coding online content-informed scaffolding of mathematical thinking

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New Ideas in Psychology 23 (2005) 152–165

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Coding online content-informed scaffolding ofmathematical thinking

K. Ann Renningera,�, Lillian S. Raya, Ilana Luftb, Erica L. Newtonc

aSwarthmore College, Department of Educational Studies, 500 College Avenue, Swarthmore, PA 19081-1397, USAbState University of New York at Albany, USA

cUniversity of Pennsylvania, USA

Available online 11 July 2006

Abstract

This article describes the Mathematical Mentoring Coding Scheme (MMCS), a coding scheme that

identifies indicators relevant to the study of content-informed scaffolding. Content-informed

scaffolding refers to the use of subject matter content in ill-defined problem spaces to focus the

learner, and provide and fade feedback so that the learner becomes autonomous. It is suggested that

the MMCS could also double as a rubric for instruction in content-informed scaffolding. Two case

examples of preservice teachers’ scaffolding of elementary students’ problem solving serve as

illustrations.

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1. Introduction

The task of scaffolding may be one of the most challenging and ill-defined aspects ofteaching and learning. Sometimes linked with coaching (West & Staub, 2003), distributedlearning (distributed scaffolding, Puntambekar & Kolodner, 1998), guided participation(Rogoff, 1990), instruction (instructional scaffolding, Hogan & Pressley, 1997), andmentoring (Zachary, 2000), scaffolding involves providing a learner with feedback andthen fading, or adjusting, this feedback in relation to the learner’s responses (Collins,Brown, & Newman, 1989; Wood, Bruner, & Ross, 1976). In addition, it involves modelingways of thinking (asking questions, revising strategies, etc.) for the learner. Thus,scaffolding is a dynamic and reciprocal collaborative process.

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dress: [email protected] (K. Ann Renninger).

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Where the human tutor excels or errs, of course, is in being able to generatehypotheses about the learner’s hypotheses and often to converge on the learner’sinterpretation. It is in this sense that the tutor’s theory of the learner is so crucial tothe transactional nature of tutoring y effectiveness depends y upon the tutor andtutee modifying their behavior over time to fit the perceived requirements and/orsuggestions of the other. The effective tutor must have at least two theoretical modelsto which he must attend. One is a theory of the task or problem and how it may becompleted. The other is a theory of the performance characteristics of his tutee.Without both of these, he can neither generate feedback nor devise situations inwhich his feedback will be more appropriate for this tutee in this task at this point intask mastery. The actual pattern of effective instruction, then, will be both task andtutee dependent, the requirements of the tutorial being generated by the interaction ofthe tutor’s two theories. (Wood et al., 1976, p. 97).

Underlying the idiosyncratic nature of scaffolding are theories that are likely to bemediated by the learner’s and the scaffolder’s interests, problem-solving beliefs, and levelsof skill (Renninger, Ray, Luft, & Newton, 2006)—not to mention the sociocultural contextin which scaffolding is extended, or the nature and context of the task (Cocking &Renninger, 1993; Collins et al., 1989; Pea, 2004; Wood et al., 1976). Building ondiscussions of psychological distance (Cocking & Renninger, 1993; Collins et al., 1989;Hunt, 1961; Piaget, 1954; Renninger & Cocking, 1993; Rogoff, 1990; Sigel, 1970, 1993;Vygotsky, 1978; Wood et al., 1976), scaffolding describes bridging the distance betweenwhat the learner understands and what still needs to be understood by focusing the learner,and both providing and fading task-informed feedback. The task of scaffolding is notsimply promotion of performance (Pea, 2004), nor does it lack subject matter content(West & Staub, 2003). The Mathematical Mentoring Coding Scheme (MMCS; seeAppendix A), described in this article, was developed in order to study a practice that isoften either assumed or overlooked in discussions of scaffolding (Pea, 2004): using subjectmatter content to focus the learner, and providing and fading feedback in ill-definedproblem spaces, so that the learner becomes autonomous. Two case examples are providedas illustrations.

