A description and characterization of student activity in an open,online

A description and characterization of student activityin an open, online, mathematics help forum

Carla van de Sande

Published online: 29 January 2011# Springer Science+Business Media B.V. 2011

Abstract Free, open, online, calculus forums are websites where students from around theworld can post course-related queries that may be viewed and responded to by anonymousothers. These sites are an emergent resource for students seeking help and have become apart of many students’ mathematical experience. The purpose of this paper is to introduceand describe the forums to the research community as a computer-mediated form of helpseeking, and to briefly characterize the forms of student activity in one popular calculushelp forum. Two hundred exchanges on limit and related rates were collected and examinedfor evidence that students are contributing ideas and proposals for action when they initiatethe exchange and following intervention. The findings justify the need for further researchsince students are using the forums in a variety of ways and sometimes participatemeaningfully. The forums have potential for testing student responses to pedagogicalapproaches.

Keywords Help seeking . Computer-mediated communication . Calculus . Discussionforums

Formal learning does not end when the bell rings at the end of class, and it is common forstudents to receive homework assignments to complete outside of school hours andboundaries (Cooper, 2007). It is also common for students to need and seek help whenworking on assignments and attempting to master the content that they are trying to learn.In doing so, students depend on a varied set of resources, including textbooks, class notes,peers, parents, and tutors. Students now also turn to free, open, online, internet help forumsto ask questions and receive help. Students access these forums and communicateanonymously with a network of volunteers as they seek help on their assignments andcoursework.

A casual search for “free homework help” on the web turns up several such forums, andshows that many students are using these forums to seek help in core science, technology,

Educ Stud Math (2011) 77:53–78DOI 10.1007/s10649-011-9300-y

C. van de Sande (*)Arizona State University, Tempe, AZ, USAe-mail: [email protected]

engineering, and mathematics courses. For instance, one such site (www.mathhelpforum.com) that offers help in arithmetic through higher mathematics has over 29,000 membersand received an average of 152 queries daily in 2009. Other sites, that cover subjects acrossthe sciences and humanities, extend to 120,000 members and receive close to 1,000requests for help daily. Within these sites, whether devoted to mathematics or broader inscope, calculus is consistently one of the most heavily trafficked areas. Yet, despite agrowing interest in computer-mediated communication and mathematics learning (e.g.,Engelbrecht & Harding, 2005), open, online, help forums have remained largely off theradar of education research. The purpose of this paper is to introduce the forums as a help-seeking resource and briefly to characterize the kinds of student activity that can occur.

Table 1 contains common terminology associated with asynchronous interaction inonline environments, where asynchronous interaction refers to the affordance of sendingand receiving messages without requiring that users be logged on at the same time.

1 Free, open, online, help forums

1.1 Availability and extent

Open, online help forums are found on websites and allow students anonymously to postcourse-related queries that are then visible to others. These forums are open in the sensethat, unlike course forums or discussion boards, access is not restricted to any particularcourse or institution. Also, instead of hosting discussions based on the curriculum from aparticular course, the forums cover broad school subject areas (such as mathematics,science, and business) at a range of course levels (from elementary to graduate). Thus, theseforums are a help-seeking resource that is currently available to any student who hasinternet access.

Of particular interest to the mathematics education community are several mathemat-ically focused forums. Students from around the world access these forums when they arein need of help completing assignments or understanding course materials. Thus, althoughthese sites may be based in different countries, participation in a given forum is by nomeans restricted to the nationality of the site. An administrator of a site based in NorthAmerica, www.mathhelpforum.com (MHF), has on two occasions published a map (createdby Google Analytics) in the MHF Community forum detailing the geographic location ofpeople who accessed the forum over a recent time period. On these maps, dots marklocations of forum activity, and the size of a dot correlates with the number of visitors.Figure 1 shows the second of these maps posted on February 27, 2006. People fromvirtually every continent are represented in this snapshot of activity on a single forum,demonstrating that these online forums are not merely open in name but also in practice. In

Table 1 Vocabulary

Term Definition

Forum (sub-forum) Web application for holding discussions and posting user-generated content

Post(ing) Contribution or message that is published on the site, either to initiate adiscussion or in response to another’s contribution

Topic, thread, exchange,or discussion

The set of contributions pertaining to a single request for help

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http://www.mathhelpforum.com



short, participation in open, online help forums spans geo-political boundaries and affords aunique opportunity for people around the world to connect with one another and discussmathematics.

1.2 Forum models

The orchestration of multilateral connections between a large number of help seekers andhelpers requires the adoption of a participation structure and monitoring policy, whereparticipation structure refers to the affordance of one-to-one, many-to-one, or many-to-many interactions (Baym, 1996). Previous research (van de Sande & Leinhardt, 2007) hasidentified two commonly used help forum models, namely Assigned Online Help (AOH)and Spontaneous Online Help (SOH), that each corresponds to a participation structure (seeFig. 2). In AOH forums, an incoming query for help is assigned to a helper according tosome selection model. For example, incoming queries can be assigned by the forum

Fig. 1 Geographical diversity of forum activity, indicated by dots

Fig. 2 AOH (left), SOH (middle), and BOH (right) help forum models

A description and characterization of student activity 55

administration to participant helpers (based on pre-determined criteria such as subject areaand helper availability), or incoming queries can be stored in a restricted-access database tobe selected by (and thereafter assigned to) one of the helpers. The helpers in AOH forumsare chosen on the basis of qualifications determined by the forum administration. Forinstance, www.mathnerds.com screens applicants for evidence of sufficient mathematicalcontent knowledge, adherence to an inquiry-based teaching philosophy, and communicationskills demonstrated through responding to five to ten practice questions. AOH forums havea participation structure that supports one-to-one interactions between students and helpers.

In contrast to AOH forums, SOH forums allow any forum member, regardless ofexpertise, to spontaneously respond to a query. SOH forums support many-to-manyinteractions since any member can take part in an ongoing thread. Whereas threads in AOHforums tend to be relatively short in length (few contributions), exchanges in some SOHforums can be extended and may contain sophisticated pedagogical moves (such as hintingand Socratic questioning; van de Sande & Leinhardt, 2007).

More recently, a forum model (BOH) that blends the participation structures of AOH andSOH sites has been implemented and researched (Puustinen, Volckaert-Legrier, Coquin &Bernicot, 2009). In the BOH model, the set of helpers is selected (like AOH sites), but theforum structure allows multiple helpers to contribute within a single thread (like SOH sites).

Thus, as depicted in Fig. 2, all help forum models accept requests from the generalpublic (represented by dashed lines as open sets). The core differences between thesemodels are realized in the participation structure and privilege. AOH forums support one-to-one interactions between students and helpers who meet certain criteria (represented as aclosed set) and requests are assigned to a helper (indicated by arrows pointing from requeststo helpers). In contrast, SOH forums support many-to-many interactions within threadsbetween anyone who chooses to help (represented as an open set) and students, and helperschoose which requests to pick up and participate in (indicated by arrows from helpers torequests). Also, helpers can interact with other helpers within SOH threads (indicated by thedashed arrow). BOH forums restrict the body of helpers to qualified participants but allowhelpers to self-select requests.

1.3 Cost

Perhaps more astonishing than the connections that forums bridge between help seekers andhelp providers around the globe (and the efficiency with which this takes place) is the factthat many sites (both AOH and SOH) provide assistance to students free of charge. The costof hosting an online forum is low, approximately $10/month (Ted Wilcox, personalcommunication, June 9, 2007); the real cost is in supplying the person-power needed torespond efficiently to the incoming queries. However, in many open, online forums, thehelpers represent a pool of volunteers who donate time, help, and experience as a service toanonymous students. Thus, the forums contribute significantly to the theme ofdemocratizing education through technology (Larreamendy-Joerns & Leinhardt, 2006), inthat any student with access to the internet can use this resource for getting help oncoursework. These free forums remove the disadvantages of students who may be (quasi-)isolated from help providers (e.g., distance education) (Bernard et al., 2004, 2009), arehesitant to ask questions in person (VanderMeij, 1988), who cannot afford to purchase helpservices, or who cannot get adequate help from others in their immediate surroundings(Solomon, Warin & Lewis, 2002).

Accounting for the beneficence that supports this public service remains to be exploredto the extent accomplished by research in fields such as business and economics (cf., Avery,

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http://www.mathnerds.com

Resnick & Zeckhauser, 1999). What do these volunteers get out of participation and whatmotivates them to become members of a particular forum community? Positing the helpersas mathematical “Good Samaritans” can account for some observed patterns ofparticipation in one particular SOH forum (van de Sande & Leinhardt, 2008b), butexplaining the motivation behind forum participation would require a more intrusivemethodology than observations of archived threads and, because of ethical considerations(e.g., the potential of disrupting the forum and making members self-conscious), has notbeen addressed at this time.

