Representation Standard of Principles and Standards

Presented by Krista Roop and Miranda Waddell

What Does the Representation Standard State?

Instructional programs from pre-kindergarten through grade 12 should enable all students to—

• create and use representations to organize, record, and communicate mathematical ideas;

• select, apply, and translate among mathematical representations to solve problems;

• use representations to model and interpret physical, social, and mathematical phenomena.

Do the grade level Standards address issues of equity?

Krista: No, I think they are implied, but I don't think equity is necessarily addressed in the Standards.

Miranda: I agree with Krista. The Standards try to provide ideal steps to achieve equal opportunity for learning, but never explicitly addresses equity.

Do the 9-12 Standards build on the ideas presented in the 6-8 Standards? What more is

expected of students?

• Yes, the 9-12 Standards do build on the 6-8 Standards. The 9-12 Standards include more "Real World Applications“. Also there is improved "Problem Solving“ sections. Students are expected to use their knowledge more.

“Developing students' representational fluency

using virtual and physical algebra balances”

This journal article addresses the use of the Representation standard using:

• Virtual and Physical manipulatives• Multiple representations and the flexibility

to translateand analyzed third-grade students and

-the use of different algebraic models to represent their relational thinking

“Developing students' representational fluency

using virtual and physical algebra balances” • The Principles and Standards for School Mathematics

(NCTM, 2000): "...create and use representations to organize, record, and communicate mathematical ideas; select, apply, and translate amongmathematical representation to solve problems; and use representations tomodel and interpret physical social and mathematical phenomena" (p.67).

• Experience, representation, and relational thinking• Synchronous vs. sequential processing• Use of Physical and Virtual Representations in

Classrooms

“Developing students' representational fluency using virtual

and physical algebra balances” • A classroom project involved two groups of third-grade students in a unit

focusing on algebraic relationships.

• Group One worked with the Virtual Balance Scale applet on the National Library of Virtual Manipulatives to solve simple linear equations. Once the beam balances to represent the given linear equation, students can choose to perform any arithmetic operation, as long as they perform the same operation on both sides of the equation, thus keeping the pans balanced.

• Group Two worked with a physical manipulative called Hands-On Equations. The teacher's edition comes with a stationary plastic balance scale, number cubes, and pawn pieces. The student edition comes with a balance scale mat, number cubes and pawn pieces to represent algebraic equations. The pawn pieces represent the unknown x value and the number cubes represent numbers in the equation.

• In the physical environment, some unique features of the Hands-On Equations manipulatives were: (a) tactile features; (b) more opportunities for invented strategies; and (c) more mental mathematics.

• The virtual environment also had unique features that promoted student thinking such as: (a) explicit linking of visual and symbolic modes; (b) guided step-by-step support in algorithmic processes; and (c) immediate feedback and self-checking system. Also, guidance would be given such as, "You can't subtract 4x from both sides unless there are at least 4xs on each side.“

“Developing students' representational fluency

using virtual and physical algebra balances” • Students in the physical and virtual manipulative environments

showed significant gains in achievement between the pre and post-test measures. Although the environments had different features, both representations were effective in supporting students' learning in different ways. Students' scores were highest on items with pictorial representations.

• These classroom projects show how the use of different representational forms can contribute to student learning and promote relational thinking.

• Students drawings, equations, and written explanations showed their fluency in translating from one representational form to another, whether they were using the virtual balance scale or the physical balance scale as their primary classroom manipulative for the unit. Moving among different representations helped to strengthen their understanding. Students expressed their understanding of numeric relationships through drawings, symbols, and written explanations.

Lesson Activity-Middle School

On TV, you see an ad for some decently priced jeans. When you arrive at the store, you see 6 pairs of jeans that interest you. The prices of the jeans are: $ 18, $ 56, $ 22, $ 28, $ 60, and $ 30. What is the average amount of money you may pay for the jeans? Find the mean, median, upper and lower quartiles, and maximum and minimum. Use a box-and-whisker plot to explain your answer.

Lesson Activity-High School

As you look around, you find a $ 5 coupon to use and everything in the store is 25% off (after the coupon is used) that day only! Find an equation using x (as the original price of the jeans) and y (as the maximum amount of money you can save on each pair of jeans) to solve how much you can save on each pair of jeans.

Lesson Activity-High School continued

Which pair of jeans will you save the most money on when the coupon and discount are applied? Create a graph to show your findings.

Draw a best fit line in your graph. Is this a negative or positive correlation?