Designing example-generating tasks for a technology-rich mathematical environment

This paper provides some insights into the use of example-generating tasks in the design of a technology-rich learning environ-menttoenhancestudents’mathematicalthinking.Thepaperreportsonanearlystageofadesign-basedresearchprojectconcerningthe designoftasksandassociatedfeedbackutilisingtheaffordancesprovidedbyacombineduseofadynamicmathematicssoftware environmentandacomputer-aidedassessmentsystem.Inexample-generatingtasks,studentsareaskedtogenerateexamplesthatfulfil certainconditions.Basedondataintermsofexamplesgeneratedby491first-yearengineeringstudents,takingafirstcourseincalculus, thepaperexaminespatternsofstudentresponsetothreeexample-generatingtasks.Asatheoreticallens,thenotionsof dimensions of possible variation and associated ranges of permissible change are used. In light of the observed patterns, the paper provides some guiding principles for designing example-generating tasks and associated formative feedback to foster students’ mathematical understanding by enriching their example spaces. For example, this paper illustrates occasions where it might be instructive to start by asking for two examples, followed by adapted feedback before requesting a third example.


Introduction
Recent decades have seen a rapid development of technologies that support teachers in the time-consuming work of giving feedback to students by offering automated correction of student responses.A common notion for this type of technology is computer-aided assessment (CAA) systems.Today, many mathematics courses in higher education utilize mathematically sophisticated CAA systems, such as STACK and Möbius (Kinnear et al., 2022).So far, CAA systems have mainly been used for assessing basic mathematical procedural skills; it is a challenge to design tasks for a CAA system that address higher-order skills in mathematics (Rønning, 2017).
In this paper, we address this challenge by investigating the potential of designing CAA tasks based on the pedagogical approach of prompting students to generate examples that fulfil certain conditions.Researchers suggest example-generating tasks as a way to engage students actively in their development of deeper mathematical understanding (Bills et al., CONTACT Maria Fahlgren maria.fahlgren@kau.se2006; Mason & Watson, 2008).Since there are no general methods for solving these types of task, students have to be creative and develop solution strategies building on conceptual understanding (Antonini, 2006).Furthermore, by prompting students to provide more than one example fulfilling certain conditions, they are encouraged to reflect on how the first example could be varied without destroying these conditions (Watson & Mason, 2005).This pedagogical approach has been adopted by researchers in the creation of novel types of tasks appropriate for CAA systems, since it allows for automatic assessment of higher-order mathematical skills (Kinnear et al., 2022;Yerushalmy et al., 2017).
We had occasion to elaborate on the design of example-generating tasks as part of a broader design-based research project aiming to investigate how the combined use of a CAA system and a dynamic mathematics software (DMS) environment can support the development and assessment of higher-order mathematical skills.The context for this project is a first-year engineering calculus course.Typically, the example-generating tasks developed prompt students to provide two examples of functions that fulfil certain conditions.As a design principle, we explicitly ask students to use a DMS environment to verify their suggested examples (in the form of function formulae).The findings reported in this paper will contribute to the development of design principles for tasks and associated feedback in a relatively new research area: computer-aided assessment of higher-order mathematical skills at the university level (Kinnear et al., 2022).
The design of tasks as well as the analysis of student responses have been guided by Watson and Mason's (2005) theory of example spaces.In particular, two theoretical ideas denoted as dimensions of possible variation (DofPV) and ranges of permissible change (Rof-PCh) have been in focus.Although the idea of using the theory of example spaces as a research tool has been introduced in earlier work (Zazkis & Leikin, 2007), there is a lack of literature on the use of DofPV and RofPCh in analysing student responses.In this sense, the paper contributes by providing a model for how these theoretical ideas can help in the analysis and design of example-generating tasks.
The paper first gives a brief overview of the literature on the technologies used in the project-that is, CAA systems and DMS environments.Next, a section on example spaces as a theoretical lens is introduced, followed by literature on the use of example-generating tasks.Then, the objective of this paper is articulated, including the research question.The paper ends with sections on method, results and discussion.

