by Barbara Oakley
Note: This A Mind For Numbers summary is part of an ongoing project to summarise The 70 Best Books on Learning.
A Mind For Numbers Overview
“A wonderful summary… one of the best we’ve ever seen.”
Barbara Oakley, author of A Mind for Numbers
Though the title promises improvements in math and science, A Mind for Numbers lessons are applicable to all forms of learning and problem-solving.
Barbara Oakley explains that learning begins with creating and internalising small chunks of information (e.g., starting a car, pressing the accelerator, changing gear). As we learn, we add to and connect these chunks and use them to solve bigger problems in the world around us (e.g., driving). This is especially useful in analytical problem solving where chunks allow our limited working memory to increase how much information we can process.
She shows that to efficiently add to and apply these chunks requires learning to use and balance two modes of thought: focussed and diffuse thinking.
What follows in the rest of the book is an excellent summary of practical tips to improve anyone’s learning and problem solving as well as some of the common pitfalls we face on the way (and simple ways to overcome them).
Enjoy this A Mind For Numbers summary? Why not check out Barb’s excellent (and FREE) Learning How to Learn Course over on Coursera!
For now though, let’s dive into the summary of the book.
A Mind For Numbers Summary
Effective learning needs Focussed and Diffuse thinking.
Chunking is at the cornerstone of learning and problem-solving…
- Learning = linking information together to create and slowly add to conceptual chunks
- Problem-solving = Identifying what chunks to use when (and how) to tackle a specific problem
… and we tackle it with two types of thinking.
- Focused thinking (requiring active attention) is conscious, analytical and serial in nature.
- Diffuse thinking (requiring passive attention) is subconscious, creative and parallel in nature.
(Diffuse thinking is what’s going on when you have that “Aha” moment whilst doing something totally different like sleeping, running errands or enjoying a shower)
Both types of thinking are involved in learning…
- Focussed: Gathering information and forming new chunks
- Diffused: Connecting different chunks together
… and both are required in using that learning effectively.
- Focussed: Identifying and loading chunks into working memory for analytical problem solving
- Diffused: Big picture and lateral thinking, sense-checking, creative out of the box thinking
The Medici Effect is the name given to creatively linking seemingly totally separate chunks together to create a new and creative solution
We often learn sub-optimally because we fail to set up and/or alternate effectively between both modes.
- We are too distracted or engaged in attentional multitasking to think deeply (focused)
- We fool ourselves into thinking following is the same as understanding
- We over champion analytical (focused) thought and fail to leverage the power of diffuse thinking
Meanwhile, almost every single successful scientist, author and artist in recent history used a daily routine that effectively set up and then alternated between focussed and diffuse thinking (see the popular book, Daily Rituals).
So, how can we learn effectively?
1. Create the best conditions for focused and diffuse thinking
Focused thinking needs meaningful stretches of undisturbed time to focus and think.
- Prioritise making distraction-free time and space to think deeply
- (SQ3R: Survey, Question, Read, Recall, Review)
- Practice purposefully (Work the hardest bits, generalise through variation)
- Think on paper, there’s magic between the hand and the brain
Diffuse thinking occurs subconsciously by temporarily loosening attention.
- Relax/disengage attention: Sleep, walk, drive, blink, exercise
- Keep information fresh: Recall and test frequently (e.g., spaced repetition techniques)
Both Dali and Eddison used a form of napping in which objects dropping from their hands would wake them up just as they were falling asleep to trigger diffuse thinking.
2. Actively build time into each day to alternate between both modes
You can’t do anything about these and you’re not alone so don’t worry!
Robust learning takes a long time (quickly learned = quickly forgotten)
- Learning has a similar gain profile and risks as physical training
- Long-term learning needs long-term physical changes in brain structure
- Be very wary of short-term cramming and illusions of confidence
Occasional knowledge collapse is inevitable, natural and temporary
- Information often outgrows initial organising structures and mental models we have built for them
- At this point, the brain needs a bit of time to break down and reshuffle chunk/model reshuffle the information we have learned
- This process can last as long as one or two weeks depending on the rate of learning
Think of it a little like defragmenting an old hard drive, or knocking down a house that’s been patched together over time to build a new and improved one from the materials.
