Quantitative Problem Solving in Applied Science, Natural Sciences Mathematics and Commerce: “Hands on, Heads up” Learning Quantitative Problem Solving What is “Problem Solving”? This is a form of learning based on discovery: to solve the problem, you must both think and compute systematically. It is different from both “exercise solving”, in which past routines are applied to solve similar problems, and a “trial and error” approach is used to match correct formula for the problem. A central idea in problem solving is the use of “concepts”, which are the fundamental general ideas on which other notions may be built. In any subject, there are usually only a few basic concepts (sometimes expressed as formula), which are applied in a variety of ways or situations. For example, basic concepts include limit of function in math, and t- test in statistics, Newton’s 2nd Law in physics, mole in chemistry, and liability in accounting. Identifying and deeply understanding key concepts, and developing an organizational structure to allow you to recall how they relate to each other are essential elements in expert problem solving The “spiral of learning” occurs when basic concepts are used repeatedly to solve a variety of problems. The central concept is the core of the spiral, and various applications spin out from, and loop back to, that concept. Frequently re-visiting those basic concepts allows you to firmly fix them in your long-term memory, where they can be quickly recalled and applied. People learn in different ways, and have different preferred styles of relating to their world, seeking sensory input, making information meaningful, and patterns of learning. It is very helpful to understand your own preferred learning style, and use methods that both mesh with and challenge your style. See the free “Index of Learning Styles” by Felder and Silverman. www.ncsu.edu/felder-public/ILSpage.html and refer to the “Working with Your Preferred Learning Style” resource on the Learning Strategies web site www.queensu.ca/learningstrategies Self-Reflection Questions Do you: 1. understand your own approach: strengths and weaknesses? 2. focus on concepts to increase understanding, and as an organizational framework? 3. learn material sequentially? 4. look for the “spiral of learning”: repetition and expansion of basic concepts? 5. develop a systematic, methodical approach, to talk yourself through each step? 6. compute accurately, and eventually… quickly 7. persist? 8. get help when needed? What is YOUR Approach to Quantitative Problem Solving? Awareness of your own attitudes and habits is a good starting point to see your strengths and areas to change. Click on the “Evidence Based Components” questionnaire to assess your approach. Characteristics of Expert Problem Solvers 1. Attitude Characteristics * Optimistic: you believe “I can do it” * Confident: the problem really does have a reasonable, but perhaps difficult, solution * Willing to persevere: you aim for a complete and well reasoned solution, not an immediate or superficial one * Concern for accuracy in reading: you concentrate, re-read and paraphrase to increase understanding, and translate unfamiliar words or terms * Concern for accuracy in thinking: you work at a moderate to slow pace initially, perform operations carefully, check answers periodically, and draw conclusions at the end not part way through. 2. Skill Characteristics * Systematic approach: you have a plan to follow, which i. reduces the panic ii. allows you to monitor your thought processes iii. helps isolate errors in logic or computation * Sound knowledge of basic concepts, which you mentally organize so you can recall and apply them * Computational skill, at a good speed * Habit of vocalizing or “thinking aloud”: you talk yourself through all thoughts i. how to start the problem ii. steps to break problems into parts iii. decisions iv. analyses v. conclusions * Awareness of your own thought processes: What did I do or learn? How did I do or learn this? How effective was my process? Typical Characteristics of Novice Problem Solvers 1. You don’t believe that persistent analysis is essential, therefore your effort and motivation to persist is weak. 2. You are careless in their reasoning. 3. You don’t break problem into component parts and go step-by-step, therefore there are errors in logic and computation. 4. You focus on individual details, and don’t see how details relate to concepts. Therefore, every problem feels new…how overwhelming! 5. Formula-memorizing is the main strategy. 6. You get behind in your learning, and then sequential learning is hampered. 7. You lose confidence in your ability to solve problems, due to lack of success. Strategies to Improve Problem Solving Skills 1. Use Time and Resources Effectively * Work on courses regularly: keep up so you can build on past knowledge (sequential learning), and get help quickly for difficulties. * Do all the questions assigned, rather than dividing questions among group members, as you will get more practice with the concepts your Professor expects you to know. Aim for accuracy, then speed. Start assignments at least a week ahead of the due date, so you have time for help if needed. * Use study groups to compare completed solutions to assigned problems. Teaching someone is a very effective learning and study technique. * Choose problems wisely: learn to apply a specific concept to solve a variety of related problems. Start with simpler ones, and work up. Identify the relevant concept and practice until you know when and how to apply it, i.e. you may not need to do all questions. * Set a time limit: attempt a new problem every @ 15-20 minutes. If you can’t complete a problem, check your “thinking strategies” and change to a new problem. Get help with the problems you couldn’t complete, at tutorial, etc. * Do some uncalculated solutions: If you are confident in your calculations-set up the solution but don’t finish the calculation. * Learn the necessary background and skills: find out from professor, course outline, etc. what the course involves and upgrade before the course begins if you don’t feel confident about the prerequisites. * Find and use help resources: use tutors, professors, TAs, friends, text, internet. For example: in accounting, economics, and finance texts, it is common to find examples that are quite similar to the problems at the end of the chapter. Work through the logic of the examples to develop a better understanding of how best to start the homework problems, if you run into trouble. 