, <italic>p</italic>. 305)</p></list-item></list></td><td>Area under a curve</td></tr><tr><td><graphic href="tmes_a_2348145_ilg0001.gif" content-type="Graph" /></td><td /></tr><tr><td><list list-type="Bullet"><list-item><p>Prove that <graphic href="tmes_a_2348145_ilm0007.gif" content-type="Graph" /><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup xmlns=""><mo>∫</mo><mi>a</mi><mi>b</mi></msubsup><mrow xmlns=""><mi>x</mi><mrow><mtext>d</mtext></mrow><mi>x</mi><mo>=</mo><mfrac><mrow><mrow><msup><mi>b</mi><mn>2</mn></msup></mrow><mo>−</mo><mrow><msup><mi>a</mi><mn>2</mn></msup></mrow></mrow><mn>2</mn></mfrac></mrow></math> (SC, p. 396)</p></list-item></list></td><td>Anti-derivative (function matching)/area under a curve</td></tr><tr><td><list list-type="Bullet"><list-item><p>Express the limits in Exercises 1–8 as definite integrals <graphic href="tmes_a_2348145_ilm0008.gif" content-type="Graph" /><math xmlns="http://www.w3.org/1998/Math/MathML"><munder xmlns=""><mrow><mo form="prefix">lim</mo></mrow><mrow><mi>p</mi><mo stretchy="false">→</mo><mn>0</mn></mrow></munder><mo xmlns="">⁡</mo><msubsup xmlns=""><mo movablelimits="false">∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><mrow xmlns=""><msubsup><mi>c</mi><mrow><mi>k</mi></mrow><mn>2</mn></msubsup><mi mathvariant="normal">Δ</mi><mrow><msub><mi>x</mi><mi>k</mi></msub></mrow><mo>,</mo></mrow></math> where <italic>p</italic> is a partition of [0, 2] (TC, p. 274)</p></list-item></list></td><td>Expressing each mathematical symbol in Riemann sum to integral</td></tr><tr><td><list list-type="Bullet"><list-item><p>Estimate <graphic href="tmes_a_2348145_ilm0009.gif" content-type="Graph" /><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup xmlns=""><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mrow xmlns=""><mrow><msup><mi>e</mi><mrow><mrow><msup><mi>x</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup></mrow></mrow></msup></mrow><mrow><mtext>d</mtext></mrow><mi>x</mi></mrow></math> using <italic>n</italic> = 5 rectangles to form a left-hand sum (HC, p. 306)</p></list-item></list></td><td>Evaluating Riemann sum or integral using finite number of rectangles</td></tr><tr><td><list list-type="Bullet"><list-item><p>Use the properties of integrals to verify the inequality without evaluating the integrals <graphic href="tmes_a_2348145_ilm0010.gif" content-type="Graph" /><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup xmlns=""><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mrow xmlns=""><msqrt><mn>1</mn><mo>+</mo><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></msqrt><mrow><mtext>d</mtext></mrow><mi>x</mi><mo>≤</mo></mrow><msubsup xmlns=""><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mrow xmlns=""><msqrt><mn>1</mn><mo>+</mo><mi>x</mi></msqrt><mrow><mtext>d</mtext></mrow><mi>x</mi></mrow></math> (SC, p. 397)</p></list-item></list></td><td>Using integral properties</td></tr></tbody></table> </ephtml> </p> <p>The first example shows that evaluating definite integral can be done with several rectangles. This task was included after the Riemann sum was introduced. Students are asked to think about what will happen if more rectangles are added, which gives them ideas of adding up small pieces to get area. As the number of rectangles increases, students have the opportunity to conceptualise the addition of infinitely many pieces. This provides them with opportunities to observe partitions of a given interval and understand the quantitative relationship between two quantities: the length of the interval and the function value (the product layer), and how to sum these quantities.</p> <p>The second example is about area under a curve. This is included after introducing Riemann sum (adding up small pieces) and defining definite integral. At this point, students have been introduced Riemann sum and adding up pieces, but similar problems are solved using the calculator to estimate the area. Students can interpret this as finding area under a curve as one undivided piece instead of small pieces. Although students may recall Riemann sum and rectangles, this example is more about understanding area as a whole piece as the area of each region (undivided piece) is provided already in the figure.</p> <p>The third example can be answered if students know how to find an anti-derivative of <emph>x</emph>. This example is included after Riemann sum and the answer represents the area under a line <emph>y = x</emph>. At this point, anti-derivative has not been formally introduced, but several properties of integral have. Students solve this with an understanding of area under a curve and realising that</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mrow><mn>2</mn></mfrac></math> </ephtml> is the result of integration. Although we can easily see that</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mrow><mn>2</mn></mfrac></math> </ephtml> is the result of integration, since anti-derivative was not formally introduced, we coded it as anti-derivative/area under a curve.