The Trump Economy

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The goal for this piece is to be comprehensive about the effects of Trump policies on the economy. That sounds rather ordinary, but in fact most discussions of the Trump economy has been myopic to the point of misrepresentation.  There are several reasons for that:

  1. To understand the effect of Trump’s policies you need to untangle Trump’s results from the inherited economy. People tend to shy away from that, but there is actually no alternative—otherwise there is no way to say where we are going from here.  As we’ve noted before, Trump has pulled a fast one on the American public.  It’s not just a matter of claiming responsibility for successes of an inherited economy.  It’s that he is substituting wildly dangerous policies for the ones that actually got us here.  When we last discussed that subject we could only talk theoretically, but now there is considerably more to say.
  2. The usual statistics (even the unemployment rate) tell you more about the business view of the world than about what it means for people. The difference between the two says a lot about the real significance of Trump policies.
  3. Finally, most discussions of Trump’s policies focus on results from the hugely expensive tax cuts that were done last December. However, it is at least as important to understand what we have NOT DONE because of Trump’s economic priorities.  As we’ll see those sins of omission are a serious part of the picture.

The discussion plays out as a series of charts.

 

  1. Effects of Trump’s policies on the economy

We begin by teasing out the effects of Trump’s own policies from what was inherited.

Our first chart shows the changes in unemployment rate over the past ten years:

te1What is most striking about this picture is the continuity:  the trend line is almost straight extending through the present.  That isn’t surprising.  Changes in employment don’t happen overnight, and after the election there were no substantive changes in economic policy until the new tax plan was passed in December of 2017. (Reduced regulation at the EPA was never a factor, since—despite the propaganda—environment regulation was positive for jobs.)

Another way of looking at the same trend is with job creation, which we see in the second chart:

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Again what is striking is continuity with the inherited trends.

For 2018, however, we’re no longer in a world where nothing had changed—Congress has just passed a tax plan with monumental business tax cuts sold to the public for effects on employment and wages.  Mitch McConnell was hardly shy in his comments: “Under the policies of this unified Republican government, American workers, families, and business owners are achieving economic growth that is unmatched in recent memory.”

Polifact, however, examined the detailed jobs data—which looks like this:

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Their conclusion was what you can see: “Since 2010 — the year the Great Recession began to wane and the recovery began — every January-to-May period saw an average monthly job gain of between 160,000 and 236,000. The performance for 2018 was slightly higher than the average, but pretty typical.”

For now, we’re not going to pronounce on whether the tax cuts will ultimately prove useful or not.  However what we have seen is that even the monumental tax cuts of December, 2017 were not enough to change the clear, multi-year trend shown by all three figures.  The decline in unemployment is NOT where to look for effects of Trump’s economic policy.

Where should we be looking?

To start with we need to look at indicators that are more immediately sensitive to economic changes than the labor market is.   The obvious one is the stock market.

The following chart shows what has been happening to the Dow.

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Starting with Trump’s election the Dow has been rising in anticipation of some form of tax cuts.  Not only were those promised in the campaign, but they were the explicitly-stated objective of the major Republican donors.  Corporate tax cuts, which eventually formed the biggest single part of the story, influence the market directly by going to the corporate bottom lines.  (The markets correctly assumed that any payout to workers would be negligible.)

The markets continued to rise until Trump’s up and down trade wars entered the picture.  Since then neither the market nor anyone else knows exactly where things are going.   There are actually two negatives.   First is the obvious uncertainty.   Second is the fact that Trump’s specific tariffs don’t instill confidence, because they don’t match the business needs or even promote employment.

The graph above shows the uncertainty, but it should be pointed out that the performance is actually worse than it appears.  The fact is that the single biggest use for the corporate tax cuts of 2017 was stock buybacks, nominally to raise share value for investors.  Despite all that extra money, there is no current evidence of a solid increasing market trend.  Business confidence has been replaced by nervousness.  Even the Koch brothers and the conservatives in Congress are worried about the consequences of the threatened trade wars.

Then there is the matter of the deficit.  With the Trump tax cuts we have undertaken massive deficit-funded stimulation of an economy at essentially full employment.  Many economists at the time noted the risk of a speculative crash with nothing left to fund recovery.  Recent data has underlined that risk.

In April the IMF issued a warning.  They see a buildup of worldwide debt to record levels with the Chinese and the US as primary actors.  This first chart shows the overall numbers; the “emerging economy” portion represents largely Chinese debt.