2. Develop Strategies to Organize Your Thinking * Quantitative Concept Summary Strategy Concepts are general organizing ideas, are there are often very few of them taught in a course, along with their many applications (ie. the spiral of learning). Key concepts may be identified by: * reading the learning objectives on the course outline or the course description, * referring to the lecture outline to identify recurring themes, * thinking about the common aspects of problems you are solving. Learn and understand the small amount of information essential to each concept. If in doubt, ask the professor what is important for you to “get”. For more information, click here for the Quantitative Concept Summary Strategy description, Concept Summary form, and an example of a Concept Summary for Ordinary simple Annuities. View the video at (click Online Resources, scroll to “Math”, select desired topic and format) * General Problem Solving Method Use a methodical, thorough approach to solve problems logically from first principles. Refer to the self-assessment questionnaire by Woods et al. (2000) in this guide to remind yourself of target activities you need to focus on. Steps involve: * Engage with the problem * Define and understand the problem- what is being asked? Express your thinking in several ways, such as verbally, graphically or pictorially, and finally mathematically * Explore links between the current problem and related ones you have previously solved. * Plan how you will solve the problem * Do it ? * Evaluate your method and result, and revise as needed Click here for more information on the General Problem Solving Strategy. * Decision Steps Strategy This strategy is a specific application of the General Problem Solving Strategy described above, and is suitable for use in statistics, accounting and other applied problem solving situations. During the lecture or when reading course notes, focus on the process of solving the problem, instead of on the computation. When your professor is lecturing, listen to their comments on how steps are inked from one to another. This helps you identify the “decision steps” that lead to correct application of a concept. Ask yourself “Why did I move from this step to this step?” Click here for more information on the Decisions Steps Strategy, and examples of Decision Steps in Calculus and Decision Steps for Rational Expressions. View the video at (click Online Resources, scroll to “Math”, select topic and format) * Problem Solving Homework Strategy Use homework as a learning tool. Effective learning of the concepts and general methods will reduce the number of problems you may need to solve to feel confident in your knowledge and computations. Click here for details on the Problem Solving Homework Strategy. * Range of Problems Strategy Exams will challenge you to apply your knowledge to new situations, so prepare by creating questions or problems that are slightly different in some variable from your homework problems. Actively think about the range of problems that are associated with a concept. Think in terms of both i. level of difficulty of the problems and ii. common kinds of difficult problems. Use this to anticipate different kinds of difficult problems for exam preparation, and solve some practice problems to test yourself. This is an excellent activity for a study group. Click here for common examples of difficult problems using the Range of Problems Strategy. View the video at (click Online Resources, scroll to “Math”, select topic and format) . Some Evidence-Based Components of Expert Problem-Solving1a Observe yourself as you solve problems, and rate how frequently you DO any of the following. Progress toward internalizing these targets, aiming for doing these activities 80-100% of the time. Targets for expert problem-solving 20% 40% 60% 80% 100% 1. I describe my thoughts aloud as I solve the problem. 2. I occasionally pause and reflect about the process and what I have done. 3. I don’t expect my methods for solving problems to work equally well for others.b 4. I write things down to help overcome the storage limitations of short-term memory (where problem-solving takes place). 5. I focus on accuracy and not on speed. 6. I interact with others. b 7. I spend time reading the problem.c 8. I spend up to half the available time defining the problem.d 9. When defining problems, I patiently build up a clear picture in my mind of the different parts of the problem and the significance of each part.e 10. I use different tactics when solving exercises and problems.f 11. I use an evidence-based systematic strategy (such as read, define the stated problem, explore to identify the real problem, plan, do it, and look back). I am flexible in my application of the strategy. 