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Bridging Curriculum Concepts through Trigonometric Representations

Bridging Curriculum Concepts through Trigonometric Representations. OCMA 28 th Annual Conference Patricia (Trish) Byers Georgian College tbyers@georgianc.on.ca. Meaningful research. Bridging Curriculum Concepts through Trigonometric Representations. Defining representations

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Bridging Curriculum Concepts through Trigonometric Representations

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  1. Bridging Curriculum Concepts through Trigonometric Representations OCMA 28th Annual Conference Patricia (Trish) Byers Georgian College tbyers@georgianc.on.ca

  2. Meaningful research

  3. Bridging Curriculum Concepts through Trigonometric Representations • Defining representations • Rationale for representations • Mapping representations through the curriculum • Trigonometric representations – preliminary findings & implications for teaching OCMA 2008

  4. Focusing the analysis • Recent secondary school mathematics curriculum changes; • Results from the College Mathematics Project 2006; • Personal college experiences teaching trigonometry and with student difficulties learning representations. OCMA 2008

  5. College Mathematics Project 2006 • Scope • more than 5000 students enrolled in 139 technology programs at 6 Ontario colleges • Steering Committee • representatives from the 6 participating colleges, 9 partner school boards, SCWI-GTA, ACAATO, MTCU, and the Ministry of Education, YSIMSTE representatives OCMA 2008

  6. College programs – A sample • Engineering technology programs • Architectural • Mechanical • tool & die • design • Construction • Electrical • Applied Science (e.g., Environmental) • Computer Science OCMA 2008

  7. Secondary school math curriculum 2007 – Pathway 1 Grade 9 Academic Grade 9 Applied T Gr 10 Academic MPM2D Gr 10 Applied MFM2P Gr 11 C Foundations for College Math MBF3C Gr 11 U/C Function Applications MCF3M Grade 11 U Functions Grade 12 U Advanced Functions Grade 12 U Data Management Gr 12 C College Technology MCT4C Gr 12 C Foundations for College Math MAP4C Calculus and Vectors 12U Course

  8. Secondary school math curriculum 2007 – Pathway 2 Grade 9 Academic Grade 9 Applied T Gr 10 Academic MPM2D Gr 10 Applied MFM2P Gr 11 C Foundations for College Math MBF3C Gr 11 U/C Function Applications MCF3M Grade 11 U Functions Grade 12 U Advanced Functions Grade 12 U Data Management Gr 12 C College Technology MCT4C Gr 12 C Foundations for College Math MAP4C Calculus and Vectors 12U Course

  9. Secondary school math curriculum 2007 – Pathway 3 Grade 9 Academic Grade 9 Applied T Gr 10 Academic MPM2D Gr 10 Applied MFM2P Gr 11 C Foundations for College Math MBF3C Gr 11 U/C Function Applications MCF3M Grade 11 U Functions Grade 12 U AdvancedFunctions Grade 12 U Data Management Gr 12 C College Technology MCT4C Gr 12 C Foundations for College Math MAP4C Calculus and Vectors 12U Course OCMA 2008

  10. Results from CMP 2006 • Summary of student data analysis • The study found that 30% to 50% of all students (all clusters) in all program areas were at risk of failing or failing. OCMA 2008

  11. Results from CMP 2006 • More than 700 students entering 1st year technology programs at 6 Ontario colleges in F04, but fewer than 25% had taken MCT4C (Mathematics for College Technology). • 69% of these students achieved an A, B or C grade in their 1st semester college mathematics course, with 31% obtaining a D, F or withdrawal from the course. OCMA 2008

  12. Results from CMP 2006 • By contrast, the Grade 12 mathematics course taken by over half of the students was MAP4C (College and Apprenticeship Mathematics). • Of this group, <35% achieved a good grade (A, B or C) in first semester college mathematics and 65% obtained a D, F or withdrawal from the course. OCMA 2008

  13. Mathematics curriculum engineering technology programs • Geometry • 2- & 3-dimensions • Linear equations • Algebraic & graphic solutions • Trigonometry • Right angle trigonometry – acute & obtuse angles; sine & cosine laws; working in all 4 quadrants, etc. • Sinusoidal waveforms & graphing • Vectors – resolving vectors; adding vectors; vectors in rectangular & polar form • Complex numbers • rectangular, polar, exponential forms OCMA 2008

  14. Representations and their role in teaching & learning trigonometry • “The ways in which mathematical ideas are represented is fundamental to how people can understand and use those ideas” (NCTM, 2000, p 67). • The AMATYC Standards for Intellectual Development (2006) refer to students learning through modeling, linking multiple representations, and, selecting, using, and translating among numerical, graphical, symbolic, and verbal representations to organize and solve problems (p. 5). OCMA 2008

  15. Representations and their role in teaching & learning trigonometry • It is suggested that mathematical sophistication develops out of a comprehensive cache of representations that support deep conceptual understanding (Pritchard & Simpson, 1999, p. 87). • Research in learning trigonometric functions reveals that a key source of student difficulty is the lack of ability to move from one representation to another. OCMA 2008

  16. Focusing teaching & learning • To investigate whether trigonometric representations are a source of difficulty as students transition from secondary to college mathematics. OCMA 2008

  17. Defining representations • “A representation is a configuration of signs, characters, icons, or objects that can somehow stand for, or “represent” something else ... • According to the nature of the representing relationship, the termrepresentcan be interpreted in many ways, including the following (the list is not exhaustive): correspond to, denote, depict, embody, encode, evoke, label, mean, produce, refer to, suggest, or symbolize (italics in the original)” (Goldin, 2003, p. 276). OCMA 2008

