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Opening Activities

Opening Activities. Add a post-it to the graph. Respond to the following “ Four Quadrants ” review of reading (fold your paper and respond in each quadrant):. Developing Early Number Concepts & Algebraic Thinking. Chapter 8 & 14. Today I will be able to….

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Opening Activities

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  1. Opening Activities Add a post-it to the graph. Respond to the following “Four Quadrants” review of reading (fold your paper and respond in each quadrant):

  2. Developing Early Number Concepts &Algebraic Thinking Chapter 8 & 14

  3. Today I will be able to… • Describe important content for young children. • Design questions to help young children articulate their mathematical thinking. • Explain the way in which algebra is a part of early childhood and elementary curriculum. • Identify key features of Transformation (KQ)

  4. Let’s Plan It Choose an activity from Chapter 8 Explore the Task How would you plan each of the three lesson phases?

  5. Diverse Learners • Accommodations & Modifications • Struggling students, special education students, English Language Learners, gifted and talented students… • Chapter 6 • CAN YOU COME UP WITH AN ACCOMODATION AND A MODIFICATION FOR YOUR LESSON?

  6. Your Turn: Plan It! Share it! • Number off from 1 - ??? at your table. • Find your partner at the matching table. • Share/compare your before, during, and after. • Revise, as appropriate.

  7. Developing Early Number Concepts & Number Sense

  8. The Knowledge Quartet: Transformation

  9. The Knowledge Quartet Contingency Transformation Connection Foundation

  10. Transformation • Choice of Examples • Use of Representations • Demonstration

  11. Transformation • Choice of Examples: Think back to last class…task selection & implementation, low vs. high cognitive demand. Purposeful use of numbers and operations. Planned ahead of time. • Use of Representations: Number line, place value grid, manipulatives. Focus students on a key mathematical concept. A common pitfall is that the math is changed to match the representation. The representation supports the math, not vice versa! • Demonstration: Demonstrates accurately and clearly how to carry out procedures. Gives clear explanations of mathematical ideas or concepts. High level demonstrations focus on a conceptual understanding of mathematics.

  12. Transformation, Choice of Representations:Using Manipulatives

  13. Round Robin • There are stations with activities for PreK-2 students. • On your recording sheet, list the following: • Name of the activity • Mathematical Purpose • Questions to ask students to ensure the math concept is learned • Possible manipulatives • You will have three minutes at each table

  14. In your color groups… • Work through each station • Discuss the mathematical purpose for each lesson • Share your ideas for student questions & choice of manipulatives • What conclusions can you draw about early number concepts and number sense?

  15. Four Categories of EarlyNumber Relationships • Spatial Relationships • One More/Two More/One Less/Two Less • Anchors to 5 and 10 • Part-Part-Whole

  16. Early Number Diagnostics How do we know they are “getting it?” • What could we ask? • What could students demonstrate? • What could students explain?

  17. Early Number Diagnostics What do we need to know?It is not enough to know if thechild can get right answers.We need to know what mathematicsthe child knows and understands.

  18. Early Number Diagnostics 9 • Use the cubes to show this number • Add a unit to your number • Now add two more units • What does the first number represent? What does the second number represent? What does this assess?

  19. Early Number Diagnostics 15 feet • Could the teacher be 15 feet tall? • Could your living room be 15 feet wide? • Can a man jump 15 feet high? • Could three children stretch their arms 15 feet? What does this assess?

  20. Counting / Number Relationships Counting Objects Changing Numbers More/Less Trains http://www.assessingmathconcepts.com/

  21. Early Number Diagnostics Counting Stories Five apples are on the tree. Three apples are on the ground. Count the apples. Tell a number story with a partner What does this assess?

  22. Graph • What do you notice about our data? • Make a statement that… • Combines two or more rows. • Involves counting on. • Compares two rows.

  23. Algebraic Thinking: What does this mean to you?

  24. Algebraic Reasoning • Generalization from arithmetic and from patterns in all of mathematics. • Meaningful use of symbols. • Study of structure in the number system. • Study of patterns and functions. • Process of mathematical modeling, which integrates the first four. J. J. Kaput, 1999, “Teaching and Learning a New Algebra,” in E. Fennema & T. A. Romberg [Eds.], Mathematics Classrooms That Promote Understanding, pp. 133–155, Mahwah, NJ: Erlbaum

  25. Algebra is generalized arithmetic! • “[Algebra] involves generalizing and expressing that generality using increasingly formal languages, where the generalizing begins in arithmetic, in modeling situations, in geometry, and in virtually all the mathematics that can or should appear in the elementary grades.” --Kaput

  26. What might be contributing to students’ difficulties with algebra? • Secondary school teaching has a considerable part of the responsibility for students’ difficulties with algebra • Elementary school teaching shares part of the responsibility for students’ failure with algebra • Look at how arithmetic is taught in primary school • Algebra can be thought of as “generalized arithmetic” HOWEVER “If you have a problem in your high school algebra classrooms, you have a problem in your elementary classrooms.” (Kathy Richardson, North Carolina Council of Teachers of Mathematics Conference, October 29, 2009).