1.4 Member status

For helpers, forum participation is a form of online information gift giving. One factor thatmay influence how helpers give involves the assignment of status, or standing, in the onlinecommunity (Kollock, 1999; Rheingold, 1993). Online help forums can assign membersstatus according to quantitative counts of participation (e.g., number of contributions), orcan employ a more elaborate system. For example, some help forums adopt a “reputationsystem” (Dellarocas, 2003) that allows community participants to rate the activities andcontributions of others in the community. This policy can be implemented by means of a“thank you” button feature that increases member status according to how many othersacknowledge a contribution (and possibly the thanker’s status as well). An even morecomplex reputation system is one that allows students to rate the quality of helpercontributions, who then receive differential increments to their forum status. Becausereputation is often the sole resource in online communities for improving one’s standing,the system by which a help forum assigns and promotes member status could shape thenature of interactions (e.g., the quality and thoughtfulness of contributions) and ultimatelydetermine the sustainability of the forum (e.g., whether helpers leave because they do notfeel adequately recognized or rewarded for their participation).

A forum community may also designate a special, elevated status, e.g., as moderator, toselect members. These participants are empowered to lock threads, and move, delete, or editposts. Moderators are responsible for maintaining the appropriateness, quality, andorganization of the forum, a task that would otherwise be unwieldy for a community ofappreciable size.

1.5 The stuff of forum interactions

The queries posted on open, online, help forums are generally from traditional textbookexercises that have become problems for the students who are seeking help. As A. Selden, J.Selden, Hauk and Mason (2000) note, the distinction between a “problem” and an “exercise”is a function of both the task and the solver. Thus, although the queries posed by students onthe help forums may represent closed, prescribed tasks or exercises from the perspective of amathematics educator, they are problems from the perspective of a student. The goal of thestudents who participate in the forum is to seek help in the construction of a solution, toreceive verification of a solution that has been constructed for coursework, or to construct anexplanation of a solution from another source (such as a solution manual or textbook).

In this way the help forums are, as an anonymous reviewer pointed out, very muchsubsidiary to “school mathematics.” The seeds of the discussions are determined by thetexts that are used and the assignments that are made in material classrooms, and stand incontrast to the meaty, challenging Math Forum’s Problems of the Week as discussed inRenninger, Ray, Luft and Newton (2005). However, the nature of the “naked” problem


statements that students bring to the forums (whether abstract, contrived, or imaginary)does not prescribe the nature of the interactions. In the classroom, the construction ofsolutions for routine tasks can represent (spontaneous) opportunities for studentengagement in mathematical analysis (Lozano Teran, 2009). For instance, in the processof constructing a solution to a routine task, the application of a rule can become the subjectof a discussion that itself contributes to a broader instructional explanation, such as whenthe validity of the rule is questioned or alternative rules are debated (Leinhardt, 2001). As isthe case for help seeking in general (Nelson-Le Gall, 1981, 1985; Karabenick & Newman,2006), the potential behind open, online help forums lies in the manner in which thestudents interact with the subject material and others (for instance, making assertions andproposals for mathematical actions, questioning and challenging others’ proposals,indicating resolution, self-explaining, etc.).

1.6 Transforming mathematics and help-seeking

Open, online, help forums represent computer-mediated discourse that, as Herring (2011)notes, is a medium of communication that is distinct from speaking and writing (distributedby electronic means). As such, the forums alter the very nature of student-initiated helpseeking as it is traditionally conducted (e.g., in walk-in help centers or tutoring sessions)and transform it from a private, face-to-face activity between helper and student into apublic activity between people who share some interest (whether intrinsic or extrinsic)1 inthe subject domain but who are otherwise unconnected. The broader social dimensionafforded by SOH forums, in particular, reframes help seeking as a collective, social activityin which the exchanges form the threads of a public conversation between individuals whoare seeking help and those who provide it.

At the same time, the online venue of the forums transforms the actual mathematics thatis the topic of discussions. The forums are a classic example of a “humans-with-media”collective (Borba, 2005) that produces a different written mathematics than is developed inface-to-face situations (Borba & Villarreal, 2005; Borba & Zulatto, 2006). Gestures andlooks have to be conveyed in alternative ways, the formality of the mathematics has to bereconciled with the informality of the communication mode, and contributions (which arelinked to a participant’s forum identity) have to be directed toward an anonymous,unknown audience. Rather than relying on memory or notes, a written record (archive) iscreated that allows all participants to view (and even review) the content (what was said),source (by whom), and timing (when) of previous entries. This paper investigatescharacteristics of student activity in this emergent, informal, and unregulated resource thatnow shapes many students’ mathematical experience.

2 Method

The methodology adopted for this investigation of free, open, online help forums wasobservational. Because help forums are a relatively uncharted territory for educationalresearch, it was appropriate to conduct an exploration rather than a confirmation (Goodyear,

1 Student interest in the subject area may be extrinsic if it is driven or motivated by ulterior motives (such asgrades). However, the fact that the students are seeking help on the forums demonstrates a level of interest orcognitive engagement (Blumenfeld, Kempler & Krajcik, 2006) that is distinguishable from making no effortto complete an assignment.

58 C. van de Sande

Jones, Asensio, Hodgson & Steeples, 2005). In addition, an observational methodologyensured that the research could be conducted without disrupting the learning environment,and perhaps violating the trust of participants (Green, 2007).

In order to strike a balance between capturing the online persona established by forummembers (Spears, Lea & Postmes, 2007) and protecting privacy and anonymity, I adoptedthe following policy for referring to participants in this paper: participants are referred to bynames that characterize and closely resemble their self-designated user names or “handles”(e.g., ihatecalc might be referred to as calc_hater), giving the reader a sense of how aparticular user elected to present him/herself to other forum members. If a user nameappeared to reveal a participant’s real world identity (such as a surname), a pseudonym wasassigned. Informed consent was not obtained because, according to the most recentguidelines for psychological research, the forums are public spaces in which participants donot operate under the assumption that their communications are confidential, and the risksposed to the participants, should their identity become known, were considered minimal.2

See Ess (2007) for an in depth discussion of Internet research ethics.The general approach to investigating student learning experiences in online help forums

was to select a target site and then track exchanges on specific topics. Several choices canbe made, for example to track all the exchanges in a given time period or until a targetedquantity of exchanges occurred.

2.1 Site choice

The popularity and availability of free, open, online, calculus help forums means that therewere several possible candidates for study. The calculus forum at www.freemathhelp.com(FMH) was selected because it has an extensive history (archives dating back to 2005),includes a search mechanism for locating exchanges that contain a keyword or phrase, andis active in terms of daily postings and membership. Between its inception in June 2002 andthe time of this study, it had attracted 10,494 members and received 85,173 total posts,contributing to 20,570 exchanges. Although there are larger, more popular forums, FMHhad the additional feature of a member status policy based on amount of participation(specifically, the number of individual threads to which one has contributed). Becausestatus seeking is known to influence social interactions in online communities (Lampel &Bhalla, 2007), it was decided that a forum with a policy in which status is dependent onnumber of contributions alone would be best suited for the exploration of the range ofstudent activity that can occur in forum interaction.

2.2 Topics

Having selected an open, online, help forum, the next step was to choose a subset of thethreads for analysis. As mentioned above, the questions that get posed in the open, online,help forums are a reflection of the curricula of the material schools and stem from theexercises that are assigned. The calculus forum of FMH contains requests for help spanningthe wide range of curricular material that is commonly covered in single and multivariablecalculus instruction. Most topics are present throughout any given time period becausestudents may be taking any course during any quarter or semester anywhere in the world. In

2 Although informed consent was not sought, when the FMH forum administrator was approached andapprised of the nature of this research, he did not have any objections (Ted Wilcox, personal communication,May 31, 2007).


http://www.freemathhelp.com

order to construct a data set in which the threads shared a discussion theme andmathematical level, two instructional topics that are generally taught in an introductorysingle variable calculus course were chosen: the limit and related rates. These two topicsreflect the diversity of the material that is characteristic of introductory calculus instruction(abstract and applied), are difficult for students to master, and surface often in the forum.

Coming to grips with the limit concept is one of the first challenges that students in anintroductory calculus course face, and research reveals that many students harbor incorrector incomplete conceptions of limits (Cornu, 1991; Cottrill et al., 1996; Tall & Vinner, 1981;Williams, 1991). Furthermore, many students fail to achieve a coherent understandingthrough instruction (Szydlik, 2000). Some of the difficulty can be chalked up to the abstractnature of the limit concept; it is based on a never-ending process and therefore requires thecontemplation of the result of an infinite process (Vinner, 1991)—a big conceptual leapfrom the mathematics encountered in algebra and precalculus instruction. In order to solvetypical calculus exercises involving limit, students must deal with infinity (What does itmean to approach infinity?), division by zero (How does this make sense, and what is themeaning of “undefined?”), and indeterminate forms (Why is it that certain special ratiosare not uniquely determined?).

