PHYSICS
Power Tools for
Problem Solving
(suggestions for reading)
This page describes some especially interesting
parts of the book.
Mathematics
Mathematical skill is essential for physics.
Students can quickly learn what they need to know about geometry and trigonometry
in Sections 1.1-1.2. {
Tips for Using PDF Files and a websurfing
option: If using links to PDF-files produces an extra "empty browser
window" you can avoid this in another version of this page, as explained
in "a note about websurfing" in the final
paragraph of this page. }
Section 2.3 explains how a "system of five equations" (from 2.2)
makes it easy to Choose a Useful Equation. Later, 4.12 explains
how to Choose an Equation from Chapters 2 (motion), 3 (F = ma), 4A (work-energy),
and 4B (impulse-momentum), while 5F encourages
you to rationally Cope with Equation Overload by understanding the equations
you're using.
4.7 shows
the conceptual and practical utility of a Many-Sided Equation.
The characteristics of motion graphs (point, slope,
shape, area) are in 2.10 and the connections with calculus are in 19.1-19.2.
Physics
3.2-3.4 explains
how to use the cause-effect relationship summarized in "F = ma" (*)
and, in 3.4, how to "play with a problem" in
order to fluently translate between thinking modes (verbal, visual, and mathematical)
and skillfully coordinate their concrete manifestations (in words, pictures,
and equations) while solving problems. {* To make a force
diagram, for example, "choose an object, look at a drawing of the problem-situation,
imagine you are the object and ask "What forces
do I feel pushing and pulling on me?", then draw and label these forces." }
Later, Chapter 8 illustrates
a combining of modes: 8.1 helps a student explore (and intuitively
understand) a cycle of simple harmonic motion, 8.2 explains how imaginary
circular motion can be used as a visual-mathematical model for real harmonic
motion, and 8.3 summarizes math-formulas and shows the difference between
constants,
constant-variables, and changing-variables. { This
distinction betwen variables is ignored in most textbooks. }
Problem 2-G
and Section 3.5 (plus a "lazy horse"
challenge in 3.91) are designed to
help students replace wrong ideas — things they know that just ain't
so — with correct ideas. { 2-G compares Aristotelian Intuition
and Galilean Relativity, while 3.5 shows why forces that are "equal and
opposite" may not be related by Newton's Third Law }
The similarities and differences between related
concepts are explained in Sections 3.7 (FRICTION:
kinetic versus static), 4.8 (FORCE: internal
vs external, and CONSERVATION: of momentum vs kinetic energy), and in 5A
and 5D (for MOTION: linear vs tangential
vs angular) and 5F (for a rotational analogy
of F = ma, and rotational applications of work-energy and impulse-momentum).
Two right-hand rules (for moving charge producing
magnetism in 12.1, and moving charge being affected by magnetism in 12.2)
are combined in 12.3.
2.6-2.8 show three
types of motion problems — involving two intervals, two objects, or
two dimensions — and the tools you'll need to solve them. Disciplined
step-by-step strategies are explained in Sections 3.7
(with a flowchart for friction force) and 5G
(for torque statics) and elsewhere. Strategies for circuit analysis,
showing similarities and differences between V=IR and Q=VC, are in 11.1-11.4.
| LINKS EARLIER IN PAGE | LINKS LATER IN PAGE |
| 1.1-1.2 (geometry & trigonometry) 2.2-2.3 (a "tvvax equation-system" ) 4.12 (equation choice from 4 chapters) 5F (coping with equation-overload) 4.7 (a "many-sided equation") 2.10 & 19.1 (motion graphs & calculus) 3.2-3.4 (Aesop's Problems for F = ma) Chapter 8 (shm: cycle, model, variables) 2-G (the "release principle" of Galileo) 3.5 (equal & opposite twice, lazy horse) 4.8 (force on system: internal & external) 5A & 5D (motion: linear, tangential,...) 5F (rotational F=ma, work-energy,...) 12.3 for combining the right-hand rules 2.6-2.8 (for two intervals, objects, or...) 3.7 (step-by-step flowchart for friction) 5G (a careful method for torque statics) 11.1-11.4 (circuit analysis: V=IR Q=VC) |
memory and problem
solving more mathematics: |
Memory and Problem Solving
— Review & Organization with Flashcards & Summaries
Yes, memory is useful because it "provides
raw materials... for creativity and critical thinking" and "although
memory is not sufficient for productive thinking, it is necessary,"
as explained in my web-page about Productive
Thinking.
Two key memory-improvers are review and
organization. At the end of each chapter is a flashcard review
that will help students review what they have learned, and an
overview-summary that provides logical organization. These
mega-tools will help students "put it all together" and master the
effectively coordinated use of their problem-solving tools. {
comments: The first time you try a flashcard review, there is a feeling of
"trying to guess what's in the teacher's mind" but this decreases
with further reviews, and the cumulative result of "putting tools into
working memory" will improve the quality of problem solving, especially
when memory-review is combined with problem-practice, as explained in Principles
plus Practice. }
Some ideas (especially concepts) are only in the
flashcard review, while some (including most equations) are only in the summary,
and some "central ideas" are in both.
Most chapters end with a summary, and all available
summaries (for 1 2 3 4 5 8 10 11 19) are collected in a file for Chapter
Summaries. Together, the summaries for Chapters 2-5 provide a nice
overview of motion physics, and Chapter 1 summarizes the geometry-and-trig
commonly used in physics, while Chapter
10 shows
a useful perspective on electrostatic relationships between F, E, V, and
W.
