Insights into Our Understandings of Large Numbers

Abstract

How much is a million? This question is familiar to many elementary teachers who have challenged their students to think about and explore large numbers. Much has been written about the importance of having children explore large numbers, think about large amounts of something, estimate large quantities of objects, and be able to gain a sense of how much is “a lot.” Explorations with these goals in mind often invite students to use a familiar context or an everyday object to think about how much a million or some other very large number is. The children's book How Much Is a Million? (Schwartz 1993) is well known in many classroom settings and frequently used as a springboard for lessons posing this kind of challenge. In fact, there may be millions of resources available to teachers who are looking for ways to help their students conceptualize large numbers.

Differential processing of “small” and “large” multidigit numbers.

“…This notion is supported by the log-to-linear shift hypothesis according to which logarithmic coding (i.e., condensing large numbers into a smaller space) is replaced by linear coding as individuals gain experience with a varied set of numbers (e.g., Hurst et al, 2014;Siegler et al, 2009;Siegler & Booth, 2004;Siegler & Opfer, 2003;Thompson & Opfer, 2010). In addition, research on mathematical learning has shown that adults, even those who teach mathematics, struggle with understanding large numbers (Brass & Harkness, 2017;Gough, 2008;Kabasakalian, 2007;Kastberg & Walker, 2008). Given that large numbers play important roles in various scientific and geopolitical contexts (e.g., Batt et al, 2008;Dorogovtsev et al, 2006;Dunning, 1997;Landy et al, 2013Landy et al, , 2017Meffe, 1994;Resnick et al, 2017), understanding how this understudied category of numbers is processed may contribute significant value to many domains.…”

Large number placements on a bounded number line: The importance of explanations

“…Additionally, it is evident from this research that PSTs need more experiences with large numbers and large number data. We agree with education researchers that many children, adults, and PSTs need more concrete experiences with scaling large numbers to help them move from abstractions (Kabasakalian, 2007;Kastberg & Walker, 2008). Enactive or iconic lessons with numbers such as a million (or more) -students or PSTs collect a million pennies for a charity or they draw a million starhelp to make the relative size of a million more concrete.…”

“…The strategy of One One‐Thousandth included explanations where PSTs emphasized the relationship inherent in the name. For example, a PST noted, “Because 1 billion is 1/1000 of a trillion so there are 999 more billions after it so basically on the line you have to leave enough room for an additional 999 dots w/ equal spacing.” This strategy demonstrated PSTs' numerical and spatial reasoning as well as their grasp of the multiplicative nature of magnitudes that increase exponentially (Kastberg & Walker, 2008; Landy et al, 2013).…”

“…As such, the review of the literature focuses on these three plausible placements. Please note: All studies referred to below were conducted by cognitive science researchers except for the three studies conducted by education researchers: Kabasakalian (2007); Kastberg and Walker (2008); Skwarchuk and Anglin (2002).…”

“…Similarly, Kastberg and Walker (2008), education researchers, highlighted the impact of student understanding of numbers as composite units; a number represents a quantity that contains other quantities. An understanding of both the additive and multiplicative nature of composite units is critical for development of connections between counting known numbers (additive) and experiences with large numbers which are abstractions (multiplicative) (Kastberg & Walker, 2008). Comparably, Landy et al (2013) noted the importance of understanding magnitudes as increasing exponentially.…”