M3 WA#1: Venn Diagram

Cognitive Development in Early Childhood Ch.7

7-1Piaget’s Cognitive Developmental Theory in Early Childhood

We asked him why a boat floats on the water whilst a little stone, which is lighter, sinks immediately. Vern reflected and then said: “The boat is more intelligent than the stone—What does ‘to be intelligent’ mean?—It doesn’t do things it ought not to do.” (Piaget, 1929/1963, p. 223)

This is an interview between Swiss researcher Jean Piaget (1896–1980) and a 6-year-old boy who believed that boats have the will to avoid doing things that they ought not to do. This is an example of animism, which is characteristic of Piaget’s second stage of cognitive development, the preoperational stage. Recall from  Chapter 3 that Piaget was a constructivist who believed that children construct their own understanding of the world by assimilating or accommodating experience into their mental schemes of the world. The tug-of-war between assimilation and accommodation in order to achieve cognitive equilibrium results in cognitive development. This process leads toddlers to progress from the sensorimotor stage to the preoperational stage in early childhood. Let’s explore this stage, with its charming misconceptions.

7-1aThe Preoperational Stage (about Ages 2 through 7)

Operations are mental actions that follow rules. You could think of an operation as an act of logic. The term  preoperational refers to Piaget’s view that children in this stage are not yet capable of thinking logically. He believed they have other limitations as well, like inability to think about how an object might look from different points of view, which requires mental manipulation. He believed they do not fully understand cause-and-effect sequences (Desrochers, 2008). According to Piaget, other cognitive deficits include those discussed next.

Animism

Animism is the belief that nonliving, inanimate objects have lifelike qualities. For example, children may think that the sun follows them, that boats are intelligent, that a single flower might get lonely, and that things that move, such as a flickering flame, are alive. They believe that objects or natural phenomena (rain, wind, snow) have intention, including intent to harm. For example, they might cry because they think that a leaf blowing toward them is chasing them.

Lack of Hierarchical Classification

Hierarchical classification means that things can be members of multiple levels of categories at the same time. For example, a collie is a dog and a mammal at the same time. Preoperational children have trouble classifying objects in a hierarchy. They have trouble understanding that all collies are dogs, but not all dogs are collies. We heard a young child correct someone who referred to her mother as a woman by saying, “That’s no woman—that’s my mommy!” which indicated trouble with hierarchical classification.

How did Piaget study children’s trouble with hierarchical classification? He used  class inclusion tests (Piaget & Inhelder, 1964). For example, he presented children with two colors of beads—10 red and 5 blue. He asked, “Are there more red beads or more beads?” Preoperational children tend to answer “more red beads” even though there are clearly more beads. Piaget argued that this is because the child cannot at the same time think of the whole class of beads and the parts (red and blue) that compose it, which is part of a general deficiency in mental flexibility.

Egocentrism

Young children are  egocentric , which is the tendency to see the world from their own point of view and to assume that other people do too (Piaget, 1926/1959, p. 9). Have you seen young children on the phone silently nod their head to say yes, when the person on the other end of the line cannot see the nod? Have you seen them close their eyes in order to hide from you? Preschoolers also show their egocentrism when they hold  collective monologues in which they speak with another child and even take turns talking so they appear to be conversing, but neither is listening to the other.

One way that Piaget demonstrated children’s egocentrism was with the “three-mountain task” (see  Figure 7.1). A child is presented with a three-dimensional model of three mountains that are distinctive because they have different objects at each peak: snow, a cross, or a house. The child is given 10 pictures of different views of the three mountains. A doll is placed at various points around the model, and the child is asked to pick the picture that shows what the  doll sees. Children in the preoperational stage tend to choose the picture that depicts what the  child sees, not what the doll sees.

Figure 7.1Piaget’s Three-Mountain Task

When preoperational children are asked to choose photos depicting the doll’s view, they tend to choose photos of their own view.

Lack of Conservation

Conservation refers to the fact that the properties of objects such as mass, volume, and number do not change just because the objects’ appearance changes. For example, if you have a ball of clay, the amount of clay in the ball does not change if you smash it down it into a pancake shape, though the appearance changes (see  Figure 7.2). Preoperational children might say that the smashed lump of clay has more mass than the ball. This is because they  center on the superficial attributes of objects. They are unable to  decenter and simultaneously consider that the smashed clay covers more surface area but is thinner than it was. You can see this when you cut an apple in half for one child, but not the other. Listen to this conversation between two brothers:

Figure 7.2Piagetian Conservation Tasks

Examples of different types of conservation tasks. Notice that children master different types at different ages.

5-year-old:

Ha, ha, I got more than you!

11-year-old:

You did not. They are the same.

5-year-old:

Nuh-uh. I got two and you only got one.

11-year-old:

It doesn’t matter. They are still the same size.

5-year-old:

They are not! I have two, so that is bigger!

11-year-old:

Look, if you put your two pieces together, they are the same size as mine.

5-year-old:

Oh, I see. But I still have more!

Piaget is probably most famous for his conservation tasks, even though they constituted a small part of his work. They are popular because they illustrate the apparent illogic of young children and are fun to replicate. The best-known task involves two identical glasses and a third taller, skinnier glass. You pour liquid carefully into the identical glasses until the child agrees that they have the same amount. You then pour the liquid from one glass into the glass that is taller and ask, “Do they still contain the same amount?” Preoperational children will say that there is more liquid in the tall, skinny glass, even though no liquid has been added or subtracted. A preschool teacher did this task with her class:

Ellery:

The tall glass has more water. It is bigger. The other one is short.

Teacher:

Does everyone agree with Ellery? (All the children agree) Why do you think this one has more?

Ong:

Because it is taller.

The children center on the height of the liquid in the glass and cannot consider the height and width of the container at the same time. In addition, the children cannot  reverse operations —that is, mentally consider that if the liquid were poured back into the glass from whence it came, the height would be the same as before.  Figure 7.2 provides examples of conservation tasks and the ages at which they are typically mastered.

Think about This

Do any of the Piagetian tasks seem like trick questions to you? Might they to a child? Explain.

Additional tasks that separate preoperational thinkers from concrete operational thinkers are the bottle- and mountain-drawing tasks (Piaget & Inhelder, 1956). In the bottle-drawing task, the child is presented with a picture of a bottle with liquid at the bottom and three other bottles that are tilted or inverted. The child is asked to draw where the water would be in the next three bottles. In the mountain-drawing task, children draw a mountain with trees or people or houses.  Figures 7.3 and  7.4 depict children’s responses while in the preoperational stage.

Figure 7.3Bottle Drawing Task

A child is presented with row A and asked, “Look at the water in the first bottle. Draw where the water would go in the next three bottles.” Row B depicts drawing by a preoperational child.

Figure 7.4Drawings of a House and Mountain

When preoperational children draw a house or mountain, they tend to place the chimney and trees perpendicular to the surface instead of vertical. You may notice this in their art.

