Wicked Problem: Using Algebra Balance Scales Virtual Manipulative

Educational Need

Solving algebraic equations represents the focus of the educational need addressed with this project. More specifically, this project targeted the following two algebra standards:

• Understand that adding or subtracting the same number to both sides of an equation creates a new equation that has the same solution (A.FO.06.12)
• Understand that multiplying or dividing both sides of an equation by the same non-zero number creates a new equation that has the same solution (A.FO.06.13)

Proposed Technological Solution

After time spent both thinking and searching for strategies to teach solving equations, the decision was made to use the National Library of Virtual Manipulatives Algebra Balance Scales applet to address the educational need identified. The free Algebra Balance Scales applet provides students with a linked visual and symbolic representation of the equation which change concurrently once students modeled the equation and begin to solve to determine the value of the variable. As students selected an operation to perform to both sides, the changes are evidenced in both the equation as well as the blocks removed from the respective sides of the scale.

Technological Pedagogical Knowledge 

The selected technology, Algebra Balance Scales virtual manipulative, supports the teaching methods and strategies intended for the intervention. The applet scaffolds solving equations, models steps to solve an equation sequentially both symbolically and visually, and provided immediate feedback both verbally and visually to student responses. The applet offered students the opportunity to compare the created model and the given equation before proceeding to solving. Additionally, the virtual manipulative allowed for connections and observations to be made regarding how changes effect all representations of the equation, an advantage over using a physical manipulative to investigate solving equations.

Technological Content Knowledge

“Meaning does not reside in tools; it is constructed by students as they use tools.” Herbert and Colleagues (1997) quoted by Suh in Third Graders' Mathematics Achievement and Representation Preference Using Virtual and Physical Manipulatives for Adding Fractions and Balancing Equations. The Algebra Balance Scales applet helps make the content accessible by providing linked, multiple representations. The visual representation of the scale and blocks helps address student misconceptions regarding coefficients. The picture links to the symbolic representation by clarifying what coefficients actually mean. “One,” is a common response from students when given an equation such as 3x+7=13 and asked how many “x's” are on the left side. The manipulative helps address this misconception by providing verbal and visual feedback during the modeling phase. Students have the opportunity to recall prior information, or experience for the first time, that multiplication is repeated addition. Simultaneous manipulation of the scale and symbolic representation contribute greatly to the applet's support of the content and increased accessibility to students. The manipulative prompts students to work between both the visual and symbolic representation. As one representation is changed, students evidence how the change effected the other representation supports the learning objectives. Students watch as blocks are added, taken away, multiplied, or divided and the scale remains balanced. The differences in the two images demonstrate the linked representations previously described. The applet provides students an image of the intended learning goals.

Pedagogical Content Knowledge

The instructional strategies used for this intervention activity support the content much like the selected technology supports both content and pedagogy. Students ability to solve equations depends largely on a developed understanding of the symbolic representation. The misconception regarding coefficients discussed previously relates to the essential understanding of symbolic representations. Scaffolding supports the content by ensuring students have appropriately modeled the given equation. The scaffolding continues after the modeling phase also. This links to the importance and understanding of order of operations. Immediate feedback then continues the support of the content. Students proceed through the scaffold to solve the equation with appropriate mathematical moves; however, the feedback redirects students with a little hint to reconsider and manipulate the equation differently. Both symbolic and visual representations of equations further support the content's accessibility to students. As noted previously, verbal and visual feedback helps connect and develop an understanding of what manipulating an equation actually does to the equation. Again, the visual feedback contributes and supports the learning objectives by showing a balanced scale emphasizing the equality of both sides of the equation. Scaffolding, providing feedback and multiple representations, along with student manipulation of the applet all assist in making the content more accessible to students. 

Click here to read the full script.

Group Leadership Project: Jing Tutorial

Our group selected Screencast-O-Matic to capture demonstrating and narrating the features of Jing.  Using the screencast allowed for the viewer to actually see what the process will look like as the different steps and features were modeled and described.  With several unsuccessful attempts combining both the narration and visual models, I decided to capture the video and audio separately using Camtasia Studio.  Audacity was used to clean up the narration using the "noise removal" feature.  The video published above was produced using Camtasia to edit together the screencast and narration.  This allowed for smoother transitions, mistakes that were edited out, and a more professional representation of what Jing offers.

