Counting errors as a window onto children’s place-value concept Winnie Wai Lan Chan ⇑ , Terry K. Au, Nathan T.T. Lau, Joey Tang Department of Psychology, University of Hong Kong, Hong Kong Special Administrative Region article info Article history: Available online 4 July 2017 abstract Ó 2017 Elsevier Inc. All rights reserved. 1. Introduction 1.1. Place-value concept and mathematical learning While numbers are arbitrary symbols, their formation relies on systematic, meaningful rules – namely the place-value concept. Such concept is essential when making sense of multi-digit num- bers. Students should understand that each position in a number represents a power of ten, and that each digit in the number carries a place value depending on its position. Take ‘‘54” as example. The digit 4 in the ones place carries a place value of 4 (4 Â 10 0 ); the digit 5 in the tens place carries a place value of 50 (5 Â 10 1 ). Anal- ogous to morphemes in alphabetical words, the place-value con- cept governs the organization of digits in a number – rendering it meaningful. Having a good grasp of the place-value concept is crucial to mathematical learning (Chan, 2014; Chan, Au, & Tang, 2013, 2014; Chan & Ho, 2010; Wearne & Hiebert, 1994). Previous studies have shown that children’s place-value understanding predicts their performance in comprehension and production of numbers (McCloskey, 1992), computation (e.g., addition and subtraction; Ho & Cheng, 1997), mathematical problem-solving (Collet, 2003; Dehaene & Cohen, 1997; Fuson, Wearne, et al., 1997), and early mathematical achievement (Miura & Okamoto, 1989). Children with subpar understanding of the place-value concept are prone to mathematical learning difficulties (Chan & Ho, 2010; Chan et al., 2014; Hanich, Jordan, Kaplan, & Dick, 2001; Jordan & Hanich, 2000). With training in the place-value concept, children improve their arithmetic performance (Fuson, 1990; Fuson & Briars, 1990; Ho & Cheng, 1997; Jones, Thornton, & Putt, 1994). Hence the place-value concept plays a vital role in early mathemat- ical learning. 1.2. Using base-ten blocks to illustrate the place-value concept One traditional approach to illustrate the place-value concept is to use base-ten blocks. For one thing, young children lack the cog- nitive capacity to understand the logic behind the arbitrary place- value system (e.g., ten units in the ones place should be traded for one unit in the tens place; Chandler & Kamii, 2009; Fosnot & Dolk, 2001), so they need concrete materials to figure out how the digits in a number relate to each other (e.g., ten cubes are equivalent to one bar; Fuson, 1990; Fuson & Briars, 1990). For another thing, the sizes of the base-ten blocks being proportional to their quanti- ties (e.g., ten cubes make one bar) helps children to relate the blocks with the base-ten grouping in the place-value numeration system (Hiebert & Carpenter, 1992). Indeed, the base-ten blocks have long been adopted as a useful tool to teach and assess chil- dren’s place-value concept across cultures. Traditional textbooks show diagrams of base-ten blocks to illustrate how the model maps onto the digits in a number (e.g., ‘‘46” means four bars and six cubes). To tap into children’s understanding of the place- value concept, children are asked to represent a multi-digit num- ber using base-ten blocks (Miura, Okamoto, Kim, Steere, & Fayol, 1993; Naito & Miura, 2001; Saxton & Cakir, 2006) or translate a model of base-ten blocks into the corresponding number (Chan et al., 2014). 1.3. Children’s counting strategies and the development of the place- value concept Systematic examination of how children count the base-ten blocks can further provide useful insights into how they under- stand the place-value structure of multi-digit numbers. Based on their classroom observations, Fuson, Smith, and Lo Cicero (1997) described children’s conceptual structures of two-digit numbers in relation to how they counted the base-ten blocks. These concep- tual structures include unitary multi-digit, sequence-tens and ones, and separate-tens and ones. Consider the quantity fifty- four. Children who conceptualize a multi-digit number as an undi- http://dx.doi.org/10.1016/j.cedpsych.2017.07.001 0361-476X/Ó 2017 Elsevier Inc. All rights reserved. ⇑ Corresponding author. E-mail address: wlwinnie@hku.hk (W.W.L. Chan). Contemporary Educational Psychology 51 (2017) 123–130 Contents lists available at ScienceDirect Contemporary Educational Psychology journal homepage: www.elsevier.com/locate/cedpsych