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Functions Based Curriculum

Functions Based Curriculum. Math Camp 2008. Trish Byers. WELCOME BACK!. Anthony Azzopardi. FOCUS: FUNCTIONS BASED CURRICULUM DAY ONE: CONCEPTUAL UNDERSTANDING DAY TWO: FACTS AND PROCEDURES DAY THREE: MATHEMATICAL PROCESSES. Grade 12 U Calculus and Vectors MCV4U.

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Functions Based Curriculum

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  1. FunctionsBasedCurriculum Math Camp 2008

  2. Trish Byers WELCOME BACK! Anthony Azzopardi

  3. FOCUS: FUNCTIONS BASED CURRICULUM DAY ONE: CONCEPTUAL UNDERSTANDING DAY TWO: FACTS AND PROCEDURES DAY THREE: MATHEMATICAL PROCESSES

  4. Grade 12 U Calculus and Vectors MCV4U Revised Prerequisite Chart Implementation Grade 12 U Advanced Functions MHF4U Grade 11 U Functions MCR3U Grade 9PrinciplesAcademicMPM1D Grade 10PrinciplesAcademicMPM2D Grade 12 U Mathematics of Data Management MDM4U Grade 11 U/C Functions and Applications MCF3M T Grade 12 C Mathematics for College Technology MCT4C Grade 9FoundationsAppliedMFM1P Grade 10 Foundations AppliedMFM2P Grade 11 C Foundations for College Mathematics MBF3C Grade 12 C Foundations for College Mathematics MAP4C Grade 9 LDCC Grade 10 LDCC Grade 11 Mathematics for Work and Everyday Life MEL3E Grade 12 Mathematics for Work and Everyday Life MEL4E

  5. Principles Underlying Curriculum Revision • Curriculum Expectations • Learning • Teaching • Assessment/Evaluation • Learning Tools • Equity Areas adapted from N.C.T.M. Principles and Standards for School Mathematics, 2000

  6. “Icebreaker” • Select a three digit number. (eg. 346) • Create a six digit number by repeating the three digit number you selected. (eg. 346346) • Is your number lucky or unlucky?

  7. Do our students see mathematics as • meaningful? • magical? • both?

  8. “Icebreaker” • 346346 = 3x100 000 + 4x10 000 +6x1 000 + 3x100 + 4x10 + 6x1 • 346346 = 3x100 000 + 3x100 + 4x10 000 + 4x10 + 6x1 000 + 6x1 • 346346 = 3 x (100 000 + 100) + 4 x (10 000 + 10) + 6 x (1 000 + 1)

  9. “Icebreaker” • 346346 = 3 x (100 100) + 4 x (10 010) + 6 x (1 001) • 346346 = (3 x 1 001 x100) + (4 x 1 001 x 10) + (6 x 1 001 x 1) • 346346 = 1 001 x (3x100 + 4x10 + 6x1) • 346346 = 1 001 x 346 AND 1 001 = 13 x 11 x 7

  10. Why is it so important for us to improve our teaching of mathematics?

  11. Underlying Principles for Revision • Equity focuses on meeting the diverse learning needs of students and promotes excellence for all by • ensuring curriculum expectations are grade and destination appropriate, • by providing access to Grade 12 mathematics courses in a variety of ways. • supporting a variety of teaching and learning strategies

  12. Identify 3 key points from your article segment. What is one idea from the classroom that reminds you of these ideas?

  13. Underlying Principle for Revision • Effective teaching of mathematics requires that the teacher understand the mathematical concepts, procedures, and processes that students need to learn and use a variety of instructional strategies to support meaningful learning;

  14. Mathematical Proficiency

  15. Mathematical Proficiency

  16. Mathematical Proficiency Representing Reflecting Reasoning and Proving Connecting Selecting Tools and Computational Strategies Problem Solving Communicating Reasoning and Proving Reflecting Communicating Representing Selecting Tools and Computational Strategies Connecting Problem Solving Mathematical Processes

  17. Reasoning and Proving Reflecting Communicating Representing Selecting Tools and Computational Strategies Connecting Problem Solving Mathematical Proficiency

  18. Teaching Mathematical Expert Pedagogical Expert

  19. Teacher Curriculum Student Student Teachers use • strong subject/discipline content knowledge • good instructional skills • strong pedagogical content knowledge

  20. Teacher: What is the area of a rectangle with length 5 unitsand width 3 units? Student: 16 Pedagogical Content Knowledge • Applying subject knowledge effectively, using concepts in ways that make sense to students Teacher: What is the perimeter of this rectangle?

  21. Student: sin 90° Teacher: What is the sin 30° + sin 60° ? Pedagogical Content Knowledge • Applying subject knowledge effectively, using concepts in ways that make sense to student Teacher: Is f(x) + f(y) always equal to f(x+y)?

