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As discussed further in Chapter 6 , early childhood classrooms typically are ill suited to helping children learn mathematics and underestimate their ability to do so. In some cases, children can even experience a regression on some mathematics skills during prekindergarten and kindergarten Farran et al. Mathematics needs to be conceptualized as more than skills, and its content as more than counting and simple shapes.

Without building a robust understanding of mathematics in the early years, children too often come to believe that math is a guessing game and a system of rules without reason Munn, Both education and experience can make a difference, as evidenced by data from the latest international Trends in International Mathematics and Science Study, which added data collection on early mathematics education Mullis et al. Students with higher mathematics achievement at fourth and sixth grades had parents who reported that they often engaged their children in early numeracy activities and that their children had attended preprimary education and started school able to do early numeracy tasks e.

Those children who had attended preschool or kindergarten had higher achievement, while the 13 percent who had attended no preprimary school had much lower average mathematics achievement Mullis et al. Children move through a developmental progression in specific mathematical domains, which informs learning trajectories as important tools for supporting learning and teaching.

Box illustrates the concept of a developmental progression through the example of subitizing , an oft-neglected mathematical goal for young children. Research shows that subitizing, the rapid and accurate recognition of the number in a small group, is one of the main abilities very young children should develop Palmer and Baroody, ; Reigosa-Crespo et al. Through subitizing, children can discover critical properties of number, such as conservation and compensation Clements and Sarama, ; Maclellan, and develop such capabilities as unitizing and arithmetic.

Subitizing is not the only way children think and learn about number. Counting is the other method of quantification.

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It is the first and most basic mathematical algorithm and one of the more critical early mathematics competencies Aunola et al. Chapter 6 includes examples from a complete learning trajectory—goal, developmental progression, and instructional activities—for counting Clements and Sarama, Subitizing: A Developmental Progression. For example, very young children possess approximate number systems ANSs that allow them to discriminate large and small sets, more Children with special needs in learning mathematics fall into two categories. Those with mathematical difficulties struggle to learn mathematics for any reason; this category may apply to as many as percent of students Berch and Mazzocco, Those with specific mathematics learning disabilities are more severe cases; these students have a memory or cognitive deficit that interferes with their ability to learn math Geary, This category may apply to about percent Berch and Mazzocco, ; Mazzocco and Myers, In one study, this classification persisted in third grade for 63 percent of those classified as having mathematics learning disabilities in kindergarten Mazzocco and Myers, Mathematics learning disabilities, while assumed to have a genetic basis, currently are defined by students' behaviors—yet with ongoing debate among experts about what those behaviors are.

One consistent finding is that students with mathematics learning disabilities have difficulty retrieving basic arithmetic facts quickly. This has been hypothesized to be the result of an inability to store or retrieve facts and impairments in visual-spatial representation.

As early as kindergarten, limited working memory and speed of cognitive processing may be problems for these children Geary et al. Many young children with learning disabilities in reading show a similar rapid-naming deficit for letters and words Siegel and Mazabel, ; Steacy et al.

Another possibility is that a lack of higher-order, or executive, control of verbal material causes difficulty learning basic arithmetic facts or combinations. For example, students with mathematics learning disabilities may have difficulty inhibiting irrelevant associations. One explanation for the difficulty students with mathematics learning disabilities have learning basic arithmetic combinations might be delays in understanding counting.

These students may not fully understand counting nor recognize errors in counting as late as second grade. Other experts, however, claim that a lack of specific competencies, such as subitizing, is more important Berch and Mazzocco, Some evidence suggests that it is possible to predict which kindergartners are at risk for mathematics learning disabilities based on skill including reading numerals, number constancy, magnitude judgments of one-digit numbers, or mental addition of one-digit numbers Mazzocco and Thompson, However, until more is known, students should be classified as having mathematics learning disabilities only with great caution and after good mathematics instruction has been provided.

Such labeling in the earliest years could do more harm than good Clements and Sarama, It can appear that language is less of a concern in mathematics compared to other subjects because it is assumed to be based on numbers or symbols, but this is not the case Clements et al. In fact, children learn math mainly from oral language, rather than from mathematical symbolism or textbooks Janzen, Vocabulary and knowledge of print are both predictors of later numeracy Purpura et al. Similarly, growth in mathematics from kindergarten to third grade is related to both early numerical skills and phonological processing Vukovic, In one study of linguistically and ethnically diverse children aged years, language ability predicted gains in geometry, probability, and data analysis but not in arithmetic or algebra controlling for reading ability, visual—spatial working memory, and gender Vukovic and Lesaux, Thus, language may affect how children make meaning of mathematics but not its complex arithmetic procedures.