7-1bAdvances in Knowledge since Piaget

Piaget’s research was brilliant and groundbreaking. However, researchers have continued to move forward. Today, Piaget’s stages are not widely accepted among researchers. Some researchers also challenge his notion that knowledge begins with experience through the five senses, which is then stored as concrete knowledge and later becomes the basis for abstract concepts (Uttal, Liu, & DeLoache, 2006). Children often acquire abstract concepts before the concrete examples of them. For example, young children apply abstract grammar rules to language when they say  foots instead of feet, even though they have never heard adults say  foots.

Underestimation of Abilities

Recent research suggests that Piaget  underestimated the cognitive abilities of young children. In  Chapter 3, you learned that infants know many surprising things about their world—scientists call this  core knowledge. In addition, in your interactions with toddlers, you will see evidence of non-egocentrism, or an ability to anticipate others’ perspectives. For example, an 18 month-old hid under the kitchen table while stuffing herself with cookies from the cookie jar. Why hide if you cannot anticipate that your dad will scold you? Similarly, young children often show good reasoning ability, such as this 3-year-old who wants crackers:

Mother:

They’re all gone.

Boy:

No they aren’t. I want some!

Mother:

Yes they are. What makes you think they aren’t?

Boy:

’Cause if they was all gone, I seed them [the empty package] in the garbage. Look, nothing in the garbage!

In the 1970s, researchers began to demonstrate that with modest changes in the tasks used by Piaget, young children could reverse, decenter, use logic, and classify hierarchically (Donaldson, 1978; Gelman & Baillargeon, 1983). For example, the “Sleeping Cows Task” is a modified test of class inclusion. In this task, some black and white toy cows are placed on their sides as though they are sleeping. The standard Piagetian approach would be to ask the child, “Are there more black cows or more cows?” (like the beads question presented earlier: “Are there more beads or more red beads?”) When asked in this way, only 25% of 6-year-olds respond correctly. If one word is added—“Are there more black cows or more  sleeping cows?”—then 48% of 6-year-olds respond correctly. Only one word is changed, but responses are substantially different. Another example is the “Police Task,” which tests egocentrism. A child is asked to hide a child doll from police dolls. Partitions are set up so that the child must consider two different points of view simultaneously (see  Figure 7.5). Most (90%) 3-year-olds can do this task correctly, whereas few can do the three-mountain task, even though both tasks require the child to consider what another person would see.

Figure 7.5Hide-from-Police-Officer Task

A child is given a doll and asked to hide it so that neither police officer can see it. About 90% of 3- to 5-year-olds get the task correct, even though results of the three-mountain task would predict a much lower proportion.

Source: Donaldson (1978).

Thus, current evidence suggests that there is less difference between the logic of children and adults and that young children are not as egocentric or illogical as Piaget believed, though there continues to be some controversy (Kagan, 2008). However, it is also true that if you carry out Piagetian tasks in the same way that Piaget did, you will tend to get the same responses. This suggests that young children do have cognitive limitations. Their abilities are fragile. Their knowledge is often implicit, meaning the child cannot reflect on and discuss the knowledge. In addition, young children do not give primacy to language, which may cause them to fail at Piagetian tasks.

Primacy of Language

When children begin to learn language, they pay more attention to context than to words. In fact, they often learn words through clues from the context. The child first makes sense of the context and then uses that to make sense of what is said. For example, a 2-year-old is taking a bath “by herself” with instruction from her mother:

Mother:

Take the washcloth and wash your mouth.

(Child does.)

Mother:

Good! Now wash your eyes.

(Child does.)

Mother:

Good! Now wash your ears (while unconsciously scratching her nose).

(The child washes her nose.)

When there was conflict between what her mother  said and what she  did, the child gave primacy to the mother’s action. The language-learning child gradually moves from understanding words in conjunction with action to understanding words in isolation. Children must have a great deal of experience with language before they can give primacy to language—that is, more weight to the meaning of words than to the meaning of the context. Children are not able to give primacy to language until they are confident about their understanding of language and have sufficient experience to know when to give primacy to language and when not to.

Many Piagetian tasks require children to give primacy to language because they require responses that conflict with contextual clues. For example, the class inclusion task “Are there more beads or more red beads?” is likely interpreted by the child as “Are there more  blue beads or more red beads?” because that is what the context suggests, even though that is not literally what is asked.

Today’s research does not support many of Piaget’s assumptions about the meaning of his research. For example, young children understand more about the world than he concluded. If Piaget was wrong about some things, why study him? For two reasons:

1. His theory is a good starting point because it caused scientists to try to understand why children respond so oddly to Piaget’s tasks. Science evolves through our quest for understanding. As a result of newer research that tests Piaget’s theory, we now understand more about children’s cognitive development.

2. Piaget’s theory continues to be widely applied in education. Let’s look at these applications next.

7-1cClassroom Implications of Cognitive Developmental Theory

There are several school-based legacies from Piaget’s theory. One legacy is the notion of school readiness (see  Chapter 1).

School Readiness

Piaget believed that little can be done to accelerate development because each child has a biologically based rate of transition from one stage to the next. In fact, Piaget was annoyed with teachers who wanted to speed up cognitive development. Thus, some educators have claimed that learners at young ages are not ready to study algebra (which requires understanding confusing variables) or the scientific method (which requires thinking logically and systematically) because the learners are not in the appropriate stage. In contrast to Piaget’s view, current research does not support stage-based readiness for school, nor does it support the notion that some concepts should not be taught to young children because they are not yet at the appropriate stage (APA, 2015). Some researchers believe that a strong readiness stance can deprive children of valuable experiences. Unfortunately, Piaget’s theory is sometimes mistakenly used as a rationale for not teaching children valued content, like history, because the children are supposedly not ready to learn it (Hinde & Perry, 2007).

Developmentally Appropriate Practice

Another concept that was influenced by Piaget’s theory is  developmentally appropriate practice (DAP), which is an approach to educating children from birth to age 8 that emphasizes the child as an active participant in learning, not just a passive receiver of knowledge. The teacher’s role is to create an environment in which the child can construct meaning from interactions with people and objects, rather than telling the child information. Young children learn through active exploration and play. A DAP position statement published by the National Association for the Education of Young Children (NAEYC, 2009) makes the following points:

1. Know children well, including their significant adults, so that you can scaffold their learning. Help them attain goals that are challenging but achievable.

2. Make instruction appropriate to children’s age and developmental level, individualized to them, and fitting their social and cultural background.

