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The Wiley Handbook of Early Childhood Care and Education, First Edition. Edited by Christopher P. Brown, Mary Benson McMullen, and Nancy File. © 2019 John Wiley & Sons, Inc. Published 2019 by John Wiley & Sons, Inc.

15

There is growing concern that preschool and kindergarten in the U.S. has become too academically oriented at the expense of play, exploration, and nonacademic subjects, such as social development (DEY, 2014; Miller & Almon, 2009; Stipek, 2006). Bassok, Latham, and Rorem (2016) noted that kindergarten teachers’ aca-demic expectations of children for the beginning and end of kindergarten in 2010 were far higher than in 1998 and that they devoted substantially more time to advanced literacy and mathematics and less to art, music, and child‐selected activities. Walker (2016) quoted a veteran teacher: “The changes to kindergarten make me sick. Think about what you did in first grade–that’s what my 5‐year‐old babies are expected to do.” Walker further noted that the teacher’s district even tried to remove play items from the room and concluded, “The implication was clear: There’s no time for play in kindergarten anymore.” Bassok et al. further found that teacher‐directed activities, including the use of textbooks and worksheets, have increased dramatically.

To provide a perspective on these concerns, we address two key, interrelated issues in early childhood mathematics education: What is the place of mathematics instruction in today’s preschool and kindergarten classrooms? How guided should the (mathematics) instruction of young children be?

What is the Place of Mathematics Instruction in Today’s Preschool and Kindergarten Classrooms?

Many early childhood educators have long resisted a teacher‐centered, academic approach to instruction because of an antipathy toward such traditional prac-tices as didactic instruction and memorizing mathematics by rote—instruction that is often ineffective, joyless, or even destructive. Instead, they have embraced a child‐centered, “developmentally appropriate” approach: fostering nonaca-demic goals (creativity, social skills) and self‐discovery that they consider more

Teaching and Learning Mathematics in Early Childhood ProgramsArthur J. Baroody1,2, Douglas H. Clements2, and Julie Sarama2

1 University of Illinois at Urbana‐Champaign2 University of Denver

Baroody, A. J., Clements, D. H., & Sarama, J. (2019). Teaching and learning mathematics in early childhood programs. In C. Brown, M. B. McMullen & N. File (Eds.), Handbook of Early Childhood Care and Education (1st ed., pp. 329-353). Hoboken, NJ: Wiley Blackwell Publishing.

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consistent with children’s developmental needs. There are, however, good rea-sons for raising early childhood educators’ mathematical expectations for chil-dren, and there is no reason why early mathematics instruction must be joyless or crowd out nonacademic instruction. In this section, we provide a rationale for early childhood mathematics instruction and recommendations for ensuring such instruction is effective, engaging, and holistic (i.e., fosters mathematical thinking and knowledge joyfully while supporting nonacademic goals).

Rationale for Early Childhood Mathematics Instruction

Reasons for involving young children in mathematics instruction are detailed in, for instance, Baroody, Lai, and Mix (2006); Bassok et  al. (2016); Clements, Baroody, and Sarama (2013); Frye et al. (2013); and National Research Council (2009), and are summarized below:

1) Young children are capable of considerable and often surprising mathematical learning and can be quite interested in learning mathematics.

2) Early informal (everyday and largely verbal and manipulative) mathematic knowledge provides an important foundation for learning school (and largely written) mathematics and other academic content.

3) Important individual and group differences in informal knowledge emerge early, and these differences can significantly affect school achievement.

4) Early intervention has been shown to be effective in closing the gap in early individual differences and leveling the playing field for children who would otherwise be at risk for formal mathematical learning difficulties.

5) U.S. children overall have lower mathematics achievement than children of many other industrialized countries putting them and the U.S. at an economic competitive disadvantage.

For such reasons, the recent Institute of Educational Studies early numeracy practice guide recommends early childhood teachers dedicate time that targets mathematics each day and look for opportunities to integrate mathematics throughout the school day and across the curriculum (Frye et  al., 2013). Consistent with the first point above, research has revealed many developmen-tally appropriate mathematics goals that can and should be targeted for instruc-tion in preschool (Clements & Sarama, 2014; Clements, Sarama, & DiBiase, 2004; Sarama & Clements, 2009) and kindergarten (Council of Chief State School Officers [CCSSO], 2010). The fact that kindergarten teacher’s expectations for beginning and ending kindergartners rose substantially between 1998 and 2010 (Bassok et  al., 2016) reflects the growing awareness of the rationale provided above and is a positive, not a negative, development, when implemented accord-ing to research and wisdom of expert practice.

Providing Educative Experiences: Implementing Early Mathematics Instruction Effectively, Engagingly, and Holistically

Implementing early childhood mathematics instruction in an effective manner—as Bassok et al. (2016) thoughtfully noted

Teaching and Learning Mathematics in Early Childhood Programs 331

need not be at odds with “play” and other types of pedagogical approaches considered developmentally appropriate in early childhood (Bassok, Claessens, & Engel, 2014; Clements & Sarama, 2014; Pondiscio, 2015)… Increasingly, developmental scientists agree that there are ways to mean-ingfully engage young children in … math learning and that the effective-ness of such efforts depends on the pedagogical approach, the quality of teaching, and the connection of the instruction to young children’s curiosity. (Katz, 2015; Snow & Pizzolongo, 2014, p. 15)

Indeed, Baroody, Purpura, Eiland, Reid, and Paliwal (2016) concluded the term developmentally appropriate is evolving from its traditional meaning of exclud-ing academic instruction (indicated henceforth as “developmentally appropri-ate”) toward one that includes academic instruction that fits young children capacities, interests, and needs (indicated throughout without the quotes). Indeed, Bredekamp (2004), the main architect of the developmental appropriate construct, stated, “The assumption [academic] goals will be inappropriate or unachievable is unfair. As long as goals are developed drawing on research and the wisdom of practice, [academic] goals can be excellent contributions” (p. 79).

