Don't memorize the complicated stuff.
I recently got to go see a talk by Chris Hadfield held during the BC Association of Professional Engineer's annual conference. Chris Hadfield has a degree in Engineering, and there were many engineering students in attendance to the talk. Someone asked him “What was the most important thing you learned studying engineering?”
He had to think about it for awhile, and answered a couple of other questions while he thought about it. A couple minutes later he came back with an answer: “Don't memorize the complicated stuff. Try to understand it, even though it will take longer, the information will stick with you for much longer.”
His statement really stuck in my mind, and the more I thought about it, the more it rings true for me. It makes me think back to when I was in 3rd year engineering. It was a very tough semester, and I had been really sick, missing close to a month's worth of classes. Although I was able to get exemptions for tests and assignments, it doesn't mean you don't have to do them, it just means you have to do twice as many the next week. I was behind, and lost. I was really struggling to understand the material in most of my classes.
Fortunately I had a friend who was a year ahead of me and working on campus. He offered to come over to help me. So with a promise of a homemade spaghetti dinner, he came over to my place and we started to work through assignments. I was particularly lost on linear circuits, and still wasn't able to work out the first assignment, partially from frustration being so overwhelmed, but also because I likely just crammed and memorized what I need the previous year without really understanding the underlying concept.
I remember coming to a point where I needed to use the voltage divider formula for doing a calculation. I said “I can never remember the formula for a voltage divider, do you remember what it is?” and he said to me “It's really easy to derive, so just derive it when you need it.”
I looked at him like he just turned into an alien. To me formula derivations were like mathematical proofs, nice to justify in your mind that what you are doing is correct, but you only do it once, and then you just memorize the formula, plug in the numbers and get the answer.
In retrospect, I am so thankful for my friend's incredible patience. He calmly explained the basics of how a voltage divider works and walked me through the derivation to get to the formula. Inside I was thinking that he was treating me like an idiot, I know basics, I didn't need them explained to me again. Turns out though, that I didn't UNDERSTAND when to apply simple basic concepts to get to the next step building towards something more complicated.
This was my first step to realizing that by trying to take the “easy” route, give me a formula and I'll plug in the values, wasn't the way to solving problems. I started using basic principals to derive all kinds of formulas, and it made everything so much easier. My cheat sheets got smaller, because I didn't need to copy out as many formulas.
This has come up in tutoring as well. Early on when I was just starting tutoring, I had a grade 10 student that just wasn't able to memorize her times tables. She told me about all the different methods teachers have had her do over the years: write them out, recite them over and over, sing them, bounce a ball to them. Nothing worked, and it seemed that all her math problems were wrapped up in the fact that she just couldn't recall her multiplication facts.
She didn't have trouble with all of them, certain ones were easy for her to recall using “tricks”. She amazed me by saying “9 times tables are the easiest.” Really? I always remember it being one of the harder ones to learn. Then she showed me the finger trick, I had never seen it before!
I spent some time in the days following my first meeting with this student, searching for ways to learn multiplication tables without memorizing, because as far as I was concerned, memorization was the only way to learn multiplication tables. I found a site that had a series of lesson's about how to learn multiplication facts WITHOUT memorization. As I read through it, I realized that most of the multiplication facts that I recall, I use various tricks and short cuts, and don't actually recall them using strictly memorization. Some facts don't have (obvious) tricks, which make them more difficult to recall, and I realized it wasn't coincidence then that I have trouble recalling those same facts myself. (I've since learned other techniques that help figure it out if something doesn't come to mind right away).
When I use the term “tricks” above, I'm not talking about cheats or anything non-mathematical. It's from exploring and applying natural patterns and interesting mathematical properties to quickly figure out what the answer is. With almost all of them, they can be expanded to be used well beyond 10x10, and can help with mental calculations of all kinds of things. In other words, taking something basic, and applying to something more complicated.
I recently had a session with a student doing volume and surface area calculations. Again, the student is shown a series of formulas used for various shapes. Many students I've had in the past struggle with recalling the formulas. I recognize myself in them, they look at me quizzically: “I don't remember the formula for that....” in hopes I'll just give it to them. So I go through with them how to derive the formula.
Of course they object, because it takes too long, and they don't have that much time on tests. I quietly take in a calming yoga breath, and patiently explain that this is the best way to recall this information, and will help them in the long run, and go through the derivation with them. I'm not sure I always entirely convince every student, but over time I can see ideas shifting in student's mind on how to effectively solve problems.
All the complicated stuff starts off simple, and bit by bit more simple things are added to it, until in the end you have something complicated. Don't start at the complicated end, start at the simple end, and work your way up, step by step, to the solution, and it won't seem so complicated any more. That's not just my thought, it was what Chris Hadfield said, and he's an astronaut! And that was his answer to a question from a room full of engineers!
Even astronauts and engineers start from the understanding of the basics. Even if you don't aspire to be an astronaut or engineer, you need basics to get through school to move on to what ever you want to do in life. So what are you struggling with? Tutors can help break it down and explain the basics so you understand each step needed to get to the solution to your problem, and the complicated stuff will start looking easier.