</p> <p>The fourth example asks students to express each symbol in the Riemann sum to the corresponding symbol of definite integral. Students are able to see the product layer (how</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mrow><mrow><msub><mi>c</mi><mi>k</mi></msub></mrow></mrow><mo stretchy="false">)</mo></math> </ephtml> are replaced</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> </ephtml> and how the subinterval widths</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Δ</mi><mrow><msub><mi>x</mi><mi>k</mi></msub></mrow></math> </ephtml> become the differential d<emph>x</emph> – these are mentioned precisely by TC), but solutions to similar worked examples show that it is possible that students just simply express each symbol in Riemann sum to each symbol in integral without thinking about adding small pieces and the product layer.</p> <p>The fifth example asks students to use five rectangles, a finite number, instead of conceptualising many rectangles to estimate</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mrow><mrow><msup><mi>e</mi><mrow><mrow><msup><mi>x</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup></mrow></mrow></msup></mrow><mrow><mtext>d</mtext></mrow><mi>x</mi></mrow></math> </ephtml> . Although this problem includes the idea of adding up pieces (partitions, product and sum), we think there is a conceptually different from the first example. Adding more rectangles can help students conceptualise the process of adding infinitely many pieces while the areas of finite number of rectangles be calculated without considering the infinite process.</p> <p>The last example asks students to verify an inequality using one of the properties of the integral. This is included after discussions of Riemann sum and definite integral. Although students have opportunities to be exposed to adding up pieces, similar examples are solved using area under a curve. This can be answered correctly once the graphs of two functions are drawn. This requires an understanding of area, but this example may not lead students to consider adding up rectangles. They will likely see this as comparing areas of undivided pieces once the graphs are drawn and students can visually confirm the inequality without considering adding small pieces.</p> <hd id="AN0187890693-11">3.4. Reliability</hd> <p>We carefully reviewed previous studies to develop our codes (Jones, [<reflink idref="bib22" id="ref108">22</reflink>]; Jones et al., [<reflink idref="bib25" id="ref109">25</reflink>]; Sealey, [<reflink idref="bib38" id="ref110">38</reflink>]) and we were able to develop a few more codes as we examined the textbooks. Once developed, two independent readers reviewed the textbooks several times and compared findings to ensure reliability of results. We read each textbook carefully and coded each worked example and exercise problems independently. The initial agreement rate for task discussions was 95%. For instance, initial disagreements were noted with the first and fifth examples in Table 2. However, by the end, consensus was reached that considering an infinite process would offer distinct learning opportunities for students compared to adding finite rectangles. Additionally, during our discussions, the fourth and sixth examples in Table 2 emerged as new codes found in the textbooks. These two codes present unique learning opportunities for students compared to the other codes in the table. The discussion was done until there was 100% agreement.</p> <hd id="AN0187890693-12">4. Results</hd> <p>In this section, we will describe how the definite integral was introduced in each textbook, how each textbook defines the definite integral, and how each worked example and exercise problem provides opportunities to learn about the definite integral.</p> <hd id="AN0187890693-13">4.1. Riemann sum, area under a curve and distance travelled</hd> <p>All three textbooks described how area under a curve can be estimated with rectangles. As the number of rectangles increases, the infinite process can be seen to conceptualise adding up small pieces. Both SC and TC describe the product layer, how</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mrow><mrow><msub><mi>c</mi><mi>k</mi></msub></mrow></mrow><mo stretchy="false">)</mo></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Δ</mi><mrow><msub><mi>x</mi><mi>k</mi></msub></mrow></math> </ephtml> can be multiplied to find the area of each rectangle. SC and TC show how the area can be estimated with 1000 rectangles. In HC, the first example is about distance travelled. HC's example shows that the area of each rectangle describes the distance travelled and with more intervals (every</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mspace width="thinmathspace" /><mfrac><mn>1</mn><mn>4</mn></mfrac></math> </ephtml> , 2 and 4 s), we can get better estimates of distance travelled. After introducing area under a curve, SC and TC show examples of distance travelled. All examples in the three textbooks provide opportunities for students to become familiar with partitioning intervals, the product layer (height times width or distance times time) and adding up pieces (adding area under a curve or distance travelled). For example, HC includes the following explanation.</p> <p>Continuing in this way and adding all the estimates, we get an estimate for the total distance traveled. In the last interval, the velocity is approximately <emph>f</emph>(<emph>t</emph><subs>n−1</subs>), so the last term is <emph>f</emph>(<emph>t</emph><subs>n−1</subs>)Δ<emph>t</emph>. (HC, p. 290)</p> <p>Introducing Riemann sum and the distance travelled provides opportunities for students to divide given interval into smaller pieces (partition of interval), see the product layer (target quantity) and add those products (the sum). The worked examples in this section use a finite number of rectangles to describe area or distance. (Further analysis of worked examples in this section is included in the following section.)