This post continues a series begun in 2014 on implementing the Common Core State Standards (CCSS).  The first installment introduced an analytical scheme investigating CCSS implementation along four dimensions:  curriculum, instruction, assessment, and accountability.  Three posts focused on curriculum.  This post turns to instruction.  Although the impact of CCSS on how teachers teach is discussed, the post is also concerned with the inverse relationship, how decisions that teachers make about instruction shape the implementation of CCSS.

A couple of points before we get started.  The previous posts on curriculum led readers from the upper levels of the educational system—federal and state policies—down to curricular decisions made “in the trenches”—in districts, schools, and classrooms.  Standards emanate from the top of the system and are produced by politicians, policymakers, and experts.  Curricular decisions are shared across education’s systemic levels.  Instruction, on the other hand, is dominated by practitioners.  The daily decisions that teachers make about how to teach under CCSS—and not the idealizations of instruction embraced by upper-level authorities—will ultimately determine what “CCSS instruction” really means.

I ended the last post on CCSS by describing how curriculum and instruction can be so closely intertwined that the boundary between them is blurred.  Sometimes stating a precise curricular objective dictates, or at least constrains, the range of instructional strategies that teachers may consider.  That post focused on English-Language Arts.  The current post focuses on mathematics in the elementary grades and describes examples of how CCSS will shape math instruction.  As a former elementary school teacher, I offer my own personal opinion on these effects.

Certain aspects of the Common Core, when implemented, are likely to have a positive impact on the instruction of mathematics. For example, Common Core stresses that students recognize fractions as numbers on a number line.  The emphasis begins in third grade:

CCSS.MATH.CONTENT.3.NF.A.2
Understand a fraction as a number on the number line; represent fractions on a number line diagram.

CCSS.MATH.CONTENT.3.NF.A.2.A
Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.

CCSS.MATH.CONTENT.3.NF.A.2.B
Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.

When I first read this section of the Common Core standards, I stood up and cheered.  Berkeley mathematician Hung-Hsi Wu has been working with teachers for years to get them to understand the importance of using number lines in teaching fractions.^[1] American textbooks rely heavily on part-whole representations to introduce fractions.  Typically, students see pizzas and apples and other objects—typically other foods or money—that are divided up into equal parts.  Such models are limited.  They work okay with simple addition and subtraction.  Common denominators present a bit of a challenge, but ½ pizza can be shown to be also 2/4, a half dollar equal to two quarters, and so on. 

With multiplication and division, all the little tricks students learned with whole number arithmetic suddenly go haywire.  Students are accustomed to the fact that multiplying two whole numbers yields a product that is larger than either number being multiplied: 4 X 5 = 20 and 20 is larger than both 4 and 5.^[2]  How in the world can ¼ X 1/5 = 1/20, a number much smaller than either 1/4or 1/5?  The part-whole representation has convinced many students that fractions are not numbers.  Instead, they are seen as strange expressions comprising two numbers with a small horizontal bar separating them. 

I taught sixth grade but occasionally visited my colleagues’ classes in the lower grades.  I recall one exchange with second or third graders that went something like this:

“Give me a number between seven and nine.”  Giggles. 

“Eight!” they shouted. 

“Give me a number between two and three.”  Giggles.

“There isn’t one!” they shouted. 

“Really?” I’d ask and draw a number line.  After spending some time placing whole numbers on the number line, I’d observe,  “There’s a lot of space between two and three.  Is it just empty?” 

Silence.  Puzzled little faces.  Then a quiet voice.  “Two and a half?”

You have no idea how many children do not make the transition to understanding fractions as numbers and because of stumbling at this crucial stage, spend the rest of their careers as students of mathematics convinced that fractions are an impenetrable mystery.   And  that’s not true of just students.  California adopted a test for teachers in the 1980s, the California Basic Educational Skills Test (CBEST).  Beginning in 1982, even teachers already in the classroom had to pass it.   I made a nice after-school and summer income tutoring colleagues who didn’t know fractions from Fermat’s Last Theorem.  To be fair, primary teachers, teaching kindergarten or grades 1-2, would not teach fractions as part of their math curriculum and probably hadn’t worked with a fraction in decades.  So they are no different than non-literary types who think Hamlet is just a play about a young guy who can’t make up his mind, has a weird relationship with his mother, and winds up dying at the end.

