Senior Mathematics Curriculum Revision

Senior Mathematics Curriculum Revision
Supporting students and teachers by keeping Ontario’s K curriculum current and relevant College Math Project Forum June 15, 2006 Anthony Azzopardi Curriculum and Assessment Policy Branch Ministry of Education 4

What is Curriculum Review?
A staged process to review Kindergarten to Grade 12 curriculum documents by discipline area that: builds on the quality curriculum currently in place ensures that the curriculum remains current and relevant

Curriculum Review Process
integrate review of elementary and secondary curriculum policy documents have parallel revision processes for English and French language curriculum involve teachers, principals, board staff, subject experts, education stakeholders, parents, students and sector representatives.

Curriculum Review Process
Sept. Sept. Sept. Sept. Sept. 2003 2004 2005 2006 2007 Grades * Grade 11 * Grade 12 * * Mandatory Implementation Revision and Feedback Consultation Analysis and Synthesis Editing, Publication and Distribution

Division Associations Feedback Consultations
Curriculum Review Process Technical Analysis Focus Groups Other Consultations and Input Subject / Division Associations Analysis / Synthesis Achievement Charts Research Revision / Feedback Feedback Consultations

Ontario Mathematics Curriculum 2000
Grade 9 Academic Grade 9 Applied Grade 10 Academic Grade 10 Applied Grade 11 U Functions and Relations Grade 11 U/C Functions Grade 11 C Personal Finance Grade 11 W Math for Everyday Life Grade 12U Geometry and Discrete Grade 12U Advanced Functions Grade 12U Data Management Grade 12C Math for College Technology Grade 12C College and Apprenticeship Grade 12 W Math for Everyday Life

Student Destinations 1999-2000 Cohort to Fall 2004
19% to College 18% OSSD to Work 30% Leave before OSSD 33% to University Grade 9 Enrolment = 100% Source: Alan King, Double Cohort Study 2005

Double Cohort Study – Phase 4 Grade 11 Achievement
Grade 11 Courses Functions and Relations (U) 11.4% 11.0% 9.2% Functions (U/C) 20.9% 19.7% 18.2% Personal Finance (C) 18.6% 17.3% 16.5% Math For Everyday Life (W) 17.0% 15.8% 15.3%

Grade 11 Student Achievement
Double Cohort Study: Phase 4, 2005

Grade 12 Student Achievement
Double Cohort Study: Phase 4, 2005

PISA 2003: Indices of Student Engagement In Mathematics (15 year olds)
Significantly higher than Canadian average Performing the same as the Canadian average Significantly lower than Canadian average Interest and enjoyment in mathematics ONTARIO NFLD, PEI, NS, NB, QU, MAN, SK, AL BC Belief in usefulness of mathematics NS, QU NFLD, PEI, MAN, SK, AL NB, BC Mathematics confidence QU, AL NFLD, BC PEI, NS, NB, MAN, SK Perceived ability in mathematics NFLD, PEI, NS, NB, SK MAN, BC Mathematics anxiety NB, QU, MAN, SK, AL, BC NFLD, PEI, NS

Double Cohort Study – Phase 4 Grade 11 Enrolment
Grade 11 Courses Functions and Relations (U) 34.3% 28% 26.8% Functions (U/C) 26.2% 27.4% 26.1% Personal Finance (C) 29.6% 32.8% 34.4% Math For Everyday Life (W) 10% 11.7% 12.7%

Consultation with Universities
Initial consultation - early 2004 Informal meetings (e.g., Mathematics Education Forum - Field’s Institute) Deans of Engineering - June 2005 Revision writing - July 2005 Feedback consultation in Fall 2005 Grade 11 consultations – Early 2006 Revision writing – July 2006

Consultation with Colleges
Heads of Technology - Spring 2004 College Math Survey – April 2004 ACAATO consultation - June 2004 Colleges Gr 9/10 feedback – Nov 2004 Revision writing - July 2005 Feedback consultation Gr 11/12 - Nov 2005 Grade 11Consultations – Spring 2006 Revision writing – July 2006

ACAATO Recommendations 2004
Create a clearer pathway from Grade 10 Applied to Grade 12 College Tech Revise Grade 11 Personal Finance course to better prepare students for Grade 12C. Address overlap in 11U and 11M to ensure 11M is more appropriate for students entering college tech programs.

