Power Tools for Problem Solving in Physics
This page describes some especially interesting
parts of my book, Power Tools for Problem Solving in Physics
(maybe it was the best of books, and the worst?) and helps you discover features that include Aesop's Problems (each designed to illustrate specific principles) and — to help students master strategies for solving problems — repetition of ideas in different contexts (as in a spiral curriculum), and flashcard reviews that help students store tool-ideas (in their long-term memory) so the ideas will be available (in their “working memory”) when the ideas are needed for solving problems, and one of my favorite parts, the chapter summaries that provide “big picture overviews” with logically meaningful visual organization. { an update-comment in 2019: I'm becoming more excited about the possibilities for using, in physics education, Strategies-and-Tools to improve Memory — by Reviewing & Organizing with Flashcards & Summaries — to improve Problem Solving. }
A Suggestion for Evaluation: As explained in the homepage, while writing this book in the late-1980s my goal was "just to explain ideas-and-skills in ways that are logical and clear. Since then, I've learned much more about thinking, learning, and teaching (while earning a Ph.D. in Curriculum & Instruction, and afterward), and now I've become more aware of the ways that effective teaching requires more than just clear explanations. ... So I'll ask you to evaluate my book based on what [in the 1980s] it was intended to achieve, by simply asking whether its explanations are logical and clear." But then and now, most educators agree that many benefits are offered by eclectic instruction — for example, by combining explanation-based instruction (as in my book) with interactive computer games or simulations like PhET (FAQ - research) or EpiGame or EGame (HomePage) — to help students achieve multiple educational goals for improving their ideas-and-skills.
This mini-section is about links-navigation. You can skip it if you want, and move on to Quantitative...Understanding below.
Here is an option: If you want links to open in a separate new window, so this page remains open in this window, it will happen in another version of this page.
I made "named destinations" within each PDF file, so clicking a link usually takes you directly to the correct location in the file. For example, the first link below (for "2.2") goes to Section 2.2. / link-problems with Macs: These links work properly in Windows, and also with inside-the-browser viewing on Mac's Safari or Chrome or Firefox, but not with Mac's Firefox when (in "Applications") it's set to open PDF files in Adobe Reader or Preview.
Quantitative (mathematical) Understanding
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. (also, check the logically organized Chapter Summaries)
Section 2.3 explains how a "system of five equations" (from 2.2)
makes it easy to Choose a Useful Equation. When students realize how easily they can choose an equation (which can be difficult without the 5-equation system) they will feel free to focus on the more important qualitative understanding in Step 1 of the equation-choosing strategy, when they "read carefully, think, draw pictures; do whatever is needed to form a clear idea of the problem situtation." 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 (in ways that are both qualitative & quantitave) the equations
you're using.
4.7 shows the conceptual
and practical utility of a Many-Sided Equation by explaining that "each of the 8 boxes is equal to every other box" so "you can equate any two of these boxes to make an equation that fits the needs of a particular problem." / a comment in 2011: Oops! Because the term "Many-Sided Equation" might encourage a bad habit, such as writing "F = ma = m(-9.8 m/s2)" if "a" is 9.8 m/s2 downward and "up" is defined to be +, instead I should have called it a "A Multi-Option System of Equations" or something similar. To discourage the bad habit of writing 3-sided equations, I emphasize (in Section 18.2) an important strategy for algebra — "USE VERTICAL SUBSTITUTION. Don't substitute horizontally; by definition, an equation has 2 (and only 2) sides" — and in this case vertical substitution is done by writing "F = ma" and then below it "F = m(-9.8 m/s2)" to keep the equation 2-sided. I emphasize the importance of vertical substitution in 3.2, and explain another reason for it, but I should have included both reasons in 3.2.
The characteristics of motion graphs (point, slope,
shape, area) are in 2.10 and their connections
with calculus are in 19.1-19.2.
Qualitative (non-mathematical) Understanding
It's also important to construct a conceptual understanding that is is accurate and extensive. Helping students improve their qualitative understanding, and connecting this with quantitative understanding, is a frequent goal throughout the book.
For example, the section above describes a 5-equation system which shows that (because “choosing an equation” is easy after you understand) in physics the key to problem solving is qualitative understanding, along with translations of qualitative-into-quantitative. This theme is continued later, with similar strategies for choosing equations in other areas, including physical situations involving work-energy and impulse-momentum relationships.
The section above ends with graphs-and-calculus connections, which illustrates my goal of helping students see “the big picture” and how different parts of it fit together to form the whole. Relationships between ideas are explained in every chapter, and are shown (verbally-visually-mathematically) in the chapter summaries.
The section below describes playing with a problem, using several thinking modes and translating between them, and imagining you're an object being pushed-and-pulled by forces. Some parts of the book explain how to avoid common misconceptions (about Newton's Third Law, and in other areas) and "the similarities and differences between related
concepts." There is a flowchart to explain the process of finding a friction force, a creative way to think about simple harmonic motion, and much more, continuing throughout the book.