In contrast, related rates represent a class of problems that involves the relationship(s)between two or more changing quantities, one of which is unknown and must bedetermined. These generally appear in introductory calculus instruction as applications ofimplicit differentiation and the chain rule, with the solution often scripted as a five to sevenstep process. Yet, students struggle with related rates as well (Martin, 2000), perhapsbecause they lack the quantitative and covariational reasoning abilities necessary toconstruct solutions (Engelke-Infante, 2004; 2007). Typical related rates exercises areframed as word problems, requiring that students engage in four general cognitiveactivities: problem translation, problem integration, solution planning and monitoring, andsolution execution (Mayer, Larkin & Kadane, 1984). Each of these stages represents apotential roadblock on a student’s path to the construction of a solution, and these factors,coupled with other commonly encountered potholes—such as the inaccessibility of anappropriate problem-solving schema (Sebrechts, Enright, Bennett & Martin, 1996) andimpoverished metacognitive skills (Schoenfeld, 1992)—make related rates exercisesanother good candidate as a topic for investigation.

2.3 Building a database and coding it

In order to canvas the type of student activity present on an online forum, 100 threads on eachtopic dating back from April 2008 were culled from the forum archives. The time periodcovered is not exactly the same (approximately 14 months for limit and 21 months for relatedrates) but the number of exchanges analyzed was. These exchanges were analyzed for evidencethat students were contributing to the construction of the solutions to the problems that theybrought to the table, both in the initial posting and following helper intervention.

Assertions and proposals for action One overt indicator of student activity and initiative isthe presence of assertions and proposals for action that contribute to the construction of asolution to a problem at hand (Greeno, 2006). Each thread was examined in two locationsfor evidence of student activity: the initial post and in posts following helper intervention.Thus, as seen in Table 2, there were four possibilities, descriptively titled coasting,slacking, sustaining, and ramping, for a given thread as the unit of analysis. Coastingdescribes the presence of activity in the initial post but no activity following helper

60 C. van de Sande

intervention. Slacking describes the complete absence of student activity within a thread.Sustaining describes the presence of student activity both in the initial post and followinghelper intervention, and ramping describes the absence of initial activity and the presence ofactivity following helper intervention.

To count as activity in the initial post, it was required that the student publish a (partial)solution or suggest a possible solution method. Because students are seeking help frommore knowledgeable others on the forum, they may well hedge these proposals (Rowland,1995). (For example, see the partial solution excerpt from a student’s initial post inTable 3). The coding of the data, therefore, allowed any partial solution or proposal as anindication of student activity, regardless of the manner in which the proposal was made.Student activity following help was defined in accordance with a framework fordistinguishing active, constructive, and interactive learning activities from the perspectiveof the learner (Chi, 2009). Two indicators of student activity were a response to scaffoldingor a correction of errors based on feedback, and these were taken as evidence that thelearner was substantively and meaningfully participating in a guided-construction activitywith a more experienced other. The accuracy of the student contributions was not assessedfor this analysis.

Cohen’s κ showed perfect inter-rater consistency on a sample of 20 exchanges forevidence of student activity. Table 3 contains the problem statement and excerpts(reproduced verbatim) of threads that were taken as instances of student activity (proposedsolution and suggested strategy) from two exchanges on limit.

3 Results

The purpose of this paper is to introduce in some depth the ideas and practices that can existin online help forums and to characterize the forms of student activity in this environment.

Table 2 Descriptive labels of student behavior in an exchange

Characterization Activity in initial post Activity following help

Coasting Yes No

Slacking No No

Sustaining Yes Yes

Ramping No Yes

Table 3 Example excerpts from limit threads showing student activity

(Partial) worked solution Suggested solution strategy

Problemstatement

lim x ! infinity 2^xð Þþ1½ �= 2^ xþ1ð Þ½ � lim of SQRT 3xð Þ � 3ð Þ= 2x� 6ð Þ as xapproaches 3.

Evidence ofstudentactivity

This is what i did. I don’t know exactly howto solve for this but, my logic is that it’ll go tozero eventually, because the bottom goes toinfinity faster than the top so it’ll go to zero.But I am not sure exactly if this is right cansomeone prove this or tell me if I’m intuitivelycorrect. Thanks

Here I know you can’t plug in 3 and when Itried manipulating the limit algebraically bymultiplying by the conjugate I got nowhere


The exploration of student activity is prefaced with an in-depth description of FMH,including its history, operational policies, expectations, and guidelines for participation(termed “netiquette,” a concatenation of network and etiquette; cf., Shea, 1994). Theanalysis of student assertions and proposals for action is augmented by the presentation of aprototypical example exchange illustrating each of the four characterizations: coasting,slacking, sustaining, and ramping.

3.1 FreeMathHelp

FMH is an advertisement-supported mathematics help portal established in 2002 by TedWilcox, an enterprising high school junior at the time. In addition to the discussion forums,the site includes lessons, games, a graphing utility, and worksheet pages. The discussionforums are subdivided into 14 subforums: 10 homework help forums organized by subjectarea (ranging from arithmetic and pre-algebra to calculus and differential equations), twoforums pertaining to calculators (games and questions), and two devoted to “importantstuff” (e.g., forum news and administrative issues).

The sole requirement for becoming a forum member is registration (which entailsagreeing to abide by terms of permissible content and conduct, providing a username ande-mail address, and selecting a password). FMH is an SOH forum in that any membercan initiate threads in a discussion forum and can respond to others’ posts. Forummembers also have access to user profiles that include self-volunteered information onoccupation, residence, contact information, as well as statistics on discussion boardactivity. However, even if members do not volunteer personal information, somefeatures of “real life” identity are revealed to others through the framing of contributions(Herring, 2011).

The status of each member is determined by the total number of distinct threads to whichcontributions have been made: new (0–49), junior (50–249), full (250–999), senior (1,000–2,499), and elite (more than 2,500). At the time of this investigation, there were several elitemembers who had contributed to more than 2,500 threads, five of whom had contributed tomore than 4,000. As discussed above, other help forums have more complicated means ofassigning status, such as a reputation system.3

The computer window for constructing posts contains traditional icons for highlightingtext (e.g., italics, boldface, underlining, and font size and color), inserting material (e.g.,external links and images), and organizing text (e.g., forming lists). A large selection ofgraphic “emoticons” (symbols) is available for expressing feelings such as :D (very happy)and :? (confused) that often accompany problem solving, and allow participants to simulatesocial presence (Gudergan, Josserand, & Pitsis, as cited in Lampel & Bhalla, 2007). Inaddition, using LaTeX, a document preparation system designed to typeset mathematicaltext, participants can use command strings and code to produce mathematical symbols(such as ∞) and vertical expressions (such as dA

dt ). In order to encourage the use of thissoftware, FMH includes a tutorial for LaTeX, as well as a link to a free equation editor thatgenerates the LaTeX code. In practice, it appears that the frequent users of the forum aremore likely to use LaTeX, and that the casual users make do with keyboard symbols (suchas ^ for exponential notation).

3 The members of FMH debated at one point whether or not to adopt such a system. In 2007, there was athread in the Administration Issues forum in which a helper proposed the incorporation of a “thank you”button that would feed member status points. However, other helpers objected, arguing that such a systemwould reward the provision of full worked solutions over other forms of help. At the conclusion of thediscussion, the forum administrator chose not to introduce such a feature.

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Unlike more sophisticated technologies, such as those used by Virtual Math Teams, achat-based environment that allows teams of students to collaboratively solve challengingproblems (Stahl, 2009), there is no “whiteboard” capability or tool that would allowparticipants to sketch graphs. The lack of a tool is a decided handicap for discussions ofmathematics, since “draw-a-figure” is one of the most frequently suggested strategies forproblem solving (Pólya, 1945; Schoenfeld, 1985) and furthermore, visualization is apowerful analytical tool (Zazkis, Dubinsky & Dautermann, 1996). However, forumparticipants can either create and upload constructions from other software or construct“ascii art.” The ways in which participants do mathematics under these conditions (e.g.,forced to render their problems and communicate meanings linguistically) remains to besystematically explored, although preliminary research suggests that students aredisinclined to include diagrams (or descriptions thereof) and that helpers may go toconsiderable lengths to incorporate them into an exchange (van de Sande & Leinhardt,2008a).

Each forum has moderators assigned by the site administrator who may lock topics andmove, delete, or edit postings. In addition, members can edit their own contributions afterthey have been posted. If this is done after the member has logged off of the forum, then amessage is appended to the altered contribution: “Last edited by [member] on [date andtime]; edited [number] times in total.” If editing takes place while the member is still loggedon to the forum, then there is no official evidence of the modification although the generalpractice appears to be for the author to indicate that the contribution has been edited.

Just as classrooms have norms and practices that describe the manner in whichparticipants interact with one another (and with the subject matter), a given online helpforum community can be characterized by its habits of discourse and interaction. In someforums, the netiquette is explicitly recorded at a location that is accessible to newcomersand available as a reference for others. In FMH, the netiquette, described in Appendix A, islocated in a “sticky4” that is the lead topic within each help forum, labeled “Read BeforePosting!!”

3.2 Population

FMH features member profiles that include information on occupation, location, andinterests. Whereas many students do not provide this information, the helpers in thecalculus forum are self-reportedly students, educators, professionals, and retired mathe-matics professors. Most helpers are from the USA, although there are representatives from avariety of other countries as well. Of course, the anonymity of the environment opens widethe possibility for role playing, so that members may be presenting an altered version ofwho they are in real or true self (McKenna, 2007).

