{"id":9365,"date":"2020-02-12T15:50:22","date_gmt":"2020-02-12T15:50:22","guid":{"rendered":"https:\/\/citejournal.org\/\/\/"},"modified":"2020-05-18T18:06:01","modified_gmt":"2020-05-18T18:06:01","slug":"why-and-how-secondary-mathematics-teachers-implement-virtual-manipulatives","status":"publish","type":"post","link":"https:\/\/citejournal.org\/volume-20\/issue-1-20\/mathematics\/why-and-how-secondary-mathematics-teachers-implement-virtual-manipulatives","title":{"rendered":"Why and How Secondary Mathematics Teachers Implement Virtual Manipulatives"},"content":{"rendered":"\n

In today\u2019s classrooms, teachers are expected to integrate technology as a means to enhance student understanding and engagement. Depending on the content area, the technology tool <\/em>can take on many forms (e.g., interactive white boards, tablets, computers, games, software, virtual manipulatives, etc.) and roles (see Cullen, Hertel, & Nickels, 2020). Yet, teachers often report that they are not prepared to integrate technology in an effective and innovative manner (Albion, Tondeur, Forkosh-Baruch, & Peeraer, 2015). <\/p>\n\n\n\n

Over the past 30 years, virtual manipulatives (VMs) are one technology tool that teachers have been encouraged to implement to enhance student learning (Association of Mathematics Teacher Educators [AMTE], 2017; National Council of Teachers of Mathematics [NCTM], 2000). A VM is an \u201cinteractive, technology-enabled visual representation of a dynamic mathematical object, including all of the programmable features that allow it to be manipulated, that presents opportunities for constructing mathematical knowledge\u201d (Moyer-Packenham & Bolyard, 2016, p. 13). <\/p>\n\n\n\n

As with any technology\ntool, the potential for VMs to support student learning lies not within the\ntool itself but in how students engage with the VM and the relevant mathematics\n(Moyer-Packenham & Bolyard, 2016). Therefore, teachers must be prepared to\ncreate opportunities that promote quality engagement with VMs, thus presenting\nopportunities for students to develop conceptual understanding.<\/p>\n\n\n\n

Unfortunately, most of the studies investigating teachers\u2019 use of VMs occur at the prekindergarten through Grade 6 levels, with minimal studies conducted at the middle school, high school, and above levels (Moyer-Packenham & Westenskow, 2013). Therefore, this article reports the findings of a study on a professional development (PD) opportunity for secondary mathematics teachers aimed at supporting their efforts to teach with VMs. (See http:\/\/bit.ly\/VirtManips<\/a> for an annotated list of VM collections and http:\/\/bit.ly\/VMActivities<\/a> for a repository of VM tasks used during the PD.)<\/p>\n\n\n\n

Virtual\nManipulatives<\/h2>\n\n\n\n

VMs\nare commonly categorized into five environments: single representation,\nmultirepresentations, tutorial, gaming, and simulation (Moyer-Packenham &\nBolyard, 2016). The single-representation<\/em>\nVM \u201cenvironment contains an interactive pictorial\/visual representation (i.e.,\nimage) of a dynamic mathematical object and is not accompanied by numerical or\ntext information\u201d (p. 14). <\/p>\n\n\n\n

Typically,\nthis environment requires teachers to design tasks to focus students\u2019 attention\non the relevant mathematical ideas and learning goals. The multi-representation<\/em> \u201cenvironment contains the interactive visual\nrepresentation (i.e., image) of the dynamic mathematical object and is accompanied\nby numerical and, sometimes, text information\u201d (Moyer-Packenham & Bolyard,\n2016, p. 15). Typically, the numeric information is linked simultaneously with the\nvisual representation, thus promoting students to make connections and see\npatterns more easily. <\/p>\n\n\n\n

The\ntutorial \u201c<\/em>environment contains the interactive\nvisual representation (i.e., image) of the dynamic mathematical object and is\naccompanied by numerical and text information in a format that guides the user\nthrough a tutorial of the mathematical procedures and processes being\npresented\u201d (Moyer-Packenham & Bolyard, 2016, p. 17). The tutoring and guiding\ncharacteristics of this environment distinguish it from the multirepresentation\nenvironment. <\/p>\n\n\n\n

In\nthe gaming<\/em> environment, the VM is\n\u201cembedded in a format that allows the user to play a game with the object to\nreach goals that are reflected in the gameplay\u201d (Moyer-Packenham & Bolyard,\n2016, p. 18). Finally, in the simulation <\/em>environment\nthe VM is an interactive image \u201cof the dynamic mathematical object\u201d and other\nrepresentations (e.g., text and numeric) embedded in a manner that enables \u201cthe\nuser to run a simulation intended to represent or draw attention to embedded\nmathematics concepts\u201d (p. 20). <\/p>\n\n\n\n

Teachers\nin the PD described here interacted with VMs from all five environments;\nhowever, VMs were primarily from the single-representation,\nmultirepresentation, and tutoring environments. Appendix A<\/a>\ncontains examples of VMs from the different environments that teachers\ninteracted with during the PD. <\/p>\n\n\n\n

Moyer-Packenham and Westenskow\u2019s (2013) meta-analysis of studies investigating the effects on student achievement when VMs are used identified five interrelated categories of affordances of VMs: <\/p>\n\n\n\n

focused constraint <\/em>(i.e., VMs focus and constrain students\u2019 attention on mathematical objects and processes), creative variation<\/em> (i.e., VMs encourage creativity and increase the variety of students\u2019 solutions), simultaneous linking <\/em>(i.e., VMs simultaneously link representations with each other and with students\u2019 actions), efficient precision (i.e., VMs contain precise representations allowing accurate and efficient use), and motivation (i.e., VMs motivate students to persist at mathematical tasks). (p. 35)<\/p><\/blockquote>\n\n\n\n

Amongst the affordances,\nthe interaction between the dynamic object, the learner, and the mathematics is\nwhat determines the actual affordance of the VM for student learning (Moyer-Packenham\n& Westenskow, 2013). Both Tucker, Moyer-Packenham, Westenskow, and Jordan\u2019s\n(2016) and Moyer-Packenham and Westenskow\u2019s (2016) more recent meta-analysis\nfound that creative variation was the least frequent affordance category\nidentified by evidence as contributing to student learning.<\/p>\n\n\n\n

Beyond these affordances, additional reasons exist for promoting teachers\u2019 use of VMs in the classroom. For example, studies indicate that VMs can be used as a tool to support students\u2019 learning by providing opportunities for immediate feedback (Edwards Johnson, Campert, & Zuidema, 2012). In today\u2019s dynamic classrooms, VMs can be used to support teachers\u2019 differentiation efforts (Bouck, Flanagan, & Bouck, 2015; Shin et al., 2017) to support students\u2019 emerging understanding, as well as a means for challenging (or encouraging) students to explore a mathematical idea from a different perspective. Bouck and colleagues (2015) also suggested that VMs benefit students with learning disabilities in both their performance, their confidence, and possibly, their development of conceptual understanding. Finally, VMs can provide equal access for students to learn content by reducing effects of students\u2019 demographics (socioeconomic status and English language learner status) as predictors of achievement (Moyer-Packenham et al., 2014). <\/p>\n\n\n\n

Providing\naccess to technology tools is \u201cnot enough\u201d in supporting teachers to teach with\nthe tools; rather teachers need to \u201ccome to know the appropriate and\nconstructive uses of technology\u201d (Wilson, 2008, p. 415). Our PD aimed to\nsupport teachers implementing VMs and tasks in appropriate and constructive\nways. Since minimal studies have been done investigating secondary mathematics\nteachers\u2019 use of VMs, this study aimed to extend the findings from studies at\nthe elementary levels investigating teachers\u2019 implementation of VMs to the\nsecondary levels. Specifically, this\nstudy explored the following research question: Why and how do secondary mathematics teachers report implementing\nvirtual manipulatives while participating in a focused professional development\nopportunity?<\/p>\n\n\n\n