Computer-aided assessment systems
One significant feature of CAA systems is that they provide the opportunity to randomise values of variables and parameters, which allows for the creation of individual tasks (Greenhow, 2015;Rønning, 2017).In relation to this, Greenhow (2015) points out the importance of considering how different values of parameters may affect the difficulty of a task.Another significant feature is the opportunity of providing instant feedback so that students receive feedback while they are still engaged with the task (Barana et al., 2018;Greenhow, 2015).However, this feedback has so far predominantly been on the correctness of a final answer because such types of feedback are the most straightforward to implement in CAA systems (Rønning, 2017).
This way of using a CAA system is in contrast with what Sangwin (2013), the developer of STACK, advocates.According to Sangwin and Köcher (2016), 'STACK was designed primarily for formative feedback purposes, not for examinations '. (p. 225).Besides indicating the correctness of an answer, this CAA system can provide bespoke feedback on a specific response, and occasionally also hints or suggestions for improvement (Sangwin, 2013).However, as Sangwin points out, creating tasks and sophisticated feedback is demanding for teachers, not least since this often requires writing code.
Drawing on the literature on formative assessment as well as their own research, Barana et al. (2018) suggest a model for automatic formative assessment using a CAA system (Maple T.A.).One key aspect of this model concerns interactive feedback.For students who fail in solving a task, they advise providing students hints and step-by-step guidance towards a solution.In this way, they argue, students can gradually acquire the knowledge required for solving the task.This way of constructing feedback is made possible by the 'adaptive' modality provided by this particular CAA system (Barana et al., 2018).However, Rønning (2017) argues that there is a risk that this type of feedback will result in a different and simpler problem, missing the intended learning goal.Sangwin (2003) highlights the possibility of embedding example-generating tasks in CAA systems.This type of task is also advocated by Kinnear et al. (2022) in their suggestion for a research agenda on computer-aided assessment of university mathematics.The idea of asking students to generate their own examples has also been used as a design principle in the development of the STEP (Seeing The Entire Picture) platform at Haifa University in Israel (Yerushalmy et al., 2017).For instance, they ask students to construct examples to justify different statements.The STEP platform is primarily designed to categorise student responses.Besides analysing the properties of mathematical expressions, the platform offers the possibility of analysing attributes of drawings constructed in DMS applets (embedded in the CAA system) as well as free-hand sketches (Yerushalmy et al., 2017).

Dynamic mathematics software
Although researchers suggest embedding DMS environments as a way to increase the learning potential when using a CAA system (Sangwin, 2013;Yerushalmy et al., 2017), there are few studies that have investigated this combined learning environment (Luz & Yerushalmy, 2019).DMS tools are well acknowledged to foster students' conceptual understanding of mathematics, not least in the context of calculus (e.g.Ferrara et al., 2006).Particularly interesting for this paper is the opportunity to directly manipulate dynamically linked representations of functions by using a slider tool (Lagrange & Psycharis, 2014).This tool provides students with the opportunity to examine the visual effect on a graph when changing the value of a parameter (e.g.Fahlgren, 2017).
As mentioned in the Introduction section, one task design principle incorporated in our project is to encourage students to utilize a DMS environment to verify that their suggested examples meet the given conditions.This way of using DMS feedback (i.e. to verify conjectures) has been recognised as helpful for students in their reflection on and, if necessary, revision of conjectures (Laborde, 2002).However, there is an identified risk that students will use the DMS environment in a different way than the intended one.For example, in a study investigating how students interact with a DMS environment when working on example-generating tasks, Haddif and Yerushalmy (2015) found that the DMS environment ' . . .can help those who are close to the correct answer, but when the correct answer is too far removed, the tool may encourage trial and error behavior' (p.2507).

Example spaces
In mathematics education, examples play a key role (Bills et al., 2006).Most often, the examples are chosen and presented by a teacher or a textbook to introduce a concept or method (Bills et al., 2006).In this paper, we are interested in examples generated by students.Watson and Mason (2005) advocate asking students to construct their own examples as a powerful pedagogical tool in the learning of mathematics.They use the construct of example spaces when referring to collections of examples that fulfil certain conditions.Moreover, they distinguish between personal example spaces, conventional example spaces and collective and situated example spaces.While personal example spaces indicate an individual's repertoire of available examples, conventional example spaces refer to examples ' . . .generally understood by mathematicians and as displayed in textbooks . . .' (Watson & Mason, 2005, p. 76).A collective and situated example space is the example space within a group at a specific time (Watson & Mason, 2005).The richness of an example space can serve as an indicator of students' mathematical understanding (Watson & Mason, 2005;Zazkis & Leikin, 2007).To describe the structure of example spaces, Watson and Mason (2005) introduce the notions of dimensions of possible variation and associated ranges of permissible change.Dimensions of possible variation (DofPV) refer to the features of an example that are possible to vary without losing the determining characteristics.The associated ranges of permissible change (RofPCh) refer to the extent to which these dimensions can be varied while still being a valid example (Watson & Mason, 2005).As an example, if a function parameter is a DofPV, its acceptable values constitute the associated RofPCh.

