A mind for numbers by Barbara Oakley
How to excel at math and science.
[I attended Professor's MOOC course and I even use some of the technique that was taught in the class]
You will be surprised at how spending a minute or two glancing ahead before you read in depth will help you organize your thoughts. You are creating little neural hooks to hang your thinking on, making it easier to grasp the concepts.
Two different types of networks that the brain switches between - highly attentive state and more relaxed state networks. We will call them the focused mode and diffuse mode respectively.
It is typical to be stumped by new concepts and problems when we first focus on them
To figure out new ideas and solve problems, it is important not only to focus initially, but also to subsequently turn our focus away from what we want to learn
The Einstellung effect refers to getting stuck in solving a problem or understanding a concept as a result of becoming fixated on a flawed approach. Switching modes from focused to diffuse can help free you from this effect. Keep in mind, then, that sometime you will need to be flexible in your thinking,. You may need to switch modes to solve a problem or understand a concept. Your initial ideas about problem solving can sometimes be very misleading.
Use the focused mode to first start grappling with concepts and problems in math and science
After you have done your first hard focused work, allow the diffuse mode to take over. Relax and do something different
When frustration arises, it is time to switch your attention to allow the diffuse mode to begin working in the background
It is best to work at math & science in small doses - a little every day. This gives both the focused and diffuse modes the time they need to do their thing so you can understand what you are learning. That is how solid neural structures are built.
if procrastination is an issue, try setting a timer for 25 minutes and focusing intently on your task without allowing yourself to be drawn aside by text message, web surfing or other attractive distractions
There are two major memory systems: working memory & long term memory.
- Spaced repetition helps move items from working memory to long term memory
- Sleep is a critical part of the learning process. It helps you:
- make the neural connections needed for normal thinking process -
- Figure out tough problems and find meaning in what you are learning
- Strengthen and rehearse the importance of what you are learning and prune away tribalistas
Practice helps build strong neural patterns - that is conceptual chunks of understanding
Practice gives you the mental fluidity and agility you need for tests
Chunks are best built with:
- Focused attention
- Understanding of the basic idea
- Practice helps you gain big-picture context.
Simple recall - trying to remember the key points without looking at the page -is one of the best ways to help the chunking process along
We procrastinate about things that make us feel uncomfortable. But what makes us feel good temporarily isn’t necessarily good for us in the long run
Procrastination can be like taking tiny amounts of poison. It may not seem harmful at the time. But the long term effects can be very damaging.
A little bit of work on something that feels painful can ultimately be very beneficial
Habits such as procrastination have four parts;
- The cue
- The routine
- The reward
- The belief
Change a habit by responding differently to a cue ir even avoiding that cue altogether. Reward and belief make the change long-lasting
Focus on the process (the way you spend time) instead of the product (what you want to accomplish)
Use the 25 minute Pomodoro to stay productive for brief periods. Then reward yourself after each successful period of focused attention
Be sure to schedule free time to nurture your diffuse mode
Mental contrasting is a powerful motivating technique - think about the worst aspects of your present or past experiences and contrast these with the upbeat vision of your future
Multitasking means that you are not able to make full, rich connections in your thinking,. because the part of your brain that helps make connections is constantly being pulled away before neural connections can be firmed up (there is a switching cost when brain changes focus from one task to another and hence complete one task before going to another task)
Chunking means integrating a concept into one smoothly connected neural thought pattern
Chunking helps increase the amount of working memory you have available
Building a chunked library of concepts and solutions helps build intuition in problem solving
When you are building a chunked library, it is important to keep deliberate focus on some of the toughest concepts and aspects of problem solving
Occasionally you can study hard and fate deals a bad hand. Bue remember the Law of Serendipity: If you prepare well by practicing and building a good mental library, you will find that luck will be increasingly on your side. In other words, you guarantee failure if you don’t try, but those who consistently give it a good effort will experience many more success.