Division is the most difficult operation to grasp for those arrested at the part-whole stage of understanding fractions.  A problem that Liping Ma posed to teachers is now legendary.^[3]

She asked small groups of American and Chinese elementary teachers to divide 1 ¾ by ½ and to create a word problem that illustrates the calculation.  All 72 Chinese teachers gave the correct answer and 65 developed an appropriate word problem.  Only nine of the 23 American teachers solved the problem correctly.  A single American teacher was able to devise an appropriate word problem.  Granted, the American sample was not selected to be representative of American teachers as a whole, but the stark findings of the exercise did not shock anyone who has worked closely with elementary teachers in the U.S.  They are often weak at math.  Many of the teachers in Ma’s study had vague ideas of an “invert and multiply” rule but lacked a conceptual understanding of why it worked.

A linguistic convention exacerbates the difficulty.  Students may cling to the mistaken notion that “dividing in half” means “dividing by one-half.”  It does not.  Dividing in half means dividing by two.  The number line can help clear up such confusion.  Consider a basic, whole-number division problem for which third graders will already know the answer:  8 divided by 2 equals 4.   It is evident that a segment 8 units in length (measured from 0 to 8) is divided by a segment 2 units in length (measured from 0 to 2) exactly 4 times.  Modeling 12 divided by 2 and other basic facts with 2 as a divisor will convince students that whole number division works quite well on a number line. 

Now consider the number ½ as a divisor.  It will become clear to students that 8 divided by ½ equals 16, and they can illustrate that fact on a number line by showing how a segment ½ units in length divides a segment 8 units in length exactly 16 times; it divides a segment 12 units in length 24 times; and so on.  Students will be relieved to discover that on a number line division with fractions works the same as division with whole numbers.

Now, let’s return to Liping Ma’s problem: 1 ¾ divided by ½.   This problem would not be presented in third grade, but it might be in fifth or sixth grades.  Students who have been working with fractions on a number line for two or three years will have little trouble solving it.  They will see that the problem simply asks them to divide a line segment of 1 3/4 units by a segment of ½ units.  The answer is 3 ½ .  Some students might estimate that the solution is between 3 and 4 because 1 ¾ lies between 1 ½ and 2, which on the number line are the points at which the ½ unit segment, laid end on end, falls exactly three and four times.  Other students will have learned about reciprocals and that multiplication and division are inverse operations.  They will immediately grasp that dividing by ½ is the same as multiplying by 2—and since 1 ¾ x 2 = 3 ½, that is the answer.  Creating a word problem involving string or rope or some other linearly measured object is also surely within their grasp.

I applaud the CCSS for introducing number lines and fractions in third grade.  I believe it will instill in children an important idea: fractions are numbers.  That foundational understanding will aid them as they work with more abstract representations of fractions in later grades.   Fractions are a monumental barrier for kids who struggle with math, so the significance of this contribution should not be underestimated.

I mentioned above that instruction and curriculum are often intertwined.  I began this series of posts by defining curriculum as the “stuff” of learning—the content of what is taught in school, especially as embodied in the materials used in instruction.  Instruction refers to the “how” of teaching—how teachers organize, present, and explain those materials.  It’s each teacher’s repertoire of instructional strategies and techniques that differentiates one teacher from another even as they teach the same content.  Choosing to use a number line to teach fractions is obviously an instructional decision, but it also involves curriculum.  The number line is mathematical content, not just a teaching tool.

Guiding third grade teachers towards using a number line does not guarantee effective instruction.  In fact, it is reasonable to expect variation in how teachers will implement the CCSS standards listed above.  A small body of research exists to guide practice. One of the best resources for teachers to consult is a practice guide published by the What Works Clearinghouse: Developing Effective Fractions Instruction for Kindergarten Through Eighth Grade (see full disclosure below).^[4]  The guide recommends the use of number lines as its second recommendation, but it also states that the evidence supporting the effectiveness of number lines in teaching fractions is inferred from studies involving whole numbers and decimals.  We need much more research on how and when number lines should be used in teaching fractions.

