The Area Enclosed by a Function Is Not Always the Definite Integral: Relearning Through Collaborative Transitioning Within a Learning-Support Module

Learning-support system is an umbrella term that we use for digital resources that assign students with mathematical questions and give automatic feedback on the inserted answers. Transitioning between questions and feedback is characteristic to students’ work with such systems. We apply the commognitive framework to explore the role of within-system transitions in students’ mathematics learning, with a special interest in what we term as “reroutinization”—a process of repeated development of conventional routines to be implemented in already familiar mathematical tasks. The study revolves around a digital module in integral calculus, which was designed to support undergraduates with finding areas enclosed by functions. The data comes from dyads and triads of first-year university students, who collaboratively interacted with the module. The analyses cast light on how transitioning within the module aided students to review familiar routines, amend them, confirm, and solidify the amendments. The transition process was not always linear and contained instances of students cycling back and forth between the assigned questions and feedback messages. We conclude with the discussion on the module’s design that afforded reroutinization and suggest paths for further research.

By offering a screen on which we and our students can express our aspirations and ideas, the computer can help to make explicit that which is implicit, it can draw attention to that which is often left unnoticed. […] The computer, as we shall see, not only affords us a particularly sharp picture of mathematical meaning-making; it can also shape and remould the mathematical knowledge and activity on view. (p. 5) On some meta-level, this quotation echoes Papert (1980), who argued that Turtle Geometry, or computers in general, can serve as "vehicles" for "mathematical ways of thinking." But the argument by Noss and Hoyles appears more nuanced. Our reading of it is that a computer can act not only as the Uber Eats that delivers "mathematical knowledge and activity" to hungry minds and then leaves, but it can also stay to "shape and remold" students' mathematics in a systematic manner.
Such shaping and remolding align with a tutoring function of a computer (Taylor, 1980), a function that has been extensively studied as part of "computer-aided instruction," "computer-based instruction," "computer-aided learning," "computerbased training," and "tutoring systems" (for a review, see VanLehn, 2011). Within this function, students interact with a computer-based system or a digital resource that presents questions and hints and provides feedback on the plugged-in answers. For instance, when asked to find the area enclosed by a graph of a function, students can work out a number on a scrap paper before entering it into the system. Then, they are either congratulated on their success or provided with feedback and asked to try again. Tutoring of this sort can be traced back to the 1980s (e.g., Anderson et al., 1985;Bierman et al., 1989;Sleeman & Brown, 1982), when the affordances of contemporary systems are substantially more advanced. For instance, the modules developed within these systems reside on-line and can be accessed from any digital device.
Today, students can be invited to work within a module, create and manipulate visuals, use natural language to communicate with an artificial agent, and get feedback on intermediate steps in different registers (e.g., Sangwin, 2013;VanLehn, 2011). Furthermore, while it is generally challenging to provide effective feedback that supports students' learning, mathematics education research has documented various ways of thinking that students often develop in particular content areas (e.g., Grundmeier et al., 2006;Kontorovich, 2020Kontorovich, , 2021Orton, 1983;Rasslan & Tall, 2002). Automatically inferring these ways of thinking from students' digital activity enables timely feedback to be given with targeted explanations and clues.
Multiple studies have reported on the positive impact of tutoring systems on students' mathematics learning (e.g., Barana et al., 2021;Steenbergen-Hu & Cooper, 2013). In their recent meta-analysis of 92 studies on learning of mathematics and science, Hillmayr et al. (2020) found that tutoring systems with adaptive features had the largest effect on students' learning outcomes, while "less intelligent" systems produced smaller but still medium gains. These positive findings summon indepth studies on how mathematics learning comes about with systems of this sort. We were able to locate only a handful of such studies in the mathematics education literature (Dorko, 2020;Fujita et al., 2018). Balacheff and Kaput (1996) propose that, in tutoring systems, student-computer interactions are, "based on a symbolic interpretation and computation of the learner Digital Experiences in Mathematics Education (2023) 9:  input and the feedback of the environment, is provided in the proper register allowing its reading as a mathematical phenomenon" (p. 470). In terms of this DEME Special Issue, Balacheff and Kaput's proposal can be read as within-system transitions are a key characteristic of students' interaction with tutoring systems. Accordingly, in the study in hand, our overarching aim is to investigate the role that transitioning within systems of this sort can play in students' mathematics learning. We embark on this investigation with the commognitive framework and data from first-year university students, who collaboratively engaged with a module in integral calculus.

Background
This section consists of three parts. The first one is concerned with tutoring and learning-support systems. The second part overviews selected approaches to learning from feedback. The last part presents the mathematics in the focus of our study. We mobilize on each part to present related aspects of the research project from which this study originated.

