Methods of Teaching Secondary Mathematics

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Methods of Teaching Secondary Mathematics

• Fall 2007
• Dr. Marshall Lassak

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Things to think about

• Q1 What is mathematics?
• Q2 What makes mathematics important?
• Q3 Why do you want to teach mathematics?
• Q4 How do you want to teach it?

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Changing ideas about how we learn
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Constructivism

• In order to construct your own understanding, you
  must be actively engaged in learning
• What students learn is connected to how they
  learn it

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Types of Knowledge

• Declarative Knowledge
• definitions, theorems
• Procedural Knowledge
• rules, procedures, algorithms
• instrumental understanding
• Conceptual Knowledge
• reasons, relationships, connections
• relational understanding

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Types of Teaching Level of Demand

• Teaching with wonderful explanations
• Teaching with problem-based tasks
• Levels of Demand
• Low-levels
• High-levels

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Have you answered questions like these before?

• Ive decided to count to one million. 1, 2, 3, 4,
  5, ,,,. How long will it take?
• If you had a million drops of water, would you be
  more likely to
• drink it?
• take a bath in it?
• swim in it?

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How about these?

• Does
• What is a logarithm?
• Which of the following is ?
• (a) 5 (b) -5 (c)
• Why?
• What is the area formula for a circle?
• Why?

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Have you considered

• What does society expect mathematics education to
  do for our students?
• Why is it okay (sometimes even funny!) to admit
  I was never very good at math?

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My Beliefs

• Mathematics must make sense to students.
• There are many ways teachers can help students
  make that sense.
• Fundamental situations, Multiple representations,
  Guided discovery, Interactive lecture,
  Appropriate use of technology
• You must have time to play with mathematics
• A wrong answer can sometimes give as much
  information as a right answer.
• Open-ended questions are typically the best type
  of question.
• Many paper and pencil methods are becoming relics
  that we continue to cling to because we have yet
  to realize the alternatives.

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The National Council of Teachers of Mathematics

• Who is the National Council of Teachers of
  Mathematics (NCTM)?
• What does the NCTM say about school mathematics?
• What does NCTM say about Secondary Mathematics
  Teachers?

NCTM Webpage
ICTM Webpage
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What Should Secondary School Mathematics Teachers
Know?

• Undergraduate mathematics
• Connections between collegiate mathematics and
  high school mathematics
• What students learn in elementary and middle
  school
• What mathematics your students will encounter
  upon leaving high school
• Mathematical maturity that enables and
  encourages continued growth in the subject

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The NCTMs Principles and Standards for School
Mathematics

• Standards
• Content
• Process

Principles
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Principles

• Equity Excellence in mathematics education
  requires equityhigh expectations and strong
  support for all students.
• Curriculum A curriculum is more than a
  collection of activities. It must be coherent,
  focused on important mathematics, and well
  articulated across the grades.
• Teaching Effective mathematics teaching requires
  understanding what students know and need to
  learn and then challenging and supporting them to
  learn it well.
• Learning Students must learn mathematics with
  understanding actively building new knowledge
  from experience and prior knowledge.
• Assessment Assessment should support the
  learning of important mathematics and furnish
  useful information to both teachers and students.
• Technology Technology is essential in teaching
  and learning mathematics it influences the
  mathematics that is taught and enhances students
  learning.

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Standards

• Content Standards
• Number and operations
• Algebra
• Geometry
• Measurement
• Data analysis and probability

• Process Standards
• Problem solving
• Reasoning and proof
• Communication
• Connections
• Representations

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The Content Standards
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Understanding about the pedagogy that the NCTM
proposes

• The Principles and Standards advocate increased
  use of contextualized problems that ground
  learning mathematics in real-world situations.
• Thus, reform-oriented teachers have students
  solve problems individually, in pairs, or in
  small groups. The students then share
  solutionsincluding representationsthat may be
  questioned and analyzed by peers (and the
  teacher).
• Such pedagogy results in a proliferation of
  representations Students create idiosyncratic
  representations, draw on conventional
  representations, and blend idiosyncratic and
  conventional representations as they solve
  contextualized problems.

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Standards-based Reform

• The bottom line is that standards-based reform
  appears to work when it is implemented as part of
  a coherent systemic effort in which curriculum,
  assessment, and professional development are
  aligned.

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Assignment

• To do by or before next class meeting
• Mathematics Education Where do I stand survey
• Read pg. 1-4 of Teaching and Learning High School
  Mathematics
• Read Grades 9-12 Reasoning Proof Standard in
  PSSM.
• Write a paragraph about something you agree with
• Write a paragraph about something you disagree
  with
• To get started on
• Your Mathematics Story
• Will be due next week on Tuesday