Physics Thinking
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." / comment in 2011: This visualizing is easier for contact forces; but students must also develop the skill of imagining non-contact forces, such as gravitational, electrical, and magnetic forces. Therefore, I should have emphasized this contact/non-contact distinction in the book, as I did with my in-person teaching, which also included "hand waving" visual gestures to show what I (as an object) was feeling, to illustrate the process-of-imagining and make it more dramatic.
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. }
Misconceptions: 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 } / Here is a comment in 2005 while writing this page: I learned much more about "misconceptions research" after writing this book, so although it clearly explains a useful way to think about many "tough concepts" it doesn't do this for all of the common misconceptions; and it doesn't explicitly use strategies of Teaching for Conceptual Change, such as those of Posner, et al (1982) who suggest first producing dissatisfaction with an alternative preconception, before showing that the corresponding scientific concept is intelligible, plausible, and fruitful.
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:
|
Improving Memory
— by Reviewing & Organizing with Flashcards & Summaries — for Problem Solving
Yes, memory is extremely useful because, as explained in my web-page about Productive
Thinking, memory "provides
raw materials... for creativity and critical thinking" and "although
memory is not sufficient for productive thinking, it is necessary." {more about productive thinking}
Flashcard Reviews & Chapter Summaries: Two key memory-improvers are review &
organization,* and both principles are used in this book: 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
two reviewing-and-organizing tools, when used in the context of personal experience with solving problems, will help students “put it all together” and master the
effectively coordinated use of their problem-solving tools. The cumulative result of Principles plus Practice — of building a working memory containing useful principles, and working with these principles during practice in problem solving — of combining memory-building with problem-practicing — is improving the quality of problem solving. { one file has all of the Chapter
Summaries for 1 2 3 4 5, 8, 10, 11, 19; these also are near the end of their chapters } { there is a FlashCard Review near the end of most chapters - in 1 2 3 4 5 6 7 8 9 13 14 15 16 17 19 - but not 10,11,12,18,20 } { more about the flashcards & summaries }
* In scientific studies of techniques-for-learning the two best were practice testing and distributed practice. Flash Cards let you do both. How? The author explains that "While practice testing is not a common strategy – despite the robust evidence supporting it – there is one familiar approach that captures its benefits: using flash cards." Yes, flash cards are practice testing, and are a quick-and-easy way to do distributed practice.
{also based on research, The Educational Value of Organization}
Here are practical tips for using these tools:
using Flash Cards: The first time you try my flashcard review for a chapter – they're near the end of most chapters – you'll have a feeling like
"trying to guess what's in the teacher's mind" to fill the blanks.* But after you think about the logic of WHY each blank is filled the way it is — and this thinking is a great way to develop deep understanding! — your later reviews will be easier and more effective for helping you understand-and-remember. Or you can use my flash cards as a starting point to make your own personally customized flash cards. / * maybe sometime I'll revise the FlashCard Reviews, so they will require less "guessing" because the cues are described in better ways; or you (especially if you're a teacher, but also as a student) can do this for yourself.
using a Chapter Summary: When you invest time in a deep study of the visually organized logic in chapter summaries, you will be rewarded with improved understanding of the concepts (the problem-solving tools) and – with "deep study" when you think about the visual organization – their inter-relationships. You will improve your understanding of the concepts (the principles of physics) and skills (in using the concepts to solve problems).
combine Memory Improving with Problem-Solving Practice: To make both tools more effective, combine your memory-improving strategies (using flashcards & summaries) with problem-solving practice. Why? To be an effective problem solver, you need to combine problem-solving knowledge (by remembering concepts and tools) with the problem-solving skills (so you're able to use your knowledge) that will be developed-and-improved by practice.
Some ideas (especially concepts) are only in a chapter's
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 an
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.
Also, elsewhere in this page: comments about summaries for geometry and trigonometry & graphs-and-calculus.
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)
all problems are in camera-ready format with text and diagrams. Unfortunately, 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. (I.O.U. - I may scan-and-post these missing diagrams in the future, maybe in late-2022.) 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,
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 (in a miniature spiral curriculum), gathered into a flashcard
review at
the end of the chapter and are "visually organized" in
a chapter summary that
follows the flashcard summary. {summaries for Chapters 1, 2, 3, 4A-4B, 5, 8, 10, 19} Memory
and Problem Solving
The nature of problem-solving tools 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.
SUMMARIES for Chapters (1 and 2-3-4-5 and 8, 10, 19)
Math (1, end of 2, 19)
Motion (2 3 4 5)
special (SHM in 8, Electrostatics in 10, [RC in 11?])
Some ideas (especially concepts) are only in a chapter's
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 an
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.
Also, elsewhere in this page: comments about summaries for geometry and trigonometry & graphs-and-calculus.
this page is http://www.asa3.org/ASA/education/teach/tools/tips.htm