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.