12. I monitor my thought processes about once per minute while solving problems. Endnotes a Problem-solving contrasts with exercise-solving. In exercise-solving, the solution methods are quickly apparent because similar problems have been solved in the past. bAn important target for team problem solving. c Successful problem solvers may spend up to three times longer than unsuccessful ones in reading problem statements. d Most mistakes are made in the definition stage! e The problem that is solved is not the problem written in the textbook. Instead, it is your mental interpretation of that problem. f Some tactics that are ineffective in solving problems include: 1. trying to find an equation that includes precisely all the variables given in the problem statement, instead of trying to understand the fundaments needed to solve the problem; 2. trying to use solutions from past problems even when they don’t apply; 3. trial and error Quantitative Concept Summary Strategy Taken from: Fleet, J., Goodchild, F. and Zajchowski, R., “Learning for Success”, 2006 See http://zackr.disted.camosun.bc.ca/ for several completed examples. Click on SFU Q Conference, used with permission. Purpose: to provide a structure for organizing fundamental, general ideas. The mental work involved in constructing the summary helps clarify the basic ideas and shift the information from working memory to long-term memory. This is an excellent study tool, for quick review. Method: The organizational elements are i. Concept Title You can identify key ideas by referring to the course outline, chapter headings in the text, lecture outline. Sometimes concepts are thought of individually, other times they are meaningfully grouped for better recall. Eg. Depreciation, Capital Cost Allowance, and Half-Year Rule; acid, base and PH.. ii. Use general categories to organize material, and then add specific details as appropriate. Sample general categories may include: * Allowable key formula- check summary page of text or ask professor * Definitions- define every term, unit and symbol * Additional important information- sign conventions, reference values, meaning of zero values, situations in which formula do not work, etc * Simple examples or explanations- use your own words, diagrams, or analogies to deepen your thinking and check your understanding * List of relevant knowns and unknowns- to help you know which concepts are associated with which problems, use crucial knowns to help distinguish among problems. QUANTITATIVE CONCEPT SUMMARY Concept Title: Allowable Key Formula: Definitions of each symbol, and its units; Additional important information: (eg. sign conventions, special characteristics, when concept doesn’t work, special cases, etc) Simple examples, explanations, cases: Relevant knowns, and unknowns: (and words or phrases from word problems that signal these) By permission from website of R. Zajchowki <> Concept Summary for Ordinary simple Annuities Used with permission General Problem Solving Strategy based on D.R. Woods, “Problem–based Learning”, 1994 A systematic approach to problem solving helps the learner gain confidence, and is used consistently as a “blue print” by expert problem solvers as a way to be methodical, thorough and self-monitoring. This model is used in life generally, as well as in the sciences. The steps are not linear, and multiple processes are happening in your brain simultaneously, but the basic template hinges on effective questioning as you carry out various steps 1. Engage * Invest in the problem through reading about it and listening to the explanation of what is to be resolved. Your goal is to learn as much as you can about the problem before you begin to actually solve it, and to develop your curiosity (which is very motivating). Successful problem solvers spend two to three times longer doing this than unsuccessful problem solvers. Say “I want to solve this, and I can”. 2. Define the stated problem…a challenging and time consuming task * Understand the problem as it is given you, ie. “What am I asked to do?” * Ask “What are the givens? the situation? the context? the inputs? the knowns? etc. * Determine the constraints on the inputs, the solution and the process you can use. For example, “you have until the end of class to hand this solution in” is a time constraint. * Represent your thinking conceptually first, by reading the problem, drawing a pictorial or graphic representation or mind map (see example attached), and then a relational representation. * Then represent your thinking computationally, using a mathematical statement 3. Explore and search for important links between what you have just defined as a problem, and your past experience with similar problems. You will create a personal mental image, trying to discover the “real” problem. Ultimately, you solve your “best mental representation” of the problem. * Guestimate an answer or solution, and share your ideas of the problem with others for added perspective. * Self-monitoring questions include: What is the simplest view? Have I included the pertinent issues? What am I trying to accomplish? Is there more I need to know for an appropriate understanding? 4. Plan in an organized and systematic way * Map the sub-problems * List the data to be collected * Note the hypotheses to be tested * Self-monitoring questions include: What is the overall plan? Is it well structured? Why have I chosen those steps? Is there anything I don’t understand? How can I tell if I’m on the right track? 5. Do it * Self-monitoring questions include: Am I following my plan, or jumping to conclusions? Is this making sense? 