</p> <hd id="AN0187890693-14">4.2. Introduction and definition of definite integral</hd> <p>Figure 1 shows the definition of definite integral in each textbook.</p> <p>Graph: Figure 1. Introduction of definite integral.</p> <p>All three textbooks describe the infinite process of dividing an interval to smaller pieces showing the hint of describing 'adding up pieces' and the product layer when Riemann sum and definite integral are introduced. For example, SC describes</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Δ</mi><mrow><msub><mi>x</mi><mi>n</mi></msub></mrow></math> </ephtml> as widths approaching zero while HC describes d<emph>t</emph> in the integral comes from the factor Δ<emph>t</emph>. TC describes the product layer more precisely. It describes what</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>⋅</mo><mrow><mtext>d</mtext></mrow><mi>x</mi></math> </ephtml> (summing all products) means and where d<emph>x</emph> came from. Here are additional explanations that we found from the textbooks. These explanations all show the product layer of Riemann sum (HC and TC) and how each notation will be replaced, but it is not so clear how <emph>f</emph>(<emph>x</emph>) · d<emph>x</emph> is related to the product layer.</p> <p>Leibniz introduced a notation for the definite integral that captures its construction as a limit of Riemann sums. He envisioned the finite</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo movablelimits="false">∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><mrow><mi>f</mi><mo stretchy="false">(</mo><mrow><mrow><msub><mi>c</mi><mi>k</mi></msub></mrow></mrow><mo stretchy="false">)</mo><mi mathvariant="normal">Δ</mi><mrow><msub><mi>x</mi><mi>k</mi></msub></mrow></mrow></math> </ephtml> becoming an infinite sum of function values <emph>f</emph>(<emph>x</emph>) multiplied by 'infinitesimal' subinterval widths d<emph>x</emph>. (TC, p. 266)</p> <p>If <emph>f</emph>(<emph>x</emph>) is positive we can interpret each term <emph>f</emph>(<emph>x</emph><subs>0</subs>)Δ<emph>x</emph>, <emph>f</emph>(<emph>x</emph><subs>1</subs>)Δ<emph>x</emph>, ... in a left- or right-hand Riemann sum as the area of a rectangle. (HC, p. 300)</p> <p>When Leibniz chose the notation for an integral, he chose the ingredients as reminders of the limiting process. In general, when we write</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mrow><mo form="prefix">lim</mo></mrow><mrow><mi>n</mi><mo stretchy="false">→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mo>⁡</mo><msubsup><mo movablelimits="false">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><mrow><mi>f</mi><mo stretchy="false">(</mo><msubsup><mi>x</mi><mi>i</mi><mo>∗</mo></msubsup><mo stretchy="false">)</mo><mi mathvariant="normal">Δx</mi><mo>=</mo></mrow><msubsup><mo>∫</mo><mi>a</mi><mi>b</mi></msubsup><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mrow><mtext>d</mtext></mrow><mi>x</mi></mrow></math> </ephtml> , we replace lim Σ by ò,</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>x</mi><mi>i</mi><mo>∗</mo></msubsup></math> </ephtml> by <emph>x</emph>, and Δ<emph>x</emph> by d<emph>x</emph>. (SC, p. 387)</p> <p>Since students struggle to understand the product layer, it will be beneficial to describe more precisely what those symbols mean in Riemann sum and in definite integral as TC precisely states that all products of the form <emph>f</emph>(<emph>x</emph>) · d<emph>x</emph> are added.</p> <hd id="AN0187890693-15">4.3. Worked examples</hd> <p>Table 3 shows the distribution of worked examples by each code. We can see the three textbooks' suggestions to solve worked examples, which can be guidelines for students when studying on their own. Figures 2 and 3 show worked examples from SC and TC.</p> <p>Graph: Figure 2. A worked example from SC (SC, p. 389).</p> <p>Graph: Figure 3. Worked example from TC (TC, p. 271).</p> <p>Table 3. Distribution of worked examples.</p> <p> <ephtml> <table><thead valign="bottom"><tr><td>Code</td><td>SC</td><td>TC</td><td>HC</td></tr></thead><tbody><tr><td>Adding up pieces</td><td char="(">2 (13.3%)</td><td>1 (11.1%)</td><td>0</td></tr><tr><td>Area under a curve</td><td char="(">2 (13.3%)</td><td>3 (33.3%)</td><td>3 (37.5%)</td></tr><tr><td>Anti-derivative/area under a curve</td><td char="(">0</td><td>0</td><td>0</td></tr><tr><td>Expressing mathematical symbols to integral</td><td char="(">2 (13.3%)</td><td>0</td><td>0</td></tr><tr><td>Evaluating Riemann sum or integral with several rectangles</td><td char="(">6 (40%)</td><td>3 (33.3%)</td><td>5 (62.5%)</td></tr><tr><td>Using integral properties</td><td char="(">3 (20%)</td><td>2 (22.2%)</td><td>0</td></tr><tr><td>Total</td><td char="(">15</td><td>9</td><td>8</td></tr></tbody></table> </ephtml> </p> <p>All three textbooks show how finite number of rectangles can be used to describe either area under a curve or total distance travelled. Then, after defining a definite integral, SC includes worked examples like the fourth example in Table 2 – converting Riemann sum to definite integral. Next, the worked example in Figure 2 shows how to set up an expression from an integral. It describes dividing an interval and adding those. At this point, students have opportunities to see adding up pieces and the product layer. The solution to this worked example provides students with opportunities to express each symbol of the integral to summation symbols. To complete this task, students refer to Theorem 4, which is about converting the definite integral to Riemann sum. They may be able to see how the product layer of Riemann sum is related to integrand and differential, but it is not clear whether they will use them. They can just match each symbol without considering what each means.