Scaffolding in the classroom is complicated, because some students may receivescaffolding while others may not, and/or some students may seek feedback while othersmay not. In addition, the person providing scaffolding may not know what had beensuggested as a previous course of action. Furthermore, scaffolding in the context of theclassroom may not be appropriately gauged to the students’ strengths and needs, and/or tothe possibilities for working with the given task, at least in part because there is so littletime for reflection, and for some teachers, due to their level of content knowledge.

Study of scaffolding in the classroom is also compromised because reliably identifyingwhat counts as scaffolding is challenging, and there is no guarantee that there are enoughrepeat instances to use for purposes of comparison. Because scaffolding in the classroom isso complex, it is difficult to assess whether its effectiveness is due to one or anothercondition. Thus, even if the complexity of scaffolding in the classroom is fully appreciated,it is hard to identify and study scaffolding in this context (see related discussion in Davis &Miyake, 2004).

In contrast, the online problem-solving environment provides a controlled setting forscaffolding when a live mentor interacts with a learner around his or her problem solving.

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Online, feedback can be provided for every problem that the learner engages. Also,because this feedback is delivered in an asynchronous context, it affords both the learnerand the person doing the scaffolding time to reflect on the problem, as well as the work ofthe other, over multiple interactions. In addition, because work online is often archived,the person doing the scaffolding can review prior work of the learner and/or his or herability to make use of feedback. Time for reflection and archived prior work allow eachnew interaction to be faded according to the learner’s needs.In contrast to the complications of studying scaffolding in the classroom, study of

scaffolding online allows agreement on the characteristics of scaffolding that might beassessed and coded. Not only does the online environment allow for more plannedscaffolding, the archiving of exchanges also permits focused analysis of scaffoldingcharacteristics, increasing the likelihood that conditions supporting effective scaffoldingcan be identified. It appears that the study of online scaffolding should inform both thepractice and study of scaffolding in the classroom.The MMCS was developed to assess the quality of mentoring support for elementary

students’ mathematical thinking as part of the National Science Foundation-funded Online Mentoring Project (OMP; see Renninger et al., 2006; Ray & Renninger,2006). In the OMP, preservice teachers (PTs) mentored students’ work with the MathForum’s Problems of the Week (PoWs). Like the Dad’s Cookies problem depicted in Fig. 1,the PoWs are non-routine challenge problems, problems that are not easily solved bysimply applying an algorithm. In submitting their solutions, learners write out the solutionpaths they used and are then scaffolded by mentors. While this type of scaffolding mayconsist of a single exchange, the learner has an open-ended opportunity to revise andresubmit his or her solution and, in this sense, assume more responsibility for learning.Typically, this type of scaffolding is handled by Math Forum staff members or trainedvolunteers, and it focuses on engaging the learners in reflecting on their strategies andrevising their work both by providing them with necessary support and with models ofhow they might think about mathematics; the mentors do not provide learners with thesolutions. In the OMP, as part of their preservice mathematics training, PTs are taught tomentor, as a way of supporting them to learn how to provide scaffolding for their futurestudents’ work.

Dad’s CookiesDad bakes some cookies. He eats one hot out of the oven and leaves the rest on

the counter to cool. He goes outside to read. Dave comes into the kitchen and finds the cookies. Since he is hungry, he eats

half a dozen of them. Then Kate wanders by, feeling rather hungry as well. She eats half as many as

Dave did.Jim and Eileen walk through next, and each of them eats one third of the

remaining cookies. Hollis comes into the kitchen and eats half of the cookies that are left on the

counter. Last of all, Mom eats just one cookie.Dad comes back inside, ready to pig out. "Hey!" he exclaims. "There is only one

cookie left!"How many cookies did Dad bake in all?

Fig. 1. Sample Math Forum Problem of the Week (PoW).


As described elsewhere, PTs readiness to engage with the elementary students variedgreatly, and it appears to be mediated by the PTs’ interest for mathematics, mathematicalbeliefs, and mathematical content knowledge (Renninger et al., 2006). Support that the PTwas providing to the student was assessed in the initial draft of the MMCS; the MMCSwas subsequently refined to address how the mentor was interacting with both the studentand the problem.