Working on this tutorial reaffirmed that the content, in many respects, is just as important as the delivery and modeling of the information.  Creating tutorials requires much the same thought process as lesson planning.  The audience must be considered.  Before creating the final product, it needs to be determined who the tutorial is intended for and the previous knowledge and experiences with technology anticipated.  Revisiting our tutorial, our group anticipated viewers to be familiar with computers and accessing the internet to download files as our tutorial began by encouraging viewers to download Jing.  Thinking through the development of the tutorial labeling these types of experiences as professional learning seems more fitting as considerations similar to those used in lesson planning occurred.  

In the future, I would be intentional about thinking through the tutorial itself as a professional learning experience.  Camtasia Studio offers different features which I would also include in the tutorial such as quizzing.  This would provide the viewer with a way to quickly self assess personal progress and understanding.  Additionally, I would consider captioning the tutorial to make it accessible to a greater population of people.  

Professional Learning Plan

Status Report:  In Progress.

Mobile Learning

The images presented above represent the poll created for my students. I experimented with the software and created both multiple choice and free response questions. Due to the approaching end of school, after the creation of the polls for students I printed and distributed the questions for students to respond to on their own. The software allows for polls to be left open. I left all three polls opened so students can submit their responses on their own time. The software allows polls to remain open for 30 days before the poll is automatically closed. Responses can be viewed as submitted allowing viewers to see the collected data immediately.


Joining the cell phones in education on Classroom 2.0, I found several posts sharing information regarding use of cell phones as well as questions other educators posed requesting feedback and advice about incorporating cell phones into the curriculum. I responded with some suggestions I had heard at a conference regarding appropriate use of cell phones in the classroom. One suggestion I had heard at the conference was to have students leave the phone on the desk until the time came to text a response. It provides a quick visual reference for the teacher to see which phones are or are not on the desk and in turn then further inquire as to what the cell phone is being used for.


In the past I've used laptops in the classroom.  The high school I taught at used Carnegie Learning Algebra which had an online software component.  Students used the laptops to complete instructional units using the software.  In addition, students used Gizmos; students completed the online simulations and then submitted work through e-mail.  My STEM geometry students were issued a flash drives for saving and storing their work.  Each student kept work on the drive so they could access it when needed.  The main issue that arose from using these drives were students forgetting to bring them or losing them. 


Mobile technology in the classroom offers many benefits.  Cell phones and iPods seem the most prevalent technologies in the hands of my students.  This seems a good place to start requiring some thought related to students participating without either technology.  I prefer the thought of cloud computing to USB or flash drives.  Work saved in a web-based program can be accessed anywhere with the internet regardless of the software installed on the computer being used.  It also helps eliminate the issue of lost drives.  Although interest in working with other technologies such as iPads or personal technology, the expense would need to be overcome.  I've heard of iPads being used in small groups to present projects to groups.  This would minimize the number necessary.  Ultimately, I'd like to incorporate mobile technology in my classroom as a method of participation for students such as cell phone polls.

Part D: Findings and Implications

Formative:  Did the project get implemented as planned?
As noted in the post, Part C:  Implementation, the implementation of the Algebra Balance Scale Virtual Manipulative intervention experienced revisions prior to the onset of implementation as well as early on during implementation.  Some of those revisions coupled with time constraints induced both by the end of the school year and absences of students selected to participate elicited further alterations to the implementation plan.  The chart below highlights the proposed implementation plan juxtaposed with the actual implementation plan.  Play the video to hear the explanation and continue on to hear about evidence of success, or read the explanation about the implementation plan below.

The proposed implementation plan provided time to model the virtual manipulative to students and provided independent time to familiarize themselves with the tool to be used for the intervention.  Additionally, students would both see demonstrations and have opportunity to practice capturing and saving screen shots.  Working with the group of students throughout the course of the year, I selected students to participate who struggled during the Algebra unit having difficulties both writing and solving equations.  Students previously participated in intervention activities during the Algebra unit to help develop a better understanding of equations.  The plan dedicated the second lab day to completing an activity on solving equations.  The plan called for the third meeting to be a reflection and discussion of the activity.  Journal prompts would provide an opportunity for students to individually think through the activity, and then process and discuss observations and thoughts with the group.  The discussion was intended to help focus student thinking, clarify identified misconceptions and questions, and help students articulate the necessary information to create the tutorial on solving equations.  I intended the remaining four meetings to provide groups time to create the tutorial.


Lending consideration to thoughts and questions of peers as well as problem solving a few issues, the actual implementation began with students completing a survey to gauge current understandings of solving equations.  As originally planned, the first day included time to model the Algebra Balance Scales applet and provided students individual exploration time.  Due to inconsistent attendance, field trips, and a lack of available computer labs, the remaining meetings for students occurred in the listed order; however, some students experienced the steps nearly a week apart.  Student's second meeting included using the Algebra Balance Scale to solve equations generated by the applet.  As explained in Part C: Implementation, the create feature in the applet had unforeseen limitations.  Students captured screen images and saved the files for future review and reflection.  After the images were captured, I created presentations for students to view the sequence of images which were used to complete a reflection using another survey created with Google Forms. 