</p> <p>After defining definite integral, TC introduces several integral properties such as order of integration and zero width interval. Figure 3 shows one worked example that describes using one of the properties of integral. Students need to understand the minimum and the maximum values of the function in the given interval and the area under the function, but it is not clear if they will consider adding small pieces to solve this problem. More likely, they will use area under a curve (one undivided piece instead of small pieces) as TC demonstrated why these properties are true using area under a curve (undivided piece). From the initial introduction to worked examples, there are opportunities to become familiar with adding up pieces, the product layer in Riemann sum, and converting Riemann sum to integral, but students have fewer opportunities to see how Riemann sum and definite integrals are related to each other.</p> <p>After defining definite integral, HC shows how to estimate</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>1</mn><mn>2</mn></msubsup><mrow><mfrac><mn>1</mn><mi>t</mi></mfrac><mrow><mtext>d</mtext></mrow><mi>t</mi></mrow></math> </ephtml> with several rectangles, demonstrating the idea of adding up pieces and the product layer. Then,</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mrow><mo>−</mo><mn>1</mn></mrow><mn>1</mn></msubsup><mrow><msqrt><mn>1</mn><mo>−</mo><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></msqrt><mrow><mtext>d</mtext></mrow><mi>x</mi></mrow></math> </ephtml> is evaluated using a calculator or computer (area under a curve). Although students may be able to make a connection between Riemann sum, the product layer, and integral, it is more likely that they will use this as an opportunity to understand as area under a curve (as an undivided piece).</p> <hd id="AN0187890693-16">4.4. Exercise problems</hd> <p>Table 4 shows percentage of each code when compared to the total number of exercise problems in each textbook. The distribution of these codes is one indication of how each textbook covers each code.</p> <p>Table 4. Percent distribution of exercise problems.</p> <p> <ephtml> <table><thead valign="bottom"><tr><td>Code</td><td>Stewart</td><td>Thomas</td><td>Hughes-Hallett</td></tr></thead><tbody><tr><td>Adding up pieces</td><td char="(">12 (10.2%)</td><td char="(">19 (12.8%)</td><td>0</td></tr><tr><td>Area under a curve</td><td char="(">14 (12.2%)</td><td char="(">41 (27.7%)</td><td>35 (35.7%)</td></tr><tr><td>Anti-derivative/area under a curve</td><td char="(">3 (2.5%)</td><td char="(">0</td><td>0</td></tr><tr><td>Matching mathematical symbols to integral</td><td char="(">25 (21.3%)</td><td char="(">9 (6.08%)</td><td>0</td></tr><tr><td>Evaluating Riemann sum or integral with several rectangles</td><td char="(">48 (41%)</td><td char="(">22 (14.8%)</td><td>63 (64.3%)</td></tr><tr><td>Using integral properties</td><td char="(">15 (12.8%)</td><td char="(">57 (38.5%)</td><td>0</td></tr><tr><td>Total</td><td char="(">117</td><td char="(">148</td><td>98</td></tr></tbody></table> </ephtml> </p> <p>Both SC and TC include adding up pieces problems (Figure 4). In the process of solving the problem in Figure 4, students need to think about the number of rectangles (adding up pieces). Thus, they can see that partitioning of given interval, the product, and adding many rectangles will give the area under a curve. They may see the product layer when they create rectangles, but it would be useful to ask students to describe the area of each rectangle in Figure 4, how each area can be expressed as a product of two quantities, and how the product is related to integrand and differential. Students will have more opportunities to understand what it means to add infinitely many rectangles, how the area of a rectangle can be formed, and how those areas are related to definite integral, which is useful in understanding integral in other contexts (Ely, [<reflink idref="bib11" id="ref111">11</reflink>]; Sealey, [<reflink idref="bib38" id="ref112">38</reflink>]).</p> <p>Graph: Figure 4. Adding up pieces problem from TC (TC, p. 277).</p> <p>For SC and HC, the most frequent task (more coverage) is evaluating Riemann sum with several rectangles (Figure 5).