Solving problems tests your ability to apply theoretical concepts. You’ll need to think theoretically as well as do the calculations to do well in math-based courses.
Spend enough time on your math courses.
- See how you do by putting in 8-10 hours per week on each course (this time includes time you spend in class, labs, etc.).
- Spread out your work; do some math every day. It will add up.
- Keep up with the homework; concepts later in the term build on concepts from earlier in the term.
- Having trouble with time management? See us at SASS; we can help!
Don’t give up.
- Expect math to take time and to be a challenge, and just keep trying.
- Mistakes and uncertainty aren’t a sign that you’re bad at math. They’re part of the process.
- Get help or take a break when you are stuck or frustrated.
- Be optimistic; the problem does have a solution.
- Don’t assume you’re “not a math person.” Everyone can improve their math skills.
Recognize repeat concepts.
- Most math courses ask you to do hundreds of problems, but the problems usually fall under only a few key concepts that you’ll revisit in different forms over the term.
- Learn to identify and understand these few concepts and their relationships to each other, and recognize them when they take different forms.
- The learning objectives of a course syllabus often tell you what the key concepts are.
Be thorough. Don’t just rush through problem sets.
- Take a systematic approach. Many mathematicians use Polya’s.
- Read and define the problem first; this takes time.
- Look for and understand the underlying concept of each question.
- Produce a complete and well-reasoned solution, not a superficial one.
- Aim for accuracy before you aim for speed.
Use your resources.
- TAs and professors want you to succeed and will generally welcome questions, even if you don’t know where to start.
- Make math more social to boost your skills, motivation and confidence. Work with others: share resources, talk through solutions with each other, and explain concepts to each other.
- Check out resources at SASS: online resources, workshops and appointments.
How to solve problems
Practice problems are for figuring out, and then practicing, new and different ways to solve a type of problem. The process is what matters.
Before you start your homework questions, review your class notes / relevant textbook chapter, and identify the key concepts that they describe. Try working a sample problem from your notes or text, without looking at the solution, to see if you understand the idea. Then try the homework problems:
- Think of problems as a way to communicate, from the problem-setter to you. Ask: what do we know (givens)? What can we do? Are there clues or buzzwords in the problem that point to a concept?
- Try to identify a key course concept that applies to the problem. See our concept summary strategy.
- Diversify your thinking; there’s often more than one way to solve a problem.
- Accept mistakes as a valuable part of the learning process.
- Identify where you get stuck and, if you can, why.
- Prepare questions to bring to your TA / prof / help desk.
- Model the problem, draw it, talk it out, use analogies, change something (e.g., the scale), or ask “what if…” to see the problem in a new way.
- Predict / explain as you go, to understand more analytically.
- Work out loud; notice what strategies you’re using and why.
- Use a two-column approach to notes (one for the solution steps and one for your explanation of why you’re taking each step, including when you are uncertain).
What good problem-solvers do
- describe their thoughts aloud as they solve the problem.
- occasionally pause and reflect about the process and what they have done.
- don’t expect their methods for solving problems to work equally well for others.
- write things down to help overcome the storage limitations of short-term memory (where problem solving takes place).
- focus on accuracy and not on speed.
- work with others.
- spend a lot of time reading and defining the problem.
- when defining problems, they patiently build up a clear picture in their minds of the different parts of the problem and the significance of each part.
- use different tactics when solving exercises and problems. Some tactics that are ineffective in solving problems include:
- trying to find an equation that includes precisely all the variables given in the problem statement, instead of trying to understand the fundamentals needed to solve the problem
- trying to use solutions from past problems even when they don’t apply
- trial and error
- use an evidence-based, systematic strategy (such as read, define the stated problem, explore to identify the real problem, plan, do it, look back). They are flexible in my application of the strategy.
- monitor their thought processes while solving problems.
Source: 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.
Problem solving strategies
- Work with a good problem solver and compare your thought process to theirs.
- Working in groups can be helpful to share ideas, but do some of each problem type yourself…exams are solo events!
- Don’t try to use solutions from other problems that don’t apply. Focus on identifying underlying concepts first.
- In labs: relate experiment or process to problems in class. Do specific equations describe phenomenon being observed in the lab?
- Check your work using a different method, if possible.
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 blueprint 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:
Engage and define the stated problem.
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”.
Define the stated problem
- 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
Explore and search
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?
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?
Do it, then look back and revise
- Self-monitoring questions include: Am I following my plan, or jumping to conclusions?
- Is this making sense?
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 steps strategy
Taken from: J. Fleet, F. Goodchild, R. Zajchowski, “Learning for Success”, 2006.
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?”
Purpose and method
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.
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.
- 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?
- Test run the decision steps on a similar problem, and revise until the steps are complete and accurate.
Note that these decision steps try to capture what and especially how each step is carried out – including possible alternatives that can be tweaked so that the student is not left wondering how to make the decision needed. Most textbook steps tend to give the what only.