Although laissez‐faire free play or “letting children be children” seems on the surface to be helpful or at least innocuous, this is not necessarily so. In his analy-sis of why child‐centered progressive education failed, Dewey (1963) concluded that educators should not simply adopt teaching methods that are the opposite of the often, unsuccessful methods used in traditional instruction (e.g., relying on laissez‐faire free play instead of teacher‐guided activities). He distinguished between educative experiences, which lead to worthwhile learning or a basis for later learning, and mis‐educative experiences, which—even if fun—do not pro-mote learning. In other words, engaging in “developmentally appropriate” activi-ties may or may not advance learning (Fuson, 2009). Educative experiences require developmentally appropriate activities in which external factors (e.g., the instructional content and practices) mesh with internal factors (e.g., a child’s developmental readiness and interest).

As Balfanz (1999) observed, such founding figures of early childhood educa-tion as Fredrick Froebel and Maria Montessori believed that young children enjoyed using mathematics to explore and understand the world around them and incorporated rich, structured mathematical experiences into their early childhood programs. In the remainder of this section, we discuss how structured play, learning trajectories, and integrated instruction can be useful pedagogical tools in providing structured mathematical experiences that ensure educative experiences.

Structured PlayIf chosen carefully with a young child’s interests, needs, and developmental level in mind, structured play, such as that involving games, can be a developmentally appropriate (enjoyable and educative—challenging but achievable) means of attaining research‐based academic goals. Mathematics educators have long underscored the value of playing games that target the learning or application of specific mathematical concepts or skills (Baroody, 1987; Baroody & Coslick,

Arthur J. Baroody, Douglas H. Clements, and Julie Sarama332

1998; Bright, Harvey, & Wheeler, 1985; Ernest, 1986; Rutherford, 2015). How math games can foster specific early mathematical concepts and skills has received increasing attention from researchers (Ramani & Siegler, 2011; Reid, Baroody, & Purpura, 2013). Math games are a common aspect of many early childhood curricula (Clements & Sarama, 2013; University of Chicago School Mathematics Project [UCSMP], 2016a, 2016b) or otherwise widely available on, for example, the internet (e.g., Griffin & Case, 1997; Public Broadcasting System, 2016; Public School of North Carolina, 2016; USCMP, 2016b). For example, observations indicate that primary‐grade children greatly enjoy the game‐based Wynroth (1986) Math Program and benefit significantly from it (Baroody & Ginsburg, 1983; Gelberg, 2008).

Learning Trajectories: Ensuring Developmentally Appropriate InstructionAny form of instruction needs to be carefully crafted or guided to build effec-tively on what children know (Fyfe, Rittle‐Johnson, & DeCaro, 2012). Learning trajectories—which include theoretically and empirically based developmental levels or steps—can provide clear and specific direction for (a) defining the goals for meaningful instruction; (b) identifying children’s current developmental lev-els and their next (developmentally appropriate) instructional step; and (c) designing instruction to help them achieve the next level (Clements & Sarama, 2014; Daro, Mosher, Corcoran, & Barrett, 2011; Sarama & Clements, 2009), regardless of the context (e.g., small or large group, play) and teaching strategy.

Integrated InstructionA strategically and carefully implemented academic approach is compatible with developing social and behavioral skills. For example, playing math games can and should involve turn taking and respecting everyone’s right to decide their own solutions. Some games can involve cooperative decision‐making such as deciding on the rules (e.g., to win a board game, do you need to roll the exact number to move to the last space of the board or any number that gets you to the last space or beyond?). Further, mathematics activities can also promote children’s development of self‐regulation, such as response inhibition (Clements, Sarama, & Germeroth, 2016; Clements et al., 2018). Moreover, mathematics and music, art, or literacy instruction—all of which involve patterning—can go and should go hand in hand. For example, toddlers are unlikely to keep a three‐beat rhythm without an exact cardinal concept of three and the ability to subitize three. Contrasting a three‐beat with a two‐beat may help children both construct an exact cardinal concept of three (subitize three) and lean to maintain a three‐beat.

Conclusions

Critics of introducing academics in kindergarten or earlier point out that “accountability pressure” can warp early academic instruction by, for example, shifting the emphasis from formative assessment, which is central to planning developmentally appropriate instruction, to summative assessment (standard-ized testing that merely indicates achievement, too often of low‐level skills). They are alarmed by pedagogical approaches to preschool and kindergarten

Teaching and Learning Mathematics in Early Childhood Programs 333

instruction indicated by, “the heightened [and over] use of textbooks, work-books, and worksheets” (Bassok et  al., 2016, p. 14). For example, Heckman, Krueger, and Friedman (2004) concluded that a focus on academics in early childhood programs is misplaced, because the priority should be fostering non-cognitive social and behavioral skills.

However, contrary to Heckman and colleagues’ (2004) conclusion, a focus on academics in early childhood programs is not misplaced and, indeed, the long‐term benefits of such programs depend on building cognitive, as well as noncog-nitive, skills. Mathematical instruction that is thoughtfully crafted to capitalize on children’s natural interest in playing games, their involvement in everyday activities and other content instruction, and their existing knowledge can be effective without sacrificing play, exploration, nonacademic learning, and social development. Indeed, such cognitive‐oriented instruction can greatly benefit social and behavioral development, and the integration of the two has additional advantages, such as building children’s positive self‐concepts of themselves as learners and motivation to learn. Addressed in the following section is how teachers can promote educative experiences with careful guidance.

How Guided Should the (Mathematics) Instruction of Young Children Be?

Although there is now general agreement that unguided discovery in the form of laissez‐faire free play recommended by some extreme constructivists is neither efficient nor effective (Alfieri, Brooks, Aldrich, & Tenenbaum, 2011; Mayer, 2004), there is far less agreement about how much guidance should be provided when introducing children to mathematical ideas and skills (Hmelo‐Silver, Duncan, & Chinn, 2007). In this section, we first try to put the debate between different degrees of guidance into perspective and then argue there is not a sim-ple answer to the question: How much guidance needs to be provided?