From Tutoring Systems to Modules That Support Students' Learning
The interest in tutoring systems can be traced back to the computer-aided instruction movement. Lawler and Yazdani (1987) maintain that such systems communicate "some knowledge through computer facilities which may employ domain-specific expertise, error analysis, and user modeling. Their user-oriented intelligence is controlled by instructional strategies, which present problems and then test for understanding of the content" (p. ix). The researchers highlight that the strength of such systems is in their completeness-the possibility to embody well-articulated curriculum and explicit theory of instruction in sets of rules that will be automatically employed by the computer through self-contained modules. Balacheff and Kaput (1996) construe tutoring systems as antipodes of microworlds. These researchers describe microworlds as opening the doors to new mathematical realms that can be explored freely, when tutoring systems "provide students with strong guiding feedback" (p. 483). Regarding the latter, Papert (1980) went as far as to suggest that "the computer is programming the child." Similar juxtapositions have been made in the broader area of computer-supported education (e.g., Lawler & Yazdani, 1987). Overall, many mathematics education researchers highlight that tutoring systems can lead students towards target performances, but they cannot ensure the type of learning that takes place on the way (e.g., Noss & Hoyles, 1996). This has been often explained using pre-designed feedback that is not sensitive to students' progress and with the fact that "the student can learn how to optimize the use of the tutor feedback instead of the knowledge the task is supposed to convey" (Balacheff & Kaput,p. 483). At that time, Taylor (1980) postulated that "tutor mode computing should not have a significant place in education" (p. 250).
Let us make three comments as a response to the above-mentioned reservations. First, we concur with Brown et al. (2005), who maintain that: ICT [information and communication technology] is just another, albeit very powerful, resource which you, as a mathematics teacher, will need to consider when planning work for your students inside and outside mathematics lessons.
[…] The test of whether it makes sense to deploy ICT is a simple one: "Does it benefit the students' effective learning of mathematics?" (p. 5) As we mentioned at the beginning, there is ample evidence pointing at the benefits of tutoring systems in various disciplines and mathematics included (for metareviews, see Hillmayr et al., 2020, andSteenbergen-Hu &Cooper, 2013). These benefits seem inseparable from the technological advancements of contemporary systems, compared with those that were criticized originally.
Second, we believe that the affordances of any instructional resource should be considered against established goals and educational circumstances (e.g., Brown et al., 2005). In the university setting, first-year mathematics courses bring together large number of students from substantially different mathematical backgrounds. By the end of these courses, which are often lecture-based, fast-paced, and contentintense, students are expected to become fluent with rather complex and abstract topics. To support students in reaching these targets, many university teachers capitalize on the affordances of ICT. More often than not, this support materializes into teachers designing modules (sometimes also called "quizzes") within the course learning management system (e.g., Canvas, Moodle) and engage their students with them as part of formative assessment (e.g., Engelbrecht & Harding, 2005;Kinnear, Wood & Gratwick, 2022a, 2022bSangwin, 2013). These modules frequently provide students not only with marks but also with explanations and hints on how particular questions should be tackled. In this way, engaging with mathematical content through tutoring-like modules is part of the learning reality for many university students these days.
Third, we note that many educators' reservations about tutoring systems revolve around how students engage with them. The tacit assumption seems to be that students work individually and act as passive tutees, who passage in silence from one screen to another. This article grows from an on-going research project, where firstyear university students engage with a digital module in small groups, work collaboratively, and articulate and reason their thinking to each other. We encourage the students to capitalize on the module's questions and targeted feedback, in order to make sense of the underpinning mathematics rather than to "game the system" (cf. Kinnear et al., 2022b). To paraphrase Papert (1980), in our project, the module is not "programming the children," but rather constitutes an element in a complex and rich environment, designed to support their learning (cf. Noss & Hoyles, 1996). The study of students' work in such learning-support modules is also consistent with a recently introduced research agenda for e-assessment in undergraduate mathematics (Kinnear et al., 2022a).
Only a handful of studies in mathematics education have explored learning as it unfolds with modules of the described sort. Fujita et al. (2018) report on a module that was designed to assist students with their first steps in deductive reasoning in school geometry. In the university setting, Kinnear, Wood and Gratwick (2022b) describe an on-line course that has been delivered in STACK-System for Teaching and Assessment using Computer algebra Kernel (Sangwin, 2013). The course focused on high-school-level fundamentals of algebra and calculus, and it was offered as a complementary option for first-year students of linear algebra. The course materials were coherently organized in digital modules that practically "put the textbook into the quiz" and presented full feedback on students' answers on each attempt. The course evaluation demonstrated that the students made large gains from the pre-to post-test, when some students commented on greater depth of knowledge from studying in this way.
We acknowledge the work of Dorko (2020), which provides arguably the most indepth glimpse that we have come across into undergraduates' engagement with what we called learning-support modules. That study characterizes the processes that nine Calculus II-students went through when completing homework assignments in the MyMathLab platform. The findings point at the cyclic nature of students' individual activity, which consisted of reasoning, submitting answers, obtaining feedback from the module, reasoning more, and submitting another answer. Notably, Dorko argues that the on-line platform supported the cycled nature of this activity by verifying answers and providing students multiple attempts for each problem.

Learning from Feedback
A distinctive feature of learning-support modules is the elaborated feedback that they can provide. Many researchers argue that such feedback is effective when it provides learners with information for altering the gap between their current and desired state (e.g., Hattie & Timperley, 2007). In turn, Sadler (1989) proposes that feedback is effective when it is used effectively. The researcher delineates three learner-centered conditions that need to be in place for this to happen: (i) the learner needs to realize the standard being aimed for; (ii) they have to be able to monitor the discrepancies between their actual performance and the standard; (iii) they should be able to engage in appropriate action which leads to a reduction of the gap. Sadler argues that these conditions must be satisfied simultaneously rather than sequentially. The importance of Sadler's perspective is in expanding the research focus to include students as key actors in learning from feedback.
Contemporary research advances in a similar direction, foregrounding the processes of students' interaction with feedback. Indeed, many researchers agree that learning from feedback involves the construction of meanings, discussing feedback with others, and connecting it with prior knowledge (e.g., Barana et al., 2021). For instance, Nicol (2019) argues that "inner feedback is inherent in any use of external feedback. Whenever external information or advice is provided by teachers, this has to be turned into inner feedback if it is to influence subsequent learning and performance" (p. 73). He notes that conceptualizing feedback as an internal process is not new, but it has not yet become central in research. We endorse Sadler's and Nicol's perspectives on feedback in the context of learning-support modules.