Moreover, there is an important bidirectional relationship between learning in mathematics and language Sarama et al. Each has related developmental milestones. Children learn number words at the same time as other linguistic labels. Most children recognize by the age of 2 which words are for numbers and use them only in appropriate contexts Fuson, Each also has related developmental patterns, with learning progressing along similar paths. In both, children recognize the whole before its parts. In learning language, this is word before syllable, syllable before rime-onset, and rime-onset before phoneme see also Anthony et al.

Similarly in mathematics, numbers are first conceptualized as unbreakable categories and then later as composites e. By 6 years old in most cultures, children have been exposed to symbol representations that are both alphabetic and numerical, and they begin to be able to segment words into phonemes and numbers into singletons e. The ability to identify the component nature of words and numbers predicts the ability to read Adams, ; Stanovich and Siegel, and to compute Geary, , Furthermore, there appear to be shared competencies between the two subject areas.

For example, preschoolers' narrative abilities i. Beginning mathematics scores have been shown to be highly predictive of subsequent achievement in both reading and mathematics although beginning reading skills such as letter recognition, word identification, and word sounds were shown to be highly predictive of later reading advanced competencies such as evaluation but not mathematics learning Duncan et al. A causal relationship between rich mathematics learning and developing language and literacy skills is supported by a randomized study of the effects of a math curriculum called Building Blocks on prekindergarten children's letter recognition and oral language skills.

Building Blocks children performed the same as the children in the control group on letter recognition and on three oral language subscales but outperformed them on four subscales: ability to recall key words, use of complex utterances, willingness to reproduce narratives independently, and inference Sarama et al. These skills had no explicit relation to the math curriculum.

Similarly, a study of 5- to 7-year-olds showed that an early mathematics and logical-mathematical intervention increased later scores in English by 14 percentile points Shayer and Adhami, Time on task or time on instruction does affect learning, which naturally leads to consideration of potential conflicts or tradeoffs between time spent on different subjects e. Indeed, a frequent concern is that introducing a mathematics curriculum may decrease the time devoted to language and literacy, impeding children's development in those areas, which are heavily emphasized in early learning goals see Clements and Sarama, ; Farran et al.

However, this assumes that mathematics activities will not have a positive effect on language and literacy. Yet as described here, evidence from both educational and psychological research suggests the potential for high-quality instruction in each to have mutual benefits for learning in both subjects. Rich mathematical activities, such as discussing multiple solutions and solving narrative story problems, can help lay the groundwork for literacy through language development, while rich literacy activities can help lay the groundwork for mathematics development Sarama et al.

For mathematics learning in children who are dual language learners, the language, not just the vocabulary, of mathematics need to be addressed Clements and Sarama, Challenges for dual language learners include both technical vocabulary, which can range in how similar or distinct terms are from everyday language, and the use of complex noun phrases.

On the other hand, bilingual children often can understand a mathematical idea more readily because, after using different terms for it in different languages, they comprehend that the mathematical idea is abstract, and not tied to a specific term see Secada, At a minimum, their teachers need to connect everyday language with the language of math Janzen, Instructional practices for teaching mathematics with dual language learners are discussed further in Chapter 6. For subject-matter content knowledge and proficiency, children learn best when supported along a trajectory with three components: 1 their understanding of the subject-matter content itself, 2 their progress through predictable developmental levels and patterns of thinking related to their understanding of the content, and 3 instructional tasks and strategies that adults who work with children can employ to promote that learning at each level.

For example:. Some principles of how children learn along a trajectory hold across subject-matter domains, but there are also substantive differences among subjects in the specific skills children need and in the learning trajectories. Both generalizable principles and subject-specific distinctions have implications for the knowledge and competencies needed to work with children. An important factor in children's learning of subject-matter content is how each of the components of learning trajectories both requires and develops aspects of learning that are not content specific, such as critical reasoning, executive function, self-regulation, learning skills, positive dispositions toward learning, and relationships.

Educators, developmental scientists, and economists have long known that academic achievement is a result of both the growth of specific knowledge and the development of general learning competencies that regulate how children enlist cognitive resources when they encounter learning challenges, motivate advances in learning, and strengthen children's self-confidence as learners. These general learning competencies have been labeled and categorized in various ways.