Summative:  Evidence of success in addressing the problem of practice.
The intended indicators of success were the explanations presented by students in the tutorial modeling and explaining how to solve equations.  The altered implementation also impacted the success indicators.  For the aforementioned reasons, students did not produce the tutorial.  The reflection and discussion involved students viewing the images captured working with the Algebra Balance Scales and responding to questions presented using Google Forms.  Here are two examples of the image sequences students used to reflect and explain their thinking. 

Reflecting further on identifying the level of success led to the conclusion success did not depend on the creation of the tutorial.  The actual implementation plan merely changed the presentation of the evidence.  Instead of listening to student explanations of solving equations, to look for evidence of success I needed to consider student reflection responses in light of the educational need which remained the same:

• Understand that adding or subtracting the same number to both sides of an equation creates a new equation that has the same solution (A.FO.06.12)
• Understand that multiplying or dividing both sides of an equation by the same non-zero number creates a new equation that has the same solution (A.FO.06.13)

 Please watch the following to hear the analysis of two student's reflections and what success can be evidenced from their responses.

How would you approach another project of this type differently given what you've learned here?
Given the opportunity to consider a similar project of this type, the first step after identifying an educational need would be to gather and analyze information regarding student prior knowledge necessary to address the identified educational need.  In the case of the Algebra Balance Scales Virtual Manipulative Intervention, the educational need identified, although relevant, provided insight as to why students struggled with solving equations.  Student responses and explanations demonstrated an insufficient understanding of equations.  Students needed to develop a conceptual understanding of equations, the necessary previous knowledge, before moving on to the standards identified for this project.  Assessing student understanding of prior knowledge essential for approaching the educational need provides opportunities to refine the educational need to better meet students where they are at and perhaps provide insight as to how to approach the subsequent educational need.  


Future approaches to similar projects will allow for more implementation time and considerations for the circumstances which made it difficult to arrange meeting time with students.  Creating a greater implementation window offers not only more time for students to work with the manipulative itself to see if that fosters a deeper understanding and progress toward the educational need, but also more time for students to reflect.  This too would provide time to discuss and question students regarding their reflections and thoughts on the activity to better gauge their level of understanding.  


What are the lessons learned that others might benefit from knowing about?
Although not a novel thought, one lesson reaffirmed during this project is the need as an educator to try the activity before asking students to complete the task.  Although I spent time working with the virtual manipulative applet and its features, I did not try solving the real-world scenarios using the create feature of the Algebra Balance Scales.  Trying the activity would have helped combat the troubleshooting on the fly when I realized with several students at computers that the virtual manipulative could not be used to solve the equations I had created.  


A second lesson, again reconfirmed throughout the project, is to ask for feedback.  Posting the activity and having others respond with questions and thoughts helped shape my final project.  Responses prompted ideas I had not considered and in turn resulted in revised or clarified plans.  


In what ways will you endeavor to do the same project again, and what will you change or not do?
The Algebra Balance Scales Virtual Manipulative provides great opportunity for impacting student understanding of solving equations.  I would consider use in the future to help students develop connections between verbal and visual representations.  In the future, I would again assess students before working with the applet.  In this instance, student responses indicated another educational need:  developing an understanding of equations.  Aspects of the implementation plan such as modeling the manipulative and offering students time to individually experiment with the applet would remain.  Using Google Form to create surveys also provided significant benefits.  Students navigated to the form from a link on my wiki page.  Also, the results were compiled in one place accessible from anywhere with an internet connection.  Additionally, Google Forms provided user-friendly options for viewing and representing the data collected.  Reflecting on the actual implementation plan, I would still have students complete an activity using context based questions for students to solve; however, the questions need modification to work with the manipulative.  Lastly, students would still reflect and discuss their experiences.  One suggestion to incorporate is VoiceThread.  Students could narrate explanations to their screen images captured step by step.  Additionally, this would provide another piece of evidence to consider when determining student understanding that allowed dialogue between myself and the student.  To facilitate discussion among students in the group, students could post responses to prompts and comment to each other using a blog.  This could be the platform used for groups to discuss ideas and concepts prior to creating the tutorial.  Another modification to the project includes a greater timeline.  Working with the manipulative over the course of a marking period or semester provides students more time and opportunities to impact their understanding of solving equations.  The tutorial then serves as a summative assessment to evaluate student understanding and progress.