2. Two case examples

Fig. 2 depicts the work of two PT mentors, one strong and one weak. PT1 and PT2

would likely be more or less effective in their scaffolding of elementary students to thinkmathematically in work with Dad’s Cookies. In order to scaffold students effectively, thePT mentor needs to: (a) connect to the mathematics of the problem in order to support thestudent who is being mentored to make this connection (Chi, Siler, & Jeong, 2004; Ma,1999; Schoenfeld, 1987; Verschaffel, De Corte, & Borghart, 1997; Vygotsky, 1978), and(b) identify the student’s strategy, evaluate it, and work with the student to possibly revisitwhat he or she has done (DeCorte, Verschaffel, & Op’t Eynde, 2000; Lampert, 1986;RAND Mathematics Study Panel, 2002; Schoenfeld, 1987). The process of scaffolding alsoinvolves developing a collaboration (Collins et al., 1989; Gauvain, 2001; Rogoff, 1990;Vygotsky, 1978) with the student that is respectful of the students’ readiness to beautonomous in problem solving (Deci, 1992; Piaget, 1954; Renninger, Farra, & Feldman-Riordan, 2000) and the students’ needs for emotional support (Neitzel & Stright, 2003;DeCorte, Verschaffel, & Op’t Eynde, 2000).

If we look closely at the work of PT1, strengths include:

�
making a connection with specific math in Student 1’s problem solving, � leading Student 1 to reflect on her strategy, in that she may not have realized
exactly why the number of cookies Hollis ate was half the number Jim or Eileen eachate,
� supporting Student 1’s autonomy by asking, rather than telling, her to explain, and � being specifically encouraging/emotionally supportive.
PT1’s response might have benefited from:

�
reflecting with the student about strategy use, as Student 1 may have needed moresupport in verbalizing her connection, and � asking Student 1 to explain a different part of the problem than the one already
discussed.

In contrast, PT2’s strengths include:

�
connecting to specific math content in Student 2’s solution (the idea that Jim andEileen’s cookies are ‘‘unknown’’), � suggesting useful problem-solving strategies, � supporting Student 2’s autonomy by not being too directive, and � using tone to provide emotional support.

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Fig. 2. Samples of mentoring.

K. Ann Renninger et al. / New Ideas in Psychology 23 (2005) 152–165156

PT2’s response might have benefited from:

�
not suggesting that trial and error are required to find out how many cookies Jim andEileen ate—(PT2 fails to connect the student’s work to the mathematics of the problem),


�

1

the

addressing Student 2’s strategy specifically—Student 2 is unlikely to realize why herwork is not considered to be ‘‘step-by-step’’,
� being more specific in order to support an increase in Student 2’s autonomy as a
problem solver, and
� using tone to provide emotional support in the second paragraph.
2.1. Coding scaffolding

Given differences between PT1 and PT2 in their readiness to scaffold elementarystudents, coding of scaffolding focuses on how the PT interacts with both the student andthe mathematics of the problem. In the first iteration of the MMCS, assessment focused onthe support that the PT provided to the student, using Connections, Strategy, Autonomy,and Emotional Support as components. However, as in the case of PT2, many of the PTsprovided support that was strong in some ways and challenged in other ways, making thecoding unclear. While the initial coding scheme addressed the spirit of the research onscaffolding (e.g. Pea, 1987; Wood et al., 1976) and the culture of mathematics (e.g.Schoenfeld, 1987; Lave, 1992), its emphasis on supporting the child did not fully addressthe reciprocal nature of the mentoring relationship (Valsiner, 1984). In particular, it didnot address what the PT brings to the collaboration. Since interest, mathematical beliefs,level of skill, sociocultural context, and nature and context of the task affect how mentorsare prepared to engage with scaffolding (Renninger et al., 2006; see related discussions inRAND Mathematics Study Panel, 2002), the coding scheme would be more theoreticallyconsistent and informative about the nature of scaffolding in a particular instance if it alsoaddressed PT engagement.1