As in many online groups (cf. Ahuja & Galvin, 2003), FMH has members who postmore frequently than others (core members) and those who temporarily drop by(peripheral members). This is true for both helpers and students in FMH, although thehelpers appear to be more committed over longer periods of time. The sample contained100 related rates exchanges initiated by 65 different students, with responses from 18different helpers and 100 limit exchanges initiated by 67 different students, withresponses from 23 different helpers. There was some overlap in participants (both

4 A “sticky” is a descriptive label for a thread that functions as a reminder in the same way that a physicalsticky note, or Post-it®, does.


students and helpers) across the two mathematical topics: 17% of these students postedqueries on both limits and related rates, and 63% of the helpers provided assistance forboth topics.

3.3 Assertions and proposals for action

The pattern of activity types was shared across topics: coasting (45% limit; 53% relatedrates) followed by slacking (31% limit, 21% related rates) as the largest categories,followed by sustaining (14% limit; 12% related rates) and ramping (9% limit; 12% relatedrates). A small percentage of the threads did not fit into any category because they wereeither unanswered (one related rates query) or because the student answered the questionher/himself without any helper intervention (one in each instructional topic).

These results indicate that students are generally contributing to the construction ofsolutions in FMH, and that contributions by students tend to occur in the initial post ratherthan later in the thread following helper intervention. Each of the categories is discussed inturn below and illustrated with a prototypical example.

Coasting There are three major origins of coasting: students seeking verification versusconstruction of a solution; helpers providing solutions or a significant amount of help; and,failure on the student’s part to carry through with the help seeking. First, students may beseeking verification of a solution to a problem with their work presented in the initial post.In this case, if a helper confirms the accuracy of the solution, there is no reason for thestudent to contribute further. If the solution is inaccurate and the helpers either point out theerror or provide a correct solution, then coasting reflects a failure of the student toresponsively reflect on errors as part of the help-seeking process

Second, when helpers provide students who have shown activity in the initial post with aworked solution (or a close approximation thereof), coasting is supported. In this case, themotivation for further student activity is stifled, since the helper has taken on the role as thepresenter of the mathematics and the student is effectively relegated to the role ofquestioning or accepting the solution. Although there are FMH helpers who object to theprovision of worked solutions by other helpers, the policy is that any helper may participatein the forum and employ any pedagogical approach.5

Finally, coasting may occur for a variety of reasons that are not discernible. Figure 3contains an example of coasting in which the student simply failed to return to the threadfollowing helper intervention. The student, semisweet, posted a problem on related ratesinvolving the rate at which the water level is decreasing in a leaky conical paper cup, andreceived a timely response from helper, squito.

Did I label everything right? In the initial post [10:40 pm], semisweet presents the problemsituation (“A conical paper cup of radius 5 cm and height 15 cm is leaking water at the rateof 2 cm^3/min. At what rate is the lvel of water decreasing when the water is 3 cm deep?”)

5 In 2008, there was an extensive (37 contributions) and heated discussion on the Administrative Issuesforum on the provision of worked solutions as a legitimate means of help, and whether FMH should adopt apolicy on this issue. Several different measures were debated, ranging from extreme measures delimitingparticipation (such as banning certain users and deleting posts) to more moderate policies (such as separationof turf, flagging posts, and issuing warnings), and also extending to constructive approaches (such asencouraging the use of skeletal worked solutions and worked solutions to analogous exercises). Theadministrator elected to adopt a live-and-let-live policy, encouraging helpers to respect and not interfere withany method being employed by another helper.

64 C. van de Sande

and defines three variables, namely r, h, and dV/dt (“So r=5 cm//h=15 cm//dv/dt=−2 cm^3/min”). Semisweet neglects to identify the quantities to which these variables refer, althoughbased on the letters used, presumably r represents some radius, h represents some height, andv represents some volume. The concept of rate is absent from these definitions, and there isno evidence that semisweet is cognizant of the difference between the radius of the physicalcup and the radius of the water level that is changing over time. One might surmise thatsemisweet is acting according to a script for solving (related rates) problems and defining avariable name for each numerical value that appears in the problem statement. This premise isstrengthened when semisweet questions whether the remaining numerical value of 3 cmshould be labeled as “v” or something else (“Is v=3 cm? or what is the 3 cm?”), and poses theproblem as one of labeling (“did I label everything right?”)

Things are changing FMH helper, squito, replies 17 min later [10:57 pm], emphasizingwith bold type how the quantities change over time: “3 cm=h is the depth of water in thecup at a specific point in time …h=15 cm when the cup is full” and “note that r, the radiusof water in the cup is changing also… r is 5 cm when the cup is full.” Although squito doesnot use function notation (e.g., h(t) and r(t)), these hints convey the distinction between theradius and height of the water level at a specific point in time and the radius and height ofthe physical cone (which corresponds to the values that the quantities assume when thecone is full). Also, an additional final hint is given that coordinates or relates the quantities:“the relationship between r and h at any time is r/h=5/15=1/3, so h=3r or r=h/3.”

The thread ends at this point. In terms of activity, this exchange is one in which thestudent is active initially (although demonstrating a flawed understanding) but fails to pickup on the hints provided by a helper, even though they were posted in short time. Inparticular, there are no corrections made in response to the feedback that the helperprovided and no further attempts on the part of the student to construct a solution. From the

Fig. 3 Exchange exhibiting coasting


perspective of forum participants, the student effectively coasts following the activitypresent in the initial post. In other forum exchanges, students fail to demonstrate even aninitial spurt of activity.

Slacking Slacking marks a thread in which the student fails to contribute anything (corrector incorrect) to the construction of the solution. In the initial post, the student posts theproblem but makes no attempt to contribute to the construction of a solution. Often,slacking is marked by a plea for help with a prevailing tone of despair and helplessness.Following helper intervention, the student still fails to contribute mathematically to thethread, either through silence (e.g., failing to return) or by pressing for further help. In eithercase, the student fails to demonstrate that s/he is interactively engaged in the process ofconstructing a solution. As with coasting, when helpers provide worked solutions, slackingis encouraged. The message to the student in such cases appears to be, as Skemp (cited inTall, 2002) noted with regard to undergraduate teaching approaches, that the product ofmathematical thought trumps the process of mathematical thinking. However, slacking canalso be attributed to the failure of the student to respond to scaffolding.

Table 4 contains the text of a slacking exchange in which the student presses for furtherhelp following helper intervention and then leaves the exchange.6 The student, June, posteda problem on limits involving the relationship between sales of a new product and theamount of money spent on advertising, and received help from two FMH helpers, Dayn and@_@.

Any help? In the initial post, June presents the problem situation but makes no attempt atconstructing a solution or proposing a strategy ([1]–[4]). After presenting the problemscenario, this curt posting ends with an emphatic plea for help and a politeness marker ([5]).

To start Three minutes later, Dayn responds with a strategy for evaluating the given limit,framed at two levels of detail ([6]–[7]). First, Dayn suggests that June multiply the fractionby the reciprocal square of the independent variable. As an alternative to carrying out thisprocedure, Dayn then mentions the general result. There is a sense here that Dayn iscovering all bases, not knowing whether or not June has learned the “rule” or is still at thestage of performing operations. The response also baldly presents strategies for reaching asolution and is unaccompanied by any form of explanation, a pedagogical move that mayfunction as a “didactical obstacle” by denying June an opportunity to learn (Harel &Sowder, 2005).

Unfortunately… Following this help, June returns to the exchange 17 min later. Afterquoting Dayn’s response, June expresses thanks (politeness [8]), but an inability to profitfrom the help ([9]). The helplessness is punctuated by the rolling eyes emoticon at the endof this posting.

Can you see what’s happening? A short time later, another helper, @_@, enters theexchange, fleshing out the hints from Dayn. First, however, note that both Dayn and @_@appear to have interpreted the sales function as S �ð Þ ¼ 25; 000�^2= 5�^2þ 5; 000ð Þ,rather than as S �ð Þ ¼ 25; 000�^2=5�^2ð Þ þ 5; 000, which is how it would be interpretedparentheses in absentia as typed by June in the initial post. Of course, the latter

6 In the interest of space, the contents of threads are from this point on reproduced verbatim in tables versusscreenshots.

66 C. van de Sande

interpretation makes no sense in the context of the problem, since it would model salesindependently of advertising funds. Neither helper takes up this issue with June but acts asthough the problem statement were simply the evaluation of the limit of a rationalfunction.7 In other words, it appears that the helpers are inured to these types of exercisestatements and view the problem scenario as a thin veil for asking students to evaluate alimit. Therefore, @_@ begins her/his post by explicitly showing June the multiplication ofthe rational function by 1/x2 ([10]) and explaining what is being done ([11]). Theexplanation continues with the mathematical principle behind this construction, first statedformally and then informally rephrased ([12]). The help continues with the resultingexpression from the suggested multiplication ([13]–[14]) and ends with a question checkingthe student’s understanding ([15]).

