Posted in Educational Reform, Inspiration

6 Realities Education Must Adapt To

1) The Entirety Of Human Knowledge Is Expanding Exponentially

This means we can forget trying to summarize it, let alone fit it all in our heads. What is “core knowledge” is entirely relative, and isn’t something that any committee or board can hope to establish for everyone.

2) The Entirety Of Human Knowledge Is Universally Accessible

This statement is virtually true already, and becoming more so all the time. This means that memorization is becoming obsolete. The two primary factors that influence memory are significance and frequency. Education used to focus on creating artificial significance and frequency (studying) in preparation for artificial scarcity (tests), but this paradigm is no longer worth pursuing. We can determine what is
significant to us, and we can expose ourselves to information as often as we wish.

3) Human Knowledge Is Increasingly Diverse

This means that we can forget standardizing. Not everyone wants or needs to know the same things, and none of us will have access to more than a minute fraction of all accumulated knowledge even with a lifetime of learning. Geeking out on something worthwhile is a perfectly acceptable way to devote one’s mental resources.

4) Human Knowledge Is Continually Being Updated

There will be more new knowledge created in the next ten years than was created in all of previous human history, and this
statement will remain true for the foreseeable future. This means that most of our knowledge will be updated fairly
rapidly, and the best ways of learning and applying it certainly will be. What was cutting-edge mathematics in
Newton’s day is now routinely mastered by high school juniors and seniors. We can expect similar advances to happen in much
shorter time frames.

5) Creativity And Conformity Are Mutually Exclusive

The importance of imagining what can be is on the rise relative to the importance of knowing what has been. This means that our educational process, however it evolves, must welcome variation and continually become more divergent than convergent.

6) People (Especially Kids) Want To Learn

And just as importantly, they want to teach; not by lecturing to a captive audience from behind a podium, but by sharing what they are passionate about with those who are motivated to learn about it. This means we can forget about enforcing education and focus on facilitating it, providing people of all ages with greater and greater opportunities to teach and learn what they want.

Posted in Educational Reform, Math

Math Is A Language (An Easy One)

I’m always amazed by how many people tell me they think they are bad at math, yet they say this in a perfectly formed English sentence! Math is a language, and a far simpler one than any natural language. English, with its thousands of rules and hundreds of thousands of exceptions, is one of the most complicated languages in existence (as any ESL student will tell you). Yet many native speakers take for granted the massive feat of intellectual prowess represented by being able to communicate and think in this language.

Mortimer J. Adler explains this in the classic guide to intelligent reading, How to Read a Book:

“We are not told, or not told early enough so that it sinks in, that mathematics is a language, and that we can learn it like any other, including our own…When we underwent our initial reading instruction in elementary school, our problem was to learn to recognize certain arbitrary symbols when they appeared on a page, and to memorize certain relations among these symbols…Since mathematics is a language, it has its own vocabulary, grammar, and syntax, and these have to be learned by the beginning reader. Certain symbols and relationships between symbols have to be memorized. The problem is different, because the language is different, but it is no more difficult, theoretically, than learning to read English or French or German. At the elementary level, in fact, it may even be easier…We are also not told, at least not early enough, how beautiful and how intellectually satisfying mathematics can be. It is probably not too late for anyone to see this if he will go to a little trouble.”

Lately I’ve been getting really into the articles that math professor Sanjoy Mahajan has written for the Freakonomics blog. They’re worth checking out, especially this one on the similarities between how math and languages are taught in schools, and how both can be improved:

“Instead of teaching physics or mathematics as we teach second languages, then blaming the victims for not doing well, and expecting them to internalize the blame…why not use physics and mathematics to ask and answer questions about the world?”

Posted in Inspiration, Math, Tips for Teachers

Why Math Is Easier Than You Think (Part 3)

What Could Be Different

So math is actually easy but we are convinced (or we convince ourselves) that it is hard. What can we do about it? There is a LOT that is wrong with the typical approach to teaching math, but I’ll try to keep this positive and reasonably short. Here are just a few guiding suggestions:

Think Creatively As Well As Critically

Math is known for having only one right answer and thus not thought to be a creative endeavor. However, while there may be only one right answer, there are usually multiple or even unlimited ways to arrive at that answer, which is where divergent thinking has a role to play. Instead of (or at least in addition to) teaching algorithms we can be teaching concepts and principles, throwing them out like legos and letting students decide how to assemble them. Different groups can be given the same objective, and then if they don’t arrive at the same answer they can figure out through discussion where the discrepancy is (this is where critical thinking comes in).