Finding Areas with Definite Integrals
The mathematical focus of this study pertains to integral calculus. Orton (1983) was probably the first study to document students' issues with the area-integral relations. Some of these issues emerged from a routine question asking students to find the area between the function y = 2x − x 2 and the x-axis over the interval [0,3] . Note that this function intersects the axis within the interval and creates two regions: one below and one above the curve. In that study, 33 out of 110 participants were convinced that the integral must be calculated in separate parts, but could not explain why. A further seven students separated between two integrals to compute the contributing areas, but suggested that they could work out a single integral instead. Five additional students maintained that a single integral ∫ 3 0 yields the answer. More recent studies point to similar issues with students' use of definite integrals (e.g., Grundmeier et al., 2006;Rasslan & Tall, 2002;Kontorovich, accepted, Kontorovich & Li, accepted).
The above question is representative of a larger class that is common to secondary school and first-year calculus-questions, asking students to compute the area formed by graphs of single-variable functions and axes. The routineness of these questions should not overshadow their mathematical significance. From Archimedes and ancient Greek mathematicians, through Valerio, Kepler, and Cavalieri, to Newton and Leibniz, calculus development has been instigated by the interest to measure "curvy" objects (for a short review, see Rosenthal, 1951, andThompson, 1994). And this significance goes beyond mathematics: computing the total distance travelled by an object given its velocity function, calculating the overall invested work based on the applied forces, and deducing the size of a bacteria population from its rate of growth-these are only a few contextualized instances where it is instrumental to find the "area" of the represented quantity in its totality and avoid a common claim that "the area is simply a definite integral." As part of our project, we analyzed answers of nearly twelve thousand students to the types of questions described above that appeared in 17 multiplechoice final exams (Kontorovich, accepted). Nearly 30% of the students' selected answers were consistent with taking a single definite integral, while this proportion varied between 14 and 64% over the exams. In five exams, answers of this sort were chosen by more students than the conventional ones. A complementary analysis of students' full solutions to homework assignments confirmed that 53% of the areas were computed with a single definite integral. These results instigated us to develop a learning-support module aimed to ensure that students employ conventional approaches to compute areas enclosed by functions.

Theoretical Framing
This section starts with an overview of the commognitive framework before using it to frame the study.

Commognition in a Nutshell
We turn to the commognitive framework (Sfard, 2008) for theoretical foundations and analytical tools. Being interested in a rather nuanced mathematical issue and its learning, it seems justifiable to capitalize on a framework that "has been developed within the field of mathematics education and is designed to address the problems arising in this field" (Morgan, 2020, p. 226). In university mathematics education, commognition has been acknowledged for its capability to account for the complexity of this context (Karavi et al., 2022;Kontorovich, 2021;Kontorovich et al., in press;Nardi et al., 2014) and has been used to study teaching and learning of calculus (e.g., Kontorovich et al., 2019;Pinto, 2018). Several recent studies have considered students' collaborative work with digital resources through the commognitive lens (e.g., Baccaglini-Frank, 2021;Ng, 2019).
Commognition maintains that mathematics as a whole and its particular disciplines (e.g., calculus) can be construed as a discourse. Discourses are defined as "different types of communication, set apart by their objects, the kinds of mediators used, and the rules followed by participants and thus defining different communities of communicating actors" (Sfard, 2008, p. 93). Operationally speaking, discourses are distinguishable through keywords (e.g., "area," "definite integral") and their use, visual mediators (e.g., a graph of a function, a diagram of an enclosed region) and their use, narratives endorsed by the mathematical community as true , and routines (e.g., computing the enclosed area). Commognition posits that one's participation in a discourse is a patterned activity, the communicational regularities of which are defined as a personal discourse. Part of this discourse unfolds in a public sphere where a person communicates with others; other parts constitute one's thinking, which is defined as self-communication. Sfard (2008) defines learning as a lasting change in one's discourse, i.e., in at least one of its four characteristics. Lavie et al. (2019) focus on the processes that learners go through when developing discursive routines and argue that "routinization of our actions is what learning seems to be all about" (p. 156). The researchers suggest that this process occurs through learners individualizing mathematical routines that are rendered conventional. Operationally speaking, they introduce the notion of a task situation to refer to "any setting in which a person considers herself bound to act-to do something" (p. 159). Then, they propose that one's capability to act in a task situation is afforded by precedents-past situations that are interpreted as sufficiently similar to the current one to replicate what was done then, either by the person themselves or by someone else. This theoretical account leads the researchers to the following operationalization of a routine: "a routine performed in a given task situation by a given person is the task, as seen by the performer, together with the procedure she executed to perform the task" (p. 161; our italics).
Commognition distinguishes between ritualistic and explorative versions of a routine (e.g., Sfard, 2008). A routine is said to be implemented ritualistically when its execution has been motivated by social reasons (e.g., to please the teacher or a peer). In such a case, the routine's outcomes are of a little interest for its implementor, and the execution of the procedure constitutes the gist of the task. In turn, a routine is considered as an exploration when it is applied for the sake of the result that it entails, for instance, a new narrative about a mathematical object. Lavie et al. (2019) argue that it is rare for routines to feature in purely ritualistic or explorative forms in a didactical context, and thus, the researchers talk about the two as edges of a scale. They propose that newcomers to a mathematical discourse develop rituals, which can later evolve into explorations, and suggest attending to six characteristics of a routine to trace its gradual transformation from a ritual to an exploration. These characteristics are flexibility-diversification of procedures that one undertakes to pursue a task; bondedness-the implementation of steps, each of which is necessary and feeds into each other; applicability-the expansion of the set of task situations in which one performs a routine; agentivity-the growth in one's autonomous activity; objectification-a quality of narratives that one generates as being about mathematical objects, rather than processes that people undertake; substantiability-one's capability not only to articulate the performed routine, but also justify its implementation.

