RESEARCH INFORMATION THE ACQUISITION OF NUMERACY FOR TEACHERS Jenny Young-Loveridge ■ Department of Education Studies, University of Waikato set one,1999 12 he New Zealand mathematics expansion and elaboration to include other counting process, including the one-to-one curriculum aims to provide students components of mathematics, such as principle (pairing off each object with a T decimal fractions. different number word while maintaining one- with the skills and understandings “to to-one correspondence between an object and a The developmental framework shows how help them understand and play a number word), the stable order principle children’s understanding of the number system responsible role in our democratic (producing the sequence of number words in a becomes increasingly sophisticated as their 1 consistent order each time), and the cardinality society”. The number system provides thinking develops. The framework is designed principle (the idea that the last number word an important and powerful tool with to help teachers differentiate among their produced during counting designates the total students on the basis of the children’s which to do that. Understanding the number of objects in the group).5 Slightly more number system is the key to teaching understanding of the number system. Each challenging than simply counting a given group stage in the framework includes three different and learning mathematics. This article of objects is making a group of objects on components: the number concept itself, the request (that is, forming sets). Research has outlines a framework that shows how spoken number word which refers to the shown that children’s competence with forming children think about the number system. concept orally, and the written numeral which sets when they begin school at 5 years of age is records the concept in written form using strongly related to their success in mathematics symbols. Each of the three components is several years later.6 Everyday life is full of A NEW FRAMEWORK FOR linked to the other two, although initially (in opportunities for children to learn how to THE ACQUISITION OF NUMERACY the years prior to school entry) knowledge of create small groups of objects, for instance written numerals may not be connected either getting a particular number of carrots or Several researchers have developed models to number concepts or to spoken number potatoes for dinner, setting the table, ensuring to explain how children’s understanding words (see dotted lines in figure 2). there are enough chairs for everyone to sit about numbers develop as they progress down, sharing out special food among friends.7 2 from beginning to competent thinking. NUMBER CONCEPTS However, each of these models has problems Another way of establishing quantity can be or limitations. For example, some focus on a Children need to build a rich network of ideas through recognition of a stylised number very narrow domain of understanding, about the patterns and relationships among pattern, such as that depicted on dice or while others are overly complicated and numbers. Central to this network of dominoes (that is, pattern recognition, difficult to apply quickly in a classroom relationships is the number concept itself, sometimes called subitising). Such recognition context, and still others lack a clear rationale which may be established in a variety of ways. is almost immediate and does not involve for progress to more advanced stages. For For example, counting is an important process counting. Board games, card games, and this reason, constructing a developmental by which the answer to a “How many?” dominoes also provide enjoyable contexts for framework to help teachers understand question can be determined. Rochel Gelman learning about quantities.8 children’s learning and thinking about the has identified several principles involved in the number system seemed important. Research in the United States and Figure 1: Developmental framework Figure 2: A variety of ways of understanding Australia has shown for the acquisition of numeracy the concept “five”: a unitary concept that teachers who were given this kind of Construction of a strong unitary concept framework were better of numbers, then a shift to multi-unit able to help their understanding NUMBER CONCEPT students build on their eg, concept of “five” mathematical 1. Unitary concept thinking.3,4 The (for single-digit and multi-digit Established by pattern by recall of a framework described numbers) by counting recognition “number fact” here is based Knowledge of number word sequences, or or predominantly on the counting processes, part-whole relation- work of Karen Fuson ships, numerals and number patterns “1, 2, 3, 4, 5” “5” “3 and 2 is 5” and Lauren Resnick in 2. Ten-structured concept or the United States, and Whole decade partitioned into units of Knowledge of familiar quantities integrates many 10 ones (eg, 5 fingers on each hand) important features of the other models. 3. Multi-unit concept ○○○○○ The framework consists Units of tens and ones counted SPOKEN NUMBER separately, and can be traded and WRITTEN NUMERAL WORD of four stages, each exchanged (eg, 10 ones for one 10, or characterised by a one 10 for 10 ones) 5 ○○○○○○○○○○ “FIVE” major shift in ways of Different versions of Rote counting thinking about 4. Extended multi-unit concept “5”(eg, on calculators, sequence “one, two, videos, letterboxes, three, four, five” (no numbers (see figure 1). Units can be any power of ten TV, money) objects) There is room within the framework for 1 Another source of knowledge about quantities is the familiar quantities DEVELOPMENTAL PROGRESSION that children come to “just know” (for example, two hands, three The developmental framework begins with a unitary (by ones) concept wheels on a tricycle, four legs on a cat, five fingers on a hand). Most of numbers, then moves through a transitional stage of ten-structured children learn very early the meaning of “two” without needing to thinking, which leads to a multi-unit (by tens and ones) concept of count (wanting to have one biscuit for each hand is a good example). numbers (see figure 1). The fourth stage in the framework is an extended Gradually, with experience, they learn to recognise larger groups of multi-unit stage in which the idea of multiple units is extended to units objects, like “three”, “four”, and “five”, without needing to count them. larger than ten (for example, hundreds, thousands, and so on). Extended Children also come to “just know” certain number combinations multi-unit understanding makes it possible for children to work with involving small numbers (that is, number facts). Children learn about large numbers efficiently using advanced understanding of place-value. combinations of small numbers first (for example, “two and one more makes three”). “Doubles” can be used to work out some of the other UNITARY CONCEPT combinations. For example, “two and two makes four, and another one Building a strong unitary concept of numbers is an essential first makes five” could eventually lead to “three and two makes five step in coming to understand the number system, particularly as it altogether” (that is, “derived number facts”). relates to small quantities (that is, those represented by single-digit numerals). The magnitude of a quantity can be ascertained by SPOKEN NUMBER WORDS unitary counting. Children’s counting gradually shifts from a As number concepts develop, so too does the language which dependence on physical materials to mental counting, and enables children to communicate their understanding of number eventually to an understanding that a number can be taken to concepts to others (spoken number words). Children learn the names of the number words which in their own culture are used to talk about and refer to quantities. Evidence from earlier research Figure 3: Example of each stage in the developmental shows that sometimes the connections between number concepts framework for numeracy acquisition and the spoken number words that refers to those concept may be slightly faulty to begin with. Some young children seem to go NUMBER through a stage of being able to hold up the correct number of CONCEPT fingers corresponding to a stylised number pattern presented on a WRITTEN dot-dice, but say the wrong number name (for example, they see a eg, concept of “twenty-five” 1 NUMERAL dot-dice pattern of three and hold up three fingers, but say “four”). 25 This idiosyncratic response pattern, which was marked among established by counting by ones Unitary 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, “25” refers to younger 4-year-olds, disappeared as the children established the concept the whole quantity correct connection between number concepts and the spoken 12, 13, 14, 15, 16, 17, 18, 19, 20, number words used to refer to those concept.9 21, 22, 23, 24, 25 The conventional sequence of spoken number words, while meaningless on its own, is an essential tool in building an understanding of the number system.10 Often children’s errors in rote NUMBER CONCEPT counting reflect their understanding of the rules by which the number WRITTEN system operates, as in the production of novel number words such as eg, concept of “twenty-five” 2 NUMERAL “twenty-ten”, and “twenty-eleven”, or consistently stopping at a Ten- 25 number ending in nine or zero. There is a huge literature on the issue 10, 20, structured “2” refers to twenty of language and culture and its impact on children’s learning about the and “5” refers 21, 22, 23, 24, 25 concept number system.11 For example, Asian languages differ from languages to five like English and French in having a very systematic and transparent structure for their number word sequence (for example, “12” in Japanese is “juni” or “ten-two”).