Order Matters

An example of how solutions can be arrived at in different ways is in the order of arithmetic operations. Look at the following numbers and try to add them in your head:

12 + 15 + 11 + 3 + 19

Now try to add them in this order:

19 + 11 + 12 + 3 + 15

It’s easier to do in the second case because of the way they are grouped: 19 + 11 = 30, 12 + 3 = 15, 15 + 15 = 30, 30 + 30 = 60. These steps are all easy to do in your head, making the answer easier to arrive at than if you tried to add them in a different order. Here’s another example, with multiplication:

2 x 7 x 3 x 5

Now multiply them in this order:

7 x 3 x 2 x 5

It’s easy to do 7 x 3 = 21, 2 x 5 = 10, and 21 x 10 = 210 in your head. Many basic arithmetic calculations can be simplified this way (including subtracting and dividing), and done more quickly and easily in your head than by writing them down and following the standard algorithmic approach, or even using a calculator.

Make Things Easier, Not Harder

You can’t blame math teachers for wanting to dazzle and stun their students by making things look more complicated than they are, but everything in math is actually simple if presented in the right way. Different students will take better to different explanations, but because math is based on logic, there is nothing about it that is not self evident if viewed from the right perspective.

Be Like Socrates

In fact, because math is so internally consistent, it is the ideal arena to apply the Socratic method. When I tutor clients in math, I do my best to avoid ever telling them anything, but rather lead them through the process of discovery by asking them the right questions at the right times in the right way to allow them to arrive at conclusions themselves. What I am teaching them is not a set of facts, but rather a method of directing their thinking in productive ways, so that eventually they come to be able to automatically ask themselves the right sorts of questions in a variety of situations.

Follow Interest, Not A Curriculum

The prerequisite for any type of learning is a thirst for knowledge, which can only come from creating and pursuing questions a student is actually interested in, and it just might not line up exactly with any pre-determined curriculum – in fact, it probably won’t. Learning does not take place on a pre-set schedule, but then neither does life. By identifying and supporting the interests of students, instead of trying to standardize them, we empower them to develop their own greatest strengths, which are, of course, unique – and thank goodness for it.

Nurture The Joy Of Discovery

There is a widespread myth that kids have to be forced to learn, but it’s simply not true; kids are, in fact, eager to learn, just not necessarily whatever happens to be put in front of them. Often in math we focus on teaching certain techniques to the exclusion of all of the interesting diversions that appear at every turn. The irony is that all we really have to do is nurture the joy of discovery, and kids will discover more for themselves than we could ever teach them. By allowing kids to ask and answer the questions that are important, meaningful, or just plain interesting to them, we empower them with the ability and the confidence to develop their own gifts, which is what will actually enable them to lead happy, meaningful, and fulfilling lives.

Posted in Educational Reform, Inspiration, Math, Tips for Students

Why Math Is Easier Than You Think (Part 2)

The Stories We Tell

I’m just no good at math.

To learn math you have to be really smart.

I don’t need math.

I don’t like math.

Do any of these statements sound familiar?  The second reason that math seems hard is because of the stories (AKA excuses) we tell ourselves about why we can’t do it.  The statements above are representative of the main math-negative stories we tell ourselves.  If any of these types of stories are part of your internal dialogue about math, it will inevitably seem hard to you, even if the story itself is false (hint: they are all false).

Before I demonstrate exactly why each of these stories is a steaming heap of malarkey, however, it will help to understand what causes all such stories to arise in the first place.  It is important to recognize that limiting beliefs like these originate from two sources, internal and external, that tend to reinforce and support each other.

External sources. From a very early age we are all constantly bombarded with messages, either directly or by implication, that we are flawed, inadequate, unlovable, and just generally not enough: not good enough, not smart enough, not strong enough, not pretty enough, not cool enough.  Think of every situation in which you have ever felt the sting of disapproval, the pain of ridicule, the bite of shame or embarrassment, and you will just be seeing the tip of the iceberg.  Whether someone explicitly tells you “You’re stupid!”, or just looks at you with a facial expression that says “You should have known better”, or even just actively ignores you, your brain easily and correctly draws the conclusion that this person views you as inferior.  Enough data points make a worldview, and given that disapproval is the stock and trade of behavior modification in all realms of our society, it is easy for a young, impressionable brain to draw the conclusion that “thousands of people can’t be wrong — I must be an ignoramus”.  The fact is, for most of us, it actually takes work to create and sustain a healthy, realistic self concept in the face of this societal
consensus that we all suck.  The truth is, we are all animals, living extensions of the natural world, and just like no squirrel is “better” than any other squirrel, and no oak tree is “better” than any other oak tree, no human being is “better” than any other.  However, we can have beliefs that are not consistent with reality, including the belief in our own inferiority.  See if you can just sit with this idea for a moment.

