How to Teach Math

Several years ago, I wrote a column explaining why math is an important subject. In it, I pointed out that the actual content, that is, the actual mathematical manipulations, didn't really matter.  What matters in math is the meta-concepts.

To be successful in math, you have to master an approach to the world that is inherently useful. College majors typically require mastery of a math course not because anyone cares about math, but because passing a math course is the easiest way to demonstrate mastery of the meta-concepts that everyone actually does care about.

Once you realize that math is about meta-concepts, and not about math at all, teaching math becomes really very easy.

1) Buy a math notebook.

Every frustrated math student, everyone who is poor at math, has one thing in common: they don't organize their work. They scribble numbers down all over the page without regard to sequence. If students learn ANYTHING in math, they must learn to break that habit.

So, beginning students should be actively discouraged from doing math in their heads. Yes, I know the Math-Bowl encourages this for the advanced students who compete, but it's not a good idea for beginners. Beginners need the external structure. So, buy a THICK, empty math notebook, with lots of empty pages.

2) Throw out the calculator

For basic math, they don't need it.

Calculators interrupt the student's concentration, forcing him to alternate between doing the procedure the problem requires and doing the procedure the machine requires, figuring out the correct sequence of key punches.

Calculators are mostly a distraction. No one needs a calculator until they start doing trigonometry or statistics. If they aren't doing either, then let them learn the multiplication tables.

If you think calculators should be allowed on basic math tests, then why shouldn't cell phones and internet access be allowed for reference on history or English tests? The Internet is the equivalent of a history or grammar calculator. Why bog down the student with memorization of useless dates and grammar rules when they could be doing higher-order stuff?

Now, watch the history and English teachers howl in outrage that you should suggest such a thing. Watch the math teachers smile sadly and say, "Yeah, well, welcome to our world, suckers."

3) Write down each step

The student must write down each problem as follows:

1. On the first line, the problem itself
2. On the second and subsequent lines, write each successive step.
3. No more than one operation (add, subtract, multiply, divide) is permitted in any step.
4. The answer is written at the bottom of the step sequence. 

No scribbling in side margins allowed at all. Don't allow multiple operations in any one step because beginning students get themselves confused easily. Each step does exactly one thing, that is all.

Ignore their whining. Even if a beginning student gets the answer correct, the problem is wrong if they haven't shown all their steps. Make that clear. Stand over them for a month, enforce it, and they will gain the habit. Ingraining into them this single, solitary little trick solves over half your math problems overnight.

4) A fresh sheet of paper for every problem 

Math is not an exercise in conserving paper. Be profligate. Paper costs less than half a cent a sheet. Splurge. Once they have successfully trained themselves to write out every step, you can alter this rule to allow more than one problem per page, but even then NEVER let them break a problem over two sheets of paper. Ever. No. I mean it, don't do it.

Beginning students get a feeling of accomplishment from seeing all their work laid out neatly at a glance. It feels restful, as the eye glides downhill through the gears of the problem and finally takes up its ease at the bottom of the sequence, peacefully resting upon the (correct) answer.

5) When they get stuck

First, if there is ANY sign of margin scribbling, turn the old sheet of paper face-down, start on a fresh sheet.
No.
Do it.
If you start on the old sheet, all the old scribblings will be a distraction. The student will wander down rabbit-trails trying to figure out what went wrong with the previous procedure. Clear his mind. Start fresh. Give him the gift of new eyes and a clean slate.

Now, math teaches a lot of (seemingly) arbitrary procedures. The student has to know all the procedures and know when to apply which procedure. Both parts of this are hard, but the second part - knowing when to apply which procedure - is the hardest. So, when he gets stuck and isn't sure what to do next, here's what you do:

• Ask him a question you are sure he can answer.
• When he answers correctly, affirm it ("That's right."), 
• Rinse and repeat. Ask a series of questions, each one of which you are confident he can answer.
• Build that series of questions so as to lead him to or through the correct procedure.
• In basic math, this question will always come up at some point: "Do you think you would add, subtract, multiply or divide?" Those are the four basic operations, and one of them is almost certainly going to be part of the path to solving the problem. 
• The student is almost always able to weed out at least a couple of the operations. That instills self-confidence, it shows partial mastery.
• Don't give the student the answer. 
• Ever. 
• Always respond with a question you know s/he can answer.
• Once s/he has gotten the answer, point out the truth: "I didn't tell you the answer. All I did was ask questions. You KNEW all the answers. You already KNEW how to do it."
• And the student DID know how to do it. He just needs to internalize how to ask himself the same series of questions you asked.
• Don't point this out. 
• Simply keep repeating this sequence with him on every problem he has, week in, week out, making sure he writes down every problem step-by-step
• He will learn to internalize the question sequence himself. He will start asking and answering his own questions. 
• At that point, you can go bake brownies.
• This whole sequence only takes a couple of months to instill. 