3. Base practices on research about how children learn and develop, not on unfounded assumptions.

4. Use practices that are likely to reduce the achievement gap.

Is DAP actually beneficial to children? Little research has directly addressed this question. One study of more than 3,000 1st-, 2nd-, and 3rd-graders found that classrooms that were more developmentally appropriate did not foster greater academic achievement in diverse, low-income learners than traditional classrooms (Van Horn & Ramey, 2003). Although children may not do better academically in DAP classrooms, they may feel less stress and anxiety, be more physically active, and be more likely to be on-task, all of which are important (Alford, Rollins, Padrón, & Waxman, 2015; Van Horn, Karlin, Ramey, Aldridge, & Snyder, 2005).

Piaget’s theory was too lean in three ways:

· (1)

He was vague about how children transition from one stage to another;

· (2)

he was silent on the issue of individual differences in cognitive abilities;

· (3)

he commented on the importance of sociocultural influences on learning, but this was not his focus.

Sociocultural influences were the focus of Vygotsky, who was also a constructivist.

7-2Vygotsky’s Sociocultural Theory in Early Childhood

One morning, a preschool class is practicing counting objects. Ms. Kelly is working with Darius. Darius can count aloud to 10, but cannot count objects. He points at the first block and says, “One.” He points at the second and third block as he says “two” and “three,” but then he points at the fourth block and says “seven.” Ms. Kelly models pointing at each block as she counts 10 blocks. Darius does not see any difference between what he did and what Ms. Kelly did. Ms. Kelly gently holds Darius’s hand and helps him count to 10 as they point to each of the 10 objects in order. After several days, he can competently count 10 objects while pointing at each one.

Darius began preschool without the cognitive ability to count objects while pointing at each one, but he developed the ability, although still fragile, after several days. This newfound ability was a result of interaction with Ms. Kelly. Vygotsky believed that social interaction with others is the primary force driving cognitive development (Clarà, 2016).

Lev Semenovich Vygotsky (1896–1934) was born in Belorussia in Eastern Europe, lived through the Bolshevik Revolution of 1917, and later became a major intellectual in Russia. He died of tuberculosis at age 37. For political reasons, his works were largely banned after his death, but in the 1960s and 1970s, some of his works began to appear in English. Since that time, his views on children’s cognitive development have had a substantial impact on education.

Vygotsky’s theory of cognitive development has been labeled  sociocultural or  cultural-historical because of his focus on how social relationships, social interaction, historical context, and culture interact to promote cognitive development. Like Piaget, he believed that knowledge cannot be directly communicated from the teacher’s head to the learner’s head; such attempts result in “meaningless acquisition of words,” not understanding (Gredler, 2012). Although Vygotsky emphasized interaction with  adults to foster cognitive growth, most current sociocultural approaches also emphasize interaction with peers.

7-2aThe Role of Social Interaction

According to Vygotsky, children grow into the intellectual life of those around them. Vygotsky wrote, “Every function in the child’s cultural development appears twice: first, on the social level, and later, on the individual level; first between people (interpsychological), and then inside the child (intrapsychological)” (Vygotsky, 1978, p. 57). That is, social interaction with a more competent person in a shared activity drives cognitive growth. The more competent person and the child first  co-construct skills and understanding out of their interaction, which then is internalized by the child.

Whether a toddler is learning to count or an adolescent is learning to solve liquid dynamics problems, the learner may initially merely observe the expert. Next, the expert does most of the work, both cognitive and physical, while guiding the learner through the task. The learner may have the appearance of doing the task, but could make no progress without the expert’s aid. As the learner gains increased competence, the expert gives more and more of the responsibility to the learner, who grows in expertise. The expert’s support is reduced. The expert may still need to give hints and reminders for a time, until at last the learner can perform the task independently.

Darius’s teacher was gradually placing more of the responsibility for counting on him, as he was increasingly able to work independently. This sequence took several days. Some skills take substantially longer to develop. Learning to read, or to solve calculus problems, requires several months or years of working with more-competent others before developing proficiency. Whether you are teaching a skill within a single class period or across several years, the same mechanism is at work: a more competent individual scaffolds learners’ performance within their zone of proximal development.

7-2bZone of Proximal Development

The  zone of proximal development (ZPD) is the level of competence between what a learner can do alone and what he or she can do with assistance. Darius’s learning to count objects occurred in his zone of proximal development. Without Ms. Kelly’s assistance, he experienced failure. Yet, with a little help, he was successful. You can probably think of many activities, such as learning to walk or to drive, in which novices struggle alone but vastly improve their performance with a little assistance or scaffolding. With assistance, the learner’s performance reveals a level of development to come. Day-to-day interactions with more-competent others are the roots of higher mental functions. Teaching children in their zone of proximal development, which challenges them, promotes growth; teaching them skills they already know, or skills that are easy, does little to promote growth (see  Figure 7.6).

Figure 7.6Teaching in the Zone of Proximal Development

In a study of kindergarteners, children scored higher (+) on mathematical knowledge at the end of kindergarten if the content level on which their teachers spent the most time was just above the level they already knew when they entered kindergarten. If their teacher focused on content that was at their existing (or a lower) level, they regressed (−) during the year unless they had unusually strong math ability to begin with. If their teacher focused on content much above their current level, it didn’t seem to have an effect (ϕ). Unfortunately, the majority of teachers focused on level 1 in kindergarten, which 95% of kindergarteners already knew.

Source: Data from Engel, Claessens, and Finch (2013).

7-2cScaffolding

Think about This

If a child is easily attaining straight As in class, is he or she operating in the zone of proximal development? Explain your answer.

Scaffolding is support for learning and problem solving that comes from outside the learner (Wood, Bruner, & Ross, 1976); it usually includes social interaction with a more competent other but could also include textbooks or prompting from a computer. Scaffolding occurs when an expert helps a novice master new skills by breaking the skill into small units and guiding performance to a higher level. Scaffolding can occur in the emotional, physical, social, or cognitive realm. Ms. Kelly was providing scaffolding when she prompted Darrius to use the counting out loud skills that he already had. So is the parent who holds the back of the bicycle while a child learns to ride. So is the coach who reminds the angry athlete to count to 10 before talking to the referee. An analysis of 37 studies found that students learning science benefited from an inquiry approach that included teacher scaffolding (Furtak, Seidel, Iverson, & Briggs, 2012). An inquiry approach invites learners to ask questions, collect and analyze data, and develop explanations for patterns in the data. Just turning learners loose to discover for themselves does not work well. They need guidance (Fisher, Hirsh-Pasek, Newcombe, & Golinkoff, 2013; Shneidman & Woodward, 2016). From the sociocultural perspective, a teacher’s primary role is to scaffold children in their zone of proximal development. You do this largely through language.

7-2dLanguage and Private Speech

Language is one of the most important tools of any culture because it provides an extremely efficient means of learning. For example, in  Chapter 12 you will learn that when adults talk about emotions with children, the children become better at perceiving others’ emotions. Below, you also learn that when adults talk about an event, the children remember the event better. That is, perceptions, memory, and reasoning are enhanced by talk.