In light of the work of PT1 and PT2, the original coding scheme only provided a senseof what each mentor did well. It did not provide clarity about the PTs’ scaffoldingbecause the scales were not parallel across the different components (Connections,Strategy, Autonomy, and Emotional Support). In the revised MMCS, each componentwas coded using similar scales; this was achieved by focusing the coding on the PTs’ability and willingness to engage with the mathematics behind each component. Briefly,PTs who successfully scaffold by assessing the student’s current level and providingfeedback that enables the student to reach a new mathematical understanding arecoded as being at the highest level, Level 4 (e.g. giving the student emotional supportthat reflects a specific mathematical achievement of the student). PTs who attemptto assess the students’ current level or give support to help them reach new understanding,but either inaccurately assess the students’ level or give inaccurate information, arecoded as Level 3 (e.g. giving emotional support that suggests the student’s math iscorrect, when in fact it is inaccurate). PTs who give suggestions, but do not attemptto tailor their suggestions to the students’ current level or the understanding they couldreach, are coded as Level 2 (e.g. saying something generic to each student, like ‘‘Great job,thanks for your submission’’). PTs who make no attempt to give useful support arecoded as Level 1 (e.g. giving the student no emotional support by being discouraging

It is likely that the control afforded by the online environment made the study of individual contributions to

scaffolding collaboration possible.


in tone throughout the response). Thus, in the revised MMCS, PT1 would probably becoded as:

Connections: 4 (succeeded in making a useful connection to the student’s math),Strategy: 4 (succeeded in supporting the student’s own strategy),Autonomy: 4 (successfully provided support at the appropriate level),Emotional Support: 4 (successfully provided specific, meaningful emotional support).

PT2 would probably be coded as:

Connections: 3 (attempted to connect to the student’s reasoning but made an error),Strategy: 2 (did not sufficiently address or support the student’s strategy),Autonomy: 2 (was not specific enough in her support),Emotional Support: 3 (attempted to give meaningful emotional support, but whatwas supported was not correct).

We see that PT1 is successfully engaging in all four components of mentoring, while PT2

has some challenges both with understanding the mathematics and engaging Student 2’sspecific work. The challenges and strengths that need to be addressed in supporting thePTs’ scaffolding are clearer in the second coding scheme.The case examples in Fig. 2 illustrate the potential of using content to focus attention,

and provide and fade feedback. PT1 had rich content at the center of each act of focusingor providing feedback, while PT2, despite providing feedback and focusing attention, is notable to anchor her response in the content, and therefore her mentoring is not as likely toengage the student in thinking and reflecting mathematically.PT1’s appreciation of Student 1’s strategy for finding the number of cookies Hollis ate is

intended to bring Student 1’s attention to an especially sophisticated aspect of hermathematical strategy in a way that encourages her to reflect and think again, rather thansimply telling her why it was mathematically interesting. This leaves an opening forreturning to the conversation. This kind of response is significantly different than asupportive but content-less response (e.g. ‘‘Great job’’), because it focuses the learner on aspecific aspect of the task, chosen to stretch her mathematical thinking. PT1’s secondchallenge for the learner also involves reflecting on the problem. In particular, the learneris encouraged to use words to explain her thinking; this is another technique for supportingthe learner to reflect on the problem, but it also clarifies the learner’s thinking for thementor. Asking the student to provide an explanation for a confused person serves asanother kind of focus or task constraint; by giving Student 1 a specific audience, thementor scaffolds Student 1’s explanation. PT1 had to recognize that Student 1’sunorthodox strategy was not only valid, but it involved an interesting insight that neededto be developed further. The focusing of learner attention is specifically informed by thecontent of the problem and requires a mentor who thinks mathematically.PT2 first attempts to draw Student 2’s focus to the claim that ‘‘trial and error’’ is

necessary to solve the problem. Either because PT2 does not know that the problem can besolved without trial and error, or because she does not think Student 2 is ready to solve itwithout trial and error, PT2 tells Student 2 that she is correct in thinking that the problemrequires trial and error. PT2’s feedback is not open-ended; instead, she tells the student thatshe is correct. Especially, since the student is not correct, this method is ineffective forfocusing the student’s attention on her strategy and does not encourage Student 2 to thinkabout other strategies besides trial and error. It could even prevent Student 2 from


thinking about other strategies. PT2’s strategy of asking Student 2 to explain her workstep-by-step is a good method of focusing Student 2 on the problem, since explaining herthinking could help Student 2 reflect on her mathematics. However, PT2 does not provideStudent 2 with a focus for how to explain her work. She asks Student 2 to explain all thework she did, step by step. Since Student 2 has already failed to sufficiently explain herwork, simply asking her to do it over again will probably not be enough to focus herreflection. PT2 may not have asked for a more specific focus, because she had a hard timeidentifying the key elements of the problem, which would have made her unable to focusStudent 2 on a single central element.