Once this basic skill set is instilled, it is now permissible to have the student walk through the steps of a failed problem to see where the mistake was made.

• If each step has only one operation, it will be relatively easy to see which step failed. 
• Now the student will see the wisdom of the step-by-step process.
• He doesn't have to re-do every problem from scratch. 
• He can find and correct his own mistakes easily.
• Once he realizes this, math becomes almost bearable.

6) When YOU get stuck

Don't be afraid to say, "I don't know. Let's Google it." You don't have to know everything in math. In fact, you don't have to know ANYTHING about math. Remember, math isn't about math. Math is about learning how to be

• organized, 
• good at documenting details, 
• good at being detail-oriented, and 
• good at following and trusting arbitrary procedures.

None of those skills require you to know the arbitrary procedures yourself. Even if you are no good at math, you will naturally be better at searching for the correct way to do it. Model how to search for the right way to do things. Have your student watch you as you bumble along, figuring it out.

The student thereby learns:
(1) it is ok to not know something,
(2) this is how you find out what you don't know,
(3) Searching for the right procedure takes time and that's also ok,
(3) Perseverance can be as important, or more important, than possessing knowledge.

That's all there is to teaching math.
Seriously.

Well, that and liberal use of Khan Academy. Yes, I have a degree in computer science, minor in math, and have taught developmental math at the college level for years, but I taught my children almost no math at all. There's no point. Khan Academy teaches the concepts as well or better than I could. I only got involved if a video was opaque (unusual) or a solution sequence unclear (also unusual).

Often-times, I would walk along through the Khan Academy solution to the problem as perplexed about the correct sequence as my child was. It's not like I remember most of the stuff I learned thirty or forty years ago. We would discover the solution together, which was rather fun.

No, the only way I have ever taught math was to follow the sequence I have described above. It works.

Update:

A friend reminds me that I have omitted an important step.  Obviously, patience on the part of both teacher and student is a developed skill that is the absolute key to the method, on both parts.

But here's the step I'm missing: constantly remind the student that math requires only one thing - it simply requires you to be as perfect as God. You can't make any mistakes. Easy, eh?

I used to regale the children with stories of mathematicians who made very simple mistakes and destroyed millions of dollars worth of equipment, or entirely killed people. Everyone makes math mistakes, even the most skilled engineers and mathematicians. As I have frequently pointed out, I am a math teacher because I have gotten thousands more problems wrong than any of my students. My students can only become math teachers if they have failed as often as I have.

Fail early and often!
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Is Wealth Inequality Bad?

Many people, including Pope Francis, rail against wealth inequality. From an economic perspective, it is not at all clear why. Wealth inequality actually corresponds quite well with the rising tide of affluence throughout the world. Right now, during a period of the most extreme wealth inequality between nations ever, we are also on the verge of wiping out extreme poverty. This is not a coincidence.

I have noted before that the office of the papacy does not include the requirement that the Pope be very knowledgeable of economic theory, nor that he be very intelligent in his comments on it. While the Church has a duty to serve the poor, the method by which the poor are best served is largely prudential - different people might legitimately choose different means to solve it. Wealth inequality, by itself, is not a sin so long as everyone's minimum requirements for food, clothing, shelter, education, medical care and basic human dignity are being met. Quite frankly, all but the last of those minimum needs are being met much better today, in a time of enormous wealth inequality, then they have ever been met in the entire course of human history.

In fact, things have gotten so much physically better precisely as a result of growing wealth inequality. Wealth inequality is correlated with EVERYONE getting out of poverty.
The "trickle-down" theory of Reaganomics is a fine example of how inequalities actually help everyone. Even Ted Kennedy admitted that much. The rising tide which brings incredible wealth to the 1% ALSO lifts the 99% out of poverty. The mechanics for how it works is very straightforward.

Take laparoscopic surgery, for instance. When I was young, that was only an option for millionaires. But, as more millionaires bought the procedure, economies of scale kicked in and semi-millionaires could afford it. There were a lot more semi-millionaires than there were millionaires, so scale kicked in again, repeatedly. As cost fell, more and more people could afford it, so more and more people did it, which dropped costs still further. The increase in scale also meant the procedure became increasingly streamlined and efficient, if only so as to handle the demand better.

Today, laparoscopic surgery is standard procedure for gallbladder and appendix removal, among a host of other applications. It provides cheap out-patient surgery that simply didn't exist 50 years ago. Why? Because millionaires volunteered to act as the guinea pigs. They were the only ones who could originally afford it. As they and their extremely well-paid doctors refined the procedure, it became increasingly available to the rest of us. Today, it is so common that it doesn't merit mention.