Steps to building a powerful chunk:
- Work a key problem all the way through on paper
- Do another repetition of hte problem, paying attention to the key process
- Take a break’
- Do another repetition
- Add a new problem
- Do ‘active’ repetition (mentally review key problem steps in your mind while doing something active, such as walking to the library or exercising).
Mental tricks can be powerful tools. The following are some of the most effective:
- Put yourself in a place with few interruptions, such as a library, to help with procrastination
- Practice ignoring distracting thoughts by simply letting them drift past
- If your attitude is troubled, reframe your focus to shift attention from the negative to the positive
- Realize it is perfectly normal to sit down with a few negative feelings about beginning your work.
- Planning your life for playtime, is one of the most important things you can do to prevent procrastination and one of the most important reasons to avoid procrastination
- At the heart of procrastination prevention is a reasonable daily to-do list, with a weekly once-over to ensure you are on track from a big-picture perspective
- Write your daily task list the evening before
- Eat our frogs first.
Procrastination is such an important topic that this summary includes key takeaway points from all this book's chapters on overcoming procrastination
- Keep a planner-journal so you can easily track when you reach your goals and observe what does and doesn’t work
- Commit yourself to certain routines and tasks each day
- Write your planned tasks out the night before so your brain has time to dwell on your goals to help ensure success
- Arrange your work into a series of small challenges. Always make sure you (and your zombies) get lots of rewards. Take a few minutes to savor the feeling of happiness and triumph
- Deliberately delay rewards until you have finished a task
- Watch for procrastination cues
- Put yourself in new surroundings with few procrastination cues, such as the quiet section of a library
- Obstacles arise, but don’t make a practice of blaming all your problems on external factors. If everything is always somebody else's’ fault, it is time to start looking in the mirror
- Gain trust in your new system. You want to work hard during times of focused concentration - and also trust your system enough that when it comes time to relax, you actually relax without feelings of guilt
- Have backup plans for when you still procrastinate No one is perfect, after all
- Eat your frogs first.
The memory palace technique - placing memorable nudges in a scene that is familiar to you - allows you to dip into the strength of your visual memory system
Learning to use your memory in a more disciplined, yet creative manner helps you learn to focus your attention, even as you create wild, diffuse connections that build stronger memories
By memorizing material you understand, you can internalize the material in a profound way. And you are reinforcing the mental library you need to become a genuine master of the material.
Metaphors can help you learn difficult ideas more quickly
Repetition is critical in allowing you to firm up what you want to remember before the ideas fade away
Meaningful groups and abbreviations can allow you to simplify and chunk what you are trying to learn so you can store it more easily in memory
Stories - even if they are just used as silly memory tricks - can allow you to more easily retain what you are trying to learn
Writing and saying what you are trying to learn seems to enhance retention.
Exercise is powerfully important in helping your neurons to grow and make new connections.
At some point, after you got chunked material well in hand (and in brain), you start to let go of conscious awareness of every little detail and do things automatically
In may seem intimidating to work alongside other students who grasp material more quickly than you do. But ‘average’ students can sometimes have advantages when it comes to initiative, ability to get things done, and creativity.
Part of the key creativity is to be able to switch from full focused concentration to the relaxed, day dreaming diffuse mode
Focusing too intently can inhibit the solution you are seeking - like trying to hammer a screw because you think it's a nail. When you are stuck, sometimes it is best to get away from a problem for a while and, move on to something else or to simply sleep on it.
Brains mature at different speeds. Many people do not develop maturity until their mid twenties.
Some of the most formidable heavyweights in science started out as apparently hopeless juvenile delinquents
One trait that successful professionals in science, math and technology gradually learn is how to chunk - to abstract key ideas
Metaphors and physical analogies form chunks that can allow ides from very different areas to influence one another
Regardless of your current or intended career path, keep your mind open and ensure that math and science are in your learning repertoire. This gives you a rich reserve of chunks to help you be smarter about your approach to all sorts of life and career challenges.
Equations are just ways of abstracting and simplifying concepts. This means that equations contain deeper meaning, similar to the depth of meaning found in poetry
Your mind’s eye is important it can help you stage plays and personalize what you are learning about
Transfer is the ability to take what you learn in one context and apply it to something else.