Extra Problems
Some "Aesop's Problems" are inside the
body of each chapter, but there are also end-of-chapter problems for
most chapters. For
three chapters (1-3)
these problems are in camera-ready format with text and diagrams, but
most chapters ( 4 5 6 7 8 9 10 11 14 15 16 17 )
have the text but — at least for awhile — they don't have
any diagrams. Although
some problems & solutions are
mainly for practice, to help students build good habits and confidence,
most problems teach principles that are not "essential" (so
they don't have to be in the main part of the chapter) but are still
very useful. Some "recommended" problems are marked
with •, and you may want to look
at Problems 1-1, 1-4, 2-5, 2-12, 2-14, 2-16, 2-17, 2-19, 2-21, 2-26,
3-6,
3-8,
3-13,
3-19,
3-21,
3-25,
3-33, and
3-35.
The Chapter 5 Introduction shows how creative structure can be used to meet the challenge of making a chapter "internally logical" and easy for students to integrate with the corresponding parts of their main text.
More Mathematics
When the same variable appears in two equations,
you can solve for it in one equation and substitute it into the other, thus
linking the equations with each other. Most equation derivations
and many problem solutions use this tool. A strategy of "linking
equations" is introduced in Section 2.2
and reinforced in 3.3 & 4.1,
and is used throughout the book. Ratio Logic (intuitive and algebraic)
is in 2.9.
Useful physics-math concepts are scattered throughout
the book, as in The Meaning of ± Signs in Section 3.6,
or the visual-math "symmetry logic" of Gauss's Law in 10.93-10.95. And
three whole chapters are devoted to math:
Chapter 1 teaches Math for Physics: geometry,
trigonometry, metric prefixes (two meanings), and conversion factors.
Chapter 18 covers
a variety of useful algebra tools, including How to Make an Equation (18.1), An Overall Equation-Solving Strategy (18.4), Exponents and Logarithms (18.6),
and Optimization Analysis of Conflicting Factors (18.10).
Chapter 19 begins with Motion
Graphs (by explaining Point, Slope, Shape, and Area, in 2.10 & 19.1) for
students in either non-calculus or calculus-based physics courses. The
rest of the chapter helps students develop an intuitive understanding of how
physical concepts are expressed in the "language" of calculus, beginning
with ideas from Chapter 2 (in 19.2) and continuing with goal-directed Aesop's
Problems (to accompany sections in Chapters 4, 5, and 10) to teach skills
that are essential for a calculus-based approach to physics: constucting
equations (either derivative or integral), making variables match, using a
tangent line approximation, setting up integrals using the logic of "mass-ratio"
and "density", and more.
This book takes time to explain math tools more clearly than in most physics books. And
it covers ideas that are valuable but aren't discussed at all in
most books and courses.
Principles for Learning-and-Thinking
Useful principles are in Sections 2.1
and 2.6, in Learning from Mistakes
(how I didn't learn to ski), Aesop's Problems, Principles plus Practice,
and The Most Important Strategy. { Since 1989, these ideas
have been expanded and revised in web-pages about Aesop's
Activities for Goal-Directed Education and Motivations
& Strategies for Learning. And general "learning skills,"
originally in Chapter 20, are now in Study
Skills for Effective Learning and Strategies
for Problem Solving. }
Two features of this book are:
1) The specific "power tools" that can
be learned from each problem are clearly stated, thus the name Aesop's Problems, by
analogy to Aesop's Fables that each have a specific, clearly
stated "lesson" to
be learned.
2) To help students remember these tools and incorporate
them into an effective system of problem-solving, essential
strategies are re-emphasized in later problems, gathered into a flashcard
review at
the end of the chapter and are "visually organized" in
a chapter summary that
follows the flashcard summary. Memory
and Problem Solving
The nature of a problem-solving tool varies
from one section to another. Some sections (like 2.3,...)
focus on "how to choose a formula" because this is a common student difficulty that, if it isn't overcome, it destroys a student's chance to become a competent problem solver. In
other sections (like 3.5)
the emphasis is on physical concepts.
Although an individual section may have its primary
focus on formula knowledge or physics intuition,
when the book is viewed as a whole it is well balanced, and will
help
the student
develop
both of these valuable skills. One goal is to help students
improve their ability to fluently translate ideas between different thinking
modes (verbal,
visual, and mathematical) in the concrete form of words, pictures,
and equations.
Because the book is intended to be supplementary, my
main goal is to give a student "added value" so the
time they invest in using the book
will be time well invested.
Many years of one-to-one tutoring conversation, plus
reading about physics teaching, has helped me develop a feeling
for
concepts that students usually understand (the book sails through these with
little comment
) and
concepts that are inherently difficult (these are explained in
greater detail).
Personal Inventions
Many ideas in the
book are, as far as I know, my own inventions. These include the tvvax
system (2.2), many-sided equations (4.7), friction flowchart (3.7), distinctions
between
constant-variables and changing-variables (8.3), and more. And many
other ideas — such as "imagining you're the object" (in 3.2) and most teaching
techniques (in 2.6-2.8, 3.5, 5D & 5F,
8.1-8.3, 11.1-11.4, 2.10 & 19.1,...)
— were developed by me, although probably most of these have
also been independently developed by others.
this page is http://www.asa3.org/ASA/education/teach/tools/tips.htm