Category Archives: Policy

commentary on public policy issues

A Different Story for Mathematics

From our parents and teachers, we learn that, when performing a mathematical or arithmetical task, the result is a right answer or a wrong answer to a question posed by an adult. We learn that our answers result from a process and that there is only one process we can use to produce them. We learn that every question has an answer. We are pleased when our work produces a right answer and pleases the adult who asked the question and we are distressed when our work produces a wrong answer and displeases that adult. We conclude as young children that Arithmetic is rigorous, deductive, computational and useful. As our parents, teachers, and peers reinforce this conclusion throughout the first twelve years of school, we internalize and preserve it. It becomes bedrock to our view of the world and the places of things, events, and actions in it.

Neither History, Fiction, Poetry, Music, Physics, Chemistry, Biology nor, well, any subject is presented exclusively in this way. Sure, each field has its jargon to learn, its principles to apply and its recipes to follow. But, we learn in every field other than Math that authors embed their stories in context and tell them in several ways, teachers employ many devices and draw from many resources in their pedagogy. They tell the stories of their subjects as fields of inquiry that have evolved with civilization and like civilizations. They battle about the course of history, the nature and limits of fiction, music or poetry, or the proper use of its elements (character, plot, narrative voice, rhyme, meter, tone, key, scale, theme) the fundamental laws of matter and energy, the material composition of things or the building blocks of living things. Our teachers and parents demonstrate and encourage experimentation and discovery in every area—except Arithmetic and Mathematics.

During our lives, some of us find by chance a few corners of Arithmetic that are fun: Rubik’s Cube, Sudoku, magic squares, and ciphers among them. Others among us discover that Mathematics is quite different from Arithmetic and attempt to cross the divide between them. But, crossing this divide is arduous; everyone who has attempted this crossing knows this. Many of us decide that it’s too arduous, reverse course, and return to the safety and certainty of the Arithmetic and Mathematics we know, to the Island of Conclusions. The remaining few travelers complete the crossing and seek roles for themselves in this new land. What do they find there? Why is this crossing so arduous?

They find a vast network of communities of people, each of whom inhabits some area in an open, limitless landscape that we can characterize as an evolving field of inquiry (a field of evolving inquiry, perhaps). Some communities inhabit developed areas; some inhabit territories on the frontiers of Mathematics and others inhabit undeveloped territories beyond such frontiers. Each generation of inhabitants, called Mathematicians, explores, settles and develops land beyond its frontiers and discovers new relationships, new networks, within the developed territories. This story is the story of Mathematics; Arithmetic and school Maths are chapters in it.

We make the crossing to Mathematics, the field of inquiry, from Arithmetic, the body of knowledge, arduous by failing to prepare our students for it. If they need to cross an ocean, they must build the boats, learn to sail them and learn to swim. If they need to cross a desert, they must build the wagons or find and train the camels, and learn to navigate by the night sky, the position of the Sun, or by compass. But, we the communities of educators and parents, have failed to provide our students and children the boats, wagons, camels, or compasses or to show them how to build, train or use them. We’ve given them boats; they need ships. We’ve taught them how to use a calculator for Arithmetic and to replicate algorithms for Algebra, Geometry, Trigonometry, and Calculus to find unique answers to manufactured problems. But, that is not enough. They need to learn how to recognize patterns and describe them with precision. They need to learn how to experiment with mathematical objects to discover patterns in their behavior. They need to learn to associate freely among concepts that appear similar or disparate, seeking connections they haven’t discovered, yet. For, although Mathematics is rigorous, deductive, computational and useful, it is, like any other subject, experimental, inductive and inferential. By framing it as the former alone, we hide the latter aspect of its nature.

