Reading Actively: Part I

This is the first in a series of posts about reading actively.

Learning to read your textbook actively is perhaps the single most important study habit for success in the mathematics major.  Recently, I spoke with a math major who had transferred from another major.  He read his math textbooks in linear order with a highlighter in hand, in a distraction-reduced environment.  He copied out key vocabulary and facts onto a study sheet, neatly colour-coded.  But he was finding his reading entirely ineffective as a study tool.  What was he doing wrong?

The key is one single principle:  in mathematics, we learn by doing.  An illustrative example is given by Professor Dancealot, whom you can find on YouTube teaching foxtrot by monotone lecture.  How do you best learn to dance?  You dance.  Professor Dancealot is doing no one a favour by droning on about foot position, when he should have his students up in the aisles, trying it out.

Mathematics is no different.  Mathematics is like dance, or language learning.  The only effective way to develop mathematical skills is to practice.  The key to active reading is to learn how to get yourself practicing, not just reading.  This practice is far from automatic — it will not happen as a side effect of pronouncing the textbook words in your head, or even copying them on another piece of paper.  It is a skill in itself.

Perhaps mathematics is better illustrated by the metaphor of yoga, which asks you to assume complicated body positions.  Mathematics asks you to assume complicated mind positions.  As you read the instruction, “Place your feet shoulder width apart,” for a full understanding, you should get up and assume the position indicated before reading the next sentence.

Mathematical texts are no different.  If your text says, “Let S be any non-empty set,” you should create such a set, or a range of possible such sets, in your mind.  This is not automatic!  You must consciously do it.  How do you do it?  You would do the following:  recall the definition of a set for yourself (say it out loud from your own understanding, in your own words); recall the definition of non-empty (out loud again, from your understanding); give a few examples of sets which are non-empty (yes, out loud); and give a few examples of sets which fail to be non-empty (once again, out loud).

Ok, you can whisper.  But the point is not to let yourself get away with anything less than a full, honest, explanation of your understanding as you go along.  If you cannot produce it, you should go back and study the relevant material before proceeding.  The best test of understanding is your ability to explain.  Don’t let yourself off the hook!  It is easy to feel honest and say, “I remember what that means,” but it is a step further to force yourself to give the definition out loud — and this step is the key to strengthening your neuronal connections.

Let us give another example.  In your text, you may find the sentence, “Notice that 3 divides A.”  This is an instruction to perform a mental move!  Remind yourself what A is (out loud), what the formal definition of “divides” is (out loud, from your understanding, in your own words), and then verify, using the definition, that 3 divides A.   In this way, you have yourself performed the yoga move of logically verifying that 3 divides A.  Only then can you move to the next sentence.

And so you continue, sentence by sentence, through the text.  It is a slow process, but by engaging actively, you are growing and strengthening neuronal connections as you read.  This is your most valuable study time.

Math Anxiety

I have math anxiety.  So, perhaps, may you.  It is the flood of stress hormone in response to a mathematical challenge.  For me, it takes the form of fear of being wrong in front of my colleagues or students.  For you, it may make its appearance when you are called upon in class, have to confront a group worksheet, when you are sitting an exam, or when you are alone with your textbook.

Are you surprised that your professor has math anxiety?  Don’t be.  The truth is, many or perhaps most mathematicians have math anxiety in some form or other.  Mathematics is challenging, and there are right and wrong answers.  That’s right, I won’t baby you by pretending there are no wrong answers.  Instead, the key to surviving math anxiety is to embrace error.  Error is your friend, and you should be proud of your mistakes.  Bear with me.

It is natural and human to be concerned about how others view you, and this is a principal source of anxiety.  We all want to be wunderkinds, who immediately and impressively jump to the right conclusions to complex problems.  I am going to try to convince you that this is not actually what you should be aiming for and comparing yourself to.  Instead, you should demonstrate, for yourself and others, your courage, perseverance and creativity in the face of challenges.  These are far more important traits to aspire to than looking like a slick genius.

Learning mathematics, and doing mathematics, is, by definition, to be confused.  Confusion is a necessary part of the process.  Even the most impressive mathematical feats, which seem to come from nowhere, have as their source a stretch of frustrating confusion.  The greatest mathematical theorems are born of many hours of hopeless head-banging, as any great mathematician will tell you.  My job description, as a research mathematician, is to spend all day, every day, hopelessly confused.

Except that it is not hopeless.  Instead, every confusion is an opportunity for enlightenment.  Every mistake is an opportunity for learning.  This is not simply a cliché.  Being confused is an opportunity to ask a question.  And asking a question brings new understanding (sometimes by itself, sometimes by its answer).

I have a great variety of students in my class every year.  One of them, let’s call her Jane, is the student who believes she understands everything I am saying.  She seemingly always has the correct answer to questions posed to the audience, and may volunteer them frequently, leaving you with the impression that she, or perhaps everyone except you, understands the lecture easily.  However, Jane does not question her understanding.  Her confidence blinds her to the opportunity to discover fuzzy corners of her understanding where she may err.  Her familiarity with the concepts leads her to mentally “check them off” instead of examining them with any dedication.  Jane finds homework easy and always sees the solution right away.  Until sometimes she doesn’t.  And then she doesn’t know how to proceed.