Individual differences in these competencies are important determinants of learning and academic motivation, and children's experiences at home and in the classroom contribute to some of these differences. This section examines these competencies as well as their interrelationships with the previously discussed subject-matter domains of language and literacy and mathematics. Several cognitive control processes are important for planning and executing goal-directed activity, which is needed for successful learning e.

These processes include, for example, short-term and working memory, attention control and shifting, cognitive flexibility changing thinking between different concepts and thinking about multiple concepts simultaneously , inhibitory control suppressing unproductive responses or strategies , and cognitive self-regulation. These processes also are closely related to emotion regulation, which is discussed later in the section on socioemotional development, and which also contributes to children's classroom success. Other theoretical frameworks exist as well.

As with the overall domains of development displayed earlier in Figure , the committee did not attempt to reconcile those different perspectives. This variation in perspectives makes it difficult to parse the literature produced by different fields of research and practice. In general, however, executive function appears to improve most rapidly in young children Best et al. Executive function processes appear to be partially dependent on the development of the prefrontal cortex the site of higher-order cognitive processes , notably through the preschool and kindergarten age range Bassett et al.

Short-term memory is the ability for short-term recall, such as of a sentence or important details from conversation and reading.

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Working memory allows children to hold in their memory information from multiple sources, whether heard or read, so they can use and link that information. Updating working memory is the ability to keep and use relevant information while engaging in another cognitively demanding task Conway et al. Attention control is the ability to focus attention and disregard distracting stimuli e.

Attention shifting and cognitive flexibility are often grouped.

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Cognitive flexibility capacities develop gradually throughout early childhood and have significant influences on children's social and academic competence. Cognitive flexibility is important, for example, for reading Duke and Block, Children who are better able to consider, at the same time, both letter-sound and semantic meaning information about words have better reading comprehension Cartwright, ; Cartwright et al.

Reading comprehension also appears to improve when children are taught about words with multiple meanings e. In addition, interventions in young children that focus on cognitive flexibility have shown significant benefits for reading comprehension Cartwright, Inhibitory control involves controlling a dominant response e. The skill of simple response inhibition withholding an initial, sometimes impulsive, response develops during infancy through toddlerhood. Later in their first year, children can resolve conflict between their line of sight and their line of reaching Diamond, By about 30 months, they can successfully complete a spatial conflict task Rothbart and Rueda, From 3 to 5 years of age, complex response inhibition and response shifting develop, with attention shifting developing at about age 4 Bassett et al.

The most rapid increase in inhibitory control is between 5 and 8 years of age, although moderate improvements are seen up to young adulthood Best et al. Inhibitory control supports children's learning across subject-matter areas. As one example of its importance for mathematics, when the initial reading of a problem is not the correct one, children need to inhibit their impulse to answer incorrectly and carefully examine the problem.

Three birds already flew away. How many birds were there from the start? Cognitive self-regulation is what helps children plan ahead, focus attention, and remember past experiences. The construct of self-regulation and related concepts have a long history in psychology e.

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Most recently, researchers and educators have used the broad term self-regulation to refer to the processes involved in intentionally controlling attention, thinking, impulses, emotions, and behavior. In this way, self-regulation can be thought of in relation to several aspects of development, including the cognitive processes discussed here and the social and emotional processes discussed later in this chapter.

Developmental psychobiological research and neuroimaging indicate that these subclasses are both neurally and behaviorally distinct while also being related and correlated Bassett et al. Together, these types of self-regulation allow children to persevere with tasks even when facing difficulties in problem solving or learning, fatigue, distraction, or decreased motivation Blair and Razza, ; Neuenschwander et al. It is thus unsurprising that kindergarten teachers believe self-regulation is as important as academics Bassok and Rorem, Both cognitive self-regulation and emotional self-regulation discussed later in this chapter contribute to socioemotional development and also play a role in learning.

Although the relationship between various features of cognitive self-regulation and academic achievement has been well documented for older students e. Children's self-regulation and their ability to successfully function in school settings are related in two ways. First, emotional self-regulation enables children to benefit from learning in various social contexts, including their capacities to manage emotions in interactions with educators as well as peers e.

It also assists them in conforming to classroom rules and routines. Second, cognitive self-regulation enables children to develop and make use of cognitive processes that are necessary for academic learning Anghel, Although most studies have focused on specific effects of either cognitive or emotional self-regulation, evidence suggests that the two are interconnected. This link is probably due to the commonality of the neurological mechanisms governing both emotional and cognitive self-regulation.