According to Vygotsky, language first arises as a social/cultural tool a child uses to communicate with others. Subsequently, as language is converted to private speech, it becomes a tool for controlling one’s own thoughts, emotions, and behavior (Day & Smith, 2013; Vygotsky, 1978).  Private speech refers to talking to oneself out loud, partially out loud like mouthing words or whispering, or silently in one’s mind. Private speech can be relevant to a task at hand (such as a preschooler mouthing instructions to herself about how to wash hands—“First I stand on the stool, then turn on the water, then …”) or irrelevant (such as a boy yawning and saying, “I’m tired” to himself).

Although all children use private speech, research shows they are more likely to use it when

· (1)

engaged in goal-directed activities, like academic work, rather than play;

· (2)

their task is challenging as opposed to easy;

· (3)

an adult is aiding, as opposed to controlling, their problem solving; and

· (4)

they are alone rather than with someone (Winsler, Carlton, & Barry, 2000).

Vygotsky’s sociocultural theory is not stage oriented, but one aspect of Vygotsky’s theory that does show age trends is private speech. Children progress from task-irrelevant out-loud talk, to relevant and self-regulatory out-loud talk, and then to partially silent inner speech, such as whispers and quiet muttering (Winsler, Diaz, Atencio, McCarthy, & Chabay, 2000). The out-loud talk increases during the preschool period, peaks around 4 to 6 years of age, and then becomes replaced by increasingly silent self-talk. Thus, private speech becomes more internal as children progress from preschool through the school years (Patrick & Abravanel, 2000).

Brain Research

Private Speech Builds Brains

In  Chapter 6, you learned about brain plasticity; brains can reorganize to compensate for deficiencies in some areas. Scientists help children with cognitive problems, such as ADHD, compensate for brain deficiencies by coaching them to talk to themselves (Bryck & Fisher, 2012). Private speech helps children regulate their thoughts and attention (Fuhs & Day, 2011). In a preschool curriculum called Tools of the Mind, which is designed to improve cognitive functioning, one self-control strategy that children are taught is to talk out loud using private speech (Diamond, Barnett, Thomas, & Munro, 2007). Some studies indicate that the approach is effective, though some others do not (e.g., Blair & Raver, 2014; Jacob & Parkinson, 2015). This approach is based on Vygotsky’s view that language guides behavior.

7-2eDiversity in Sociocultural Theory

According to Vygotsky,  what a child learns is determined by the culture in which the child lives. That is, children learn what is valued in their culture, such as how to grow crops, compute algebra, or recite poems. In addition,  how a child learns and the scaffolding the child receives is also influenced by culture. For example, in one culture children may be directly instructed or formally trained, whereas in another they may simply participate in activities with no direct instruction (Chavajay & Rogoff, 2002). This is relevant to teachers who work with immigrant children whose parents may have had very little formal schooling. Such children may be used to a collaborative, whole-group approach to work more than children of formally schooled parents.

The tools children have available for thinking are determined by the culture.  Cultural tools are concrete objects and symbolic tools that allow members of a culture to think, record, problem solve, and communicate. They can be concrete objects such as rulers, books, or computers ( Photo 7.1). They can also be tools of the mind that are symbolic, such as written language or counting systems. Children’s competence depends on the cultural tools available to them. For example, multiply 578 times 264. Is it difficult? If the equation were written in a vertical format with 578 below 264, you would probably have an easier time as you first multiply 8 times 4, then 8 times 6, and so forth. You would be benefiting from a cultural tool, an algorithm for multiplication that was developed by others and made available to you through the culture of schooling.

Photo 7.1

How many cultural tools can you spot in this photo? Support your label of  cultural tool for each.

Richard Hutchings/PhotoEdit

Writing is a key cultural tool. It allows individuals to record and remember with more accuracy and less effort than was possible before its invention. Genres or types of writing are also cultural tools. For example, a story is a narrative that usually follows chronological order and often uses suspense, whereas a science report focuses on analysis of processes and seldom has elements of suspense. Young children lack these cultural tools of genre writing and often produce a story when asked to write a science report (Kamberelis & Bovino, 1999). They must be taught how to write a story versus a science report.

Written language is a school-learned cultural tool that transfers widely across many different contexts because it is seen in many contexts, not just school. Writing is used in stores, restaurants, street signs, and magazines. However, learners tend to see most cultural tools embedded in a single place and tend not to transfer those tools elsewhere. For example, our young daughter was baking cookies and asked, “Do two quarters make a half? I know they do in math class, but do they in cooking?” In another example, a high school chemistry teacher found that her learners only used the study strategies that she taught when she was present; they would not even use them with the student teacher in the same classroom (Moje, 1996). If you want your learners to transfer the use of a cultural tool from one context to another, you will have to help them see how the tool is used in other settings.

A newer cultural tool is the smartphone, which has many uses. For example, GPS functions can improve your ability to find places but may also undermine your spatial skills if you just follow the instructions without thinking about where you are. There are apps for photography, audio and video production, finding information, identifying animals, finding constellations, and many more functions. They change the ways in which we interact with one another and the world, and the sorts of problems that we can solve on the run.

In summary, culture influences what children learn, how they are taught, what tools they acquire, and where those tools are applied. One lesson for you is to be careful not to jump to conclusions about a learner’s capability based on a single cultural context. Learners may appear unskilled in some contexts but be quite skilled in others.

7-2fClassroom Implications of Sociocultural Theory

There are at least four general implications of Vygotsky’s perspective on cognitive development for your role as a teacher:

1. Use language as a tool to help learners organize their thoughts and to consolidate memories. Private speech should be tolerated and encouraged, particularly for young children or during difficult problem solving in older students.

2. Teach learners in their zone of proximal development, using appropriate scaffolding. It is not easy to determine each learner’s ZPD, and it is a moving target, always changing. It takes attention and insight on the part of the teacher to continually adjust to each learner’s abilities.

3. Help learners actively observe and participate in activities with adults and peers through  apprenticeship and  guided participation . In an apprenticeship, a novice develops competence through interaction with a more expert person who guides or scaffolds participation in the developing activity. Children are apprentices who engage in guided participation in your classroom. The apprenticeship is directed by you, the teacher, as you plan specific learning experiences and help your learners understand the experiences.

4. Think about This

How might attachment to teachers and bonding to school affect cognitive development from a sociocultural perspective?

Work together as a  community of learners in which everyone contributes to the learning process. The teacher is not the only person who has knowledge, but rather expertise is distributed among the members of the group. A community of learners experiences  distributed cognition, in which thinking and knowledge exist not only in the minds of individuals but also in their social interaction and the artifacts that they use and create, like books and computers (Salomon, 1993). Learners can become experts in certain areas and know more than even the teacher in those areas; other learners and the teacher may look to them for help.

These broad implications for classrooms stem from sociocultural theory. In addition, you may want to apply a specific style of teaching that stems from this theory—social constructivism.