Theoretical Background<\/h2>\n\n\n\n

Rather than studying teachers\u2019 implementation efforts in isolation, teachers\u2019 implementation of VM tasks are considered to be mediated by their mathematical goals for the task, the tools available related to their implementation (i.e., technology and otherwise), and the students they teach (Zbiek, Heid, Blume, & Dick, 2007). A VM task refers to a VM and all accompanying instructional materials (e.g., prompts and directions) whether on screen or in printed form. The VM task could include more than one task (Sinclair, 2003) focused on investigating a particular concept (e.g., through alternative exploration paths), but it may include only one task.<\/p>\n\n\n\n

To\nunderstand why and how teachers implemented VMs, the following study drew from the\nthird strand of activity theory (Engestr\u00f6m, 1987, 1999; Nardi, 1996). Activity\ntheory takes a multidimensional approach to investigating activities in which\npeople are engaged and acknowledges that activities are mediated by the context\nof the activity, the subjects\u2019 experiences, how they use tools and how tools\nare transformed through activity and so forth. Engestr\u00f6m (1987, 1999) discuss\nhow this strand draws and differs from the Russian strand of activity theory\noriginating with Vygotsky, Leont\u2019ev, and Luria.<\/p>\n\n\n\n

An\nactivity consists of a subject, object, and actions. In this study, the\nteachers engaged in the PD are the subject,\n<\/em>as they are the learners. The object,\n<\/em>which motivates the activity and gives the activity specific direction are\nthe teachers\u2019 implementations of VMs and VM tasks. <\/p>\n\n\n\n

Teachers\u2019 practices related to planning and preparing to implement VMs and tasks constitute the actions <\/em>(the goal-directed processes undertaken by the subject to achieve the object) of an activity system in this study. \u201cActions are chains of operations\u201d (Jonassen & Rohrer-Murphy, 1999, p. 63) and require conscious effort. Through repeated practice, actions can become operationalized (subconscious). Examples of actions include critiquing a VM task, process of developing an instructional guide to accompany a VM, and so forth. Other aspects of an activity system (see Figure 1) include the tools\/mediating artifacts (the VMs and tasks, the task analysis framework), rules (curriculum, instructional style), community (their students and other teachers in the school), and division of labor (do teachers work collaboratively or primarily individually). The focus of this study was on why and how teachers implemented VMs and tasks (the object). <\/p>\n\n\n\n

\"\"
Figure 1<\/strong>. Structure of an activity system for this PD.<\/em><\/figcaption><\/figure>\n\n\n\n

Methods<\/h2>\n\n\n\n

To address the gap in literature about secondary mathematics teachers\u2019 use of VMs and tasks, the aim of this study was to investigate why and how secondary mathematics teachers reported they implemented VMs and tasks while participating in a PD opportunity. Teachers\u2019 conversations and responses during the PD sessions, as well as interviews with some of the participants, were used to investigate teachers\u2019 reported implementation of VM tasks. <\/p>\n\n\n\n

Professional\nDevelopment Opportunity<\/h3>\n\n\n\n

The design of the PD was based on five components of effective PD (e.g., Desimone, Porter, Garet, Yoon, & Birman, 2002). That is to say, content focus<\/em> (grounded in the teachers\u2019 mathematics curriculum), active learning <\/em>(teachers engaged with VMs and tasks related to their identified learning goals), and duration<\/em> (at least 20 contact hours spread across 6 months). The PD was coherent with the district initiatives regarding teacher and student technology use and promoted collaboration amongst participants through collective participation (teaching pairs from the same school). <\/p>\n\n\n\n

In phase I, teachers reflected on the role of technology in the classroom and their current technology use. Additionally, tools (guiding questions and a task analysis framework) were introduced to support teachers\u2019 integration efforts and knowledge growth (see Appendix B<\/a>; Reiten, 2018). The task analysis framework drew from work by Trocki (2014) and Sinclair (2003) with dynamic geometry software tasks and was intended to help teachers critique and develop tasks aimed at promoting students\u2019 development of conceptual understanding of mathematics through reflection and communication (Hiebert et al., 1997), as well as through using and connecting mathematical representations (NCTM, 2014). <\/p>\n\n\n\n

During this phase, teachers completed, critiqued, and compared three VMs\/VM tasks (i.e., Modeling and Solving Two-Step Equations from ExploreLearning, https:\/\/www.explorelearning.com\/index.cfm?method=cResource.dspDetail&ResourceID=226<\/a>; Algebra Tiles from Illuminations, https:\/\/www.nctm.org\/Classroom-Resources\/Illuminations\/Interactives\/Algebra-Tiles\/<\/a>; and Virtual Algebra Tiles from Michigan Virtual University, http:\/\/media.mivu.org\/mvu_pd\/a4a\/homework\/applets_two_step.html<\/a>) for solving two-step equations. As teachers progressed from the first phase through the third phase, they took on more responsibility for finding and selecting the VMs and tasks. <\/p>\n\n\n\n

Beginning with the second PD session, teachers recorded their responses to the guiding questions and how they were using the task analysis framework using Google Docs. <\/strong>The Google Docs encouraged teachers to use the task analysis framework to critique VMs and tasks. Additionally, the Google Docs included a prompt that encouraged teachers to reapply the task analysis framework after they made or thought about potential modifications\/adaptions to the VMs and tasks that they were exploring. (Appendix C<\/a> contains an example of a Google Doc from the latter portion of the PD).<\/p>\n\n\n\n

Phases\nII and III focused on teachers\u2019 learning goals and refining their instructional\npractices related to implementing VM tasks. In Phase II teachers were given VM\ntasks to critique, whereas in Phase III teachers found the tasks to critique\nand modify. During the PD sessions in these two phases, teachers spent the\nfirst part of each session completing and critiquing VM tasks related to their\nidentified learning goals. <\/p>\n\n\n\n

For\nthe remaining part of the PD session, teachers selected a VM or task and then\nspent time preparing the task to use with their students (e.g., modifying or creating\ninstructional guides). Due to the attention given to individual teacher\u2019s\ngrowth, some teachers were at Phase II while other teachers were at Phase III. Teachers\nused the guiding questions and task analysis framework throughout the three\nphases of the PD as they critiqued and modified\/developed instructional guides\nto accompany VMs. Therefore, teachers\u2019 use of these tools constituted some of\nthe actions investigated during\nthe study that supported teachers\u2019 implementations of VM tasks (the object) of an activity system. <\/p>\n\n\n\n

Participants<\/h3>\n\n\n\n

Fourteen\nteachers in a suburban district participated in a PD opportunity aimed at\nfostering their use of VMs and tasks. Middle school teachers were originally\nencouraged to participate due to all middle school students having Chromebooks.\nTen middle school teachers, three high school teachers, and one fifth-grade teacher\ncomprised the participants. Table 1 contains additional information about the participants.\nIn the remainder of this article, \u201cteacher\u201d or \u201cteacher(s)\u201d refers to the\nteachers who participated in the PD and \u201cstudents\u201d refers to the teachers\u2019\nstudents. <\/p>\n\n\n\n