Perspective: Partially Guided Versus Fully Guided Instruction

The debate about the relative merits of traditional didactic or direct instruction and reform instruction based on discovery learning dates back to the time of the ancient Greeks and typically has been heated and unproductive (Baroody, 2003). Alfieri et  al. (2011, p. 1) concluded from their meta‐analysis of 164 studies: “Unassisted discovery does not benefit learners” and is less effective than more guided forms of instruction. Concerned that traditional direct instruction did not promote conceptual understanding and mathematical thinking (e.g., prob-lem‐solving and reasoning ability), constructivists have long recommended par-tially guided discovery learning as a compromise approach (Ambrose, Baek, & Carpenter, 2003; Kilpatrick, 1985). However, proponents of direct instruction have long been skeptical of the indirect methods advocated by reformers (Clark, 2009; Rosenshine, 2009). Mayer (2004) observed that, although research from 1950 to 1980 indicated that the then widely used unguided, problem‐based

Arthur J. Baroody, Douglas H. Clements, and Julie Sarama334

approach was unsuccessful, constructivists kept adducing discovery approaches under a new name. Kirschner, Sweller, and Clark (2006) concluded that the research shows that discovery methods are ineffective and inefficient compared to direct or fully guided instruction.

If the debate about instructional approach is to be profitable, a fair comparison of traditional direct instruction and discovery learning is needed. Hmelo‐Silver et al. (2007) argued, for instance, that Kirschner and colleagues’ (2006) category of discovery learning was extremely broad or undifferentiated and, thus, did not provide a fair comparison between direct instruction and unguided discovery. Similarly, given the growing consensus that instruction needs to focus on both procedural and conceptual competencies and indeed fostering these in an inter-twined way, it is unfair to compare conceptually oriented discovery learning with procedurally focused traditional didactic instruction.

Definition of Instructional ApproachesA fair comparison of instructional approaches involves direct methods that focus on both procedures and concept and discovery learning with some degree of guidance.

Direct (Fully Guided) InstructionAs Table 15.1 indicates, fully guided instruction involves explicit procedural and conceptual instruction by a teacher. A worked examples approach epitomizes this type of instruction (Clark, Kirschner, & Sweller, 2012). Such an approach entails completely explaining how to solve a problem and why a procedure works and then providing practice, feedback, and gradually reduced explanations to ensure internalization.

Partially Guided Discovery LearningAlfieri et al. (2011) defined discovery learning as not providing learners with the target information, but creating the opportunity to, “find it independently … with only the provided materials” (p. 2). Two caveats for evaluating discovery learning are:

1) Discovery learning is a broad category that includes vastly different ways of promoting independent insight and might best be viewed as part of a contin-uum of instructional strategies (see Table 15.1). Unlike unguided discovery learning in which students choose their own tasks, partially guided discovery at the very least entails a task chosen by a teacher for the purpose of achieving a particular instructional goal. Whereas negligibly guided discovery provides no direction (structure or scaffolding) for engaging with the teacher‐chosen task, other forms of partially guided discovery in Table 15.1 provide increas-ing structure or scaffolding (cf. Schmidt, Loyens, van Gog, & Paas, 2007). The direction may only be implicit as with minimally or moderately guided dis-covery or explicit as with highly guided discovering (again see Table 15.1).

2) Bruner (1961) cautioned that discovery learning can be effective only when pupils are developmentally ready for a constructive activity (constructing ideas or strategies that go beyond provided information). Partially guided discovery theoretically takes into account a child’s existing level of development and presents a problem one level above it.

Table 15.1 A Continuum of Approaches for Using Games to Teach the Near‐Doubles Reasoning Strategy and Its Rationale

Feature Example of how a computer program might teach the near‐doubles reasoning strategy (e.g. 4 + 5 = [4 + 4] + 1 = 8 + 1).

Degree of guidance

Ung

uide

d

Partially directed/guided

Fully

gui

ded

Neg

ligib

ly g

uide

d

Min

imal

ly g

uide

d

Mod

erat

ely

guid

ed

Hig

hly

guid

ed

Hig

hly

+ fu

lly

guid

ed

Unprescribed activities such as laissez‐faire free play: Near‐doubles are accidentally practiced without feedback about correctness

✓

Prescribed activities (program‐/teacher‐chosen set of games that include the near‐doubles) involving… No structure or scaffolding: Unstructured practice in which near‐doubles are mixed with other types of basic sums, and items are presented in random order. Answers are chosen from an unordered array of numbers. No hints about the near‐doubles strategy are provided and feedback focuses on correctness only

✓

Some structure or scaffolding implicitly underscores relation or strategy: Only near‐double and double items presented but still in random order. This increases somewhat the chances of a child recognizing that 4 + 5 can be solved using 4 + 4 = 8. Child enters sum of a near‐double by clicking on a number list in which the odd numbers are highlighted—increasing the chances that a child will notice that sums of near‐doubles are always odd (whereas as the sums of doubles are always an even number). a Feedback focuses on correctness only

✓

Structure or scaffolding implicitly underscores relation or strategy: Near‐double items such as 4 + 5 immediately follow related (smaller) double 4 + 4 = 8, and feedback includes showing 4 + 4 = 8 and 4 + 5 = 9 one underneath the other. Such structuring increases the chances a child will notice that a near‐double is 1 more than its related (smaller) double. Feedback focuses on correctness only

✓ ✓ ✓ ✓

Structure + semi‐explicit scaffolding: Game involves explicitly asking if a particular double such as 4 + 4 = 8 helps to solve a particular near‐double such as 4 + 5 =?. Feedback explains why a particular response was correct or not

✓ ✓ ✓

(Continued)

Table 15.1 (Continued)

Feature Example of how a computer program might teach the near‐doubles reasoning strategy (e.g. 4 + 5 = [4 + 4] + 1 = 8 + 1).

