By Mindy Robison Email: Mindy.Robison@CanyonsDistrict

1. CRA:Concrete-to-representational-to-abstractHow does the new Common Core as well as Big Ideas support this type of instruction? By Mindy Robison Email: Mindy.Robison@CanyonsDistrict.org

2. Website: • http://robisonlovesmath.wikispaces.com/ • On the right hand side, click on • CRA Training – New Core

3. Learning Task: • What is ½ ÷ ¼ ? • What is ¼ ÷ ½ ? • Explain how you got your answers? • Draw a picture to prove your answer. • Can you draw another picture to show your answer? • How can you concretely explain this to a student?

4. The Session Objectives: • Find ways to effectively and efficiently implement CRA using the Common Core and Big Ideas Learning. • Learn how to and why it is important implement Instructional Shift #3: • Picture It, Draw It! • Use multiple representation of mathematical entities.

5. The Plan! • Review CRA • Review Instructional Shift #3 • Review the basics of Manipulatives • Look at specific lessons from Big Ideas Learning that use manipulatives. • Look at virtual manipulatives. • Specifically look at Dynamic Classroom and the Interactive Activities. • Look at algebra tiles and counters. • Discuss ways to improve our math classes through the use of manipulatives.

6. Common core Standards for mathematical practice • #4 Model with mathematics • #7 Look for and make use of structure.

7. Quick Review • Why CRA? • “ The purpose of teaching through a concrete-to-representational-to-abstract sequence of instruction is to ensure students develop a tangible understanding of the math concepts/skills they learn. When students are supported to first develop a concrete level of understanding for any mathematics concept/skill, they can use this foundation to later link their conceptual understanding to abstract mathematics learning activities. Having students represent their concrete understandings…by drawing simple pictures that replicate or mimic their use of concrete materials, students are supported in the process for transferring their concrete understanding to the abstract level.” • http://www.specialconnections.ku.edu/cgi-bin/cgiwrap/specconn/main.php?cat=instruction&subsection=math/cra

8. Concrete: • “The concrete level of understanding is the most basic level of mathematical understanding. It is also the most crucial level for developing conceptual understanding of math concepts/skills. Concrete learning occurs when students have ample opportunities to manipulate concrete objects to problem-solve. For students who have math learning problems, explicit teacher modeling of the use of specific concrete objects to solve specific math problems is needed.

9. Representational • “At the representational level of understanding, students learn to problem-solve by drawing pictures. The pictures students draw represent the concrete objects students manipulated when problem-solving at the concrete level.

10. Abstract • “At this stage, the teacher models the mathematics concept at a symbolic level, using only numbers, notation, and mathematical symbols to represent the number of circles or groups of circles. The teacher uses operation symbols to indicate addition, multiplication, or division.”

11. Instructional Shift #3: Picture it, draw it! • “The failure to capture the mathematics being taught with a picture that helps students visualize what is going on is one of the most serious missed opportunities I observe.” p.19 • “The simple reality every teacher faces . . . is that very rarely do more than ½ of the students process the math being taught, see the math being taught, or feel the math being taught in the same way their teacher is seeing it.” p. 21 • “Valuing of alternative approaches enriches our instruction and provides new levels of access to mathematical understanding” p. 25.

12. So what should we see in an effective mathematics classroom (p.25) • Frequent use of pictorial representations to help students visualize the mathematics they are learning • Frequent use of the number line and bar models (see www.thesingaporemaths.com )to represent numbers and word problems. • Frequent opportunities for students to draw or show and then describe what is drawn or shown.

13. Manipulatives? • Discussion: • How do you define a math manipulative.

14. Advantages & Disadvantages to manipulatives in partners or small groups: • Please list at least 2 advantages to using manipulatives in your math classroom. • Please list at least 2 disadvantages to using manipulatives in your math classroom.

15. Suggestions for using manipulatives: • Talk with your students about how manipulatives help to learn math. • Set ground rules for using manipulatives, reiterate those rules every time you use manipulatives. • Develop a system for storing manipulatives. • Allow time for your students to explore manipulatives before beginning instruction. • Encourage students to learn names of the manipulatives they use. • Provide students time to describe the manipulatives they use orally or in writing. Model this as appropriate. • Introduce manipulatives to parents.