Table 1<\/strong>
Information About Participants in the Professional Development Opportunity<\/p>\n\n\n\n\n\n\n\t\n\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t
Pseudonym<\/strong> <\/strong><\/th>School<\/strong> <\/strong><\/th>Grade\/Course Taught<\/strong> <\/strong><\/th>\n<\/tr>\n<\/thead>\n
Kelly <\/td>Prospect ES <\/td>5th <\/td>\n<\/tr>\n
Josh [a] <\/td>Wetlands MS <\/td>6th <\/td>\n<\/tr>\n
Mark [a] <\/td>Wetlands MS <\/td>6th <\/td>\n<\/tr>\n
Tracy [a] <\/td>Wetlands MS <\/td>6th\/7th Interventionist <\/td>\n<\/tr>\n
Curt <\/td>Plains MS <\/td>7th <\/td>\n<\/tr>\n
Karen <\/td>Plains MS <\/td>7th <\/td>\n<\/tr>\n
Randy <\/td>Plains & Summit MS <\/td>6th\/7th\/8th Interventionist <\/td>\n<\/tr>\n
Erin <\/td>Summit MS <\/td>8th <\/td>\n<\/tr>\n
Stan <\/td>Summit MS <\/td>8th <\/td>\n<\/tr>\n
Mari <\/td>Summit MS <\/td>8th <\/td>\n<\/tr>\n
Pam <\/td>Summit MS <\/td>8th Interventionist <\/td>\n<\/tr>\n
Daron [a] <\/td>HS <\/td>Geometry, AP-Stats <\/td>\n<\/tr>\n
Jake [a] <\/td>HS <\/td>AP-Calculus, Pre-Calculus, Consumer Math <\/td>\n<\/tr>\n
Mike<\/td>HS<\/td>AP-Calculus, Pre-Calculus, Transition to College Math<\/td>\n<\/tr>\n
[a] Teachers who volunteered for interviews during the PD.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n

Data\nSources and Analysis<\/h3>\n\n\n\n

Data collected throughout the PD included initial background and final surveys (both of which were online), video and audio recordings of the PD sessions, as well as teachers\u2019 responses to Google Docs (Appendix C<\/a>) containing a set of guiding questions and encouragement to use the task analysis framework. The task analysis framework and guiding questions are in Appendix B<\/a>. <\/p>\n\n\n\n

Four\nteachers volunteered to participate in audiorecorded semistructured interviews\nat the beginning and end of the PD. Additionally, these teachers and an\nadditional teacher volunteered for brief interviews before and after\nimplementing VM tasks with their students. Transcripts were created from all\nrecordings. <\/p>\n\n\n\n

An\nopen coding approach (Salda\u00f1a, 2013)\nusing NVivo was initially applied to the transcripts and teachers\u2019 responses on\nthe Google Docs to identify themes regarding why and how teachers implemented\nVMs and tasks. Transcripts and Google Docs were initially coded based on components\nof the activity system. For example, discussions about students and fellow\nteachers were coded as community<\/em>; statements\nabout implementing VMs and tasks were coded as object.<\/em> Then, transcripts and Google Docs were coded again in their\nentirety. <\/p>\n\n\n\n

Drawing from the literature, some codes investigating why teachers implemented VM tasks included differentiation, feedback, exploration, and engagement. Some emergent codes investigating how teachers reported implementing VM tasks included supplement curriculum, solidify understanding, and the types of modifications. These emergent codes were used rather than Moyer-Packenham, Salkind, and Bolyard\u2019s (2008) seven categories [a]<\/a> articulating how K-8 teachers used mathematics tools to better capture how teachers in this study articulated using VMs and tasks. <\/p>\n\n\n\n

Additional\nrounds of analysis occurred using the constant comparative method (Glaser & Strauss, 1967) to refine\ncode definitions based on themes between data within categories and then\nfocused coding proceeded. Connections in the data between components of the\nactivity system (tools, community, and\ndivision of labor) gave insight\ninto how teachers implemented VM tasks (the object). <\/p>\n\n\n\n

All data sources were coded using the same codes (see Tables 3, 4 and 5 for codes). A data excerpt consisted of at least a complete sentence in each data source. To minimize the potential of segmenting the data too much, at times a datum excerpt contained multiple sentences if all the sentences were related to the same code (e.g., see the example for Practice <\/em>in Table 2). When more than one code applied to a datum excerpt (see Daron\u2019s response to a question on the final survey in Table 4), the excerpt was coded for each relevant code.<\/p>\n\n\n\n

Findings\npresented are based only on the data collected during the PD sessions,\ninterviews, and online surveys. Due to other afterschool commitments, one eighth-grade\nteacher stopped participating in the PD after the first two sessions. Teachers\u2019\nimplementation of VMs and tasks between PD sessions is not included in the\npresented findings unless teachers specifically talked about their implementation\nefforts during a PD session, interview, or survey. Therefore, it is possible\nthat teachers implemented more VMs and tasks than indicated in the data.  <\/p>\n\n\n\n

Findings<\/h2>\n\n\n\n

Teachers\u2019 appropriation of tools (i.e., guiding questions, task analysis framework, annotated VM list, and folder of tasks) introduced during the PD supported their implementation efforts (Reiten, 2018). Teachers\u2019 responses on the initial background survey indicated that only three out of 14 teachers felt comfortable integrating VMs before the PD. However, at the conclusion of the PD, 10 teachers stated that they regularly implemented VMs and tasks and felt comfortable doing so. <\/p>\n\n\n\n

The\npresented findings and discussion highlight teachers\u2019 conversations and\nresponses to exemplify why and how they implemented VM tasks with their\nstudents. Although the goal of the PD was to support teachers in implementing VM\ntasks, it was not known what influence the PD and support tools may have on\nteachers\u2019 practices. <\/p>\n\n\n\n

Why Teachers Implemented VM Tasks<\/h3>\n\n\n\n

Two themes emerged from the data regarding why teachers implemented VMs and tasks. The first theme relates to the various instructional benefits teachers identified that supported their students\u2019 emerging understanding as to why they chose to implement VMs and tasks \u2013 meaning that teachers\u2019 actions related to preparing to implement VM tasks were influenced by how they thought their students\u2019 (part of their community) emerging understanding was supported. Additionally, teachers identified implementing VMs and tasks (object) because the VM tasks contained instructional guides and questions that further supported their learning goals (rules). Both themes are discussed in more detail in the following subsections. <\/p>\n\n\n\n

Instructional benefit<\/strong>.<\/em> Teachers implemented VMs and tasks because they thought their students benefited or their learning was enhanced due to students\u2019 engagement with the VM tasks. The following subsection discusses the benefits teachers identified and then provides examples of how Tracy (a middle school teacher) and Daron (a high school teacher) thought their students benefited from interacting with VMs and tasks. <\/p>\n\n\n\n

Table 2 highlights the different types of potential benefits teachers described related to why they used VMs and tasks. The number of data excerpts related to each category are included in the table for descriptive purposes only, as multiple data sources may have contributed to the number of data excerpts. As displayed in the table, teachers identified the visual nature of VMs, opportunities for student exploration, and assisting in differentiating instruction as the most common benefits to instruction and student understanding. Teachers also cited students benefiting from the use of VMs and tasks: Students could offload computational aspects of a task, receive feedback, practice problems (including extending beyond the VM), and connect multiple representations. The VM also encouraged student engagement in the lesson and learning goal. <\/p>\n\n\n\n

