A Mind for Numbers

{2} Easy Does It

This chapter focuses on the two types of thinking that we employ when learning and solving problems: focused and diffuse
Focused thinking refers to the effortful, concetrated type of thinking we employ when first trying to solve problems.
- Characterized by being logical, sequential, grounded.
- Often the result of previously built neural circuits in the brain - we are essentially comparing what we are currently working on to things we’ve learn in the past.
Diffuse thinking refers to more zoned-out, intuitive, creative, and effortless “thinking” that occurs when we stop consciously concentrating on a problem using preconceived notions and information.
- The is the epitome of “ah-ha” moments. It is playful and imaginative, in a sense.
Both types of thinking work together and communicate with one another
- Focused thinking provides raw material for diffuse thinking to come up with a solution.
- Being able to consciously shift back and forth between the two modes of thinking yields better problem solving skills.
Einstellung Effect - results from using flawed modes of initial thinking when solving a problem initially in a focused manner. Our first approach is not necessarily correct, so don’t get stuck on this and instead try approaching it from a completely new angle.
Procrastination is a bad idea in subjects like math or science which require an adequate amount of time and diffuse thinking to digest and understand, since studies show it’s inherently more difficult for the human brain to process than language or social interaction.
Main idea: Don’t approach problems in the same exact way if you’re not getting results, and don’t try so damn hard all of the time.
At the end interview with the student, the student mentioned that she thinks better when she’s occupied doing something like driving - consciously sitting down and trying to think about something doesn’t work. I find this is often the case with me - some of my best insights and solutions come when I am doing something else and thoughts flow freely in and out.

{3} Learning is Creating

The first half of this chapter was a reiteration of the concepts of diffuse and focused thinking covered in the last one.
The focused mode of thinking is extremely important for learning, because it is what shuttles the raw material in your brain. Diffuse thinking alone will not help you learn.
When you spend a long time on a problem and even start becoming frustrated, this is a very good opporunity to completely step away from the problem so the diffuse thinking part of your brain can start working on a solution.
What are some great activities to encourage diffuse thinking?
- Walking, going to the gym, socializing, meditation, listening to music or drawing.
Research has shown that spaced repetition is the key to learning.
- In the ideal situation, you would study a little bit of material each and every day. Consistency is key.
- Studying for an hour a day 7 days per week is far more effective than studying for 12 hours in one day.
Sleep is the ultimate mode of diffuse thinking and is repsonsible for moving information from your short term memory to long term memory.
Recalling things in your long term memory strengthens the memories. This is why quiz and recall type of studying is so important.
One of the suggestions at the end of the book was to put the book down, walk around, and recall everything I’d learned in the chapter. After doing this, I found that I did not have to even look at the chapter again to write these notes.
- A great skill to implement in whatever I learn is walking away and trying to recall what the fuck just happened.
Some things to think about:
- What can you do to be absolutely sure you haven’t explored other alternate perspectives for solving some problems? I think that completely forgetting about the problem and doing something else is key.
- What can you do to make sure you review a bit of the material every day? I think keeping a journal with TO-DO’s is great for this.
- There is a negative correlation between levels of creativity and agreeableness! Yet another reason to dissent and go against the grain.