It is important to grasp chunked essence of a mathematical concept, because then it is easier to transfer and apply that idea in new and different ways
Multitasking during the learning process means you don’t learn as deeply - this can inhibit your ability to transfer what you are learning.
Learning on your own is one of the deepest, most effective ways to approach learning:
- It improves your ability to think independently
- It can help your answer the strange questions that teachers sometimes throw at you on tests
In learning, persistence is often far more important than intelligence
Train yourself to occasionally reach out to people you admire., You can gain wise new mentors who with a simple sentence, can change the course of your future. But use your teachers’ and mentor’ time sparingly
If you are not very fast at grasping the essentials of whatever you are studying, don’t despair. Surprisingly, often, slower students are grappling with fundamentally important issues that quicker students miss. When you finally get what’s going on, you can get it at a deeper level
People are competitive as well as cooperative. There will always be those who criticize or attempt to undermine any effort or achievements you make. Learn to deal dispassionately with these issues.
The focused mode can allow you to make critical errors even though you feel confident you have done everything correctly. Rechecking your work can allow you to get a broader perspective on it, using slightly different neural processes that can allow you to catch blunders.
Working with others who are not afraid to disagree can:
- Help you catch errors in your thinking
- Make it easier for you to think on your feel and react well in stressful situations
- Improve your learning by ensuring that you really understand what you are explaining to others and reinforcing what you know
- Build important career connections and help steer your reward better choices
Criticism in your studies whether you are giving or receiving it, shouldn’t be taken as being about you. iIt is about what you are trying to understand
It is easiest of all to fool yourself.
Not setting up enough sleep the night before a test can negate any other preparation you have done
Taking a test is serious business. Just as fighter pilots and doctors go through checklists, going through your own test preparation checklist can vastly improve your chances of success
Counter-intuitive strategies such as the hard-start-jump-to easy technique can give your brain a chance to reflect on harder challenges even as you are focusing on other more straightforward problems
The body puts out chemicals when it is under stress. How you interpret your body’s reaction to these chemicals makes all the difference. If you shift your thinking from ‘this test has made me afraid’ to ‘this test has got me excited to do my best’ it helps improve your performance.
If you are panicked on a test, momentarily turn your attention to your breathing relax your stomach, place your hand on it and slowly draw a deep breath. your hand should move outward and your whole chest should expand like a barrel
Your mind can trick you into thinking that what you have done is correct, even if it isn’t.. This means that whenever possible, you should blink, shift your attention, and then double-check yours answers using a big-picture perspective asking yourself ‘Does this really make sense?’
10 Rules of Good Studying
1. Use recall. After you read a page, look away and recall the main ideas. Highlight very little, and never highlight anything you haven’t put in your mind first by recalling. Try recalling main ideas when you are walking to class or in a different room from where you originally learned it. An ability to recall—to generate the ideas from inside yourself — is one of the key indicators of good learning.
2. Test yourself. On everything. All the time. Flash cards are your friend.
3. Chunk your problems. Chunking is understanding and practicing with a problem solution so that it can all come to mind in a flash. After you solve a problem, rehearse it. Make sure you can solve it cold—every step. Pretend it’s a song and learn to play it over and over again in your mind, so the information combines into one smooth chunk you can pull up whenever you want.
4. Space your repetition. Spread out your learning in any subject a little every day, just like an athlete. Your brain is like a muscle—it can handle only a limited amount of exercise on one subject at a time.
5. Alternate different problem-solving techniques during your practice. Never practice too long at any one session using only one problem-solving technique—after a while, you are just mimicking what you did on the previous problem. Mix it up and work on different types of problems. This teaches you both how and when to use a technique. (Books generally are not set up this way, so you’ll need to do this on your own.) After every assignment and test, go over your errors, make sure you understand why you made them, and then rework your solutions. To study most effectively, hand-write (don’t type) a problem on one side of a flash card and the solution on the other. (Handwriting builds stronger neural structures in memory than typing.) You might also photograph the card if you want to load it into a study app on your smartphone. Quiz yourself randomly on different types of problems. Another way to do this is to randomly flip through your book, pick out a problem, and see whether you can solve it cold.
