1) Reflect and Summarize
at the end of each class, at the end of each week, at the end of each month. Review your notes and/or think back over the material that has been covered, then decide which skills or ideas you think are most important. Summarize the material you are learning as concisely as you can, because summarizing helps you learn. Identify any skills or ideas that you are not confident about. Write your reflections and summarizations as part of your notes, with reminders about what needs more work.
2) Use scrap paper
Using scrap paper removes a source of anxiety when solving problems. You will not have erase mistakes (unless you wish to), and nobody else will see this work but you. Plus, you will wish to copy your work over (see below).
Doodling all relevant information from a problem (organizing the information in an illustration such as the one above this post, or just restating it “your way”):
– restates the problem in your preferred notation
– helps you verify you fully understand every part of the problem
– helps organize the problem’s details in your working memory
– helps engage your intuitive thinking
Doodle the problem’s information before beginning to try to solve it. Scrap paper is a great tool for doodling, as you may often find that your first attempt at doodling a problem’s information ends up not being your favorite way of organizing the information, or was not a correct interpretation of the information.
4) Use Both Intuitive And Procedural Approaches
Don’t abandon your intuition by focusing only on procedures when faced with a complex problem. Both intuitive and procedural approaches are important, and one can assist the other. Think about the context and nature of the problem regularly as you solve it, as they may help you discover an approach to solving it that you have not tried yet.
5) Copy your work onto the page you will hand in
after solving a problem. You will have to do this if you used scrap paper. I think I have learned more math by re-thinking my solution as I copy my work over than I have via any other method. Copying your work onto another sheet of paper gives you an opportunity to reflect by:
– re-thinking the “big picture” that drove your solution process
– finding better ways to present your work
– checking your work for the errors you typically make, which is one of the few times you will ever be able to “practice” finding your own mistakes
6) Divide and conquer
It is often helpful to break problems into a series of smaller questions which are easier to answer. This can reduce the load on your working memory and reduce the number of errors made. If you worry about losing your place in the problem, organize the problem on paper in a way that makes it clear what will still need to be done. Even a very small task within a larger problem, such as combining like terms, can often be broken down further in ways that make it easy to be confident about getting the step right:
– what will the sign of the combined term be?
– what will the coefficient be?
– what variable(s) must follow the coefficient?
When working on a series of sub-problems, I tend to:
– focus on the relevant details
– determine the answer
– write the result
– check the result
– “step back” and decide what the next step will be
– repeat… until the problem has been solved
This process can be repeated many, many times when solving a complex problem.
7) Paper = Working Memory
Let your work on paper serve as most of your working memory. Organizing your work well on paper, starting with a doodle, can make problems feel simpler to solve because you don’t feel stressed by trying to retain all their details in your working memory. Having a problem organized in writing also makes it easier to focus completely on smaller parts of the problem (divide and conquer) without overlooking steps that need to be completed later.
8) Work On Paper Engages Our Visual Memory
Our brains devote a lot of their capacity to visual memories. Take advantage of this by finding consistent ways to organize your work on paper. The overall visual appearance of a problem can help remind you of what your options are at that stage in working a problem.
9) Errors? Re-Solve The Problems
If you made a mistake solving a problem on homework, a quiz, or a test… write the problem down on a separate piece of paper, and re-solve it a day or two later to verify for yourself that you can now solve it confidently and without hesitation. If not, figure out how to solve it, or what was slowing your solution process down, and write the problem down on a new piece of paper to be solved again in a day or two. This repetition constitutes “frequent quizzing” which research has shown to be the best way to transfer stuff from short term memory to long term memory. It also offers an opportunity for reflection if you seek to answer the question: “what is it about this problem that I find challenging?”
10) Ask “Why?”
It is often all too easy to approach math class as an endless series of instructions for “how” to do things. Don’t neglect to address “whys” such as:
– Why is this true?
– Why should I solve it this way instead of another?
– Why am I confident that my solution is correct?
These types of questions lead to a better understanding of a topic over time.
11) If it is assigned, be able to solve it confidently… even if it is not graded.
If you hesitate or struggle to complete part of an assignment, re-do that part a day or two later to verify you really do know it. You will be glad you did when quiz or test day arrives.