{4} Chunking and Avoiding Illusions of Incompetence

Chunks are pieces of information that are bound together through meaning, and allow us to store things in conceptual packages instead of memorizing every small detail without context to other information.
- Once you chunk a piece of information, there’s no need to know all the underlying details - you can make intuitive leaps in math and science with chunking.
There is a process of turning raw information to chunks in order to solve any problem
- First, focus your attention very carefully and intensely on what you want to understand. It’s hard to make good connections in your brain if some parts are focused on other things.
- Understand the main conceptual ideas. Have it “click” - this is that “aha” moment.
- Gain context to you see not just how, but when to apply the conceptual chunk. This is done through a lot of practice and failing.
“Top down” big picture learning and “bottom up” chunking provide context for any problem you need to solve.
The best overall strategy for learning is recall. This is quizzing yourself or trying to actively recall information and main ideas in your mind or out loud. This is far more effective than often useless exercises such as highlighting, underlining, and rereading multiple times.
- This is a great habit to have when reading non-fiction books. I will personally try to get more out of them by closing the book for a bit after each chapter and recalling all of the key ideas and pieces of information. It’s helpful to look back if you need to in order to check understanding. Writing short summaries for pages is also helpful.
- When learning something new and challenging, try to come back to it every day. Spaced and consistent repetition, along with recall, is the best learning method.
- Research has found recall to be vastly superior to things like trying to create “mind maps”. Most of the time, brute rote memorization and chunking is essential to understand a subject and become skilled in it.
The larger amount of chunked information you have in your mental library, the better you will perform and the more creatively you will be able to think in a subject.
- This is why it’s really important for me to hammer down those core CS concepts whenever I get the chance.
Interestingly enough, research shows that recalling material outside of the environment you typically study it in helps learning as you view it from a different perspective. This is why many breakthroughs have taken place during walks.
To avoid overlearning, it’s essential that you mix a variety of different problems together. It’s not really helpful to continue working on the same type of problems after you’ve gotten one completely correct.
My most common illusion of competence in learning stems from my laziness. I go through flashcards or problems, have a vague intuition of the answer, and then just move on without actively writing down or solving anything. I’ve blanked on tests before because of this. I’m now trying to make it a habit to really write everything down out loud or talk to myself, verbalizing understandings and information.
- I also find myself too focused on trying to write good notes to pay attention to lectures or books sometimes. It’s more helpful to read and understand chapters or watch lecture videos, and then skim back over it a second time, taking notes, I think. Before you do this, however, you can try actively recalling whatever you remember.

{5} Preventing Procrastination

We procrastinate things which make us feel pain or discomfort. However, it’s the anticipation that it uncomfortable. When it comes to actually doing the thing, there is no discomfort.
Procrastination is sort of like an addiction. It offers relief from a boring or painful reality, which is why looking at youtube videos instead of going over complex math problems makes us feel better temporarily, but even shittier and more anxious at the end.
The end of the chapter asks if procrastination has had an impact on my life and it what ways. It absolutely has - for the majority of my college career, I was depressed, lazy, and constantly procrastinated. I dug myself into a hole and a self-imposed hell where I was constantly anxious about my grades and performance because my procrastination got so bad I wouldn’t start the homework or studying until like fucking 15 minutes before class started. I also had horrific studying techniques of rereading something or looking at solutions and thinking I understood it. Fuck man, I was bad. Luckily, I made a dramatic shift in my studying and learning habits towards the end of my college career and finished with a 4.0 my last semester while taking 300 level CS courses. Better late than never, I suppose, and these habits have served me extremely well in my professional life and my personal life of self-learning and development.

{6} Zombies Everywhere

Habitual actions save energy. You don’t have to think about them - they simply happen. Therefore, habits are much more powerful than willpower.
- Cue: What you perceive and then respond to, because it is associated with your habits. An item on a TO-DO list, your computer, a meditation pillow.
- Routine: Seeing a cue and going into zombie mode. Can be used for self development or destruction, depending on the habit.
- Reward: Habits typically have a stronger chance of continuing if you strengthen them. Finding ways to reward yourself for study and problem solving is a great way to end procrastination.
- Belief: To change habits, you must change the underlying belief.
You need to create a systenm in order to crush bad habits and develop new ones.
- Willpower is only required initially, to respond to the action of a cue. This is “catching yourself in the act” so to speak, and then deciding to resist what you’re about to do. It is also required for taking the first steps to establish a habit.
- The key to rewiring your habits is to have a plan and developing new rituals. Going to certain places to study, listening to a certain playlist, working at a certain time of the day, such as right after work in my case.
- Rewards are very important. Your brain will be more likely to perform a set of tasks if it knows some good shit is about to happen right after. I personally reward myself by getting on Steam after study sessions and playing whatever the fuck I want to for awhile while I forget about everything.
A good motivator is found in mental constrasting. Where do you truly desire yourself to be at, and how does that compare to your current state?
Important: Product vs. Process
- Product is the end result of something. Finishing a homework assignment. Writing a chapter of a book. Getting promoted.
- Process is the flow of time and the habits associated with that flow of time.
- It’s extremely important to focus on the process rather than the product when it comes to habit formation and productivity. Focusing on the product causes the pain of procrastination. I find it helpful to, instead of saying get to Chapter 10 of a book, for example, to just say work for 3, 25-minute sessions. You can stay in the moment and achieve a state of flow without worrying about accomplishing some set goal the entire time. Process and flow go hand in hand - with flow, you aren’t judging yourself with thoughts like “am I close to the goal yet?”
Chapter summary mentions scheduling free time for diffuse nurturing. I think everyone should be reminded of this. Free time and recreation are absolutely vital for learning and energy levels.