Professor Wu states the following, “The shift of emphasis from models of a fraction in the initial stage to an almost exclusive model of a fraction as a point on the number line can be done gradually and gracefully beginning somewhere in grade four. This shift is implicit in the Common Core Standards.”^[5]  I agree, but the shift is also subtle.  CCSS standards include the use of other representations—fraction strips, fraction bars, rectangles (which are excellent for showing multiplication of two fractions) and other graphical means of modeling fractions.  Some teachers will manage the shift to number lines adroitly—and others will not.  As a consequence, the quality of implementation will vary from classroom to classroom based on the instructional decisions that teachers make.  

The current post has focused on what I believe to be a positive aspect of CCSS based on the implementation of the standards through instruction.  Future posts in the series—covering the “bad” and the “ugly”—will describe aspects of instruction on which I am less optimistic.

The American people deserve to have a federal government that is both responsive and effective. That simply isn’t the case for more than 1 million people who are awaiting the adjudication of their applications for disability benefits from the Social Security Administration.

Washington can and must do better. This gridlock harms applicants either by depriving them of much-needed support or effectively barring them from work while their cases are resolved because having any significant earnings would immediately render them ineligible. This is unacceptable.

Within the next month, the Government Accountability Office, the nonpartisan congressional watchdog, will launch a study on the issue. More policymakers should follow GAO’s lead. A solution to this problem is long overdue. Here’s how the government can do it.

Congress does not need to look far for an example of how to reduce the SSA backlog. In 2013, the Veterans Administration cut its 600,000-case backlog by 84 percent and reduced waiting times by nearly two-thirds, all within two years. It’s an impressive result.

Why have federal officials dealt aggressively and effectively with that backlog, but not the one at SSA? One obvious answer is that the American people and their representatives recognize a debt to those who served in the armed forces. Allowing veterans to languish while a sluggish bureaucracy dithers is unconscionable. Public and congressional outrage helped light a fire under the bureaucracy. Administrators improved services the old-fashioned way — more staff time. VA employees had to work at least 20 hours overtime per month.

Things are a bit more complicated at SSA, unfortunately. Roughly three quarters of applicants for disability benefits have their cases decided within about nine months and, if denied, decide not to appeal. But those whose applications are denied are legally entitled to ask for a hearing before an administrative law judge — and that is where the real bottleneck begins.

There are too few ALJs to hear the cases. Even in the best of times, maintaining an adequate cadre of ALJs is difficult because normal attrition means that SSA has to hire at least 100 ALJs a year to stay even. When unemployment increases, however, so does the number of applications for disability benefits. After exhausting unemployment benefits, people who believe they are impaired often turn to the disability programs. So, when the Great Recession hit, SSA knew it had to hire many more ALJs. It tried to do so, but SSA cannot act without the help of the Office of Personnel Management, which must provide lists of qualified candidates before agencies can hire them. SSA employs 85 percent of all ALJs and for several years has paid OPM approximately $2 million annually to administer the requisite tests and interviews to establish a register of qualified candidates. Nonetheless, OPM has persistently refused to employ legally trained people to vet ALJ candidates or to update registers. And when SSA sought to ramp up ALJ hiring to cope with the recession challenge, OPM was slow to respond.

In 2009, for example, OPM promised to supply a new register containing names of ALJ candidates. Five years passed before it actually delivered the new list of names. For a time, the number of ALJs deciding cases actually fell. The situation got so bad that the president’s January 2015 budget created a work group headed by the Office of Management and Budget and the Administrative Conference of the United States to try to break the logjam. OPM promised a list for 2015, but insisted it could not change procedures. Not trusting OPM to mend its ways, Congress in October 2015 enacted legislation that explicitly required OPM to administer a new round of tests within the succeeding six months.

These stopgap measures are inadequate to the challenge. Both applicants and taxpayers deserve prompt adjudication of the merits of claims. The million-person backlog and the two-year average waits are bad enough. Many applicants wait far longer. Meanwhile, they are strongly discouraged from working, as anything more than minimal earnings will cause their applications automatically to be denied. Throughout this waiting period, applicants have no means of self-support. Any skills applicants retain atrophy.