Nearly all students learn to calculate, deduce and apply arithmetical and mathematical tools readily enough to get them through the days of their lives. But, the mathematics required for daily living is even more elementary than the reading required for it. Barcodes and computers total our groceries purchase and we use a plastic card to pay; no computation is required. Most daily math tasks are counting tasks. Calculators, hand-held or online, can calculate mortgage payments and their elements and other tasks that require more than counting. Students learn enough math for their daily living before they complete the seventh year of school. Demonstrations of and practice at applications of Algebra, Geometry, Trigonometry, Matrices, Vectors or Calculus convince our students that these tools are useful, indeed. But, such demonstrations and practice don’t convince them that they will ever use those tools. Mathematics’ usefulness in the world fails to motivate students to learn it after they’ve completed the sixth year. For the first three or four years, they needed only the approval of adults to motivate learning it; its usefulness didn’t matter to them. Once they’ve learned enough of the math they need to use daily, they lose interest in learning any further uses. Teaching them, in subsequent years, for example, how to use techniques in solving systems of linear equations in linear programming applications does not suffice to impel them to learn either how to solve systems of linear equations or how to apply those techniques in the world. The answer, “Because it’s useful,” to the question, “why do we have to learn this stuff?” is true, but it fails to stimulate students’ interest and impel them to learn such techniques and their applications. Only a few students will use them, and those students are unaware of their futures and don’t know, yet, that they will. WYSIATI (What You See Is What There Is) dominates their perceptions and their conceptions.

Mathematics is a body of knowledge, a compendium of results (truths) developed over at least seven millennia. Its method of verification is most rigorous of the methods of the sciences and its facts live longest of them. Logical deduction plays a unique role in its practice. Yet, Mathematics is more than a static body of knowledge. It’s an open, evolving field of inquiry into the nature of numbers and their relationship to the world. Its practitioners, Mathematicians, pursue answers to questions that arise in their endeavor to locate and understand the fundamental elements of the mathematical universe and explain their existence and behavior because such questions interest and challenge them and because they feel themselves continue to grow and learn, which they like to do. A Mathematician enjoys feelings similar to the feeling an author feels when he’s written a satisfying paragraph or an effective stanza or to a musician when he’s executed a passage “perfectly” or his performance jells with an ensemble.

Importance and applicability are reasons necessary to the pursuit of mathematical knowledge, and they are reasons necessary to the pursuit of any and all other knowledge. But, they are insufficient. In this regard, too, Mathematics is like all other fields of inquiry. We pursue our inquiries into them because we enjoy the process, and our success in discovering and answering questions amplifies this enjoyment a thousand times. The pursuit of mathematical knowledge is interesting, gratifying and creative. These factors impel practitioners to practice. These factors will impel students to learn to practice it, too.


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Three Myths of Mathematics and Mathematics Education

To understand current Math teaching practice, IWY (I-do, We-do, You-do), we need to recognize at least three myths about Mathematics and Mathematics education—stories about what it is, whether and how it differs from other fields of inquiry, why we need to learn it, and how we learn it:

  • Myth 1: Mathematics is unique among all fields of inquiry in its structure and practice. Mathematics is deductive and rigorous—and nothing else;
  • Myth 2: Logical rigor and deductive method—and nothing else—must frame Mathematics pedagogy;
  • Myth 3: Mathematics is a tool for use in the construction of other things such as buildings, rockets, theories and submarines—and is nothing more.

Math is different, right? There’s always a unique, right answer to every problem. Math is rigorous: you can always verify whether your answer is correct or incorrect with absolute certainty. When you read a topic section in a Math textbook or an academic paper, you encounter definitions, first, then theorems and their proofs. The theorems use the definitions to assert some claims about the world. These theorems follow by strict rules of formal logic from other theorems that were proven previously. A machine could, in principle, prove the theorems in your textbook. New research results are always published in essays in academic journals and rarely included (as new results) in textbooks. Although there is little or no exposition in an academic paper, there is almost always some exposition in a textbook, usually pictures, diagrams or graphs of a situation, or examples of algorithms, or all of them. The pictures, diagrams or graphs are heuristic devices; they motivate and depict the concept you’re supposed to learn to help you retain them for use. The examples are heuristics, too. They show you how to use the algorithms you learned to solve problems posed in the book, that is, how to do the homework problems. By telling the story of Mathematics only in this way in books, classrooms, and lecture halls, textbook authors, publishers, professors, and teachers present an incomplete picture of it. They create the myths that: Mathematics consists of definitions, theorems, proofs, and problems; theorems are deduced rigorously from other theorems by strict, formal rules of logic; and these theorems are deployed as algorithms to solve problems—and nothing else.

