A Reflection: Base Blocks Addition and IXL

Description

The lesson in review was designed for third grade students participating in an after school program.  Students selected for the program demonstrated need for intervention in the areas of both mathematics and literacy.  The program coordinator admitted students to the program based on reading level and GMADE scores.  Homogeneous student groups consisted of up to twelve students who attend four forty-five minute sessions divided between two days.  The intended standards addressed in the lesson from the MA Curriculum Framework included: 

·         “Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones.” (P.42)

·         Fluently add within 1000 using strategies and algorithms based on place value.” (P.46)

The lesson design incorporated the use of two specific technology programs.  The Base Blocks Addition applet intended to allow students a visual representation of the addition process.  The applet selected provided a picture explaining commonly used terminology such as carrying to the next place value.  The second program, IXL, offered students an opportunity to apply the addition process illustrated with the Base Blocks Addition applet.  Additionally, IXL collected data on student performance recording the amount of time spent working on problems, the number of problems answered both correctly and incorrectly, and student responses to specific questions.  The lesson plan allotted time for whole-group and individual time to demonstrate and investigate the Base Block Addition applet.  Students used IXL individually and the data collected by the program used to assess student understanding of the indicated standards.  No changes were made to the original lesson plan prior to implementation.

Narrative

The initial two student groups who participated in the lesson described above comprised students with the lowest reading level and GMADE scores of all the student groups.  The lesson and objectives were introduced to the group using the SMART Board to demonstrate the applet and discuss place value and addition strategies using place value.  Students expressed interest in the applet and all wanted an opportunity to use the manipulative on the SMART Board.  Students were selected throughout the demonstration of how the applet worked to both explain the next step in the process and perform the described step using the SMART Board.  After seeing student interest in using the applet with the SMART Board, the decision to modify the original plan was made.  Instead of sending students to individual computers to investigate the Base Blocks Addition applet, the group remained together taking turns explaining steps of the addition process and using the manipulative.  After modeling the features of the applet and students collectively demonstrating how to use the applet to model and solve addition problems, students proceeded to individual computers and began applying the strategies for solving addition problems.  Allocating more time for the introduction of the applet as a whole group, the lesson ended with student work on IXL.  The closing discussion described in the original lesson plan was rescheduled to begin the next lesson before allowing additional time for students to complete problems using IXL.

Reflection

The intended objectives for the lesson included:  modeling numbers with three digits using base blocks, modeling and solving addition problems with numbers up to three digits, and explaining how place value can be used to solve addition problems.  Underlying assumptions about knowledge involved connecting representations to help develop concepts.  Affordances and constraints of the knowledge presented are discussed in conjunction with the benefits and drawbacks of selected technologies in a following paragraph.  The planned demonstrations of both Base Blocks Addition and IXL represented information processing.  Although student led, the group discussion falls into the information processing category.  Constructivism was evidenced as students investigated the Base Blocks Addition applet individually and figured out how to use the tool to solve addition problems.  

The lesson designed was used as an intervention for low-achieving students.  The lesson’s objectives and goals align with those specified in the curriculum for third grade mathematics.  Student differences in learning were addressed offering the option of using the Base Blocks Addition manipulative to solve problems.  IXL also differentiates difficulty level by increasing difficulty as students correctly solve problems.  The virtual manipulative also addressed differences in learning styles by including numeric and visual representations.  The significance of the tool remains in that the applet highlights the connection between the numeric and visual representation offering insight into the addition process.

The class discussion provides an opportunity for instructors to gauge student understanding of place value and the process of adding numbers with up to three digits in addition to the feedback individualized IXL reports offer.  To be successful, IXL reports would demonstrate increased progress toward mastering the skill and students could explain mathematical ideas such as grouping, carrying, and decomposing numbers.

The technology identified for the lesson on addition of two numbers up to three digits served a variety of purposes.  First, the virtual manipulative played a critical role in helping students develop an understanding of the addition process by connecting common numeric representations with a visual representation.  The visual representation additionally offered students an image to refer to when explaining or discussing addition problems.  The applet communicated the content visually for students and allowed student manipulation of the blocks, both of which an advantage.  One disadvantage of the applet related to grouping or decomposing blocks.  Students were intrigued by this feature of the applet yet the feature remains limited.  For example, students wanted to move a block from the thousands place to the ones place; the applet only decomposed blocks one place-value lower.  The design of the blocks which represented specific place values were also advantageous for helping define each place value.  Many students referred to blocks in the tens column as “longs.”  When asked how long those blocks were, students could count the unit block markings to help develop an understanding of place-value name meanings. 

Second, IXL offered a platform for students to apply the strategies using grouping for addition problems as well as an assessment tool for the instructor.  Affordances of IXL included individualized problem sets which progress in difficulty based on student performance.  Additionally, the program provided a variety of reports for analysis of student progress indicating information such as incorrect responses and mastery of skills.  One drawback of coupling the Base Blocks Addition applet and IXL involved the reporting feature.  Creating and solving the given problem with the applet increased the amount of time taken on each problem.  IXL uses length of time to solve a problem as an indicator of student progress toward mastering a skill.  Looking forward, the Base Block Addition applet may be better used early on in presenting grouping strategies for addition and encouraging students to rely less on the tool as their understanding develops.  

