What does understanding in mathematics mean, and how do the Standards for Mathematical Practice lead to understanding? Essential Question
How do you currently support students in becoming effective mathematical practitioners? Teaching the Standards for Mathematical Practice
How can teachers support students as they develop the mathematical practices? Supporting Students As They Develop the Mathematical Practices
Brainstorm some ways that teachers can develop the mathematical practices through various classroom activities: 1.Teacher modeling 2.Task selection 3.Relevant student discourse 4.Questioning strategies 5.Classroom routines
1.At what point in time do teachers select a task? 2. When do teachers model the Standards for Mathematical Practice? 3. When do teachers support student discourse? 4. When do teachers use questioning strategies? 5. What does it look like when teachers integrate the Standards for Mathematical Practice into classroom instructional processes? Reflect for a moment on the following questions: Bringing the Practice Standards to the Classroom
What is a comparison? Comparing Quantities Why do we compare? Bringing the Practice Standards to the Classroom
(Americas Choice 2009) Compare the number of stars to the number of triangles using subtraction. Compare the number of stars to the number of triangles using division. Comparing Quantities Bringing the Practice Standards to the Classroom
Attend to precision. Look for and make use of structure. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. How would these mathematical practices emerge if students worked this same problem set? Bringing the Practice Standards to the Classroom
List 3 new ideas related to teaching the Standards for Mathematical Practice that you are going to implement in the classroom. List 2 strategies you have previously used that support the Standards for Mathematical Practice. List 1 question you still have about the Standards for Mathematical Practice. Teaching the Standards for Mathematical Practice Reflection: 3-2-1
How can strategies and classroom routines help students develop the mathematical practices of mature mathematical thinkers and learners?
Modeling and Problem Solving How many different ways can you find to count the border tiles of an 8 x 8 pool without counting them one at a time? Modeling and Problem Solving: Border Problem (Van de Walle, Karp, and Bay-Williams 2010, 257)
Numerical ExpressionsGeometric Explanation 10^2-8^2 Area of large (exterior) square minus area of the small (interior) square. 4(10)-4 There are four sides of length 10 minus the four overlapping squares. 2(10)+2(8) The are two sides of length 10 and two sides of length 8. Notice that for the two sides of length 10 that all corners of the border are counted. 4(8)+4 There are four sides of length 8. We must add 4 for the four corners. 4(9) There are four sides of length 9. Each side includes a different corner. Each of the four corners is counted exactly once. 10+9+9+8 There is one side of length 10 (which includes two corners). There are two sides of length 9 (which includes a total of 2 corners). There is a side of length 8 (which includes no corners)
Video Discussion Why without talking? Why without writing? Why without counting one by one? Why not give them each a grid to facilitate their thinking? Why did the teacher act as the recorder for the arithmetic expressions? Boaler, J. & Humphreys, C. (2005). Building on student ideas: The border problem, part I. Connecting mathematical ideas: Middle school video cases to support teaching and learning (pp.13-39). New Hampshire: Heineman Publications.
What about a 6 in by 6 in grid? What about a 15 in by 15 in grid? What about a 253 in by 253 in grid? What about an n inch by n inch grid? Create a verbal representation Use the verbal representation to introduce the notion of variable If n represents the number of unit squares on one side, give an algebraic expression for the number of unit squares in the border. Develop understanding of function, variables (independent and dependent) and graphing.
Modeling and Problem Solving Your summer job is to repair tiles on the corner of roofs. The tiles you will use are 1 foot by 1 foot squares. Jobs are identified as follows: # 2# 4# 5 (Pearson Education, Inc. 2008, B-25) Modeling and Problem Solving: Tile Problem
a.Draw Jobs #3 and # 6. b.What do you need to know to draw the figures? c.After laying the tiles, you need to apply some waterproof caulking around the outside of the pattern. The caulking is sold by the foot. Job #2 requires 8 feet of caulking. Job #5 needs 20 feet of caulking. Your supervisor assigns you to do Job #12. How many tiles and how much caulking will you need to complete the job? Explain how you figured this out. d.What would you hope your students will understand as a result of solving problem c? e.What difficulties might your students have with problem c? (Pearson Education, Inc. 2008, B-25) Modeling and Problem Solving
# 2# 4# 5 f.Write a rule that would allow you to determine how many tiles and how much caulk you would need for any job. Can you identify any patterns between the two rules? g.What would you hope your students will understand as a result of solving problem f? h.What difficulties might your students have with problem f? (Pearson Education, Inc. 2008, B-25) Modeling and Problem Solving
or The Tile Problem builds on content knowledge and conceptual understanding of perimeter and area to develop general expressions. The length of caulking is represented by a linear function c = 4s, and the number of tiles is represented by a quadratic function with the formula where c is the length of caulking, t is the number of tiles, and s is the length of the side of the figure. The quadratic formula could also be expressed by the equivalent equations. Modeling and Problem Solving (Pearson Education, Inc. 2008, B-25)
Reason abstractly and quantitatively. Make sense of problems and persevere in solving them. Use appropriate tools strategically. Model with mathematics. Look for and express regularity in repeated reasoning. Connections to Modeling and Problem Solving (Common Core State Standards Initiative 2010a, 10)
(Bay-Williams and Karp 2008) Promoting the Standards for Mathematical Practice Reflection: Symbolic Prompts