For example, children lacking emotion regulation are likely also to have problems with regulating cognitive processes, such as attention Derryberry and Reed, ; LeDoux, Moreover, earlier patterns in the development of emotion control have been shown to be predictive of children's later ability to exercise control over their cognitive functioning Blair, Several studies have shown positive correlations between self-regulation and achievement in young children e.

Preschoolers' cognitive self-regulation, including inhibitory control and attention shifting, were found to be related to measures of literacy and mathematics ability in kindergarten Blair and Razza, In another study, children with higher self-regulation, including attention, working memory, and inhibitory control, achieved at higher levels in literacy, language, and mathematics McClelland et al. Interventions in the area of self-regulation have shown positive effects for reading achievement Best et al. Among struggling first graders in an effective reading intervention, those who were retained in grade showed significantly weaker self-regulation skills Dombek and Connor, Cognitive self-regulation appears to be strongly associated with academic learning Willoughby et al.

In addition, both cognitive and emotional self-regulation contribute to variance in attention, competence motivation, and persistence Bassett et al. In addition, differences in self-regulation competencies raise important issues related to disparities in educational achievement. Children in poverty can have lower self-regulation competencies e.

One reason is the effect of chronic stress on behavioral and biological capacities for self-control see discussion of chronic stress and adversity later in this chapter. This risk is exacerbated for children who are also dual language learners Wanless et al. Students with special needs are another population who may require focused interventions to develop self-regulation competencies Harris et al. Students who are gifted and talented may also have exceptional needs in this domain e. Adults who work with children have the opportunity to provide environments, experiences, and curricula that can help develop the competencies needed, including for children whose skills were not optimally developed in the earliest years.

Rather, effective educators and programs provide learning activities and environments that increase children's capacity and disposition to set a goal e. The science of how children develop and learn indicates that integrating academic learning and self-regulation is a sound approach. As already noted and shown in several examples, executive function processes are closely related to achievement in both language and literacy and mathematics Best et al. In some research, executive function has been correlated similarly with both reading and mathematics achievement across a wide age span 5 to 17 years , suggesting its significant role in academic learning Best et al.

In contrast, some studies have found that executive function is more strongly associated with mathematics than with literacy or language Barata, ; Blair et al. A strong relationship between executive function and mathematics may reflect that mathematics relies heavily on working memory and attention control, requiring the ability to inhibit an automatic response to a single aspect of a problem, to hold relevant information in mind, and to operate on it while shifting attention appropriately among different elements of a problem Welsh et al.

This relationship is especially important given that mathematics curricula increasingly require higher-order skills, which executive function competencies provides Baker et al. Some research indicates that most executive function competencies correlate significantly with mathematics achievement Bull and Scerif, , while other studies suggest a greater role for particular executive function competencies in the learning of mathematics for young children—especially inhibitory control Blair and Razza, or working memory Bull et al.

Neuenschwander et al. These latter two competencies have been shown to predict success in mathematics in primary school students Toll et al. Working memory tasks have also been shown to predict mathematics learning disabilities, even more so than early mathematical abilities Toll et al. Several studies have identified lack of inhibition and working memory as specific deficits for children of lower mathematical ability, resulting in difficulty with switching to and evaluating new strategies for dealing with a particular task Bull and Scerif [] and Lan and colleagues [] found similar results.

Persistence, another learning skill that is interrelated with cognitive processes, also has been linked to mathematics achievement for both 3- and 4-year-olds Maier and Greenfield, Executive function competencies may be differentially associated with distinct areas of mathematics. For example, executive function was found to be correlated more with solving word problems than with calculation Best et al. Different aspects of working memory also may be related to different mathematical areas Simmons et al.

Parallel observations have been made for executive function and reading, with executive function playing a larger role in reading comprehension than in decoding. In addition to the role of executive function in learning mathematics, mathematics activities also contribute to developing executive function. Some mathematics activities may require children to suppress prepotent responses, manipulate abstract information, and remain cognitively flexible.

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Importantly, neuroimaging studies suggest that executive function may be developed through learning mathematics in challenging activities but not in exercising mathematics once learned Ansari et al. Some students with special needs may have a specific lack of certain executive function competencies Harris et al. Most of the research on executive function deficits in relation to disabilities that affect young children has focused on specific disorders, particularly attention deficit hyperactivity disorder ADHD.