These two case examples, along with application of the MMCS to the work ofapproximately 50 additional PTs, illustrate that the PTs’ levels of mathematical contentknowledge greatly influence their ability to scaffold (Renninger et al., 2006). In fact, PT1 isa mathematics major, and PT2 is a special major in education and sociology. The power ofnon-routine challenge problems for students (and PTs and teachers) is that they do notallow you to simply apply an algorithm to solve the problem. Thus, if mentors are weak inmathematics or believe that mathematics is only about getting the right answer, they mayoverlook the portion of the solution that contains exciting math, they may not recognize acorrect or nearly correct solution path, or they may reduce the process of problem solvingto whether the answer is correct or not, preventing the learner from reflecting on theirmathematics. In fact, the PTs with weak mathematics skills who were studied did notattempt to stretch the mathematical thinking of students with well-developed solutions,and they had an even harder time supporting the work of students with satisfactory, butincomplete, solutions to those students whose mathematics was at a level similar to thePTs’ (Renninger et al., 2006).

3. Discussion

Scaffolding online in an asynchronous context can support the person doing thescaffolding and the learner to reflect. It allows the person doing the scaffolding to gaugethe strengths and needs of the learner and to respond to him or her in ways that couldstretch mathematical thinking. The MMCS directs attention to (a) the types ofConnections that the mentor makes with the learner’s work and supports the learner tomake with the mathematics in the task, (b) the ways by which the mentor supports andextends the learner’s Strategies, (c) how the mentor supports the learner’s increasingAutonomy in problem solving, and (d) the mentor’s provision of Emotional Support forthe learner. Aggregating coded data from each exchange within an initial period of time,and comparing this to data from a second time period, permits analyses of change. Becausethe rubric allows assessment of four components (Connections, Strategy, Autonomy, andEmotional Support), each can be studied and needed remediation can be determined.

The components of the MMCS would make little sense if the MMCS were not contentinformed; in fact, identification of the components was informed by the literaturedescribing mathematical thinking as well as that on scaffolding. A content-informedcoding scheme gives definition to the process of scaffolding. That each component reliablyincludes four levels of response allows assessment of scaffolding and also provides a rubricfor understanding scaffolding processes.

In its present form, the MMCS can also be used to identify the ability and willingness ofPTs to engage as mentors with mathematics content. Low ratings on any one component


indicate a need for additional support. For example, the MMCS pointed to the PTs’ needfor additional support to develop their interest for the subject content, subject contentbeliefs, and subject content knowledge (Renninger et al., 2006).Interestingly, the MMCS might also serve as a rubric (coupled with detailed knowledge

and analysis of the task) for persons learning to mentor, in that it outlines possibilities forcontent-informed scaffolding.2 In Wood et al.’s (1976) pyramid task, for example, theability to articulate the possibilities for the task provided the tutor with clearly definedparameters within which to work. Because the pyramid tutor understood both the mosteffective (only) strategy for constructing the pyramid and could anticipate thecomplications learners might have, he was able to concentrate on when and how to focusthe learners, and provide and fade feedback. In an ill-defined problem space, like solving anon-routine challenge problem, it is much harder for the mentor to anticipate all of thepossible ways to work with the task. As a result, the content of the non-routine challengeproblem, or an open-ended task, becomes a central focus for scaffolding and provides thementor with a challenging task. The mentor needs to consider when and how to focus andgive feedback, and the mentor needs to worry about the content, which ideas to extend,and the range of possibilities for effectively stretching the learner.The mentor using the MMCS as a rubric should realize that he or she must be able to

identify the mathematics in the problem in order to recognize the learner’s mathematicalthinking, and be prepared to provide the learner with a well-developed solution so thatmathematical understanding can be stretched. Facilitating that movement requiresunderstanding the necessary mathematics. It also involves considering likely strategies ofthe learner and ways in which to respond to these strategies. Imagining possible levels ofthe learner’s work and the support needed at each level could help the mentor support thelearner’s autonomy. Finally, the mentor might consider the kinds of mathematical thinkingor insights that would demonstrate the learner’s understanding of the problem, so thatemotional support can be provided. Given this kind of preparation, the task might feelmore defined, and like the pyramid tutor, the mentor could begin to focus on how he or shewill deliver his or her support—when to model, when to focus attention, and when to givefeedback. Although the impact of using the coding scheme as a rubric for scaffoldingremains to be tested, it provides details about the components of scaffolding that increaseunderstanding and should allow improved assessment of the scaffolding process bothonline and in the classroom.