This story of Mathematics became the dominant narrative of Mathematics as the result of cognitive processes developed to promote our survival as a species over millions of years of evolution by trial and error. WYSIATI combines with representation, cognitive ease, and repetition to embed this myth in our memories and enable its instant, subconscious recall. WYSIATI is an acronym for What You See is All There Is. It is a slogan for the fact that the associative machine—our minds operating in “current awareness only” mode (which is 99% of the time)—dominates our awareness at any and all given moments; we are “hardwired” by evolution to jump to conclusions based on limited information. If the only story about Mathematics that you’ve been told is that it is rigorous, deductive, and computational, you will jump to the conclusion that it is only rigorous, deductive, and computational. This jump occurs early in school, in the first grade when we learn to add and before we’ve learned to reflect on and filter our immediate experience. Jumping to the Island of Conclusions is easy. But, you can’t jump back and there are no bridges or boats; you must swim back to the mainland, which is hard, so you are unlikely to try. Soon, we develop a stereotype of Mathematics that we use to represent it in all of our thoughts and emotions about it. This stereotype is reinforced by the repetition of IWY pedagogy over the first twelve years of school. It becomes the easy representation that we invoke subconsciously whenever we think about Mathematics or perform Mathematical tasks. By the end of first grade, we use this stereotype to represent Math throughout the school years and for the remainder of our lives, unless and until we learn another story and stereotype with which to replace it. Go on, try it! Try to think of Mathematics in some other way. How did you do?

The myth that Mathematics consists only of definitions, theorems, proofs, and problems (applications) frames our pedagogy and thereby limits our practice to stocking our student’s inventory of facts and algorithms to use them. We employ IWY as the default pedagogy because we believe this myth. We learned this convenient, easy representation of Mathematics from our professors, teachers, and textbooks, and we use it in our classrooms and lecture halls. It “worked” for us; it should work for everyone, shouldn’t it? We want our children, our students, to succeed, don’t we? After all, success is good; it reinforces what they’ve learned. Practice makes perfect. Homework is the opportunity to practice and, thereby, perfect and make permanent what they’ve learned. Solve the problems, achieve success, feel good about what you’ve done and who you are. Of course, we use heuristics to motivate topics. Thus, we use the rectangle to illustrate the commutative property: A rectangle’s area is the product of its length and width; the order of multiplication doesn’t matter (32 = 8 x 4 = 4 x 8). Or, we use a pair of scissors to illustrate the Hinge Theorem: the length of the side of a triangle is proportional to the measure of the angle opposite it; in a triangle, the side opposite a 60º angle is longer than the side opposite a 30º angle (√3 times longer). The hinge of the scissors represents the vertex of the relevant angle and the opening between the tips of the blades represents the side opposite the angle. As you open the scissors, the angle between the blades increases and the distance between their tips increases; as you close them, the distance between the tips shrinks as the angle between the blades decreases. We intend such heuristics to point to the path to understanding and retention, but they don’t.

Perhaps the most surprising and wonderful fact about Mathematics is its fruitful application to nearly every aspect of the world we inhabit. Its applicability is integral to the prevailing story of Mathematics. Applicability would,  of course, be integral to any story of Mathematics. But, its role in this story is to play the sole reason to learn and do Mathematics. We learn Mathematics because it’s useful. Its value lies in its instrumentality. In at least one widely used high school textbook, at the beginning of each section there is a short list: what the section is about, what the student is going to learn how to do, and why the student should learn it. The third item in this list is, invariably, “It’s useful.” And, naturally, the section includes examples of its use in the world. Thus, we learn algorithms to perform such counting tasks as totaling the cost of a basket of groceries or the addition to your house or the size of your farm, or to perform such engineering tasks as determining the thrust required for a rocket of a given size to achieve escape velocity. We also learn algorithms to help us tell which algorithm(s) to apply to which circumstances. We call it “cookbook math” or “engineering math”. Our students—our children—do not meet “real” Mathematics until they’ve completed a twelve-to-fourteen-year apprenticeship that fails utterly to prepare them for this meeting.

