Twenty years ago few preschools (or parents, for that matter), paid much attention to teaching mathematics to four-year-olds. In 1998, for example, only four percent of a nationally representative sample of American children entering kindergarten could add or subtract. Today, math is firmly entrenched in the pre-K curriculum. And the Common Core State Standards, which are the new instructional guidelines for K-12 math instruction in 40+ states, and which require kindergartners to engage in algebraic thinking, are being extended downward into pre-K in many locales. New York state, for example, has pre-K standards aligned to the Common Core that require four-year-olds to “demonstrate an understanding of addition and subtraction by using objects, fingers, and responding to practical situations (e.g., If we have three apples and add two more, how many apples do we have all together?).” Thus, in about 15 years we’ve moved from virtually no preschoolers being able to add and subtract to the goal of all four-year-olds being able to do so.

There is a scientific basis for the growth in the emphasis on math in pre-K, as researchers in early mathematics have followed previous work by early literacy researchers by investigating the longitudinal associations between early skills and later development. As a result of these efforts, we now know that early math skills are the strongest early predictors of children’s math achievement years later (Aunola, Leskinen, Lerkkanen, and Nurmi, 2004; Duncan et al., 2007; Geary, Hoard, Nugent, and Bailey, 2013; Jordan, Kaplan, Ramineni, and Locuniak, 2009; Siegler et al., 2012). This finding, in turn, provides an empirical basis for devoting a considerable chunk of the pre-K curriculum to math instruction, since math skills and course taking during the high school years are related to important life outcomes such as college success (Lee, 2013). Unfortunately, the effects of early math interventions found in experimental studies clearly diminish over time. My recent work suggests that differences in children’s math achievement are influenced by a combination of differences in both earlier math achievement and the relatively stable factors affecting children’s math achievement across development— with the effects of stable factors being several times larger than the effects of children’s earlier math achievement. This suggests that pre-school level math instruction alone will not be sufficient to substantially boost long-term math achievement outcomes, and raises important questions about what kinds of interventions are likely to produce the longest lasting effects on children’s math achievement.

## Examining the relation between early and late mathematics skills

The strong association between children’s measured math skills in preschool and later, during the school years, is only a warrant for an emphasis on math instruction in pre-K if the relationship between earlier and later skills is causal, e.g., teaching children to add and subtract as four-year-olds leads directly to increased math learning for those children in elementary school. If the correlation between early and later math skills is fully driven by other variables that affect both early and later math skills, such as children’s intelligence or interest in learning, then teaching preschoolers to add and subtract would not have a direct impact on later math skills.

Researchers have tried to rule out other variables that might explain the correlation between earlier and later mathematics ability by controlling statistically for some of the factors that might affect children’s math learning both early and later in their development, including: family characteristics; children’s cognitive abilities such as intelligence and working memory; and reading achievement. These statistical controls are used to reduce bias in the estimates of the effect of early math achievement on later math achievement.

To the extent that we are able to make causal inferences from these studies, the implication is clear: improving children’s early math skills should produce sizable effects on their much later math achievement. This would be good news, as we know of some effective ways to increase children’s math achievement in preschool; for example, Doug Clements’s and Julie Sarama’s early math curriculum produces impressive effects on children’s early math achievement (Clements, Sarama, Spitler, Lange, and Wolfe, 2011; Clements and Sarama, 2008).

Further, there is a promising logic underlying the idea that effective early math interventions will have long-lasting effects. In math, earlier skills are often repurposed as subroutines of later skills. For example, children use counting when they learn single digit arithmetic, they use single digit arithmetic when they learn multi-digit arithmetic, and they use whole number arithmetic when they learn fraction arithmetic. Failing to learn earlier skills disadvantages children trying to learn later skills. Additionally, sometimes knowing one mathematical principal can help children learn another. This phenomenon is known as transfer of learning and has been demonstrated in many studies of children’s math learning. For example, having an accurate understanding of where numbers fall on a number line facilitates preschoolers’ learning of simple addition (Siegler and Ramani, 2009).

To summarize, there are strong empirical relations between children’s school-entry math achievement and their math achievement many years later, and there is a sensible theoretical framework for understanding how differences in early math skills might cause differences in later math skills. But what do experimental studies on the effects of early childhood interventions on children’s later math outcomes—those that compare children who received some early math intervention to those who did not—find? Unfortunately, these studies show a different pattern. Effects of early interventions on children’s math achievement reliably diminish over time, a finding known as the “fade-out” effect.

