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 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 B: Application of TPACK

Technological Pedagogical Knowledge
The Algebra Balance Scales  virtual manipulative, an instructional 
technology, supports the teaching methods and strategies intended for this intervention activity.  The applet scaffolds solving equations, an important strategy for the students participating in the intervention.  The solving process is modeled step by step for students both visually and symbolically simultaneously.  Student response followed by immediate feedback demonstrates another strategy supported by the manipulative.  Students begin by placing the blocks on the scale to model the equation.  When students believe the model is correct, a continue button at the bottom moves students on to the next phase of solving the equation or states, "The two sides don't match the equation," pictured above.  Additionally, students have the visual feedback that the scale lacks balance.  Students have the immediate opportunity to compare the created model and the equation before proceeding to solving.  This virtual manipulative allows for connections and observations to be made regarding how changes effect all representations of the equation, an advantage over a physical manipulative to investigate solving equation.  As blocks are moved to keep the scales balanced, a visual representation, the equation(s) in the boxes above evidence the operations symbolically.  The technology selected also offers the flexibility to create equations to be solved.  This supports the teaching method of giving the equations context.  Students will investigate solving equations in a real world context.  The "create" feature allows the technology to support the instructional decision to provide students with real world problems.  The manipulative also offers an opportunity for students to create a scenario for the given equations.  As an extension or future use of the applet, students could create the context for the given equation and then solve.


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

Not only does the Algebra Balance Scales virtual manipulative support the instructional strategies, the manipulative also works with the content.  The two learning objectives for the activity include:

• 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 solutions (A.FO.06.13)

The Algebra Balance Scales 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.  As students enter in the appropriate steps to solve the equation, the visual 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 below show the linked representations described.  The applet provides students an image of the intended learning goals. 

To solve for x, subtract 4 from both sides.

The scales remain balance yet four less unit blocks are present on each side

Pedagogical Content Knowledge
Lastly, the instructional strategies used for this intervention activity also 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.

Part A: Description of Need or Opportunity

I.  Educational Need or Opportunity
Solving algebraic equations represents the focus of the educational need to be addressed with this project.  Algebra represents a significant number of grade level content expectations for the sixth grade students I work with.  Specifically, this project targets the following 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)

Reflecting on my students performance through the unit on algebra and solving equations, revisiting the content and representing the ideas in a way different than students experienced in their respective math classroom would benefit student understanding and skill development.  I intend to use this project as an intervention to address student misconceptions related to solving equations as well as promote a deeper understanding of algebraic equations.


II. Proposed Technological Solution to Educational Need
I intend to use the Algebra Balance Scales applet from The National Library of Virtual  Manipulatives to address the educational issue articulated above.  The balance scales applet will be used to investigate solving equations of the first degree with whole number coefficients.  The applet provides students with a visual representation of the equation; students model the given equation on the scale by placing blocks representing the variable (x) and the number of unit blocks on each side.  Student users receive visual and verbal feedback from the applet.  If students attempt to continue to solve before the equation is modeled correctly, a message appears noting that the scales are not yet equal.  Visually, the scales shift left and right as blocks are added to both sides until equal also indicating to students the need to adjust the blocks on the scales.  After correctly modeling the equation, students proceed  to solving the equation.  Students must decide which operation to perform and the amount to add, subtract, multiply, or divide in order to keep the scales balanced.  Again, verbal feedback prompts student thinking lending consideration to how equations can be manipulated.  The Algebra Balance Scales applet offers students a link between visual and symbolic representations.  As students manipulate the blocks on the scale below, the display box above manipulates the equation.  Students have the opportunity to see the connection between what happens on the scale and what happens to the equation.  The applet is free to use and requires only an internet connection to access.  Additional technology to be used includes audacity and windows movie maker for students to produce a tutorial describing how to solve equations.  This use of technology encourages students to process and reflect on the activity and additionally create a new resource collaboratively with a group demonstrating new understandings about solving equations. 


III.  Logistics
The Algebra Balance Scales applet will be used during one hour sessions in the computer lab by two student groups on my caseload.  The eight students identified for the intervention will be provided time to work independently with the applet after brief instruction of how to manipulate the applet.  The technology will be used to reteach solving equations by providing and connecting multiple representations.  Screenshots will be captured as students work through solving equations keeping the scales balanced for use in the final student created tutorial.  The students on my caseload are sixth graders identified as at-risk at a public middle school in Kentwood, MI.  Students will document responses on an assignment sheet and will also complete journal reflections related to the use of the applet.


IV.  Relevant Research and Resources
Research suggests the use of virtual manipulatives increases student understanding of mathematical ideas and  concepts.  I tagged resources "Wicked" on my delicious page relevant to the educational need identified.  Initial findings helped clarify what constituted a virtual manipulative.  Moyer, Bolyard, and Spikell (2002) articulated that virtual manipulatives are, "an interactive, web-based visual representation of a dynamic object that presents opportunities for constructing mathematical knowledge" (p. 373).  The Algebra Balance Scales applet does fit this definition in representation as well as opportunity for students to construct and model mathematical concepts.