6. Take breaks. It is common to be unable to solve problems or figure out concepts in math or science the first time you encounter them. This is why a little study every day is much better than a lot of studying all at once. When you get frustrated with a math or science problem, take a break so that another part of your mind can take over and work in the background.
7. Use explanatory questioning and simple analogies. Whenever you are struggling with a concept, think to yourself, How can I explain this so that a ten-year-old could understand it? Using an analogy really helps, like saying that the flow of electricity is like the flow of water. Don’t just think your explanation—say it out loud or put it in writing. The additional effort of speaking and writing allows you to more deeply encode (that is, convert into neural memory structures) what you are learning.
8. Focus. Turn off all interrupting beeps and alarms on your phone and computer, and then turn on a timer for twenty-five minutes. Focus intently for those twenty-five minutes and try to work as diligently as you can. After the timer goes off, give yourself a small, fun reward. A few of these sessions in a day can really move your studies forward. Try to set up times and places where studying—not glancing at your computer or phone—is just something you naturally do.
9. Eat your frogs first. Do the hardest thing earliest in the day, when you are fresh.
10. Make a mental contrast. Imagine where you’ve come from and contrast that with the dream of where your studies will take you. Post a picture or words in your work-space to remind you of your dream. Look at that when you find your motivation lagging. This work will pay off both for you and those you love!
Ten Rules of Bad Studying
Avoid these techniques—they can waste your time even while they fool you into thinking you’re learning!
1. Passive rereading—sitting passively and running your eyes back over a page. Unless you can prove that the material is moving into your brain by recalling the main ideas without looking at the page, rereading is a waste of time.
2. Letting highlights overwhelm you. Highlighting your text can fool your mind into thinking you are putting something in your brain, when all you’re really doing is moving your hand. A little highlighting here and there is okay—sometimes it can be helpful in flagging important points. But if you are using highlighting as a memory tool, make sure that what you mark is also going into your brain.
3. Merely glancing at a problem’s solution and thinking you know how to do it. This is one of the worst errors students make while studying. You need to be able to solve a problem step-by-step, without looking at the solution.
4. Waiting until the last minute to study. Would you cram at the last minute if you were practicing for a track meet? Your brain is like a muscle—it can handle only a limited amount of exercise on one subject at a time.
5. Repeatedly solving problems of the same type that you already know how to solve. If you just sit around solving similar problems during your practice, you’re not actually preparing for a test—it’s like preparing for a big basketball game by just practicing your dribbling.
6. Letting study sessions with friends turn into chat sessions. Checking your problem solving with friends, and quizzing one another on what you know, can make learning more fun, expose flaws in your thinking, and deepen your learning. But if your joint study sessions turn to fun before the work is done, you’re wasting your time and should find another study group.
7. Neglecting to read the textbook before you start working problems. Would you dive into a pool before you knew how to swim? The textbook is your swimming instructor—it guides you toward the answers. You will flounder and waste your time if you don’t bother to read it. Before you begin to read, however, take a quick glance over the chapter or section to get a sense of what it’s about.
8. Not checking with your instructors or classmates to clear up points of confusion.
Professors are used to lost students coming in for guidance—it’s our job to help you. The students we worry about are the ones who don’t come in. Don’t be one of those students.
9. Thinking you can learn deeply when you are being constantly distracted. Every tiny pull toward an instant message or conversation means you have less brain power to devote to learning. Every tug of interrupted attention pulls out tiny neural roots before they can grow.
10. Not getting enough sleep. Your brain pieces together problem-solving techniques when you sleep, and it also practices and repeats whatever you put in mind before you go to sleep. Prolonged fatigue allows toxins to build up in the brain that disrupt the neural connections you need to think quickly and well. If you don’t get a good sleep before a test, NOTHING ELSE YOU HAVE DONE WILL MATTER.