Example-generating tasks
The literature highlights the importance of encouraging students to extend their existing and accessible example spaces by asking for several examples (Goldenberg & Mason, 2008;Zaslavsky & Zodik, 2014).When students are prompted to construct several examples that fulfil certain conditions, they are encouraged to explore DofPV and the associated RofPCh (Mason, 2011).Yerushalmy et al. (2017) found, in their early trials of the STEP platform, how students most often responded with prototypical examples.With the inspiration of the idea of asking for another, and then another, example to encourage students to elaborate further on their first example (Watson & Mason, 2005;Zaslavsky & Zodik, 2014), they decided to ask for three different examples.In tasks where students were asked to construct three different examples to support a specific claim, they found that some students expanded their example space by providing significantly different examples.At the same time, they point out that students interpret 'different examples' in various ways.Consequently, although student responses collected by the STEP platform are marked as 'correct', the complexity of the student-generated examples can vary considerably (Yerushalmy et al., 2017).
To further increase the opportunities for students to generate examples beyond those that first come to their minds in a specific situation, researchers suggest asking for examples that differ as much as possible (Watson & Mason, 2005;Zaslavsky & Zodik, 2014).For example, Zaslavsky and Zodik asked for 'notable variation' between the examples to encourage students to move from familiar and prototypical examples to more sophisticated ones.To do this, they argue, students need to make comparisons to identify similarities and differences between various examples (Zaslavsky & Zodik, 2014).
Another way to encourage students to enrich their example spaces is by adding constraints to the initial conditions (Watson & Mason, 2005).In many cases, this ' . . .opens up new possibilities for the learners and promotes creativity ' (p. 11).If an added constraint makes prototypical examples invalid, students need to be creative and think along novel lines to create valid examples.Watson and Mason (2005) suggest various types of examplegenerating tasks.In the context of a CAA system, Sangwin (2003) suggests using a sequence of prompts that progressively add more constraints.

The objective
The aim of this paper is to develop design principles for example-generating tasks and associated feedback in a technology-rich mathematical environment.To achieve this, the theoretical constructs of DofPV and RofPCh will be used to discern patterns of student responses to three example-generating tasks.The guiding research question for this paper is: What is important to consider when designing example-generating tasks and associated feedback, appropriate for the combined use of a DMS environment and a CAA system, to enrich students' example spaces?

Method
This section starts with a description of the research context, followed by a presentation of the tasks in focus, including conventional example spaces.The section ends with information about data collection and analysis.

Research context
The study was conducted at a Swedish university during autumn 2020 and autumn 2021.It involves two cohorts of first-year engineering students taking a first course in calculus -256 students in 2020 and 235 students in 2021.During the course, the students were asked to perform small group activities in the form of task sequences focusing on function understanding.The activities are designed for a DMS environment (GeoGebra), and the learning environment is intended to provide students with the opportunity to explore and communicate mathematics with peers.These types of activity have constituted mandatory parts of the first-year engineering mathematics courses at Karlstad University since 2015.
To further increase the potential of the DMS activities, we decided in 2020 to adapt them to a CAA system, in this case Möbius.One reason for this was, besides reducing the workload in terms of correction work, to utilize the affordance of using randomisation to give different variants of the question to different students.To ensure active involvement by all students, it was important to include tasks with individual elements requiring an individual answer from each group member.Another reason for implementing the tasks in a CAA system is the opportunity to provide instant formative feedback.However, to design feedback, we need to know more about what typical student responses we can expect.So far, students have only received CAA feedback after they have finished the whole activity.
The empirical data used in this paper are student responses to three example-generating tasks.The tasks were deliberately chosen in that they offer different opportunities in terms of dimensions of possible variation (DofPV).Since example-generating tasks constitute a novel type of task used in the course, we decided to only ask for two examples in these first trials (the two cohorts).The tasks were developed in collaboration with the course teacher.In all three tasks, students were prompted to provide two examples of functions satisfying certain conditions.Before students submitted their answers into the CAA system, they were encouraged to check their suggestions in GeoGebra.In the first two tasks, students were supposed to agree on and submit a joint group answer.The intention of encouraging students to work in small groups was to promote productive interactions among students, something that can be supported by DMS environments (e.g.Manouchehri, 2004).Due to technical constraints, only the first and the third task were automatically corrected by the CAA system (the second task was manually corrected).The responses to the group tasks constituted the basis for an oral group examination at the end of the course.

