## Math problem solving

Math can be challenging and takes time to master. To keep going, even when it’s difficult, you need a plan and you need to use your resources. You don’t have to figure this out on your own!

Getting stuck and making mistakes are not a sign that you’re bad at math; they’re a necessary part of the process.

## Problem solving process

Follow this flowchart and its associated tools (the **course learning tracker**, **concept summaries**, and **decision steps**) to establish your course learning goals and progress toward them while also building your self-regulation and thinking strategies.

*Click on flowchart to expand (PDF).*

This procedure begins with making a plan for getting unstuck.

## Make plan for getting *un*stuck

In any class, including math class, you *should *get stuck. If you’re not getting stuck, you’re not learning.

#### Instructions

**Start a list** of where you will go to find information about the concepts you are (or will be) working on.

Considering the course you’re in, take some time to think about and write down resources and supports for finding information and for helping you to move your learning forward.

**Where to find information:**Figure out where the key places are to find information about what you’re working on (notes, textbook, different textbooks, problem solution books (resources containing many worked examples of problems), use the internet to find sites that display solved problems in a variety of ways.**Who can help:**Figure out when the prof’s office hours are, build a bank of people in your class that you might want to study with, ask their prof early on for suggestions for additional resources they may be able to use for helping to learn content or study for the exam.

**Prepare to ****ask questions**: TAs, professors, and peers generally want you to succeed and will welcome questions.

- If you don’t know where to start with a problem, you can still explain in general what you know about the concept, and what you’re thinking of doing.
- If you’re stuck in the middle of a problem, but know what to do next, make up an answer for the step you’re stuck on and use it to solve the rest of the problem. Then get help. Your attempt at a solution will get you better feedback from your TA/professor and will mean more than no attempt at all.

**Make math more social** to boost your skills, motivation, and confidence. Work through problems as a group, share resources, talk through solutions, and explain concepts to each other.

**Check out other SASS resources**: academic skills resources, subject-specific academic resources, workshops, and appointments.

Next, organize and track your learning of the key concepts and problem types.

## Organize and track your learning

### Why focusing on key concepts is important

Many students try to jump straight into solving problems without initially (or concurrently) working on understanding the concepts that are being applied in those problems. They try to look for specific formulas that match specific problems and end up memorizing too much information with almost no hierarchy or connection.

This matching of formulas with problem-types can work in the short term. It is possible to solve a number of problems very quickly by memorizing a few formulas and solutions, and you may be able to keep this up for a unit or two, but it won’t work over the course of an entire semester, and that’s what we care about in University.

In University math courses:

- You have way too many formulas and solutions to remember
- Professors will often choose exam questions that cannot be solved by referring simply to a specific memorized formula and solution. They want to test whether you really can understand and apply the concept.
- Courses, and the units within those courses, are sequential. Like a set of building blocks, today’s concept is actually built on previous concepts. To solve problems in a new concept today, often requires that you know how to apply previous concepts. If one of those concepts is missing, the whole structure falls down.

The mental work involved in understanding key concepts helps clarify the basic ideas and shift the conceptual information from working memory to long-term memory.

#### Instructions

Organize and track key concepts and problem-types using the ‘course learning tracker’ table (or similar format). Getting this table ‘finished’ is not the goal. What’s important is that you’re thinking about what the key concepts are in the course, and how they relate to the types of problems you’re being asked to solve.

Determine the key concepts and the problem types for each key concept by:

- reading the learning objectives on the course outline or the course description
- checking the syllabus and weekly reading list
- referring to the textbook’s table of contents and headings within a chapter (focus on how concepts and practice problems are organized and related to each other)
- thinking about the key formulas (which are often related to key concepts), and common aspects of the problems you are solving
- referring to the lecture outline to identify recurring themes
- referring to learning objectives and topics from lectures and other materials
- past years’ final exams, if available.

Course learning tracker (PDF).

Next, build your understanding of key concepts and problem types.

## Build your understanding

Build your understanding of key concepts (e.g., slope of a line) using the **concept summary** tool, and build your understanding of problem-types/applications (e.g., calculate the slope from a graph) that fall under each concept using the **decision steps tool**.

While you use the **concept summary **and **decision steps **tools, make a note in the “notes” section of the course learning tracker of where you get stuck or what you don’t understand. You can use these notes as a springboard for conversations with your profs, TAs, tutors, and peers that are targeted specifically to your learning needs.

Keep in mind that while the final products of these learning tools (completed concept summaries and decision steps) are important and will make valuable study aids, the *process* of thinking, struggling, and questioning while working with these tools is essential for achieving a deep understanding.

### Concept summary

(Fleet, Goodchild, & Zajchowski, 2006)

Concepts are general organizing ideas. Often, a course will cover just a few key concepts, along with their many applications. 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.”

#### Instructions

Choose a key concept from your course (potentially identified while filling in the course learning tracker), and use the following five categories as a guide for building your understanding of that concept.

- Title
- Key equations
- Definitions of each term in the equations
- Additional information (e.g. sign conventions, reference values, the meaning of zero values, cases where the concept doesn’t apply, relevant knowns/unknowns)
- Your own example or explanation

For a concept summary example, see https://sass.queensu.ca/wp-content/uploads/2013/09/Quantitative-Concept-Summary.pdf.