Commognitive Framing
As noted at the outset, the main aim of our study is to explore the role of students' transitions within a learning-support module in relation to their learning of mathematics. Now, we are in the position to elaborate on this aim in commognitive terms. We view within-module transitions as a dynamic process where students shift between different task situations. The mathematical components of these situations are pre-set by the module, and we distinguish between their two kinds: questions that summon implementation of routines and circumstances where feedback on the performed routine is given. Both kinds locate routines at the center of students' attention and engender learning opportunities-"circumstances that call for, and support, a change in the learner's participation in a discourse, a transformation that would bring him or her closer to the [conventional mathematical] discourse" (Chan & Sfard, 2020, p. 3). Indeed, each question requires students to interpret the assigned task situation, detect relevant precedents, and adapt a procedure to the specifics of the task in hand. An iterated implementation may trigger nuanced changes in the routine performance, gradually shifting it from the ritual to the explorative edge of the scale. In this way, working on the sequence of similar questions has the potential to promote routinization.
In turn, feedback provides students with opportunities to consider the match between the assigned task situation and the implemented routines. Corrective feedback (i.e., information on how well a question has been completed) may be useful to assess whether the applied procedure fits the assigned task. Elaborated feedback can point at possible points of confusion and suggest an alternative course of action. Accordingly, internalization of external feedback can entail revision of familiar routines, trigger the development of new ones, and give rise to broader insights regarding why changes are required. In this way, students' transitioning within a learningsupport module can be construed as a passage through pre-designed opportunities to change their mathematical discourse and, in particular, routines as its smallest manifestation (Lavie et al., 2019). In this study, we explore how students mobilize the design features of the learning-support module and interactions with each other to seize and miss these learning opportunities.
In tune with the last sub-section of the previous section, we focus on routines in Integral Calculus that are needed to compute areas enclosed by graphs of functions that change their signs within the assigned intervals. In the next section, we elaborate on why we expected these routines to be familiar to our participants. Notwithstanding, international research and our own analysis of the local context suggested that it is likely that many students will not perform conventional routines autonomously. Accordingly, our investigation is focused on a particular type of learning, in which students redevelop routines already existing in their mathematical repertoire; we refer to this process as reroutinization.

Method
In terms of IES and NSF (2013), this is an early-stage exploratory research that aims to contribute to core body of knowledge in mathematics education. We embarked on it with the abduction methodology (Peirce, 1955). This mode of inquiry requires identification of a phenomenon of interest and giving rise to an initial theoretical account. Then, the account is supported and refined through a purposeful corpus of evidence (Svennevig, 2001). Svennevig argues that, while being a less than certain mode of inference, abduction compensates with a vengeance by providing new ideas and developments. Moreover, the methodology relies on contextual judgments, which seem necessary to analyze collaborative learning with digital resources.
The data comes from a broader project involving small groups of undergraduates who took first-year courses in Calculus and Linear Algebra. The data was collected in a laboratory setting, where each group sat in front of a single computer screen and worked together on learning-support modules developed in STACK (Sangwin, 2013). Each module consisted of a sequence of questions related to the mathematical content earlier discussed in the courses. Each question appeared on a separate screen, asking students to enter their final answers. Once the students clicked the "check" button, the module automatically assessed their answer and presented feedback that was designed based on previous research.
Specifically, we extracted common issues and mistakes reported in the existing literature, considered what they may look like in our mathematical context, and prepared feedback messages communicating what went wrong and how students may revisit their work (see examples in the next section). Drawing on Dorko (2020) and Kinnear, Wood, and Gratwick (2022b), we designed the modules in such a way that students could progress to the next question without submitting their answer, return to the previous question, and attempt each question several times. The modules were piloted with three individual students who reflected on the modules' structure and clarity of feedback messages. This phase led to another round of redesign, where the formulations of feedback messages were refined to draw students' attention to confusions that could underpin their answers and provide clearer hints regarding conventional approaches.
In this article, we use data from students' engagement with the module on areas. After this topic had been discussed in the relevant courses, an announcement was made in the course learning management system, inviting students to our study. The announcement explained that the students will be provided with an opportunity to enhance their grasp of the course content through engaging with purposefully developed digital materials. Seven students responded to the invitation, and we put them in three groups: a triad-Ely, Ali, and Betty-who studied towards a major in commerce; a dyad-Leah and Manny-who enrolled in a business program, and a pair of future mathematicians.
The first two groups were taking a course for non-mathematics majors that combined topics in Calculus and Linear Algebra. Regarding the routine in the focus of our study, nearly a third of the course syllabus was dedicated to Integral Calculus: anti-derivatives, the Fundamental Theorem of Calculus, and integration methods. The coursebook introduced definite integrals via Riemann sums, explaining that the definite integral gives the displacement between the x-axis and a function's graph. The integral of the absolute value of a function was proposed as the advised method to find areas, "since it makes all the negative parts positive again." This method was illustrated with a worked example featuring a function that changes its signs in the assigned interval. Being aware of students' common confusion with areas and definite integrals, the course lecturer made explicit attempts to support the course cohort in developing a conventional grasp of the area-integral relationships.
Each group session lasted for 40-55 min, when we video-recorded students' interactions, digital activity, and written artifacts. Then, we transcribed the recordings and embarked on their analysis with two questions: (i) What developments in students' routines for determining areas enclosed by the graphs of functions can be observed? (ii) How do these developments relate to students' interactions with questions, feedback, and each other as part of their work with the learning-support module?
To analyze the data, we systematically traced patterns in students' actions and the use of keywords, attended to their engagement with the assigned questions and feedback messages, as well as to the changes in communication that followed.
Students' work on each question was construed as an implementation of some routine. Different implementations were associated with the same routine based on students' words, the undertaken steps, and resulting narratives. Then, similar implementations were examined to characterize the underlying procedures and the tasks. The procedures were constructed by generalizing the commonalities of the executed steps. In some cases, the students explicated the aim of their actions to their peers, which was associated with their task. In other instances, we inferred the tasks based on the constructed procedures, while paying special attention to the end-results (Kontorovich, 2021). Such an inference was inspired by Schoenfeld's (2011) methodological approach to the construct of goal: not necessarily as something that a student consciously wished to achieve, but as an analytical statement with which the observed performance is consistent.
Digital Experiences in Mathematics Education (2023) 9:255-282 Each time a group performed a routine for determining an area, we characterized its implementation on a ritual-exploration scale. Specifically, we attended to the sequences of steps undertaken to pursue the assigned questions (flexibility), the connections between the steps (bondedness), the questions in which the routine was implemented (applicability), students' independent decision-making (agentivity), linguistic features of mathematical narratives (objectification), and the reasoning behind the implementation (substantiation). Tracing the location of routines on the ritual-exploration scale afforded spotting routinization developments as the groups progressed through the module.