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The next two charts show the US contribution.  The first shows that the US is an anomaly among developing nations in deciding to run the risk of large deficits in good times.

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The next slide adjusts the width of the vertical bar by the size of the national economy, which makes the impact of the US behavior very clear.

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The overall message here is that the amount of debt makes the world’s economic system increasingly fragile.  Further it should be remembered that in the 2008 crash the US and China were the major players who prevented a depression by stimulus spending.  They will be hard-pressed to do it again.  Just to be clear, this is not an argument about international responsibilities.  It is an argument about the risk to us.

As to what would provoke such a crash, we have an obvious candidate in Trump’s trade wars.  However the risks are not necessarily so exotic.  Business cycles end (ten years is a long time for one), and they end just when things seem to be going so swimmingly that people get giddy—like Mitch McConnell in the quote given earlier or the many critics of Dodd-Frank.  The following chart (which we’ll revisit in a minute for other reasons) shows when the past few recessions occurred.

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On that basis you have to say that 4% unemployment can be dangerous territory (particularly after a multi-year stock market boom).  We are quite literally going in blind with no backup.  And you can be sure that if we have a problem there will be scant sympathy from an administration that hates “losers”.

  1. Trump’s policies and the workforce

The cheers for the Trump economy have been based mostly on the unemployment numbers.  However, the chart just presented shows that is only part of the story.  As is evident from the chart, each of the previous business cycles was accompanied by wage growth in recovery.  But you can see that for the current cycle that wage growth hasn’t occurred.  (From this morning’s NY Times: “The rise in consumer prices over the last year has effectively wiped out any wage increases for nonsupervisory workers.”)

Otherwise stated, the benefits of the recovery have most emphatically NOT trickled down to existing workers.  Even in “Trump country” most people were not unemployed—the problem was replacing good union jobs with Walmart.  That hasn’t gotten better.

The reasons for lack of wage growth have been much studied.  The primary factors are

  • Globalization
  • Rise of Non-Standard Employment
  • Rise of Non-Compete Agreements
  • Automation
  • De-unionization

All of these items reduce the power of workers in dealing with management.  We won’t go through the items in detail (the referenced article does an excellent job), but the bottom line is that the Trump administration is actively engaged in making this problem worse.  The anti-globalization trade wars have been of little value for workers, and on non-compete agreements and de-unionization the administration has aggressively taken the side of management against workers.

Overall the current state of the Trump economy is NOT good for workers.

  1. Lost opportunities

The final topic is what has not been done because of the priorities established by Trump’s economic policies.  The point of departure is the following figure showing a breakdown of the financial impact of the business tax plan.

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The most obvious element on the chart is the loss of income due to the corporate tax cut, alone estimated to cost $1,349 B over ten years.  Since the decrease is from 35% to 21%, we get a nice round number for the effect—$100 B for each percentage point.  Note that this occurred as business was doing well, the economy was close to full employment, and the actual corporate tax rate (all benefits included) was a competitive 24%, (and real tax reform could have been close to revenue neutral).  Further we now know that the main beneficiaries thus far have been investors seeing results of corporate stock buybacks.

What could that money have bought?

Two obvious target areas are

  • Education, where school financing has never recovered from the 2008 crash, state budget cuts have teachers out on strike, and increased public college costs have led to a generation trapped in debt.
  • Infrastructure, a problem area identified by both candidates for President but unaddressed in the budget, because there was just no money left.

Estimates are available online for many projects in both areas, so with a hypothetical set of projects we can make the answer concrete:

  • The estimated backlog for repairs of existing highways is $430 B.
  • The estimated cost of upgrading US airports over ten years is $48 B.
  • Ten years of K-12 school repair to upgrade fair and poor facilities nationwide is $380 B.
  • Ten years’ worth of the estimated cost of a federal program to provide free tuition to all the US public colleges and universities is $470 B.

With $1,349 B we could have done all of that.

 

We can now summarize the state of the Trump economy.

