A number of historically “good” math students seem to reach a point during their High School years where their feeling of mastery starts to fade away. While teachers usually expect more from a student with each passing year, this alone does not explain the frustration these students experience. I believe it arises because a familiar study habit, memorization, is no longer enough to assure mastery.
I used to read a math or science textbook in pretty much the same way I read anything: as quickly as I could. In fact, for math I often skipped the reading entirely as I had been shown how to do the new types of problems in class, so all I had to do was sit down and follow the procedure I had been shown – no need for all the verbiage.
However, this approach stopped working when I got to college. If my notes from lecture did not help me figure out how to solve a problem, I had to rely on the text in the textbook for almost the first time. I learned that “believing I understood everything that happened in class” was a very different thing from being able to solve the problems assigned for homework.
After skimming through my math text, I often found that I still could not do the problem that faced me. So I went back to re-read relevant sections more closely. When that still didn’t help enough, I went back and worked some of the example problems myself, then tried some of the easier problems in the book (even though they were not assigned), and only then did I begin to have some ideas for how to approach the problem I had been assigned.
Reading a math text for comprehension is NOT like reading a summer novel. Whereas I might spend 20 seconds reading one page of a book I am reading for fun, I often spend 20 minutes (or more) reading one page about something I wish to master. I cover up all but the beginning of a sample problem with a piece of paper, and try to work it myself. I verify that the author did not “make any mistakes in the book” by checking that I can derive every line of their work on a sample problem. This approach led to homework taking me much longer to complete, but at least I could (eventually) complete it…
But there was a bonus! When I sat down to begin studying for the big test, I discovered that I knew the material much better. In fact, my study time for tests dropped from multiple hours to less than an hour. I became better at predicting what types of questions would be on the test, and better at picking out which problems I should practice to test my knowledge.
What others say and do
Years ago I was given a wonderful little book called “Study Is Hard Work”. As I recall, the author recommends a process of distilling each week’s classes in a single subject onto a single sheet of paper, then once per month distill the four weekly one page summaries into a one page monthly summary. Then, once per semester distill the monthly summaries into a one page semester summary.
This approach is similar to teachers telling students they are welcome to bring one 3″ x 5″ card of notes into the final exam. Most students discover during the exam that they don’t need most (or all) of the information they managed to cram onto the card. Why? Because creating such a small card required them to summarize: reflect on the material, find connections within the material, decide what was most important, then write the essence of it all down.
The process of summarizing and finding connections, then writing them down and/or talking about them, is essential to “understanding” material. “Taking notes” may seem similar, but it all too often does not involve summarizing or finding connections. Summarizing and finding connections require reflection – you must think about the new material and how it relates (if at all) to experiences you have had and/or other topics you are familiar with. The discovery of connections and/or relationships are what lead to both long term recall and understanding.
As to the traditional study technique of highlighting and rereading later, a September 2012 Scientific American article titled “How Science Can Improve Teaching” states: “Studies have shown that highlighting and rereading text is among the least effective ways for students to remember the content of what they have read. A far better technique is for students to quiz themselves. In one study, students who read a text once and then tried to recall it on three occasions scored 50 percent higher on exams than students who read the text and then reread it three times.” A July 2014 New York Times Sunday Review article titled “How Tests Make Us Smarter” provides additional evidence that low or no-stakes quizzing over time (not high stakes testing) leads to stronger retention of information.
What you can do
So, how can you get to the point where you understand something well?
- At the end of each class, spend 30 seconds (or more if you can) replaying in your mind what took place in this class. What was most important, or significant, or interesting? Write these thoughts down at the end of your notes for the day, even if they duplicate some of what you already wrote in your notes. Reflection is an important part of summarizing.
- When doing homework, read the text closely, deliberately, and slowly.
- Check for understanding at the end of every sentence fragment, and if in doubt, re-read and think about it until you are certain you understand exactly what the author intended to say (the same as my recommendation for how to start word problems). Only then should you proceed to the next sentence.
- After reaching the end of a paragraph or section, reflect on what has been described and ask (“quiz”) yourself the most important questions for understanding:
What was just described (in my own words)?
What concepts is it connected to or based upon?
Why am I being asked to learn this?
Why is this idea being brought up at this point?
How does this relate to what we learned earlier?
How do I know it’s true?
How does this work in the world I live in?
- When you reach an example in the text, do not read through it – do it yourself by covering up the solution until you have solved it. If you get stuck and have tried every approach you can think of, use the author’s work to “cheat” and figure out the step you were stuck on… but don’t look beyond that! Only uncover as much of the solution as you need to get yourself un-stuck. Solving a problem yourself helps develop a stronger understanding – reading through someone else’s solution is seldom as effective.
- After working on something completely different for a while, and perhaps again a day later, quiz yourself on what you just learned. What was presented? Why is it relevant? What else is it connected to? What major and minor points were made? What kinds of problems does it help solve? Then find a few such problems and solve them.
- Remember that “thinking you understand the ideas” or the formula is often a very different thing from being able to solve a problem… you must both understand the ideas and be able to solve problems based on them before you are ready to take a test.
- If you find yourself seeking to memorize all possible problem types and their solution patterns, you are not trying to understand the concepts. Stop memorizing, and focus on describing the problem visually as well as verbally. Sketch (or “doodle”) the situation if possible, and annotate the sketch with key facts from the problem. Write a list of needed variables with a brief description of each. Then describe the problem using mathematics notation. Coming up with a complete (solvable) mathematical description of a problem is often the most challenging part… from there the solution process is often less challenging.
- Teach and explain the topic to someone else… either in person or by writing out what you would say. When you stumble across things that are hard for you to explain, you have discovered an area you do not understand well (yet). Writing, then editing, an explanation of the topic until you are happy with it is a great way to improve your understanding of the topic.
If you can find the time and the patience to approach homework and studying in this manner, quizzes and tests should be much easier to prepare for, and your grades should rise as well. Why? Because you are:
- spending more time on the subject
- giving yourself time to identify and reinforce connections between ideas before you are tested on them
- practicing recalling new material by answering questions you ask yourself (quizzing)
- reducing pre-test stress by avoiding the need to “cram” for a test
- increasing your self-confidence by mastering material soon after it is presented, then confirming your mastery to yourself over the next few classes
Two other posts of mine may be of interest:
And, some wonderful study tips can also be found at:
- Make It Stick, by Brown, Roediger, McDaniel, describes effective learning techniques that are supported by research evidence.
- Stan Brown’s web site, and in particular his How to Read a Math Book.
- Improving Classroom Performance by Challenging Student Misconceptions About Learning. A great description of what students or teachers can do to improve understanding and retention.
- Mastering Linear Algebra in 10 Days: Astounding Experiments in Ultra-Learning. A great description of the techniques an MIT student used to master material four times faster than the normal rate. Highly recommended.
- the Little Warrior blog.