The tasks
Below is a detailed description of each of the three tasks, including the intended learning goals expressed by key ideas.Moreover, the conventional example space in terms of DofPV and associated RofPCh is introduced.Note that the identified conventional example space is our interpretation of what DofPV and RofPCh the tasks might activate.

Task 1
This task was intended to be solved in small groups.Having been asked to investigate how the parameters a, b and k affect the graph of the function f (x) = k(x + a) 2 + b by using the slider tool, the student groups were requested to provide two examples of quadratic functions with a specified local minimum point.The key ideas addressed in these tasks are (i) shifting a graph horizontally (parameter a) and vertically (parameter b), and (ii) scaling a graph by compressing or expanding it horizontally (parameter k).

Conventional example space
Since a quadratic function formula can be written in three different forms: a standard form (f we regard the form of the function formula as a DofPV with these three forms as the associated RofPCh.However, since the function formula in the preceding sub-task is written in a vertex form to draw students' attention to the key ideas of shifting the graph (horizontally and vertically), we expect students to respond with this type of function form.
In this task, students using the vertex form must realize how the values of a and b affect the location of the minimum point, and that these parameters must have specific values and are not possible to change.Consequently, it is only the parameter k that can be varied without destroying the given conditions.Concerning the RofPCh for this DofPV, the value of k can be any positive real number.

Task 2
This task, which aims to consolidate students' understanding of trigonometric functions, was also intended to be solved in small groups.Having been asked to use GeoGebra to investigate how the parameters A, B, C and D affect the graph of the function f (x) = A sin(B(x + C)) + D, students were asked to consider what has to be fulfilled for a trigonometric function to have a specific range, for example −2 ≤ y ≤ 4.Then, they were asked to provide two examples of trigonometric functions that fulfil the given condition.The key ideas addressed in these tasks are (i) shifting a graph horizontally (parameter C) and vertically (parameter D), (ii) horizontal scaling of a graph by compressing or expanding it horizontally (parameter B), and (iii) vertical scaling of a graph by compressing or expanding it vertically (parameter A).

Conventional example space
When introducing the conventional example space, we assume that the domain consists of all real numbers.The possibility of restricting the domain will be discussed in the section on possible redesign of the task.In this task, students have to realize that the parameter D must have a specific value.The other parameters could have more than one value, which in turn gives three DofPV.Concerning the parameters B and C, the RofPCh are all real numbers (except for B = 0).In contrast, the RofPCh of parameter A is restricted to two values: one specific (real) number and its negation.
We also regard the form of the trigonometric function -that is, sine or cosine -as a DofPV.However, the functions provided when changing the sign of parameter A or the form of the function (sine or cosine) could also be obtained by shifting a graph horizontally -that is, changing the parameter C.

Task 3
Task 3 is a follow-up task to a task where students were asked (in small groups) to find the formula of a rational function, with one horizontal and two vertical asymptotes, given by its graphical representation.Having realized that it must be a rational function, the students were supposed to utilize the asymptotes to construct the function formula.
In Task 3 (see Figure 1), the students were requested to provide two examples of functions with specified asymptotes: two vertical asymptotes and one horizontal asymptote.In contrast to the preceding task, students received different values of the asymptotes, and they were prompted to provide individual answers.The main key idea addressed in these two tasks is the relationship between asymptotes (in a graph) and the corresponding function formula.

