OCMA Conf 08 May 17

OCMA Conf 08 May 17 - Bridging Curriculum Concepts through...

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Unformatted text preview: Bridging Curriculum Concepts through Trigonometric Representations Representations OCMA 28th Annual Conference Patricia (Trish) Byers Georgian College tbyers@georgianc.on.ca Meaningful research Meaningful Bridging Curriculum Concepts Bridging through Trigonometric Representations Representations Defining representations Rationale for representations Mapping representations through the Mapping curriculum curriculum Trigonometric representations – Trigonometric preliminary findings & implications for teaching teaching OCMA 2008 Focusing the analysis Recent secondary school mathematics Recent curriculum changes; curriculum Results from the College Mathematics Results Project 2006; Project Personal college experiences teaching Personal trigonometry and with student difficulties learning representations. learning OCMA 2008 College Mathematics Project 2006 Scope Scope more than 5000 students enrolled in 139 more technology programs at 6 Ontario colleges technology Steering Committee representatives from the 6 participating representatives colleges, 9 partner school boards, SCWIcolleges, GTA, ACAATO, MTCU, and the Ministry of GTA, Education, YSIMSTE representatives Education, OCMA 2008 College programs – A sample Engineering technology programs Architectural Architectural Mechanical Mechanical tool & die tool design design Construction Electrical Applied Science (e.g., Environmental) Computer Science OCMA 2008 Secondary school math curriculum 2007 – Pathway 1 Grade 9 Academic Grade 9 Applied T Gr 10 Academic MPM2D Grade 11 U Functions Grade 12 U Advanced Functions Calculus and Vectors 12U Course Gr 10 Applied MFM2P Gr 11 U/C Function Applications MCF3M Grade 12 U Data Management Gr 12 C College Technology MCT4C Gr 11 C Foundations for College Math MBF3C Gr 12 C Foundations for College Math MAP4C Secondary school math curriculum 2007 – Pathway 2 Grade 9 Academic Grade 9 Applied T Gr 10 Academic MPM2D Grade 11 U Functions Grade 12 U Advanced Functions Calculus and Vectors 12U Course Gr 10 Applied MFM2P Gr 11 U/C Function Applications MCF3M Grade 12 U Data Management Gr 12 C College Technology MCT4C Gr 11 C Foundations for College Math MBF3C Gr 12 C Foundations for College Math MAP4C Secondary school math curriculum 2007 – Pathway 3 Grade 9 Academic Grade 9 Applied T Gr 10 Academic MPM2D Grade 11 U Functions Grade 12 U Advanced Functions Gr 10 Applied MFM2P Gr 11 U/C Function Applications MCF3M Grade 12 U Data Management Calculus and Vectors 12U Course OCMA 2008 Gr 12 C College Technology MCT4C Gr 11 C Foundations for College Math MBF3C Gr 12 C Foundations for College Math MAP4C 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. % At Risk or Failing First Sem ester College M ath 90 80 70 60 % 50 %-ROG 40 %-*ROG 30 20 10 0 Mech Const Elect Cluster OCMA 2008 App Sci Comp 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 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 Mathematics curriculum Mathematics engineering technology programs engineering Geometry 2- & 3-dimensions Linear equations Algebraic & graphic solutions Trigonometry Right angle trigonometry – acute & obtuse angles; Right sine & cosine laws; working in all 4 quadrants, etc. sine Sinusoidal waveforms & graphing Vectors – resolving vectors; adding vectors; vectors in Vectors rectangular & polar form rectangular Complex numbers Complex rectangular, polar, exponential forms OCMA 2008 Representations and their role in Representations teaching & learning trigonometry teaching “The ways in which mathematical ideas are represented The represented is fundamental to how people can understand and use those ideas” (NCTM, 2000, p 67). those The AMATYC Standards for Intellectual Development The Standards (2006) refer to students learning through modeling, linking multiple representations, and, selecting, using, multiple and, and translating among numerical, graphical, symbolic, numerical graphical symbolic and verbal representations to organize and solve verbal problems (p. 5). OCMA 2008 Representations and their role in Representations teaching & learning trigonometry teaching It is suggested that mathematical sophistication develops It out of a comprehensive cache of representations that representations 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. move OCMA 2008 Focusing teaching & learning To investigate whether trigonometric To representations are a source of difficulty as students transition from secondary to college mathematics. mathematics. OCMA 2008 Defining representations “A representation is a configuration of signs, representation characters, icons, or objects that can somehow stand for, or “represent” something else ... stand According to the nature of the representing According relationship, the term represent can be interpreted in relationship, represent 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). (Goldin, OCMA 2008 Systems of representations External systems Structured by the conventions underlying them No longer arbitrary No Accepted by the mathematics community waiting Accepted to be “discovered” by the student to Internal systems Demonstrate how a student understands a Demonstrate mathematical concept mathematical Verbal/syntactic; Imagistic; Formal notational; Verbal/syntactic; Affective Affective Dimensions in representations Horizontal: between external systems Vertical: with external & internal systems OCMA 2008 Ways to represent Ways a function function Stewart, Redlin, & Watson (2002, p. 150) 1. 2. 3. 