ACAATO Recommendations 2004
Grade 12 College Tech should be more appropriate for college bound students; Improve how the curriculum helps students develop concepts, basic numeric and algebraic skills and the ability to apply processes such as problem-solving, estimation and communication.

ACAATO Research 2004 Math-related Program Clusters: Applied Arts
Business Health Sciences Hospitality Human Services Technology Skilled Trades

Review Process: Synthesis
Revisions address: Curriculum Expectations Equity Learning Teaching Assessment and Evaluation Learning Tools

Goals Of Revision: Reduce the density of the curriculum Provide more opportunities for students to develop and apply important life-long process skills Provide clearer pathways Incorporate more grade and destination appropriate topics and skills

Goals Of Revision: Enhance curriculum coherence and concept development over the grades Improve student achievement and graduation rates Improve access to higher mathematics, attitudes towards mathematics, student retention in mathematics

MATHEMATICAL PROCESSES OVERALL/SPECIFIC EXPECTATIONS
INTRODUCTION PATHWAYS REVIEW EXAMPLES STRANDS SAMPLE PROBLEMS Mathematics Grades 11 and 12 SUBHEADINGS ACHIEVEMENT CHART OVERALL/SPECIFIC EXPECTATIONS 42

Review Process: Feedback Consultations
feedback consultation on proposed revisions to Grades 11 and 12 occurred in the fall of 2005 Day 1 - information provided on curriculum review process and the proposed revisions in the draft Day 2 - participants share feedback on the draft of proposed revisions gathered through a consultation process within their board or organization information from the consultations and feedback sessions informs further revisions

Review Process: Feedback Consultations Grade 11: Math for Work and Everyday Life
Strengths: excellent; students will have success the entire course is about real-life mathematics the intended depth and breadth of the expectations is much clearer there is an appropriate balance between conceptual and procedural development given the destination, this is a very appropriate course.

Review Process: Feedback Consultations Grade 11: Math for Work and Everyday Life
Suggestions and Considerations: more specific references to technology more “Sample Problems”

Review Process: Feedback Consultations Grade 11: Foundations for College Math
Strengths: destination appropriate expectations clearer; examples and sample problems clarify the intended depth, breadth, and level of difficulty better results; expanding on topics introduced in grade 10 provides better preparation for grade 12 College course

Review Process: Feedback Consultations Grade 11: Foundations for College Math
Suggestions and Considerations: more examples and “Sample Problems” identify use of technology in more specific places consider impact of availability of local technology to support implementation students from 10 Applied without a strong foundation may find this course challenging

Review Process: Feedback Consultations Grade 11: Functions and Applications
Strengths: provides good grounding for broad range of math applications revised version is more destination appropriate for college bound students many expectations call for investigation; the smaller number of expectations should help support this change clarity of expectations

Review Process: Feedback Consultations Grade 11: Functions and Applications
Suggestions and Considerations: more examples and sample problems more examples of the use of technology should ‘radians’ have been removed? the course is better preparation for Gr. 12C Math for College Tech than for the Gr. 12U Data Management course students from 10 Applied may find the course challenging

Media Response to Revision
Public response to the proposed DRAFT Senior Mathematics revisions focused primarily on the issue of Calculus.

Questions Raised: ? ? Does COMPLEX = DIFFICULT? Does RIGOROUS = HARD ?
How many are served better by a DENSE CURRICULUM? For whom is a DENSE CURRICULUM developmentally appropriate? Who is marginalized by a DENSE CURRICULUM? What is the relationship between content density and curriculum quality? CONTENT DENSITY CURRICULUM QUALITY CONTENT DENSITY CURRICULUM QUALITY ? ?