{7} Chunking vs. Choking: Increasing Expertise and Reducing Anxiety

Here are the steps to building a powerful chunk:
- Work problem all the way through - Each step must make sense, nothing should be skipped.
- Do another repetition of the problem, noting key steps
- Take a break - Study a different aspect of a subject or go work out etc.
- Sleep - Do another repetition of problem before you sleep to consolidate it.
- Do another repetition - The next day, solve it again to engage in deliberate practice, which is essential to learning and consolidation.
- Add a new problem - Pick another key problem and begin solving it the same way. Be mesmerized as you realize having other chunks in your library you’ve created previously makes learning new chunks even easier and more efficient.
- Do active repetitions - At random moments during the day, try to recall key steps and concepts
If you’re pressed for time, just learn a few key problems and concepts to form the framework of your understanding.
Retrieval practice is the most effective learning strategy, period. Rereading material or looking at solutions on paper does basically nothing. It is difficult, but stick with it and always work things out from beginning to end and spend large amounts of effort recalling everything you can remember so it embeds itself further.
Sometimes, you hit a wall in your learning, where it seems like you are no longer progressing. This is temporary, and a result of your mind restructuring the knowledge. Often, you come out of these ruts much better than before you came in them, so chill the fuck out and give it time, man.
Chunking is important because it frees up space in your working memory to make room for important details when solving a problem. This is why crammin doesn’t work - premade chunks do not exist and your working memory is filled up entirely.
Creating and taking mini-tests are the best way to form chunks and learn material. Of course, there are a form of retrieval practice.
- It is vital that you check your understanding by looking at solutions and answers to questions, and understanding why things went wrong.
There is a good definition in the summary about what a chunk really is: “One smoothly connected neural thought pattern integrated into place.”
- Learning progresses much more quickly when you compress things into a few key chunks instead of rote memorization of all details.

{8} Tools, Tips, and Tricks

Reframe your mind. Do not think of all of the reasons why you cannot do something - time is better spent thinking how you can accomplish it given your unique circumstances.
Self-experimentation is the key to change, and observing the results of random experiments in your behavior can have profound change.
- A good recommendation is keeping a journal of your daily habits so you can see what leads you to procrastination.
I’ve been doing this for quite some time after seeing it in Newport’s book, but it’s a good habit to rwite all of your tasks for the next day the night before.
Do your most important and/or disliked tasks first as soon as you begin your block of work.
Developing a schedule provides a large amount of freedom and control over your life.
- Additionally, having a system of goal/task keeping and measurement allows you to get a thrill out of making strides every day, almost like a video game. Crossing an important task off of a list is much like getting an achievement in a video game tbh.
A daily shut down is one of the most important things to have.
- This is where you set a goal time of being done every day. You stop all work regardless of where you are, as long as time permits it, you write down tasks for the next day, and then you shut down and forget about work.
- Not only does this create a sense of urgency towards productivity, but it ensures that you get the vital relaxation each day so you don’t burn out.

{9} Procrastination Zombie Wrap-Up

This chapter once again highlights the importance of a work-life balance
- People who are extremely productive in short binges are often less productive after these binges are over. A better approach is slow and steady daily productivity, making sure to include relaxation each day because you trust the system you’ve developed.
Pausing and reflecting are key not only in procrastination, but math, science, and problem solving.
- The difference between professors and graduate students versus undergraduates is that the professors and grad students actually take longer to categorize a physics problem before they begin creating a solution.
- Always step back, close your eyes, and reframe yourself to see problems from new perspectives instead of barreling in on a single solution. This is certainly a problem of mine - I try to get things done as fast as possible instead of playing with it for longer.
Students who blindly followed their passions without rational analysis or practical consideration end up less happy than students who couple passion with rational analysis.
- As Newport has also said, following your passion is bad advice. Passion is often derived from spending a lot of time in a subject, working hard, and subsequently becoming very good at it. You can’t wait for this shit to just come up and strike your fancy out of the blue. You have to go after it.
Deliberately delay rewards until you’ve finished a task, and make sure to savor the happiness and feelings of accomplishment that come along with accomplishing goals instead of bemoaning yourself for not getting more done.