Internal sources.  Spiritually speaking, we are all branches of the same tree of life, but we are also distinct individuals.  We have a physical body that is defined by a physical boundary known as skin, and we have a psychological identity that is demarcated by a psychic boundary known as ego.  The purpose of both “membranes” is the same: to delineate a distinction between “self” and “not-self”, allowing for existence and action as an independent being.  When the physical skin is cut, bruised, and punctured, it heals itself by constructing scar tissue, and the body learns to recoil from the source of the harm.  All of the affronts to our psychological self that we encounter from a very early age, every instance of “make-wrong” we are afflicted by, results in the buildup of egoic defenses.  Though egoic defense mechanisms vary widely in appearance, they all serve the same basic function: making things not be our fault.  However, any idea that anything is anybody’s “fault” is just a manifestation of make-wrong, an ineffectual judgment about what should or should not be the case.  Creating blame of any kind and taking action to improve things are two separate, and in fact, mutually exclusive activities. Chew on that for a bit, if you will.

Now we can address each of the above stories in terms of how it arises from internal and external sources, and provide iron-clad proof that each one is a blatant falsehood.

I’m just no good at math.

The subtext of this story is that “Some people are inherently good at math, and I’m not one of them.”  The external sources of this belief are all of the messages we receive that either compare us negatively to others or outright denigrate our ability: “You should have gotten this by now”; “You’ll never get this”; “Why can’t you be more like so-and-so?”  The internal source of this belief is the protection mechanism that “Since I’m not one of the lucky chosen ones, I can’t be expected to be good at math, so, naturally, it’s not my fault.”

Fact: Your brain is composed of approximately 86 billion neurons, just like everybody else’s, and they all work the same way.

Fact: The English language is far more complex and confusing than any system of mathematics, yet you were probably able to read and speak it fluently by the time you were six years old.

Now, humor me with a thought experiment.  I toss you a softball.  You catch it and toss it back.  Guess what – your brain just calculated two independent parabolic trajectories without so much as breaking a sweat.  Congratulations, you can do math.

To learn math you have to be really smart.

This is a variation of the previous story that explicitly invokes genetics.  The external sources of this belief are all the messages we receive that we are not “smart”.  The internal source is the blame-avoiding contention that you just missed out on the genetic lottery – naturally through no fault of your own. The truth is that math is an innate ability common to all humans; it is a skill that can be learned and invokes no more intelligence than tying your shoes or writing your name.  Thinking mathematically is as natural to humans as flying is to birds, swimming is to fish, and climbing is to monkeys.

Also, “smartness”, commonly known as “intelligence”, is a nebulous, socially constructed concept with no clear attributes that not even experts can agree how to define or measure.  So, even if there is such a thing as generalized, innate, differential intelligence, nobody can say definitively how much of it anybody has.

I don’t need math.

This story is technically true, but pointlessly so.  It is promoted externally by well-intended people who either want to make you feel better about not learning math, or who want you to think that you do need math, but can only come up with lame, flimsy arguments when pressed for details, or just demand that you should just take their word for it.  Being innately clever, your brain concludes, “Ha! If there were a good reason why I need math then they could explain it to me, so since they can’t it must mean that there isn’t.”  Which is a perfectly valid logical inference (the contrapositive).  Internally, of course, if you don’t need math then it doesn’t matter whether you are good at it or not, so there is nothing wrong with not learning it, leaving you, naturally, blameless.

It is true that you do not need math.  The fallacy lies in the conclusion that it is useless.  By way of analogy, I remember receiving my first cell phone while I was in college.  My parents gave it to me.  I said “Okay, but I don’t think I’ll have much use for it”, because, of course, I was used to not using a cell phone.  But, of course, once I had it, I found all kinds of ways to use it, even though I had gotten along without it just fine before.  Furthermore, once you get used to having a cell phone, it does come to seem like a real need, especially if everybody else has one.

There are people who grow into adulthood without learning how to read.  They survive, perhaps even thrive.  But if they learn how to read (which it is never too late to do, BTW), they find all sorts of uses for this skill.  Likewise, while you obviously won’t have a use for math you don’t know, whatever math you do know you will invariably find fascinating, life-enriching uses for.

I don’t like math.

“Only dweebs like math. You’re not a dweeb, are you?” So says the voice of external reinforcement for this limiting belief.  Internally, it’s not that you can’t learn math.  You just don’t want to, and no one can make you.  So therefore you aren’t dumb or incapable, you are exercising your right to do what you want as a free human being, which, naturally, no one can blame you for.  So there.

So you say you don’t like math.  Sounds pretty inviolable.  Surely you know what you like and what you don’t, right?  Yet I ask, “Are you sure? How do you know?” Show me a person who doesn’t like math and I’ll show you a textbook case of operant conditioning. Were you forced to learn math when you didn’t want to? Were you punished for not learning it when you were expected to? Were you shamed or embarrassed when you tried to learn? These are the things you actually don’t like. If you get a headache whenever you see equations written on a page, this is a conditioned response to psychological trauma.  It might feel real and automatic to you (in fact it probably does), yet like all variable emotional responses, it had to be trained, and it can be untrained just as well; not by getting positive feedback when you succeed and negative feedback when you fail, but by getting consistent positive feedback every time you try, and letting your natural sense of curiosity, consistency, and completion lead you to the “right” answers.