4. Verbally – in words Algebraically – by an explicit formula Visually – with a diagram or figure Numerically – by a table of values OCMA 2008 Ways to represent Ways a trigonometric function trigonometric Algebraic/symbolic Algebraic/symbolic formulas for trigonometric ratios & formulas trigonometric functions trigonometric Numeric tables Visual Visual right triangle, circle, sinusoidal waveform OCMA 2008 Ministry of Education Ministry curriculum expectations Gr. 10 Academic-Trigonometry By the end of this course, By students will: students 1. 2. 3. 4. use their knowledge of ratio use and proportion to investigate similar triangles and solve problems related to similarity; problems solve problems involving right solve triangles, using the primary trigonometric ratios and the Pythagorean theorem; Pythagorean solve problems involving solve acute triangles, using the sine law and the cosine law. law Under Analytic Geometry, Under properties of the circle given by the equation x2 + y2 = r2 Gr. 10 Applied-Measurement Gr. & Trigonometry Trigonometry By the end of this course, By students will: students 1. 2. 3. use their knowledge of ratio use and proportion to investigate similar triangles and solve problems related to similarity; problems solve problems involving right solve triangles, using the primary trigonometric ratios and the Pythagorean theorem; Pythagorean solve problems involving the solve surface areas and volumes of three-dimensional figures, and use the imperial and metric systems of measurement. measurement. Ministry of Education curriculum expectations Gr. 11M-Trigonometric Functions Gr. By the end of this course, students By will: will: 1. 2. 3. solve problems involving solve trigonometry in acute triangles trigonometry in using the sine law and the cosine law, iincluding problems arising law ncluding from real-world applications; from demonstrate an understanding of demonstrate periodic relationships and the sine function, and make connections between the numeric, graphical, and algebraic representations of sine functions; sine iidentify and represent sine dentify functions, and solve problems involving sine functions, including involving sine problems arising from real-world applications. applications. Gr. 11C-Geometry & Gr. Trigonometry Trigonometry By the end of this course, By students will: students 1. 2. represent, in a variety of represent, ways, two-dimensional shapes and threeshapes dimensional figures arising dimensional from real-world applications, and solve design problems; and solve problems involving solve trigonometry in acute triangles using the sine law triangles and the cosine law, iincluding ncluding and problems arising from realproblems world applications. Ministry of Education curriculum expectations Gr. 12(MCT) –Trigonometric Gr. Functions Functions By the end of this course, By students will: students 1. 2. 3. determine the values of the determine trigonometric ratios for angles less than 360º, and solve problems using the primary trigonometric ratios, the sine law, and the cosine law; law, make connections between the make numeric, graphical, and algebraic representations of algebraic sinusoidal functions; sinusoidal demonstrate an understanding demonstrate that sinusoidal functions can be used to model some periodic phenomena, and solve related problems, including those arising from real-world applications. applications. OCMA 2008 Gr. 12C(MAP)-Geometry & Gr. Trigonometry Trigonometry By the end of this course, students By will: will: 1. 2. solve problems involving solve 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; figures; solve problems using primary solve trigonometric ratios of acute and obtuse angles, the sine law, and the cosine law, iincluding problems ncluding cosine arising from real-world applications, and describe applications of trigonometry in various occupations. trigonometry Mapping the right triangle representation representation Right Triangle (10 Applied & Academic) 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 Mapping the right triangle Mapping representation 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 (12 Mixed) OCMA 2008 Sine & Cosine Laws for oblique triangles Primary trig ratios of obtuse angles (12 College) Preliminary findings 1. The mapping resembles a hypothetical The learning trajectory (HLT) learning Components: “the learning goal, the developmental progressions the of thinking and learning, and a sequence of instructional tasks” (Clements & Sarama, 2004, p. 85). 85). OCMA 2008 A relationship with HLT Learning goals Progressions of thinking & learning Primitive characters or signs Configurations Representations OCMA 2008 Learning tasks Preliminary findings 2. Representations are arbitrary but are Representations established through use becoming signs and configurations for newly developing representations. OCMA 2008 Implications Potential discrepancies exist between types of Potential 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 . trigonometric Strategies to help students with potential Strategies difficulties learning trigonometric representations need developing. need OCMA 2008 Implications A point of focus to share various representations point 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 bridging not taught in secondary school but required for college studies. college Further research to unpack discrepancies in Further other college mathematics concepts. other OCMA 2008 References Clements, D.H., Sarama, J. (2004). Learning trajectories in mathematics education. Mathematical Thinking and Learning, 6(2), 81-89. Mahwah, NJ: Thinking (2), Lawrence Erlbaum Associates. Lawrence 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 School Reston, Council of Teacher of Mathematics. pp. 275-284. Council OCMA 2008 Bridging Curriculum Concepts using Trigonometric Representations Representations OCMA 28th Annual Conference Patricia (Trish) Byers Georgian College tbyers@georgianc.on.ca ...
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