Teaching and Learning High School Mathematics
November 2009, ©2010
Ball and Bass (2000), as well as other mathematics educators, have recognized that mathematics teachers not only need to know mathematics content and mathematics pedagogy (i.e., teaching strategies) but they also need to know how these ideas are integrated. This mathematical knowledge for teaching is the knowledge that teachers of mathematics need and it differs from the knowledge that research or applied mathematicians must know. This text is designed to provide teachers with insights into this mathematical knowledge for teaching.
Teaching and Learning High School Mathematics is likely different from many other texts that you have used. It integrates both content and pedagogy to help you develop and build your own understanding of teaching. The text is designed to help you develop “deep conceptual understanding of fundamental mathematics” (Ma 1999) so that you are able to approach mathematics from multiple perspectives with many tools. Such flexibility in teaching is essential if teachers are to help all students become mathematically proficient.
Throughout this book, you are encouraged to work in cooperative teams. This strategy is designed to help you develop a mathematics learning community and build a professional network that will be a valuable resource during your professional career. Hopefully, you will experience the benefits of engaging in rich mathematical discussions with peers and consider how to encourage such learning environments in your own classrooms.
Lesson planning is another element pervasive throughout this text. To help teachers plan for effective student-centered lessons, the Question Response Support (QRS) Guide is introduced in Lesson 1.1 and used throughout the remainder of the lessons. The QRS Guide is a tool on which teachers may record tasks or questions (Q) for students, expected and observed student responses (R), and teacher support (S) in the form of additional “just enough” questions to support students in their progress on the task. In each unit, teachers expand their repertoire of teaching and learning elements and strategies and incorporate these elements as they plan additional lesson segments. In Unit 4 lesson planning is formally introduced as teachers put together elements from previous units into complete, cohesive lesson plans.
Mathematics Education: Where Do I Stand?
Unit One. Encouraging Communication in Mathematics Classrooms
(Mathematics Strand: Logic and Reasoning)
Unit One Team-Builder: Carpet Square Maze
Preparing to Observe Mathematics Classrooms: Focus on Equity
Listening to Students Reason About Mathematics
1.1 Developing Questioning Strategies: Conjecturing and Reasoning
1.2 Exploring Mathematical Concepts Cooperatively:
1.3 Using Representations to Investigate
Mathematics: Reasoning with
Conjunctions, Disjunctions, and Negations
1.4 Learning from Students: Valid and Invalid Arguments
1.5 Summarizing Classroom Observations and
Listening to Students:
Focus on Equity
Synthesizing Unit One
Unit One Investigation: Carpet Square Mazes
Unit Two. High School Students and How They Learn
(Mathematics Strands: Geometry and Measurement)
Unit Two Team-Builder: Transformed Snowflakes
Preparing to Observe Mathematics Classrooms: Focus on Learning
Listening to Students Reason About Geometry
Understanding Geometry Learning: Coordinate Geometry
2.2 Building Conceptual Understanding: Congruence and Similarity
2.3 Learning Mathematics through Multiple Perspectives: Quadrilaterals and Constructions
2.4 Using Physical Tools and Technology: Circles
2.5 Tasks with High Cognitive Demand: Measurement in the Plane and in Space
2.6 Doing Mathematics: Axiomatic Systems
2.7 Summarizing Classroom Observations and Listening to Students
Synthesizing Unit Two
Unit Two Investigation: Transformations
Unit Three. Planning for Instruction
(Mathematics Strands: Algebra and Functions)
Unit Three Team-Builder: Find Your Function Family
Preparing to Observe Mathematics Classrooms: Focus on Curriculum
Listening to Students Reason about Functions
3.1 Building on Students' Knowledge and
Variables and Linear Functions
3.2 Thinking about Learning Outcomes: Exponential Functions
3.3 Active Learning: Modeling Data Through Experiments
3.4 Teaching with Technology: Geometry of Functions
3.5 Increasing Challenge or Accessibility of Problems: Polynomial Functions
3.6 Accommodating Different Learning Styles: Rational Functions
3.7 Summarizing Observations and Listening to Students
Synthesizing Unit Three
Unit Three Investigation: Families of Functions
Unit Four. Lesson Planning
(Mathematics Strands: Data Analysis and Probability)
Unit Four Team-Builder: A Dream Team in Hockey
Preparing to Observe Mathematics Classrooms: Focus on Teaching
Listening to Students Reason about Data Analysis and Probability
4.1 Planning a Lesson Launch and Explore: Data Analysis
4.2 Planning a Lesson Share and Summarize: Probability
4.3 Blending Direct Instruction into a Lesson: Variability and Distributions
4.4 Planning for Alternative Schedules: Statistical Decision Making
4.5 Summarizing Observations and Listening to Students
Synthesizing Unit Four
Unit Four Investigation: Build Your Own Dream Team
Unit Five. Assessment of Students' Learning
(Mathematics Strand: Precalculus)
Unit Five Team-Builder:
Preparing to Observe Mathematics Classrooms:
Focus on Assessment
Listening to Students Reason about Precalculus
5.1 Daily Assessments: Limits
5.2 Rubrics: Rates of Change
5.3 Designing and Aligning Tests with Instruction
5.4 Alternative Assessments: Accumulations
5.5 Summarizing Observations and Listening to Students
Synthesizing Unit Five
Unit Five Investigation: Conic Sections
Unit Six. Collaborating with Educational Partners
Listening to Educational Partners about Issues in Mathematics Education
6.1 Evaluating Curriculum Materials
6.2 Coordinating Curricula Beyond the Classroom
6.3 Continued Professional Development
6.4 Summarizing Interviews on Educational Issues
Synthesizing Unit Six
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