The shortage of ALJs is not the only problem. The quality and consistency of adjudication by some ALJs has been called into question. For example, differences in approval rates are so large that differences among applicants cannot plausibly explain them. Some ALJs have processed so many cases that they could not possibly have applied proper standards. In recognition of both problems, SSA has increased oversight and beefed up training. The numbers have improved. But large and troubling variations in workloads and approval rates persist.

For now, political polarization blocks agreement on whether and how to modify eligibility rules and improve incentives to encourage work by those able to work. But there is bipartisan agreement that dragging out the application process benefits no one. While completely eliminating hearing delays is impossible, adequate administrative funding and more, better trained hearing officers would help reduce them. Even if OPM’s past record were better than it is, OPM is now a beleaguered agency, struggling to cope with the fallout from a security breach that jeopardizes the security of the nation and the privacy of millions of current and past federal employees and federal contractors. Mending this breach and establishing new procedures will — and should — be OPM’s top priority.

That’s why, for the sake of everyone concerned, responsibility for screening candidates for administrative law judge positions should be moved, at least temporarily, to another agency, such as the Administrative Conference of the United States. Shortening the period that applicants for disability benefits now spend waiting for a final answer is an achievable goal that can and should be addressed. Our nation’s disabled and its taxpayers deserve better.

Editor's note: This piece originally appeared in Politico.

If you’re reading this article, chances are you know people who are still getting married. But it’s getting rarer, especially among the youngest generation and those who are less educated. We used to assume people would marry before having children. But marriage is no longer the norm. Half of all children born to women under 30 are born out of wedlock. The proportion is even higher among those without a college degree.

What’s going on here? Most of today’s young adults don’t feel ready to marry in their early 20s. Many have not completed their educations; others are trying to get established in a career; and many grew up with parents who divorced and are reluctant to make a commitment or take the risks associated with a legally binding tie.

But these young people are still involved in romantic relationships. And yes, they are having sex. Any stigma associated with premarital sex disappeared a long time ago, and with sex freely available, there’s even less reason to bother with tying the knot. The result: a lot of drifting into unplanned pregnancies and births to unmarried women and their partners with the biggest problems now concentrated among those in their 20s rather than in their teens. (The teen birth rate has actually declined since the early 1990s.)

Does all of this matter? In a word, yes.

These trends are not good for the young people involved and they are especially problematic for the many children being born outside marriage. The parents may be living together at the time of the child’s birth but these cohabiting relationships are highly unstable. Most will have split before the child is age 5.

Social scientists who have studied the resulting growth of single-parent families have shown that the children in these families don’t fare as well as children raised in two-parent families. They are four or five times as likely to be poor; they do less well in school; and they are more likely to engage in risky behaviors as adolescents. Taxpayers end up footing the bill for the social assistance that many of these families need.

Is there any way to restore marriage to its formerly privileged position as the best way to raise children? No one knows. The fact that well-educated young adults are still marrying is a positive sign and a reason for hope. On the other hand, the decline in marriage and rise in single parenthood has been dramatic and the economic and cultural transformations behind these trends may be difficult to reverse.

Women are no longer economically dependent on men, jobs have dried up for working-class men, and unwed parenthood is no longer especially stigmatized. The proportion of children raised in single-parent homes has, as a consequence, risen from 5 percent in 1960 to about 30 percent now.

Conservatives have called for the restoration of marriage as the best way to reduce poverty and other social ills. However, they have not figured out how to do this.

The George W. Bush administration funded a series of marriage education programs that failed to move the needle in any significant way. The Clinton administration reformed welfare to require work and thus reduced any incentive welfare might have had in encouraging unwed childbearing. The retreat from marriage has continued despite these efforts. We are stuck with a problem that has no clear governmental solution, although religious and civic organizations can still play a positive role.

But perhaps the issue isn’t just marriage. What may matter even more than marriage is creating stable and committed relationships between two mature adults who want and are ready to be parents before having children. That means reducing the very large fraction of births to young unmarried adults that occur before these young people say they are ready for parenthood.