  1. The declines in unemployment associated with recovery from the 2008 crash have continued in the same way through the Trump Presidency. No policy act of the Trump administration, including even the 2017 tax cuts, has produced a significant impact on that trend.
  2. Trump’s own policies have been disruptive. His repeated and ever-changing threats of trade wars have unsettled markets and businesses.  Even more importantly he has embarked on massive deficit-based stimulation in good times, contributing to an IMF-identified risk of repeating 2008 or worse.
  3. The decline in unemployment has not produced the usual accompanying increase in wages. Some of the reasons are structural, but the Trump administration has exacerbated this effect by systematically attacking labor’s leverage in dealing with management. Despite the decline in unemployment, the Trump economy has not been good for the existing workforce.
  4. Trump’s tax cuts have essentially stifled any action to address the pressing problems of education and infrastructure in this country. We have become in effect a poorer country.

 

On the North Korean Deal

Comment to the NY Times:

You have to say Trump got what he wanted—something that he could sell as an accomplishment for the mid-term elections. Kim of course had a stronger negotiating position, since he didn’t need a deal in the same way.  What we’ve got is a repeat of the way this process started: Trump gives in to Kim and gets lauded as a peacemaker. Maybe the press will grow up.

Kim got a cancellation of military exercises—a clear signal of US disengagement in the area. Good news for him and for China. The US got nothing, since sanctions were effectively abandoned by China when the negotiation process started. Kim’s statement of principles is no more of a commitment than what he’s said all along.

It also needs to be noted that the whole affair amounts to a ringing endorsement of nuclear proliferation.

Best quote thus far (from the Guardian):

Beatrice Fihn, the executive director of the International Campaign to Abolish Nuclear Weapons (Ican), tweeted: “We support diplomacy and peaceful solutions. But there is no agreement on nuclear disarmament and this all looked more like a big welcome party to the nuclear-armed club.”

Mid-Term Diplomacy

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Given all the publicity around our trade wars and the North Korean negotiations, it’s worth taking a step back to look at what’s going on.  Let’s check each one.

Why have we started the trade wars?

It’s a good question, since

– The steel and aluminum tariffs have nothing to do with the real issue.

– We gave up half our leverage with China by going it alone, and there have been no significant achievements beyond what China was prepared to do anyway.

Given those facts, there is only one conclusion here—it’s reality TV for the mid-term elections.  “I’ve fought for you like no one else has ever done.”  That way every single news story is a win.  Who cares about the details anyway?

That also fits with Wibur Ross’ explanation to the Europeans of the peculiar idea that the best path to negotiation is to declare war:

“China are paying their tariffs …. China hasn’t used that as an excuse not to negotiate… It’s only the EU that is insisting we can’t negotiate if there are tariffs.”

Otherwise stated:  why can’t you people be like the Chinese?  They let us play boss when we think it looks good.  You people just have to learn.

What’s going on with North Korea?

Trump needs a deal, and he’ll get one on Kim’s terms.   It will be just like all the other agreements with North Korea—phased (and easily reversed) build-down of nuclear weapons in exchange for benefits.  Except this time it will probably include phased withdrawal of US forces from the South—a bigger concession than any other American president has every made.

However, that will be enough for the Trump propaganda machine to get going, and the rest of the press will be so relieved that Trump’s worst impulses were contained that they’ll probably agree.  And so, for public consumption, Trump can be peacemaker for the mid-terms!

 

Back in the real world, though, there are two different questions worth asking:

What do these trade games mean for our future?

If there’s any policy, it’s that we’ve decided it’s no fun winning unless everyone else loses.  So we’re not interested in alliances or trade agreements.  Prosperity is only achieved at others’ expense, so conflict is good—a philosophy that is historically disastrous and particularly inapt today.

Where is the outcry about “national security”?

Our President’s “national security” trade wars are already a constitutional crisis—arbitrary control of trade is NOT an executive power.  And it’s just going to get worse until someone finally does scream.  Trump has been increasingly willing to make up a “national security” excuse for almost anything he wants to do.  Unless the Supreme Court (or public outcry) stops him, there is no limit to where that leads.

 

Trump’s Fabulous Foreign Policy Triumphs

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Trump’s foreign policy has been a media success. David Brooks has decreed that Trump’s “Lizard Wisdom” is far superior to liberal elitism.  Others have called attention to Trump’s fiendishly-clever strategy of brutal attack followed by pull-back to less crazy positions.  That approach hasn’t done much for the real world, but it sure has worked with the press.

With North Korea the press turned from hysteria at the “rocket man” rhetoric to admiration when Trump decided to cool it by accepting Kim’s meeting proposal.  All that  happened, though, is that Kim received a gift that North Korean leaders have wanted for years—certified international status—with no preconditions, which is to say with no commitment to do anything at all.