  27. Elementary school arithmetic tends to be separated from ways of thinking that would support algebra learning • The teaching of arithmetic in elementary school tends to emphasize doing calculations, without paying much attention to helping students think about the properties of number that make the calculations possible • Consequence 1: When students study algebra in secondary school, they have difficulty to see the algebraic procedures as being based on the same properties of number they used in elementary school arithmetic • Consequence 2: The conceptions about arithmetic that many students bring to algebra get in the way of their learning of algebra

  28. The main idea It is important to teach arithmetic in elementary school in a way that not only enhances students’ understanding of arithmetic but also provides a solid foundation for pupils’ learning of algebra. This does not mean using the current secondary algebra curriculum in elementary school. To the contrary, rather than teaching algebra procedures to elementary school students, the goal should be to support them in developing ways of thinking about arithmetic that are more consistent with the ways that students have to think in order to learn algebra. These ways of thinking both pave the way for learning algebra and enhance the learning of arithmetic.

  29. An example of the relationship between arithmetic and algebra • Arithmetic problem: Calculate 50 + 30. • Pupil reasoning: “50 plus 30 equals 80, because 5 plus 3 equals 8, so 5 tens plus 3 tens is 8 tens, or 80.” • Representation using number sentences: 50 + 30 = 5 x 10 + 3 x 10 = (5 + 3) x 10 = 8 x 10 = 80 • Algebraic problem: Simplify the expression 5b + 3b. • Pupil reasoning: “5b plus 3b equals 8b, because 5b plus 3b means 5 times b added to 3 times b, or 8 times b.” • Representation using symbols: 5b + 3b = 5 x b + 3 x b = (5 + 3) x b = 8 x b = 8b

  30. Common Core State Standards Examine the Standards for Mathematical Thinking & the “Operations and Algebraic Thinking” items. What do you notice about the connections between arithmetic & algebra? • Understanding Equality • Using relational thinking • Making conjectures • Justifying conjectures

  31. Primary sources • Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic & algebra in elementary school. Portsmouth, NH: Heinemann. • Carraher, D. W., Schliemann, A. D., Brizuela, B. M., & Earnest, D. (2006). Arithmetic and algebra in early mathematics education. Journal for Research in Mathematics Education, 37(2), 87-115. • Kaput, J. J., Carraher, D. W., & Blanton, M. L. (Eds.) (2008). Algebra in the early grades. Mahwah, NJ: Erlbaum. • Linchevski, L. & Herscovics, N. (1996). Crossing the cognitive gap between arithmetic and algebra: Operating on the unknown in the context of equations. Educational Studies in Mathematics, 30(1), 39-65. • Stylianides, A. J. (2007a). The notion of proof in the context of elementary school mathematics. Educational Studies in Mathematics, 65, 1-20. • Stylianides, A. J. (2007b). Proof and proving in school mathematics. Journal for Research in Mathematics Education, 38, 289-321.

  32. 1. Generalizations: The Border Problem • Find all possible ways to determine the border of this shape without counting the squares one by one.

  33. 2. Meaningful Use of Symbols 8 + 4 = + 5

  34. The Meaning of the Equal Sign= • Turn to a partner • Discuss where we go wrong in teaching the equal sign • What activities/ strategies could we implement to increase understanding?

  35. The Equal Sign as a Balance • What number goes in the box? How do you know?

  36. Equal Sign as a Balance AND Meaningful use of Variables • http://illuminations.nctm.org/ActivityDetail.aspx?id=33 • http://illuminations.nctm.org/ActivityDetail.aspx?id=26 • http://illuminations.nctm.org/ActivityDetail.aspx?id=10 • http://nrich.maths.org/4725

  37. More Balance Practice Can you solve these?

  38. 3. Making Structure in the Number System Explicit • Domino Lesson (flipping the Domino) • Broken Calculator: Can You Fix It? (page 272) • Explore these two challenges, and afterward ask students for conjectures they might make about odds and evens. • If you cannot use any of the even keys (0, 2, 4, 6, 8), can you create an even number in the calculator display? If so, how? • If you cannot use any of the odd keys (1, 3, 5, 7, 9), can you create an odd number in the calculator display? If so, how?

  39. 4. The Study of Patterns & Functions Two of Everything: A literature connection Modeling of RAP 4

  40. Summary Video: Dr. Weston teaching Kindergarten Keep in mind • Before/During/After lesson planning • Knowledge Quartet: Transformation • Big ideas in Algebra & Early Number

  41. Choice of ExamplesUse of RepresentationsDemonstration Transformation: • Generalizations • 2. Meaningful use of symbols • 3. Making structure in the number system explicit • 4. The study of patterns and functions

  42. What do we need to keep in mind? • Emphasize appropriate vocabulary • Independent “input” and dependent variables “output” • Discrete and continuous • Domain and range • Multiple representations • Context • Table • Verbal Description • Symbols • Graphs • Connect representations • Algebraic thinking across the curriculum

  43. Exiting Thoughts… • Draw three balloons in your notebook. • In each one, write something you want to remember from today’s topics & activities.

  44. For Next Time…

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