Studying (math focus)

This winter I’ve been teaching math with my dad (how lucky to get to try that!). In his teaching, he stresses the importance of reading. When we are young the majority of the math books are problems to solve. We learn that math is about solving problems. Later on, the textbooks change and become more and more text. This is when more people start having a problem with math. By only trying to solve the problems, first of all, the majority of the book is skipped (ever read one-third of a history book and expected to do well?). Secondly, it’s as hard as trying to read the text in a foreign language without learning the words first. Intuition can help to understand things, but there is no intuition for knowing that a right angle is 90° or that a rectangle has four right angles or what absolute value is. It’s just a concept to learn and memorize. I was teaching 12-year-old girl math and asked her what she did if she didn’t know or remember some math concept. She was not sure. Then I asked her what she did if she didn’t know some word in English. The answer was easy, look it up and repeat it 100 times!

“Look it up and repeat it 100 times!”


Being on the other side of the table, now teaching, has made me think a lot about how to study. I already wrote one article before about studying (Studying vs studying) so I had given this some thought but this time I am focusing more on math and the overall studying process, step by step. I gave a lecture about it to our students and they were happy about having this discussion. It also helped later on when I could remind them of this tip or that tool that would help them in some situations. Here is the summary of my thoughts:

I decided to divide the studying process into

READ -> UNDERSTAND -> REMEMBER -> USE

These steps don’t happen completely linearly, they mix but what is sure is that you have to start by reading(or hearing) the material and the main goal is to be able to use it. For each step, I will give a few points and thoughts on how to make it easier and more effective.

READ

New material, ouch. You know the feeling (or if you don’t I’m happy for you!). If you want to learn something new there will always have to be the time when you see it for the first time and it’s hard. But there is nothing wrong with that. Be kind to yourself, if you don’t know it, it’s because you are learning something new (well done you!). That first time when you are reading about something new it’s okay to not understand or remember everything, that’s what the other 99 times(or so) are for. It just has to be done (eat the frog). It might help to get that painful first time over with before class. Or at least I could imagine that doing it at home, on the sofa, with a warm cup of coffee in hand will make it a little bit more pleasing.

UNDERSTAND

This is most often the next step for me but sometimes it comes with repetition of the other steps. Also known as learning by doing. But I think it’s still important to always spend some time beforehand to learn the main concepts. Another thing that help with understanding is to write it down. By making notes the learning is tripled. All at once you read the material, you write it and you see it. To follow up on that it also helps a lot to say it out loud, or even explain it to someone else. If no one has an ear to lend, simply talk to a rubber duck. When you write something down or say it out loud you are forced into taking that chaos in your head and put it into words that (hopefully) make sense. Another thing that works well is to rephrase the material. Are you able to say the things with your own words? That’s a good proof that you understand the material.

REMEMBER

This is the thing that should be most straight forward but it requires a lot of work. Just because it’s not common to think about math as a subject that focuses on remembering don’t skip the part that everything else is built on top of. Use ALL the tools you already know to get this step done. I am talking about repetition over a longer period (forgetting curve), mind maps, flashcards, put the notes on your fridge or make a friend quiz you at the gym. Another thing that I like to do when re-reading the example problems is to only look at one line at a time and look away and see if I remember what comes next before continuing (active reading). It’s also possible to take notes from your notes to slim down the essential part.

USE

Finally, time to make use of all the other work. While going through the examples for the first time, make use of the time by putting X in front of the examples that need another repetition or write a side note with a colorful pen regarding that new trick or repeated errors. Create a routine around solving problems, here is an example:

  1. Read the question and try to understand what is being asked for
  2. Highlight keywords and numbers
  3. Write down the things you know and will probably have to use (equations, concepts or extract the essential information from the question)
  4. One step at a time, derive an answer (no need to know the answer, to begin with, just know the next step)
  5. Using your answer calculate backward to see if it makes sense
  6. Read the question again to confirm you gave the answer asked for

Do fewer calculations in your head, get it down on the paper. The human brain can only hold up to 7 things in short-term memory so don’t waste it on things you could have already written down. Also writing every step down can prevent clumsy mistakes because they are more likely to happen in our heads. Once it’s written down you automatically review it and can spot it. Having your calculations readable also makes it easier to read over later for review. When showing up for tests also get all those thoughts you squeezed in last minute down on the paper to clear up your 7 things and to be able to review later. Then test the test situation. The idea well known from sports is to try the competition situation beforehand. Don’t make the first time you solve problems without any material be during the test itself. Knowing that you’ve done it before gives confidence and limits uncertainty.

Some of the steps come more naturally than others. What I struggle with the most is getting that first reading off, memorization and underestimating practicing solving the problems (one time doing it correctly is not enough). So if you’ve realized your weakness pick a tip or two that might help and just start with that. Once that’s in the routine (if it works for you) you can add to it. Next fall I will start my master’s in Computer Science at ETH in Zurich (looking forward to joining @hannaragnars and @thorhildurthorleiks!!) and planning to keep a close eye on myself, that I use some of this myself!

tl;dr
READ
– Get that first reading done (even though is almost physically hurts)
– No need to understand everything the first time
– Not understanding means learning something new (woohoo!)
UNDERSTAND
– Write it down
– Say it out loud (rubber duck)
– Rephrase
REMEMBER
– Repetition
– Mind maps and Flashcards (and all the other tools you already know)
– Active reading
USE
– One step at a time (routine)
– The brain can only hold 7 things in short-term memory so write EVERYTHING down
– Test the test situation beforehand

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