Review Process: Minister’s Task Force
February 16, 2006 – Minister announces extended review and organization of the Ministry of Education’s first Curriculum Council: Task Force on Senior High School Mathematics February/March 2006 – Task Force Consultations April 2006 – Task Force submits report June 2006 – Task Force report released (

Task Force Recommendation:
That the Grade 12 courses Mathematics for Work and Everyday Life, Foundations for College Mathematics and Mathematics for College Technology be implemented essentially as currently planned.

Key Message: Curriculum
The revised curriculum is more coherent, focused on important mathematics and well articulated across the grades.

Summary of Key Changes address concerns regarding an overcrowded curriculum: reduced the number of expectations (e.g., removed “Conics ” strand from Grade 11U Functions course) address high failure rates: (e.g., concepts developed in a more developmentally appropriate manner and link better with Grade 10) create clearer pathways to Grade 12 from Grades 9 and 10 Applied Mathematics courses ( e.g., revised 11U/C to articulate with both 10 Academic and 10 Applied)

Summary of Key Changes create clearer pathways for students not entering mathematics or science programs at university (e.g., created a more focused pathway through Grade 11U/C Function Applications to Grade 12); revise expectations to reflect a better balance between the development of procedural fluency, deeper conceptual understanding and the ability to apply key mathematical processes like problem solving, communication and reasoning. improve curriculum coherence (e.g., reorganized strands in college destination courses to improve concept development);

Summary of Key Changes reduce or eliminate overlap (e.g., reduced overlap between Grade 11U and Grade 11U/C mathematics courses); engage students in a more relevant high school learning experience by increasing emphasis on connections within mathematics and between mathematics and the real-world (e.g., stronger connections between topics related to functions from Grade 9 through to Grade 11, increased career connections in the Grade 11 workplace course) encourage the use of a broad range of learning tools to support meaningful student learning in mathematics (e.g., revised specific expectations to include more references to the use of technological tools like graphing technology, calculators, statistical software)

Clear Pathways (DRAFT)
Grade 9 Academic Grade 9 Applied Grade 9 L.D.C.C. T Grade 10 Academic Grade 10 Applied Grade 10 L.D.C.C. Grade 11 U Functions Grade 11 M Function Applications Grade 11 C Foundations for College Math Grade 11E Work and Everyday Life Revised (draft) pathway chart demonstrates some of the key changes. The new Grade 12U mathematics course will address concepts from Calculus and Vectors; Grade 12U Advanced Functions Grade 12U Data Management Grade 12C College Technology Grade 12 C Foundations for College Math Grade 12E Work and Everyday Life Calculus and Vectors U Course

Comparing Strands: Grade 11U
2000 Curriculum Financial Applications of Sequences and Series Arithmetic/Geometric Sequences and Series Compound Interest and Annuity Problems Financial Decision Making Trigonometric Functions Sine Law/Cosine Law for Oblique Triangles Understanding and Applying Radian Measure Graphs and Equations of Sinusoidal Functions Models of Sinusoidal Functions Tools for Operating and Communicating with Functions Polynomials/Rational Expressions and Exponential Expressions Inverses/Transformations/Function Notation Mathematical Reasoning Loci and Conics Loci Equations Solving Problems Revised 2006 Curriculum Characteristics of Functions Representing Functions Solving Problems Involving Quadratic Functions Determining Equivalent Algebraic Expressions Exponential Functions Representing Exponential Functions Connecting Graphs and Equations of Exponential Functions Solving Problems Involving Exponential Functions Discrete Functions Representing Sequences Investigating Arithmetic and Geometric Sequences and Series Solving Problems Involving Financial Applications Trigonometric Functions Determining and Applying Trigonometric Ratios Connecting Graphs and Equations of Sinusoidal Functions. Solving Problems Involving Sinusoidal Functions

Revision Highlights: 11U
Increased focus on: characteristics of functions; transformations; exponential functions; discrete functions; modelling; rate of change; radical expressions; reciprocal trig identities; periodic functions; Decreased focus on: conics and loci; annuities and mortgages; solving exponential equations; solving trig equations; complex roots; radians; tangent function; solving linear inequalities Return