Conventional example space
In comparison to Task 1 and Task 2, the DofPV in Task 3 are more complex.First of all, there are several types of function that could be adjusted to fit the conditions, which we regard as a DofPV.However, since the preceding task concerns a rational function, we expect students to respond with this type of function.Moreover, the degree of the numerator and the denominator of a rational function satisfying the given conditions can vary but must be equal (when expressed as a single quotient).Hence, we suggest this as a further DofPV Table 1.Three main ways of expressing the function formula as a response to Task 3.

Form of function formula Example
A Single quotient B Partial fraction, reduced quotients, and a constant term (i.e. the horizontal asymptote) C Reduced quotient, and a constant term (i.e. the horizontal asymptote) with all integers greater than one as the associated RofPCh.Most probably, students would provide rational functions with numerators and denominators of degree two.Moreover, a rational function could be expressed in various ways.In an earlier paper (Brunström et al., 2022), we identified three main ways of expressing the function formula among the student responses, as shown in Table 1.Although these are just different ways to express rational functions, we regard the choice of expression as a DofPV.Notice that we use the same letters, a and b, to indicate the parameters that could be varied without destroying the given conditions, even if they affect the function in different ways (depending on the form of function formula A, B or C).Since both parameter a and parameter b could take on various values, each of them corresponds to a DofPV.Concerning the associated RofPCh, the values of parameters a and b could be any real number (except for zero in some of the forms of the function formula).

Data collection and analysis
The data consists of student responses, in the form of function formulae, to the three tasks, collected through the CAA system.In total, the study involved 491 students divided into small groups (usually 3 students per group).
The data analysis process was guided by Watson and Mason's (2005) theory of example spaces, in particular the notions of DofPV and RofPCh.The goal of the analysis was to identify the collective and situated example space for each task.In the first stage of the data analysis process, we categorised the first and the second example for each student response.In addition to indicating correctness, we categorised the type and/or form of function formula provided.This analysis was guided by the conventional example space described earlier.For example, concerning the form of the function formula used, Task 1 generated two categories (standard and vertex), Task 2 generated two categories (sine and cosine) and in Task 3 all categories introduced in Table 1 (A, B and C) were used.Note that, in Task 3 we decided not to distinguish between responses where the numerator and/or denominator was written in a standard form or a factored form.
Next, we compared the two examples provided by each group/student to identify which dimensions of possible variation had been utilized.This analysis resulted in initial codes for the strategies used to generate a second example.These codes were then organised into categories to discern general patterns in the data material (Saldaña, 2013).For example, in Task 3 we initially had three codes in relation to the parameters a and b, indicating a change of a or b or both.These codes were then grouped together into one single category.

Results
The outline of this section is as follows.The findings in terms of the collective and situated example spaces are reported task by task.Then, in light of the findings, possible redesigns of the task, including formative feedback, are discussed.

Task 1: collective and situated example space
In total, 210 groups (out of 212) provided a first example, among which 199 were correct.Most of the student groups (159 groups) provided the first example of function formula in a vertex form (f (x) = k(x + a) 2 + b), among which 60 groups used a value of k = 1.There were also 48 groups giving the first example in a standard form (f (x) = ax 2 + bx + c), predominantly by assigning the parameter a the value 1 (36 groups).When constructing the second example, most of the students (199 groups) used the DofPV concerning the scaling of the graph (parameter k).Seven groups switched the form of the function formula to construct their second example.Regarding correctness, the number of correct answers decreased from 199 (first example) to 191 (second example).

Task 1: possible redesign
Although most of the students used the vertex form, there were students responding with a function in standard form.Of course, some of these students might have started with a vertex form, which they then expanded into a standard form.Since it is the vertex form of the function that clarifies how the parameters a and b affect the horizontal and vertical shift, respectively, we suggest adapted feedback that encourages these students to provide another example in the vertex form.
In the current version of the task, both a specified local minimum point and the type of function (i.e.quadratic) are given.One way to increase the opportunity for a richer example space would be to specify an extreme point, instead of a local minimum point, as the given condition.In this way, a further DofPV is added -the type of extreme point -where the associated RofPCh consists of two values (maximum or minimum).Probably, this revision will encourage many students to switch the sign of the parameter k to construct their second example.In this way, they are not prompted to think in terms of scaling, which is one of the key ideas addressed by the task.Therefore, we think it would be instructive for these students to receive feedback consisting of a request for a third example (without further instructions since a third example can only be created by using scaling).On the other hand, for those students who have provided two functions with a local minimum (or maximum) point, the feedback could prompt students to provide one further example with the other kind of extreme point.This feedback encourages them to activate a further DofPV.
Another way to introduce a further DofPV might be to not specify the type of function.However, then there is a risk that several students will solve the task without considering one or several of the key ideas addressed in the task.