**PDF**: Example of a concept summary for Equilibrium of a Rigid Body (Physics).

### Decision steps

(Fleet, Goodchild, & Zajchowski, 2006)

This tool is helpful for any applied problem-solving situation (e.g. mathematics, physics, statistics, accounting). 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.

#### Instructions

**To build your understanding of the various applications (or problem-types) of key concepts**

- Choose a problem-type in your course (potentially identified while filling in the course learning tracker), and analyze/build decision steps for a solved example (from a lecture, from your homework, from a study guide) by answering what was done, how it was done, and why was it done for each step.
- During the lecture or when you read 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 linked 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?” - Test run these decision steps by using them to solve a similar problem. Usually your initial decision steps will be incomplete and require revision.

**PDF**: Example

### Track your progress weekly

Over the course of the semester, as you attend lectures, complete homework, problem sets, and assignments, continue to track your progress:

- monitor key concepts and add, merge, or separate concepts as necessary
- make/add to existing concept summaries and decision steps.

Next, take on some of the approaches and habits that will ensure your success.

## General suggestions

### Helpful habits

Math is about creativity and making sense of the world. It’s also about connections and communication. It’s not just about getting the right answer. One of the most effective things you can do is to try to shift your thinking–about math and about your own ability. These habits will help.

#### Spend enough time on your math.

- See how you do by putting in 8-10 hours per week on each course (this time includes time you spend in class, labs, doing homework, etc.).
- Don’t do this 8-10 hours straight.
- Spread out your work; do some math every day. It will add up. Try using 2-3 hour blocks, where you’re working for 50 minutes and then taking a 10 minute break each hour.
- Keep up with the homework. Concepts later in the term build on the ones from earlier in the term.
- Having trouble managing your time? Make an appointment with a learning strategies advisor; we can help!

#### Be thorough. Don’t just rush through problem sets.

- Take a systematic approach (e.g., Polya’s problem solving techniques).
- Read and define the problem first; this takes time, but it’s worth it.
- Look for and understand the underlying concept (the “why” or “big picture”) of each question, not just the procedure for solving it.
- Produce a complete and well-reasoned solution, not a superficial one.
- Aim for accuracy before you aim for speed.
- Spend time on challenging questions, not just familiar ones.

#### Recognize repeat concepts.

- Most math courses ask you to do hundreds of problems, but the problems usually fall under a handful of 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 (e.g., how are the concepts similar, how are they different?).
- The learning objectives of a course syllabus often tell you what the key concepts are.

#### Self-assess and reflect.

- Monitor your thought process while solving problems (e.g., by using decision steps).
- Reflect on how you present your thinking. Is it clear and purposeful?
- Monitor your progress; change your approach if you need to.
- Use incorrect answers and failures to motivate a change in strategy.
- Ask, “does this make sense?” and, “did I solve the problem/answer the question?”
- Check the reasonableness of your answer.

#### Don’t give up.

- Expect math to be a challenge and to take time. Keep trying.
- Mistakes are valuable! They aren’t a sign that you’re bad at math; they’re a necessary part of the process.
- Questions are important. Get help when you are stuck. (And take a break when you feel frustrated.)
- Be optimistic. The problem
*does*have a solution. - Don’t assume you’re not a math person. Everyone can do math at a post-secondary level.

### Solving 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.** Thinking through a problem will deepen your understanding and help clarify the questions you will ask peers, TAs, or profs.

Before you start your homework questions, **review** your class notes and/or the 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 keywords in the problem that point to a particular concept? - 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…” as ways to see the problem in a new way.**Work out loud**; notice what strategies you’re using and why.

### Studying for tests and exams

In math-based courses, your goal should be to **focus on solving problems**, not on reading. For example, if you have six hours a week to study math, spend one hour reading and five hours doing problems. Then be sure you practice problems effectively.

### Use problem sets effectively

**Do problems to mastery.**Once you’ve mastered one kind of problem, don’t worry if you haven’t finished every single problem in the set–move on to the next type, or apply what you’ve learned in a different context.**Use the answer key strategically.**Avoid looking at the answer key while you work on a problem, but don’t do problems without checking to see if your answer is correct.**Ask for help when you need it.**Use the decision steps and concept summary tools to help communicate what you know and where you got stuck so that you can ask specific questions.**Work backwards.**For problems where you are given the answer but don’t know the starting point, begin at the end and work backwards to undo the problem step by step.**Use images.**What can you draw to help yourself understand and solve the problem? Can you make a mental picture or otherwise visualize this problem?

### Study techniques

**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; see interleaved mathematics practice.**Self-testing**(including the ) 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 will be.**Explaining to and/or 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.

For more study strategies, see our test and exam preparation section.

## Resources

Boaler, J. (2016). *: Unleashing students’ potential through creative math, inspiring messages and innovative teaching*. San Francisco, CA: Jossey-Bass, Wiley.

McMaster University’s academic resources website, which features three videos on problem solving:

- Problem Solver I, general ideas
- Problem Solver II, differences in applying concepts vs. formula chasing
- Problem Solver III, applying the Decision Steps strategy

Fleet, J., Goodchild, F., & Zajchowski, R. (2006). *Learning for success**: Effective strategies for students (4th Edition).* Thomson Nelson.