Table 2
<\/strong>Categories, Definitions, and Examples of Teacher Identified Student Benefits When Using VMs and Tasks<\/p>\n\n\n\n\n\n\n\t\n\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t
Definition<\/strong><\/th>Number of Data Excerpts [a]<\/strong><\/th>
\n Example<\/strong><\/th>\n<\/tr>\n<\/thead>\n
Connect Multiple Representations: <\/strong>opportunities for students to connect multiple representations<\/td>8<\/td>Mark (DecGoogleDoc): <\/strong>\u201cThe connection between the area model, the measurements, and the formula for area and perimeter of rectangles makes this a meaningful lesson.\u201d <\/strong><\/td>\n<\/tr>\n
Differentiation: <\/strong>opportunities for personalized learning by differentiating instruction based on student needs.<\/td>16<\/td>Josh (MarchPDSummary): <\/strong>Josh shares his thoughts on the assessment questions at the end (i.e., Activity B) one of the questions he would get about 50% of the students and Q5, he will get about 2% of the students, which would be perfect. Josh says it would push the students in different directions. <\/strong><\/td>\n<\/tr>\n
Engagement: <\/strong>promotes student engagement (e.g., through motivation and\/or piquing their interest). <\/td>9<\/td>Mark (FinalSurvey): <\/strong>\u201cA larger number of students are engaged. All students have an opportunity to interact with the problem(s) they are attempting to solve.\u201d <\/td>\n<\/tr>\n
Exploration: <\/strong>promotes opportunities for students to explore concepts through manipulation, interaction and so forth.<\/td>27<\/td>Jake (FinalSurvey): <\/strong>\u201cOne advantage is, kids, you-they can manipulate. I mean, they can move things around, they can change parameters, they can change\u2026\u201d<\/td>\n<\/tr>\n
Feedback: <\/strong>provides opportunities for students to receive some type of feedback.<\/td>11<\/td>Jake (FinalSurvey): <\/strong>\u201c\u2026some of them they can get feedback instantly or very close to instantly. So-it allows, when you are doing something, if they are getting that feedback, that you don\u2019t have to be at, 20 different 20 different people or 20 different desks or whatever.\u201d<\/td>\n<\/tr>\n
Modeling: <\/strong>provides opportunities for students to use or develop models.<\/td>8<\/td>Kelly (NovGoogleDoc): <\/strong>\u201cBeing able to model division situations and having a better understanding of how remainders happen and what they mean.\u201d<\/td>\n<\/tr>\n
Offload: <\/strong>due to built in features (e.g., calculations, drawing graphs) of the VM, students can focus on the content rather than the menial <\/em>or tedious <\/em>components of a task.<\/td>12<\/td>Pam (MarchGoogleDoc): <\/strong>\u201cThe students would benefit from this because it\u2019s less time-consuming than creating the intersecting graphs with a pencil and paper. They would be able to practice with more solid examples.\u201d<\/td>\n<\/tr>\n
Practice: <\/strong>provided opportunities for students to practice problems.<\/td>11<\/td>Tracy (OctPD): <\/strong>\u201cIt\u2019s is a-it\u2019s got a lot of sequential steps. I mean, it starts with the very very basic of just learning how to build them. And then gets you going, like you can keep clicking to the next level to get more difficult, more difficult, more difficult. So, umm, it\u2019s great practice for kids who, just don\u2019t even understand what Algebra tiles are.\u201d<\/td>\n<\/tr>\n
Visual: <\/strong>provides a visual of the concept being investigated.<\/td>30<\/td>Mari (DecGoogleDoc): <\/strong>\u201cStudents would benefit with the sliding aspect of the gizmo. Seeing the function change as the y-intercept changes or as the growth\/decay rate changes.\u201d <\/strong><\/td>\n<\/tr>\n
[a] Number of excerpts is included for descriptive purposes only.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n

Teachers identified other student benefits as to why they chose to use VM tasks that are not included in the table due to the low occurrence of these benefits being mentioned: communication (promoted students\u2019 discussion about what they were doing and learning), conceptual understanding (promoted opportunities to develop students\u2019 conceptual understanding), concrete (potential for making an abstract concept more concrete), make and test conjectures (provides opportunities for students to make and test conjectures), and using technology (using technology is a benefit in and of itself). These benefits occurred six or fewer times in the dataset. Although not included in the table, these benefits are still noteworthy and may contribute to why other teachers consider integrating VM tasks in their instruction. <\/p>\n\n\n\n

During a final interview, Tracy shared her thoughts\nas to why she chose to implement VM tasks:<\/p>\n\n\n\n

It piques their interest, in all of it. All of a sudden we\u2019ll [i.e., the students] TRY because we\u2019re not writing on pencil and paper. Like the one that we did with the area and the perimeter when they\u2019re making the gardens [i.e., Fido\u2019s Flower Bed [b]<\/a> from ExploreLearning] all of a sudden, \u201cOH! A man walked by.\u201d<\/em> Or \u201cOh, did you see that dog?!\u201d <\/em>You know, I mean \u2026 they get excited about it and they can compare [with each other]. \u2026 They had a little liberty as to HOW they could make it. \u2026 So, instead of drawing it, they can click. So I think it piques their interest a little bit, too. [Capital letters used to signify emphasis in speech; italics used to signify change in tone.] <\/p><\/blockquote>\n\n\n\n

Tracy used VMs and tasks because the tasks gave her students opportunities to explore content in a more dynamic way and with less restrictions than if the task had been done with paper and pencil. Students were able to offload the drawing aspects of creating the lawns to the VM. Additionally, students\u2019 engagement was increased because students\u2019 interests were piqued. The interactive environment of the VM helped to promote students\u2019 engagement with the task as well as reflecting on the mathematics through comparing solution strategies with their peers, thus promoting students\u2019 development of conceptual understanding. <\/p>\n\n\n\n

On Daron\u2019s final survey, he said that he used the VM tasks because \u201cstudents can discover and explore concepts in a dynamic way. [The VM tasks] allow for a deeper level of engagement and more efficient way to investigate deep mathematical ideas.\u201d Daron highlighted student explorationdue to the dynamic nature of the VM and student engagement as reasons why he used VMs and tasks. <\/p>\n\n\n\n

Daron\u2019s comment about a \u201cmore efficient way to investigate\u201d may be due to the ability to offload computational\/drawing aspects of a task to the VM. Additionally, the statement could refer to the task providing some guidance to support students\u2019 exploration rather than students choosing how and what to explore. Due to the different possibilities of interpreting Daron\u2019s comment, this part of the comment was not coded.<\/p>\n\n\n\n

Instructional guides.<\/strong><\/em> Besides the benefits to their students\u2019 developing understanding, teachers implemented VM tasks due to instructional support offered by accompanying instructional guides or questions. Twelve teachers mentioned purposefully looking for accompanying questions or instructional guides when searching for VMs to implement with their students. Many of the VM tasks that teachers chose to implement had accompanying guided questions or instructional guides to help students engage in the task and support their developing understanding. During Mark\u2019s final interview in May, he said, <\/p>\n\n\n\n

The thing that I got a LOT out of and it \u2014 I thought this was early on, I think we talked about it \u2014 was just having a study guide that kind of hits prior knowledge and then kind of guides the kids specifically through activities. And then kind of releases them to do, to solve problems based on what they, what they did with the virtual manipulative. The way the Gizmos work. I really like that progression. Umm, and so that\u2019s why I think it\u2019s easy to pick those, \u2018cause you know you\u2019re going to get that. Whereas other ones, you have to, you have to make stuff up. Umm, and it\u2019s not like that\u2019s super hard, once we\u2019re working together on things. [Capital letters used to denote emphasis in speech.]<\/p><\/blockquote>\n\n\n\n

Due\nto the belief that instructional guides were beneficial to students in activating\nprior knowledge and offering guidance, Mark was drawn to VMs that had\naccompanying questions or instructional guides. At times, the accompanying\ninstructional guide or questions provided additional support related to how Mark\nmay want to use the VM (e.g., launching a particular topic or providing more\npractice). <\/p>\n\n\n\n