Among single women under the age of 30, 73 percent of all pregnancies are, according to the woman herself, either unwanted or badly mistimed. Some of these women will go on to have an abortion but 60 percent of all of the babies born to this group are unplanned.

As I argue in my book, “Generation Unbound,” we need to combine new cultural messages about the importance of committed relationships and purposeful childbearing with new ways of helping young adults avoid accidental pregnancies. The good news here is that new forms of long-acting but fully reversible contraception, such as the IUD and the implant, when made available to young women at no cost and with good counseling on their effectiveness and safety, have led to dramatic declines in unplanned pregnancies. Initiatives in the states of Colorado and Iowa, and in St. Louis have shown what can be accomplished on this front.

Would greater access to the most effective forms of birth control move the needle on marriage? Quite possibly. Unencumbered with children from prior relationships and with greater education and earning ability, young women and men would be in a better position to marry. And even if they fail to marry, they will be better parents.

My conclusion: marriage is in trouble and, however desirable, will be difficult to restore. But we can at least ensure that casual relationships outside of marriage don’t produce children before their biological parents are ready to take on one of the most difficult social tasks any of us ever undertakes: raising a child. Accidents happen; a child shouldn’t be one of them.

At an event this past week, Hillary Clinton announced that, if elected, she planned to put Bill Clinton in charge of creating jobs. If he becomes the “First Gentlemen” -- or as she prefers to call him, the “First Dude,” – he just might have some success in this role. The country’s very strong record of job creation during the first Clinton administration is a hopeful sign. (Full disclosure: I served in his Administration.)

But assuming he's given the role of jobs czar, what would Bill Clinton do? The uncomfortable fact is that no one knows how to create enough jobs. Although about 50 percent of the public, according to Pew, worries that there are not enough jobs available, and virtually every presidential candidate is promising to produce more, economists are not sure how to achieve this goal.

The debate centers around why we think people are jobless. Unless we can agree on the diagnosis, we will not be able to fashion an appropriate policy response.

Some economists think that an unemployment rate hovering around 5 percent constitutes “full employment.” Those still looking for jobs, in this view, are either simply transitioning voluntarily from one job to another or they are “structurally unemployed.” The latter term refers to a mismatch, either between a worker’s skills and the skills that employers are seeking, or between where the workers live and where the jobs are geographically. (The decline in housing values or tighter zoning restrictions, for example, may have made it more difficult for people to move to states or cities where jobs are more available.)

Another view is that despite the recovery from the Great Recession, there is still a residue of “cyclical” unemployment. If the Federal Reserve or Congress were to boost demand by keeping interest rates low, reducing taxes, or increasing spending on, say, infrastructure, this would create more jobs – or so goes the argument. But the Fed can’t reduce interest rates significantly because they are already near rock-bottom levels and tax and spending policies are hamstrung by political disagreements.

In my view, the U.S. currently suffers from both structural and cyclical unemployment. The reason I believe there is still some room to stimulate the economy is because we have not yet seen a significant increase in labor costs and inflation. Political problems aside, we should be adding more fuel to the economy in the form of lower taxes or higher public spending.

High levels of structural unemployment are also a problem. The share of working-age men who are employed has been dropping for decades at least in part because of outsourcing and automation. The share of the unemployed who have been out of work for more than six months is also relatively high for an economy at this stage of the business cycle. One possibility is that the recession caused many workers to drop out of the labor force and that after a long period of joblessness, they have seen their skills atrophy and employers stigmatize them as unemployable.

The depressing fact is that none of these problems is easy to solve. Manufacturing jobs that employ a lot of people are not coming back. Retraining the work force for a high-tech economy will take a long time. Political disagreements won’t disappear unless there is a landslide election that sweeps one party into control of all three branches of government.

So what can Bill Clinton or anyone else do? We may need to debate some more radical solutions such as subsidized jobs or a basic income for the structurally unemployed or a shorter work week to spread the available work around. These may not be politically feasible for some time to come, but former President Clinton is the right person to engage communities and employers in some targeted job creation projects now and to involve the country in a serious debate about what to do about jobs over the longer haul.

Editor's note: This piece originally appeared in Inside Sources.

