WELCOME The webinar will begin at 3:30. While you are waiting, please mute your sound. During the webinar please type all questions in the question/chat box in the go-to task pane on the right of your screen. As always, this webinar and the supporting materials will be available on our wikispace.

Making Mathematics Accessible Department of Public Instruction Mathematics Consultants

Mathematics Section Contact Information 3 Kitty Rutherford Elementary Mathematics Consultant Amy Scrinzi Elementary Mathematics Consultant Robin Barbour Middle Grades Mathematics Consultant Johannah Maynor High School Mathematics Consultant Barbara Bissell K-12 Mathematics Section Chief Susan Hart Program Assistant

Our teachers come to class, And they talk and they talk, Til their faces are like peaches, We don’t; We just sit like cornstalks. Cazden, 1976, p. 64

ALL students can generate mathematical understandings!

“Whenever a teacher reaches out to an individual or small group to vary his or her teaching in order to create the best learning experience possible, that teacher is differentiating instruction.” -- Carol Tomlinson

Instructional Strategy Concrete-Representational-Abstract (CRA) -Concrete “doing” stage -Representational “seeing” stage -Abstract “symbolic” stage

Division of Fractions 5 ÷ ⅓ = ? 5 3 = 15 Why?

Division of Fractions 5 ÷ ⅓ = ? 5 3 = 15 Why?

Division of Fractions 5 ÷ ⅓ = ? 5 3 = 15 Why?

Algorithms Algorithms without understanding –Errors practiced and hard to break –Extensive practice time –Limited retention

Algorithms Algorithms with understanding –Conceptual development –Reduction in practice time –Extended retention and application Deborah Ball, Secretary’s summit on Mathematics, Washington, D.C., 2003, (accessed November 12,

Can all students explain the WHY-TOs not just the HOW-TOs?

When planning, “What task can I give that will build student understanding?” rather than “How can I explain clearly so they will understand?” Grayson Wheatley, NCCTM, /2/2015 page 14

The Border Problem Sue is tiling a 10 by 10 patio. She wants darker tiles around the border. How many tiles will she need for the border? Show the arithmetic you used to get your solution. Describe your method and explain why it makes sense. Use algebraic expressions to write a rule for each method you found.

Sue is tiling a 10 by 10 patio. She wants darker tiles around the border. How many tiles will she need for the border? Show the arithmetic you used to solve the problem. Describe your method. Explain why your method makes sense. Use algebraic expressions to write a rule for each method you found.

Sue is tiling a 10 by 10 patio. She wants darker tiles around the border. How many tiles will she need for the border? Show the arithmetic you used to solve the problem. Describe your method. Explain why your method makes sense. Use algebraic expressions to write a rule for each method you found.

The Border Problem Several Possibilities: = x 4 = = 36

The Border Problem

1.Make sense of problems and persevere in solving them. 2.Reason abstractly and quantitatively. 3.Construct viable arguments and critique the reasoning of others. 4.Model with mathematics. 5.Use appropriate tools strategically. 6.Attend to precision. 7.Look for and make use of structure. 8.Look for and express regularity in repeated reasoning. Standards for Mathematical Practices

12/2/2015 page 21 What is the area and perimeter of this shape? How do you know?

12/2/2015 page 22 Make sense of problems and persevere in solving them.

12/2/2015 page 23

Instructional Task With a partner, using color tiles solve the task below: What rectangles can be made with a perimeter of 30 units? Which rectangle gives you the greatest area? How do you know? What do you notice about the relationship between area and perimeter?

Compared to… What is the area of this rectangle? What is the perimeter of this rectangle?

When thinking about the concept of area and perimeter what mathematical terms come to mind? In two minutes list all terms you can think of in the center box.

When thinking about the concept of area and perimeter what mathematical terms come to mind? In two minutes list all terms you can think of in the center box. area perimeter multiply array

Strategies for Developing Mathematical Understanding 1.Allow mathematics to be problematic for students. All students need to struggle with challenging problems Teacher must refrain from doing too much of the mathematics Problem solving leads to understanding!

Strategies for Developing Mathematical Understanding 2.Focus classroom activity on the methods used to solve problems. Opportunity for students to share one’s own method Hear alternative methods of solving a problem Examine the advantages and disadvantages of these different methods (efficiency) Class discussions should revolve around sharing, analyzing, and improving methods. Mistakes become opportunities for learning.

Strategies for Developing Mathematical Understanding 3.Determine what mathematical information should be presented and when this information should be presented. Presenting too much information too soon removed the problematic nature of problem Presenting too little information can leave the students floundering

ENCOURAGING MATHEMATICAL DISCOURSE Teachers: Use effective questioning Be nonjudgmental about a response or comment Let students clarify their own thinking Require several responses for the same question Require students to ask a question when they need help. Never carry a pencil Mathematical Teaching in the Middle School, April 2000, Never Say Anything a Kid Can Say.

But Most Importantly… Never Say Anything a Kid can Say!!

Our teachers come to class, And they talk and they talk, Til their faces are like peaches, We don’t; We just sit like cornstalks. Cazden, 1976, p. 64 Please don’t let student sit like Cornstalks!

12/2/2015 page 35

Mathematics Section Contact Information 36 Kitty Rutherford Elementary Mathematics Consultant Amy Scrinzi Elementary Mathematics Consultant Robin Barbour Middle Grades Mathematics Consultant Johannah Maynor High School Mathematics Consultant Barbara Bissell K-12 Mathematics Section Chief Susan Hart Program Assistant