An early theory posited that ADHD is a lack of the behavioral inhibition required for proficiency with executive functions such as self-regulation of affect, motivation, and arousal; working memory; and synthesis analysis of internally represented information Barkley, Shuai et al. A meta-analysis of studies of one measure of executive function, the Wisconsin Card Sorting Test, suggests that the performance of individuals with ADHD is fairly consistently poorer than that of individuals without clinical diagnoses Romine et al.

In another study, children with ADHD were found not to have learning problems but rather problems in a measure of inhibitory control, which affected arithmetic calculation as well as written language Semrud-Clikeman, Other evidence suggests that children diagnosed with ADHD may have deficits not in executive processes themselves but in motivation or response to contingencies, that is, the regulation of effort allocation Huang-Pollock et al.

Having ADHD with deficits in executive function, compared to ADHD alone, is associated with an increased risk for grade retention and a decrease in academic achievement Biederman et al. The relationship between ADHD and executive functions may also depend on subtype. One study found that children with an inattention ADHD subtype showed deficits in several executive function competencies Tymms and Merrell, , whereas children with the hyperactive-impulsive ADHD subtype may have fewer executive function deficits Shuai et al.

Deficits in executive function have been studied in other developmental disorders as well, albeit often in less detail. Romine and Reynolds, ; and mathematics learning disabilities Toll et al. Other learning skills that are important to early academic achievement include persistence, curiosity, self-confidence, intrinsic motivation, time perspective e.

The growth of emotional and cognitive self-regulation is also fundamentally related to many of these developing learning skills. In addition, social experiences, discussed later in this chapter, are important for the growth of these learning skills. Note also that although these skills are referred to sometimes as dispositions, they are fostered through early experience and can be supported through intentional caregiving and instructional practices; they are not simply intrinsic traits in the child.

A capacity for focused engagement in learning is apparent from very early in life, although it is also true that these learning competencies develop significantly throughout early childhood as processes of neurobiological development interact with children's social experiences to enable greater persistence, focused attention, delayed gratification, and other components of effective learning and problem solving. As a consequence, very young children are likely to approach new learning situations with enthusiasm and self-confidence but at young ages may not necessarily bring persistence or creativity in confronting and solving challenging problems.

Older preschoolers, by contrast, are more self-regulated learners. They approach new learning opportunities with initiative and involvement, and they are more persistent and more likely to solve problems creatively, by proposing their own ideas NRC, Considerable research confirms the importance of these skills to early learning. In preschool-age children, learning skills that include motivation, engagement, and interest in learning activities have been found in longitudinal studies to predict children's cognitive skills at school entry Duncan et al.

Similarly, these characteristics continue to be associated with reading and mathematics achievement in the early elementary grades Alexander et al.

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Differences in these learning skills are especially associated with academic achievement for children in circumstances of economic disadvantage who face various kinds of self-regulatory challenges Blair and Raver, ; Howse et al. Much of school success requires that children prioritize longer-term rewards requiring current effort over immediate satisfactions. The classic demonstration of this skill comes from a series of studies led by Walter Mischel beginning in the s. Young children were offered the option of choosing an immediate, smaller reward or a larger reward if they waited to receive it later.

For several years developmental outcomes for these children were tracked, which revealed that children who were better able to delay gratification at age 4 scored higher on measures of language skills, academic achievement, planful behaviors, self-reliance, capacity to cope with stress and frustration, and social competence measured in adolescence and adulthood Mischel et al.

Other studies have reported consistent findings. Early development in the ability to prioritize future, long-term goals over short-term lesser gains improves children's chances of academic achievement and securing and maintaining employment Rachlin, Conversely, the inability to delay gratification is associated with young children's aggressive behavior, conduct problems, poorer peer relationships, and academic difficulty during preschool and the transition to elementary school Olson and Hoza, as well as later outcomes, including academic failure, delinquency, and substance abuse in adolescence Lynam et al.

The ways that children view themselves as learners are also important. Young children's self-perceived capability to master learning challenges develops early and exerts a continuing influence on their academic success. Early self-evaluations of competence are based on the positive and negative evaluations of children's behavior and competence by parents Stipek et al. Parent and educator expectations for children's success remain important.

High parent expectations for children's school achievement are associated with children's later academic performance, and this is also true of educator expectations. In one longitudinal study, teacher expectations for children's math achievement in grades 1 and 3 directly predicted children's scores on standardized achievement tests 2 years later, and expectations for reading achievement had indirect associations with later reading scores.

There was also evidence in this study that expectations were especially influential for academically at-risk students Hinnant et al. Messages from parents and educators are also important in shaping how children attribute their own success and failure which, in turn, predicts their future effort and expectations of success.

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