Comparing Strands: Grade 11C
Revised 2006 Curriculum Mathematical Models Connecting Graphs and Equations of Quadratic Relations Connecting Graphs and Equations of Exponential Relations Solving Problems Involving Exponential Relations Personal Finance Solving Problems Involving Compound Interest Comparing Financial Services Owning/Operating A Vehicle Geometry and Trigonometry Representing Two-Dimensional Shapes and Three-Dimensional Figures Applying the Sine Law and the Cosine Law in Acute Triangle Data Management Working With One-Variable Data Applying Probability 2000 Curriculum Models of Exponential Growth Nature of Exponential Growth Mathematical Properties of Exponential Functions Manipulating Expressions Compound Interest/Annuities Arithmetic/Geometric Sequences and Series Compound Interest and Annuity Problems Effect of Compounding Personal Financial Decisions Owning/Operating A Vehicle Renting/Buying Accommodation Designing Budgets Making Informed Decisions Career Opportunities

Revision Highlights: 11C
Increased focus on: quadratic relations; modelling; exponents; two-dimensional shapes; three-dimensional figures; sine and cosine laws; one variable statistics; probability; Decreased focus on: sequences and series; annuities and mortgages; financial decision making; career opportunities; Return

Comparing Strands: Grade 11E
Revised 2006 Curriculum Earning and Purchasing Earning Describing Purchasing Power Purchasing Saving, Investing and Borrowing Comparing Financial Services Saving and Investing Borrowing Transportation and Travel Owning and Operating a Travelling by Automobile Comparing Modes of Transportation 2000 Curriculum Earning, Paying Taxes and Purchasing Earning Money Describing Forms of Taxation Purchasing Items Saving, Investing and Borrowing Calculating Simple and Compound Interest Understanding Saving and Investing Understanding Borrowing Transportation and Travel Understanding the Costs of Owning and Operating a Vehicle Understanding the Costs of Travelling by Automobile Comparing Travel Costs

Revision Highlights: 11E
Increased focus on: connections to workplace; gathering and interpreting information; Decreased focus on: personal income tax; monitoring value of investments; Return

Comparing Strands: Grade 11M
2000 Curriculum Financial Applications of Sequences and Series Arithmetic/Geometric Sequences and Series Compound Interest and Annuity Problems Financial Decision Making Trigonometric Functions Sine Law/Cosine Law for Oblique Triangles Understanding and Applying Radian Measure Graphs and Equations of Sinusoidal Functions Models of Sinusoidal Functions Tools for Operating and Communicating with Functions Polynomials/Rational Expressions and Exponential Expressions Inverses/Transformations/Function Notation Mathematical Reasoning Revised 2006 Curriculum Quadratic Functions Solving Quadratic Equations Connecting Graphs and Equations of Quadratic Functions Solving Problems Involving Quadratic Functions Exponential Functions Connecting Graphs and Equations of Exponential Functions Solving Problems Involving Exponential Functions Solving Financial Problems Involving Exponential Functions Trigonometric Functions Applying the Sine Law and the Cosine Law in Acute Triangles Connecting Graphs and Equations of Sine Functions Solving Problems Involving Sine Functions

Revision Highlights: 11M
Increased focus on: characteristics of functions; quadratic functions; exponential functions; modelling; rate of change; periodic functions; Decreased focus on: sequences and series; rational expressions; annuities and mortgages; solving exponential equations; solving trig equations; complex roots; radians; cosine/tangent functions; inverse functions; transformations; solving linear inequalities; Return

Grade 11M: Functions and Applications Connections to Other Courses
MFM2P MCF3M MBF3C MPM2D MCR3U

Concept Development: Looking at Financial Concepts

Concept Development: Looking at Functions

Revising the Expectations
some expectations were revised by: combining similar expectations folding expectations into processes reducing overlap of content among expectations removing inappropriate expectations some expectations were expanded for clarity