When Franklin Roosevelt delivered his second inaugural address on January 20, 1936 he lamented the “one-third of a nation ill-housed, ill-clad, ill-nourished.” He challenged Americans to measure their collective progress not by “whether we add more to the abundance of those who have much; [but rather] whether we provide enough for those who have too little.” In our new paper, One third of a nation: Strategies for helping working families, we ask a simple question: How are we doing?

In brief, we find that:

• The gulf in labor market income between the haves and have-nots remains wide. The median income of households in the bottom third in 2014 was $24,000, just a little more than a quarter of the median of $90,000 for the top two-thirds.
• The bottom-third households are disproportionately made up of minority adults, adults with limited educational attainment, and single parents.  
• The most important reason for the low incomes of the bottom third is a “work gap”: the fact that many are not employed at all, or work limited hours. 

The work gap

The decline in labor force participation rates has been widely documented, but the growing gulf in the work gap between the bottom third and the rest of the population is truly striking:

While the share of men who are employed in the top two-thirds has been quite stable since 1980, lower-income men’s work rates have declined by 11 percentage points. What about women?

Middle- and upper-income women have increased their work rates by 13 percentage points. This has helped maintain or even increase their family’s income. But employment rates among lower-income women have been flat, despite reforms of the welfare system and safety net designed to encourage work.

Many on the left point to problems like low pay and lack of access to affordable childcare, and so favor a higher minimum wage and more subsidies for daycare. For many conservatives, the problem is rooted in family breakdown and a dependency-inducing safety net. They therefore champion proposals like marriage promotion programs and strict work requirements for public benefits. Most agree about the importance of education.

Time to debate some more radical solutions 

It may be that the standard solutions to the problems of the bottom third, while helpful, are no longer sufficient. A debate about whether to make safety net programs such as Food Stamps and housing assistance conditional on work or training is underway. So are other solutions such as subsidized jobs (created by some states during the Great Recession as a natural complement to a work-conditioned safety net), more work sharing (used in Germany during the recession), or even a universal basic income (being considered by Swiss voters in June).

As the United States and Cuba rebuild formal relations, certain challenging topics remain to be addressed. Among these are outstanding U.S. property claims in Cuba. In this report, Richard E. Feinberg argues that it is in both countries’ interests to tackle this thorny issue expeditiously, and that the trauma of property seizures in the twentieth century could be transformed into an economic opportunity now.

The report looks closely at the nearly 6,000 certified U.S. claims, disaggregating them by corporate and individual, large and small. To settle the U.S. claims, Feinberg suggests a hybrid formula, whereby smaller claimants receive financial compensation while larger corporate claimants can select an “opt-out” option whereby they pursue their claims directly with Cuban authorities, perhaps facilitated by an umbrella bilateral claims resolution committee. In this scenario, the larger corporate claimants (which account for nearly $1.7 billion of the $1.9 billion in total U.S. claims, excluding interest) could select from a menu of business development rights, including vouchers applicable to tax liabilities or equity investments, and preferred acquisition rights. Participating U.S. firms could also agree to inject additional capital and modern technology, to ensure benefits to the Cuban economy.

Though it is often argued that Cuba is too poor to pay some $2 billion of claims, the paper finds that Cuba can in fact manage payments if they are stretched out over a reasonable period of time and exclude interest. The paper also suggests a number of mechanisms whereby the Cuban government could secure funds to pay compensation, including revenues on normalization-related activities.

The Cuban government does not dispute the principle of compensation for properties nationalized in the public interest; the two governments agree on this. Cuba also asserts a set of counter-claim that allege damages from the embargo and other punitive actions against it. But a grand bargain with claims settlement as the centerpiece would require important changes in U.S. sanctions laws and regulations that restrict U.S. investments in Cuba. The United States could also offer to work with Cuba and other creditors to renegotiate Cuba’s outstanding official and commercial debts, taking into account Cuba’s capacity to pay, and allow Cuba to enter the international financial institutions.

Feinberg ultimately argues that both nations should make claims resolution the centerpiece of a grand bargain that would advance the resolution of a number of other remaining points of tension between the two nations. This paves the way for Cuba to embrace an ambitious-forward-looking development strategy and for real, notable progress in normalizing relations with the United States.

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