  18. Systems of representations • External systems • Structured by the conventions underlying them • No longer arbitrary • Accepted by the mathematics community waiting to be “discovered” by the student • Internal systems • Demonstrate how a student understands a mathematical concept • Verbal/syntactic; Imagistic; Formal notational; Affective • Dimensions in representations • Horizontal: between external systems • Vertical: with external & internal systems OCMA 2008

  19. Ways to represent a function • Stewart, Redlin, & Watson (2002, p. 150) • Verbally – in words • Algebraically – by an explicit formula • Visually – with a diagram or figure • Numerically – by a table of values OCMA 2008

  20. Ways to represent a trigonometric function • Algebraic/symbolic • formulas for trigonometric ratios & trigonometric functions • Numeric • tables • Visual • right triangle, circle, sinusoidal waveform OCMA 2008

  21. Gr. 10 Academic-Trigonometry By the end of this course, students will: use their knowledge of ratio and proportion to investigate similar triangles and solve problems related to similarity; solve problems involving right triangles, using the primary trigonometric ratios and the Pythagorean theorem; solve problems involving acute triangles, using the sine law and the cosine law. Under Analytic Geometry, properties of the circle given by the equation x2 + y2 = r2 Gr. 10 Applied-Measurement & Trigonometry By the end of this course, students will: use their knowledge of ratio and proportion to investigate similar triangles and solve problems related to similarity; solve problems involving right triangles, using the primary trigonometric ratios and the Pythagorean theorem; solve problems involving the surface areas and volumes of three-dimensional figures, and use the imperial and metric systems of measurement. Ministry of Educationcurriculum expectations

  22. Gr. 11M-Trigonometric Functions By the end of this course, students will: solve problems involving trigonometryin acute trianglesusing the sine law and the cosine law, including problems arising from real-world applications; demonstrate an understanding of periodic relationships and the sine function, and make connections between the numeric, graphical, and algebraic representations of sine functions; identify and represent sine functions, and solve problems involvingsinefunctions, including problems arising from real-world applications. Ministry of Educationcurriculum expectations • Gr. 11C-Geometry & Trigonometry • By the end of this course, students will: • represent, in a variety of ways, two-dimensional shapes and three-dimensional figures arising from real-world applications, and solve design problems; • solve problems involving trigonometry in acute trianglesusing the sine law and the cosine law, including problems arising from real-world applications.

  23. Gr. 12(MCT) –Trigonometric Functions By the end of this course, students will: determine the values of the trigonometric ratios for angles less than 360º, and solve problems using the primary trigonometric ratios, the sine law, and the cosine law; make connections between the numeric, graphical, and algebraic representations of sinusoidal functions; demonstrate an understanding that sinusoidal functions can be used to model some periodic phenomena, and solve related problems, including those arising from real-world applications. Gr. 12C(MAP)-Geometry & Trigonometry By the end of this course, students will: solve problems involving measurement and geometry and arising from real-world applications; explain the significance of optimal dimensions in real-world applications, and determine optimal dimensions of two-dimensional shapes and three-dimensional figures; solve problems using primary trigonometric ratios of acute and obtuse angles, the sine law, and the cosine law, including problems arising from real-world applications, and describe applications of trigonometry in various occupations. Ministry of Educationcurriculum expectations OCMA 2008

  24. Mapping the right triangle representation connecting structures 9 Applied & Academic: Equivalent ratios Ratios & proportion Proportional reasoning Pythagorean theorem Interior & exterior angles of triangles Angle measurement & polygons 10 Applied & Academic: Similar triangles Pythagorean theorem 10 Academic: Proportional reasoning Sign Configuration Representation Right Triangle (10 Applied & Academic)

  25. Mapping the right triangle representation Right Triangle Representation (10 Applied & Academic) Sine & Cosine Law for acute triangles (10 Academic) Sine & Cosine Law for acute triangles (11 Applied & Mixed) Sine & Cosine Laws for oblique triangles Primary trig ratios of obtuse angles (12 College) Sine & Cosine Laws for oblique triangles (12 Mixed) OCMA 2008

  26. Preliminary findings • The mapping resembles a hypothetical learning trajectory (HLT) • Components: • “the learning goal, the developmental progressions of thinking and learning, and a sequence of instructional tasks” (Clements & Sarama, 2004, p. 85). OCMA 2008

  27. Learning goals Progressions of thinking & learning Primitive characters or signs Configurations Learning tasks Representations A relationship with HLT OCMA 2008

  28. Preliminary findings • Representations are arbitrary but are established through use becoming signs and configurations for newly developing representations. OCMA 2008

  29. Implications • Potential discrepancies exist between types of trigonometric representations & depth to which these are taught (hence, disruptions in a hypothetical learning trajectory from secondary school to college). • Potential student difficulties learning trigonometric representations can be identified . • Strategies to help students with potential difficulties learning trigonometric representations need developing. OCMA 2008

  30. Implications • A point of focus to share various representations used in the others’ classrooms of each educational sector beginning the conversation on student difficulties in college mathematics. • A point of departure to build a destination bridging sequence to address representations not taught in secondary school but required for college studies. • Further research to unpack discrepancies in other college mathematics concepts. OCMA 2008

  31. References • Clements, D.H., Sarama, J. (2004). Learning trajectories in mathematics education. Mathematical Thinking and Learning, 6(2), 81-89. Mahwah, NJ: Lawrence Erlbaum Associates. • Goldin, G. (2003). Representation in school mathematics: A unifying research perspective. In J. Kilpatrick, W.G. Martin; D. Schifter (Eds.), A Research Companion to Principles and Standards for School Mathematics. Reston, VA: The National Council of Teacher of Mathematics. pp. 275-284. OCMA 2008

  32. Bridging Curriculum Concepts using Trigonometric Representations OCMA 28th Annual Conference Patricia (Trish) Byers Georgian College tbyers@georgianc.on.ca Thank you

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