{10} Enchancing your Memory

Joshua Foer, who wrote Moonwalking with Einstein, went from being someone with relatively bad memory to becoming a top memory champion by employing certain techniques.
The primary technique mentioned in this chapter is the concept of a memory palance.
- This involves using a well known place in real life, and then as you walk through it, you begin seeing in vivid and strange detail the things that you want to remember.
- Visual-spatial memory is one of the most powerful parts of the human brain.
The main idea is that we can employ creative and strange associations between pieces of information in order to remember it better.
Although there is some value in rote memorization, it is far more powerful to use devices which stick for a long time in our head, including songs, acronyms, and visualizations.
An excerpt from a professor of astrophysics at the beginning of the chapter explained how memorization of things is key for playing around with “chunks” and forming good understanding. I couldn’t agree more. Because of instant access to factual information via the internet, many of us think there is no use in memorizing when we can just “look it up.” However, this ignores the fact that memorization of information is usually key to large conceptual understanding.

{11} more memory tips

Creating analogies and metaphors, often visual ones, are powerful tools for learning and memory.
- This is why it’s helpful to learn a broad range of things as well - you have more information in your head to draw associations with.
- Additionally, it is sometimes even helpful to put yourself in the place of what you’re learning, for example, what would it be like to be an electron?
- Importantly, trying to create a metaphor helps break you out of einstellung.
Spaced repetition is vital in the learning process, as it helps transfer information in short term memory to long term memory. If you don’t repeat information frequently, the neural networks associated with them in your brain weaken.
- When learning a new topic, it is helpful to practice and study it a little bit each day, sometimes multiple times per day. Doing this before sleeping is also helpful.
Grouping information is another technique, often paired with memorable sentences or sayings.
Writing is a great technique as well. When writing something, our brain is forced to make connections and hone understanding. Additionally, we may be able to come to new insights - writing out a physics equation may make it come to life.
Overall, understanding mathematics and science in a visual way is one of the most effective ways to learn and understand.
- People can often become experts faster by utilizing a multitude of memory techniques involving visualization and association.
The end of the chapter asks to create a doodle or sketch of something you’re trying to currently learn. I think this is something I will try to implement more often. No one has to know, but perhaps I will learn a lot more information in programming by making weird drawings and strange associations in my head.

{12} learning to appreciate your talent

The first part of this chapter explains how we should work towards an intuitive understanding. In other words, there comes a point when you reach a state of flow and no longer have to think about rules and procedures, which are simply constraints.
- True understanding comes from deep personal understanding and making sense of information.
More traditionally “intelligent” individuals often struggle with procrastination and getting lost in complexity instead of finding simple solutions. They imagine every possible scenario and overthink things, to some extent.
- The author mentions that there is creative value in being someone with a lower working memory capacity/being someone more pronde to ADHD-like diffuse thinking.
  - Having a somewhat smaller working memory means you can generalize your learning into new, creative combinations.
  - Chess has some elite individuals with relatively average IQs.
Deliberate practice can lift average brains into the realm of those who are deemed as “gifted.”
One of the main ideas of this chapter was to focusing on improving what you have to the best of your ability instead of getting hung up on the autistic geniuses in your class/work who seem to understand everything instantly I guess lul.

{13} sculpting your brain

The neuroscientist Cajal grew up as a delinquent child who completely neglected his studies and was horrible at rote memorization. However, as he matured, he began to work very hard at neuroscience to become successful, often developing strong “chunks” by creating detailed drawings and art of the cells that he studied.
We can make significant changes in our brain by changing how we think, which has a direct effect on the neural structure.
Good chunks form abstractions that can carry over to other aspects of our lives, which is my physics and engineering people are often sought after in the field of finance.