Degree of guidance

Ung

uide

d

Partially directed/guided

Fully

gui

ded

Neg

ligib

ly g

uide

d

Min

imal

ly g

uide

d

Mod

erat

ely

guid

ed

Hig

hly

guid

ed

Hig

hly

+ fu

lly

guid

ed

Structure + explicit scaffolding—Strategy and its rationale (relations) explicitly summarized after the strategy is discovered Program structured to provide both implicit and explicit opportunities to discover the near‐doubles strategy and then have it explicitly specified

✓

Strategy and its rationale (relations) explicitly taught Program explicitly specifies and illustrates the near‐doubles strategy: “As 5 = 4 + 1, you can think of 4 + 5 as 4 + 4 + 1, and as 4 + 4 is 8 and 8 + 1 is 9, then the answer to 4 + 5 is 9 too”

✓

Note . ✓  =  a defining or critical attribute—a characteristic of all cases of an approach. a True to the next three rows as well.

Teaching and Learning Mathematics in Early Childhood Programs 337

Differences in the ApproachesFully and partially guided approaches are based on fundamentally different psy-chological assumptions.

Fully Guided ApproachKirschner et al. (2006) concluded that fully guided instruction, but not discovery learning, is consistent with what research has revealed about how human brains learn. Cognitive Load Theory (Sweller, 1988) suggests, “working memory limita-tions dictate high levels of instructional guidance initially for domain novices” (Matlen & Klahr, 2013, p. 622). According to this theory, discovery learning is presumed to require a novice to search long‐term memory for a strategy or solu-tion‐relevant information, and the resulting heavy burden on working memory precludes learning. In contrast, they claimed that fully guided instruction, such as the worked examples approach, minimizes demands on working memory.

Partially Guided ApproachesIn comparison to the worked examples approach based on a behavioristic model of problem solving in which a learner is merely passive and reactive, guided dis-covery learning is based on an “insight” model of problem solving in which a learner is active and creative. Guided discovery learning does not underestimate children’s capacities when provided developmentally appropriate problems. Moreover, a fully guided approach is based on the questionable assumption that conceptual understanding can typically be imposed on children, whereas par-tially guided approaches are based on assumption that such learning requires a children’s active reflection (Baroody, 2003). As an example of such reflection, Alfieri et al. (2011) hypothesized that actively and independently constructing knowledge (e.g., exploring a phenomenon, noticing patterns or relations, and generalizing such regularities) produces a “generation effect”: enhancing under-standing and thus learning, retention, and transfer. Chi (2009) similarly hypoth-esized that constructive activities are more effective than active activities (doing something physically), which in turn, are more effective than passive activities (e.g., listening or watching). An effective constructive activity is providing chil-dren with examples and nonexamples of a concept so that they can discover for themselves the critical (defining) attributes of the idea (Hattikudur & Alibali, 2010; Prather & Alibali, 2011; Rittle‐Johnson & Star, 2011; Schwartz, Chase, Chin, & Oppezzo, 2011).

Comparative ResearchGiven developmental readiness and the opportunity, children can invent their own solution strategies—including those as, or even more, efficient than school‐taught strategies (Baroody & Coslick, 1998; Clements & Sarama, 2014; Ginsburg, 1977). Problem exploration and strategy invention can lay the founda-tion for understanding more advanced school‐taught procedure concepts and procedures (Schwartz, Lindgren, & Lewis, 2009; Schwartz & Martin, 2004). Several experiments that involved comparing the impact of exploration and direct instruction provide direct evidence of a generation effect for exploration (DeCaro & Rittle‐Johnson, 2012; Schwartz et al., 2011; Shafto, 2014).

Arthur J. Baroody, Douglas H. Clements, and Julie Sarama338

Alfieri and colleagues’ (2011) meta‐analysis indicated that guided discovery is more effective than unguided or fully guided instruction. In their review of the research, Hmelo‐Silver et  al. (2007) concluded that discovery learning in the form of problem‐based or inquiry‐based instruction with at least modest scaf-folding (i.e., moderately or highly guided discovery) is effective. For instance, Mix, Moore, and Holcomb (2011) compared relatively unguided and relatively guided discovery methods for fostering an understanding of one‐to‐one corre-spondence with 3‐year‐olds. Both conditions involved providing participants with toys for home play. The relatively unguided condition involved providing two unrelated sets of toys of equal number for home play (e.g., six wiffle balls and six plastic frogs). The equal number of toys in each set implicitly scaffolds the discovery of one‐to‐one correspondence and thus qualifies as minimally guided discovery in Table 15.1. The relatively guided condition entailed providing a set of toys and a container with an equal number of inserts (e.g., six whiffle balls and a six‐muffin muffin tin in which the wiffle balls were nested). Although one‐to‐one correspondence was still only implicitly suggested, nesting the toys in the spaces of the container more clearly underscored the correspondence. Children in the moderately guided condition outperformed those in the minimally guided condition on a challenging equivalence task, namely a cross‐mapping version of number matching (e.g., shown two rose stickers, chose two turtle stickers rather three roses stickers or one racecar sticker).

One possible reason guided discovery has been shown to be more effective than direct instruction is that, in the latter, many pupils become so procedure‐oriented they lose interest in understanding (the “just tell me how to do it” syn-drome). Moreover, sidestepping constructive activity and attempting to impose conceptual understanding on children may result in partial understanding or even confusion (DeCaro & Rittle‐Johnson, 2012; Rittle‐Johnson, 2006). Murata (2004) found that Japanese primary‐grade children taught a decomposition strategy (e.g., 9 + 8 = 9 + 1 + 7 = 10 + 7 = 17) and its rationale with larger‐addend‐first combinations did not exhibit strategy transfer when smaller‐addend‐first items (e.g., 8 + 9) were introduced. In other words, the Japanese student appar-ently memorized an adult‐imposed procedure for problems in which the 9 was the first addend and could mechanically apply it to similar problems, such as 9 + 5 and 9 + 7. However, as is often the case with a procedure learned by rote, they failed to apply it to modestly novel problems in which the 9 was the second addend, such as 5 + 9 and 7 + 9. Children who are guided to discover/reinvent such reasoning strategies theoretically would understand a strategy and apply it effectively to moderately novel items.