Kim is running this show for his own benefit.  Whether there will be advantages or disadvantages to the US remains to be seen, but Chinese president Xi sure doesn’t look unhappy.  And the presumption of success adds pressure on Trump to get an agreement under whatever assurances Kim will accept.  The only success in this picture is Trump’s convincing the press of an accomplishment.

Next about China.  David Brooks was crowing about a new opening for American cars—ignoring that Xi had already announced an opening for American cars before the trade war.  Trump’s trade war with China seems to be following the pattern of North Korea:  bluster followed by an agreement that can be trumpeted as “great”.  And the press is likely to fall for it again, overjoyed that the trade war has been replaced by “reason”.

In fact the dual trade wars (China and the EU) have greatly weakened the negotiations with China, and Xi can be quite happy with the cards Trump has dealt.  The U.S. represents 18% of China’s exports; the EU is almost the same.  Trump took half his leverage off the table with the attack on the EU.  As far as Xi is concerned—only in his dreams!

The third issue is the cancelling of the Iran nuclear deal.  This isn’t a case of bluster and retreat, but it’s another pretty story for public consumption.  Trump, Brooks and others talk about the moral imperative (ignoring the nuclear weapons consequences) of imposing sanctions for Iran’s other transgressions.  However, not only is it clear that the sanctions strengthen the hand of the fanatical clerics, but also by turning on the sanctions we have just played our last card.  We’re simply out of the game in Iran, waiting for rescue by regime change.   It’s interesting that we expect ordinary Iranians to love us and hate the Mullahs, because we choose to starve the poor and bankrupt the middle class.

Iran now has no reason (short of war) to care about US policy,  a position they succinctly expressed with the immediate rocket attack on Israel.  We have taken one more step to complete irrelevancy in the Middle East, this time leading to a possible war and a nuclear Iran.

As Trump has said over and over again:  he can’t lose because he owns the press—he’s just too good as copy.  On foreign policy you can push that one step farther: the copy is all you get.

The Trade Wars Are a War on Trade

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The impending trade war with China has already generated the drumbeat of preparations for a real war:  “cheating”, “usurping”, “impoverishing”, “existential threat”, …  Of course with Trump you never know what’s just posturing, given how quickly the North Korean “rocket man” became “very honorable”. So there’s hope that the Chinese trade war will somehow wind down.

But even if it does, we have to recognize there is now a real, ongoing threat to international prosperity.  Trump is attacking the system of fair trade that underlies the world’s rise in prosperity since the second world war.  His notion of sovereignty means he refuses to acknowledge any limits (international or constitutional) on his ability to use trade as a weapon.  That is new, and if it wins we all lose.

 

We start by quoting a position from the Business Roundtable of corporate CEO’s on the trade negotiations with China.

“During negotiations with China, the Administration’s objective should be to secure lasting economic reforms that will curtail China’s unfair trade practices and allow U.S. businesses to compete on a level playing field. Negotiations that focus on temporarily reducing the trade deficit would make this a wasted effort. Working in coordination with our allies, the U.S. should set deadlines on those economic reforms and outline the consequences China would face if reforms aren’t made. This approach will provide an opportunity for the Chinese to produce results and for the Administration to protect the interests of U.S. businesses and workers effectively.”

The important thing about the quote is that it sees the problems with China within the context of internationally-defined fair trade.  And it emphasizes the importance of working with our allies to make the system successful.   That’s as opposed to “reducing the trade deficit”—which seems to be at the top of the administration’s list both for China and for Mexico, Canada, and others.  (This is despite the fact that our balance of payments deficit with China has decreased greatly from peak, is not a financial problem, and is not the reason for the stresses on the American middle class.)

It is important to recognize that regulating deficits is worse than “wasted effort”; it actually subverts the real objectives of fair trade.  Not only does it make China responsible for something it doesn’t completely control (we’re the ones pumping up the federal deficit), it is a rule we would never accept for ourselves.  The whole idea of fair trade is that it should be a system of known rules by which everyone can play; here we’re just imposing whatever we think we can get away with.

The quote is of course coming from businessmen, but the issue is one for everyone.  The downsides of international trade exist, but most cases the problems are of our own doing.  Further the most effective way to impose standards for labor and environmental issues is to work through the definition of fair trade.

We have already sinned against WTO fair trade once, by invoking “national security” as a blanket excuse for unilateral tariffs.   We are going beyond that here by setting rules for others we have no intention ever to obey.