16. “Manipulatives in middle grades Mathematics”, Research Summary.” • “Although manipulatives have an important place in learning, their physicality does not carry the meaning of the mathematical idea. They often can be used in a rote manner . . . Students may require concrete materials to build meaning initially, but they must reflect on their actions with manipulatives to do so.” • http://www.nmsa.org/Research/ResearchSummaries/Mathematics/tabid/1832/Default.aspx

17. “Manipulatives in middle grades Mathematics”, Research Summary.” • Student performance with manipulatives may exceed student performances without manipulatives. • Student achievement levels are related to teachers’ experience and expertise with manipulatives. • The “relation between manipulatives and their intended referents may not be transparent to children.” • Time of interaction with manipulatives affects success for elementary and for middle students as they assume responsibility for their use.

18. “Manipulatives in middle grades Mathematics”, Research Summary.” • Teacher use is related to teachers’ prior experience with manipulatives. • Children’s comprehensions of manipulatives depend on instruction. • Manipulative use alone cannot be expected to improve mathematics education. • Teachers reported manipulatives were used for demonstrations, problem solving, change of pace, rewards, “fun,” and better understanding. • Computer manipulatives and physical manipulatives have different affordances, and both types should be used in middle grades classrooms.

19. 7th Grade Book: Lesson 1.2 – Let’s figure out how to teach it! • Turn to page 8 in your textbook and read through page 8. • Your goal with your group is to create a basic lesson plan that you could teach. • Questions to consider: • What do you and your students need to know about integer counters to understand this section? • How do you help the students understand the relationship between the counters and the concept? • What examples will best help your students understand this concept?

20. 7th Grade (Red Book) Interactive Component • Now go to www.bigideaslearning.com and login. • Click on the Teachers, and then click on the Red Book. • On the left side click on Dynamic Classroom. • Click on the 7th Grade Book. • Click on Lesson 1.2. • As a group investigate the Algebra Tiles application. • Create at least 5 problems that you could use to help teach section 1.2.

21. Summarize Lesson 1.2 • Why does the book using integer counters? • What do you and your students need to know about integer counters to understand this section? • Does the order of your examples matter? • What problems could arise from using integer counters? • What problems could arise if a student doesn’t understand the integer counters? • What did you learn from talking and sharing with other teachers?

22. SmartBoard • How could you use (if available) a SmartBoard to help strengthen lesson 1.2? • Examples from my classroom.

23. 7th Grade Book: Lesson 2.4. Solving Equations Using Addition or Subtraction. • Now that we have figured out 1.2 – lets look at page 70 and figure out how we would teach Lesson 2.4. • Read through the section and then create a basic lesson plan. • You may use the online resources. • Come up with at least 5 examples you could use to help your class understand the basics of solving equations. • Carefully consider the examples and the order that you chose to present them.

24. Summarize lesson 2.4 • How does the textbook use manipulatives? • Will using the tiles guarantee concrete understanding? • After sharing, do you want to make any changes to the lesson you created?

25. SmartBoard • How could you use (if available) a SmartBoard to help strengthen lesson 2.4? • Examples from my classroom.

26. Now Do What will help you the most! • In the 7th grade book look at the following sections: • Lesson 1.3 (Subtracting Integers) • Lesson 1.5 (Dividing Integers) • Lesson 2.4 (Solving Equations Using Addition or Subtracting) • Lesson 2.6 (Solving 2-Step Equations) • In the 8th grade book look at section 1.1. What do students already have to know in order to fully understand this section? • Look at the red book Chapter 2 to help answer the above question. • Spend time looking at Dynamic Classroom and the interactive activities that are available to you.

27. Summary & Goals • Understand CRA and the purpose it has in your classroom. • Understand Instructional Shift #3 and find at least 1 specific way you can use it this upcoming school year. • Recognize that Big Ideas Learning supports the use of manipulatives. • Be familiar and use at least a few of the virtual manipulatives or Interactive Activities (online resources) from Dynamic Classroom. • Try algebra tiles and counters and see what happens. • Use what is available for you- don’t create more work for yourself!

28. Additional Resources: • National Library of Virtual Manipulatives: • http://nlvm.usu.edu/