When Should Guidance Be Provided?Other questions debated by educators concern the relative effectiveness of dif-ferent approaches to integrating conceptual and procedural instruction, including which should be introduced first (cf. Matlen & Klahr, 2013). Having children explore problems and devising their own informal strategies before explicit scaffolding, such as summarizing such strategies and their rationale (e.g., highly + fully guided instruction in Table 15.1), may better prepare them for explicit instruction and facilitate learning (Fyfe et al., 2012). DeCaro and

Teaching and Learning Mathematics in Early Childhood Programs 339

Rittle‐Johnson (2012) found this approach helped children, “more accurately gauge their competence, attempt a larger variety of strategies, and attend more to problem features” (p. 552). Self‐discovery learning and then explicit teacher scaffolding helps ground instruction in children’s existing informal knowledge and fosters sense making of new problems and solution strategies (Clements & Sarama, 2014; NCTM, 2006)—a process that supports transfer (Watts, Duncan, Clements, & Sarama, 2017).

In contrast, initial direct/explicit instruction on procedures or even concepts and procedures may undermine a pupil’s curiosity or autonomy, forestall a nov-ice’s exploration of a domain, and inhibit invention of procedures and conse-quently limit their understanding and interfere with transfer (Rittle‐Johnson, Fyfe, Loehr, & Miller, 2014; Schwartz et al., 2011). For example, Shafto (2014) concluded that, with direct instruction (a demonstration by a teacher perceived as knowledgeable), 4‐ and 5‐year‐olds assumed there was nothing more to learn, engaged in less subsequent exploration, and learned less than peers who did not see a demonstration from a knowledgeable teacher before exploration.

The Need for a Nuanced Approach Based on Conceptual Readiness

As the analysis of the previous subsection indicates, clearly, there is a need to go beyond the simple dichotomy of direct instruction versus discovery learning or even fully guided instruction versus partially guided discovery learning. Specifically, its implication is that partially guided discovery with more scaffold-ing (e.g., moderately guided discovery, highly guided discovery, and highly + fully guided instruction in Table 15.1) is more efficacious than discovery learning with less scaffolding (unguided, negligibly, and minimally guided discovery in Table  15.1) or fully guided instruction. However, there are three reasons care needs to be exercised in applying even this more specific conclusion to particular aspects of early childhood mathematics education:

1) An often‐overlooked finding of Alfieri and colleagues’ (2011) analysis is that their outcomes were moderated by domain and age. Specifically, the magnitude of effects was smaller for certain academic domains—notably mathematics—and for different ages—notably for younger children. Put differently, the rela-tively little research done in the area of early childhood mathematics education makes general conclusions about instructional approach less certain.

2) Relatively speaking, the domain of early childhood mathematics offers many relatively salient patterns and relations that children may be able to discover without much guidance, whereas other domains and higher‐level mathemat-ics involve patterns and relations that are more abstruse and require more guidance to notice. In other words, partially guided instruction with relatively little scaffolding may be more appropriate in early, rather than later, mathe-matics instruction.

3) Previous research frequently has not taken into account developmental readi-ness—particularly, conceptual prerequisites. This leaves open the possibility that less scaffolding may be needed in cases where children are developmentally ready than in cases where they are not.

Arthur J. Baroody, Douglas H. Clements, and Julie Sarama340

Consider three cases that, taken together, suggest a nuanced view about the degree of guidance in early childhood mathematics education is needed.

Counting‐onA counting‐on strategy for determining sums (e.g., for 5 + 3, counting “5; 6, 7, 8”) is a key primary‐grade goal. Although some research seems to indicate that chil-dren with only a basic informal understanding of addition can understand such a strategy and readily benefit from seeing it modeled (Siegler & Crowley, 1994; Tzur & Lambert, 2011), a careful analysis and other research indicates that the success of such instruction depends on a child’s developmental level on a learn-ing trajectory (see Baroody & Purpura, 2017).

A Key Developmental DistinctionIt is critical to differentiate between Level 1 strategies such as concrete counting‐on and Level 2 strategies such as abstract counting‐on (Fuson, 1988). Concrete counting‐on involves three steps: (a) representing an addend with objects, (b) stating the cardinal value of the other addend, and (c) then continu-ing the sum count from this cardinal number until all the objects are exhausted. For example, for 3 + 5, a child could put up five fingers, state the cardinal number “three,” and say (as the five fingers are counted): “four, five, six, seven, eight.” Abstract counting‐on entails two steps: (a) stating the cardinal value of an addend and (b) then continuing the sum count from this cardinal number a number of counts equal to the other addend. For example, for 3 + 5, a child could state the cardinal number “three,” and count: “four, five, six, seven, eight.” Unlike concrete counting‐on for which for there is a well‐defined stopping point (the last item previously put out), a child who uses abstract counting‐on in summing 3 + 5 must keep track of five more counts (e.g., “Three, four is 1 more [than three], five is 2 more, six is 3 more, seven is 4 more, eight is 5 more”).

In terms of conceptual prerequisites, both concrete and abstract counting‐on require what Fuson (1988) called the embedded cardinal‐count concept: In the context of computing a sum, stating the cardinal value of an addend is equivalent to counting from “one” up to this number and eliminates the need to represent one addend with objects before the sum count. However, only abstract count-ing‐on requires a keeping‐track process, which entails an additional conceptual prerequisite that Fuson (1988) called the numerable chain of counting: the insight that counting words themselves, like objects, can be counted.

Research Indicating that Developmental Readiness is CriticalMoomaw and Dorsey’s (2013) results underscore that achieving even the con-crete counting‐on strategy is a major hurdle for many preschool children and that relatively indirect and minimally guided instruction (simply providing a manipulative) may not be sufficient. Their 3‐ to 5‐year‐old participants were sig-nificantly more successful adding with two dot cards (e.g., ooo + o o) than with two numeral cards (e.g., 3 + 2) or even with a numeral card and a dot card (e.g., 3 + oo). Indeed, qualitative analyses revealed that only 2 of 40 preschoolers used a (concrete) counting‐on strategy and error analyses revealed the common error of treating the numeral card as one item regardless of the numeral showing, even when a child could read numerals.