Social constructivism shares with Piagetian constructivism the assertion that knowledge is not poured into children’s brains, but rather must be constructed. It uses the term  social because it emphasizes social interaction as the source of knowledge construction. Social constructivist instruction uses scaffolding.

Think about This

Compared to low-socioeconomic (SES) preschoolers, middle-SES preschoolers are twice as likely to play board games like Chutes and Ladders™ and card games like Uno, whereas low-SES preschoolers are twice as likely to play video games (Ramani & Siegler, 2008). How might this partially explain lower academic achievement of low-SES children? Most cultures have children’s games that promote such basic skills. Why might low-SES children not engage in them? Defend your argument using the family investment and family stress models and cultural capital from  Chapter 1.

Recall that scaffolding involves a more competent individual (like a teacher) helping novices master new skills. You’ve already seen how Ms. Kelly scaffolded Darrius’s beginning math skills. For another example, imagine that you want to teach preschoolers how to play number-oriented board games, like Chutes and Ladders™, which foster children’s understanding of numbers. Initially you might model how to count and move the pieces, pointing to each square in turn. After several episodes of scaffolded play, children may be able to play with one another with no support from you.

Learners at all ages, from preschool to graduate school, can benefit from scaffolding. One study found that when learners revealed their lack of understanding and asked for help, successful teachers carefully scaffolded them through a solution process, sometimes by having peer learners help with the instruction (Turner et al., 2002). Teachers did not tell learners how to do the problem, but asked questions and gave hints until the learners understood what was being taught.

Vygotsky’s sociocultural theory does not specify how growth occurs in the zone of proximal development. In contrast, the information-processing model outlines how learners become better at processing information.

7-3Information Processing in Early Childhood

Ms. Wadsworth is teaching her 4- and 5-year-olds how to tie shoes. She begins demonstrating on Nick’s shoes by pulling both laces straight up, one lace in each hand. Then she crosses them over and points out the teepee/triangle that they form. Then she crosses one over the other and pulls the laces apart to tighten them. She now begins the actual knot. “We make a loop with each lace. Look, they look like bunny ears.” When Nick tries, he can get the laces tight on the shoe, but keeps forgetting how to make the loops: “I can’t do it,” he moans as he drops the laces.

Why did Nick forget the instructions? Nick had to remember a sequence of steps and the relevant finger movements—which overloaded his working memory. Recall from  Chapter 3 that the capacity of working memory is relatively small and of short duration. Adult working memory capacity is about one to four chunks of information that can only be retained a few seconds, and for young children it is even less (Cowan, 2010). That is, you can keep a few items active in working memory by repeating the items or by using them in some way. For example, you might remember a new phone number long enough to dial it, but only if nothing interrupts you. You will find it easier if you  chunk the seven digits into fewer items. Imagine you want to remember 882-2015. If 882 is a common prefix in your town, and if you graduated from high school in 2015, the phone number effectively consists of only two chunks—882 and 2015. It is much easier to hold two chunks in working memory than seven items. Chunking helps you remember other types of information as well. For example, memorizing a list of vocabulary words is easier if you place them in a sentence that makes sense.

How do researchers know the capacity of working memory? One approach uses  memory-span tasks. Memory span refers to the number of items, usually presented rapidly, that you can recall in exact order. When digits are used (e.g., 5, 1, 3) as the thing to remember, the measure is called  digit span. Some tests use nonsense words like  woog, spleg, and  symo. Spatial working memory is tested by tasks like recalling a route through a pictured maze. A complex task for 3- to 6-year-olds requires them to remember a list of one-syllable words (e.g.,  nest, fire, hole, hand) in backward order (Noël, 2009). The most complex tasks, known as  storage-and-processing tasks, ask you to remember numbers or words while also processing other information such as counting to 100 by 5s. Your executive functions control the shifting of attention back and forth from storage to processing. For Nick, the shoe-tying task in the opening vignette is equivalent to a storage-and-processing task.

7-3aExecutive Functions

In  Chapter 3, you learned an information-processing model that is composed of the sensory register, long-term memory, and executive functions. Executive functions include working memory and inhibitory control. Two additional executive functions are cognitive flexibility and metacognition.

Cognitive Flexibility

Cognitive flexibility (also called attention shifting) is the ability to change how you think about something, switch perspectives, and adjust to changing demands. Cognitive flexibility supports creativity and problem solving. One way scientists measure this is the Dimensional Change Card Sorting task. Children are given cards that can be sorted by either color, object, or number (e.g., three red trucks on one card, two blue trucks on another). They are asked to sort by one dimension, and then to switch and sort by another dimension. They are scored on accuracy and speed. The task can be made more difficult by increasing the frequency of switching.

Metacognition

Sometimes you  think about your thinking, which is an executive function called metacognition.  Metacognition refers to your knowledge of your own learning processes and how to regulate them. It takes metacognition to choose a strategy to apply to a problem. It takes metacognition to answer the question “What do you know and how do you know it?” When you finish reading a page of this textbook and suddenly realize that you have not processed a single word, that is metacognition at work. Planning and using effective learning strategies is part of metacognition.

Brain Research

Brains Can Be Trained

Can you improve your learners’ brains by training their executive functions? Scientists are trying to through two approaches, both of which have been used with typical children and children with ADHD, language delays, or low SES. One approach is to intensively drill children in a narrow skill, using a computer program. For example, programs may ask children to remember where an object is on a 4 × 4 grid, or find matching figures among many figures. This approach has been used with 4-year-olds to adults. Does it work? It improves performance on laboratory tests of executive functions, with impressive effect sizes ranging from 0.40 to 1.80. However, there is no robust evidence that it improves classroom performance or that it lasts over the long run, though scientists are working on it (Bryck & Fisher, 2012; Jacob & Parkinson, 2015; Shipstead, Hicks, & Engle, 2012).

The second approach is to improve the structure, discipline, and emotional supportiveness of the classroom. Does it work? It has only been tested in early childhood (3- to 7-year-olds), but it has been linked to better executive functions and academic skills (Bryck & Fisher, 2012; Jacob & Parkinson, 2015).

Two types of metacognition are important for learning.  Metacomprehension refers to judging when you have understood something. Even college students have difficulty with this. You can improve metacomprehension by practicing summarizing what you have read (Dunlosky & Lipko, 2007).  Metamemory refers to what you know about your own memory and how to store or retrieve information from it. Children have metamemory when they know that learning precise historical facts takes more effort than learning the gist of history or that stories are easier to remember than lists. Children with better metamemory are better at recall and use better memory strategies (Pierce & Lange, 2000).

You may have noticed overlap between all these executive functions—working memory, inhibitory control (including attention control), cognitive flexibility (or attention shifting) and metacognition. Scientists currently disagree about how distinct different executive functions are, but agree that they are strongly intertwined (Diamond, 2013; Jacob & Parkinson, 2015). For example, working memory capacity is linked to attention control. Preschoolers who can hold more in working memory are particularly good at controlling their attention and are likely to become teenagers who are fast and accurate at inhibitory control tests (Eigsti et al., 2006). As you read this text, if you have larger working memory, you will more quickly disengage from distractions and get your attention back on task (Fukuda & Vogel, 2011).