Task 2: collective and situated example space
In total, 202 groups (out of 212) responded to the task.Most of the student groups ( 181) gave a sine function as their first example, among which 173 were correct.For example, when the range was −2 ≤ y ≤ 4, a common student response was f (x) = 3 sin x+1.Among the rest of the responses, six groups provided a (correct) cosine function.Since this task, due to technical constraints, allowed students to respond in text, 13 groups responded in more general terms without providing a specific example.While all of these groups provided specific values for parameters A and D, only four groups explained that parameters B and C could have any value.
According to the identified conventional example space (described earlier), there are four DofPV that students might use to construct the second example: the type of function (sine/cosine) and the values of the parameters A, B and C.More than half of the student groups (116) used the strategy to (only) change the value(s) of the parameters B and/or C.There were 40 groups that switched from a sine function to a cosine function (or vice versa) to create the second example.The possibility to change the sign of parameter A was utilized by 17 student groups.

Task 2: possible redesign
Since this task is intended to consolidate students' understanding of trigonometric functions, it is not an option to introduce a further DofPV in terms of the type of function.One way to encourage students to reflect further on the RofPCh of the parameters B and C would be to add a constraint by specifying the domain of the functions.For example, the domain could be restricted to− π 6 ≤ x ≤ π 2 .However, since the students are using GeoGebra when working with these tasks, they probably will realize that large values of B will always work, independent of the value of C. Hence, there is a risk that the use of the slider tool reduces the epistemic value of this possible redesign.
As in Task 1, one way to encourage students to consider a further DofPV could be to first ask for two examples and then provide adapted feedback depending on the features of these examples.For those students who switch from a sine function to a cosine function (or vice versa) or change the sign of the parameter A, a request for a third example (without further instruction) might prompt them to extend their example space by considering a further DofPV.On the other hand, students who change the value of the parameter B and/or C require adapted feedback, directing their attention to other DofPV before the request for a third example.For example, if a student uses sine functions in both examples with different values of B and C (as many students did), the feedback could prompt students to provide a cosine function as a third example.
Another way to activate a further DofPV would be to give students the opportunity to choose the domain.This will make it possible to obtain the given range (e.g.−2 ≤ y ≤ 4) also by adapting the domain.However, there is probably a need for adding a constraint to compel students to consider this opportunity.One way to do this might be to require that the third example should have a different amplitude than the first two.

Task 3: collective and situated example space
In total, 479 students (out of 491) gave a first example, among which 465 were correct.As expected, all student responses consisted of rational functions.Most of the students provided a response with the horizontal asymptote as a constant term, formula C (282) or formula B (80), as their first example.Among the rest of the student responses, there were 109 formula A. Concerning the construction of a second example, 11 students (out of 479) either did not provide a second example or gave an incorrect answer (although their first example was correct).Moreover, 25 students provided a second example that was a rewritten form of the first example.
In this task, the identified conventional example space (described earlier) includes five DofPV that students might use to construct the second example: the type of function, the degree of numerator and denominator, the form of function formula (A, B or C) and values of the parameters a and b (see Table 1).
To construct the second example, most of the students (397 out of 479) changed the parameter a and/or the parameter b, although in slightly different ways depending on the form of formula used (A, B or C).This is illustrated in Table 2 by some examples of student responses in terms of a first and a second example.
The response from Student 1 indicates that s/he realized that the parameters a and b could take on any real number since they seem to be randomly chosen.Although Student 2 chose to change both parameters a and b, s/he did this in a simple way, by changing both parameters from 1 to 2. Student 3 and Student 4 provide a simple version of formula C in their first example (a = 0 and b = 1).To construct their second example, they use quite Formula A x = −6, x = 1 and y = 5 diverse strategies: while Student 3 changes both parameters a and b, Student 4 demonstrates a rich RofPCh within one specific DofPV by choosing an extreme value of parameter b.
Instead of sticking to the same form of formula, 34 students (out of 479) switched the way they expressed the function formula.Most of these students changed from formula A (11) or formula C (21) to another form.
With respect to the focus of the paper, it is particularly interesting to look more in detail at responses indicating DofPV that are beyond the expected ones.In relation to the DofPV concerning the degree of the numerator and denominator, there were 20 students who activated this DofPV to construct the second example.Table 3 shows four of these unexpected student responses.
Predominantly, these students squared one or two of the factors in the denominator, and, if necessary, adjusted the numerator.