Eliminating Redundancy
2006 REVISED CURRICULUM Grade 11U: Functions Grade 11M: Functions and Applications Understanding Functions Exponential Functions Discrete Functions Trigonometric Functions Quadratic Functions

Grade 11E: Mathematics for Everyday Life
Improving Clarity 2000 CURRICULUM Grade 11E: Mathematics for Everyday Life 2006 REVISED CURRICULUM Grade 11E: Mathematics for Work and Everyday Life calculate compound interest by using the simple-interest formula and a given spreadsheet template; determine, through investigation using technology, the compound interest for a given investment, using repeated calculations of simple interest for no more than six compounding periods. (Sample problem: Someone deposits $5 000 at 4% interest per annum, compounded semi-annually. How much interest accumulates in 3 years? );

Real-world Connections
2006 DRAFT REVISED CURRICULUM Grade 11M: Functions and Applications solve problems arising from real-life situations, given the algebraic representation of quadratic relationship (e.g., given the equation of a quadratic function representing the height of a ball over an elapsed time, answer questions that involve finding the maximum height of the ball, the length of time needed for the ball to touch the ground, and the time interval when the ball is higher than a given measurement) (Sample problem: The relationship between power dissipated in a load resistor, P (in Watts, W), electrical potential (in Volts, V), current (in amperes, A) and resistance , R (in Ohms, Ω) is described by the formula P = EI – I2R. If the electrical potential is fixed at 24 V, and the resistance is fixed at 1.5 Ω , determine graphically and algebraically the current that results in the maximum power dissipated.) < NEW >

2006 REVISED CURRICULUM Grade 11M: Functions and Applications collect data arising from applications that can be modelled as an exponential relation, through investigation with and without technology, from primary sources using a variety of tools (e.g., concrete materials; measurement tools such as electronic probes) or from secondary sources (e.g., web sites such as Statistics Canada, E-STAT), and graph the data (Sample problem: Collect data and graph the cooling curve representing the relationship between temperature and time for hot water cooling in a porcelain mug. Predict the shape of the cooling curve when hot water cools in an insulated mug. Test your prediction.)

There was a time when some said the national debt increased exponentially. Determine if there is a domain over which the graph of the National Debt could be modelled by an exponential curve.

Grade 11E: Mathematics for Everyday Life
More Examples 2000 CURRICULUM Grade 11E: Mathematics for Everyday Life 2006 REVISED CURRICULUM Grade 11E: Mathematics for Work and Everyday Life < NEW > describe the effects of different remuneration methods (e.g., hourly rate, overtime rate, job or project rate, commission, salary, gratuities) and remuneration schedules (e.g., weekly, biweekly, semi-monthly, monthly) on decisions related to personal spending habits (e.g., the timing of a major purchase, the scheduling of mortgage payments and other bill payments.);

Key Message: Equity The revised curriculum supports equity by promoting excellence in mathematics education for all students.

Equity – NCTM Perspective
All students, regardless of their personal characteristics, backgrounds, or physical challenges, must have opportunities to study and support to learn mathematics All students need access each year they are in school to a coherent, challenging mathematics curriculum taught by competent and well-supported mathematics teachers. Too many students, especially students who are poor, not native speakers of English, disabled, female, or members of minority groups, are victims of low expectations in mathematics.

Equity – Feedback Revisions must meet the needs of the students entering mathematics-related university programs. Equal access to senior mathematics courses across the province is very important. The current curriculum is too dense resulting in a reduction of students engaging in senior mathematics and a decrease in the chance of success for some students. Access to Mathematics for College Technology from 10 Applied course was one issue raised.

What Factors Contribute Most To Students’ Success in Mathematics?
active participation in meaningful mathematics; in-depth understanding of mathematics is supported by active involvement in mathematical modelling, problem solving and reasoning through application ample time to perform investigations and to revise work; classroom practices that encourage discussion among students and between students and teachers; student reflection on their work; an appreciation of student diversity. Ed Thoughts 2002 – Research and Best Practice.