{14} developing the minds eye through equation poems

This chapter primarily focused on utilizing various creative abilities of the mind to help understand what you are learning.
- Einstein imagined himself as a photon, while another biology researcher imagined herself interacting at a molecular level and what that would be like.
- Visualization is a powerful tool for understanding, and can be applied to equations and the relationships governing them.
Equations themselves are not simply something else to memorize. It turns out that this is the way novice physicists and mathematicians see them. Conversely, graduate students and professors see the relationships behind the equation.
Mathematical equations and concepts are compared to potry, as they express meaning and various modes of interpretation.
Personalization is a large factor when it comes to learning ability. Explaining a concept via your own explanations and visualizations is the key to learning.
- Feynman and Darwin were both fans of trying to explain things as simply as possible, getting to their very essence.
When you learn the general concepts behind subjects, as you do in an abstracted field like mathematics, you are given the tools to apply these concepts elsewhere.
- This is called transfer learning. This is the reason it is easier to learn a third language after you learn a second.
As I’ve read many other places, this chapter mentions that interruptions in the form of checking email or texts inhibits understanding and transfer learning by impeding the chunk forming process.
- You need significant stretches of focused concentration without interruptions to learn effectively.

{15} renaissance learning

Self learning is often the most effective learning. You learn best when you are engaged with the material. This is why sitting there and passively listening to a lecture doesn’t do much.
- Author provides example of a surgeon who never went to class because he learned best through textbooks.
- Self learning also means that you are taking the initiative yourself, so you are often more motivated and more interested in the material that you’re studying.
You need to try to make the effort to learn and understand the material on your own as much as possible. Relying on teachers for all of the answers, or anyone else for that matter, is a great way to end up not learning anything and instead develop a sense of helplessness.
- I was guilty of this during my times of laziness my first few semesters of college. I was so tired and drained, and I couldn’t make the effort to do work or learn, so I simply got answers wherever I could to prevent my grades from becoming too bad. I felt like I had zero control. I found that taking responsibility for my learning and putting in the effort significantly improved my mood and my levels of confidence. And to some extent, it became fun and enjoyable.
Avoid thinking that you’re answer is always absolutely correct, especially if you’re new to a subject or area.
Learn through multiple different mediums via the internet, lectures, books, and articles to see things from new perspectives.
There will always be people in life who undermine your successes and doubt your capabilities.
- Hunker down and either ignore them, or use this as emotional fuel to prove them wrong and prove yourself right.
- I’ve found it easier to simply ignore these types of things in my current life. I simply don’t give a fuck anymore and realize how fleeting everything is, and I realize how the thoughts in their minds are nothing more than a series of invisible neurochemical impulses and do not mean anything in particular. Realizing this puts everything into perspective.
There is great value in being an autodidact and an independent mind. It is something that I strive for myself. Learning how to learn has been one of the most beneficial skills that I’ve picked up, and they never taught it in school.
- Follow your intuition, respectfully dissent, and if you see everyone else heading a certain way, go the opposite way.
  - I think getting off of the echo chamber that is the internet really helps to foster this mindsed.

{16} avoiding overconfidence

There is a lot of evidence that the right hemisphere of the brain is responsible for playing sort of a “devils advocate” role and allowing us to see the big picture perspective, while the left side of the brain is for focused thinking that wants to stick with its own reasoning.
- There is a potential for rigidity, egocentricity, and dogmatism if you utilize the left side of your brain too much.
- When you blow through homework problems and don’t take the time to double check solutions or see problems from a different perspective, you’re not using the full capabilities of all brain regions.
There is tremendous value in working, studying, and solving problems with others. As long as criticism is not taken personally (it is not you being criticized, it is what you are trying to understand), working together can be very productive.
- Bohr relied on Feynman for brainstorming sessions. Feynman would call his bullshit out because he was the only one not intimidated by the “genius” of Bohr, and Bohr greatly appreciated it.
- Furthermore, explaining your understanding to others can aid in your own learning.
It is easy to fool yourself and pretend that you have the correct answer because it gives you comfort and peace of mind.
- Try to break away from this mindset and double check your solutions to everything. Play the devil’s advocate to make sure things are air-tight.
- Develop the patience to check your solutions and make sure your quantitative answer and intuition are aligned. I often lacked any patience at all to double check my solutions when I was in college because I loathed seeing something again once it was already finished. Mindfulness and developing patience really help with this. It’s worth the time.