Acknowledgments

We gratefully acknowledge editorial support and feedback from Vanessa Gorman, NiraGranott, and Cynthia Lanius. This coding scheme was developed during work on theOMP (NSF Grant no. 0127516). The findings and recommendations expressed in thisarticle are those of the authors, and do not necessarily reflect the views of the NationalScience Foundation.

2It is expected that the MMCS could be adapted for use with tasks other than mathematics. This expectation is

based on two premises. First, the coding scheme was developed based on theories and methods from a number of

literatures, including work on scaffolding, learning, mathematical thinking, children’s play, and motivation.

Second, its revision was undertaken as part of a research and development project focused on supporting the

development of mentoring.


Appendix A. Mathematical Mentoring Coding Scheme (MMCS)3

A.1. Overview

The MMCS was developed to assess PTs’ support for elementary students’mathematical thinking in work with online challenge problems. A form of content-focused coaching (West & Staub, 2003), it consists of four independent but relatedcomponents: Connections, Strategy, Autonomy, and Emotional Support. The codingscheme builds on the three components used to code mathematical thinking in problemsolving: Connections, Strategy and Autonomy (Renninger et al., 2000); although in theMMCS, these components are adapted to evaluate a mentor’s response to a learner’ssolution rather than to evaluate the solution itself. The additional component ofEmotional Support measures the level and type of encouragement that mentors provide(Neitzel & Stright, 2003).

Each of the MMCS components consists of a 4-point scale. Briefly, mentors whosuccessfully scaffold by assessing the learner’s current level and providing feedback thatenables the learner to reach a new mathematical understanding are coded as Level 4 (e.g.giving the learner emotional support that reflects a specific mathematical achievement ofthe learner). Mentors who attempt to assess the learner’s current level or give support tohelp them reach new understanding, but either inaccurately assess the learner’s level or giveinaccurate information, are coded as Level 3 (e.g. giving emotional support that suggeststhe learner’s math is correct, when in fact it is inaccurate). Mentors who give suggestions,but do not attempt to tailor their suggestions to the learner’s current level or theunderstanding they could reach, are coded as Level 2 (e.g. saying something generic to eachlearner, like ‘‘great job, thanks for your submission’’). Mentors who make no attempt togive useful support are coded as Level 1 (e.g. giving the learner no emotional support bybeing discouraging in tone throughout the response).

A.2. The MMCS

A.2.1. Connections

The Connections a learner makes during problem solving refers to his/her readiness toidentify links between mathematical concepts and real-world situations. In addition, theyindicate the learner’s present ability to interpret a problem and develop multiple solutionstrategies for solving it.

Applied to in mentoring, Connections describe the importance a mentor places on usingreal-world links to help illustrate a mathematical concept or preserve mathematicalmeaning (Verschaffel et al., 1997). Connections also assess the mentor’s acceptance ofdifferent kinds of strategies and his or her ability to interpret a learner’s line of thinkingbased on the solution provided (Fennema & Loef, 1992; Fennema et al., 1996).

A.2.1.1. Level 1. Does not bring in outside information, cannot succeed in finding aconnection. Only addresses the accuracy of the learner’s work.

3Use of the MMCS should be undertaken by two persons with a high level of pedagogical content knowledge of

mathematics.


A.2.1.2. Level 2. Brings in outside information, but it is not math (e.g. suggests they fixspelling or units). Usually accurate with outside information. Does not appear to havetried to bring in mathematical connections or to look at ‘‘deeper’’ or ‘‘broader’’ math thatthe learner is doing.