## Explaining the discrepancy between correlational and experimental findings

What can account for the apparent discrepancy between results from correlational analyses of longitudinal datasets and results from experimental studies? The answer is likely to be that correlational studies do not adequately control for all of the relatively stable factors underlying children’s math learning throughout development. Because of this, correlational studies will over-estimate the effects of improving children’s early math achievement on their later math achievement. There are two reasons to favor this explanation:

1) Many child characteristics are statistically associated with and may plausibly cause children’s math outcomes. These characteristics include commonly used statistical control variables, such as socioeconomic status, working memory, and intelligence. Deary, Strand, Smith, and Fernandes (2007) found that children’s intelligence measured at age 11 accounted for 59 percent of the variance in their math achievement at age 16. Further, other characteristics, such as children’s motivation, attention, processing speed, and particular facets of working memory, may also influence children’s math learning. Longitudinal datasets often contain measures of *some* of these stable characteristics, but do not contain complete, high-quality measures of *all* of them for the same children and therefore cannot statistically control for all of them. Therefore, causal estimates generated from these datasets may yield upwardly biased estimates of the effect of early math achievement on later math achievement.

2) Critically, the association between early and later math achievement remains surprisingly stable as the time between the “early” and “late” measurements increases. If the correlation between early and later math achievement primarily reflects the causal effect of the former on the latter, then this correlation should diminish over time. As an analogy, if a dog is walking around a field, we should have a more accurate idea of where he is at any given time the more recently his previous location is known. As this time interval increases, we will become less and less sure of where in the field the dog is located. To the extent that the correlation between measures of math achievement is stable as the distance in time between the measurements increases, relatively stable factors that influence math achievement similarly over time are likely responsible for the correlation between early and later math achievement. Again using the analogy of the dog in the field, if the dog is leashed to a post somewhere in the field, then knowing his location at any previous time should be similarly helpful for predicting the dog’s subsequent location (the dog will be by the post).

Children’s math achievement is likely both influenced by previous knowledge and stable factors (the dog is leashed to a post, but the leash has some slack), but there is reason to think that stable factors might account for a substantial part of the correlation between early and much later math achievement: Though the correlation between early and later math achievement does not remain completely unchanged as the distance in time between the measurements increases, it remains surprisingly stable. For example, in a longitudinal dataset containing data from 1,124 children, the correlation between children’s first grade math achievement and their third grade math achievement was .72, and the correlation between children’s first grade math achievement and their math achievement at age 15 was .66. If these correlations primarily reflect the stability of factors underlying math learning throughout children’s development, and secondarily reflect smaller effects of early math achievement on later math achievement, then effects of early interventions that affect early math achievement but not the stable factors influencing learning across time will fade out.

My collaborators— Tyler Watts, Andrew Littlefield, Dave Geary— and I used a statistical model to partition the correlation between individual differences in children’s math achievement measured at different times into two parts: the part caused by direct effects of children’s earlier math achievement on their later math achievement, and the part caused by relatively stable factors that affect children’s math learning similarly across their development (Bailey, Watts, Littlefield, and Geary, 2014). Using data from two longitudinal studies of children’s math achievement, our model suggests that children’s math achievement is influenced by a combination of both earlier math achievement and the relatively stable factors affecting children’s math achievement across development. However, the effects of stable factors are several times larger than the effects of children’s earlier math achievement. Further, a set of common statistical controls, such as intelligence, working memory, socioeconomic status, and reading achievement accounted for a large amount of the variance in these stable factors (approximately 2/3). This means that estimates of the effects of early math achievement on later math achievement based on correlational data over-estimate the direct effect of early mathematical knowledge on the acquisition of later mathematical knowledge, and that this bias increases with the distance in time between these two measurements. Consistent with data from experimental studies on early interventions, our model predicts that the effects of increasing young children’s early math skills on their later math achievement will fade over time.

## Conclusion

The primary implication of our study for mathematics is that increasing children’s school-entry math achievement alone will not be sufficient to substantially boost their math achievement outcomes many years later. This does not mean that early interventions cannot affect other important aspects of children’s lives (particularly for children living in the poorest environments), some of which may even affect their later academic achievement, nor does it mean that preschoolers shouldn’t be taught math. It does mean that the yield from preschool math instruction on children’s much later math achievement will be less than is often assumed. Investing more in later math interventions may be a more effective approach. Though nearly all older children eventually learn how to count and add single-digit numbers, many U.S. children never develop the ability to efficiently compare the sizes of different fractions (Schneider and Siegler, 2010). Further, researchers have identified effective interventions for teaching children how to do this (Fuchs et al., 2013). It seems plausible that teaching older children information they are at risk for never learning may have more persistent effects on their math achievement than teaching younger children information they very probably otherwise would have learned in kindergarten or first grade. Finally, to the extent that effective early math instruction is implemented, our model and common sense predict that children will need higher quality later math instruction to sustain their higher math achievement trajectories into the long-term.

In the broader context of preschool policy, our results suggest that we need much more knowledge than is presently available with respect to which children need what kinds of instruction when.

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