Power Tools for Problem Solving in Physics

by Craig Rusbult, Ph.D.

This page describes some especially interesting parts of the book, Power Tools for Problem Solving in Physics (it was the best of books, and the worst) and helps you discover features that include Aesop's Problems (each designed to illustrate principles) and — to help students master strategies for solving problems — repetition of ideas in different contexts (as in a spiral curriculum), flashcard reviews that help students store ideas in long-term memory so these ideas will be available when they are needed for solving problems, and one of my favorite parts, the chapter summaries that provide "big picture overviews" with logically meaningful visual organization. 
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... 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 the basics remain similar.  Then and now, 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 navigation.  You can skip it if you want, and move on to Quantitative...Understanding.
    Here is an option:  If you want all links to open in this window, so only one window is open and you can navigate using your browser's back-button, it will happen in another version of this page.
    Also:  I made "named destinations" within each PDF file, and used these for most links in this page, so clicking a link should take you directly to the appropriate location within the file.  For example, the first link below (for "2.2") goes to Section 2.2.  If instead the link takes you to the beginning of Chapter 2 (this happens with Mac's Preview, unfortunately) you'll have to manually scroll down to the appropriate place.  Sorry.  This inconvenience is a result of Apple's failure to make their program (Preview) do what it should.

        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.
        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 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.
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
      with flashcards and summaries
extra problems (for Chapters 1-3,...)
Chapter 5: Introduction & Summary

more mathematics:
2.2 (linking equations, plus 3.3 & 4.1
2.9 (ratio logic, by intuition & math)
3.6 (the meaning of + and signs)
10.93-10.95 (visual-math symmetry)
Chapter 1 (geometry, trig, prefixes)
Chapter 18 (algebra for physics)
Chapter 19 (calculus for physics),
    2.10 & 19.1 (graphs: points,...) 

2.1 & 2.6 (principles for thinking)

        Memory and Problem Solving — Review & Organization with Flashcards & Summaries
        Yes, memory is extremely 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,* 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.   /   In scientific studies of learning techniques the two best were practice testing (and "there is one familiar approach that captures its benefits: using flash cards") plus distributed practice (and flash cards are an excellent way to do this).  {also, The Educational Value of Organization}
        Here are practical tips for using these two tools:  1) The first time you try the flashcard review for a chapter, you'll have a feeling of "trying to guess what's in the teacher's mind" to fill the blanks;  but if you think about the logic of WHY each blank is filled the way it is (this is a great way to learn!) 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.   2) 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 and their inter-relationships, with better ideas (about the physics) and skills (in using the ideas to solve problems).
        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 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. (but I may scan-and-post these missing diagrams during the summer of 2011)   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 (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.

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