Findings
The central finding of this study is that transitioning within the learning-support module assisted reroutinization of our student participants. To be consistent with the abduction methodology, we start with a case that ignited our thinking in this direction (the first sub-section) and continue with elaborations on the central finding: the identified constituents of reroutinization (the second sub-section) and deritualization of revised routines through progression within the module (the third sub-section). For illustrative purposes, we include exceptions from the data and analyze them with a focus on the findings at the center of each section.

Deroutinization and Transitioning Within a Learning-Support Module
As all students in our project, Ely was planned to engage with the learning-support module collaboratively. However, her intended peers were running late, and we decided to let her start working on her own. In this sub-section, we delve into Ely's individually implemented routines and, in the third sub-section, we describe how they evolved once her peers joined her.
The first question in the module asked Ely to find the area between the graph of the function f (x) = 2x − 6 and the x-axis over the interval [0,7] . She read the question for nearly twenty seconds, before reaching for the pen and paper on the desk in front of her. Her written work is captured in Fig. 1a. The figure shows that Ely started with a schematic visual of the assigned interval and what may stand for a graph of a linear function, without delving into the specifics. Then, she turned to calculating the area with the definite integral. Once the answer of "7" emerged, Ely entered it into the module and clicked "check." Fig. 2 presents the feedback message that was triggered. Ely read it for nearly five seconds and said, "Ah, yes, it is positive. OK." Then, she clicked the "next-page" button, which took her to the second question.
In question 2, Ely was given the function f (x) = 48 − 3x 2 and asked to find the area between it and the x-axis over the interval [−3, 10] . She read the question in silence for nearly half a minute and then clicked "Previous page." This page displayed question 1 and the feedback message from before. This time, Ely got closer to the screen, reread the message for nearly twenty seconds, and clicked "try again." Once question 1 reappeared, Ely took her scrap paper, crossed her previous writing as incorrect, and generated a new solution (see Fig. 1b).
The presented account tells the story of Ely who revisited her solution to question 1. From the commognitive standpoint, she engaged in the same task situation twice, each time implementing a different procedure to compute the area enclosed by the graph of the given function and the x-axis over the assigned interval. Similar to the participants of the studies discussed in the last sub-section of the second section, in her first attempt, Ely simply calculated the corresponding definite integral and submitted the emergent result. A different procedure was implemented in her second trial. There, Ely found the function's point of intersection with the x-axis, used this point to divide the interval into two parts, and computed the sum of the absolute values of corresponding definite integrals over each sub-interval. Ely implemented this procedure in the consequent question and built on it in the questions that followed (we elaborate on this in the third sub-section). Accordingly, we propose that the change in her routine was not only profound but also lasting, at least for the time of her engagement with the learning-support module.
To recall, this episode took place after the relations between definite integrals and areas had been discussed in Ely's course. Thus, there are no grounds to claim that we witnessed Ely's learning, in the sense of expanding her discursive "toolkit" with a new routine. Instead, it seems more accurate to suggest that this is a case of reroutinization, where a routine for computing areas was elicited, assessed, and revised. Given that this process unfolded in the context of Ely's work with the learning-support module, it appears reasonable to propose that transitioning through the module, i.e., questions and feedback, played a key role in her reroutinization.
On the face of it, the proposed account appears almost self-evident. Indeed, question 1 put Ely in a task situation where she implemented a familiar routine. The module assessed the entered result and displayed feedback that assisted Ely to correct her routine. Yet, this account is silent regarding how the within-module transitioning supported her reroutinization. Moreover, it does not account for the nuances of Ely's digital path. For instance, she engaged with the feedback message twice: immediately after the submission of her first answer and after encountering question 2. Why would she not attempt to revise her routine right after the module sanctioned her first attempt as a "Good try, but not quite right?" Similarly, what was going through Ely's mind after her careful reading of the second question that made her return to the previous screen? These questions are doomed to remain unanswered. Yet, they showcase that reroutinization can be not a straightforward process brought about by the linear progressions through the learning-support module. In the next two sub-sections, we elaborate further on the relations between reroutinization and students' within-module transitions.

Constituents of Reroutinization
In this sub-section, we present three constituents of reroutinization as they emerged from the analysis of the groups' engagement with the learning-support module. These constituents are reviewing the implemented routine in light of feedback, amending the familiar routine, confirming the amendment, and solidifying it. These constituents emerged from all three groups, but they were especially visible in the collaboration of Leah and Manny.