Because June failed to return to the exchange, we do not know if s/he saw what washappening, or not. We do know that the two helpers jointly provided progressively moredetailed, directed help, a practice that individual expert tutors adopt when it becomesobvious that the student is not profiting from the current level of assistance (Hume,Michael, Rovick & Evens, 1996). We also know that June slacked throughout the entireexchange and failed to contribute to the construction of the solution by exudinghelplessness and not responding to the scaffolding provided by the helpers. At the otherextreme, are instances in which student activity is evident both in the initial post andfollowing helper intervention.

Table 4 Text of slacking exchange: help with a limit function

Posted by [forumparticipant] at [time]

Text [line numbers corresponding to line breaks in the online threads]

June (new member) at5:07 am

[1] The sales of a new product over a period of time are expected to follow therelationship

[2] SðxÞ ¼ 25; 000x^2=5x^2þ 5; 000

[3] where x is the amount of money spent on advertising.

[4] Evaluate lim (x →inf) S(x)=_________

[5] Any help with this is much appreciated! Thanks!!

Dayn (full member) at5:10 am

[6] To start, multiply top and bottom of the fraction by 1/x^2, OR try toremember the rule for a limit of a ratio of polynomials

[7] Having equal degree.

June (new member) at5:27 am

[8] (Quotes [6]–[7])

[9] Thank you Dayn. Unfortunately I never received the rule and I don’t knowhow to multiply by 1/x^2 :roll: [rolling eyes].

@_@ (full member) at7:37 am

[10] 25;000x2

5x2þ5;000 �1x21x2

[11] Really, when you multiply both top and bottom by 1/x2, you’re justmultiplying by 1. But the idea is to use the fact that:

[12] limx!1

1xn ¼ 0 for n>0. (one over a really big number is pretty close to 0)

[13] So continuing on with multiplying by “1”,

[14] limx!1

25;0005þ5;000

x2

� �[15] Can you see what’s happening?

7 Other instances of ambiguity in communicating mathematical expressions have been addressed by forumhelpers. The need for parentheses, in particular, appears to be poorly understood by students and, onoccasion, is brought up within an exchange (van de Sande & Leinhardt, 2007).


Sustaining Sustaining describes how students can be interactive with helpers throughout anexchange. In some of these exchanges, there were discernable actions (such as questioning)from helpers that promoted continued student activity. However, there were more cases inwhich sustaining was attributable to student initiative. For example, consider the exchangein Table 5 between a student, Chrissie, and tutors, Honey and Galan, involving the rate atwhich a boat is approaching a dock. Chrissie participates in the joint construction of thesolution by making assertions and proposing mathematical actions throughout thediscussion, making connections to previous help received on the forum, and evaluatingthe state of her understanding.

I’m trying In this exchange, Chrissie begins by posting the problem statement ([1]–[3])and expressing her attempt to present a diagram of the problem situation ([4]–[5]). Ashort time later, she edits this post ([11]) with a verbal description of the diagram thatshe has constructed ([7]–[9]). In this initial contribution, Chrissie is assigning valuesto quantities and attempting to draw a diagram, despite the shortcomings of the userinterface.

What I’ve learned Prior to the publication of Chrissie’s verbal description of her diagram, ahelper, Honey, enters the discussion and poses a hint in the form of a question ([12]). Bydrawing attention to the presence of a right triangle, Honey, is positioning Chrissie tocontinue to contribute to the solution by broadly hinting that the quantities are related bythe Pythagorean theorem. Chrissie replies in the affirmative and describes the right triangle([15]–[16]). In this exchange, she also expresses a desire to make use of help that she hasreceived on another problem ([27]–[28]), proposes an assignment of variables ([29]–[30]),and presents a meta-analysis of her understanding of the connection between the problems([31]–[34]). With these actions, Chrissie not only establishes herself as an active participantin the discussion of this particular problem but also positions herself more broadly as acalculus learner who is trying to build connections between problem situations: The “lastproblem” was posted by Chrissie on the forum 47 min earlier and refers to a problem (alsoabout ships, hence the current thread title “Another moving ship problem I’m strugglingwith”) in which two ships are sailing away from one another (north- and eastbound) withgiven velocities and the rate at which the distance between them is changing at a certaintime is sought.

I’m making headway A short time later in this thread, Chrissie, posts another contributionin which she specifies the rates of change of the length of the rope and the height of thedock ([36]–[37]); establishes a relationship between the variables ([38]–[40]); differentiatesthe equation that relates the variables ([41]–[42]); and, proposes a solution goal ([44]–[46]).In this update on her understanding of the problem, Chrissie provides a large portion of thesolution concerning the dynamic aspect of the problem. There are correct values assigned tothe rates of change, and the relationship that connects them is also described. The piece ofthe solution that is lacking concerns the static aspect of the problem, that is, the values ofthe variables at the time in question: The length of the rope (represented by z) needs to bespecified at the time when the boat is 8 m from the dock, which is 1 m above the boat’sbow. In response to this contribution, another helper, Galan, enters the exchange and refersto the similarity between this problem and one for which Chrissie previously received help([47]–[48]). After specifying the relationship in terms of the same variables that Chrissieintroduced ([49]), Galan differentiates the equation ([50]–[51]); uses the relationship todetermine the length of the rope at the moment in question ([52]); and, coordinates the

68 C. van de Sande

Table 5 Text of sustaining exchange: another moving ship problem I’m struggling with


Text

Chrissie (junior member) at2:03 am

[1] A boat is pulled into a dock by a rope attached to the bow of the boat andpassing through a pulley on

[2] The dock that is 1 m higher than the bow of the boat. If the rope is pulledin at a rate of 1 m/x, how fast

[3] Is the boat approaching the dock when it is 8 m from the dock?

[4] I’m trying to get my work to show up, I tried this way to draw it and itlooked terrible so just a minute

[5] and you’ll be seeing another post with more work.

[6] Edit:

[7] Okay, the vertical distance is 1 m and is fixed

[8] The horizontal distance between the dock and the boat is 8 m and isdecreasing

[9] Then I have the velocity for the hypotenuse as decreasing by 1 m persecond.

[10] (I’m still trying to work on it….)

[11] Last edited by Chrissie on [date] 2:14 am, edited 1 time in total.

Honey (elite member) at2:12 am

[12] In your drawing, did you find a right triangle?


[13] (Problem statement)

[14] (Quotes [12])

[15] Yes.

[16] The base is 8 m, and the height is 1 m.

[17] Edit:

[18] ……………….|

[19] ……………….| 1 m fixed

[20] ……………….|

[21] ……………….|

[22] __________________

[23] 8 m

[24] This program is so frustrating. I had spaces because the line is supposedto be to the right, and it

[25] completely removed my spaces.

[26] 8 m and changing. Okay, ignore the… those are just to fill in space.

[27] So yes I do have a right triangle and I’m just trying to figure out how touse the stuff I used on the last

[28] problem to solve this.

[29] Let x be the distance between the boat and the dock.

[30] 8-x at any particular time.

[31] Okay here is where I’m getting confused applying the previous problem’sknowledge. I think I should

[32] take 8−x and take the derivative, but I’m obviously missing something. Iknow the rate of change of the

[33] rope, and I know the vertical distance isn’t changing. Somehow, thehorizontal distance must be related

[34] to the hypotenuse rope pulling in. I’m lost though. ☹ [Sad]


Table 5 (continued)


Text


[35] Still struggling (new topic heading)

[36] dzdt ¼ �1

[37] dydt ¼ 0

[38] z2 ¼ ðxÞ2 þ y2

[39] z2 ¼ ðxÞ2 þ 12

[40] z2 ¼ ðxÞ2 þ 1

[41] Differentiate:

[42] 2z dzdt ¼ 2x dxdt þ 0 this is Eq. [1]

[43] Okay, well I managed to make something look okay by copying andpasting from someone else using

[44] that “tex” notation. But I haven’t gotten much further understanding this. Ihave an idea that I should

[45] Solve for

[46] dxdt ¼ ??? at x=8

Galan (elite member) at3:05 am

[47] This is a lot like the other problem.

[48] You have another Pythagoras deal.

[49] z2 ¼ x2 þ y2

[50] Differentiate:

[51] z dzdt ¼ x dxdt þ y dy

dt

[52] z ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi82 þ 12

p ¼ ffiffiffiffiffi65

p

[53]ffiffiffiffiffi65

p ð1Þ ¼ ð8Þ dxdt þ ðyÞð0Þ[54] Note dy/dt remains constant, so it is 0.

[55] Solve the above equation for dx/dt.


[56] (Quotes [47]–[51])

[57] So this equation z=is more to help me know what to put on the next line(like the line above and below are more

[58] related?

[59] (Quotes [52]–[55])


[60] (Quotes [47]–[55])

[61]ffiffiffiffiffi65

p ð1Þ ¼ ð8Þ dxdt þ ðyÞð0Þ[62]

ffiffiffiffiffi65

p ¼ ð8Þ dxdt[63] sqrt65

8 ¼ dxdt

[64]=8.062257748/8

[65]=1.007782219

[66] Does this seem like I’m doing the right thing at all????

