Functions Based Curriculum Math Camp 2008

Trish Byers Anthony Azzopardi

Try to establish a one-to-one correspondence with at least five people in the group. Does this relationship represent a function? Functions Icebreaker

Learning Needs Survey

FOCUS: FUNCTIONS BASED CURRICULUM DAY ONE: CONCEPTUAL UNDERSTANDING DAY TWO: FACTS AND PROCEDURES DAY THREE: MATHEMATICAL PROCESSES Why focus on functionsWhy focus on functions?

What is Mathematics?

The study of relationships The science of patterns The science of patterns

What is Mathematics? Mathematics is the study of patterns, abstracted from the world around us – so anything we learn in mathematics has literally thousands of applications, in arts, sciences, finance, health and leisure. Professor Ruth Lawrence University of Michigan

Mathematics makes the invisible visible. Keith Devlin

Why do we learn mathematics?

Why Do We Learn It? Mathematics helps us solve problems.

Anticipation Guide Purpose Help students to activate their prior knowledge and experience Encourage students to make personal connections with a topic so they can integrate new knowledge with their prior knowledge Payoff Students will: engage with topics, themes and issues at their current level of understanding become familiar and comfortable with a topic before reading unfamiliar text or investigating a concept

Anticipation Guide Statement Strongly Agree Grades 7 and 8 teachers Grades 9 and 10 teachers All students do not learn in the same way. 86%77% Students learn mathematics most effectively when given opportunities to investigate ideas and concepts through problem solving. 41%17% Current technologies bring many benefits to the learning and doing of mathematics. 38%24% Certain aspects of mathematics need to be explicitly taught. 60%59% Problem solving helps students develop mathematics understanding and gives meaning to concepts and skills in all strands. 55%41% Through discussion students are able to reflect upon and clarify ideas and relationships between mathematics ideas. 46%33% Manipulatives are necessary tools to support effective learning of mathematics for all students. 48%16% Collaborative learning activities enhance student learning of mathematics. 47%20%

A Vision of Mathematics Mathematics is something a person does. Mathematics has broad content encompassing many fields. Mathematical power can, and must, be at the command of all students in a technological society. NCTM 1989

Revised Secondary Curriculum Grade 12 U Calculus and Vectors MCV4U Grade 12 U Advanced Functions MHF4U Grade 12 U Mathematics of Data Management MDM4U Grade 12 C Mathematics for College Technology MCT4C Grade 12 C Foundations for College Mathematics MAP4C Grade 12 Mathematics for Work and Everyday Life MEL4E Grade 11 U Functions MCR3U Grade 11 U/C Functions and Applications MCF3M Grade 11 C Foundations for College Mathematics MBF3C Grade 10 LDCC Grade 9 Foundations Applied MFM1P Grade 11 Mathematics for Work and Everyday Life MEL3E Grade 9 LDCC Grade 10 Principles Academic MPM2D Grade 10 Foundations Applied MFM2P Grade 9 Principles Academic MPM1D T

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

Learning mathematics involves the meaningful acquisition of concepts, skills and processes and the active involvement of students building new knowledge from prior knowledge and experience; Underlying Principle for Revision

Any fool can know. The point is to understand. Conceptual Understanding

Conceptual understanding refers to an integrated and functional grasp of mathematics. It is more than knowing isolated facts and procedures.

Knowledge that has been learned with understanding provides the basis for generating new knowledge and for solving new and unfamiliar problems. Bransford, Brown and Cocking 1998 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 Conceptual Understanding

Activate Your Memory Try This!

A

B

C

D

E

F

G

H

I

Write DIG

Write HAD

Write AGE

How Did You Do?

Making a Connection! ABC DEF GHI

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. Developing Understanding John Van de Walle

Understanding Requires Making Connections: RATIO John Van de Walle

A Focus On Concepts If you could only memorize one area formula, which one would it be?

Parallelograms Do these 3 pieces make a parallelogram?

Area of a Parallelogram height base Area = base x height

Area of a Triangle height base A = base x height 2

Area of a Trapezoid a b + ba+ Area = (a + b)h height

Procedural Fluency Conceptual Understanding “ Learning with understanding is essential to enable students to solve the kinds of problems they will inevitably face in the future”

“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

Teaching: Through Investigation Discovery Learning Again? Learning through Investigation

Painted Cube Problem A 3 x 3 x 3 cube made up of small cubes is dipped into a bucket of red paint and removed. (a) How many small cubes will have 3 faces painted? (b) How many small cubes will have 2 faces painted? (c) How many small cubes will have 1 face painted? (d) How many small cubes will have 0 faces painted?

What if a 10 x 10 x 10 cube were dipped? What additional strategies would help to develop the 10 X 10 X 10 solution? Can you generalize to an n x n x n cube? Explain your thinking. Painted Cube Problem...

2 Faces Painted N 2 = 12(n – 2) Painted Cube Problem: Using Finite Differences

1 Face Painted N 1 = 6(n – 2) 2 Painted Cube Problem: Using Finite Differences

0 Faces Painted N 0 = (n – 2) 3 Painted Cube Problem: Using Finite Differences

Graphically, using Fathom... Painted Cube Problem: Graphically

Graphically, using Excel... Painted Cube Problem: Graphically

Geometrically, using cubes and patterns... 2 faces painted1 face painted Painted Cube Problem: Graphically 3 faces painted N 2 = 12(n – 2)N 1 = 6(n – 2) 2 N 3 = 8 8 Corners 12 Edges 6 Faces (n – 2) (n – 2)X(n – 2) “square”

Learning Tools

Investigating Functions: Pendulum Activity Dice Activity Geoboard Activity Ball Bounce Activity

Give me a fish and you feed me for a day. Teach me to fish and you feed me for life. Chinese Proverb A Vision of Teaching Mathematics

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