Reviewing a Routine and Internalizing Feedback
In the first question, Leah and Manny were presented with a linear function f (x) = 4x − 4 and asked to find the area between its graph and the x-axis over the interval [0, 2] . Manny took a scrap paper and embarked on the question in a manner that was not very different from Ely: he wrote the definite integral ∫ 2 0 (4x − 4)dx , computed it to be zero and after receiving Leah's approval, entered "0" as the answer. The module displayed a feedback message similar to the one presented in Fig. 2. Then, the following conversation took place: The turns 1-2 indicate that the dyad interpreted the module's rejection of their answer as evidence of an issue with their implemented routine. This interpretation cannot be taken for granted. Indeed, the rejection could stem from students making a calculation mistake or a technical error on the module's side. Yet, we find the students engaging with the feedback message (see 2b), criticizing their previous work (see 2c and 2e), and even proposing an alternative course of action (see 2d).
The above closely relates to what we term as feedback internalization (cf. Barana et al., 2021)-a discursive process of transitioning to the previous task situation and reflecting on it with an eye to received feedback. We construe this process as an intertwining of three related attributes: • Feedback and its components come to the center of students' physical actions and verbal narratives. For instance, Leah's "there's a difference between area under the curve and a definite integral" (in 2b) appears as an individualized version of "Be careful not to confuse the area between curves with the definite integral" from the feedback message. In this way, we see the group actively searches for "blind spots" in their previous discursive activity. • When working on question 1, the students generated utterances that concerned the assigned function, its variables, and numerical calculations (e.g., "four x squared over two minus four x, which will be two x squared minus four x"). However, as part of the reflection, the particularities of the assigned problem are ignored, and the use of the keywords "areas," "definite integrals," and "curves" shifts Leah's talk to a higher discursive level. This can also be seen in the utterances of Manny, who refers to the particular question as an instance of a larger class (see turn 5). Note that this talk is compatible with how the keywords of "definite integrals" and "areas" feature in the feedback message. Thus, we propose that the students attuned their discursive level to match with feedback. • Leah uses the feedback terminology and narratives to address the dyad's previous work and explain why their answer was sanctioned incorrect (e.g., see 2c-2e). Indeed, while the answer for area emerged from computing a definite integral, this is the first time that the corresponding notions feature in students' communication.

Amending a Routine
In the second question, the dyad was asked to find the area between the graph of the parabola f (x) = 225 − 9x 2 and the x-axis over the interval [−5, 8] . They embarked on this question as follows: The episode ends with Manny adding the two positive numbers and Leah entering the obtained sum into the module.
The turns 1-2 suggest that the dyad recognized the similarity between questions 1 and 2. But their current task situation is qualitatively different from the previous one described. Indeed, now, the students are aware that the answer obtained with their previously implemented routine was rejected, and they have already had a rather elaborated discussion on the possible reasons for this. Notwithstanding, the students implement the same routine once again, wishing that "the right value comes out this time" (see turns 3-6).
Expectedly, the wish does not come true, and in turns 7-14, we witness the dyad redeveloping a procedure for finding areas. This process does not occur at once but unfolds gradually. In turn 7, Leah proposes to compute the focal area in three steps: to calculate the two components with definite integrals and then add the results. Manny comments that this step requires knowing where "the function equals to zero first," finds this point (see turn 8), and Leah uses it to outline the limits of each of the two integrals (see turns 9 and 11). When a negative result is obtained, the students apply the absolute value to make it positive (see turn 12).
In this way, in the process of carrying out an initially outlined procedure, the dyad also refines and unpacks its steps further.
Another noteworthy feature of the redevelopment in-play pertains to the mixture of general and concrete elements in students' utterances. Indeed, the students are grappling with a specific question, and thus, they could be expected to talk about how its specific parameters can be used to reach the answer (like they did in the previous one). However, we find the dyad using phrases and words that stand for rather general mathematical objects, when the particularities of the question are often implied or signaled through deictic "this" and "it" (e.g., "you calculate this definite integral" in turn 7, "where does it cross x" in turn 9, "we'll make an absolute value of that" in turn 12). Such mixed talk can be explained with the students realizing that while they are coping with a certain task situation, they also seize an opportunity to revise their discourse and develop a routine that will be useful in a range of similar tasks. In other words, this talk can be viewed as a preparation for coping with similar task situations in the future.
Overall, we refer to the described discursive process as routine amendment. We use "amendment" rather than "developing a new routine," for example, for two complementary reasons: first, while the steps that the students implemented are different from the ones they took earlier, their task remained intact-to determine the measure of the enclosed area. Second, note that the students still implement their previous procedure as part of the new one. In other words, the steps embedded in each of the two procedures overlap to a considerable extent.

Confirming and Solidifying an Amended Routine
A careful look at the previous episode suggests that the students were not very sure in the correctness of the amendments that they introduced to their routine. This is visible in Manny's questioning intonation in turn 12 and in Leah's comment in turn 13, implying that there is no harm in letting the module assess their answer. Once their answer has been approved, both students enthusiastically exclaimed, "Yes!" and Manny added that, "Now we know how to do it!". Recall that, in the previous reroutinization phase, the dyad interpreted the module's rejection of their answer as a marker of a problem with their routine. Here, we witness an antecedent phenomenon, where the dyad views the sanctioning of the entered answer as confirmation of their amended routine. 1 Interestingly, after this confirmation, the dyad decided not to progress to question 3 but transition back to the first question "to see if we can fix it." There, we find Leah and Manny implementing the amended routine in a considerably more confident manner. This is evident in Manny quickly finding the line's point of intersection with the x-axis, Leah subdividing the interval into two parts, and Manny promptly computing the definite integrals. There are almost no instances of them consulting one with another, and their communication appears efficient and mathematics-focused. 2 The module sanctions the students' answer, and the students sound satisfied. Leah adds "we are on the roll now" and "well, this is going to save a whole lot of revision time for the exam." We suggest that through transitioning to their previously attempted question, the students solidified their amended routine in a two-fold sense: (i) they became more confident in the applicability of their routine to the whole class of task situations that questions 1 and 2 instantiated. Indeed, in spite of feedback, their own substantiations, and the module's confirmation of their revisited answer to question 2, no mathematical authority has confirmed their amended procedure. Hence, its repeated implementation and confirmation raises the chance that the amendment is mathematically valid. (ii) To use an amended routine, its procedure needs to be routinized, i.e., to become come a part of students' mathematical discourse. Hence, by implementing the routine once again, the students realized the opportunity to rearticulate its steps and tailor them to the particularities of another task situation.
Lastly, let us stress a key role of students' cycling back to question 1: to mobilize the affordances (i) and (ii), the dyad needed a question where this amendment would be applicable. Given that the connections between the two questions appeared selfevident to Leah and Manny, it only makes sense for them to revisit a task situation where their previous routine failed them.

