Such are statistics. What almost tripled was the absolute number of unmarried cohabiting couples with children under it. What went down from 68% to 64% was the proportion of US children under 18 who lived with their two parents who were married and living together. Unchanged: children living with only the mother (24%), only the father (4%), or neither parent (4%).
Why are there such dramatically different ways of telling the same story? First, a smaller proportion is subject to change a lot more radically, relative to its former size, than a larger proportion is. When the number of states with a black senator doubled from 1 to 2, the proportion of states without one only went down by 2%. Also, the first number used absolute numbers, which could also go up as population, and particularly the child population, increases.
Another big caveat: This study, like most statistics, substantially undercounts family disruption because it does not distinguish between the child's two actual parents, still married and/or still cohabiting, and one actual parent who's married to someone else who's referred to as a stepparent. "For this indicator, unless otherwise specified, a two-parent family refers to parents who are married to each other and living in the same household. They may be biological, adoptive, or stepparents. The Current Population Survey identifies all parents who are family or subfamily heads. Where cohabitants are concerned, until 2007, the CPS did not ask whether that person was also the parent of the child." Also, people are more likely to call themselves stepparents if they are married, not just cohabiting, so that is yet another distorting factor which increases the measurement of married parents versus cohabiting ones.
Ironically, the report's introductory section, titled "Importance", emphasizes how important the differences are between children living in truly intact married families and those who experience divorce or unwed birth and then live with stepparents. But in most cases, the government simply does not collect the data on that extremely important distinction, adding that insult to all the injuries done to marriage, families and children over the decades.
With that in mind, here are some excerpts. The article also has many useful links to many current sources of family statistics, the full study, and the underlying data.
The proportion of children living with both parents, following a marked decline between 1970 and 1990, has fallen more slowly over the most recent two decades, dropping from 69 percent in 2000 to 64 percent in 2014. ...
Between 1960 and 1996, the proportion of all children under age 18 who were living with two married parents decreased steadily, from 85 to 68 percent. This share was stable during much of the late 1990s and into the 2000s, but by 2012 it had decreased to 64 percent. The rate was stable between 2012 and 2014.
In 1960, the proportion of children living in mother-only families was eight percent, but by 1996 that proportion had tripled, to 24 percent. Since then, it has fluctuated between 22 and 24 percent, and was at 24 percent in 2014. Between 1990 and 2013, the share of children living in father-only families has fluctuated between three and five percent, and was at four percent in 2014. The proportion living without either parent (with either relatives or with non-relatives) has remained steady, at approximately four percent.
In 2014, there were 3.1 million cohabiting couples (unmarried) with children under 18. This number has been steadily increasing: in 1996, it was 1.2 million. ... Compared with married couples with children, cohabiting couples with children tend to be younger, less educated, lower–income, and with less secure employment. ... In eight percent of unmarried couples with children, neither person was employed in 2014, compared with only four percent among married couples with children.
In 2014, 34 percent of black children were living with two parents, compared with 85 percent of Asian children, 75 percent of white children, and 58 percent of Hispanic children.
[The following are from the introductory section, titled "Importance"]
Both mothers and fathers play important roles in the growth and development of children. The number and the type of parents (e.g., biological, step) in the household, as well as the relationship between the parents, are strongly linked to a child’s well-being.  Among young children, for example, those living with no biological parents, or in single-parent households, are less likely than children with two biological parents to exhibit behavioral self-control, and more likely to be exposed to high levels of aggravated parenting, than are children living with two biological parents. Children living with two married adults (biological or adoptive parents) have, in general, better health, greater access to health care, and fewer emotional or behavioral problems than children living in other types of families.
Among children in two-parent families, those living with both biological parents in a low-conflict marriage tend to do better on a host of outcomes than those living in step-parent families. Outcomes for children in step-parent families are in many cases similar to those for children growing up in single-parent families., Children whose parents are divorced also have lower academic performance, social achievement, and psychological adjustment than children with married parents.
Single-parent families tend to have much lower incomes than do two-parent families, while cohabiting families fall in-between. Research indicates, however, that the income differential only partially accounts for the negative effects on many areas of child and youth well-being (including health, educational attainment and assessments, behavior problems, and psychological well-being) associated with living outside of a married, two-parent family.,
This New York Times article, passed on by my New York divorce lawyer friend Chaim Steinberger, reports and beautifully illustrates the conclusions of recent studies from Child Trends and Brookings on how stable marriage helps children by improving their families' economic welfare, the amount of time adults devote to them, their educational and extracurricular opportunities, and behavioral and health outcomes. Its tone is evenhanded and dispassionate, yet its information and insight is passion-stirring -- in a much more constructive way than a lot of that "mommy wars" and "culture wars" stuff that people so often write about marriage, work, and single parenthood.
"Two Classes, Divided by ‘I Do’" By JASON DePARLE, New York Times July 14, 2012
In 2009, the poverty rate for children in married-couple families was 11.0 percent. By contrast, the poverty rate for children in female-headed families was 44.3 percent.  The difference between these two poverty rates is a specter haunting American social policy because the percentage of American children who live in female-headed families has been increasing relentlessly for over five decades. In 1950, 6.3 percent of families with children were headed by a single mother. By 2010, 23.9 percent of families with children had single-mother heads.  That a higher and higher fraction of children live in the family type in which they are about four times as likely to be poor exerts strong upward pressure on the poverty rate. One way to think of the shift to female-headed families is that even if government policy were successful in moving people out of poverty, the large changes in family composition serve to offset at least part of the progress that otherwise would be made. In fact, a Brookings analysis shows that if we had the marriage rate we had in 1970, the poverty rate would fall by more than 25 percent. 