7-3bDevelopment of Information Processing

Compared to infancy,  processing speed continues to improve in the preschool years. However, more dramatic growth occurs in  executive functions. In fact, you can measure improvement within a year’s time (Clark et al., 2013). This is apparent in the Dimensional Change Card Sort that was described above. Recall that children are asked to first sort by one dimension (e.g., color). After sorting several cards, they are asked to switch and sort by a different dimension (e.g., shapes). You will find that 3-year-olds across the world will usually continue sorting by the old rule (color), even though they can tell you they should be sorting by the new rule (shape). Why is this task so difficult? They must pay attention to instruction, keep the new rule in working memory, and inhibit their original response. By age 4 to 5, most children become successful at this task, and the day/night task ( Chapter 3) in which they must say “day” when shown a picture of night (Best & Miller, 2010). See  Figure 7.7. Yet, inhibitory control continues to be challenging for young children (Diamond, 2013).

Figure 7.7Cognitive Flexibility by Age

Children perform better on the Dimensional Change Card Sort test with age. At what age is there more rapid growth (i.e., where is the line steepest)?

Source: Zelazo et al. (2013).

7-3cIndividual Diversity in Information Processing

Some learners process information faster and have better executive functions than same-age peers. In  Chapter 3, you learned that individual differences in information processing are apparent in infancy and that the gap in information processing gets wider from preschool through adolescence.

What Do Individual Differences in Information Processing Predict?

Although the laboratory tasks used to measure information processing, like repeating digits in reverse order, can seem trivial, performance on the tasks predicts a remarkable range of real-world outcomes over the long term. Faster processing speed and better executive functions have been linked to academic success. For example, learners who have better information-processing abilities are better at solving math problems and comprehending what they read; they also get higher standardized test scores. This pattern has been found for multiple ages and in multiple countries.     Indeed, executive functions form the core of “school readiness” skills such as the ability to concentrate, pay attention, and follow instructions. They may be more important to school success than IQ or prereading skills (Diamond, 2013). Children who start kindergarten with these skills will make greater gains over the years (Li-Grinning, Votruba-Drzal, Maldonado-Carreño, & Haas, 2010). Learners with poor information-processing ability may struggle more as they progress through school because school tasks will increasingly demand these abilities.

Information-processing abilities are also linked to emotional and social skills. Learners with poor attention control and other executive functions tend to have more problems like anxiety, depression, impulsiveness, aggression, and acting without thinking (e.g., Khurana et al., 2015).

What Predicts Individual Differences in Information Processing?

Genes predict differences in information processing. Executive functions have a very high heritability index in childhood (Engelhardt, Briley, Mann, Harden, & Tucker-Drob, 2015). However, this does not mean executive functions are not influenced by experience. Heritability is an estimate of how much variation among the children tested at a particular point in time might be attributed to genes, but says nothing about whether it can be changed. As you learned in  Chapter 6, the brain is shaped by experience.

An important experience is the quality of a child’s home (see  Figure 7.8). Mothers and fathers who are sensitive, supportive, and provide activities that stimulate their children’s cognitive and language development tend to have children who have better information-processing abilities (Fay-Stammbach, Hawes, & Meredith, 2014; Meuwissen & Carlson, 2015). For example, a national study found that quality of parenting when children were 4 years old predicted their executive functions and memory at that time and reading and mathematics test scores when they were in 3rd grade (Friedman et al., 2014). There is likely a gene–environment interaction at work. That is, children with a genetic predisposition tend to develop low executive functions primarily when there is also insensitive parenting, household chaos, or more hours in child care (Deater-Deckard, 2014).

Figure 7.8Individual Differences in Executive Functions

A child’s executive functions are influenced by an interaction of genes and environment, which in turn influence school success. In other chapters, you will learn about several aspects of parenting quality that have been linked to executive functions. These include sensitivity (vs. hostility) and security of attachment ( Chapter 4), reading and talking to children, scaffolding children in challenging tasks, providing a stimulating home, and authoritative parenting ( Chapter 8). See Deater-Deckard (2014). Can you explain how genes and environment might interact to influence children’s executive functions?

7-3dMemory

From an information-processing point of view, cognitive growth is the result of greater  knowledge—or long-term memory—as well as greater processing speed and executive functions. Greater knowledge enhances these other components of information processing because knowledge that is overlearned and automatic frees resources and is processed more quickly. In this section, we discuss errors, development, and individual differences in long-term memory. In  Chapter 11, you will learn strategies to help your students remember more.

Memory Errors

Memory is not an exact replica of an object, event, or experience. There are two types of memories: verbatim traces, which are detailed accurate memories, and  fuzzy traces , which are general, vague memories, or the gist of an experience. Can you remember a lecture word for word from last week? Probably not, but you might remember the gist of it. Most of your remembering involves fuzzy, not verbatim, traces. This may not seem ideal, but in fact fuzzy traces are adequate for most endeavors.

You may have worked hard to teach learners something—the names of colors, a piece of music, vocabulary words—which they appear to learn at one time, but later forget. Verbatim memories are forgotten more readily than gist memories. For example, if you read  Chapter 1 and created a verbatim trace of the fact that the 2010 census showed 64% of the U.S. population is White, non-Hispanic, this would soon deteriorate to something like “a recent census found that more than 50% of the U.S. population is White.” Like you, children remember details only if they use them frequently.

There are at least three reasons learners forget things:

1. Decay. The memory decays over time and loses strength if it is not used.

2. Retrieval  failure. Learners may know something but then go blank when asked. They cannot retrieve the information when they need it.

3. Interference . New knowledge can make retrieval of old knowledge difficult, and vice versa. For example, Kevin is an English speaker who knows the word  embarrassed (old knowledge). In Spanish class, he learns that  embarazada means  pregnant (new knowledge) but has trouble remembering the new definition because of interference from the word  embarrassed. When he does something embarrassing, he says, “ Estoy embarazado,” which unfortunately means “I am a pregnant boy” (adding to his embarrassment).

Although forgetting information you want to remember is frustrating, scientists argue that forgetting is good for you, to some extent. Forgetting painful experiences or failures helps you feel happier, and forgetting irrelevant, incorrect, or out-of-date information unclutters your mind for more efficient thinking (Nørby, 2015). It can help you see the forest rather than the trees.

Another type of memory error, besides forgetting things that you once knew, is remembering things that never happened, or  false memories. Our youngest daughter remembered family vacations that occurred before she was born. Apparently, she was a precocious child. (Actually, she saw photos and then created a “memory” of the vacation.) Other children do this too. A sneaky researcher intentionally told another adult at a preschool—in front of some of the preschoolers—that an escaped rabbit was eating carrots in another classroom. Later 55% of the  classmates of those children, not the children who overheard the story, reported having actually seen the rabbit, which did not exist (Principe, Kanaya, Ceci, & Singh, 2006)—that is, rumor-mongering caused false memories.

