Approximately how much is 192 times 12? About how much will each teammate have to pay to buy a $50 present for the coach? Roughly how many marbles are in this jar?
Learning how to estimate is important, not only because estimating is something we need to do all the time, but also because proficiency at estimation is substantially correlated with many aspects of numerical understanding and with overall math-achievement-test scores (Booth & Siegler, 2006; Siegler & Booth, 2005).
Yet little is known about the development of estimation skills. To obtain a more detailed understanding, we conducted a series of studies on number-line estimation. Children are presented a blank line, on which only a zero is printed at the left end and the number 10, 100, or 1,000 is printed at the right end. Children estimate the positions of various numbers, one number per line. This number-line task lets us see how children of various ages make estimates independent of specific entities (such as marbles or dollars) whose qualities are being estimated. It also allows us to examine any range of numbers, and allows for the examination of relations between actual and estimated numerical magnitudes.
The research has revealed that children progress through a consistent developmental sequence. Young children generate logarithmic patterns of estimates, in which estimated magnitudes rise more quickly than actual magnitudes (e.g., the number 15 is estimated as being around where the number 60 should be on a zero – 100 number line). Older children generate linear functions (e.g., the number 15 is estimated as being around where 15 should be.) The same logarithmic to-linear sequence has been observed among kindergartners through second graders for zero – 100 number lines and for second through sixth graders for zero – 1,000 lines (Siegler & Booth, 2004; Siegler & Opfer, 2003). The linearity of estimates has also proven to be highly correlated with overall math-achievement scores among students in kindergarten through fourth grade (Siegler & Booth, 2004; Booth & Siegler, 2006).
IES’s emphasis on translating educationally relevant research into evidence-based curricula encouraged us to apply these findings to help low-income preschoolers improve their numerical understanding. Large discrepancies in numerical knowledge between children from lower- and middle-income families are already present before children enter school. Consistent with this general finding, Siegler and Ramani (2005) found that on zero – 10 number lines, the estimates of 4-year-olds from lower-income backgrounds were only weakly correlated with the actual magnitudes of the numbers, whereas the estimates of peers from middle income backgrounds were substantially correlated with the actual magnitudes and fit a linear function quite well.
These differences between the numerical knowledge of preschoolers from different socioeconomic backgrounds seem likely to reflect their differing experiences with informal number-related activities in the home environment. Board games with linearly arranged sequences of numbers seem likely to play an especially important role in promoting numerical understanding, because they provide multiple cues to numerical magnitudes. When a child moves a token across a horizontal row of numbered squares, the higher the number that the token has reached, the greater the distance the child will have moved the token, the greater the number of discrete moves he or she will have made, the greater the number of number names he or she will have spoken, the greater the amount of time the moves will have taken, and so on.
Therefore, we randomly assigned children from urban Head Start centers to play a board game four times over a two-week period, 15 minutes per session, either with 10 consecutively numbered squares or with 10 differently colored squares. On each turn, the child would spin a spinner, obtain a “one” or a “two,” and then move the token one or two squares forward, saying either (for example) “four, five” or “red, blue.” Before the first session and after the last session, children were presented the number-line estimation task.
Children who used the numbered boards made substantial progress. By the posttest, their number-line estimates were as accurate and as linear as those of middle-income children who had not played the game. By contrast, children whose boards varied in color rather than number showed no improvement in numerical knowledge.
With the support of IES, we are following up these findings to determine whether they are stable over a three-month period and to determine whether they generalize to other numerical tasks. If so, we plan to determine whether such instruction is effective at the level of entire classes as well as in one-on-one interactions. Through encouraging such research, IES is helping to bridge the gap between educationally relevant research and classroom practices.
- Booth, J. L. & Siegler, R. S. (2006). Developmental and individual differences in pure numerical estimation. Developmental Psychology, 41, 189-201.
- Ramani, G., & Siegler, R. S. (2005). It’s more than just a game: Effects of children’s board game play on the development of numerical estimation. Poster presented at Society for Research in Child Development Conference, Atlanta, GA.
- Siegler, R. S. (2006). Microgenetic analyses of learning. In W. Damon & R. M. Lerner (Series Eds.) & D. Kuhn & R. S. Siegler (Vol. Eds.), Handbook of child psychology: Volume 2: Cognition, perception, and language (6th ed., pp. 464- 510). Hoboken, NJ: Wiley.
- Siegler, R. S. & Booth, J. L. (2004). Development of numerical estimation in young children. Child Development, 75, 428- 444.
- Siegler, R. S. & Booth, J. L. (2005). Development of numerical estimation: A review. In J. I. D. Campbell (Ed.) Handbook of mathematical cognition (pp. 197-212). New York Psychology Press.
- Siegler, R. S., & Opfer, J. E. (2003). The development of numerical estimation: evidence for multiple representations of numerical quantity. Psychological Science, 14, 237-243.