</p> <p>Figure 6 shows how a similar problem to the problem in Figure 5 can be solved by SC. This worked example shows how two quantities are multiplied and added to get the area of each rectangle and approximate the sum. However, the solution is presented without how the summation of five rectangles is translated to integral notation. Students will more likely be able to see partitions and the product layer if they are asked to describe how each product,</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi mathvariant="normal">Δx</mi></math> </ephtml> is formed, how to get better estimates with more rectangles and what each rectangle means in other contexts (if it is a contextualised problem).</p> <p>For TC, the most frequent task (more coverage) is using integral properties. These properties are the order of integration, zero width interval, constant multiple, sum, and difference and additivity. Although these properties are important, it is not clear if students consider adding small pieces or the product of two quantities when solving this. It is likely they will use area under <emph>y</emph> = <emph>x</emph> as these properties are demonstrated by comparing areas (as one undivided piece).</p> <p>In summary, these three widely-used textbooks provide students with opportunities to explore adding up pieces, using a finite number of rectangles, understanding area under a curve, and learning integral properties from the initial introduction to exercise problems. Although students have opportunities to become familiar with the product layer and adding infinitely many rectangles, it will be more useful to describe those ideas more precisely.</p> <hd id="AN0187890693-17">5. Summary and discussion</hd> <p>Limit, derivative, and integral are three main topics in Calculus 1; however, far fewer studies have been published about integral compared to limit and derivative (Jones, [<reflink idref="bib22" id="ref113">22</reflink>]; Larsen et al., [<reflink idref="bib26" id="ref114">26</reflink>]). With the results from earlier studies, we can understand students' learning challenges and think about how to support students to learn integral concepts productively. Recently published studies described productive cognitive resources to understand integrals (Jones, [<reflink idref="bib22" id="ref115">22</reflink>], [<reflink idref="bib23" id="ref116">23</reflink>]; Sealey, [<reflink idref="bib38" id="ref117">38</reflink>]). Since the product layer is the key idea to understanding the adding up pieces symbolic form, having opportunities to explore the product layer and adding up pieces provides additional unit of knowledge needed to students. When students are able to use adding up pieces, they are often able to identify what integrand and differential mean and how they are related to each other (Jones, [<reflink idref="bib22" id="ref118">22</reflink>]). Students often struggle with the product layer when tasks are presented in various contexts (Sealey, [<reflink idref="bib38" id="ref119">38</reflink>]). Those who are able to use adding up pieces often consider small pieces of area and domain using partitions, target quantities and the sum while those who use area under a curve often consider area as an undivided piece (Jones, [<reflink idref="bib22" id="ref120">22</reflink>]).</p> <p>We referred to previous studies to develop our codes and selected three widely-used calculus textbooks. Although all three textbooks provide opportunity to become familiar with adding up pieces and the product layer in Riemann sum, these opportunities are provided with limited number of tasks. It appears that the three textbooks place more coverage and emphasis on other units of knowledge. As a result, what d<emph>x</emph> means when definite integral is introduced and the multiplicative relationship between the differential and the integrand are briefly described. The textbooks use 'replaced' or 'comes' to describe differential but TC's explanation, summing all products of the form <emph>f</emph>(<emph>x</emph>) · d<emph>x</emph>, is clearer. Worked examples in Figures 2 and 3 allow students to see how to set up Riemann sum and use different rules of integrals but it is likely that they are exposed to calculation or area under a curve. The multiplicative relationship between two quantities (the product layer) and how infinitely many rectangles can be formed and added to represent the area (adding up pieces) are presented with a few problems. As a result, when students study on their own by reading the textbook or when the calculus instructor uses these books for lessons or homework, it is less likely that they will focus on <emph>adding up pieces</emph> to interpret definite integral because other units of knowledge have more coverage or emphasis.</p> <p>We can consider the learning opportunities that these three textbooks offer to students. For students who use SC and HC, they are likely to be more exposed to solving Riemann sums with a finite number of rectangles, while students who use TC would likely encounter more exposure to integral properties. To enhance students' familiarity with adding small pieces, it could be beneficial to include more tasks similar to those in Figure 4. Additionally, tasks that precisely describe multiplicative relationships between two quantities would be beneficial for a deeper understanding of integral concepts. By incorporating such tasks, instructors can help students develop a more comprehensive understanding of calculus concepts across different textbooks, which allows students to expand their understanding that goes beyond area under a curve.