The lesson also included other technologies such as the internet and SMART Board.  The SMART Board enabled an interactive demonstration of the applet’s features and the internet integral with the use of web-based technologies; however, the applet could have been demonstrated without the SMART Board.  Both SMART Board and internet were technologies used as a medium to access the applet and IXL but were not intended to significantly impact learning.  The most unique and significant contribution of the selected technologies remained the connections between numeric and visual representations for modeling addition.  

Students were excited to use the Base Blocks Addition applet.  The applet helped facilitate a whole-group discussion about place value and the addition process.  Student questions predominantly hinged upon features of the applet.  For the student who wanted to decompose the thousand-block, I asked how many blocks would be in each of the remaining columns to redirect and help connect how many groups of ones, tens, and hundreds make a thousand.

Digital Storytelling: Learning to Share

Understanding Student Understanding

Here's a link to an audio file of a student interview.  The student was asked to define a function.

 Student Interview: What is a Function?

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.

Web Conferencing

Our group used WebHuddle to conduct our web conference.  One appeal of WebHuddle was that the program is web based requiring no downloads.  Investigating further, WebHuddle offered ease of registration and creating meetings.  WebHuddle offered options for creating meetings such as naming the meeting, describing the meeting, the inclusion of audio, and recording.  The site also provided a link to easily invite others to join the conference.  The complications began once everyone entered the conference.


Watching our recording, one quickly observes the level of experimentation the conference involved.  As the initial moderator, I attempted to give group members microphone rights.  As it turns out, WebHuddle only allows one person to have microphone rights at a time.  Additionally, two of our group members were unable to be heard in the conference even with microphone rights.  The group spent some time troubleshooting the microphone issues and decided, after a significant amount of time passed, to share the microphone between the two who could be heard in the conference and all depended on the chat feature to communicate.  Our group found and referenced a user guide as we worked through complications during the conference.  Options were given for all members to experience the conference as a moderator and as a team we experimented with the question feature, blank slide screen, and screen share.  WebHuddle allows for uploading of powerpoint files as well as .gif and .jpeg image files.  The question feature allows  the moderator to select from yes/no, true/false, multiple choice, or essay.  Questions, once submitted, are displayed for all participants.  Results can also be viewed within the program.  The blank slide screen acts similar to a paint program allowing for color, shapes, and writing to be shared.  WebHuddle does store recorded conferences on the user homepage under "recordings."  The only option other than "delete" and "merge audio file" is to download the zip file.  Unfortunately, the file is not saved to each participants account.  To share the recording, the file was uploaded to an ftp server and the link distributed.


For future use as a conferencing tool, the microphone dilemma requires addressing.  One microphone produced a significant amount of static while mine an incredible amount of feedback.  To use the program in the future, I would need more time to figure out the audio settings.  I later realized my microphone produced so much feedback because the level was turned so high to hear another group member.


WebHuddle could be used by student groups.  Since non-moderating participants possess little control over any of the features, this tool could be used in situations where one student is responsible for communicating information to the rest of the group.  WebHuddle could be used in the classroom for project presentation and evaluation.  Conference participants could offer constructive feedback, suggestions, or concerns through the chat feature as well as the Q&A feature.  The Q&A feature allows participants to ask questions of the moderator.  The moderator can then share the questions with the entire conference if desired.  Another suggestion for use might be as a medium for tutorials.  For similar reasons as those listed above, students could view presented information as well as ask questions and provide feedback.

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. 

A brief introduction,

Screencast Introduction

The above links to a brief introduction.  The introduction begins with my website and then transitions through a few pictures.  The first image is of my family at my brothers wedding in January.  The next image captures a moment in my former classroom.  Students learned vocabulary for geometry in a charade fashion using themselves to represent the term.  The third picture highlights a few of the things I enjoy most.  The picture of the trees represents my love of photography and all things outside.  The race bibs show some of the races I've run in the past few years, the passport representative of my love of traveling, and lastly my team the Detroit Red Wings.

The Google Earth clip shows a view of the middle school campus I work at.  The wiki created for my sixth grade students appears after that with pages to each of the core content areas.  Each content page links to students respective teacher pages as well as provides access to some interventions created for specific instructional units.  Lastly, a link to two technologies of interest are posted for reference.