False memories occur in adolescents and adults as well. In fact, you may be more prone to false memories now compared to when you were a child (Brainerd, 2013). For example, researchers doctored family photographs into a picture of a hot air balloon ride, and family members remembered taking the ride, which never occurred (Garry & Gerrie, 2005). Just imagining doing something, or watching someone else do it, can create false memories that you actually did it (Lindner, Echterhoff, Davidson, & Brand, 2010).

Why do we have false memories? Memory contains pieces of reality mixed with creation. False memories are intelligent errors—that is, your mind makes sense of a situation and recalls details that were not there but logically fit your expectations. The memory learners construct is influenced by their previous experience, so different learners construct different memories of the same event. Just because learners express a memory with confidence, detail, and emotion does not necessarily mean it is factual.

One type of false memory you’ll need to help learners with is the  source monitoring error, or a false memory of the source of their information. For example, preschoolers may think they learned facts from their teacher when they actually learned non-facts from cartoons (Riggins, 2014). Source monitoring is crucial for critical thinking about the meaning and accuracy of information. A supposed “fact” is evaluated differently based on whether it came from a supermarket tabloid, a research journal, or an Internet advertisement.

Development of Memory

Preschoolers experience each of these memory errors. Young children’s memories are fragile and susceptible to interference. That is, learning something new may override existing memories (Darby & Sloutsky, 2015). Nevertheless, with adult support they can display good long-term memory. For example, in a classic study scientists asked children who were 3 and 4 years old when they visited Disneyworld to recount their trips several months later (Hamond & Fivush, 1991). The children remembered a great deal of accurate information about their trip. However, the 3-year-olds needed cues to help them remember, such as “What rides did you go on?” and “What did you eat?” The 4-year-olds recalled more spontaneously, without cues, and they provided more details. In reminiscing, adults provide more scaffolding with younger children because they need more than older children (Schneider & Ornstein, 2015).

Young children typically use only one memory strategy—rehearsal. That is, saying something over and over, whether aloud or “in their head.” For example, our 4-year-old son went to play at the neighbor’s house. He was told to be home by five for dinner. All the way from our door to the neighbor’s door we heard him repeating, “I have to be home by five. I have to be home by five.” There are more effective memory strategies (see  Chapter 11), but preschoolers are not likely to use them before age 5 or 6. You can teach kindergarteners memory strategies, but they are unlikely to apply them to situations outside the training context, may not see the value of using the strategies, and will need more time to learn the strategies than older children (Kuhn, 2000; Schneider, 2000).

Individual Diversity in Memory

Two children of the same age can differ in their ability to remember information. Children who remember more have higher achievement. This is obvious because achievement tests measure how much children know. Less obvious is that knowing more also makes children better problem solvers because knowledge can be applied to the problem at hand, like figuring out word meanings while learning to read.

Think about This

The fact that talking aids memory suggests that memory is a social event. How does this fit with Vygotsky’s sociocultural theory?

What predicts better memory? Conversation is a powerful tool in helping learners store information and create schemas. If you converse about novel objects with learners as they are handling them, learners are more likely to remember the objects (Haden, Ornstein, Eckerman, & Didow, 2001). If you talk about an event, like a visit to a museum, as it is happening or shortly afterward, learners recall the event better.

How you converse matters. When adults  elaborate as learners recall events, they develop better memories (Schneider & Ornstein, 2015). Ideally, you should ask open-ended questions (e.g., “Tell me about …” rather than “Is this a cat?”) and follow the learner’s lead by talking about things the learner brings up. For example, in one study, toddlers whose mothers elaborated, not merely repeated, what they talked about could then remember things years later, at age 12 to 13, better than could other teens (Jack, MacDonald, Reese, & Hayne, 2009).

Talking about things helps your learners retain a memory longer. Talking about things also helps them understand the information better. Thus, to help your learners remember, talk with them about what you want them to remember.

7-3eReasoning and Problem Solving

In  Chapter 3, you learned that infants are capable of induction. However, preschoolers become better at  inductive reasoning, which refers to making generalizations from observed examples, than they were at earlier ages (Fisher, 2015). For example, if a child sees various dogs bark, she might infer that all dogs bark. Preschoolers are also capable of  deductive reasoning in their everyday behavior. Deduction is a form of reasoning in which a conclusion follows logically from a set of premises. An example was given earlier in this chapter of the 3-year-old who deduced that there must still be crackers based on two premises:

· (1)

when crackers are gone, Mom throws the box away, and

· (2)

there is no box in the garbage.

Preschoolers ages 4 and 5 are capable of determining  when evidence is sufficient for drawing conclusions. For example, imagine you draw a flower on white paper with a purple marker. Then you show a child three boxes with lids. Ask the child which box contains the marker used to draw the flower, without opening them. After the child guesses, ask: “Do you know for sure, or do you have to guess?” Then open the boxes, one at a time, revealing a green, then purple, then red marker, repeating the question. Only when all three boxes are open can the child “know for sure” which box contains the marker used to draw the flower. A more complex version uses four boxes. The fourth box will contain another purple marker. Even when all four boxes are open, the child cannot determine which box contains the marker used to draw the flower. Most preschoolers can do the three-box task, and about 70% can do the four-box task (Klahr & Chen, 2003).

Preschoolers’ logic is not flawless, however. When two boxes are open, and one reveals the purple marker, preschoolers often say that purple marker must be the one—even though the third, still-closed box may also contain a purple marker. The single positive instance captures the children’s attention and blinds them to the fact that the third box might render the problem unsolvable. Even adults have this bias to some extent, which is why advanced scientific reasoning requires extensive education.

Reasoning can be improved through direct instruction. When children are told  why their response is correct or incorrect, they improve in reasoning on the purple-marker task (Klahr & Chen, 2003). Five-year-olds, but not 4-year-olds, will improve with simple experience, even without feedback. They also learn faster from feedback, improve more dramatically, and transfer their improved ability to other similar tasks more than 4-year-olds. This is probably the result of better working memory and executive functions.

In summary, although preschoolers have trouble talking about it, their behavior shows that they reason and problem solve (Lucas, Bridgers, Griffiths, & Gopnik, 2014). They gradually become more reliable, systematic, and efficient in their reasoning with age, but there is not a stage-like shift. They are substantially more logical than Piaget believed them to be (Wellman, 2011).