. . .
Reducing Nonmarital Births. One of the engines driving poverty in the U.S. is the fragmentation of families. Around a quarter of children are living in female-headed families at any given moment and about half experience at least some time during their childhood in a female-headed family. When children live in female-headed families, they are at least four times as likely to be poor as when they live in a married-couple family.  But poverty is not the only risk faced by these children. Since Sara McLanahan and Gary Sandefur published Growing up with a Single Parent in 1994,  social science research has repeatedly shown that children reared in female-headed families are more likely to fail in school, more likely to be arrested, more likely to get pregnant as teens, more likely to have mental health problems and to commit suicide, more likely to get a divorce when they grow up, and more likely to experience other negative outcomes.  In addition, as Kathy Edin has shown, these parents tend to separate within a few years, whereupon both the mother and father usually go on to form new relationships. Thus, their children experience a series of changes in household composition as their mothers form new cohabiting relationships.  The mother might even have a baby with one or more of these new men, creating a household with complex and often difficult relationships among the adults and usually making it hard for the children to establish a close relationship with their fathers.  The point is that life in female-headed families imposes both a high likelihood of poverty and of household instability that can produce negative impacts on child development.  If the share of children born into and living in married-couple families could be increased, poverty and childhood education, health, and mental health problems would decline, increasing the human capital of the nation’s children and having a long-term impact on the nation’s poverty rate. 
Several programs have proven successful in reducing teen pregnancy.  In part because of the prevalence of these programs, the U.S. teen birthrate has declined in all but three years since 1991.  It is difficult, however, to be too optimistic about the declining teen birthrate because as the teen birthrate has declined, the nonmarital birthrate for young women in the twenties and early thirties has increased more than enough to offset the decline in the teen rate. 
Even so, investments in programs aimed at reducing nonmarital births have been shown not only to actually reduce such births among women in their 20s and 30s, but to save government money.  The programs are a mass media campaign that encourages men to use condoms, a program for teens that both encourages abstinence and instructs on the proper use of contraceptives, and expansion of family planning services provided by Medicaid, mostly birth control for low-income females. Similarly, the Obama administration has initiated a number of new evidence-based initiatives that could reduce the number of nonmarital births even more. Additional investments in these programs would reduce the number of nonmarital births and in doing so reduce the nation’s poverty rate. but with 72 percent of black babies, 53 percent of Hispanic babies, and over 40 percent of all babies born outside marriage, there is a long way to go.  The nonmarital birth machine that expands poverty and produces children with less human capital than their peers being reared in married-couple families is disrupting and will continue to disrupt the nation’s drive to curb poverty.
 Ron Haskins and Isabel Sawhill, Creating an Opportunity Society (Washington, DC: Brookings, 2009).
. . .
 Census Bureau, Families, “Table C2: Household Relationship and Living Arrangements of Children Under 18 Years, by Age and Sex: 2010,” available athttp://www.census.gov/population/www/socdemo/hh-fam/cps2010.html;” Katherine Magnuson and Lawrence M. Berger, “Family Structure States and Transitions: Associations with Children’s Well-Being During Middle Childhood,” Journal of Family and Marriage 71, no. 3 (2000): 575-591.
 Sara McLanahan and Gary Sandefur, Growing up with a Single Parent: What Hurts, What Helps (Cambridge: Harvard, 1994).
 Paul R. Amato, “The Impact of Family Formation Change on the Cognitive, Social, and Emotional Well-Being of the Next Generation,” The Future of Children 15, no. 2 (2005): 75-96; Mitch Pearlstein, Shortchanging Student Achievement: The Educational, Economic, and Social Costs of Family Fragmentation (Forthcoming); Haskins and Sawhill, Creating an Opportunity Society, Chapter 10.
 Reference Edin study of transitions in Fragile Families study
 Sara McLanahan and others, “Strengthening Fragile Families,” Future of Children Policy Brief, Princeton-Brookings, Fall 2010.
 Adam Thomas and Isabel Sawhill, “For Richer or Poorer: Marriage as an Antipoverty Strategy,” Journal of Policy Analysis and Management 21, no. 4 (2002): 587-599; Ron Haskins and Isabel Sawhill, “Work and Marriage: The Way to End Poverty and Welfare,” Welfare Reform and Beyond Brief 28, The Brookings Institution (September 2003).
 Douglas Kirby, Emerging Answers 2007: Research Findings on Programsto Reduce Teen Pregnancy and Sexually Transmitted Diseases (Washington DC: National Campaign to Prevent Teen and Unplanned Pregnancy, 2007).
 Stephanie J. Ventura, “Changing Patterns of Nonmarital Childbearing in the United States,” National Center for Health Statistics Data Brief, No. 18, May 2009.
 Adam Thomas, "Three Strategies to Prevent Unintended Pregnancy,” Journal of Policy Analysis and Management 31, No. 2 (2012).
 Brady Hamilton, Joyce Martin, and Stephanie Ventura, “Table 1: Total Births and birth, fertility, and total fertility rates and nonmarital births, by race and Hispanic origin of mother: United States, final 2008 and preliminary 2009,” from National Vital Statistics Reports: Births: Preliminary Data for 2009 59, no. 3 (2010); available athttp://www.cdc.gov/nchs/products/nvsr.htm.