Teaching and Learning Mathematics in Early Childhood Programs 341

Baroody, Tiilikainen, and Tai (2006) distinguished between children who used Level 1 (concrete) strategies and those who used Level 2 (abstract) strategies. The former considered concrete counting‐on, but not abstract counting‐on, as valid, whereas the latter considered both concrete and abstract counting‐on valid. Contrary to Siegler and Crowley’s (1994) conclusion that only a basic understanding was necessary to understand and evaluate abstract counting‐on, only children who had achieved the numerable chain level of counting and learned an abstract strategy could do so. The educational implication is that, for Level 1 strategy users, modeling a concrete counting‐on strategy may promote learning of this moderately novel Level 1 strategy, but modeling abstract count-ing‐on will not foster learning of this conceptually advanced Level 2 strategy.

Teaching Counting‐on EffectivelyDifferent methods may be needed for teaching concrete and abstract counting‐on.

● Concrete counting‐on: Fuchs et al. (2013) described a highly guided and poten-tially useful approach for introducing concrete counting‐on: Have the child represent the larger addend with tokens and hold the tokens in one hand, rep-resent the smaller addend on the other hand with fingers, have a child announce the number of tokens “in hand,” and—starting with this number—count up the number the fingers on the other hand. Creating and holding a concrete repre-sentation of the larger addend (Step 1) and announcing its cardinal value (Step 2) makes the corresponding numeral representation less fragile and makes constructing the requisite (embedded‐cardinal‐count) concept more likely.

● Abstract counting‐on: Research indicates a promising approach for helping children achieve the conceptual knowledge necessary to invent the abstract counting‐on strategy. Children often appear to invent this advanced Level 2 strategy for adding 2 and then 3 shortly after discovering the number‐after rule for adding with 1 (e.g., the sum of 4 + 1 is the counting number after four in the counting sequence: five; Baroody, 1995; Bråten, 1996). It appears that this rule provides scaffolding for constructing the conceptual and procedural prerequi-sites for general abstract counting‐on strategy, namely the numerable chain level of counting and a keeping‐track strategy, respectively.

Successor PrincipleThe successor principle involves the insight that each number in the counting sequence is exactly one more than its predecessor (Izard, Pica, Spelke, & Dehaene, 2008). This big idea would seem to be the conceptual basis for transforming the counting sequence into the integer sequence, its linear representation (e.g., the difference between 3 and 4 is the same as the difference between 8 and 9, specifi-cally one unit), and, ultimately, recognizing the counting numbers as infinite (Baroody, 2016). A hypothetical learning trajectory for the development of the principle is summarized in Table 15.2.

Although the successor principle is a key goal for early childhood mathe-matic education (CCSSO, 2010; Frye et al., 2013), early childhood curricula currently do relatively little to target this key aspect of numeracy, and very little is known about how to effectively teach the principle. A review of four early childhood mathematics curricula (see https://www.researchgate.net/

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Table 15.2 Hypothetical Learning Trajectory for the Development for the Successor Principle

Level 1. Cardinal concept of small numbers ± subitizing (1 and 2 first, then 3, and—in time—4 to 6)

By seeing different examples of a number labeled with a unique number word and nonexamples labeled with other number words, children construct of precise cardinal concepts one, two, and three (Baroody, Lai, & Mix, 2006). For example, seeing various pairs labeled “two” can help a child recognize that this number word refers to number (as opposed to a particular shape, color, or other feature irrelevant to number) and multiple items at that (as opposed to a singular item), and nonexamples (“take one cookie, not two”) can help a child understand that “two” refers only to pairs. These concepts permit subitizing: the ability to immediately recognize and label small collections with an appropriate number word

Level 2. Ordinal concept of small numbers

Subitizing enables children to see that “two is more than one” item and that “three is more than “two” items (understand the term “more” and that numbers have an ordinal meaning

Level 3. Meaningful object counting (including the count‐cardinality principle)

Subitizing enables children to understand the principles underlying meaningful counting: stable order, one‐to‐one, and cardinality principles. For example, by watching an adult count a small collection a child can recognize as “three,” s/he can understand why the last number word in the count is emphasized or repeated—it represents the total or how many (the cardinal value of the collection)

Level 4. Increasing magnitude principle + counting‐based number comparisons (especially collections larger than 3)

Subitizing and ordinal number concept permit discovery of the increasing magnitude principle: the counting sequence represents increasingly larger quantities. This enables them to use meaningful object counting to determine the larger of two collections (e.g., 7 items is more than 6 items because you have to count further to get to seven than you do for six)

Level 5. Number‐after knowledge of the counting sequence

Familiarity with the counting sequence enables a child to enter the sequence at any point and specify the next number instead of always counting from oneA child is ready for the next step when he or she can answer questions such as “What comes after five?” by stating “five, six” or simply “six” instead of, say, counting “one, two, … six”

Level 6. Mental comparisons of close/neighboring number(number after = more)

The use of the increasing magnitude principle and number‐after knowledge enables children to determine efficiently and mentally compare even close numbers such as the larger of two neighboring numbers (e.g., “Which is more seven or eight?—eight”)

Level 7. Successor Principle(“Number after” = 1 more)

Subitizing enables children to see that “two” is exactly one more than “one” item and that “three” is exactly one more than “two” items, knowledge that can help them understand the successor principle: each successive number in the counting sequence is exactly one more than the previous number

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publication/319253121_Types_of_Successor_or_Related_Training_in_Four_Current_Curricula_Pre‐K_Mathematics_Bridges_in_Mathematics_Big_Math_for_Little_Kids_Building_Blocks), revealed but one program (Building Blocks, Clements & Sarama, 2007, 2013) that directly targets this principle. No research, though, has examined the efficacy of current curricula in pro-moting the successor principle—whether they target the successor principle directly or not.