Another student is Anne, who is intimidated by Jane.  Anne doesn’t understand things immediately in lecture and is often afraid to ask the question that is buzzing about her head.  However, she forces herself to ask (sometimes in office hour instead of lecture).  Before asking, she carefully and slowly formulates the question.  In so doing, she often discovers the cognitive dissonance that has caused her misunderstanding and resolves it.  But if she doesn’t, she then produces a careful, precise question which will elicit exactly the missing information she needs to keep building.  Anne forces herself to face her fear of speaking up, she perseveres if she doesn’t understand something at first, and she tries many approaches to a problem.  She often seeks help with homework, but she always works on it until she’s understood it all.

Anne is an example of “slow and steady wins the race.”  Jane eventually comes across something she doesn’t understand and her reaction is to bluff a little, or comfort herself with a surface understanding.  She doesn’t know how to proceed.  Anne, by contrast, is an expert in resolving her own misunderstandings by now, and she has become meticulous in her thought processes.  She is able to call on her sense of creativity and ability with precision to face a new challenge.

My moral is this:  Anne is your role model, not Jane.  Focus on being a model of courage, perseverence and creativity in the face of challenges.  Don’t focus on the outcome (knowing the answer), but instead on the process (the skills of learning).  They are simply more important.

After all, your trajectory in life will inevitably and eventually test your limits, wherever they may be.  You need to be prepared.


Who is that professor at the board, and what does she want?

This spring I will be teaching MATH 2001:  Introduction to Discrete Mathematics.  This is our introduction to the major here at the University of Colorado, Boulder, and most students find it a wild ride after their experience in Calculus.

What do I want out of this course?  I want to impart to my students the experience and skills of a research mathematician.  This may seem like the wrong goal.  After all, our majors are mostly headed to other careers.  Is it not a little bit of hubris, near-sightedness, or perhaps even isolation-induced narcissism, that pushes me to evangelize about my profession?

I prefer to think otherwise, but there is probably some truth to that assessment of my motivations.  However, less-than-laudable motivations does not a bad goal make.  So the purpose of this post is to explain what is valuable about the skill set I aim to teach in Math 2001.

First and foremost, my goal is precision in thought and communication.   My goal is to harness and control fuzzy, associative thinking.   Associative thinking begets creativity and intuition, two extremely valuable things in mathematics, but it must always be tempered by precision and logic, or it brings us only daydreams.  I find that students usually arrive in 2001 with great creativity, but less ability to harness their creative urges to align with reality, mathematically speaking.

Phrased this way, one might expect this task to be somewhat depressing.  But there’s another way to view the dynamic, empowering symbiosis between creativity and precision.  Mathematics is not unlike art, in a very fundamental way.  Reflect for a moment on the sonnet, a very rigid poetry form, with prescribed metre, rhyme and structure.  One has a fixed total number of syllables to work with, and it is designated which of these may be given emphasis and which may not.  The first eight lines set up or describe a problem; the ninth line is a turn in tone or mood.  It is very demanding.  And yet, there are an astronomically large (although finite) number of sonnets that could be written.  And more importantly, struggling to match the form actually drives the creative process.  It is actually much more challenging to write beautiful free verse.  Mathematics is much the same.  The mathematical artist has total freedom of subject, but she must hew to the rules of logic, like the rules of sonnet form.

This symbiosis is hardly unique to mathematics.  An engineer must balance vision with the physical laws of his building materials.  A scientist must balance the resources of his laboratory with the size of his motivating questions.  But for some reason few people expect it in mathematics.

Perhaps this is because few people realize what a research mathematician does.  A mathematician proves theorems.  Theorems are logical inferences:  if A is true, then B is true.  To even imagine the statement of a theorem is a creative act, often born of long experimentation and exploration.  The creative spark of a theorem statement may be presented by natural phenomena in our world, or by computational experiment.  To prove the theorem also requires great creativity, in the form of logical arguments.  The logic isn’t creative itself:  it is simply a set of rules, like sonnet form.  The creativity is in choosing which of infinitely many possible logical inferences is the right one.  (We are a long way now from the recipe-like solutions taught in most high school calculus texts!)

Theorems are, like the bridges built by engineers, actually useful.  They tell us how to program the scheduling of the rooms at the university, and guarantee the optimality of the outcome.  They tell us how to successfully plan a trajectory to mars, and how to statistically support or refute the dangers of smoking or cell phones.  The ability to create correct logical inferences — theorems — is at the heart of many of the jobs our mathematics majors will have in industry and elsewhere.  To correctly design software to run a medical scanner, to decipher the scientific literature and make recommendations for treatment, to analyse cosmic radiation for signs of alien life, for any of the many jobs our majors have in their futures, one needs to think with precision and creativity.

My goal is to impart exactly those skills.


Postscript:  I highly recommend the wonderful book How Not to be Wrong, by Jordan Ellenberg.  This is aimed at a popular audience, and goes a long way to explaining how precise mathematical thinking can be useful in every sphere of life.