</p> <p>Our findings from the exercise problems in these three textbooks are very similar to what we found from their initial introduction and worked examples. Tasks in Figures 5–7 show that what these widely-used textbooks demonstrate is understanding integral properties and using a finite number of rectangles. Based on our findings, we do not suggest that what these textbooks demonstrate is unimportant, nor do we recommend emphasising adding up pieces and the product layer for every task. As a matter of fact, calculus students should have sufficient opportunities to master these different interpretations of definite integrals as they are also important in calculus. However, it is challenging for students to expand their understanding of definite integral in other contexts as using the product layer is important to interpret integral in other contexts (Sealey, [<reflink idref="bib38" id="ref121">38</reflink>]). TC and SC include learning opportunities to explore adding up pieces (exercise problems) but students are asked to use a calculator or computer to think about 1000 rectangles. Although TC's definition shows how</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mrow><mrow><msub><mi>c</mi><mi>k</mi></msub></mrow></mrow><mo stretchy="false">)</mo></math> </ephtml> are replaced by <emph>f</emph>(<emph>x</emph>), how the subinterval widths</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Δ</mi><mrow><msub><mi>x</mi><mi>k</mi></msub></mrow></math> </ephtml> become the differential d<emph>x</emph>, and what <emph>f</emph>(<emph>x</emph>)·d<emph>x</emph> means, the exercise problems provide students with a limited number of opportunities to understand the meaning of these symbols. To provide more precise learning opportunities for students, it is useful for calculus instructors to consider asking students to describe the area of each rectangle in Figure 4, how each area can be expressed as a product of two quantities, and how the product is related to integrand and differential. Moreover, it will be useful for students to describe how those products can be interpreted in other contexts. As students attempt to answer these questions, they can be more familiar with three layers needed to use adding up pieces, partitions of given interval, target quantities and sum (Stevens & Jones, [<reflink idref="bib42" id="ref122">42</reflink>]). Students can be also asked to describe how finite rectangles in Figures 5 and 6 can be turned into infinitely many rectangles to be more familiar with three layers needed and infinite process. These opportunities do not need to be added to every task but adding them can provide OTL to expand their understanding, as suggested by other studies (Jones, [<reflink idref="bib22" id="ref123">22</reflink>]; Sealey, [<reflink idref="bib38" id="ref124">38</reflink>]). With these opportunities, students understand more clearly what it means to partition interval, add infinitely many rectangles (in addition to thinking about it as on undivided piece), how the area of a rectangle can be formed and what those products mean in different contexts. As other researchers indicated, such learning opportunities help students expand their understanding of definite integral in other contexts (Jones, [<reflink idref="bib22" id="ref125">22</reflink>]; Sealey, [<reflink idref="bib38" id="ref126">38</reflink>]). These findings reveal a mismatch among what experts describe as the main goals in teaching definite integrals (Sofronas et al., [<reflink idref="bib40" id="ref127">40</reflink>]), what widely-used textbooks cover and emphasise, and what research indicates about the types of knowledge students need. This misalignment highlights the need for further investigation and potential revisions in curriculum and instructional practices to better align with the intended learning outcomes and student needs in the teaching of definite integrals.</p> <p>Graph: Figure 5. Exercise problem from SC (SC, p. 395).</p> <p>Graph: Figure 6. Worked example from SC (SC, p. 390).</p> <p>Graph: Figure 7. Exercise problem from TC (TC, p. 275).</p> <p>In comparing three widely-used calculus textbooks, we can understand the learning opportunities that each textbook provides to students. Among three textbooks, TC more precisely describes the product layer and also, opportunities to conceptualise adding up pieces. On the other hand, with SC and HC, students would have more opportunities to conceptualise understanding of Riemann sums with a finite number of rectangles. It's crucial to address certain aspects that might have been overlooked. Both SC and HC did not clearly describe the product layer when the definite integral was introduced. With these opportunities in textbooks, it is important for calculus instructors to recognise the learning opportunities that each textbook provides. While it's understood that not every task will be selected and included in calculus lessons, when planning calculus lessons, it will be beneficial to consider how to incorporate the concepts of adding up pieces and the product layer into the curriculum. This approach can enrich and expand students' understanding of Riemann sums and definite integrals that might go beyond the area under a curve.