[67] It just seems like an approximately right answer (the height isn’t verymuch), and so it would make

[68] sense that the boat is moving toward the dock at a pretty similar rate asthe rope is pulling it in, but I

[69] just don’t know if I’ve done everything right.


[70] Is this right? I feel fairly certain that I did do the right thing (after beingconfused on how to set it up

[71] initially.)

70 C. van de Sande

dynamic and static information ([53]). With this presentation, Galan, explicitly provides theinformation that Chrissie was lacking, namely the length of the rope when the boat is 8 mfrom the dock, and shows how this information should be coordinated with the relationshipbetween the variables and their rates of change.

Is this why? Six minutes later, Chrissie, returns to the discussion and (rhetorically)questions precisely this aspect of the solution, setting off her inserted remark in red ([57]–[58]). That is, following Galan’s intervention, Chrissie appears to be cognizant of the wayin which the static and dynamic aspects of the problem situation are coordinated andpositions herself as someone seeking to understand the help being received in addition toarriving at the correct answer to the problem.

Sense-making One minute later, Chrissie, posts another contribution that takes up themathematics where Galan left off ([61]–[65]). In addition, Chrissie responds to her ownquestion of whether she is “doing the right thing at all” by assessing the plausibility of thisnumerical value with respect to the physical situation ([66]–[69]). This sense-making movein a patently contrived problem statement is remarkable since interpreting the result is notmentioned as part of the instructions for the exercise. However, Chrissie is placing herselfwithin the problem situation (either as an actor or spectator), playing out the scenario, andrecognizing constraints on the solution space, while at the same time hedging herpresentation of the solution.

Am I right? Eighteen minutes later, having received no reply, Chrissie posts anothermessage seeking confirmation of the accuracy of the solution ([70]–[71]). Here,Chrissie demonstrates an increase in confidence as she assesses the construction of thesolution, which she takes ownership of (“I feel fairly certain that I did do the rightthing…” [italics added]), and attributes her confusion to the initial stage of problemsolving. Several hours later, Galan responds and confirms Chrissie’s conclusion ([72]).The final posting in this exchange is made by Chrissie as she expresses appreciation forthe forum helpers and the effective help that she has received ([73]), together withhappiness at the outcome ([74]).

In terms of activity, this exchange is one in which the student is active throughout,making assertions and proposing mathematical actions, working to understand thecontributions of others, repeatedly assessing the state of her understanding, and engagingin sense making once a numerical result is produced. Although some learning theoristsmight quibble with the amount of help that Galan provided and the manner in which thiswas done (which resembles direct instruction), the interaction did prove instrumental inmoving Chrissie from being “lost” ([34]) to an acknowledged better understanding of thisrelated rates exercise, reflected in the progression of emoticons from :([sad] to :) [smile]

Table 5 (continued)


Text

Galan (elite member) at1:11 pm

[72] Seems OK to me.

Chrissie (junior member) at9:37 pm

[73] Thanks everyone for all the help in learning & understanding these.

[74] ☺ [Smile]


over the course of this exchange.8 Largely due to the persistence and efforts of the student,the exchange represents sustained interactive activity. In other instances, the presence ofstudent activity only occurs following helper intervention.

Ramping Ramping occurs when student activity grows over the course of the thread, fromno activity in the initial post to the presence of activity following helper intervention. Therewere two features of helper intervention that appeared to stimulate ramping. First, whenhelpers remonstrated with students for not showing work when the problem was posted,some students responded with demonstrable activity. In these situations, students may havebeen unaware of (or may have chosen to ignore) FMH netiquette, but were open to revisingtheir practice in order to procure help. Ramping was also a behavior associated withquestioning on the part of the helpers. When helpers posed questions in response tostudents who posted a problem without any work shown, some students became activelyengaged in constructing the solution. In either case (in response to prompting orquestioning from helpers), ramping can be viewed as an alternative outcome to whatotherwise would be slacking.

Table 6 contains an example of ramping that occurred in an exchange between a studentand three helpers. The helpers collectively worked with the student, Oneiroi, in theconstruction of a solution to a trigonometric limit exercise.

I don’t know what to do When Oneiroi posts the exercise, s/he baldly presents the problemstatement as the title of the thread ([1]). In lieu of showing work, Oneiroi describes why thisexercise is a problem and how it is different than previously encountered limit exercises([2]).

Okay, what would you have done? Four minutes later, FMH helper, Eliz, responds byquoting Oneiroi’s claim and presses Oneiroi to demonstrate exactly what s/he would havedone in the suggested situation ([4]). This command, that violates the rules of politenessand saving face (Brown & Levinson, 1987), is softened somewhat by the inclusion of“Thank you!” ([5]) and the :D emoticon ([6]).

Tired, grumpy, and off to bed At the same time, another FMH helper, John, posts someadvice to Oneiroi. John proposes that the expression will have to be transformed intodeterminate form ([7]–[8]), and proposes two strategies ([9]). This contribution, however,does not take Oneiroi’s suggestions and demonstration of understanding into account, asneither strategy is consistent with the rearrangement that this student appears to be hintingat. In other words, it is a nonadaptive tutoring move (Chi, 2004) that is framed at providinghints to solve the exercise rather than coaching the student. In John’s favor, this advice wasgiven under suboptimal conditions that underscore the human aspect of online forumexchanges ([10]). Notice also that John includes a “signature” ([12]), which is oftencharacteristic of core online community members and reveals something about himself andhis online persona.

I would have… Oneiroi responds to both helpers in a post 5 min later. S/he first providescontextual information ([13]) and then provides the solution to the alternative exercise asprompted by Eliz ([14]–[16]). It is now evident that Oneiroi is framing this alternative

8 Recent research on collaboration considers affective, as well as cognitive and social, dimensions ofcollaborative learning (Andriessen, Baker & van der Puil, 2011; Baker, 2010).

72 C. van de Sande

exercise as a rearrangement that relies on a previously learned result but that s/he does notsee how this principle can be generalized to the current exercise.

You can apply that exact same thing At this point, another helper, @_@, enters the thread witha next step hint based on Oniromancy’s presentation ([17]–[18]). The post concludes with thegeneral statement of the limit result and its application to the reciprocal expression ([19]–[21]).

Questioning After Oneiroi receives this hint, s/he still has questions ([22]), which are posedin a follow up post. Apparently, Oneiroi is unsure how this procedure will be useful,anticipates a problematic outcome, and therefore questions it.

Table 6 Text of ramping exchange: lim x→>0 sin5x/sin4x


Text

Oneiroi (new member) at1:36 am

[1] What the title says, find the limit.

[2] If the denominator was say, 4x, I would have no problem doing this but thesin x function messed up I don’t know what to do.

Eliz (elite member) at1:40 am

[3] (Quotes portion of [2], up to “doing this”)

[4] Okay, what would you have done, were that the limit?

[5] Please be complete. Thank you. :D

[6] Eliz.

John (full member) at1:40 am

[7] When the limit can’t be evaluated by simply plugging in the value, you willhave to do some manipulation to get it into

[8] determinate form.

[9] Look into trig identities or use L’Hopitals rule.

[10] Tired, grumpy, and off to bed….. best of luck!

[11] john

[12] (signature) “If I have seen further it is by standing on the shoulders ofgiants.” –Isaac Newton


[13] I’m only in Cal I so L’Hopital’s rule is off limits.

[14] If the problem were sin 5x/4x I would have multiplied by a factor of oneby using (5/4)/(5/4) so that it become sin 5x/5x

[15] so the problem becomes

[16] 1.25×limit x>0 sin 5x/5x=5/4


[17] You can apply that exact same thing to this problem as well. Trymultiplying by:

[18] 5x�4x5x�4x

[19] Can you see what you can do with this using the identity:

[20] limx!0

sin xx ¼ 1

[21] limx!0

xsin x ¼ lim

x!0

1sin xx¼ 1

1 ¼ 1


[22] 5x×4x wouldn’t get rid of the denominator?


[23] sin 5xð Þsin 4xð Þ � 5x4x5x4x ¼ sin 5xð Þ

5x � 4xsin 4xð Þ � 5x4x


[24] You gave it away. The answer is 5x/4x, right? 4x/sin 4x=1/(sin 4x/4x)

[25] Thanks.

K_S (senior member) at2:34 am

[26] (Quotes portion of [24], ending with “Right?”) → yes and no!!

[27] (Quotes remaining portion of [24])


More detail In response, @_@ elaborates on the suggested strategy by presenting the threeterms that result from the rearrangement ([23]). As in the slacking exchange (Table 4), thehelp has become more detailed when confusion is detected.

You gave it away Oneiroi returns to the thread 5 min later with a contribution that advancesthe construction of the solution to the exercise based on the help received from @_@ ([24])and an expression of gratitude ([25]). Here, Oneiroi has partially applied the advice from@_@ and uses this as the explanation for the result. However, by applying the limit to twoterms in the product (sin5x/5x and 4x/sin4x) and not to the 5x/4x term, Oneiroi demonstratesthat s/he may still lack an established understanding of the solution. This sense isunderscored by the uncertainty and request for verification that follows the assertion of theanswer ([24]).