What Factors Contribute Most To Students’ Success in Mathematics?
learning experiences that involve a range of activity from short whole-group instruction to longer times engaged in problem solving positive student-teacher relationships “user-friendly” classroom environments in which prior knowledge is identified and built upon, and where instruction is developmentally appropriate Ed Thoughts 2002 – Research and Best Practice.

Equity: Developmentally Appropriate
A developmentally appropriate curriculum is challenging but attainable for most students of a given age group preparing for a given destination allows enough flexibility to respond to inevitable individual variation is consistent with the students’ ways of thinking and learning (Adapted from Clements, Sarama & DiBiase, 1997)

How do Students’ Attitudes Affect Their Performance and Future Opportunities?
Students’ attitudes toward mathematics have a great effect on student achievement. Students who enjoy mathematics tend to perform well in their mathematics course work and are more likely to enrol in the more advanced mathematics courses. Students who dislike mathematics tend not to do well in these classes, and/or do not attempt the more advanced mathematics classes in secondary school. Ed Thoughts 2002 – Research and Best Practice

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

Students with a productive attitude find sense in mathematics, perceive it as both useful and worthwhile, believe that steady effort in learning mathematics pays off view themselves as effective learners and doers of mathematics. Ed Thoughts 2002: Research and Best Practice

Students experience frustration when they are not making progress towards solving a problem. Therefore, it is important that instruction provide appropriately challenging problems so students can learn and establish the norm of perseverance for successful problem solving. Ed Thoughts 2002: Research and Best Practice

Equity Students can be considered to be “at-risk” when they are in peril of not reaching their learning potential. CMESG Work Group

Personal Reflection Reflection: Most students who take mathematics do not pursue post secondary destinations that have an emphasis on mathematics. What are the important skills you believe these students should develop through senior mathematics?

Key Message: Learning The revised curriculum supports students learning mathematics with understanding and actively building new knowledge from experience and prior knowledge. o

Developing Understanding
We use the ideas we already have (blue dots) to construct new ideas (red dot). The more ideas we use and the more connections we make, the better we understand. John Van de Walle

Conceptual Understanding
Conceptual understanding supports retention. When facts and procedures are learned in a connected way, they are easier to remember and use and can be reconstructed when forgotten. Hiebert and Wearne 1996; Bruner 1960, Katona 1940

Improving Articulation Across The Grades
Academic Pathway Applied Pathway Grade 9 Linear Relations Grade 10 Quadratic Relations Modeling Linear Relations Draft Grade 11 Understanding Functions Exponential Functions Discrete Functions Trigonometric Functions Mathematical Models Exponential Relations Proposed Grade 12 (Nov 2005) Polynomial Functions Trigonometric, Exponential and Logarithmic Functions Rates of Change Mathematical Models: Solving Exponential Equations Interpreting and Analyzing Graphical Representations Interpreting and Analyzing Algebraic Representations

Improving Articulation Across The Grades
Draft Revised Gr. 11 Foundations Proposed Draft Gr. 12 C (Nov 2005) Mathematical Models Investigating Graphs and Equations of Quadratic Relations Understanding Exponential Growth and Decay Investigating Graphs and Equations of Exponential Relations Solving Exponential Equations Interpreting and Analyzing Graphical Representations Interpreting and Analyzing Algebraic Representations Personal Finance Solving Problems Involving Compound Interest Investing and Borrowing Owning and Operating A Vehicle Understanding Annuities Renting/Buying Accommodation Designing Budgets Measurement and Trigonometry Representing Two-Dimensional Shapes and Three Dimensional Figures Applying the Sine Law and the Cosine Law in Acute Triangles Optimization Problems Solving Problems Involving Trigonometry Reasoning With Data Working with One Variable Data Applying Probability Two Variable Analysis Evaluating Validity

2006 DRAFT REVISED CURRICULUM Grade 11U: Functions and Applications
Developing Concepts Through Investigation 2000 CURRICULUM Grade 11M: Functions 2006 DRAFT REVISED CURRICULUM Grade 11U: Functions and Applications define the term function; explain the meaning of the term function, through investigation of linear and quadratic relations using a variety of representations (i.e., tables of values, mapping diagrams, graphs, functions machines) (Sample problem: give examples of linear and quadratic relations that are functions and that are not functions using a variety of representations);