 

The second example is the administration’s other major issue in the Chinese negotiations: “Made in China 2025”.

For high-tech, China today is primarily building products for western companies.  Generally most of the intellectual content and profit goes to the parent company (e.g. Apple) as the top of the heap.  Unsurprisingly, China would like to move up the value chain to get more of the benefit.  Also, China today sources most of the IC chips in the products it builds from other countries—a fact that China views as a risk to its success.  Made in China is the plan to move up.

Made in China 2025 covers just about any technical field you can think of (except AI, with its own plan), and the government expects to spend money to make progress happen.  As an idea, this isn’t terribly different from what is going on in many other countries (see here for a summary of national spending on AI).  But the Trump administration has decided that the whole idea of Chinese government involvement in technological advancement is suspect.

While Mnuchin and others use the language of fair trade to attack the Chinese plan, those attacks have lacked much specificity.  And in fact if the administration is worried about abuse, they could take the whole affair to the WTO.  What makes the case even weirder is that, as we know, the Trump administration has proposed severe cuts in US government funding for research in essentially all fields, claiming the private sector does it better.

So one has to conclude that what is going on is trade warfare pure and simple.  The Trump people (with their zero-sum view of the world) are afraid the Chinese might catch up, and their goals is to throw as many nails on the road as possible to slow them down.  That’s what passes for economic policy.

This is arrogant foolhardiness of the sort the world hasn’t seen since the geniuses of the Iraq war.  As many have pointed out, the companies most hurt will be American.  And the message for the rest of the world is clear.  The US, with quite a lot to gain, has decided it doesn’t need free trade.

The end here, as in our last piece on trade, is constitutional.  It becomes more urgent each time.

Normally, without the seldom-used national security ploy, tariffs are a matter for Congress.  When Trump got away with it on the aluminum and steel tariffs it was a scary first step.  We’re now fighting a whole trade war with China, and no one is questioning that it can be done purely by fiat.

So we no longer need to argue about whether Trump will or won’t try to make himself a dictator.  Unless something happens, he is already in position to wreck our economy all by himself.

Tesla and Ice

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This is a short note on a couple of issues related only in that they say something relevant about the future we should be planning for.

The first relates to Tesla and its production difficulties with the new, lower-priced model 3.  The highly-automated production of the model 3 is well-behind schedule, to the point where it is a big hit to the cash flow of the company.  We mention it here, though, because the delay is an indication that mass production of electric cars is something fundamentally new.

An electric car is a much simpler machine than an ordinary, gas-powered vehicle.  In principle the construction should be both cheaper and easier to automate.  Current production of Teslas is intrinsically a low-volume operation.  The model 3 will be the first indication of what newly-imagined electric car production is like.

I don’t know if we’re in for a shock or not (this is after all a first go at it), but this could be another big change to conventional middle-class employment.  And there will be follow-on effects for gas stations, and especially maintenance and repair.  This is another of many indications that broad, technology-based disruption of jobs is going to happen.

 

The other story is about the commissioning of a new class of Russian icebreaker—targeted at clearing northern ship lanes freed up by the retreat of polar ice with global warming.  The phenomenon is already clear, although the amount of traffic is still small.  The Russians are preparing for the opportunity with multiple classes of new machines planned for release up to 2025.  The Chinese have announced cooperation with the objective of reducing shipping times to Europe by a third.

The US is of course uninterested in consequences of climate change.  The only Coast Guard ice breaker is 40 years old, and they have a hard time getting authorization to get a new one.  The Bering strait, however, could be a shipping lane.

This is a very small example, but climate change affects many things, and as a country we’re trying to avoid finding out about them.

 

The current federal budget is put together for a world where the private sector will take care of everything.  That has always been a fantasy—the efficiency of the private sector comes in large part from its ability to ignore everything not relevant to immediate financial success.  It is particularly false for a world undergoing fundamental change.  We either recognize it and help people through it, or we fall behind and revert to the nightmares of the nineteenth century.

Keeping Score on North Korea

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Now that Fox News has awarded the Nobel Peace Prize to Trump, it’s time to go over what has happened thus far with North Korea and what is likely to happen from here on out.  Here’s the timeline.

1. Trump and Kim Jong-un exchange rocket launches, military maneuvers, nuclear tests, insults, and other hysterics, so the world gets worried something disastrous might happen. This enhances the domestic position and international profile of both participants.