Reid et al. (2013) compared an intervention that semi‐directly targeted suc-cessor knowledge with an intervention that indirectly did so. The semi‐direct successor intervention consisted of what is described in Table 15.1 as a highly guided approach—that is, it entailed playing games structured to promote the discovery of the successor principle using semi‐explicit scaffolding (see Baroody, 2016, for details and illustrations). More specifically, one game required determining how many candles had to be added to a birthday cake if the candles present represented the birthday child’s current age. Another game involved two animals building a staircase to retrieve a prize and determining how many additional blocks were needed for the next step (e.g., making a 5‐step requires how many more blocks than a 4‐step?). Yet another game entailed predicting how many more branches a hungry monkey needed to swing to arrive at the branch with a banana (e.g., how many more branches must the monkey swing to get from 4 to 5?). Importantly, a game also targeted a key developmental prerequisite for the successor principle (Level 7 in Table  15.2), namely fluency with number‐after relations such as what is the number after 3 (Level 5 in Table 15.2)? The indirect intervention involved what is described in Table  15.1 as a minimally guided approach—that is, entailed playing a game somewhat structured to promote the discovery of the successor principle and only implicitly underscored this concept. Specifically, The Great Race is a linear board game that requires rolling a 1 or 2 die and moving a race car by counting‐on (e.g., rolling a 1 and moving a car 1 space from 4 by count-ing‐on: “4, 5”; Ramani & Siegler, 2008, 2011). The direct successor intervention was significantly more effective. Further research is needed to evaluate, for instance, the importance of the hypothesized successor prerequisites listed in Table  15.2 and the degree of guidance and what components are needed to most efficaciously improve successor knowledge.

Fluency with Basic Sums and DifferencesA series of training experiments revealed that the degree of guidance needed to promote learning of a reasoning strategy or combination fluency varies among combination families. Minimally guided discovery (unstructured practice, enter-ing a sum on a number list, and feedback about correctness only) was as effective in promoting learning and fluent application of add‐1 rule (the sum of 7 and 1 is the number after 7) as was highly guided discovery (Baroody, Eiland, Purpura, & Reid, 2013; Baroody, Purpura, Eiland, & Reid, 2015; Purpura, Baroody, Eiland, & Reid, 2016). However, only highly guided discovery effectively taught the near‐doubles (e.g., 3 + 4 = 3 + 3 + 1 = 6 + 1 = 7) and subtraction‐as‐addition (e.g., 7 – 4 =? can be thought of as 4 +? = 7) reasoning strategies (Baroody, Purpura, Eiland, & Reid, 2014; Baroody et al., 2016).

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Highly guided instruction may not have been needed in the case of the add‐1 rule for several reasons: (a) The connection between adding with 1 and number‐after relations is relatively straightforward and thus salient. (b) Children typically have the prior knowledge needed to notice the connection and implement the add‐1 rule. As Kirschner et  al. (2006) noted, although guided instruction is generally more effective than unguided instruction, this advantage recedes, “when learners have sufficiently high prior knowledge to provide ‘internal’ guid-ance” (p. 75). (c) Entering a sum of an add‐with‐1 combination on a number list may underscore, if only implicitly, the connection between the relatively novel experience of adding 1 and the relatively familiar knowledge of number‐after relations.

Summary

Even for a novice in a domain, the degree of guidance need for meaningful learn-ing depends on such intertwined factors as a child’s existing knowledge, how readily a new relation can be assimilated to this knowledge, and the salience of new relations. For example, for children just learning a counting‐based strategy to add or for those who use a basic concrete strategy, fully guided, or direct instruction in the form of modeling concrete counting‐on may well be helpful in learning the procedure, because no new conceptual understanding is required. However, such an approach is not likely to be effective in teaching abstract count-ing‐on, which requires new conceptual insight. With this more advanced form of counting‐on, discovery learning of the underlying conceptual basis of the strat-egy (e.g., the add‐1 rule) may be appropriate once a child has mastered the devel-opmental prerequisites that make the relations salient (i.e., number‐after relations and the successor principle; Levels 5 and 7, respectively, in Table 15.2).

Indeed, the degree of partial guidance children may need to construct different concepts may vary. For example, minimally guided discovery (The Great Race) seems to be less effective in promoting the successor principle than highly guided discovery (semi‐direct successor instruction; Reid et al., 2013). In contrast, mini-mally guided discovery learning has proved time and again to be as effective as highly guided discovery learning in promoting the relatively palpable add‐1 rule (Baroody et al., 2013, 2015; Purpura et al., 2016). In brief, there is not a simple answer to the question of how much guidance is needed for early childhood mathematics education. Depending on such external factors as the complexity of the new topic and internal factors such as the child’s developmental level, one of a wide variety of approaches may be appropriate.

Moreover, the quality of instruction (e.g., scaffolding provided) is a key factor and may be more important than the amount of guidance provided (Clements & Sarama, 2012). For example, although children frequently spontaneously invent abstract counting‐on as described previously by about 7 years of age, disadvan-taged children and children with learning difficulties often do not, and this can impede arithmetic progress (Baroody, 1988; Baroody, Berent, & Packman, 1982). Particularly with such children, teachers can play an important role in ensuring the timely development of this relatively efficient strategy. For example, they can ensure a child has mastered perquisite knowledge for counting‐on such as fluent

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knowledge of number‐after relations. A teacher can catalyze children’s invention by presenting a series of +1 problems to pairs of students. Once a child responds using the add‐1 rule (e.g., 5 + 1 is the number after five, which is six) instead of counting, a teacher might act amazed and say, “How did you do that so fast?” and have children share their strategy with their partner and other pairs. Making the strategy explicit can help solidify the strategy in the child’s mind and may prompt other students to adopt it. After children can confidently apply the add‐1 rule (e.g., 5 + 1 is the number after five, which is six), a teacher can present a +2 prob-lem, with a smile and a warning, “I’m going to give you a harder one—watch out!” to both alert and motivate children. Similarly, the highly guided discovery teach-ing of near‐doubles may be more effective if the teacher (a) ensures readiness (i.e., all doubles are known fluently), (b) has children work in pairs on solving doubles‐plus‐one problems, and (c) scaffolds discovery of the double‐plus‐one strategy by, for instance, presenting 5 + 5 and then 5 + 6. A teacher could also prompt reflection by asking if children noticed anything about such pairs of numbers and encouraging children to share their ideas.