In Progress: Final Reflections

Brophy stated, "Effective teachers allocate most of the available time to activities designed to accomplish instructional goals."  This quote reflects current learning and personal perspective with regard to effective strategies for integrating technology in the classroom.  The statement seems a given and the thought taken for granted about the importance of beginning by articulating instructional goals.  No doubt most educators would agree on the significance of starting the planning process with goals and objectives.  Adding technology to the equation and reflecting back it seems, although the obvious place to begin, in some instances the emphasis turns to the technology and away from the instructional objectives.  As an educator, I need to be mindful of technology's place:  as a tool used to support content, instruction, and objectives.  Thus to effectively incorporate technology within the curriculum and classroom, educators must know what needs to be accomplished to help select the best tool (technology) to support student learning.  With respect to UDL principles, technology remains a strategy and tool to provide students with multiple representations of concepts, options for action and expression, as well as increasing student engagement.  Technology tools serve to enhance lessons in the previous three areas while helping eliminate barriers for all students to engage in the curriculum.


Recent work with WebQuests, http://www.merlot.org/, http://www.wikispaces.com/, http://www.weebly.com/, and the creation of a StAIR helped process through which of the listed technologies and tasks constitute an online learning experience and which might be used in a face to face environment to enhance lessons.  Creating projects and pages with the technology listed above provided opportunity to think through ways to accomplish technology standards and requirements.  One use of the created wiki which served an excellent purpose was storage for the created StAIR to reteach fractions for some students.  The project, created with Powerpoint in kiosk mode on the computer in my office, was unavailable to the group of students I needed to work through the activity.  Uploading the program to the class wiki provided an access point from any computer lab with internet in the school building.  WebQuests and StAIR projects reinforced the importance of the educator testing and working through the activity prior to asking students to complete the task.  Additionally, I thought more about how to use the resource in class with students (individually, small group, pairs, etc.).  Also, evaluating a WebQuest confirmed how important clear, thorough directions and modeling are, especially when students are expected to complete the task individually.  When considering evaluating a technology, the question which initially comes to mind is, "How does this activity help accomplish the objectives?"  MERLOT offered a structure and method for evaluating submitted resources based on three considerations:  quality of content, potential effectiveness as a tool, and ease of use.  These three categories provided multiple lenses to view a web-based technology; however, these three areas of consideration transcend technology tools used in education.  "Internet brought about 'information pumps' and lessened effectivity of lesson/instruction," stated Merrill.  MERLOT's evaluation guide offers educators a way to sort through the overwhelming number of resources on the internet to find a resource that will effectively help accomplish instructional objectives.


Part of my decision to complete the MAET program emerged from an excessive amount of available technology and a fear the technology was not being used to its potential as a tool in my classroom.  In progress describes the status of my current pursuit of learning how to effectively integrate technology.  In truth, I would not mind "in progress" describing that goal for a significant amount of time indicating continued learning about how to implement technology effectively in the classroom.  Exposure to UDL principles helped see areas to include technology within lessons to help eliminate barriers for all students.  Effectively using technology in the classroom requires experience with the tools in order to discover the technology's limits and capabilities.  I had the opportunity to create a wiki, website, and StAIR.  This exposure either introduced a new program or feature of a program such as kiosk or interaction mode.  Discovering more options such as the ones presented provide opportunities to assess and evaluate when a technology will be most effective when considering instructional goals.  Additionally, the instructional design format helped organize key aspects of planning in order to utilize technology as an effective instructional tool. 


New goals at the conclusion of this course relate to specific technologies.  One goal I have is to familiarize and experiment with glogster, voice thread, web-based quiz and feedback platforms, and look into a platform for WebQuest creation.  I identified some of the technologies while working through weekly session material.  In other cases, I encountered the technology by checking the resources provided by peers in StAIR projects.  The goal remains to explore these technologies to become familiar with features and limitations; additionally to form an idea of how the technology could effectively be implemented and enhance curriculum, instruction, and student learning.

Online Experiences: WebQuests

One online experience I would use with students in a math classroom are WebQuests.  A previous textbook used by the district I taught in provided WebQuests related to unit content.   The WebQuests were not so much created to teach content but provided an opportunity for students to apply content from the course.  More specifically, the WebQuests worked for students to apply and demonstrate an understanding of algebra content. 

McGraw Hill WebQuest Projects 


WebQuests for a math classroom serve as a summative assessment at the end of a unit.  Depending on the WebQuest, such a task could be used as a quarterly project or assessment to evaluate student understanding of the implications of the content. 


WebQuests offer opportunities for students to work collaboratively in small groups as well as individually.  The WebQuest itself provides scaffolding for students walking them through the process and detailing specific information and tasks to complete.  Simulations could be incorporated into the WebQuest that students complete to demonstrate knowledge and apply the information from class.  Additionally, WebQuests could be extended to include individual or group presentations of the found information.  The WebQuests linked above could also be used as an exam review and in turn as a study tool for students.


For a mathematics class, I think online field trips would be more challenging with students.  The technology itself wouldn't be the challenge, ensuring that it was used effectively presents the greater challenge.  Objectives would need to be carefully crafted and the final product reviewed to make sure the objectives were met by the activity.  RSS feeds represent another online experience I think would be difficult for a mathematics classroom.  The reasoning here is similar, the technology itself not being the difficulty but finding ways to effectively incorporate the use of the technology might take more thought.