Representations f(x) = 2x - 1 Numerical Representation
Graphical Representation Numerical Representation Algebraic Representation Concrete Representation f(x) = 2x - 1

Culminating With Solving Problems
2000 CURRICULUM Grade 11: Mathematics of Personal Finance 2006 DRAFT REVISED CURRICULUM Grade Grade 11: Foundations for College Mathematics < NEW > solve design problems that satisfy given constraints (e.g., design a rectangular berm that would hold all the oil that could leak from a cylindrical storage tank), using physical models (e.g., built from popsicle sticks, cardboard, duct tape) or drawings (e.g., made using design software) (Sample problem: Design and construct a model boat that can carry the most pennies, using one sheet of 8 ½” x 11” card stock and no more than five popsicle sticks)

Balancing Conceptual and Procedural Learning
Reflection: Does the balance vary depending on the students? Does the balance vary depending on the course? Is there an order? Does the balance vary depending on whether the concept is new or an extension?

Personal Reflection Balanced Activity Reflection: What does an appropriate balance mean to you and how does this impact on your students’ long term success in senior mathematics?

Key Message: Teaching The revised curriculum supports effective mathematics teaching that requires understanding what students know and need to learn and do.

Teaching Learning mathematics … requires understanding and being able to apply procedures, concepts and processes. In the twenty-first century, all students should be expected to understand and be able to apply mathematics. NCTM, Principles and Standards, 2000.

Mathematical Processes: Research
Mathematical proficiency, as we see it, has five components, or strands: procedural fluency—skill in carrying out procedures flexibly, accurately, efficiently, and appropriately conceptual understanding—comprehension of mathematical concepts, operations, and relations strategic competence—ability to formulate, represent, and solve mathematical problems adaptive reasoning—capacity for logical thought, reflection, explanation, and justification productive disposition—habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy. (Kilpatrick, Swafford, &Findell, 2001)

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

Mathematical Processes:
the Actions of Mathematics ways of acquiring and using the content, knowledge and skills of mathematics interconnected not New !!

Representing “pose and solve problems related to models of sinusoidal functions drawn from a variety of applications, and communicate the solution with clarity and justification, using appropriate mathematical forms … pp. 24, Gr 11, 1999 pose solve problems models variety of applications communicate clarity and justification mathematical forms Connecting

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 The Mathematical Processes are the actions of doing mathematics. These are presented in the Revised Mathematics Curriculum of <Name them out loud> These processes form the primary gear of the metaphor. The interconnected processes of Representing, Reflecting, Reasoning and Proving, Connecting and Selecting Tools and Computational Strategies make up disk of the gear. Communicating and Problem Solving are closely linked to each of the other 5 processes. It is essential that our students are capable of communicating their mathematics therefore it is essential that they are given opportunities to practice. Communicating is integral in each of the other processes. Problem Solving is the driver of all mathematics. The solution of rich mathematics questions requires the application of skills and understanding of concepts and incorporates some or all of the mathematical processes. <<Click to next slide>> Connecting Problem Solving

Mathematical Proficiency
Representing Reflecting Reasoning and Proving Selecting Tools and Computational Strategies Connecting Communicating Problem Solving As the gears mesh, the three components come together to build our model. Interacting with the mathematics through the mathematical processes provides students the opportunity to connect facts and procedures to conceptual understanding. For example, the volume of all right prisms, including cylinders, is found by multiplying the area of the base by the height. Students extend this conceptual understanding of the volume of a right prism to develop the procedure for finding the volume of pyramids by interacting with some of the mathematical processes. The 3 gears are clearly connected and working together to enhance students overall understanding and ability to do mathematics. <<Click to next slide>>

Key Message: Assessment and Evaluation
The revised curriculum supports assessment for the learning of important mathematics and to furnish useful information to both teachers and students.