2. The South Korean President—elected on a platform of better relations with the North—takes the occasion of the Olympics to try to calm the situation.

3. Kim seizes the opportunity and opens a full-scale charm offensive toward the South. Kim’s sister easily bests Mike Pence in the Olympic charm event.

4. Trump reverses decades of US policy and agrees to a one-on-one meeting with Kim without pre-conditions. Score 1 for Kim.

5. South Korea’s president, seeing Trump’s threats and how little South Korea’s welfare figured into US policy, realizes he can’t count on the US as a reliable ally. This is a strong incentive to conclude a peace process agreement with the North.  Score 1 more for Kim.

6. For US consumption Kim has now been declared “very honorable” and Trump is a hero for resolving the situation. The negotiation is considered a success before it has begun.  On the basis of negotiating position, score 1 more for Kim.

7. The most probable outcome is a staged denuclearization of North Korea in exchange for a draw-down of US military personnel and equipment in the South. The “staged denuclearization” we’ve seen many times before; the US military disengagement is new.  Trump exults over the money saved.  Score 2 for Kim.

8. The final situation essentially replaces US influence in South Korea with North Korean influence. Further this is one more step in the US disengagement from Asia to the benefit of China.  Score 1 more for Kim, 4 for Xi Jinping.

9. Final score:  Kim 6, Xi 4, US 0

Figure Skating and Math Textbooks

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This odd coupling of topics links two separate frustrations which seem to come from the same core problem.

For figure skating the problem is one particular feature of the new scoring system—the extra points given to jumps in the second half of a program.  There’s no question that those are harder to do—anything is harder to do when you’re dead tired.  The question is why that extra stamina should be rewarded as an achievement!  In the last Olympics, those extra points were primarily an opportunity to game the system.  What was missing in artistic achievement could be made up in pure stamina.  The winner of the woman’s gold medal had all of her jumps in the second half of the program.

The fundamental problem is that from within a system, people are tempted to put a value on anything hard, regardless of whether or not it means something to the outside world.  People who have spent their careers doing and teaching figure skating have a tendency to think that any hard jumps are achievements.

Having just had reason to work my way through a high school algebra textbook, I have to say (based on an admittedly small sample) that the textbook publishers have that problem in spades.  The book was filled with minutia that need to be memorized for exam questions.  Getting through all of that is certainly hard, but it’s not the stuff that will make kids confident and successful in mathematics.  And that certainly won’t make them like it or remember it.  (One more reason it’s tough to be a teacher!)

It seems to me that high school math has a pretty good story to tell.  Core algebra is based on a really brilliant idea, that an answer you are looking for can be handled just like a number.   That lets you create a simple kind of manual computer.  If you can write down the problem, then the manual computer can solve it.  Sort of like doing your taxes with turbo-tax.  The core ideas are simple and don’t need to become a challenge.  From there it’s straightforward to do linear systems, which are part of the vocabulary in science, engineering, finance, or just about anything else.  “Vocabulary” is actually the right word.  For most students, mathematics is a language in which they need to gain confidence and fluency.

Then with quadratic equations you hit a wall.  All the techniques you just learned fail.  But you’re saved by another really good idea you would (probably) never have come up with yourself.  You do a few exercises with that idea, working toward an answer that is basically all you need to know—the quadratic formula.  Any further messing around with higher-order polynomials and rational functions is only supplementary, examples of what you can do with other kinds of problems you might run into.  No more rules to learn.

Going beyond that you have topics with really exciting applications.  Conic sections are geometrically interesting and tied to the whole story of the Copernican revolution, Kepler’s laws (“equal areas swept out in equal time”), and Newton’s gravitation.  (You don’t need calculus to tell the story.)  Sequences and series cover ideas behind every financial analysis students will see.  Permutations and combinations are important for probability and statistics and introduce the first non-trivial example of a group.  Modular arithmetic is a useful feature of every programming language.   Trigonometry is important but can become unnecessarily confusing.   Sine and cosine functions are important parts of the vocabulary, but you can get lost forever in trigonometric identities and formulas.

Mathematics shouldn’t be a drag.  There are few core principles and very little actually to memorize.  You get to see some remarkable solutions to tough problems.  One famous mathematician said that mathematics should be “like floating down a river on your back.”  My father-in-law put it differently: “Mathematics is for people too lazy to do real work.”