A Mind for Numbers by Barbara Oakley, Ph.D.
How to Excel at Math and Science
A Mind for Numbers explores new ways of thinking about how you learn. It is written primarily for students. It has a lot of specific advice related to studying for homework, projects, and working with classmates.
My Favorite Quotes
- Those who are the most disagreeable tend to be the most creative. This is because creative people challenge existing answers and assumptions.
- Merely glancing at the solution to a problem and thinking you truly know it yourself is one of the most common illusions of competence in learning.
- The dread of doing a task uses up more time and energy than doing a task itself.
- 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.
- One might say that work substitutes for talent, or better yet, that work creates talent.
- Simple explanations are possible for almost any concept, no matter how complex.
- How can you avoid falling into the trap of thinking that quicker people are automatically more clever?
- Can you find ways to turn your disadvantages into advantages?
Two Modes of Thinking:
- Focused Mode
- Diffused Mode
When you procrastinate [when studying], you are leaving yourself only enough time to do superficial focused-mode learning. The resulting neural patterns will be faint and fragmented, and will quickly disappear.
Use the focused mode to first start grappling with concepts and problems in math and science. After you’ve done your first hard-focused work, allow the diffuse mode to takeover. Relax and do something different.
Those who are the most disagreeable tend to be the most creative. This is because creative people challenge existing answers and assumptions.
Focused attention can often help solve problems, but it can also create problems by blocking your ability to see new solutions.
One of the first steps towards gaining expertise in math and science is to create conceptual chunks. Mental leaps that unite separate bits of information through meaning.
Basic Steps to Forming a Chunk
- Step 1: Focus your attention on the information you want to chunk.
No television on in the background, don’t look at your phone, etc.
- Step 2: Understand the basic idea you are trying to chunk.
- Step 3: Gaining context so you see not just how, but also when to use this chunk.
When learning a new concept, look at sample problems with worked-out solutions.
One disadvantage of looking at worked-out examples to form chunks is that it can be easy to focus too much on why an individual step works and not on the connection between steps.
When you first begin to learn something, you are making new neural patterns and connecting them with pre-existing neural patterns that are spread through many areas of your brain.
If you chunk without understanding, it is a useless chunk that won’t fit in with other material you are learning.
Learning takes place in two ways:
- A bottom-up chunking process.
Practice and repetition which helps you build and strengthen each chunk, so you can easily gain access to it when needed.
- A top-down big-picture process.
It allows you to see where what you are learning fits in.
Context is where bottom-up and top-down learning meet.
Chunking = how to use a problem-solving technique.
Context = when to use that technique instead of some other technique.
Illusions of Competence and the Importance of Recall
Attempting to recall the material you have learned (retrieval practice) is far more effective than simply re-reading the material.
Many students experience illusions of competence when they are studying. When you have the book open in front of you, it provides the illusion that the material is already in your brain.
You don’t want to wait too long for the retrieval practice. Try to recall the material you’ve learned within a day.
If at all possible, re-write your notes in the evening after a lecture.
Once you have something down, you can expand the time between upkeep repetitions to weeks or months.
Getting a concept in class, versus being able to apply it to solving a genuine physical problem, is the difference between a student and a full-blown scientist or engineer.
Your brain has roughly four spots available in working memory. When you are first chunking a concept, it takes up all four spots. Once a concept is chunked, it takes up only one slot in working memory.
Merely glancing at the solution to a problem and thinking you truly know it yourself is one of the most common illusions of competence in learning.
Pick a mathematical or scientific concept from your notes, read it over, then look away and see what you can recall. Working toward understanding what you are recalling at the same time. Then glance back, re-read the concept, and try again.
The ability to combine chunks in novel ways underlies much of historical innovation. Reference the book, Where Good Ideas Come from by Steven Johnson.
The difference between creative scientists and technically-competent but non-imaginative ones is their breadth of interest.
The bigger your chunked mental library, the more easily you will be able to solve problems.
As you gain more experience in chunking, the chunks you are able to create become bigger.