6. Look back and revise the plan as needed. Significant learning can occur in this stage, by identifying other problems that use the same concepts (remember the spiral of learning?) and by evaluating your own thinking processes. This builds confidence in your problem solving abilities. * Self-monitoring questions include: Is the solution reasonable? Is it accurate? (you will need to check your work to know this!) Does the solution answer the problem? How might I do this differently next time? How would I explain this to someone else? What other kinds of problems can I solve now, because of my success? If I was unsuccessful, what did I learn? Where did I go off track? Decision Step Strategy: Applying the General Method to a Specific Problem Taken from: J. Fleet, F. Goodchild, R. Zajchowski, “Learning for Success”, 2006 See for a completed example. Click on SFU Q Conference Purpose: to help learners focus on the process of solving problems, rather than on the mechanics of formula and calculations. The focus is on correct application of concepts to specific situations. This strategy helps you to increase your awareness of the mental steps you make in problem solving, by “forcing” you to articulate your inner dialogue regarding procedure. Method: Identify the key decisions that determine what calculations to perform. In lecture, try to record the decision steps the professor uses but may not write down or post. i. Analyze solved examples, using brief statements focusing on steps you find difficult: * What was done in this step? * How was it done; what formula or guideline was followed? * Why was it done? * Any spots or traps to watch out for? ii. Test run the decision steps on a similar problem, and revise until the steps are complete and accurate. Decision Steps in Calculus Used with permission Decision Steps for Rational Expressions Used with permission Problem Solving Homework Strategy This strategy encourages a deep understanding of concepts and procedures in calculation. The time you spend on this will reduce the amount of time you may spend in “plug and chug” attempts to do the homework, and reduce the amount of time you will need for studying later on. 1. Prepare for the homework questions.. * review class notes and understand the concepts in the examples. This might take 30 - 45 minutes. * write the first line of a sample problem, close the book, and work as far as you can without looking. * refer back to notes, and then again attempt sample * repeat over again until you can solve the sample problem both accurately and quickly. You will have memorized the rules in the process. This might take 1 hour. 2. Start the homework questions. Interrogate your problem solutions: ask questions about the problem and your method of solving it. E.g. 1. What are the givens? Can the givens be classified as Assets, Liabilities, Owner’s Equity, Income, Expenses, etc? Is there any Depreciation? 2. What is required? 3. Can I diagram this? 4. What concepts are referred to? Theorems? Operations? 5. Is the problem similar to others I solved/How? 6. What more do I need to understand this? 7. Are there any “tricks” to the question? If so, how do I deal with them? 3. Keep track of problems you have trouble solving, isolate the particular difficulty, and get help to figure it out. Drill these problems until you are both accurate and fast in solving them. Range of Problems Strategy: Common Types of Difficult Problems Taken from: J. Fleet, F. Goodchild, R. Zajchowski, Learning for Success, 2006 Expand your thinking in preparation for exams, where problems are not exactly the same as you have previously solved. Work from an existing problem, and make it more challenging by adding or changing: Hidden knowns: needed information is hidden in a phrase or diagram Eg. “at rest” means initial v = 0 in physics. Multipart-same concept: a problem may comprise 2 or more sub-problems, each involving the same concept. This type of problem can be solved only by identifying the given information in light of these sub-problems Mulitpart-different concepts: same idea as above, but the sub-problems involve the use of different concepts Multipart-simultaneous equations: same idea as above, but no single sub-problem can be solved by itself. You may have 2 unknowns and 2 equations or 3 unknowns and 3 equations, and you will need to solve them simultaneously, eg. using substitution, comparison, addition and subtraction, matrices, etc. Work backwards: some problems look different because to solve them you have to work in reverse order from problems you have previously solved Letters only: when known quantities are expressed in letters, problems can look different. If you follow the decision steps, they are not usually as difficult. “Dummy variables”: sometimes a quantity that you think should be a known is not specified because it is not really needed- that is, it cancels out. Eg. mass in work-energy problems, temperature in gas-law problems. Red herrings, unnecessary information: a problem may give you more information than is needed, which is confusing if you think you should use everything provided. Resources Online: www.nscu.edu/felder-public/ILSpage.html, last accessed May 2010. Use this free inventory, the Index of Learning Styles, to assess preferred learning styles, and get additional information on interpretation of your profile http://csd.mcmaster.ca/academic , last accessed May 2010. click “Online Resources”, scroll to “Math”, select topic and format There are 3 videos on Problem Solving illustrating general ideas (Problem Solver I), differences in applying concepts vs. formula chasing (Problem Solver II), and applying the Decision Steps strategy (Problem Solver III). http://zackr.disted.camosun.bc.ca , last accessed May 2010. Click on SFU Q Conference. The personal web site for Richard Zajchowski, with examples of completed Concept Summaries, Decision Steps and other strategies Books: Fleet, J, Goodchild, F, Zajchowski, R Learning for Success: Effective strategies for students, Thomson Nelson, 4th ed, 2006 Whimbey, A, Lockhead, J, Problem Solving & Comprehension, New Jersey: Lawrence Erlaum Associates, 5th ed., 1991 Woods, DR, Problem-based Learning: How to gain the most from PBL, Waterdown, ON: DR Woods, 1994 Mar. 2009 1 Woods, D.R., Felder, R.M., Rugarcia, A., Stice, J.E. (2000). The Future of Engineering Education III: Developing Critical Skills. Chemical Engineering Education, 34 (2), 108-117. --------------- ------------------------------------------------------------ --------------- ------------------------------------------------------------ Learning Strategies Development Queen’s University www.queensu.ca/learningstrategies