However, not all VMs had accompanying questions, especially VMs related to AP Calculus, AP Statistics, and at times Geometry. Many of the VMs related to these courses are categorized as the single-representation or multirepresentation environments. Therefore, to implement these VMs, teachers of these courses (i.e., Daron, Jake, and Mike) at times developed their own instructional guides to accompany a VM. During the January PD, Daron said, <\/p>\n\n\n\n

Some of the ones that I had been looking at don\u2019t really have a guide, so \u2026 I kind of keep these things in mind [i.e., the affordances in the task analysis framework] as I try to write a guide or put together some questions\u2026 to try to make sure as many of these were covered so it was a meaningful activity for them as well. Where otherwise you might just throw the questions out, you just, I think it allows you to think more like, \u201cHow do I move them up to those levels that we\u2019re trying to get them to?\u201d <\/p><\/blockquote>\n\n\n\n

Therefore, Daron\u2019s efforts to develop instructional guides that further supported his students\u2019 learning and engagement were supported by a task analysis framework (Reiten, 2018), a tool in this activity system that was introduced during the PD. <\/p>\n\n\n\n

How Teachers Implemented VM Tasks<\/h3>\n\n\n\n

While investigating how teachers implemented VM tasks during and after the course of the PD, two themes emerged. The first theme related to the instructional role for which the teachers used VM tasks. Therefore, how teachers implemented VM tasks (object) was influenced by their instructional goals and curricula (rules). The other theme related to teachers\u2019 implementation efforts (object). Analysis of teachers\u2019 modifications of VM tasks (actions) drew from their discussions and responses on the Google Docs about the modifications they made. Both themes and teachers\u2019 modifications are discussed in the following subsections. <\/p>\n\n\n\n

Instructional role<\/em><\/strong>.<\/strong> Teachers in the\nPD used VM tasks for a variety of purposes or instructional roles. The following subsection discusses the\ninstructional role that the teachers reported VMs and tasks had within their\nclassrooms over the course of the PD. <\/p>\n\n\n\n

Table 3 highlights the different instructional roles\nteachers described related to their implementations of VM tasks. Teachers\nprimarily reported using VMs and tasks to enhance their students\u2019 emerging\nunderstandings, as a reteaching tool, as a tool for enabling students to\nreceive in-the-moment feedback, and as a tool for collecting information about\nstudent learning that could be used to inform future instruction.\nUnfortunately, as described in more detail in the next subsection, teachers did\nnot often share their implementation efforts. Teachers implemented VMs and tasks\nmore frequently than depicted in the initial data based on their final\ninterviews and surveys. <\/p>\n\n\n\n

Table 3<\/strong>
Categories, Definitions, and Examples of Teacher Identified Instructional Roles of Virtual Manipulatives and Tasks<\/p>\n\n\n\n\n\n\n\t\n\n\t\n\t\n\t\n\t\n\t\n\t
Definition<\/strong><\/th>Number of Data Excerpts [a]<\/strong><\/th>Example<\/strong><\/th>\n<\/tr>\n<\/thead>\n
Enhance: <\/strong>used to enhance students\u2019 understanding of concepts and skills through practice, communication, feedback, connecting representations and so forth<\/td>13<\/td>Josh (FinalInt):<\/strong> \u201cI mean more to enhance some of the skills we\u2019re working on. We\u2019ve used a lot of the Gizmos\u2026 at the end of that particular unit where we\u2019ve talked about adding and subtracting fractions. And then it\u2019s, \u201cWell, let\u2019s do this Gizmo and see how it looks in a different light. \u2026<\/em>So more to enhance and follow-up and build on some of the stuff that that they\u2019ve already been learning about in class.\u201d <\/td>\n<\/tr>\n
Feedback: <\/strong>provide feedback to students and\/or teachers during the lesson<\/td>8<\/td>Stan (OctPD): <\/strong>\u201cWell, we\u2019re getting feedback too. I think that\u2019s the MOST important.\u201d<\/td>\n<\/tr>\n
Re-teach: <\/strong>used as a tool to re-teach a particular concept in a different way <\/td>11<\/td>Tracy (FinalInt): <\/strong>\u201cI\u2019ve used them with my Math Plus kiddos, more as another way to learn, another re-teaching. Because it\u2019s hard for them to, in their SECond math class of the day, to have to sit and do things pencil and paper. So, I try to use it more as some sort of instructional technology where they can see it. More, you know, just a different way\u2026so it\u2019s different from what they did in their first math class of the day.\u201d <\/td>\n<\/tr>\n
Solidify understanding: <\/strong>used as a tool to help solidify student understanding by having students interact with different representations.<\/td>7<\/td>Josh (JanPD): <\/strong>\u201cKids are used to it, they like it. \u2026we\u2019ve been using it more to solidify some of the things that we are doing in class [Kelly: <\/strong>Umm hmm]. When we did this activity, you know look on the Gizmo and try it on this.\u201d<\/td>\n<\/tr>\n
Supplement curriculum: <\/strong>used as a tool to supplement curriculum by providing models of word problems, investigations and so forth<\/td>7<\/td>Kelly (JanPDSummary): <\/strong>Kelly said that her current textbook (i.e., Everyday Mathematics) does not have a lot of examples for what they are doing (i.e., adding fractions with unlike denominators). If they had the word problems in a ConceptuaMath module, then they could use the tools to model the problems, it would be helpful. Once they know how to use the tool, then it could be used to show different things. <\/strong><\/td>\n<\/tr>\n
[a] Number of excerpts is included for descriptive purposes only.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n

Other instructional uses of VM tasks existed that are not included in the table. For example, teachers also reported using VMs to introduce or launch (Moyer-Packenham et al., 2008) and review particular concepts and units. However, these instructional uses only occurred three and four times in the dataset. Using VMs for demonstration purposes or modeling (Moyer-Packenham et al., 2008) was not a role that teachers identified. This lack of use is most likely due to the prevalence of technology in the classroom (e.g., most students had Chromebooks). Additionally, the PD focused on using VMs and tasks to promote student exploration as opposed to being used only for demonstration purposes. However, for teachers with more limited access to technology, using VMs for demonstrations or modeling with VMs may have been a means for enhancing student understanding or engagement. <\/p>\n\n\n\n

During the January PD session, Daron explored a\nconfidence interval for proportions VM from Flash & Math. He specifically\nwas looking for a VM to use in his AP Statistics class that might enhance student understanding by\nsupporting students\u2019 development of conceptual understanding through connecting\nthe confidence interval and the graph of a distribution (see Figure 2). <\/em><\/p>\n\n\n\n

During the spring months, Tracy often looked for VM\ntasks that allowed students to investigate and develop area formulas (e.g.,\ncalculating the area of a parallelogram) rather than simply being told the area\nformula. Therefore, Tracy, too, used VMs and tasks to enhance her students\u2019 emerging understandings. <\/p>\n\n\n\n

\"Figure
Figure 2. <\/strong>Confidence Intervals for a Sample Proportion VM from Flash & Math (Note: This VM is no longer available through www.flashandmath.com<\/a>.)   <\/em><\/figcaption><\/figure>\n\n\n\n

Implementation efforts.<\/strong><\/em>  The following subsection describes teachers\u2019 implementation efforts during the course of the PD. The findings are based on teachers\u2019 discussions and responses to the Google Docs during the PD sessions, as well as teachers\u2019 responses on the final survey and final interviews. Although the PD aimed to support teachers in teaching with VM tasks, there was no requirement for teachers to implement the VMs and tasks that they were critiquing and modifying during the PD session. <\/p>\n\n\n\n