Further Development of an Already Confirmed Routine
Based on the episodes presented up to this point, one may over-hastily assume that reroutinization ends with students solidifying their amended routine. To illustrate that this is not necessarily the case, we return to Ely and specifically to her work with Ali and Betty. Both joined Ely when she just started working on the third question.
In that question, the triad was asked to find the area enclosed by the function f (x) = −sinx and the x-axis over the interval − 2 , 2 . Consistent with the first subsection, Ely proposed to go with her recently amended routine. In turn, Ali instantly asked whether they need to sketch a graph. Ely responded, "I was just thinking about doing it like I did before. But I suppose if we draw a graph it will work too". After a collaborative attempt to sketch the graph of −sinx , the group delegated the computation of the focal area to Ely. Her written work is captured in Fig. 4.
The module sanctioned the obtained answer and the students turned to question 4. It asked about the area between f (x) = cosx and the x-axis over the interval − , 3 2 . The students instantly dubbed this question as "the same" and Ely started considering where the function intersects the horizontal axis. Ali revoiced the graphical approach and sketched the graph of cosine, and the group denoted the function's zeros (see Fig. 5). Then, the following conversation occurred: Then, Ali writes down the solution that Ely dictates to her. Ely is the one to compute the definite integral and multiply the result by five.
In the presented exchange, Ely plays a leading role in finding the focal areas, which tempts us to consider how her routine changed compared with what we saw in the first sub-section. Before delving into details, let us acknowledge that doing mathematics in solitude and with others constitute qualitatively different task situations. Thus, one could reasonably argue that no significant developments could take place if Ely continued working on her own. Indeed, at the beginning of this exchange, we see her sharing that she planned to use her routine "as before." If no calculational mistakes were to occur, the module would sanction her answers, offering Ely no external prompt to change her deeds. That said, we propose that by transitioning through a sequence of similar task situations, Ely and her peers were provided with an opportunity to develop their routine further.
The central development in Ely's routine pertains to her use of diagrams. In the first sub-section, we saw her constructing images as part of her solution, but these appeared as rough sketches rather than attempts to accurately depict the mathematics in the focus of the questions. It was also unclear whether and how Ely used her diagram in question 1. This is slightly different from question 3, where the triad generates a relatively accurate graph of the assigned function. Yet, Ely takes a marginal part in this generation and most of her utterances revolve around finding the function's points of intersects with the x-axis-a step that was also part of her routine beforehand (see Fig. 1). When computing the area, Ely's discursive activity contains no evidence of drawing on the constructed diagram. In commognitive terms, we propose that Ely's participation in diagram-building was ritualistic. Indeed, there is only a loose bond between the diagrammatic part of the solution and the routine that she implemented to compute the focal area. Ely's half-hearted agreement that "if we draw a diagram it will work too" appears to be driven by social factors (e.g., not to contradict Ali) rather than by seeing a mathematical value in generating an image.
Question 4 marks a substantial change in Ely's use of diagrams. There, we find her shading the focal area (see turn 1) and narrating about the diagram's elements ("agentivity," in terms of Lavie et al., 2019). Specifically, she capitalizes on the visual of the area to compute its measure. This capitalization occurs in two steps: in turns 1-8, Ely notices that the whole area can be computed through drawing on its single part (e.g., "from negative pi over two to pi over two" in turn 6) and "then times it by two and a half" (see turn 3). In the second step, Betty points out that this procedure can be optimized further by drawing on an even smaller part of the area (see turn 9). Ely instantly endorses this suggestion and focuses on the area over the interval − , − 2 . This step is bonded to the new procedure that Ely dictates to Ali: instead of computing the sum of absolute values of three definite integrals-as she would as part of her amended procedure, she obtains the answer with a single definite integral times five. Moreover, note that Ely not only narrates about the relations between different constituents of the area but also substantiates them (e.g., turns 6, 11, and 12). To summarize, in this task situation, we see Ely using the diagram in a way that is more agentive, bonded, and substantiated, which we interpret as the diagram turning into an explorative component of her routine.