So, Mathematics is, according to these three myths, in fact, and indeed, a body of knowledge consisting of definitions, theorems, their proofs and the algorithms that we use to apply those proofs to solve problems. Its value and, therefore, our motivation to learn it lie in is instrumentality. Together, these three myths comprise the meta-myth of Mathematics.

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The “Chicken(expletive) Club”


The Baseline Scenario

By James Kwak

The only “Wall Street” “executive” to go to jail for the financial crisis was Kareem Serageldin, the head of a trading desk at Credit Suisse, according to Jesse Eisinger in a recent article. Serageldin pleaded guilty to—get this—holding mortgage-backed securities at artificially high marks in order to minimize reported losses on his trading portfolio. 

Now if that’s a crime, there are a lot of other people who are guilty of it. In fact, a major premise of the federal government’s crisis response strategy was exactly that: allowing banks to keep assets at inflated marks in order to pretend they were solvent when they weren’t. FASB changed its rules in April 2009 in order to make it easier for banks to inflate their marks. And the Obama administration’s “homeowner relief program” was designed to allow banks to delay realizing losses on their mortgage loans by dragging out—but generally not preventing—foreclosures. (Remember…

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Why people accept the things they do

I just read this outstanding post on Phil Ebersole’s blog. Everyone should read it.

Phil Ebersole's Blog


I don’t know when, where or if this experiment was actually carried out, but it is a good parable of why bad customs persist.

Hat tip to Carol Avedon (who is listed on my Blogs I Like page).

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It’s obvious, “Those who own the country should run the country.” – John Jay, first chief justice of the SCOTUS

John Jay, the first Chief Justice of the Supreme Court of the United States (“SCOTUS”) wrote this sentence.  In 1800, this claim was axiomatic to governance in the U. S.  The population of the country was 3,000,000; agriculture was the primary source of income for its citizens and land ownership was the primary store of value (source of wealth).

They understood the meanings of “own”, “run” and “country”, as well.  “Own” was a legal term:  legal title was necessary and sufficient.  “Run,” meant, “to manage as a business or farm.”  And, “country” referred to the 13 original colonies.  Everyone knew what Jay meant:  land and business owners should vote and no one else should vote.  Thus, women, slaves, indentured servants (none of whom could own property), tradesmen and merchants who rented their place of business from a landlord couldn’t vote.

Around the time of the founding of the republic (circa 1770), agricultural land constituted about 50% of the total stock of public and private capital assets in the U. S.  Today, it constitutes less than 1% of total capital.  Housing constituted about 25% of capital and other domestic capital (commercial buildings, equipment, private harbors, roads, etc.) constituted about 25%.  Net foreign capital (capital owned by U. S. citizens or corporations) was tiny and remains tiny.  Today, housing amounts to about 40% of domestic capital and other domestic capital constitutes about 60% (I know this list totals to 101%, but these are estimates and I’m not quoting the exact figures, anyway).  In 1800, the population of the United States was about 3,000,000.  Today (2010), it’s about 350,000,000.  Population grew by a factor of 117 during the intervening 210 years.  The United States in 2010 is different from the United States in 1800 in practically every meaningful way.  Its population is larger and more diverse, its territory is larger and more diverse, its sources of income are different, its relative standing in the world is different, forms of ownership and “personhood” are different (corporations are “moral persons”, whatever that means), women can vote, children and women have rights and there are no slaves (legally) here, not to mention the technology gap (essentially, the entire industrial revolution and a good chunk of the information revolution).