{17} test taking

There are quite a few good testing techniques in this chapter that I wish I would’ve known back when I first got to college.
Testing itself is a great learning experience and should be used heavily during studying, especially in closely simulated scenarios like going to a classroom, and timing yourself while note looking at your notes.
Oakley advises to follow a strategy of trying to solve the hardest problems on a test first.
- You look at these problems for 1-2 minutes, and if you cannot solve them, you move on to the other questions on the test.
- Jumping back and forth between problems you cannot immediately solve allows for diffuse mode thinking to work on solutions. These actions are compared to a chef switching between tasks to bring a meal together.
A professor made a good comment about relieving testing anxiet by having some sort of back up-plan for a career if your current one does not work out. “Oh, this is okay, I can just go with plan B if I fail at plan A” counterintuitively helps you do better at plan A by relieving anxiety.
Practice both deep breathing and mindfulness to combat testing anxiety.
- Making a habit of calmly monitoring thoughts as they arise allows you to change your emotional reactions to the thoughts and see them for what they are.
A simple change of attitude goes a long way in improving performance on tests.
- Going from “this test makes me nervous and scared” to “this test is actually making me excited as fuck to see how much I can do” really changes the game.
When finished answering a question, take time to blink and see the big picture to see if what you did makes sense.
- The mind is good at fooling itself to think it’s correct. Shift attention to gain new perspectives.
If you’ve prepared well, don’t do much the day before an exam. Conserve your mental energy, let it all go, and just go in and see how many questions you can answer the next day during the test.

{18} unlock your potential

Sometimes, your desire to figure things out right now is the thing that inhibits you from figuring things out. Once again, patience comes into the equation.
Reshaping your brain is under your control. Utilize your own strengths and weaknesses.
- Concentrate intently to get a problem into your mind, and switch to diffuse mode to help work things out.
The results of weeks and months of sustained effort with periods of relaxation results in fine tuned neural structures and allows us to have fun and see the big picture at the same time, because we are not spending all of our time cramming.
- Big picture, long term persistence is the key to success in any domain. It allows us to overcome any obstacles or people who may tell us otherwise.
Learning is paradoxical in many ways.
- Focused attention, necessary for learning and problem solving, can also inhibit learning and problem solving if carried too far.
- Memorization is necessary to some extent, but memorization without conceptual understanding and chunking doesn’t allow us to see the big picture.
As Feynman did, always challenge yourself to put your understanding into the simplest possible terms.

Lady Luck favors the one who tries

Here are ten “chunked” rules to learning effectively.

Recall. Recall as often as possible. Recall the concepts in class the best you can. Recall a few paragraphs or a chapter after you have read it. Bring as much information to your mind from pure memory as you can to help ingrain it.
Test Yourself. On everything and all the time. Flashcards will become your best friend.
Chunk your problems. Do key problems over and over. Rehearse them until they become effortless so you can quickly grab the chunk from your brain.
Space your repetition. The brain can only handle so much at one time. Give yourself breaks and do a bit each day.
Alternative different problem solving techniques during sessions. Don’t stick to one type of problem or concept. Mix it up and quiz yourself randomly.
Take breaks. Go play Dead Space 3 for an hour or 7.
Use explanatory questioning and simple analogies. Explain by analogy and try to break things down to their very essence. Practice writing things and talking them out loud to more deeply encode the information.
Focus. Turn off phone, block distracting shit on the internet etc, and then focus intently for a set amount of time, giving yourself a break afterwards. Do a few of these each day. This will move along studies surprisingly quickly.
Eat your frogs first. Do the hardest thing the earliest in the day when you’re fresh.
Make a mental contrast. Occasionally visualize where you want to be compared to where you are now. Have motivational wallpaper, quotes, pictures etc. - anything that helps remind you of your goal and helps you to keep pushing forward for the long term reward.

Here are ten ways to ruin everything.

Passive rereading
Highlighting/underlining everything
Glancing at solutions and thinking you understand how to solve it
Waiting until the last minute to study
Repeatedly solving problems of the same type that you already know how to solve
Letting study sessions with friends turn into social hour
Neglecting to read the textbook before you start working problems
Not checking with classmates/instructors to clear up confusion
Thinking you can learn deeply when you are constantly distracted
Not getting enough sleep