However many common pitfalls are easily avoidable.
The difference between great and average thinkers is the way in which they frame and approach many of the following avoidable pitfalls.
Procrastination (stress also further inhibits learning)
- Focus on process instead of product (Pomodoro technique)
- Use to-do lists (weekly into daily, only add if urgent and important)
- Get organised in advance (make productivity the course of least resistance: layout clothes, tidy workspace etc…)
- Eat your frogs first (do the hardest task of the day first)
- Set a quitting time (work backwards, avoid Parkinson’s law)
Distraction (including multitasking – has big switching costs and depletes limited willpower resources)
- Eliminate cues (disable phone notifications, delete apps, block websites)
- Find a quiet space (early) / buy noise cancelling headphones
- Learn to note and then ignore cues (mindfulness / meditation)
Getting stuck (see Einstellung effect often as a result of too much focused thinking)
- Consciously alternate diffuse and focused thinking within your day
- Work with others who are honest and aligned with your best interests
- Set a quitting time each day (also good for your health!)
Confirmation bias (overconfidence in your own solution without checking)
Again, find and work with others who are honest and aligned with your best interests
Illusions of confidence (following as opposed to understanding)
- Work the problem yourself first (avoid solution viewing)
- Recall frequently (at the end of each chapter, how would you teach this)
- Avoid passive re-reading
- Avoid excessive over-learning (working same problem type over and over)
- Test yourself frequently
Fatigue (increasingly proven to be caused by build up of toxins in brain)
- Refuel (short-term, the brain consumes 25% of glucose in our body at rest)
- Exercise (short-term, increases blood flow, promotes diffuse thinking)
- Sleep (mid-term, flushes toxins from brain, promotes diffuse thinking)
- Take holidays (long-term, allows recovery, time for big-picture thinking)
Here are a few more books I’d recommend if you enjoyed this A Mind For Numbers Summary:
“Deep Thinking: What Mathematics Can Teach Us About the Mind”, William Byers: A great use of an experience we all remember (learning basic math) to show the importance of creativity in making cognitive learning leaps.
“The Art of Learning: An Inner Journey to Optimal Performance”, Josh Waitzkin: A superb first hand account showing the overlap of basic deep learning principles in two very different fields (chess and martial arts) from the protagonist of “Searching for Bobby Fisher”.
“Getting Things Done: The Art of Stress-Free Productivity”, David Allen: A cornerstone and very practical work on productivity from the world renowned David Allen – easily one of my top most life-changing reads.
“Deep Work: Rules for Focused Success in a Distracted World”, Cal Newport: Another excellent book that emphasises the importance of making (as opposed to just finding) sufficiently long undistracted periods of time to engage in focussed, deep thought.
“Daily Rituals: How Great Minds Make Time, Find Inspiration, and Get to Work”, Mason Currey: A wonderful book full of examples of the daily routines of some of history’s most famous scientists, authors, poets and artists. Perfect for dipping in and out of and giving great real life examples of how some of our best minds have used the principles explained in Barbara’s book.
A Mind For Numbers Quotes
These A Mind For Numbers quotes come from TANQ – The Art of Living‘s growing central library of thoughts, anecdotes, notes, and quotes.
The Einteilung Effect: “An idea you already have in min, or your simple initial thought, prevents a better idea of solution from being found.”
“Articulating your question is 80 percent of the battle. By the time you’ve figured out what’s confusing, you’re likely to have answered the question yourself!”
“Surrealist painter Salvador Dali like Thomas Edison, also used a nap and the clatter of an object falling from his hand to tap into his diffuse-mode creative perspectives.”
“Learning is often paradoxical. The very thing we need in order to learn impedes our ability to learn. We need to focus intently to be able to solve problems – yet that focus can also block us from accessing the fresh approach we may need. Success is important, but critically, so if failure. Persistence is key – but misplaced persistence causes needless frustration.”
“Intention to learn is helpful only if it leads to the use of good learning strategies.”
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Enjoyed this A Mind For Numbers summary? You might enjoy the rest of the books on this list of The 70 Best Books on Learning.