A.2.1.3. Level 3. Brings in outside mathematics, but the mathematics is inaccurate,unclear, or not at a useful level.

A.2.1.4. Level 4. Brings in outside mathematics/real-world connections in a meaningfulway, and is accurate in these connections and in the mathematics employed.

A.2.2. Strategy

Strategy in problem solving refers to a learner’s ability to explain mathematicalconcepts, work with mathematical terms, and explain decision-making processes inworking with a problem.Applied to in mentoring, Strategy describes a mentor’s ability to (a) provide cognitive

support, in the form of metacognitive information and goal orientation, (b) providecontent support, and (c) successfully communicate mathematical concepts and terms tolearners. Metacognitive information includes providing learners with ideas for taskmanagement techniques or strategies, serving to advance the learner’s understanding ofhow the task works, or providing a rationale for the use of a particular strategy (Neitzel &Stright, 2003). Goal orientation refers to a mentor’s ability to keep the learner on trackwith his or her chosen strategy and help the learner to focus on the problem. Contentsupport refers to a mentor’s ability to communicate the mathematical concepts involved inthe problem.

A.2.2.1. Level 1. Focuses on the accuracy of the learner’s final solution. Provides nocognitive support or minimal content support. Communication is incomprehensible.

A.2.2.2. Level 2. Centers feedback on advice and suggestions, but does not address thelearner’s work. Provides minimal cognitive and content support. Communication isunclear or convoluted.

A.2.2.3. Level 3. Provides the learner with a minimal description of his/her work.Feedback points out areas of content problems but does not necessarily constructivelyaddress the problem or the work already completed. Communication may be slightlyconvoluted or long-winded.

A.2.2.4. Level 4. Provides the learner with a clear, useful, and non-judgmentaldescription of his/her work, supporting work completed, and calling attention to whathas been accomplished. Feedback either points learner to specific activity, an alternativestrategy, and/or encourages the learner to relate information to key concepts.

A.2.3. Autonomy

Autonomy in problem solving refers to a learner’s ability to be independent in problemsolving. It is interpreted by the need for mentor feedback, which supports the developmentof learner responses in future submissions, focuses the learner’s mathematical thinking


through the use of tools such as models, examples, and scaffolding, and emphasizesproblem solving.

Applied to in mentoring, Autonomy describes a mentor’s ability to transferresponsibility for problem solving to the learner (Neitzel & Stright, 2003). Mentors whoencourage a high degree of cognitive involvement will use prompts, questions, and hintswhen responding to the learners’ solutions, rather than providing the learner with theanswer or with direct instructions.

A.2.3.1. Level 1. Assumes no responsibility for supporting learner’s problem solving.Does not address the learner’s problem-solving work.

A.2.3.2. Level 2. Takes full responsibility for supporting learner’s problem solving.Proposes general problem-solving strategies that are not specific to that learner’s work.Responses reflect suggestions from the online mentoring guide instead of reflections thatmatch the learner response level.

A.2.3.3. Level 3. Takes responsibility for supporting the learner’s problem solving. Usesprompts, questions, and hints in order to help learners understand concepts on their own,but also suggests specific strategies for the learner to try or concepts the learner shouldthink about. However, the number of prompts and/or suggestions is either too many ortoo few to address the learner’s specific needs.

A.2.3.4. Level 4. Limits control to the minimum necessary for the learner to progress.Uses prompts, questions, and hints that allow the learner to formulate strategies and drawconclusions on his or her own.

A.2.4. Emotional Support

In mentoring, Emotional Support refers to the amount and type of encouragement thata mentor provides in his or her feedback to learners (Neitzel & Stright, 2003). Applied tomentoring, encouragement can either be general or specific in nature. Generalencouragement consists of non-specific praise, whereas specific encouragement focuseson the particular strengths and work of learners.

A.2.4.1. Level 1. Does not provide learner with encouraging response.

A.2.4.2. Level 2. Provides general encouragement that is either undeserved or irrelevantto the learner’s work.

A.2.4.3. Level 3. Provides either general encouragement that is meaningful, or specificencouragement that is not correct and/or not in proportion to the learner’s work.

A.2.4.4. Level 4. Provides learners with meaningful, specific encouragement appropriateto their work describes an action or step the learner used.


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