Task 3: possible redesign
The findings indicate that most of the students used the same form of formula in both their examples.This is not surprising, but we think that it might be instructive for students to realize that there are different ways of expressing a rational function.Consequently, for these students, adapted feedback followed by a request for a third example may extend their example space.So, for example, if a student uses formula C (in both examples), the adapted feedback could be something like: 'Great, the answers are correct.Now, give one more example expressed as a single quotient'.
Since this task was an individual follow-up task to a group task on rational functions, it is not surprising that all student responses were examples of rational functions, although this was not required in the task.However, there might be a risk that students associate asymptotes only with rational functions.One way to prevent this and to extend students' example space could be to offer adapted feedback to students that have submitted two rational functions.To activate a further DofPV, the feedback could prompt students to reflect on other types of function -that is, non-rational functions, and then ask for one more example.Actually, this additional constraint was used at the follow-up examination seminar at the end of the course.At this seminar, students were asked to give (in small groups) two examples of non-rational functions that have a given horizontal asymptote, and two examples of non-rational functions that have a given vertical asymptote.The student responses obtained at this seminar indicate that it might be instructive to prompt students to also provide examples of non-rational functions.Indeed, this request activated a further DofPV in Table 4. Examples of non-rational functions provided by students at the follow-up seminar.

Given condition
First example Second example terms of types of function, and the associated RofPCh turned out to be ample.To illustrate the variety among student responses, we provide some examples in Table 4.
Concerning Task 3, we now have discussed two different alternatives for adapted feedback, provided when a student has submitted two answers.If the main key idea addressed by the task concerns rational functions, then probably the first alternative discussed is more relevant than the second one.On the other hand, if the main key idea concerns asymptotes, the second alternative might be more relevant.

Discussion
The aim of this paper is to provide guidance in designing example-generating tasks and associated feedback appropriate for the combined use of a DMS environment and a CAA system.In the following, we start by discussing the main findings in relation to the design of tasks (and associated feedback).Then, as an answer to the research question guiding the paper, we suggest a sequence of questions to consider when designing example-generating tasks.Since very little attention has been paid to the use of the theoretical ideas of DofPV and RofPCh in the design and analysis of example-generating tasks, we also discuss this issue.The section ends with reflections and some implications, both for research and practice.

Student performance
Generally, students performed well on the three tasks.One reason for this might be the encouragement of using DMS to verify their answers before submitting them into the CAA system.We could see from student responses on a questionnaire that they most often utilized the opportunity to verify their conjectures in GeoGebra and that they appreciated the visual feedback from GeoGebra.In line with Haddif and Yerushalmy (2015), we propose that students may adjust their conjectures when their initial conjecture almost meets the given conditions as a result of the instant feedback provided by the DMS environment.Besides being an instructive tool for verifications in the example-generating tasks, the DMS environment served as a tool for exploration in the tasks preceding Task 1 and Task 2. In these tasks, students were encouraged to investigate how different parameters affected the graph of a function by using the slider tool.Since some of the parameters in these tasks correspond to a DofPV, it can be argued that these tasks served as (implicit) guidance for Task 1 and Task 2. In addition, the slider tool might facilitate the investigation of the associated RofPCh.