Who, then, “owns the country” and “ought to run it”, today?  One unstated and widely believed answer to this question is something that resembles, “the U. S. citizens who own the financial-economic assets located within the borders of the United States own the country and should run it”, an echo of Jay’s pronouncement 214 years ago.  This principle excludes some obvious groups:  U. S. citizens living abroad who own no property in the U. S. and no financial instruments that evidence ownership of U. S. entities; U. S. citizens living in the U. S. who own no real or financial assets located in the U. S.; and U. S. residents who are not U. S. citizens.  This principle includes whom?  It includes me, my spouse and two of my adult children (both of whom own homes), but it excludes my other two adult children (who are college or university students of voting age).  It includes the Koch Brothers, Rupert Murdoch (not a citizen), Barack Obama, Ted Turner and George Soros (these folks are just some of the well-know owners).  It includes homeowners, the investor class and owners of private businesses and farms.  In the current demographic, the owners are the middle class, upper middle class and wealthy, while the indigent, working poor and lower middle class are generally excluded from the owner class and, therefore, from the voting class.

Yet, who, then, owns public property and public assets?  Does every citizen own, by proxy, a share of the national parks, monuments and forests, undeveloped or unincorporated land, wilderness, government buildings, or the oceans within the 12-mile boundary?  Do local residents of a municipality own its public parks, public jails, wastewater treatment plants, public landfills and streets and curbs as well as such service organizations by proxy?  Are all citizens, then, property owners?

What could “run the country” mean, today?  What could “run the country” mean in any period during a democratically elected, representative governing body?  Leading an elected body is not coaching a team, managing a commercial enterprise or growing grain (see my blogpost on this topic).  To “run the country” seems to mean that elected, representative officials would comprise people who own domestic assets (financial, productive or real).  Citizen-owners would direct or influence their elected officials to pass laws that those owners want passed and erect administrative structures of which such owners approve for their implementation.  If the class of owners constitutes all citizens (some as indirect owners of government assets, then, this definition seems to be consistent with the U. S. Constitution.  If, however, the class of owners constitutes a subset of the citizenry and their influence exceeds their portion of the population, this definition describes a situation that is inconsistent with the U. S. Constitutional principle of universal suffrage.  This idea, universal suffrage, expresses the idea that each vote counts equally and connotes its extension to the idea that each voice is heard and understood on the same basis as every other voice.  The day laborer’s vote, the doctors’ vote and the CEO’s vote count once each; the laborer’s voice, the doctor’s voice and the CEO’s voice should be heard equally well.

In the United States, today, there seems to be disconnect between the idea of universal suffrage and its implementation in the political election process.  The SCOTUS majority seems to have forgotten that the world and the U. S. have changed significantly and meaningfully since its founding 238 years ago.


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It’s obvious: “Government should be run like a business.”

It’s obvious:  “Government should be run like a business.”

This assertion has a corollary, which has a strong form and a weak form:

  • Strong form:  “Business leaders should govern.”
  • Weak form:  “Government officials, elected or otherwise, should have at least some business experience.”

It also has an analogous assertion:  “Government finances should be managed like family finances.”

First, we should try to understand the phrase, “like a business”.  How is a business “run”, and, to achieve its goals, how is a business structured?  Economists, sociologists and business people propound more than one “Theory of the Firm”, but, this article is a blog post, and I see no need and insufficient space to discuss this entire topic here.

We can, however, point out some salient characteristics of management structures that are found in almost all companies and non-profit organizations.  I don’t claim that these characteristics are essential for effective management, only that they are endemic to it.  I also claim that these characteristics are a product of design, evolution.  They are:

  1. Command and Control management structure:  The CEO, with the assistance of his or her small circle of senior lieutenants, determines policy unitarily; members of the organization implement it uniformly throughout the organization.
  2. Tightly Restricted speech:  Organization policy determines message content to other organizations (public relations, transaction content, professional and commercial communication) and among the members of the organization (informal and formal speech and writing norms).
  3. Non-representative governance:  Leaders at every level are chosen to lead by their superiors or peers within the organization (or its governing Board, a very small circle of “friends”).  Individual producers have no direct influence on the appointments of leaders, nor do policy makers consider their personal life interests or requirements when forming policy (except as an afterthought).
  4. Opaque operations, structure and decision processes:  The company attempts to hide its operations and specific intentions about many aspects of operations from competitors, observers (the press, the government, the tax authorities, for example) and employees to gain competitive advantage.
  5. The stated goal of for-profit corporations:  to maximize the financial wealth of company owners (shareholders, members, as in LLCs, partners).
  6. Singular focus on achieving this goal, i. e., “winning the game by any means ”.