Two Ways to Solve Problems
- Through sequential step-by-step reasoning (focused mode)
- Through more holistic intuition (creative diffused mode)
In building a chunked library, you are training your brain to recognize not only a specific problem but different types and classes of problems, so that you can automatically know how to quickly solve whatever you encounter. You will start to see patterns that simplify problem-solving for you.
The retrieval process itself enhances deep learning and helps us begin forming chunks.
It can help to recall material when you are in a physically different location than where you learned the material.
Doing something physically active is especially helping when you are trying to grasp a key idea.
Interleaving Versus Overlearning
Interleaving means practice by doing a mixture of different kinds of problems requiring different strategies.
Mastering a new subject means learning to select and use the proper technique for a problem. You learn this by practicing problems that require different techniques to solve them.
Emphasize interleaving instead of overlearning.
Overlearning means continuing to study or practice a problem immediately after some criterion has been achieved. For example, correctly solving a certain type of math problem and then immediately solving several more problems of the same kind.
Overlearning provides diminishing returns.
Techniques for Limited Study Time
- Read, but don’t yet solve assigned homework and practice exams.
- Review lecture notes.
- Rework example problems presented in lecture notes.
- Work assigned homework and practice exam questions.
The dread of doing a task uses up more time and energy than doing a task itself.
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.
Zombies Everywhere – Digging Deeper to Understand the Habit of Procrastination
Chunking is intimately related to habit. Habit is an energy saver.
Four Parts of a Habit:
I didn’t take many notes on the habits section. See my summary of Atomic Habits for a deeper look at habits.
Mental Contrasting: think about where you are now and contrast it with what you want to achieve. The contrast of where you want to be with where you are now that makes the difference.
Get into the flow by focusing on process not product.
Pomodoro Technique: break your work into bite-sized pieces, then work intently but briefly. This is a technique to help you focus your attention over a short period of time.
Chunking Verses Choking – How to Increase Your Expertise and Reduce Anxiety
This section discusses how to enhance and refine chunks.
Steps to Building a Powerful Chunk
- Work a key problem all the way through on paper.
- Do another repetition of the problem, paying attention to the key processes.
- Take a break. (gives your diffuse mode time to internalize the problem)
- Do another repetition.
- Add a new problem. Repeat steps 1-5 on the second problem.
- Do active repetitions. Mentally review while doing something active.
Testing in itself is a powerful learning experience, it changes and adds to what you know, also making dramatic improvements in your ability to retain the material. This is known as the testing effect.
Build a mental solution library. A key to building mental flexibility and expertise is to build your library of chunked solution patterns.
Summary of Chunking:
- 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 building a chunked library it is important to keep deliberate focus on the toughest concepts.
We develop a passion for what we are good at. It is a mistake to think that if we are not good at something we do not have, and can never develop, a passion for it.
- Keep a planner journal so you can easily track when you reach your goals and observe what does and doesn’t work.
- Write your planned tasks out the night before.
- Arrange your work into a series of small challenges.
- Don’t blame all your problems on external factors.
Enhancing Your Memory
We have outstanding visual and spatial memory systems.
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. See this article on Litemind for more information.
Learning to use your memory in a more disciplined yet creative manner helps you learn to focus your attention. By memorizing material you understand, you can internalize the material in a profound way.
One of the best things you can do to remember and understand concepts in math and science is to create a metaphor or analogy for them.
Use spaced repetition to help lodge ideas in memory.
Create meaningful groups that simplify the material you are trying to remember.
If you are memorizing something commonly used, search online to see if someone has already come up with a particularly memorable memory trick.
Beware of mistaking a memory trick for actual knowledge.
There is a direct connection between your hand and your brain. The act of organizing and re-writing your notes is essential to breaking down large amounts of information into smaller digestible chunks.
Writing and saying what you are trying to learn seems to enhance retention.
Several recent studies have shown that regular exercise can make a substantive improvement in your memory and learning ability.
Learning to Appreciate Your Talent
People learn by trying to make sense out of information they perceive. They rarely learn anything complex simply by having someone else tell it to them.
The idea you are already holding in mind blocks you from fresh thoughts. A superb working memory can hold its thoughts so tightly that new thoughts can’t easily peek through.
Your ability to solve complex problems may make you overthink simple problems.