At least 13 teachers implemented one or more VM tasks. Unfortunately,\nteachers did not always share their implementation efforts during the PD. For\nexample, Tracy, Mark, and Josh mentioned during their final interviews that they\nhad implemented at least 12 ExploreLearning gizmos alone, in addition to a few\nother VM tasks. Other common VM collections used by teachers in the PD included\nShodor Interactivate, NCTM Illuminations, preconstructed GeoGebra worksheets, and\nFlash & Math, to name a few.<\/p>\n\n\n\n

During\nthe January PD session, Daron brought up his experience taking his AP\nStatistics students to the computer lab to do the Binomial Probability gizmo\nfrom ExploreLearning (https:\/\/www.explorelearning.com\/index.cfm?method=cResource.dspDetail&ResourceID=126l<\/a>;\ngizmo was not compatible with Chromebooks). Two days later, when he opened the\ngizmo during a class discussion, he and the class realized that it had now been\nconverted to HTML5 (therefore, compatible with Chromebooks). According to\nDaron, after doing the gizmo, students had less questions about the binomial\ntheorem compared to years past. Additionally, they were not relying on their\ncalculators (rather, they were directly substituting into the formula) and were\nnot making some of the common errors that previous students had made (leaving\noff the beginning part of the binomial formula). After the PD session, Daron\nemailed the following thoughts related to his implementation of the Binomial Probabilities\ngizmo:<\/p>\n\n\n\n

I think students were much more engaged in learning the binomial pattern. They were able to see where the pattern was coming up from the most basic situation to more complex examples. I think students were able to see the pattern in the factorials and Pascal\u2019s triangles quickly, then were able to adapt them to more difficult problems. I asked for student feedback and they said they really liked using the gizmo and it helped them to see the pattern clearly.<\/p><\/blockquote>\n\n\n\n

Daron\u2019s thoughts highlight\nhow both he and his students found the VM task helpful for supporting students\u2019\ndeveloping understanding. This early positive experience implementing a VM task\nencouraged Daron to continue implementing VM tasks during the school year. By\nthe end of the PD, Daron had implemented at least four VM tasks.<\/p>\n\n\n\n

Modifications.<\/strong><\/em> Due to critiquing VMs and tasks, teachers began modifying VMs and tasks to better fit their learning goals and student needs. Additionally, critiquing and modifying VM tasks enabled teachers to focus on the value of the VM rather than a fun thing to try (Reiten, 2018). <\/p>\n\n\n\n

Table\n4 highlights the different types of modifications to VM tasks that teachers\nmade during the course of the PD. Once again, the number of data excerpts\nrelated to each category are for descriptive purposes only and should be interpreted\nwith caution. However, as displayed in the table, most of the modifications\nthat teachers suggested related to helping students engage in the learning\ngoal, creating an instructional guide, and addressing student needs.<\/p>\n\n\n\n

Table 4<\/strong>
Categories, Definitions, and Examples of Teacher Modifications to VM Tasks<\/p>\n\n\n\n\n\n\n\t\n\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t
Definition<\/th>Number of Data Excerpts<\/th>Example<\/th>\n<\/tr>\n<\/thead>\n
Assessment: <\/strong>Modifications related to using the activity as an assessment tool (at least in part).<\/td>6<\/td>Daron (DecGoogleDoc): <\/strong>\u201cI would also like to see some assessment questions to see if they understood the point of the applet.\u201d <\/td>\n<\/tr>\n
Curricula: <\/strong>Modifications related to teachers\u2019 curriculum.<\/td>13<\/td>Josh (JanGoogleDoc): <\/strong>\u201cWe would eliminate the Lateral Area and put more focus on exploring what surface area is and how it is found (thinking about adding the area of all faces).\u201d<\/td>\n<\/tr>\n
Instruction: <\/strong>Modifications related to how the activity would be implemented.<\/td>19<\/td>Curt (JanGoogleDoc): <\/strong>\u201cWalking through a unit rate to help with writing ratios and labeling measures. This will help with labeling a unit when setting up proportions.
\n Having them check in with the teacher after Activity A, #7.\u201d<\/td>\n<\/tr>\n
Instructional Guides: <\/strong>Modifications related to developing an instructional guide.<\/td>33<\/td>Jake (NovGoogleDoc): <\/strong>\u201cCreate an activity guide.\u201d<\/td>\n<\/tr>\n
Learning Goal [a]: <\/strong>Modifications related to the learning goal.<\/td>51<\/td>Curt (talking with Karen in Nov about a modification): <\/strong>\u201cAdd the question \u2018How do the image and preimage compare when the scale factor is one?\u201d\u2019<\/td>\n<\/tr>\n
Miscellaneous: <\/strong>Modifications did not fit into one of the other defined categories (e.g., related to technology constraints or non-specific modifications).<\/td>8<\/td>Josh and Mark (after Obs3): <\/strong>\u201cWe would make changes based on the constraints (#3), but that can't be done to our knowledge.\u201d <\/td>\n<\/tr>\n
None: <\/strong>Teachers said they would not modify the activity.<\/td>8<\/td>Erin (FebGoogleDoc): <\/strong>\u201cI wouldn\u2019t change it, I think it fits the student's needs.\u201d<\/td>\n<\/tr>\n
Practical Details: <\/strong>Modifications related to reformatting an instructional guide (e.g., to reduce space on a page or adding screenshots), clarifying directions, or correcting an error.<\/td>15<\/td>Mari (talking with Pam in Nov about a modification): <\/strong>\u201cMaybe we would just edit and only have two (pause) examples and save trees.\u201d <\/td>\n<\/tr>\n
Student Needs: <\/strong>Modifications related to student needs.<\/td>25<\/td>Jake (NovGoogleDoc): <\/strong>\u201cMore basic functions. \u00a0There were some pretty crazy functions that showed up that I wouldn\u2019t want my students to be getting frustrated with.\u201d <\/td>\n<\/tr>\n
[a] In this district, the learning goal<\/em> was referred to as the essential question<\/em>.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n

Unlike earlier codes that were grounded in the\nliterature, the codes related to the types of modifications arose from the\nresponses of the teachers. Previous studies (e.g., Moyer-Packenham, et al.,\n2008; Moyer-Packenham, Salkind, Bolyard, & Suh, 2013) have investigated how\nteachers implemented VMs, but have not investigated how teachers modify VMs and\ntasks to implement in their classrooms. Due to the introduction of the guiding\nquestions and task analysis framework supporting teachers\u2019 efforts to critique\nVMs and tasks, this study intentionally investigated how teachers modified VMs\nand tasks to implement with their students. <\/p>\n\n\n\n

Often, teachers\u2019 modifications spanned more than one\ncategory. For example, as an intervention teacher, many of Tracy\u2019s students\nstruggled with reading as well as mathematics (as also discussed in Vukovic,\n2012). Therefore, Tracy\u2019s modification for the VM task that she was critiquing\nin December was to <\/p>\n\n\n\n

read each portion to the kids and have them answer before moving on. Give the kids screenshots of what each portion should look like. Possibly adjust the exploration guide to eliminate some of the language to make it a little more user friendly.<\/p><\/blockquote>\n\n\n\n

These modifications align with the instruction (reading each portion to the students), practical details (adding the screenshots), and student needs (eliminate some of the language to make it more user friendly) categories defined earlier. <\/p>\n\n\n\n

Tracy\u2019s modifications before implementing the VM task (her actions) demonstrate how teachers began to think carefully about how to implement VMs and tasks to promote their students\u2019 understanding (part of their community) rather than assuming the VM already met their student needs. While intentionally thinking about how her students might engage in the VM task that was initially selected, Tracy determined that her students may struggle with the reading level. Therefore, to promote student engagement in her learning goal (rule), rather than students\u2019 struggling with the reading, Tracy decided to modify the task so that she would read the instructional guide to the students and reduce some of the language demands on the students. <\/p>\n\n\n\n