Quantitative concept summary
Taken from: Fleet, J., Goodchild, F. and Zajchowski, R., “Learning for Success”, 2006
Concepts are general organizing ideas, and there are often very few of them taught in a course, along with their many applications. 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.”
Purpose and method
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.
The organizational elements are
- 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 (e.g., Depreciation, Capital Cost Allowance, and Half-Year Rule; acid, base and PH).
- 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.
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
- level of difficulty of the problems
- 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.
What to do when you’re stuck
Common types of difficult problems
Taken from: J. Fleet, F. Goodchild, R. Zajchowski, Learning for Success, 2006
See if the problem you’re stuck on falls into one of these categories. Recognizing the type of difficulty you’re facing can be helpful.
- Hidden knowns: needed information is hidden in a phrase or diagram (e.g., “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
- Multipart-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, e.g. 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.
- 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.
Use questions to support your learning
Effective problem solving requires thinking about how you think! It’s helpful to know the difference between metacognitive strategies (i.e., “thinking about how you best learn mathematical concepts/skills”) and cognitive strategies (“interacting with the specific information to understand it”). Next time you start to solve a problem, see if thinking through your responses to these questions can help you focus your efforts.
|Advance organization||What’s the purpose in solving this problem? What is the question? What is the information for?|
|Selective attention||What words or ideas cue the operation or procedure? Where are the data needed to solve the problem?|
|Organizational planning||What plan will help solve the problem? Is it a multi-step plan?|
|Self-monitoring||Does the plan seem to be working? Am I getting the answer?|
|Self-assessment||Did I solve the problem/answer the question? How did I solve it? Is it a good solution? If not, what else could I try?|
|Elaborating prior knowledge||What do I already know about this topic or type of problem? What experiences have I had that are related to this? How does this information relate to other information?|
|Taking notes||What’s the best way to write down a plan to solve the problem? Table, chart, list, diagram…|
|Grouping||How can I classify this information? What is the same and what is different (from other problems I have encountered, from other concepts in the class…)|
|Making inferences||Are there words I don’t know that I must understand to solve the problem?|
|Using images||What can I draw to help me understand and solve the problem? Can I make a mental picture or visualize this problem?|
Many students find these types of questions boring or irrelevant and simply want to blast through all the problems, but it’s important to remember the actual purpose of solving problems (at least in homework, if not on a test): figuring out and then practicing new and different ways to solve a type of problem. The process is what matters, not getting the result as quickly as possible. Focusing on the process helps you to become more accurate and efficient, and it will save you time in the long run.
Diagnose the problem and connect it to a misconception
Sooner or later, you will run into a practice problem that stumps you. This is actually a good thing! It allows you to refine your understanding of the material, so you’ll be better prepared for the exam. At this point, it’s helpful to diagnose why you don’t understand this problem—what about your thought process isn’t working?
Here are steps to follow for diagnosing a misconception:
- Return to your notes and review course material on the topic. Try sketching the overall concept or explaining it to someone else without looking at your notes. Is your sketch or explanation accurate?
- Review your steps to the question. Look at each step individually: Was this step correct? Why did I do this part? (Think back to your sketch or explanation of the overall concept when trying to answer “why?”).
- When you have found the step where you first made an error, identify exactly why you made the error. Did you not read the question carefully? Did you use incorrect data? Did you misunderstand the purpose of the question? Did you misunderstand the concept?
- Try to think of other approaches, or find a similar practice problem and see if you can mirror the steps. Ask, “Why is this step correct? How will I modify my Concept Summary, analogy, etc. of the concept in light of this new information?”
Inspired by Chapter 4: Misconceptions as Barriers to Understanding Science from Science Teaching Reconsidered, A Handbook (1997).
Put a star next to this type of problem and be sure to practice this type again before any tests. This is exactly why practice problems are so helpful!
Common issues and problems
- Self-doubt and isolation
- Learning outside a real-world context
- No clear method
- Giving up too early; not putting in the time
- Spending time on familiar problems, instead of challenging ones
- “Siloed” thinking; not using what you’ve learned in other courses
- Rushing through problems
- careless errors
- not thinking about underlying concept, theorem, proof
Here are some study strategies particularly suited to math-based courses. For more study strategies, see our test and exam preparation section.
- Interleaving: Mixing up problem types supports your learning. The aim is to arrange problems so that consecutive problems cannot be solved by using the same strategy. Retrieval Practice has a guide that can help you get started: Interleaved mathematics practice.
- Self-testing (including the range of problems strategy) helps you anticipate different kinds of difficult problems for exam preparation, and solve some practice problems to test yourself. Don’t wait until the night before the exam! The more frequently you self-test, the better your learning.
- Explaining to / teaching others are great ways to make sure you’re thinking aloud, describing the problem, and working with others. Use study groups to compare completed solutions to assigned problems. Teaching someone is a very effective learning and study technique.
McMaster University’s academic resources website. 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).
Fleet, J, Goodchild, F, Zajchowski, R Learning for Success: Effective strategies for students, Thomson Nelson, 4th ed, 2006