The same thing happened with cell phones. The original radio phones and cell phones were the size of bread boxes and cost more than any average Joe could possibly afford. But millionaires needed to stay in constant contact with their businesses and with the stock market, so they bought the tech. They served as the guinea pigs. As scale increased, price dropped, efficiencies improved. Now, literally everyone in the US can afford cell phones. And not just cell phones. Today's phones are the 1960s equivalents of super-computers that fit in our pocket and put us in touch with most of humanity's combined store of knowledge. The rich people and their richly rewarded tech outfits worked out the kinks. We have all benefited.

THAT is what wealth inequality does. It allows millionaires to act as guinea pigs for new tech. If it doesn't work, they waste their money and/or die. If it DOES work, then I get the tech about fifteen years later, because by then, the cost has dropped down to where even I can afford it.

Wealth inequality is an extremely efficient way to utilize resources. If I have a new tech idea, I can either try to convince tens of thousands of middle-class people to fund my idea, at great personal risk to each of their wealth stores, OR I can convince one extremely wealthy person to fund it at minimal risk to his wealth store. It was easier to do the latter than the former. Even today, being on Shark Tank will bring you funding in literally 20 minutes, while promoting the same project on Kickstarter will take days, weeks, or months to accomplish the same level of funding. It is far easier for Elon Musk to guide a project like the Falcon Heavy or BFR to successful completion than it is for ten thousand people to agree on how to do so, and that assumes you could get the funding from that ten thousand at all.

Wealth inequality is not the problem many people make it out to be. In fact, wealth inequality has historically been the solution which has made us all fabulously wealthy by any historical standard you care to name. Because of wealth inequality, the rich are willing to serve as the experimental guinea pigs necessary to bring functioning solutions to the masses. That's not a bad thing.

We have a knee-jerk reaction against wealth accumulation because we innately see the world as a zero-sum game. And the fallen world often acts and reacts as if it were. But the whole point of Christianity is to change our world-view. A Christian understands that God's grace and power are infinite, therefore the zero-sum game view can never be correct. Insofar as we image God, we have the ability to change our world from being zero-sum to being infinitely resourceful and wealthy.

Can we create Utopia? Physically, sure. Spiritually, not a chance. We are still fallen creatures, and that will always prevent us from establishing any real paradise on earth. We are slowly solving the problem of physical poverty. It really is going to disappear, possibly in our lifetimes.

But spiritual poverty? As long as anyone in the world is not Catholic, then the world still suffers from extreme spiritual poverty. THAT is the wealth inequality which we, as Catholics, need to remedy. Fortunately, from that viewpoint, Catholics are the rich one-percent. We have infinite resources that can be delivered to the poor among us, and make them all wealthy too. As physical riches percolate out into the world, that is the only wealth inequality we really need to be concerned about.

Hepatitis C: We Pay More, Thank God

This is a popular meme floating around the Internet. Sometimes it picks India, sometimes Egypt, but the idea is the same. The US pays 84 times or 250 times or some ungodly percentage more than these lousy third world countries! Why are US citizens being gouged?

Well, to put it bluntly, US citizens are NOT being gouged. When all things are considered, those prices are all pretty fair. To begin with, let us simply accept the numbers above without argument. For this analysis, we shall add some numbers of our own.

Country	Price / pill	% of US cost	Median Income	% of US Income	Hep C Prevalence
USA	$1000	100%	$51,700	100%	3.2 million
Egypt	$    11	1%	$  5,680	11%	38 million
India	$      4	0.4%	$  3,800	7%	10-15 million

So, the average American earns 10x as much as the average Egyptian and roughly fifteen time as much as the average Indian. One would think, based just on this, that the average Egyptian treatment would cost one-tenth as much. That is, it would seem the price should be $100 per pill in Egypt, for instance.

But, then we have to factor in the prevalence of the disease. Egypt has 10x as many cases of hepatitis C, India has 3-5x times as many cases of the disease. So, we have to knock down the Egyptian price by another factor of ten (due to volume) and lo! The price is pretty much correct. When you factor in both India's disease prevalence and income, even the Indian price is not that far off.

There's no reason to stop with the table above, though. Try factoring in the sub-populations that actually get infected with hepatitis C and therefor have to pay these different prices. The "disparity" becomes even more interesting.

In the United States, hepatitis C is primarily the scourge of IV drug users and, to a lesser extent, homosexuals. Homosexuals tend to be richer than the average American. They are narcissists who have no children, and who claw their way up the corporate ladder quite efficiently, that is, they tend to get paid more. Conversely, in Egypt and India, hepatitis C most adversely effects the poorest of the poor. It is the poor in these countries who are the least likely to have access to clean water.

So, let's summarize those numbers with fresh eyes. In general, American IV drug users and homosexuals are subsidizing the medical treatment of the poorest of the poor in the Third World.

Personally, I can find no reason to object to this arrangement.
Can you?