Discussion<\/h2>\n\n\n\n

Teachers have been implementing VMs and tasks for the past three decades. Despite the benefits to student engagement and understanding when teachers use VMs, studies indicate that teachers\u2019 use of VMs decreases as students progress through school (Moyer-Packenham & Westenskow, 2013). Unfortunately, few studies have investigated professional development opportunities aimed at supporting teacher\u2019s use of VMs (Driskell et al., 2016). Therefore, this study aimed to address both of these needs in the field by investigating why and how secondary mathematics teachers implemented VM tasks (object) while participating in a focused professional development opportunity. <\/p>\n\n\n\n

Teachers in this study identified several benefits to students\u2019 emerging understanding as reasons why they implemented VM tasks. For example, teachers identified the visual nature of VMs <\/strong>(Moyer-Packenham et al., 2013a; Reimer & Moyer, 2005), opportunities for student exploration, and assisting in differentiating instruction (Bouck et al., 2015; Shin et al., 2017) as the most common benefits to instruction and student understanding. <\/p>\n\n\n\n

Consistent with the field, teachers also cited students benefiting\nthrough the use of VMs and tasks due to the ability for students to offload\ncomputational aspects of a task (Moyer-Packenham et al., 2015), receive\nfeedback (Edwards Johnson et al., 2012;\nMoyer-Packenham & Westenskow, 2013), practice problems (including extending beyond the\nVM), connect multiple representations (Moyer-Packenham et al., 2015;\nMoyer-Packenham & Westenskow, 2013), and encourage student engagement\n(Moyer-Packenham & Westenskow, 2013). Important to note, however, most of\nthese benefits were identified for elementary students. Thus, this study\nextends the list of benefits in the current literature base to the secondary\nlevel.<\/p>\n\n\n\n

New to this study, teachers also identified modeling (using or creating models) as benefit to students\u2019 emerging understanding and a reason why they chose to use VMs and tasks. Others have identified and discussed the remaining top eight teacher identified benefits. As attention to modeling increases, for example as teachers integrate the standards of mathematical practices (Common Core State Standards Initiative, 2010), this benefit of implementing VMs and tasks may become more prominent. It is not known, however, whether the identification of modeling is due to the teacher population (i.e., secondary teachers) or due to the increased focus of modeling in curricula.<\/p>\n\n\n\n

These teacher-identified benefits also highlight an important factor when investigating why teachers choose to implement a VM task. That is to say, when investigating why secondary mathematics teachers implement a specific technology tool (in the case of this PD VM tasks), considering how they view their students\u2019 emerging understanding to be impacted is a mediating factor that needs to be considered. After all, teachers\u2019 students are part of their community (see Figure 1) and, hence, influence their efforts to teach with VM tasks. <\/p>\n\n\n\n

If\nteachers do not think that their students will benefit from engaging with a VM\ntask, then they usually will not implement the task. Daron\u2019s early positive feedback from students\u2019 interaction\nwith the VM task from ExploreLearning encouraged him to keep looking for ways\nto integrate other VMs and tasks. Therefore, it is important to consider the\ninfluence teachers\u2019 early implementation efforts have on their future desire to\nimplement new tools.<\/p>\n\n\n\n

 Due to the prolonged duration of the PD\n(October to May), teachers were able to see the impact the VMs and tasks had on\nstudent learning. Often, for changes in teachers\u2019 instructional practices to\nendure, teachers need to have opportunities to see improvements in student\nlearning (Guskey, 2002).<\/strong><\/p>\n\n\n\n

During the PD, teachers identified instructional guides as being important components of VM tasks. In fact, teachers in this PD typically implemented VMs that had accompanying instructional guides or VMs that were in tutorial environments. Teachers in the PD stated that the instructional guides were beneficial because they provided guidance for their students and supported their students\u2019 developing understanding. <\/p>\n\n\n\n

Since the focus of this\nPD was using VMs to promote student exploration as opposed to using VMs for\ndemonstration only, it is not surprising that teachers gravitated to VMs in the\ntutorial environment or VMs that had accompanying instructional guides.\nHowever, this PD provided teachers with a tool (a task analysis framework) that supported their efforts to\ncreate instructional guides when guides did not already exist. Therefore the\nteachers in the PD were supported in developing VM tasks that aligned with how\nthey thought their students\u2019 emerging understanding would best be supported. <\/p>\n\n\n\n

Hence, when investigating why teachers choose to implement VM tasks, it is important to consider how teachers are supported in developing accompanying materials that they find helpful for supporting students\u2019 developing understanding. In this PD, the task analysis framework became an important tool that supported teachers\u2019 implementation of VM tasks (object).<\/p>\n\n\n\n

The secondary mathematics teachers in this study identified implementing VMs for a variety of instructional roles. The findings of this study further support Moyer-Packenham et al.\u2019s (2013b) findings that teachers in the PD were able to move beyond using VMs and tasks because they were a \u201cfun thing to try.\u201d <\/p>\n\n\n\n

Teachers commonly\nidentified using VMs and tasks to enhance their instruction. However, teachers\n(see Table 3) also used VMs and tasks as a reteaching tool. For example, Tracy\nfound it helpful to use VMs to help her students in the math intervention class\nto visualize the concepts in a different way than they had seen earlier in the\nday. The roles that teachers identified using VMs and tasks, however, may have\nbeen influenced by the emphasis in the PD, the access to technology (tool), and the teachers\u2019 curriculum (rule). <\/p>\n\n\n\n

In this district, all\ngrade 6-9 students had Chromebooks and some high school classes had class sets\nof Chromebooks. The following year, all students in grades 5-12 had\nChromebooks. Thus, student access to technology was readily available. Most of\nthe teacher-identified roles for using VMs and tasks are different from the\nuses identified by Moyer-Packenham et al. (2008; i.e., investigation, skill\nsolidification, introduction, game, and other). It is not certain if the uses\nof VMs in this study were due to grades represented by the teachers, the access\nto technology, the focus or the PD, or the type of data collected (i.e.,\ninterviews, discussions, surveys, and responses to Google Docs vs. teachers\u2019\nlesson summaries). <\/p>\n\n\n\n

In terms of the activity system described earlier, these factors influencing how teachers used VMs and tasks relate to the rules (the curricula) and tools (Chromebooks) components of the system. Additionally, the district initiative (rules) to use technology as a means to support student engagement and understanding was a factor in how the teachers chose to implement VMs. For teachers with limited access to technology or without the expectation to implement technology for particular purposes, using VMs to launch or review units of study may be relevant to supporting their students\u2019 developing understanding. Furthermore, using VMs for demonstration or during whole class instruction may be other relevant uses even though these uses were not identified by teachers in this PD. <\/p>\n\n\n\n

Division of labor is another component of the activity system that is important to consider when investigating why and how teachers implemented VM tasks. Teachers in the PD were supposed to sign up for the PD with their teaching partner to increase collaboration efforts. Only two teachers (Daron and Kelly) did not have someone in the PD with whom they could collaborate. The other teachers had one and sometimes more teachers with whom they could regularly collaborate. These teachers would jointly investigate potential VM tasks to implement with their students and work together to modify instructional guides to accompany the VM tasks. <\/p>\n\n\n\n

Daron, however, worked\nalone to explore potential VM tasks. Additionally, many of the VMs for AP\nStatistics did not have accompanying instructional guides. Therefore, Daron\nspent time individually working to create instructional guides or guiding\nquestions to accompany the selected VMs. <\/p>\n\n\n\n

Another teacher, Tracy,\noften had one or two fellow teaching partners to work with during the PD. Tracy\u2019s\nopportunity to distribute work amongst her two collaborators may have\ncontributed to the number of VM tasks that Tracy and her collaborators were\nable to implement during the PD (more than 12) compared to Daron (at least\nfour). <\/p>\n\n\n\n