</p> <p>Although it is critical to think about how we teach calculus, we can realistically think about what really happens in calculus classrooms. As Wagner ([<reflink idref="bib49" id="ref128">49</reflink>]) mentioned, when students are able to correctly evaluate definite integral with anti-derivative, they may not feel the need to use and understand the product layer or the adding up pieces idea. As Wagner ([<reflink idref="bib49" id="ref129">49</reflink>]) pointed out, the learning opportunities we found from these three textbooks might be widespread in other calculus classes as well. This is especially true when students are not often assessed to demonstrate their understanding of how different quantities in definite integrals are related to each other (Tallman et al., [<reflink idref="bib45" id="ref130">45</reflink>]). It can also be true that with limited exposure and coverage, students are not asked to demonstrate their understanding of these units of knowledge about integral. With what students experience in class, how they are assessed, and what they learn from these textbooks, it is highly likely that they pay less attention to accumulation, adding up pieces, and the multiplicative relationship.</p> <p>It is well known that students' learning opportunities are a combination of several different areas – class experiences, textbooks, and instructors' knowledge and intentions. Thus, what calculus instructors do in their classes critically impacts how students learn (Hong, [<reflink idref="bib17" id="ref131">17</reflink>]; Hong & Choi, [<reflink idref="bib18" id="ref132">18</reflink>]; Hong & Lee, [<reflink idref="bib21" id="ref133">21</reflink>]). When teachers use curriculum materials, they do not necessarily choose every task, they can interpret, select, and modify items in curriculum materials for their lessons (Dietiker et al., [<reflink idref="bib7" id="ref134">7</reflink>]). It is Then, being able to incorporate adding up pieces or the product layer to interpret and introduce definite integrals might be an important knowledge that calculus instructors need. As we mentioned earlier, it is important for students to interpret definite integral as an area under a curve or anti-derivative. Thus, in addition to exposing students to these interpretations, it is beneficial to include the <emph>adding up pieces</emph> concept and interpretations of integral in other contexts (physics and engineering) in the calculus curriculum to expand their understanding to other contexts (Ely, [<reflink idref="bib10" id="ref135">10</reflink>]). Our study reveals similar trends to other calculus tasks (Ellis et al., [<reflink idref="bib9" id="ref136">9</reflink>]; Tallman et al., [<reflink idref="bib44" id="ref137">44</reflink>]) that limit students' learning opportunities to be familiar with important calculus concepts.</p> <p>While we acknowledge that the analysis of textbooks can provide meaningful information about students' learning opportunities, it is important to point out that this study also has limitations. Although we selected three widely-used textbooks, it might be useful to examine calculus textbooks from other countries. How do they introduce definite integrals? How do they conceptualise adding up pieces and product layers? Exploring these questions could yield valuable insights into learning opportunities in other textbooks. Moreover, we did not examine calculus instructors' use of these textbooks. Thus, it would be meaningful to investigate how calculus instructors utilise these textbooks. What tasks do they select and implement? Do they modify any tasks to more clearly conceptualise adding up pieces and product layers? These are also critical questions to explore in order to understand more about calculus students' learning opportunities.</p> <p>Since the contents of textbooks can impact teachers' lessons and students' learning opportunities, limited opportunities in widely-used textbooks can be one contributing factor to learning challenges found in previous studies. As a previous study pointed out, important calculus concepts are not assessed often (Tallman et al., [<reflink idref="bib45" id="ref138">45</reflink>]). Limited emphasis and coverage of concepts in textbooks, coupled with the absence of assessment items for those concepts, can result in reduced attention to them in classes. Examining OTL widely-used textbooks can be an important initial step to think about issues in teaching and learning of calculus. As other researchers recently pointed out that successful curricula require interactions among a triad of elements: (a) production of national mathematics curriculum documents, (b) mathematics education research, and (c) teachers' understanding of content, cognition, and pedagogy (Thompson & Harel, [<reflink idref="bib47" id="ref139">47</reflink>]). Adding what we know about these textbooks to the results of existing studies is an initial step to think about possible ways to revise integral lessons in calculus textbooks.</p> <hd id="AN0187890693-18">Disclosure statement</hd> <p>No potential conflict of interest was reported by the authors.</p> <ref id="AN0187890693-19"> <title> References </title> <blist> <bibl id="bib1" idref="ref23" type="bt">1</bibl> <bibtext> Bowen, G. A. (2009). Document analysis as a qualitative research method. 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Items – Name: Title
  Label: Title
  Group: Ti
  Data: How Do Widely-Used Calculus Textbooks Introduce the Concepts of Definite Integrals?
