TEACHING AND LEARNING MATHEMATICS IN THE SECONDARY SCHOOL

Daniel BochicchioNeag School of Education

Introductions

Tell us a bit about yourselfWhy do you want to teach math?What do you want to learn about teaching

math?

Prior Learning

Describe the work you’ve done in your previous classes.

What piece of work so far are you most proud of?

Course Overview

Class Topics Course Assignments

Principles for School Mathematics

Equity Curriculum Teaching Learning Assessment Technology

Equity

High expectations and worthwhile opportunities for all.

Accommodating differences to help everyone learn mathematics.

Curriculum should

Be coherent Focus on important mathematics Well articulated across the grades.

Teaching requires

Knowing and understanding mathematics, students as learners, and pedagogical strategies.

A challenging and supportive classroom learning environment.

Continually seeking improvement

Learning must

Allow students to understand. Actively build new knowledge from

experience and prior knowledge.

Assessment should

Enhance students’ learning. Be a valuable tool for making

instructional decisions. Furnish useful information to students.

Technology

Enhances mathematics learning. Supports effective mathematics teaching. Influences what mathematics is taught.

NCTM Content Standards

Number and Operations Algebra Geometry Measurement Data Analysis and Probability

Number and Operations

Understand numbers, ways of representing numbers, relationships among numbers, and number systems.

Understand meanings of operations and how they relate to one another.

Compute fluently and make reasonable estimates.

Algebra

Understand patterns, relations, and functions.

Represent and analyze mathematical situations and structures using algebraic symbols.

Use mathematical models to represent and understand quantitative relationships.

Analyze change in various contexts.

Geometry

Analyze characteristics and properties of 2- and 3-dimensional geometric shapes and develop mathematical arguments about geometric relationships.

Specify locations and describe spatial relationships using coordinate geometry and other representational systems.

Apply transformations and use symmetry to analyze mathematical situations.

Use visualization, spatial reasoning, and geometric modeling to solve problems.

Measurment

Understand measurable attributes of objects and the units, systems, and processes of measurement.

Apply appropriate techniques, tools, and formulas to determine measurements.

Data Analysis & Probability

Select and use appropriate statistical methods to analyze data.

Develop and evaluate inferences and predictions that are based on data.

Understand and apply basic concepts of probability.

NCTM Pedagogical Standards

Problem Solving Reasoning and Proof Communication Connections Representation

Problem Solving

Build new mathematical knowledge through problem solving.

Solve problems that arise in mathematics and in other contexts.

Apply and adapt a variety of appropriate strategies to solve problems.

Monitor and reflect on the process of mathematical problem solving.

Reasoning and Proof

Recognize reasoning and proof as fundamental aspects of mathematics.

Make and investigate mathematical conjectures.

Develop and evaluate mathematical arguments and proofs.

Select and use various types of reasoning and methods of proof.

Communication

Organize and consolidate their mathematical thinking through communication.

Communicate their mathematical thinking coherently to peers, teachers, and others.

Analyze and evaluate the mathematical thinking and strategies of others.

Use the language of mathematics to express mathematical ideas precisely.

Connections

Recognize and use connections among mathematical ideas.

Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.

Recognize and apply mathematics in contexts outside of mathematics.

Representation

Create and use representations to organize, record, and communicate mathematical ideas.

Select, apply, and translate among mathematical representations to solve problems.

Use representations to model and interpret physical, social, and mathematical phenomena.

DERG A

DERG I

From

to

Curriculum should

Be coherent Focus on important mathematics Well articulated across the grades.

Guiding Principles

Design authentic learning experiences that integrate skills and knowledge.

Use different types of instruction to teach skills and knowledge.

Develop fluency through variation within lessons.

Organize instruction into patterns for maximum effectiveness.

Teach a range of skills and types of knowledge.

Authentic Learning Experiences

Embed skills and knowledge instruction within context of an authentic, purposeful assignment.

Have students identify and solve problems they encounter in the context of their work by teaching them the steps in the problem solving process.

Organize the learning experience around essential questions that drive inquiry.

Ask yourself, “What do I want students to know?”. Use context-rich activities such as simulations,

case studies, performances, investigations, projects, and productions.

Develop Fluency

Add complexity as students begin to show initial mastery.

Vary the means and materials students use.

Vary the duration of the activity or assignment.

Support your students in a variety of ways.

Skills and Types of Knowledge

Ask yourself what skills, of any type, are necessary to do what you ask.

Model and incorporate different types of skills for students.

Monitor the skills and knowledge you have taught.

Post certain declarative and procedural knowledge.

Move beyond basic knowledge into advanced critical thinking.

Curriculum Map

Curriculum Units

Standards Content Essential Understanding/Questions Performance Outcomes Instructional Tools/Methods Assessments

Essential Questions

Cause genuine and relevant inquiry into the big ideas and core content.

Provoke deep thought, lively discussion, sustained inquiry, new understanding, and more questions.

Create opportunities for transfer to other situations and subjects.

Require students to consider alternatives, weigh evidence, support their ideas, and justify their answers.

Essential Questions Examples

How is trigonometry used to solve real world problems in engineering and science?

How does knowing the basic trigonometric ratios simplify finding distances and angles?

What project have you or your family done, or plan to do, that might involve trigonometry?

Performance OutcomesAngles And Degree Measurement   Determine the length of the sides of a right-triangle using the Pythagorean Theorem. Measure and identify positive and negative angles using a protractor. Indicate the number of degrees in an angle formed by rotating the terminal side. Convert an angle measurement in a decimal degree to degrees-minutes-seconds and vice versa. Similar Triangles Determine if two triangles are similar. Find the lengths of unknown sides in similar triangles. Use the relationships in a 45-45-90 triangle to find the lengths of unknown sides. Use the relationships in a 30-60-90 triangle to find the lengths of unknown sides.  

Trigonometric Ratios Calculate the value of trigonometric functions for a given triangle. Evaluate trigonometric functions using a calculator. Solve a right triangle using trigonometric ratios. Evaluate inverse trigonometric functions using a calculator.  

Right-Triangle Applications Use trigonometric ratios to find the angle of elevation and the angle of depression. Use trigonometric ratios to find inaccessible distances.  

Angles And Arc Length Draw an angle in standard position. Find an angle coterminal to a given angle. Find the reference angle of a given angle. Find the length of an intercepted arc when given the central angle and the radius. Find the value of a central angle when given an arc length and a radius.

Instructional Tools/MethodsAngles And Degree Measurement   Class discussion Presentation  Paired activity - Pythagoras' Pool  Paired activity - What Goes Around? Dance - The Trig Angle Dance Activity - Going Back And Forth  Similar Triangles Class discussion Presentation Project - Height Of An Object Using Mirrors And Shadows  Trigonometric Ratios Class discussion Presentation  Activity - The Tangent Ratio Activity - Creating The Trig Table Paired activity - Building A Ramp To OSHA Standards  Right-Triangle Applications Class discussion Presentation  Activity- Coming In For A Landing Paired activity - Finding The Impossible  Angles And Arc Length Class discussion Presentation  Activity- What's Your Position? Activity - Check Your References Paired activity - Measuring The Globe Cumulative project -Using A Clinometer To Measure Heights

Assessments

Written results Informal observations  Activity results Written activity results Oral questioning Completed worksheet Student responses Written responses Completed lab worksheet Project rubric Test on right-triangle trigonometry

Preview Your Curriculum Unit

Select a unit of study

What are your essential questions? What content will you teach? What are your performance outcomes for the

unit?