Task design
The main ideas that emerged from this study concern the number of examples that students should be prompted to generate, the pros and cons of adding or removing constraints and how adapted feedback might be utilized.The choice of asking for two examples was a first step towards using this, for most of the participating students, unfamiliar type of task in mathematics.The results from all three tasks indicate that there are instances where asking for three examples might enrich students' example spaces.This observation aligns with previous research (Watson & Mason, 2005;Zaslavsky & Zodik, 2014).
However, our findings indicate that asking for three (or more) examples is not enough.For instance, considering the typical student responses to Task 3 (see Table 2), a request for another example would most probably result in a similar example to the previous ones.One way to tackle this could be to ask for examples that differ as much as possible (Watson & Mason, 2005) or with 'notable variation' (Zaslavsky & Zodik, 2014).In this study, we could observe some responses like the one from Student 4 in Table 2. Most probably, this student interprets the second example as notably different from the first one.Indeed, the student shows a relatively rich example space in terms of RofPCh, and we agree with the suggestions above to encourage more students to extend their RofPCh.However, to activate more than one DofPV, it might not be enough to ask for a third example that differs as much as possible.Therefore, we suggest starting by asking for two different examples.Depending on the features of these examples, students receive adapted feedback inviting them to activate a further DofPV in a third example.For example, as in Task 3, this feedback could consist of an additional constraint encouraging students to construct a significantly different example, such as another type of function.This illustrates the usefulness of adding constraints to promote students' creativity, as suggested by Watson and Mason (2005).At the same time, as discussed in relation to Task 1 and Task 2, sometimes when the RofPCh is restricted to two values, it is enough to simply ask for a third example to activate one more DofPV.In cases where it is instructive to activate more than two DofPV, it might be useful to provide the type of example-generating task, with progressively added constraints, suggested by Sangwin (2003).
In light of the findings, we suggest a sequence of questions to consider when designing example-generating tasks and associated feedback, appropriate for the combined use of a DMS environment and a CAA system, to enrich students' example spaces: (1) What key ideas should the task address?This question is crucial since it should guide subsequent steps in the design process.
In order to enrich students' example space while also ensuring that the key ideas are addressed, there is a need for careful consideration concerning what conditions the student-generated examples should meet.For example, in relation to Task 1, we discuss how the removal of a constraint (that it should be a quadratic function) to open up for a further DofPV might result in students generating examples without considering the key ideas.To address the key idea in Task 2, one condition was that it should be a trigonometric function.Consequently, the type of function was not a suitable DofPV in this task.
(3) How many examples should students be asked to generate?
Where the task allows for students to give a correct answer consisting solely of examples that all come from the same DofPV, it might be beneficial to first ask for two examples, and then provide adapted feedback encouraging the student to activate a further DofPV in a third example.Instances when we think that this could be helpful were discussed in relation to all three tasks.
(4) What feedback is needed to activate relevant DofPV?
When it comes to adaptive feedback, based on the first two examples provided by the students, it can be in the form of additional constraints or hints/suggestions to activate further DofPV, followed by a request for another example.However, in cases where the RofPCh is restricted to two values, it may be sufficient to simply request an additional example without further instructions.

Reflection on the use of DofPV and RofPCh
In the following, we will discuss the usefulness of the constructs of DofPV and the associated RofPCh.In this paper, we have used these constructs for various purposes.The constructs were used to describe our understanding of the conventional example space.We found the identification of the conventional example space useful since it captures all the possible (correct) student-generated examples.This, in turn, facilitated the anticipation of likely student responses.The identified conventional example space, in terms of DofPV and RofPCh, also supported the analysis process, resulting in the collective and situated example space.In addition, the constructs were helpful when discussing possible redesign of the tasks.Not least, how the addition or removal of a constraint might influence the opportunity to address a specific key idea while also allowing for a rich and varied example space.Hence, the DofPV and the key ideas were crucial in deciding which conditions the examples generated by the students should satisfy.Also, the associated RofPCh turned out to be useful, not least when discussing the number of examples requested and the need for feedback to activate further DofPV.Finally, we found the theoretical constructs useful to communicate the design principles, in terms of a sequence of questions to consider when designing example-generating tasks and associated feedback.
In summary, we suggest the constructs of DofPV and the associated RofPCh as useful in the design of example-generating tasks as well as when analysing student responses.

Concluding remarks
In this paper, we have used patterns of student responses to three example-generating tasks to provide some tentative principles on how tasks and associated feedback, appropriate for the combined use of a DMS environment and a CAA system, can be designed to enrich students' example spaces.The next step is to try out and refine these principles.

Figure 1 .
Figure 1.Task 3 as it is presented in Möbius.

Table 2 .
Examples of student responses in terms of a first and a second example for the three forms of function formula (A, B and C).

Table 3 .
Examples of unexpected student responses.