But that’s enough about the problems.  Next, we will look at some solutions…

Posted in Educational Reform, Inspiration, Math, Tips for Students

Why Math Is Easier Than You Think (Part 1)

Using Utensils With Your Feet

Imagine that you were raised in a secluded enclave by an eccentric but harmless cult.  Imagine further that the cult leader was born without arms, so had to do everything with his feet, including eating with utensils, and all the cult members are required to do the same.  So you are raised being taught to use utensils with your feet, that this is the only good and proper way to use utensils, and in fact it never even enters your mind that there might be any other way to use them. You do it so much, and from such an early age, that it even becomes easy and eventually feels “natural”.

Eventually you choose to leave the cult and discover to your amazement that people in the outside world use utensils with their hands.  Once you get used to this heretical notion you decide to give it a try.  Since it is new to you, and you have old habits to overcome, at first it feels awkward and uncomfortable, even a little embarrassing.  You are plagued by the sense that you are cheating.  But because you want to fit in you keep trying, and find that not only do you master this skill amazingly quickly, it actually feels even easier and more natural than using your feet did.  It soon gets to the point where trying to use utensils with your feet
would feel like an almost unbearable restriction.

This is an allegory for the first reason why math seems hard: because it is taught in ways that make it seem hard. There are two reasons for this.

The first reason stems from the fact that while learning (the accumulation of knowledge and experience) can and does take place under any and all circumstances, teaching (directed, outcome-specific learning) functions best as a dialogue.  The way we as a society structure education virtually eliminates the possibility of dialogue, due to two faulty assumptions:

1) teaching is necessary for learning
2) teaching and learning are separate activities

Since my purpose here is to explain why math is easier than you think, not to deconstruct modern educational theory*, rather than unpacking these assumptions I’ll just give you the simplified implication: one teacher can’t have a dialogue with twenty or more students.  (I actually think the maximum number is around five.)  Therefore, the constraints of the teaching environment itself inherently limit the effectiveness any teacher can have.

The second reason is a vicious cycle particularly endemic to math itself, which basically boils down to this: most teachers don’t teach it very effectively.  That’s not their fault, though, because they tend to teach it the same way they were taught.  But that’s not their fault either — it’s because they don’t have a very good understanding of it themselves, so it’s easier for them to teach what they learned than to teach what they know.  But that’s not their fault — it’s because the way they were taught was not very effective.  Which isn’t their teacher’s fault either, because…  And so you can see how this cycle perpetuates itself.  Couple this with the institutional and societal change resistance factor, and you get an educational process that is very slow to adapt and an educational industry with a very long lag time.

However, there are better ways…

*But stay tuned!

Posted in Inspiration, Math

There’s No Such Thing As Being Bad At Math

Too often, rather than revealing our true abilities and potential, the educational process that most of us experience convinces us that we have innate weaknesses that we can’t do anything about.  Rather than leaving school feeling empowered, motivated, intelligent, and capable, many (most?) students leave school feeling degraded, demoralized, and disempowered.

One of the most common manifestations I see of this phenomenon is the belief that, by virtue of the universal lottery, some people are “good at math” and others aren’t, often held by those who think they aren’t.

In all my years of tutoring math and related subjects, I have yet to come across a student who is “bad” at it.  In fact, based on my personal experience as an educator, I believe that math is an innate human ability, just like language or walking, and comes to us just as naturally.

What distinguishes individuals is not their mathematical ability, but the variety of cognitive processing styles they employ, as well as the variety of areas of interest they display.  An effective educational process is not one that sorts students based on ability, but one that adapts itself based on style and is motivated by interest.

Learning has nothing to do with evaluating and everything to do with instilling.  The story that you are bad at math is just that, a story.  When I tutor a student, I am evaluating my effectiveness as a teacher, not their effectiveness as a learner, and I measure this not by the quantity of knowledge the student gains, but by the quality of the beliefs they develop.

Posted in Books, Resources, Tips for Parents

Family Homework

Three education experts teamed up to write a book called From Surviving to Thriving: Mastering the Art of the Elementary Classroom. One of the revolutionary ideas they share is called Family Homework, described below:

“Family homework provides real life situations for students to apply concepts and skills they are learning at school. These types of assignments are designed to be integrated into a family’s weekly routine. It involves parents in their children’s education. Create a monthly calendar of family homework assignments that connect to current class curriculum.”

The authors discuss this idea, along with how teachers can continually improve their teaching methods, in a post on their blog:

Keeping Current or Becoming a Lifelong Learner!

I believe education is a process that should bring families together rather than pull them apart.  Family homework is a great way for parents to enjoy quality time with their children while learning along with them!