Part B: Storyboard and Script

During our web conferencing session, our group divided up the aspects of Jing to include in the tutorial.  The four main features of Jing to explain included:  screen capture, editing images, capturing video, and sharing images and videos.  In addition, an introduction to include information pertaining to where and how to download Jing would be presented initially.  Each group member left the conferencing session with the task to create the storyboard for one of the features of Jing.


Our storyboard was created using Google Documents.  Google Documents offered our group the opportunity to collaborate and provide feedback on each member's contributions.  Additionally, using Google Documents provided the group with a "big picture" view of our tutorial since each member worked individually to present one aspect of Jing.  A template created for our storyboard provided a space for the visual description, narration, interactions and transitions, as well as notes for each individual slide or scene to be recorded in the tutorial.  Additionally, the slide provided a space for the title or description of the storyboard scene as well as a slide number to arrange the order of the information for our tutorial.  Each page includes the corresponding script and narration for each slide.  Our final tutorial will be created using screencast-o-matic.  Notes under the visual area will allow Rachel to display and demonstrate the necessary steps listed.  Each slide notes the inclusion of a transition between slides consistent with those in the tutorial.  After Rachel compiles the visual information, I will use screencast-o-matic to record the tutorial with the narration.


My contribution to the storyboard include the three introduction slides which demonstrate where to download Jing for free as well as running the installation and registering for a screencast account.  I also contributed the slides titled editing images.  These four slides demonstrate to users where to find Jing's editing tools, what the editing tools include, and how to use the tools on a captured image.  Rachel contributed the storyboards titled: how to start capturing the desired screen area, selection being made, and screen area has been made and will be captured.  Spike created the storyboard titled video screen shot.  Bill contributed the four slides titled sharing with Jing.


The remaining tasks include the creation of the slides for the tutorial.  Slides will be created using powerpoint and saved as image files for use in the tutorial.  Rachel will compile the images for powerpoint slides from the content contributed to the storyboard.  I am recording and narrating the tutorial and will then distribute the tutorial.

Part C: Implementation

Please feel free to watch the above video or read the implementation updates below.

Surprises

After reviewing feedback and responses to previous blog posts, I revised the implementation plan for using the Algebra Balance Scales Virtual Manipulative.  Initially, the plan consisted of modeling the features of the applet to demonstrate for students how to create the equations using the blocks and scale.  Students would then model equations to real life situations and use the manipulative to solve, discuss the activity to make connections among representations, and create a tutorial demonstrating how to solve equations.  The following questions helped prompt further thinking about the plan and resulted in some revisions to the implementation plan: 

"Are you planning on modeling the website before students work on it alone and then create a tutorial?"
"I wonder how you will collect data before and after the technology intervention to see if it worked?"

Although working with the selected students previously during the unit on equations and noting their struggles, I created a survey for students to complete using Google Forms before working with the manipulative to address the second question.  The first day students submitted the survey and explored with the Algebra Balance Scale individually to familiarize themselves with the applet.  Thus the surprises began…

Student responses demonstrated incorrect, non-mathematical understandings of equations.  Some responses require further questioning to determine the student’s level of understanding.  This suggests another educational need for students pertaining to developing an understanding of equations, perhaps one to be tackled prior to the selected need linked to specific standards related to solving equations.  Most students correctly identified a couple of the listed equations, yet did not categorize others as equations prompting the question of if students know what equations are and moreover how the Algebra Balance Scales manipulative could be used to support and develop student definitions of equations.  I anticipated incorrect responses to questions about first steps to solve equations as that misunderstanding was the educational issue to be addressed by using the Algebra Balance Scales.  The pie charts display a variety of answers concerning steps to solve equations.  No student selected the anticipated answer in two out of three of the questions.

Unexpected Bumps...

In order to work with any of the students on my caseload, I must schedule them out of another class limited to either an elective class, lunch time which in this case doesn’t offer enough time, or during Channel 1 news and reading time in the morning.  Difficulties arose finding time in a computer lab for students to work.  The second lab opportunity included internet connection difficulties.  For the remaining implementation as well as future use of the intervention, an alternative needs to be thought through so the time is not wasted and students can still make progress toward understanding how to solve equations in the midst of internet complications.

A second Google Form was created with real-world scenarios for students to model and solve using the manipulative with the intent of students completing the task the second day in the lab.  When I initially experimented with the applet’s “create” feature, I input values to create equations and worked through to solve in order to confirm the applet worked similarly to when users solved equations generated by the manipulative itself.  For the brief moments the internet cooperated, I quickly realized a new limitation of the “create” feature.  The applet limited the numerical entries to one digit.  The scenarios created for students could not be entered in the applet, thus the scenarios will need to be revised in order to use the Algebra Balance Scales to model and solve the equation.  Instead, students worked on solving problems generated by the applet as opposed to the activity.