Extremely smart people are more likely to procrastinate than people of normal intelligence. Because procrastinating always worked when they were growing up. This means they are less likely to learn critical life skills early on.
One might say that work substitutes for talent, or better yet, that work creates talent.
Teachers can easily underestimate their students and students can underestimate themselves.
When we grasp a chunk, it takes on new life in our own minds. We form ideas that enhance and enlighten the neural patterns we already possess, allowing us to more readily see and develop other related patterns. Once we have created a chunk as a neural pattern, we can more easily pass that chunked pattern to others.
One important key to learning swiftly in math and science is to realize that virtually every concept you learn has an analogy, a comparison, with something you already know.
Analogies and metaphors can help you use an existing neural structure as a scaffold to help you build a new and more complex neural structure.
Developing the Mind’s Eye Through Equation Poems
Thoughts can be visual as well as verbal.
Scientists often see equations as a form of poetry.
Simplification is important. Simple explanations are possible for almost any concept, no matter how complex. When you cultivate simple explanations by breaking down complicated material to its key elements, the result is that you have a deeper understanding of the material.
Understanding often arises as a consequence of attempting to explain something to others, rather than the explanation arising out of your previous understanding.
Transfer is the ability to take what you learn in one context and apply it to something else.
Multitasking during the learning process means you don’t learn as deeply, this can inhibit your ability to transfer what you are learning.
The Value of Learning On Your Own
Learning on your own is one of the deepest most effective way to approach learning. It improves your ability to think independently.
Persistence is often more important than intelligence.
Taking responsibility for your own learning is one of the most important things you can do.
Avoiding Overconfidence and the Power of Teamwork
The number of acquaintances you have, not the number of good friends, predicts your access to the latest ideas as well as your success in the job market. The comes from a sociology paper, The Strength of Weak Ties by Mark S. Granovetter.
People that you study with should have a critical edge to them. Research on creativity in teams has shown that non-judgmental agreeable interactions are less productive than sessions where criticism is accepted and even solicited as part of the game.
Working with others who are not afraid to disagree with you can:
- Help you catch errors in your thinking.
- Make it easier for you to think on your feet and react well in stressful situations.
- Improve your learning by ensuring that you really understand what you are explaining to others.
- Build important career connections and help steer you toward better choices.
Testing is an extraordinarily powerful learning experience.
Test Preparation Checklist
- Did you make a serious effort to understand the text?
- Did you work with classmates on homework problems? Or at least check your solutions with others?
- Did you attempt to outline every homework problem solution before working with classmates?
- Did you participate actively in homework group discussions, contributing ideas and asking questions?
- Did you consult with the instructor or teaching assistants when you were having trouble with something?
- Did you understand all of your homework problem solutions before they were handed in?
- Did you ask in class for explanations of homework problems that weren’t clear to you?
- If you had a study guide, did you carefully go through it before the test and convince yourself that you could do everything on it?
- Did you attempt to outline lots of problem solutions quickly, without spending time on the algebra and calculations?
- Did you go over the study guide and problems with classmates and quiz one another?
- If there was a review session before the test, did you attend it and ask questions about anything you weren’t sure about?
- Did you get a reasonable night’s sleep before the test?
Not getting enough sleep the night before a test can negate any preparation you’ve done.
Start with hard problems then quickly jump to the easy ones. This allows your diffuse mode to start working on the problems.
Unlock Your Potential – Closing Summary
Ten Rules of Good Study
- Use recall.
- Test yourself. On everything, all the time. Use flash cards.
- Chunk your problems.
- Space your repetition.
- Alternate different problem-solving techniques during your practice.
- Take breaks.
- Use explanatory questioning and simple analogies.
- Focus. Turn off all interrupting beeps and alarms.
- Eat your frogs first. Do the hardest thing early in the day when you are fresh.
- Make a mental contrast.
Ten Rules of Bad Study
- Passive re-reading.
- Letting highlights overwhelm you.
- Merely glancing at a problem’s solution and thinking you know 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 chat sessions.
- Neglecting to read the textbook before you start working problems.
- Not checking with your instructors or classmates to clear up points of confusion.
- Thinking you can learn deeply when you are being constantly distracted.
- Not getting enough sleep.