Assessment and Evaluation
Do overall expectations have to be evaluated? YES Do all specific expectations have to be evaluated? NO Do all specific expectations have to be taught? Note to Presenter: Have participants discuss this question in pairs; allow 2 minutes for discussion. YES. The overall expectations must be evaluated. How is this accomplished? The overall expectation is evaluated by the student’s achievement of the related specific expectations. The overall expectations are not evaluated in and of themselves. Why? They are far too broad. If the student has achieved the related specific expectations, then the student has achieved that particular overall expectation.

Knowledge and Understanding
Factual/Procedural Knowledge Relationships (e.g. Pythagorean Relationship) Procedural Fluency (e.g. multi-digit computation) Meanings of terms in mathematics (e.g., property, parallelogram) Conceptual Understanding Reflecting an understanding of mathematical concepts (e.g. place value, area, rate)

Thinking Use of planning skills
understanding the problem (e.g., formulating and interpreting the problem, making conjectures) making a plan for solving the problem Use of processing skills carrying out a plan (e.g., collecting data, questioning, testing, revising, modelling, solving, inferring, forming conclusions) looking back at the solution (e.g., evaluating reasonableness, making convincing arguments, reasoning, justifying, proving, reflecting) Use of critical/creative thinking processes (e.g., problem-solving, inquiry)

Application Application of knowledge and skills in familiar contexts
Transfer of knowledge and skills to new contexts Making connections within and between various contexts (e.g., connections between concepts, representations, and forms within mathematics; connections involving use of prior knowledge and experience; connections between mathematics, other disciplines, and the real world)

Communication Expression and organization of ideas and mathematical thinking using oral, visual and written forms Communication for different audiences and purposes in oral, visual, and written forms Use of conventions, vocabulary, and terminology of the discipline in oral, visual, and written forms

Key Message: Learning Tools
The revised curriculum supports the use of technology and manipulatives as tools for teaching and learning mathematics. Technology and manipulatives are essential learning tools needed to help students see relationships that make the mathematics more meaningful

Learning Tools: Dynamic Geometry Software/Spreadsheets
2000 CURRICULUM Grade 11M: Functions 2006 REVISED CURRICULUM Grade 11M: Functions and Applications <NEW> verify, through investigation using technology (e.g., dynamic geometry software, spreadsheets) the sine law and the cosine law (e.g., compare, using dynamic geometry software, the ratios of a/sin A, b/sin B and c/sin C in triangle ABC, while dragging one of the vertices);

Learning Tools: Dynamic Statistics Software/Spreadsheets
2000 CURRICULUM Grade 11C: Personal Finance 2006 REVISED CURRICULUM Grade 11C: Foundations of Mathematics < NEW> collect one-variable data from secondary sources (e.g., internet databases) and organize and store the data using a variety of tools (e.g., spreadsheets, dynamic statistical software);

Learning Tools: Calculators and Manipulatives
2000 CURRICULUM Grade 11C: Mathematics of Personal Finance 2006 CURRICULUM Grade 11C: Foundations of Mathematics expand and simplify polynomial expressions involving the multiplying and squaring of binomials; expand and simplify, using a variety of tools (e.g., paper and pencil, algebra tiles, computer algebra systems) quadratic expressions in one-variable, involving multiplying and squaring of binomials (e.g., ½ x + 1)(3x – 2) or 5(3x – 1)2)

Learning Tools: Cooling Curve

Learning Tools: Fuel Consumption Calculator

Algebra Tiles: Completing the Square
Learning Tools: Algebra Tiles: Completing the Square a

TVM Solver: Doubling Time
Learning Tools: TVM Solver: Doubling Time

Learning Tools: Half-Life Activity

Next Steps 1

Working Toward Alignment
INTENDED CURRICULUM Ministry Curriculum Expectations DELIVERED CURRICULUM Instructional Program In The Classroom ACHIEVED CURRICULUM What Is Being Assessed We are seeing changes in the way mathematics is being delivered to students as well as improvements in the way we assess and evaluate. The revisions align the curriculum to support the changes in these areas.

Senior Mathematics Curriculum Revision

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