Yes and No!! The last post in this thread is from a third FMH helper, K_S who responds byquoting Oneiroi’s previous contribution ([26]–[28]) and interjecting an emphatic qualifi-cation (in red font) following the proposed answer ([26]). With this curt critique, K_S iscommunicating rather obliquely to the student that the term 5x/4x is part of the solution butthat it is not the final answer.

In terms of activity, this exchange is one in which the student becomes active followingintervention from forum helpers. Although we have no way of ascertaining the outcome ofthe thread, since Oneiroi does not return following the last helper intervention, we do haveevidence that the forum interaction was instrumental in promoting activity and encouragingthe student to make inferences and contribute to the construction of the solution of theexercise at hand. Ironically, it seems as though the student wished to contribute more thanwas expected by the helpers in this exchange ([24]). This example illustrates, furthermore,how a help trajectory can be achieved through the cooperative effort of multiple forumhelpers on an SOH site (van de Sande & Leinhardt, 2007).

4 Conclusions

Free, open, online, help forums have emerged, presumably in response to a prevailing anduniversal need for accessible, efficient, cost-effective, and personable homework assistancethat is theoretically available around the clock. The forums have transformed help seekingin a grass roots fashion from a private and individual encounter into a public and socialactivity. Through participation in a forum, students who seek help completing theirassignments are connected with others around the world willing to contribute their time andexpertise. More broadly, given the popularity of “lurking,” vicarious learning (cf.,Rosenthal & Zimmerman, 1978) through examining archived threads is also a strongpossibility. In short, the forums currently supply help on demand to a large populationof geographically distributed students and are worthy of further attention fromeducational researchers as a twenty-first century Vygotskian (1978) embodiment ofsocial learning and support.

In particular, open, online, help forums represent accessible and natural environments inwhich student responses to pedagogical approaches can be evaluated and assessed. Theforum discussed in this paper does not sanction any single pedagogical approach and theresult is a myriad of help tactics, ranging from the provision of worked solutions to guidingand scaffolding. Student activity on this site also ranges from slacking to sustaining.

74 C. van de Sande

Observing this forum raises questions about the relationship between the type of studentactivity in a given thread and the (perceived) helpfulness of the activity, the ways in whichhelpers decide which approach to take in a given situation, and the factors that influence thenature of student activity.

Open, online, help forums represent a natural environment in which student responses toalternative pedagogical approaches for constructing solutions can be tested, and, in whichhelp seeking in general can be explored. Help seeking, although once equated with lazinessand incompetence, is now recognized as a legitimate part of the learning process dependingon how it is carried out and the performance goals of the participants (Ames, 1983;Karabenick, 1998; Karbenick & Newman, 2006; Nelson-Le Gall, 1981; 1985; Newman,1994). Next steps include mapping out the relationship between help seeking and forumactivity, and learning how best to support students in making use of help forums as aresource that makes them more autonomous, self-regulated learners.

Acknowledgments The author wishes to acknowledge Gaea Leinhardt for her thoughtful comments andfeedback on earlier versions of this paper, as well as the reviewers for their helpful suggestions.

Appendix A

FMH netiquette

The netiquette represents a set of guidelines for student activity that become practice in asmuch as it is enforced through the actions of helpers and moderators. FMH netiquette,which is directed at students, covers administrative issues such as organization (Post to anappropriate category) and abuse (Don’t spam), and also presents guidelines for the effectivecommunication of mathematical text (Preview or edit your posts for clarity) and requests(Try to use halfway-decent English). The two guidelines addressing politeness (Havepatience and Be nice) emphasize the fact that the helpers are volunteers, and that helpseekers should be cognizant of them as genuine people.

Finally, there are three FMH guidelines that specifically address the contents of requestsfor help. Don’t post a list of homework problems. FMH discourages students from postingmultiple problems in a single thread and posting multiple threads containing individualproblems in a short amount of time. The intent is that students seek help on a specificproblem and then apply this help to similar exercises.

Post the complete text of the exercise This guideline addresses the conversationalimplicatures (Grice, 1989) in open, online, help forums. Conversations between helpseekers and helpers require simultaneously more and less information than instructionalconversations in other venues. First, helpers need more specificity and detail in the questionbeing posed because the forums are open so that they do not have much information aboutthe instructional context in which the student is working. At the same time, forumconversation is sparse and stripped down from other instructional discourse because it cutsright to the construction of a solution of a particular problem situation.

Show all of your work The rationale provided for directing students to show all of theirwork is to assist helpers in targeting errors. If no work or only partial work is shown, then itmay be difficult to discern where, and the extent to which, a student needs help. At thesame time, having this directive in the netiquette makes a statement about student


responsibility, accountability, and expectations—critical and controversial issues in anunregulated help-seeking environment.

References

Ahuja, M. K., & Galvin, J. E. (2003). Socialization in virtual groups. Journal of Management, 29(2), 161–185.

Ames, R. (1983). Help-seeking and achievement orientation: Perspectives from attribution theory. In B.DePaulo, A. Nadler, & J. D. Fisher (Eds.), New directions in helping (Vol. 2, pp. 165–186). New York:Academic.

Andriessen, J., Baker, M., & van der Puil, C. (2011). Socio-cognitive tension in collaborative workingrelations. In A. Lund, I. Rasmussen, & R. Saljö (Eds.), Learning across sites: New tools, infrastructuresand practices. London: Pergamon.

Avery, C., Resnick, P., & Zeckhauser, R. (1999). The market for evaluations. The American EconomicReview, 89(3), 564–584.

Baker, M. (2010). Close collaboration, dialogical thinking, and affective regulation. In G. Stevens (Ed.),International reports on socio-informatics: Workshop proceedings of 9th international conference on thedesign of cooperative systems (Vol. 7, Issue 1, pp. 57–64). Aix-en-Provence, France: IISI.

Baym, N. (1996). Agreements and disagreements in a computer-mediated discussion. Research on Languageand Social Interaction, 29(4), 315–345.

Bernard, R. M., Abrami, P. C., Lou, Y., Borokhovski, E., Wade, A., & Wozney, L. (2004). How does distanceeducation compare with classroom instruction? A meta-analysis of the empirical literature. Review ofEducational Research, 74(3), 379–439.

Bernard, R. M., Abrami, P. C., Borokhovski, E., Wade, C. A., Tamim, R. M., & Surkes, M. A. (2009). Ameta-analysis of three types of interaction treatments in distance education. Review of EducationalResearch, 79(3), 1243–1289.

Blumenfeld, P. C., Kempler, T. M., & Krajcik, J. S. (2006). Motivation and cognitive engagement in learningenvironments. In R. K. Sawyer (Ed.), The Cambridge handbook of the learning sciences (pp. 475–488).New York: Cambridge University Press.

Borba, M. C. (2005). The transformation of mathematics in on-line courses. In H. L. Chick & J. L. Vincent(Eds.), Proceedings of the 29th conference of the International Group for the Psychology of MathematicsEducation, Vol 2 (pp. 169–176). Melbourne: PME.

Borba, M. C., & Villarreal, M. E. (2005). Humans-with-media and reorganization of mathematical thinking:Information and communication technologies, modeling, experimentation and visualization. New York:Springer.

Borba, M. C., & Zulatto, R. B. A. (2006). Different media, different types of collective work in onlinecontinuing teacher education: Would you pass the pen, please? In J. Novotná, H. Moraová, M. Krátká, &N. Stehlíková (Eds.), Proceedings of the 30th Conference of the International Group for the Psychologyof Mathematics Education (pp. 201–208). Prague: PME.

Brown, P., & Levinson, S. C. (1987). Politeness: Some univerals in language use. Cambridge: CambridgeUniversity Press.

Chi, M. T. (2004). Can tutors monitor students’ understanding accurately? Cognition and Instruction, 22(3),363–387.

Chi, M. T. (2009). Active-constructive-interactive: A conceptual framework for differentiating learningactivities. Topics in Cognitive Science, 1, 73–105.

Cooper, H. (2007). The battle over homework: Common ground for administrators, teachers, and parents(3rd ed.). Thousand Oaks: Corwin Press.

Cornu, B. (1991). Limits. In D. Tall (Ed.), Advanced mathematical teaching (pp. 153–166). Boston: Kluwer.Cottrill, J., Dubinsky, E., Nichols, D., Schwingendorf, K., Thomas, K., & Vidakovic, D. (1996).

Understanding the limit concept: Beginning with a coordinated process scheme. Journal ofMathematical Behavior, 15, 167–192.

Dellarocas, C. (2003). The digitization of word of mouth: Promises and challenges of online feedbackmechanisms. Management Science, 49(10), 1407–1424.

Engelke-Infante, N. (2004). Related rates problems: Identifying conceptual barriers. Paper presented at the26th Annual Conference of the North American Chapter of the International Group for the Psychologyof Mathematics Education, Toronto, Ontario, Canada.