Summary and Discussion
This study was concerned with reroutinization as it unfolded in the context of a STACK-based module, which was designed to support first-year students' learning of Integral Calculus. Focusing on routines in relation to STACK is in tune with the broader literature on tutoring systems (e.g., VanLehn, 2011) and mathematics education research (e.g., Sangwin, 2013). For instance, Kinnear, Wood, and Gratwick (2022b) reflect that they were able to design a course with extensive usage of STACK-based quizzes, at least partially due to the procedural nature of the course. In a somewhat similar fashion, we used this system to support students' development of routines for finding areas enclosed by functions. The literature suggested that these routines may not be a part of students' mathematical repertoire, even after formal classroom instruction (see the end of the second section). To paraphrase Noss and Hoyles (1996) from the beginning, we capitalized on digital affordances of the system, "[to] help to make explicit that which is implicit […and] draw attention to that which is often left unnoticed" (p. 6).
We are not apologetic about our interest in students' routines for two reasons. The first one stems from the practice of university mathematics education. Tallman et al. (2016) showed that routine exercises dominate final exams all across the USA. Similar observations can be made on a broader scale, based on the studies from other countries that provide a glimpse of typical questions in undergraduate classrooms in different mathematics areas (e.g., Kontorovich, 2018aKontorovich, , 2020. Thus, whereas the situation where students are expected to implement conventional routines is far from desirable for many educators, it nevertheless constitutes a mathematical reality for many undergraduates. The second reason stems from our theoretical framework. To borrow from Lavie et al. (2019), "an important advantage of [routine] is that it is defined operationally. Unlike the notion of concept, the idea of routine has been described in terms of features accessible to the observer, at least in principle" (p. 172). In other words, from the commognitive standpoint, routines constitute the smallest manifestations of one's mathematical discourse, and thus, analyzing their development is tantamount to getting access to students' learning. This is also the place to acknowledge noncommognitive research that has identified close and reciprocal relations between procedural skills and conceptual understanding (e.g., Kieran, 2013;Rittle-Johnson & Koedinger, 2009).
This study enriches a rather thin line of research, tapping into mathematics learning that occurs when students engage with what we termed learning-support systems (e.g., Dorko, 2020;Fujita et al., 2018;Kinnear, Wood & Gratwick, 2022b). The main thesis that we put forward suggests that collaborative transitioning within a learning-support system can aid students' reroutinization. Before discussing this thesis in detail, let us note that all the reroutinizations paths that we observed in our project unfolded without the groups turning to external resources, such as the web or course materials. This observation is at odds with Dorko's (2020) laboratory study, where students engaged with lecture notes, instructional videos, and forums when working on on-line homework.
One explanation for our students' willingness to transition within the module, but not beyond it, may relate to what Sfard (2008) terms "meta-rules" that students tacitly impose on their own mathematical activity in a particular setting. While we did not discuss the use of external resources explicitly with our participants, it is possible that they operated under the assumption that the offered module would provide all the help and feedback that they needed. Indeed, recall how in their first problem, Leah and Manny looked for the "help thingy" in the module, but did not take the textbook out of their backpacks, because "this would be cheating." Such meta-rules can relate to students' broader image of the activity that the laboratory setting comes to emulate.
In Dorko (2020), the activity was a homework assignment, where drawing on external resources appeared legitimate and expected; in our case, the module was designed to assist students with their preparation for the forthcoming closed-book exam (see the Method section). Future research can explore meta-rules that students impose upon their mathematical activity within a learning environment, with an eye to the use of digital resources as elements in this environment.
The commognitive analysis of students' collaborative work resulted in the delineation of three constituents of the reroutinization process: reviewing a familiar routine and internalizing feedback; amending the routine; confirming the amendment and solidifying it. These categories echo Dorko's model of students' work on online homework, with a special focus on changes in students' mathematics alongside their transitions between the elements of the learning-support module. For instance, after the module rejected Ely's and Manny and Leah's answers, the students revisited their routines, drawing on the displayed explanations and cues (see the Findings section). Speaking about feedback, we draw attention to the students' interpretation of the module's confirmations and rejections of final answers as a testimony to their routines (see Dorko, 2020 for a similar finding). In most of our cases, such interpretations were appropriate and supported students' development of conventional mathematics. Nonetheless, the students in our project rarely checked their own calculations, went over each other's work, or expressed doubts in the module's assessment of their answers (see Kontorovich, 2019, for the notion of epistemological status of students' solutions and Misfeldt & Jankvist, 2018, for techno-authoritarian external conviction proof schemes). And while more research is needed to establish how common such interpretations are among students, it is not difficult to imagine a scenario where a perfectly viable routine is renounced, due to corrective feedback that the learning-support module provided on the routine's outcome (see Kontorovich, 2018b, for a materialization of this scenario in an interview setting). This scenario calls for special care when designing corrective and elaborated feedback (e.g., Barana et al., 2021;VanLehn, 2011). It also raises questions about how students can be prepared to act on automatic feedback, in ways that support their learning (e.g., Jivet et al., 2020).
Another noteworthy finding pertains to the non-linearity of students' passages between the elements of the learning-support system-non-linearity which appears inseparable from the system's design. This design allowed moving between questions without submitting answers, returning to previously presented feedback and attempting questions several times. We see possible connections between these affordances and the constituents of the reroutinization process. Indeed, the importance of internalizing feedback and the willingness to amend an existing routine are both likely to increase once the students realize that they will need to apply the same routine more than once. This is what we saw in the case of Ely, who did not appear to revise her routine instantly after the module rendered her answer incorrect, but who returned to the same feedback after an encounter with a similar question. Somewhat similarly, Manny and Leah expressed doubts about their routine, but still implemented it in the subsequent question before embarking on the project of routine amendment. Furthermore, we earlier argued that cycling back to a previously attempted question affords students to solidify their routine's amendments and become more certain that these amendments are mathematically valid.
The findings on students' non-linear passage within a learning-support system cast light on the design features of the system that afforded these passages. Accordingly, studying the role of passage affordances within digital resources in the context of students' mathematics learning may be an interesting avenue for future research. From the practical perspective, we encourage teachers to be attentive to this design feature and consider setting up learning-support systems in ways that enable students to navigate their digital paths agentively. That said, we acknowledge circumstances where it is didactically reasonable to structure students' digital paths.
Our last point focuses on reroutinization that Ely went through. On the face of it, the project of her reroutinization can be rendered complete on her second attempt of the first question, after she internalized feedback and amended her initial routine. However, the third sub-section unpacks further developments in her amended routine, which can be associated with its growth in mathematical efficiency and sophistication. We saw similar developments among other students as well. We view these cases as a reminder that reroutinization pertains not only to a change in the steps one takes to execute a procedure but also to qualitative shifts in how a routine is implemented. These shifts can be associated with a gradual development of a routine from a ritual to an exploration (cf. Lavie et al., 2019).
We are convinced that these developments are inextricable from Ely switching from individual to collaborative work with the module. The affordances of collaborative learning have been widely recognized (e.g., Resnick et al., 2015), and we are looking forward to future research on how learning of this sort comes about with learning-support modules. Furthermore, it appears almost self-evident that the developments in Ely's routine took place thanks to the module opening the space for them to happen, i.e., by offering a sequence of questions in which students could implement the same routine. This module's design stemmed from our commognitive standpoint, positing that "repetition is the gist of learning" (Lavie et al., 2019, p. 153). That said, we are also mindful of the criticism of tutoring systems, including Taylor's (1980) position on the place that tutor-mode computing should, or more precisely "should not have" in education. And while this study is too early-stage and explorative to offer suggestions for implementation in teaching, it opens the door for further research into the potential of advancing students' reroutinization with learning-support systems in small groups.
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