Therefore, when investigating how and why teachers implement technology-based tasks, it is important to consider their collaborators and whether they can distribute the time to select and then modify the tasks to fit their learning goals and student needs. Teachers working individually or in isolation may need additional resources to make their selection of tasks more efficient. In this PD, the task analysis and guiding questions, repository of tasks and guidance related to where to search for potential tasks (Reiten, 2018) are example of such supports. <\/p>\n\n\n\n

In general, teachers\u2019 experience and support resources to create instructional guides or guiding questions is something to consider. For example, many high school level VMs are single representation environments. For these environments, teachers (Daron, Jake, and Mike) typically created instructional guides or guiding questions to accompany the VM. Many of the VMs and tasks geared to the middle school level often had accompanying instructional guides or guiding questions. Teachers valued the role of the instructional guides and questions in supporting their students\u2019 emerging understanding and the time it could take to create the instructional guides, so the middle school teachers often implemented in their classroom only VMs in nonsingle-representation environments (see Mark\u2019s comment earlier about why it was easy to choose VMs from ExploreLearning). <\/p>\n\n\n\n

Since the effectiveness of a VM or a VM task is contingent upon how it is implemented in the classroom (Ladel & Kortenkamp, 2016; Suh, 2016), it is important to consider how teachers are supported in creating instructional guides or guiding questions. Teachers need to be supported in modifying and developing VM tasks that support students\u2019 emerging understanding through communication, reflection, and connecting multiple representations. The linking of multiple representations as well as students\u2019 actions (i.e., simultaneous linking of VMs) can promote the development of conceptual understanding.  Currently, however, minimal resources exist that may support teachers in creating instructional guides to accompany VMs in a single representation environment.<\/p>\n\n\n\n

Although the PD could not affect the instructional time teachers had in their classrooms, the PD did provide teachers time to find, critique, and modify VM tasks specifically related to their upcoming units. As demands on teachers\u2019 time continue, integrating supported time within PD for teachers to find and develop VM tasks directly connected to their curriculum can help to support teachers\u2019 integration efforts by strengthening the connection between the PD and teachers\u2019 instructional practices (Wilson, 2008). <\/p>\n\n\n\n

Anecdotally, structuring teachers\u2019 work time during the PD proved to be more productive than teachers choosing how to use the work time that was given. Allocating time to search and select tasks, then modify and prepare instructional guides for use with their students, then providing discussion\/reflection time was more productive compared to when teachers were not given structure for how to use the time given. Additionally, it is important to note that teachers came to each PD session with a predetermined learning goal. This learning goal helped to focus their searching efforts and therefore use the allocated time more efficiently. <\/p>\n\n\n\n

Implications for Teacher Education<\/h2>\n\n\n\n

Although VM tasks were new to over half of the PD participants, for a majority of the teachers in the PD, VM tasks became a regular tool that they implemented in their classrooms. Therefore, the findings indicate that secondary mathematics teachers can be supported to implement VM tasks; hence, potentially addressing the decrease in VM use as students progress though the grades. <\/p>\n\n\n\n

Improvement in teachers\u2019 implementation of VM tasks, is a process (not an event) and one that must be continually supported as teachers strive to improve student understanding (Guskey, 2002). Future work needs to be done to investigate how teachers, especially secondary mathematics teachers, implement VMs and tasks to support students\u2019 development of conceptual understanding; for example, investigating teachers implementing the tasks, as well as investigating the instructional guides and guiding questions that teachers use to accompany VMs. When investigating how and why teachers use a particular technology tool, it is important to consider how they are supported in doing so; for example, considering how they receive training in the tool (PD) and support resources (e.g., repository or VMs or task analysis framework). <\/p>\n\n\n\n

Additionally, it is important to acknowledge the influence that various components of teachers\u2019 activity systems have on their implementation efforts when investigating why and how teachers implement technology tools and tasks. Teachers most frequently modified VMs and tasks to better support their learning goals and student needs (modifying\/developing instructional guides, modifying the implementation strategy, etc.). Students are an important component of teachers\u2019 activity systems and influence why and how teachers\u2019 implement particular technology tools and tasks. <\/p>\n\n\n\n

Since students\u2019 achievement levels are related to teachers\u2019 experience using the manipulatives (Moyer-Packenham et al., 2013b), how teachers are supported in engaging with VMs is important. Therefore, it is not enough to only provide access to a tool or make teachers aware of a particular tool. Rather, teachers need to be supported in learning how to critique technology based tools and tasks so that they are able to modify and\/or design quality opportunities for their students to engage with the tools and tasks. Quality opportunities are ones in which students\u2019 emerging understanding is supported through communicating, reflecting, and connecting multiple representations. Therefore, PD opportunities are needed that provide teachers opportunities to interact with technology tools situated within quality opportunities.<\/p>\n\n\n\n

VMs and tasks (as well as other technology tools and tasks) can be used to address issues of inequity and support students in developing rich understandings of mathematical concepts. However, to realize the full potential of VMs (or any technology tool for that matter) teachers need to be supported in critiquing technology tools and tasks based on their learning goals and student needs. Additionally, teachers need to be supported in moving beyond choosing a particular tool or task because it is a fun thing to try<\/em>. Focused PD opportunities, similar to the one investigated in this study, are one means to support and prepare teachers to use technology in an effective and innovative manner. <\/p>\n\n\n\n

Linking technology tools and tasks directly to teachers\u2019 curricula units helps to strengthen the connection to teachers\u2019 practice and the PD. How teachers are supported to engage in and implement (new) technology tools has the potential to transform the learning opportunities for their students. As the expectations for teachers to implement technology tools and tasks continues to increase, how we support teachers is a critical issue.<\/p>\n\n\n\n

Notes<\/h3>\n\n\n\n

[a] Through their review of 580 lesson summaries of 95\nK-8 teachers, Moyer-Packenham, Salkind, & Bolyard (2008) identified \u201cseven\ncategories describing how mathematical tools were used in lessons\u201d (p. 207). The\ncategories were (a) investigate concepts, (b) skill solidification, (c)\nintroduce, (d) game, (e) aid, (f) model, and (g) extend. The investigate\nconcepts (which relates to exploration) and extend (which relates to\ndifferentiation) codes in why teachers implemented VM tasks. <\/p>\n\n\n\n

[b] Fido\u2019s\nFlower Bed (https:\/\/www.explorelearning.com\/index.cfm?method=cResource.dspDetail&ResourceID=1011<\/a>) is a gizmo from ExploreLearning exploring\nperimeter and area. Using squares representing sod, students are supposed to\nconstruct lawns and then fence the lawn to explore perimeter and area. They can\nuse prebuilt lawns of 36 squares to explore lawns containing the same area but\ndifferent perimeters or create their own lawns by clicking where they want to\nplace the sod.
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Appendix A
Examples of VMs Used by Teachers During the PD Opportunity <\/h2>\n\n\n\n
pdf<\/a>Download<\/a><\/div>\n\n\n\n

Appendix B
Task Analysis Framework<\/h2>\n\n\n\n
pdf<\/a>Download<\/a><\/div>\n\n\n\n

Appendix C
<\/h2>\n\n\n\n
pdf<\/a>Download<\/a><\/div>\n
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In today\u2019s classrooms, teachers are expected to integrate technology as a means to enhance student understanding and engagement. Depending on the content area, the technology tool can take on many forms (e.g., interactive white boards, tablets, computers, games, software, virtual manipulatives, etc.) and roles (see Cullen, Hertel, & Nickels, 2020). Yet, teachers often report that […]<\/p>\n

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