  22. A Problem Solving Moment Problem: What is the sin 50° ? Answer: Wrinkles, Grey Hair, Memory Loss

  23. Teaching: Student Engagement • Students develop positive attitudes when they • make mathematical conjectures; • make breakthroughs as they solve problems; • see connections between important ideas. Ed Thoughts 2002: Research and Best Practice

  24. PISA 2003: Indices of Student Engagement In Mathematics (15 year olds)

  25. gains-camppp.wikispaces.com

  26. “The concept of function is central to understanding mathematics, yet students’ understanding of functions appears either to be too narrowly focused or to include erroneous assumptions” (Clement, 2001, p. 747). Conceptual Understanding

  27. Frayer Model • 3 Groups • Grade 7/8 • Grade 9/10 • Grade 11/12 FUNCTIONS

  28. “Conceptual understanding within the area of functions involves the ability to translate among the different representations, table, graph, symbolic, or real-world situation of a function” (O’Callaghan, 1998). Conceptual Understanding

  29. Teaching: Multiple Representations Graphical Representation Numerical Representation Concrete Representation Algebraic Representation f(x) = 2x - 1

  30. < < 5 5 1 x + 1 1 x + 1 • 4 • 5 x > Multiple Representations MHF4U – C4.1 (x + 1) (x + 1) 1 < 5x + 5 - 4 < 5x

  31. < 5 f(x) = 1 x + 1 1 x + 1 Multiple Representations Use the graphs of and h(x) = 5 to verify your solution for

  32. Real World Applications MAP4C: D2.3 interpret statistics presented in the media (e.g., the U.N.’s finding that 2% of the world’s population has more than half the world’s wealth, whereas half the world’s population has only 1% of the world’s wealth)……. 1% 49% 50% 48%

  33. Real World Applications Classroom activities with applications to real world situations are the lessons students seem to learn from and appreciate the most. Poverty increasing: Reports says almost 30 per cent of Toronto families live in poverty. • The report defines poverty as a family whose after-tax income is 50 percent below the median in their community, taking family size into consideration. • In Toronto, a two-parent family with two children living on less than $27 500 is considered poor. METRO NEWS November 26, 2007

  34. Should mathematics be taught the same way as line dancing?

  35. A Vision of Teaching Mathematics • Classrooms become mathematical communities rather than a collection of individuals • Logic and mathematical evidence provide verification rather than the teacher as the sole authority for right answers • Mathematical reasoning becomes more important than memorization of procedures. NCTM 1989

  36. A Vision of Teaching Mathematics • Focus on conjecturing, inventing and problem solving rather than merely finding correct answers. • Presenting mathematics by connecting its ideas and its applications and moving away from just treating mathematics as a body of isolated concepts and skills. NCTM 1989

  37. Traditional Lessons Direct Instruction: teaching by example. The “NEW” Three Part Lesson. • Teaching through exploration and investigation: • Before: Present a problem/task and ensure students understand the expectations. • During: Let students use their own ideas. Listen, provide hints and assess. • After: Engage class in productive discourse so that thinking does not stop when the problem is solved.

  38. Teaching: Direct Instruction Investigation “ Effective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well”

  39. The problem is no longer just teaching better mathematics. It is teaching mathematics better. Teaching Adding It Up: National Research Council - 2001

  40. Underlying Principles for Revision • Curriculum expectations must be coherent, focused and well articulated across the grades;

  41. Identifying Key Ideas about Functions • Same groups as Frayer Model Activity • Using the Ontario Curriculum, identify key ideas about functions. • Describe the key ideas using 1 – 3 words. • Record each idea in a cloud bubble on chart paper.

  42. Learning Activity: Functions

  43. Grade 7 and 8 Patterning and Algebra Grade 9 Academic Linear Relations Grade 9 Applied Linear Relations Grade 10 Academic Quadratic Relations Grade 10 Applied Modelling Linear Relations Quadratic Relations Grade 11 Functions Exponential, Trigonometric and Discrete Functions Grade 11 Foundations Quadratic Relations Exponential Relations Grade 12 Advanced Functions Exponential, Logarithmic, Trigonometric, Polynomial, Rational Grade 12 Foundations Modelling Graphically Modelling Algebraically

  44. University Destination Transition

  45. College Destination Transition

  46. College Destination Transition

  47. Workplace Destination Transition

  48. Links to Post Secondary Destinations: UNIVERSITY DESTINATIONS: University Mathematics, Engineering, Economics, Science, Computer Science, some Business Programs and Education – Secondary Mathematics Grade 12 U Calculus and Vectors MCV4U University Kinesiology, Social Sciences, Programs and some Mathematics, Health Science, some Business Interdisciplinary Programs and Education – Elementary Teaching Grade 12 U Advanced Functions MHF4U Grade 12 U Mathematics of Data Management MDM4U Some University Applied Linguistics, Social Sciences, Child and Youth Studies, Psychology, Accounting, Finance, Business, Forestry, Science, Arts,

  49. Links to Post Secondary Destinations: COLLEGE DESTINATIONS: Grade 12 C Mathematics for College Technology MCT4C College Biotechnology, Engineering Technology (e.g. Chemical, Computer), some Technician Programs General Arts and Science, Business, Human Resources, some Technician and Health Science Programs, Grade 12 C Foundations for College Mathematics MAP4C WORKPLACE DESTINATIONS: Steamfitters, Pipefitters, Sheet Metal Worker, Cabinetmakers, Carpenters, Foundry Workers, Construction Millwrights and some Mechanics, Grade 12 Mathematics for Work and Everyday Life MEL4E

  50. Concept Maps • Groups of three with a representative from 7/8, 9/10 and 11/12 • Use the key ideas about functions generated earlier to build a concept map. INPUT OUTPUT CO-ORDINATES Make a set of

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