– Name: Language
  Label: Language
  Group: Lang
  Data: English
– Name: Author
  Label: Authors
  Group: Au
  Data: <searchLink fieldCode="AR" term="%22Dae+S%2E+Hong%22">Dae S. Hong</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0003-0529-6448">0000-0003-0529-6448</externalLink>)<br /><searchLink fieldCode="AR" term="%22Dennis+Kwaka%22">Dennis Kwaka</searchLink>
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  Label: Source
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  Data: <searchLink fieldCode="SO" term="%22International+Journal+of+Mathematical+Education+in+Science+and+Technology%22"><i>International Journal of Mathematical Education in Science and Technology</i></searchLink>. 2025 56(9):1669-1689.
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  Label: Availability
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  Data: Taylor & Francis. Available from: Taylor & Francis, Ltd. 530 Walnut Street Suite 850, Philadelphia, PA 19106. Tel: 800-354-1420; Tel: 215-625-8900; Fax: 215-207-0050; Web site: http://www.tandf.co.uk/journals
– Name: PeerReviewed
  Label: Peer Reviewed
  Group: SrcInfo
  Data: Y
– Name: Pages
  Label: Page Count
  Group: Src
  Data: 21
– Name: DatePubCY
  Label: Publication Date
  Group: Date
  Data: 2025
– Name: TypeDocument
  Label: Document Type
  Group: TypDoc
  Data: Journal Articles<br />Reports - Research
– Name: Audience
  Label: Education Level
  Group: Audnce
  Data: <searchLink fieldCode="EL" term="%22Higher+Education%22">Higher Education</searchLink><br /><searchLink fieldCode="EL" term="%22Postsecondary+Education%22">Postsecondary Education</searchLink>
– Name: Subject
  Label: Descriptors
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22Mathematics+Education%22">Mathematics Education</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+Instruction%22">Mathematics Instruction</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+Materials%22">Mathematics Materials</searchLink><br /><searchLink fieldCode="DE" term="%22Textbook+Evaluation%22">Textbook Evaluation</searchLink><br /><searchLink fieldCode="DE" term="%22Calculus%22">Calculus</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+Concepts%22">Mathematical Concepts</searchLink><br /><searchLink fieldCode="DE" term="%22College+Mathematics%22">College Mathematics</searchLink>
– Name: DOI
  Label: DOI
  Group: ID
  Data: 10.1080/0020739X.2024.2348145
– Name: ISSN
  Label: ISSN
  Group: ISSN
  Data: 0020-739X<br />1464-5211
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: This study explores opportunities to learn definite integrals in three widely-used textbooks in the US, Definitions, worked examples, and exercise problems were coded using research based cognitive resources in definite integrals to examine if widely-used textbooks provide students with opportunities to explore how two quantities are related to show area under a curve or other contexts. The results show that initial introduction, worked examples, and exercise problems all provide limited opportunities for students to explore the multiplicative relationship between two quantities (integrand and differential) and adding small pieces to understand definite integrals. Although students may be able to solve integral problems using anti - derivatives, it is possible that they may not understand the definite integral conceptually (knowing why the definite integral gives the area under a curve) or be able expand their understanding to other contexts. Implications of these results are also discussed.
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  Data: As Provided
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  Label: Entry Date
  Group: Date
  Data: 2025
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  Data: EJ1483296
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      – Type: doi
        Value: 10.1080/0020739X.2024.2348145
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      – Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 21
        StartPage: 1669
    Subjects:
      – SubjectFull: Mathematics Education
        Type: general
      – SubjectFull: Mathematics Instruction
        Type: general
      – SubjectFull: Mathematics Materials
        Type: general
      – SubjectFull: Textbook Evaluation
        Type: general
      – SubjectFull: Calculus
        Type: general
      – SubjectFull: Mathematical Concepts
        Type: general
      – SubjectFull: College Mathematics
        Type: general
    Titles:
      – TitleFull: How Do Widely-Used Calculus Textbooks Introduce the Concepts of Definite Integrals?
        Type: main
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      – PersonEntity:
          Name:
            NameFull: Dae S. Hong
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            NameFull: Dennis Kwaka
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          Dates:
            – D: 01
              M: 01
              Type: published
              Y: 2025
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              Value: 0020-739X
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              Value: 1464-5211
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            – TitleFull: International Journal of Mathematical Education in Science and Technology
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