A study by Suh and Moyer entitled Developing Students' Representational Fluency Using Virtual and Physical Algebra Balances used the Algebra Balance Scales applet to compare student understanding of solving linear equations based on instruction using either physical or virtual manipulatives.  The authors state, "The use of multiple representations and the flexibility to translate among those representational forms facilitates students' learning and has the potential to deepen their understanding" (p. 155).  This study showed manipulatives enhanced students understanding and concluded, "Different representations, including those increasingly available through technology, can facilitate the teaching of these fundamental ideas." (p. 172).  In this case, research supports the use of the Algebra Balance Scales applet to help increase student understanding of solving equations.  This article also provides suggestions for implementing the technology effectively as the students evidenced increased understanding when following the implementation method described in the study.


Further support for using the Algebra Balance Scales applet comes from Suh's dissertation (2005) entitled, Third Graders' Mathematics Achievement and Representation Preference Using Virtual and Physical Manipulatives for Adding Fractions and balancing equations, summarized from (Dorward, 2002):

"The creators of the National Library of Virtual Manipulatives designed the virtual manipulatives in hopes that it would add to some of the benefits of using physical manipulatives in the classroom and eliminate some of the drawbacks such as:  classroom management, structuring activities with manipulatives, connecting manipulative use with symbolism, and lack of resources and professional development."

The Algebra Balance Scales applet, contained in the National Library of Virtual Manipulatives, was created to address some of the concerns with using physical manipulatives in the classroom.  The applet was designed with the intent of fostering connections among representations for students encouraging a conceptual understanding of solving equations and promoting transfer of ideas to be applied in a variety of scenarios.


An identified resource, Computer Technology for Math Excellence, cited research answering the two questions below which offered perspective related to virtual manipulatives as a solution to the defined educational need:

• What is a virtual manipulative
• What role can virtual manipulatives play in the classroom?

The site noted, "Virtual manipulatives provide that additional tool for helping students at all levels of ability "to develop their relational thinking and to generalize mathematical ideas"" (Moyer-Packenham, Salkind, & Bolyard, 2008, pg. 204).  With specific regard to the Algebra Balance Scales, this statement suggests the applet will assist in developing student understanding of solving equations.  


Further research exposed a theory called Dual Coding.  According to the Dual Coding Wiki, this theory states that verbal and visual representations are processed and stored differently.  The Algebra Balance Scales applet provides both verbal and visual representations for students while engaging with the mathematical content.  Quoting Suh and Moyer (2007) who referenced (Rieber 1994):

"...it is easier to recall information from the visual processing codes than the verbal codes because visual information is accessed using synchronous processing rather than sequential processing," (p. 158). 

Again, this lends support to using the virtual manipulative Algebra Balance Scales to help students learn how to solve equations since it provides visual representations and additionally connects the different representations present.


The findings described above and the list of resources linked above were found initially using Google Scholar.  The search began with keywords such as:  solving equations with virtual manipulatives, using technology to teach solving equations, and examples of teaching math with technology.  After the preliminary searches, a few articles seemed worth investigating further.  Dissertations and research studies provided a wealth of literature and research reviews.  I used the reviews attached to the studies in order to find additional resources specific to the defined educational need.  To locate online versions of articles and documents, I often copied the citation and entered that in the search engine.  Locating articles used within studies and dissertations proved difficult at times to find a viewable copy.  Often times the articles were available for purchase of either the specific article or a membership to a database housing the articles.  In these instances, I resorted to using the synthesized information as found in the research review.  In the future, I would utilize similar strategies.  I would additionally add the search engines and subscriptions available through Michigan State University's library.




V.  Implementation Plan 
Students participating in this intervention to readdress solving equations will participate in two computer lab opportunities prior to the end of the school year.  During those times, students will work with the Algebra Balance Scales applet in order to increase and solidify their understanding of solving equations.  Additional meeting time will be used for students to reflect on and discuss the completed tasks using the virtual manipulative.  Also, students will respond to journal prompts regarding their experiences working with the applet.  Time permitting, students will compile screenshots of working with the manipulative, reflections from journal entries, and thoughts from discussion to create a tutorial to share with both math classes.  The tutorial will include an explanation of how to solve equations including links between the representations.  The tutorial may be created after the course depending on availability of lab time and permission from my students' teachers.  Additionally, the tutorial will be shared with the different math classes after the completion of the course.  


VI.  Indicators of Success
Successful learning of solving equations with the Algebra Balance Scales applet will be evidenced ultimately by the created tutorial.  The tutorial will present accurate methods for solving algebraic equations as well as mathematically correct explanations of the steps.  Additionally, students will correctly connect and link the different representations modeled by the applet.  Student discussion of their experiences in small groups prior to creating the tutorial will also be an indicator of success as well as offering another opportunity to clarify and discuss any remaining misconceptions.  The discussion group also serves to help students articulate the learning which took place while using the applet.