Delights

During the implementation of the intervention, one thing that went well was students taking screenshots while working with the manipulative.  I demonstrated for students how to use the “print screen” feature and paste the image in paint.  The images can then be saved for future use.  Students did an excellent job remembering to both take and save the images.  As I monitored student progress, I wanted to help students see the connections between representations.  I determined, given the circumstances, the images students captured would be put together in a presentation for them to review.  This combines the discussion and reflection sections from the initial plan.  In addition to students discussing noticings and wonderings, questions related to the changing representations would be strategically asked of students.

In some instances, students received feedback from the applet offering hints and suggestions to try something different to solve.  This provided opportunity to dialogue with students to hear more of their thought process and ask questions prompting students to think.  For future implementations, I’d like to think through a way to record those conversations also.

One Student's Experience 

Despite internet complications leaving most student work saved to the network and inaccessible outside of school, I had been able to upload one student’s images to Flickr.  The following represents one example of solving a problem using the Algebra Balance Scales and the images students have opportunity to review and discuss.  

Here the student correctly modeled the equation by placing the blocks on the scale.  Visually, the student made both sides of the scale equivalent since the scale is balanced.  Although possible to balance the scale by modeling a different equation, the applet does not allow students to proceed without correctly constructing the given equation.

The student attempted to solve the equation by subtracting seven from both sides.  The prompting below the scale in red indicated adding seven was incorrect.

Next, the student took away an “x” from the right side of the equation since there was at least one on both sides to subtract.

After that, although not captured as an image, the student took away the unit block on both sides.  Lastly, each side was divided by three to create three equal groups of unit blocks and determine the value of each “x” block.  

That’s all for the current update, check back for more!

PART A-Brainstorm Session

For our group leadership brainstorm session, we opted to use Adobe Connect.  Only a fraction of our recorded session is displayed above, work is being done to remedy that and display the full recorded session.  Although brief, an advantage of Adobe Connect appears immediately in the video above.  Adobe Connect offers a video feature enabling participants to see each other while conducting the meeting.  Our group did not utilize this tool for the duration of the meeting, but did experiment at the beginning.  The meeting space (window) provided a designated area for video, discussion notes, and a chat feature.  The chat feature worked well to initially communicate until the meeting host enabled microphone privileges.  Additionally, it served to discuss microphone and speaker issues.  As soon as audio was figured out, our group refrained from using the chat feature and solely communicated using microphones.  Another advantage of Adobe Connect was that all participants had microphone privileges at the same time.  The program also features a mute option which helped eliminate feedback when not speaking.  As our meeting progressed and discussion continued regarding our group leadership project, the discussion notes feature proved useful.  Our group used the discussion notes to break down the different areas and tasks for our tutorial video as well as assign an individual responsible for the task.  All meeting participants could both view and edit the notes concurrently.  The discussion notes also helped to define due dates for specific aspects of the project. 


One disadvantage of the session was my inability to hear one participant.  When my microphone was turned up so Spike was barely audible the feedback in the meeting increased.  During the meeting, I relied on others' responses to Spike to figure out what was being discussed when he talked.  Another disadvantage of the program relates to the video.  The host is the only person with access to the saved recordings.  In this situation, it's difficult since the file only contained seventeen seconds of the recording.  The rest of the video can't be referenced until the host receives the sent e-mails to attempt resending the session.  This is also a disadvantage since our timeline and tasks are documented later in the recording.  A last disadvantage pertaining to the conferencing tool as a whole relates to cost.  After our group's experiences with WebHuddle, Adobe Connect seemed a much smoother web conferencing option with the exception of its cost.  Fortunately, a free trial is available; however, I see the cost being a deterrent for use as a web conferencing tool after the trial period expired.


With regard to the content of our group discussion, our group decided the following:

• Jing will be the software we create the tutorial about.

• Powerpoint and Screencast O Matic will be used to produce the tutorial.

• Each member will take a section to complete.  I am responsible for the introduction as well as interactions.  Additionally, I will be recording the scripts created for each section.  Rachel will explain taking screenshots as well as putting together the final edit of the powerpoints and audio.  Spike has the responsibility of explaining how to make a video, and Bill will cover how to share both screenshots and videos.  Bill also will compile the storyboards.

• As for a timeline for completion, storyboards will be due to Bill June 6.  Powerpoints and the recorded audio are to be submitted to Rachel by June 15.