I was shocked!
It was my first session with a guy who’d been an A student all his life. He was starting Calculus and sensed he was going to need some help. Here was the shocker: he could not tell me the most basic facts about linear functions, even though he had studied them in at least four previous classes — Math 1, 2, 3, and Precalculus! Here’s what’s sad: he is not alone. Many smart, high-achieving students are not retaining even the most basic material from one course to the next.
How does this happen?
Here’s what I think: students are doing homework in a superficial way just to get the answers. Their short-term memory absorbs just enough to do well on the next test, but they’re not forming long-term memories. They are not digesting the material, thinking about it deeply, and talking to themselves about it. What they have memorized is mostly rote, so they don’t know how and when to use it. E.g. many people can parrot \(a^2+b^2=c^2\) but can’t tell you what it’s about.
Sooner or later, the knowledge gaps and superficial understanding catch up to them and they hit a wall. What’s really, really sad is when they decide that they’re “not good at math” and give up on it. The truth is that they’re not good at learning math — YET! They need to learn a new approach.
How can you avoid learning nothing from your homework?
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Begin each homework set by writing down all the new details.
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Then work the assigned problems.
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Refer to the details as you work.
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Identify all the problem types and write down the procedure for solving each type.
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Get all the answers right.
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Learn all the notes you’ve made for the new content. Review them again at bedtime.
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Prepare for the next class session by previewing upcoming material.
Do you see that “doing the problems” is not the point? It’s just a focused activity that gives you an opportunity to build the knowledge you need to keep.
The Process in Depth
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Write down all the new details.
Before you begin working homework problems, write down all the important new details from class, handouts, and any related reading or video content. Do this in an outline if you prefer, but I recommend using notecards for easy reviewing. What do I mean by “new details”?
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Vocabulary
Write each new term and its definition and draw an illustration if possible. Sometimes it’s helpful to give examples of things that are not the term being defined and point out why they do not satisfy (match) the definition.
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If you have a good textbook, it should have exactly what you need for each term. Otherwise, consult handouts or your class notes, Wikipedia, or an online math dictionary.
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If there’s another term with a similar meaning, you might want to make a note of how this term is different from the other one. “Compare and contrast” is an important analytic skill.
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Definitions should be precise enough that any math object either fits the definition or doesn’t. If you write a definition in your own words, be careful that it isn’t worded in such a way as to accidentally include and exclude the wrong things.
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It’s important to distinguish between definitions of objects and properties that are true about them. The definition is a bare minimum, not a detailed description. Properties follow from the description and can be proved.
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Notation
Math has lots of symbols and shorthand that allow us to write complicated ideas in a very condensed format. Write all the new notation and what it is called. Tell what the notation means and write down how to read the notation out loud. Make sure you can form any symbols correctly. Some take practice to do well. Curly braces { } were tricky for me.
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Facts
By facts, I mean things like theorems and properties. These are typically statements that boil down to something like this:
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“In [this situation], [this statement] is always true.”
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“For [this kind of math object], [such and such] is always true.”
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“[These givens] imply [this conclusion].”
Write the conventional name of the fact if there is one or make up one that will remind you what it’s about (don’t just use “Theorem 2”!). Draw an illustration whenever helpful. Always pay attention to the context in which the fact is true. Examples:
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“Two negatives make a positive” only when you are multiplying or dividing, not when you’re adding or subtracting.
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The Pythagorean Theorem: \(a^2+b^2=c^2\) applies only to right triangles.
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Procedures
Math lectures often include demonstrations of procedures and techniques for solving problems. For each new method, write down its name and the steps involved. Note when you would use it and why. If there’s a similar technique, you might want to note the differences between the two. Draw a sketch whenever it will help.
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Work the assigned problems.
The thinking you did in Step 1 prepares you to learn deeply while you do Step 2.
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Refer to the details as you work.
Talk to yourself, using the vocabulary, the conventional pronunciation for symbols, the names and specifics of theorems and properties, the types of problems, the steps to solve, etc. Talk out loud if you can. These key details are the important takeaway from your homework, not the correct answers. The more you repeat them, the better you will remember them.
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Identify all the problem types and write down the procedure for solving each type.
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Often, the instructions for a set of similar problems will suggest a name for a problem type, though you might need to add more details. E.g. “Solve the following equations” could become “Solve quadratic equations.” “Rationalize the following expressions” could become “Rationalize expressions with radicals in the denominator.”
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Note all the key steps, using the appropriate vocabulary. Avoid giving details about sub-steps you have already learned separately. Imagine there’s a hyperlink from your new card to old ones it references. At some point, “simplify” will cover all sorts of things you needed to mention specifically in previous procedures, and after a while, you won’t list “simplify” as a step because that’s just part of finishing any problem.
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Get all the answers right.
All homework problems must be done correctly, with understanding. If you don’t have answers in your book or from your teacher, compare with a study buddy. If there’s something you don’t understand and can’t figure out, make a note and ask about it as soon as possible.
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Learn all the new material and review it at bedtime.
Every detail described above needs to be stored in short-term memory before the next class. No kidding! Learning math is like building a brick wall. Each brick needs to be cemented in the appropriate place as soon as it is presented. New bricks are delivered every class session, and they need strong lower levels to support them. Reviewing right before you sleep is the way to tell your brain’s secretary to file this carefully!
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Preview upcoming material.
You’ll get much more from a class session if you are already familiar with the topic. At least skim the next section, focusing on the vocabulary and facts and thinking through the diagrams. If you have time, read more carefully and work through problems in the text.
Final Comments
Remember, your goal is to completely learn each lesson’s material before the next class. Tomorrow’s material will be built on the foundation you lay today.
The notes you make in Steps 1 and 2 should contain everything you need to know from that day’s work. Review them regularly until they are committed to long-term memory.
If you follow this procedure, I promise you will not arrive at your next course knowing nothing from your current one!
Notes
Notecards
You might wonder about using notecards someone else has made. That would be better than nothing, but you’d lose out on the learning that happens when you make your own. The thinking, planning, writing, choosing colors, flipping cards, talking to yourself all help create the neural pathways your brain needs to hang on to what you are learning.
Thomas Frank is the creator of The College Info Geek. His whole website is full of helpful, well-written material. He has an article and a video about making and using flashcards. The content is overlapping, so you can use either one. Note: The video narration is pretty fast; it might help to turn on CC and/or use the video player settings to slow down the video.
Textbooks
A textbook is a treasure! If you don't have one, I recommend you get one that covers the material in your course. I have found lots of math textbooks dirt cheap (less than $5) at thrift stores, for courses in middle school through college. Used book stores also have them, especially stores that cater to homeschoolers. Sometimes teachers have a ratty old classroom set they’re willing to loan from. The nice thing about math texts is that the content is pretty much the same for books at the same level. Get one from 20 years ago and the theorems and problem types will be the ones you’re studying. The fashions on the people in the illustrations may be good for a few laughs. Some things don’t change, and that’s a good thing! Some things do, and that’s good too.
Copyright © 2020 Alie Benson