Finally, even if partially guided discovery is only equally as effective as fully guided instruction in promoting learning of concepts and procedures (Matlen & Klahr, 2013), partial guidance engages children in the inquiry process and pro-vides cognitive and affective benefits that fully guidance does not. For example, actual problem‐solving experience would seem essential to fostering mathemati-cal problem solving, reasoning, and communicating skills, and a positive disposi-tion toward problem solving. For instance, fostering fluency with basic sums and differences provides numerous opportunities to discover patterns and relations (inductive reasoning), using these mathematical regularities to devise reasoning strategies (logical or deductive reasoning), and using reasoning strategies and other known knowledge to deduce unknown sums or differences (using logical reasoning to solve problems). See Baroody (2016) for a detailed discussion of how the meaningful memorization of basic sums and differences can serve to promote mathematical thinking and vice versa. Note that a discovery‐based approach to learning the basic sums and differences is more likely to be interest-ing to young children than nondiscovery approaches. Furthermore, unlike direct instruction or drill, fostering a disposition to look for patterns and relations and to use logical reasoning to solve problems reinforces a propensity to do so with new families of basic combinations and other aspects of mathematics.

Conclusions: Implications for Professional Development

Key to successfully implementing developmentally appropriate and effective mathematics education with young children is providing high‐quality interac-tions that provide “just enough” guidance at the right time. As this chapter sug-gests, this is not a simple matter. Teaching is a complex enterprise, and teaching mathematics is particularly complex. Further challenges confront teaching mathematics in preschool, as settings and organizational structures vary far

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more than do those at any other age level in the U.S. (National Research Council, 2009). The workforce in those settings, their backgrounds, and their professional education are similarly diverse. Research suggests that the most critical feature of a high‐quality educational environment is a knowledgeable and responsive adult and that high‐quality professional development is essential to innovation (e.g., National Research Council, 2009; Sarama & DiBiase, 2004; Schoen, Cebulla, Finn, & Fi, 2003). Therefore, professional development must meet challenges to raise the low level of mathematics content and pedagogical content knowledge of most preschool teachers (Sarama & DiBiase, 2004).

To do so, professional development must be based firmly on research, such as that presented in this chapter. Further, the professional development must be comprehensive, ongoing, intentional, reflective, goal‐oriented, focused on con-tent knowledge and students’ thinking, and grounded in research‐based peda-gogical strategies. Teachers need to be competent in all three components of learning trajectories for all major mathematical topics, as follows:

1) They must first understand the goal—the mathematical content at the level they teach as well as knowledge of how these foundational mathematical ideas connect to subsequent learning. High‐quality professional develop-ment provides teachers with broad and deep understandings of these ideas through active engagement in mathematics experiences and discoveries for themselves. The most important of the topics is the domain of number and the related concepts of quantity and relative quantity, counting, and arith-metical operations. Also important are the domains of geometry and meas-urement, through which people mentally structure the spaces and objects around them. Connections and coherence among mathematical ideas are enriched when teachers can apply number concepts and processes to these spatial structures.

2) Teachers must learn the learning trajectories’ developmental progressions. Effective teaching requires meeting children where they are on the progres-sion. Teachers must ask and answer these questions: Is this child or group of children on the learning trajectory as expected for their ages? If not, where are they on the trajectory? Where do they need to move next mathematically? Teachers need to group children by developmental level on a learning trajec-tory and, to do so, conduct careful formative assessment of both existing informal and formal mathematical knowledge. Teachers need to study videos of children at different level, and conduct their own interviews of children, noticing, comparing, and analyzing different student responses to tasks.

3) The third component of a learning trajectory is instructional tasks and strate-gies that utilize tools, tasks, and talk to support advancement in children’s mathematical understanding and skills from one developmental level to the next. As illustrated in this chapter, those strategies are fine‐tuned to tasks and levels of thinking. Teachers need to learn effective techniques, again by read-ing, studying classroom practice, and trying new approaches, with individual children, as well as small and large groups.

Simultaneously, high‐quality professional development must instill positive atti-tudes toward mathematics as a discipline and productive dispositions toward the

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teaching and learning of mathematics. Teachers must be committed to issues of equity and social justice and to the learning of all children.

To accomplish the previously outlined goals, effective professional develop-ment must be conducted by instructors who have early childhood experience with schools, teachers, and students and deep understanding of the mathematics content and the research and practice in this age range. The professional devel-opment they provide includes training outside of the classroom focused on and connected to classroom practice emphasizing all three components of learning trajectories. Importantly, it also is complemented by classroom‐based enact-ment with coaching and designed to encourage sharing with colleagues.

High‐quality professional development helps teachers learn a problem‐solving approach toward teaching (Baroody, 1987). In summary, to be an effective educational problem solver, teachers need to learn to understand the mathemat-ics they are teaching and for which they are preparing their students; the mathematical development of young children; and how to flexibly select or adapt instruction, including the use of meaningful analogies, manipulatives, and symbolic representations to an individual student’s needs.

Acknowledgments

Preparation of this manuscript was supported by a grant from the Institute of Education Science, U.S. Department of Education, through Grant R305A150243 (“Evaluating the Efficacy of Learning Trajectories in Early Mathematics”) and the National Science Foundation, through Grant 1621470 (“Development of the Electronic Test of Early Numeracy”). The opinions expressed are solely those of the authors and do not necessarily reflect the position, policy, or endorsement of the Institute of Education Science or the Department of Education or the National Science Foundation.

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