76 C. van de Sande

Engelke-Infante, N. (2007). Students’ understanding of related rates problems in calculus. Tempe, AZ:Unpublished Doctoral Dissertation, Arizona State University.

Engelbrecht, J., & Harding, A. (2005). Teaching undergraduate mathematics on the internet. Part 2:Attributes and possibilities. Educational Studies in Mathematics, 58, 253–276.

Ess, C. (2007). Internet research ethics. In A. Joinson, K. McKenna, T. Postmes, & U.-D. Reips (Eds.), TheOxford handbook of internet psychology (pp. 487–502). New York: Oxford University Press.

Goodyear, P., Jones, C., Asensio, M., Hodgson, V., & Steeples, C. (2005). Networked learning in highereducation: Students’ expectations and experiences. Higher Education, 50(3), 473–508.

Green, M. C. (2007). Trust and social interaction on the internet. In A. Joinson, K. McKenna, T. Postmes, &U.-D. Reips (Eds.), The Oxford handbook of internet psychology (pp. 43–51). New York: OxfordUniversity Press.

Greeno, J. G. (2006). Learning in activity. In R. K. Sawyer (Ed.), The Cambridge handbook of the learningsciences (pp. 79–96). New York: Cambridge University Press.

Grice, H. P. (1989). Studies in the way of words. Cambridge, MA: Harvard University Press.Harel, G., & Sowder, L. (2005). Advanced mathematical thinking at any age: Its nature and its development.

Mathematical Thinking and Learning, 7, 27–50.Herring, S. C. (2011). Computer-mediated discourse. In D. Tannen, D. Schiffrin, & H. Hamilton (Eds.),

Handbook of discourse analysis. Oxford: Blackwell.Hume, G., Michael, J., Rovick, A., & Evens, M. (1996). Hints as a tactic in one-on-one tutoring. The Journal

of the Learning Sciences, 5, 23–47.Karabenick, S. A. (1998). Help seeking as a strategic resource. In S. A. Karabenick (Ed.), Strategic help

seeking: Implications for learning and teaching (pp. 1–11). Mahwah, NJ: Erlbaum.Karabenick, S. A., & Newman, R. S. (Eds.). (2006). Help seeking in academic settings: Goals groups, and

contexts. Mahwah, NJ: Erlbaum.Kollock, P. (1999). The economics of online cooperation: Gifts and public goods in cyberspace. In M. A.

Smith & P. Kollock (Eds.), Communities in cyberspace (pp. 220–239). London: Routledge.Lampel, J., & Bhalla, A. (2007). The role of status seeking in online communities: Giving the gift of

experience. Journal of Computer-Mediated Communication, 12(2), article 5.Larreamendy-Joerns, J., & Leinhardt, G. (2006). Going the distance with online education. Review of

Educational Research, 76(4), 567–605.Leinhardt, G. (2001). Instructional explanations: A commonplace for teaching and location for contrast. In V.

Richardson (Ed.), Handbook of research on teaching (pp. 333–357). Washington, DC: AmericanEducational Research Association.

Lozano Teran, G. (2009). Teachers’ management of mathematics tasks associated to routine and non-routineproblems: A look at 5th and 6th grade classrooms in a quality Mexican school in the US–Mexico border.Paper presented at the ASU Mathematics Education Colloquium.

Martin, T. S. (2000). Calculus students’ ability to solve geometric related-rates problems. MathematicsEducation Research Journal, 12(2), 74–91.

Mayer, R. E., Larkin, J., & Kadane, J. B. (1984). A cognitive analysis of mathematical problem solvingability. In R. J. Sternberg (Ed.), Advances in the psychology of human intelligence (pp. 231–273).Hillsdale, NJ: Lawrence Erlbaum Associates, Inc.

McKenna, K. Y. A. (2007). Through the internet looking glass: Expressing and validating the true self. In A.Joinson, K. McKenna, T. Postmes, & U.-D. Reips (Eds.), The Oxford handbook of internet psychology(pp. 205–221). New York: Oxford University Press.

Nelson-Le Gall, S. (1981). Help-seeking: An understudied problem-solving skill in children. DevelopmentalReview, 1, 224–246.

Nelson-Le Gall, S. (1985). Help-seeking behavior in learning. Review of Research in Education, 12, 55–90.Newman, R. S. (1994). Adaptive help seeking: A strategy of self-regulated learning. In D. H. Schunk & B. J.

Zimmerman (Eds.), Self-regulation of learning and performance: Issues and educational applications(pp. 283–301). Hillsdale, NJ: Erlbaum.

Pólya, G. (1945). How to solve it. Princeton, NJ: Princeton University Press.Puustinen, M., Volckaert-Legrier, O., Coquin, D., & Bernicot, J. (2009). An analysis of students’

spontaneous computer-mediated help seeking: A step toward the design of ecologically valid supportingtools. Computers & Education, 53(4), 1040–1047.

Renninger, K. A., Ray, L. S., Luft, I., & Newton, E. L. (2005). Coding online content-informed scaffoldingof mathematical thinking. New Ideas in Psychology, 23, 152–165.

Rheingold, H. (1993). The virtual community: Homesteading on the electric frontier. New York: Addison-Wesley.Rosenthal, R. L., & Zimmerman, B. J. (1978). Social learning and cognition. New York: Academic.Rowland, T. (1995). Hedges in mathematics talk: Linguistic pointers to uncertainty. Educational Studies in

Mathematics, 4, 327–353.


Schoenfeld, A. (1985). Mathematical problem solving. San Diego, CA: Academic.Schoenfeld, A. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making

in mathematics. In D. A. Grows (Ed.), Handbook of research on mathematics teaching (pp. 334–370).New York: Macmillan.

Sebrechts, M. M., Enright, M., Bennett, R. E., & Martin, K. (1996). Using algebra word problems to assessquantitative ability: Attributes, strategies, and errors. Cognition and Instruction, 14(3), 285–343.

Selden, A., Selden, J., Hauk, S., & Mason, A. (2000). Why can’t calculus students access their knowledge tosolve non-routine problems? In E. Dubinsky, A. Schoenfeld & J. Kaput (Eds.), Research in collegiatemathematics education (pp. 128–153). American Mathematical Society.

Shea, V. (1994). Netiquette. San Francisco: Albion.Solomon, Y., Warin, J., & Lewis, C. (2002). Helping with homework? Homework as a site of tension for

parents and teenagers. British Educational Research Journal, 28(4), 603–622.Spears, R., Lea, M., & Postmes, T. (2007). CMC and social identity. In A. Joinson, K. McKenna, T. Postmes,

& U.-D. Reips (Eds.), Oxford handbook of internet psychology (pp. 253–272). Oxford: OxfordUniversity Press.

Stahl, G. (2009). Interactional methods and social practices in VMT. In G. Stahl (Ed.), Studying virtual mathteams (Computer-Supported Collaborative Learning Series) (pp. 41–56). New York: Springer.

Szydlik, J. E. (2000). Mathematical beliefs and conceptual understanding of the limit of a function. Journalof Research in Mathematics Education, 31, 258–276.

Tall, D. (2002). The psychology of advanced mathematical thinking. In D. Tall (Ed.), AdvancedMathematical Thinking (pp. 3–21). New York, New York: Springer.

Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular referenceto limits and continuity. Educational Studies in Mathematics, 12(2), 151–169.

VanderMeij, H. (1988). Constraints on question asking in classrooms. Journal of Educational Psychology,80, 401–405.

van de Sande, C., & Leinhardt, G. (2007). Online tutoring in the calculus: Beyond the limit of the limit.Éducation et Didactique, 1(2), 115–154.

van de Sande, C., & Leinhardt, G. (2008a). Drawing conclusions about diagram use in an online help forum.Paper presented at the 11th Conference on Research in Undergraduate Mathematics Education(CRUME), San Diego, CA.

van de Sande, C., & Leinhardt, G. (2008b). The Good Samaritan effect: A lens for understanding patterns ofparticipation. In P. A. Kirschner, F. Prins, V. Jonker, & G. Kanselaar (Eds.), Proceedings of the 8thInternational Conference for the Learning Sciences ICLS 2008, Vol II (pp. 240–247). The Netherlands:ISLS.

Vinner, S. (1991). The role of definitions in the teaching and learning of mathematics. In D. Tall (Ed.),Advanced mathematical thinking (pp. 65–81). Dordrecht, The Netherlands: Kluwer.

Vygotsky, L. S. (1978). In M. Cole, V. John-Steiner, S. Scribner, & E. Souberman (Eds.), Mind in society:The development of higher psychological processes. Cambridge: Harvard University Press.

Williams, S. R. (1991). Models of limit held by college calculus students. Journal for Research inMathematics Education, 22, 219–236.

Zazkis, R., Dubinsky, E., & Dautermann, J. (1996). Coordinating visual and analytic strategies: A study ofstudents’ understanding. Journal for Research in Mathematics Education, 27(4), 435–457.

78 C. van de Sande

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