Promised outcomes, accordance to economic theory of the firm, of this structure include:

  1. Superior (“economic”) returns to shareholders;
  2. Nothing else.

A representative governance structure and management has at least the following salient characteristics:

  1. Distributed Power:  Decisions are taken as a result of collective action; the most frequent form of collective action in governments is voting under the principles, one person, one vote and a majority decides the issue in its favor.
  2. Loosely restricted speech:  Members of the governing body are permitted to participate in framing, arguing and voting issues equally; they make speak to non-members as they wish about any subject they want to discuss.
  3. Constituent representation:  Presumably, representatives express the preferences of their constituents and act on them by voting consistently with them.
  4. The stated goals of the government are several and the representatives and their constituents dispute their composition continuously.
  5. Transparent operations, structure and decision processes.  Admittedly, no government is completely transparent, but citizens require a significant minimum transparency of their governments.
  6. Foci on various means and limitations on means to achieving such goals.

By comparing these lists, item-by-item, it should be obvious that economic entities and political entities differ materially along at least these five dimensions.  Indeed, almost no citizen of any Western democracy would want government according to business principles.  It would be tantamount to wanting governance in the style of Somalia, Syria, Egypt, Saudi Arabia, Senegal, Burma, or any other totalitarian dictatorship that you want to name.  What, then, can one mean by asserting, “Government should be run like a business?”

To run a government like a business means, apparently, that its managers should devise a budget that represents the planned implementation of strategic policy, they should be allowed to deviate from it only by special approval of strategic-level management, and total expenditures should not exceed total operating cash inflow in any single year.  This position is naïve.  The federal and state government have budgets, and expenditures that would exceed budgeted amounts require approval of their legislative bodies.  Private companies do require budgets, and upside deviations from budget do require approval by more senior managers.  But, total expenditures frequently exceed total cash operating cash inflow for companies in a given period and they may do so for several periods (years) without damage to the future viability of the company.  To the argument that this operating cash outflow must turn positive at some time (but no particular time) and over the life of the firm cash inflows must exceed cash outflows, I note that so long as lenders are willing to lend and investors are willing to invest in it, a given company may never achieve positive net operating cash flows during their lifetimes.  Many companies never do achieve positive cash flow.

So, the proposition, “Government should be run like a business,” seems quite dubious to me, at least from the perspective of a citizen of a country with a representative form of government.  No two organizations are structured or operate less dissimilarly than a private-sector company and a representative body.  Representative government is a recent development in the history of civilization; private enterprise is ancient.

Having pointed out some considerable de facto differences between representative governments and private-sector companies, I haven’t addressed the question as to whether governments should be run like businesses or vice versa, i. e., whether businesses should be run like representative governments.  There are successful cooperative private companies that are run and/or owned by their employees in the United States and in Europe.  There was, at least, one in Brazil, documented in two articles in The Harvard Business Review in 1983 and 1985.  The degree of employee (vs. owner) participation in management processes varies among companies and the incidence of companies that include such participation varies from country to country.

One could draw an analogy between the primacy of corporate stockholders and the primacy of the citizenry and claim that as corporations seek to maximize the wealth of their shareholders so should governments seek optimize the well-being of their citizenry.  This analogy only shifts the comparison from goals to methods.  Businesses are managed from the top down with great flexibility and little accountability for any particular actions; governments are managed from the top down according to strict rules developed within the framework of laws passed by their legislative bodies.  The managers of government agencies are accountable to their managers, their legislators and, ultimately, their constituents.

I just don’t see the advantage of business experience versus experience as a professional, non-profit manager or other non-government employee.  I do believe that a political career should be a second career.

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Econ 101: What is a free market? (it’s obvious, isn’t it?) – a short quiz

Econ 101: What is a free market? (it’s obvious, isn’t it?) – a short quiz.

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