7-3fClassroom Implications of Information Processing

Information-processing ability affects learners’ success in your classroom. Academic tasks—such as learning to connect squiggles on a page with words that have meanings—require strong information processing. One hypothesis used to explain why low-income children tend to have low achievement is outlined in  Figure 7.8. That is, poverty may reduce quality of parenting due to stress and limited resources (see  Chapter 1), which may compromise executive functions in children, which interferes with academic achievement. For example, in one study, poverty at 1 and 24 months predicted executive function problems in 3rd grade, which predicted low math and reading test scores in 5th grade, after controlling for IQ (Crook & Evans, 2014). Have you noticed that you have trouble thinking clearly or remembering things when your life is not going well? Executive functions are compromised when you are stressed, sad, sleep-deprived, or physically unfit, just as they are for your learners (Diamond, 2013).

If you have learners who have difficulty shifting between tasks, forget lengthy instructions, forget letters in words or words in sentences, are easily distracted or poorly organized, cannot complete a multistep task, or raise their hand but forget the response when called on, they may have limited information-processing ability. Here are a few things you can do as a teacher to help such learners be successful.

Reduce Working Memory and Executive Load

Your learners cannot process new information when their working memory capacity is overloaded. To reduce the load on working memory:

1. Limit your talking. If you keep talking after presenting important information, it will be forgotten. Present information at a speed that allows learners to fully process it.

2. Reduce distractions in your classroom. Distractions can be comings and goings, announcements, and even decorations that visually bombard learners. For example, in an experiment, kindergarteners were off-task and learned less in a highly decorated, versus spartan, room (Fisher, Godwin, & Seltman, 2014).

3. Increase your learners’ expertise. The more automatic their processing, the less space is consumed by executive functions. If shoe tying is automatic for you, you would not be overwhelmed by the task that Nick faced.

4. Provide external storage. A preschool teacher might post informational pictures on the wall. An elementary teacher might write on the board: “Read for 20 minutes. Write a summary of what you read. Look at the sample on the wall, if you need to.” A secondary teacher might provide partial notes (i.e., a rough outline of the lesson, but not details), which frees learners’ working memory to attend to the information.

5. Carve problems into smaller subtasks that can be performed sequentially. In writing, this might include having a preschooler make a straight line and then attaching two half circles to it to make a “B.” In math and science, this includes providing formulas or algorithms.

Focus Attention

Selective attention is considered a gateway to learning (Bahrick & Lickliter, 2014). The better learners can control their attention, the higher their academic achievement (e.g., Claessens & Dowsett, 2014). You want your learners to concentrate, resist distractions, and not let their minds wander during instruction. This simple skill exists in infants but is strengthened through exercise. Over time, this simple skill develops into complex skills, such as computer programming or reading Dickens (Sörqvist & Marsh, 2015).

To be an effective teacher, you need to attract learners’ attention and maintain it on important information, particularly if you teach learners who have difficulty controlling their attention. It helps to make learning goals explicit and remind learners of the goal. Children often do not know the goal of a learning task; without a goal, they cannot accurately direct their attention and assess whether they are reaching the goal (Barker & Munakata, 2015; Chevalier, 2015). It also helps to give learners a break for physical activity. After 20 minutes of exercise, learners behave more attentively (Pontifex et al., 2014).

Strengthen Executive Functions

Using executive functions makes them stronger, just as using muscles makes them stronger. You can strengthen your learners’ executive functions by following these guidelines:

1. Promote healthy habits—especially adequate sleep, good nutrition, and physical fitness. Executive functions require a large amount of glucose (the brain’s fuel, as you undoubtedly remember from  Chapter 2). Part of why you don’t think clearly when you are tired or hungry is because your brain is depleted of fuel. The brain’s store of glucose is replenished during sleep and after eating (Gailliot, 2008). Physical fitness is also linked to better brain functioning. More fit children have faster processing speed, better memory, and better executive functions, including attention control (Chaddock-Heyman, Hillman, Cohen, & Kramer, 2014). Aerobic and mindful exercise, like martial arts and yoga, may improve information processing in children (Diamond & Lee, 2011).

2. Help learners practice using their executive functions through mundane, daily activities like requiring them to sit up straight or persist in activities even when they want to stop. However, be aware that you can overtax executive functions. Learners may occasionally need a break (Kaplan & Berman, 2010).

3. Help young children improve their verbal abilities, which are linked to executive functions. In this chapter, you learned that self-talk helps children regulate their thoughts and attention. Music training has also been shown to improve both verbal ability and executive functions, presumably because they share the same brain resources (Moreno et al., 2011).

4. Ask learners to think about their thinking, or practice metacognition, with questions such as: “How did you know …? How have you improved your thinking about math (or history, or science)? How would you do it differently? What did you learn from doing this work? I noticed you erased a lot—how did you know it needed fixing?”

You have learned that language ability is inextricably linked to cognitive development in each of the theories discussed in this chapter. Children pass Piaget’s conservation tasks when they are able to give primacy to language, they control their own thought through private speech and acquire knowledge through the cultural tool of language according to Vygotsky, and memory is enhanced when children talk about events and objects. Let’s turn our attention to the impressive development of language and emerging literacy in early childhood.

7-4Language and Literacy Development in Early Childhood

In order to understand and facilitate the development of language and literacy in your learners, you need to know the components of verbal language. There are five key components of verbal language:

· (1)

phonemes,

· (2)

morphemes,

· (3)

semantics,

· (4)

syntax, and

· (5)

pragmatics.

At the most basic level is a  phoneme , or speech sound. The word  dog has three phonemes: /d/, /o/, and /g/. The letter  g expresses two phonemes: the hard (e.g.,  get) and the soft (e.g.,  gin) pronunciations. There are a limited number of phonemes—roughly 50 in English and 100 to 800 in all the world’s languages (Beatty, 2001; Gibbs, 2002).  Phonological awareness is the ability to identify phonemes or the sounds of language. Learners have phonological awareness if they can do tasks like say  plig without the  l, or tell which word doesn’t rhyme among  bat, pad, had, or tap the number of sounds in mat: three taps for /m/, /ae/, and /t/.

At the next level,  morphemes are the smallest language unit that contains meaning. Morphemes refer to units of meaning rather than units of sound. Morphemes can be word roots, suffixes, and prefixes. The word  dogs has two morphemes: /dog/ and /s/. The /s/ adds the meaning “plural” to the meaning dog. The word  unhelpful has three morphemes: /un/, /help/, and /ful/. If you change just one phoneme in “dog” to “fog” you get a different morpheme. Learners have morphological awareness when they know about the structure of words and how to manipulate them, like changing words from present tense (John feeds the fish) to past tense (John fed the fish).

Semantics refers to making meanings, or the way you use words and word combinations to express ideas. When you teach vocabulary, you build your learners’ semantic skills.  Syntax refers to the way words are organized into phrases and sentences, such as a verb followed by a noun, or preceded by an adverb. “He reads the book” is typical syntax in English, but “The book he reads” is typical in Turkish.  Pragmatics refers to using language appropriately according to sociocultural rules. For example, you adjust your speech based on whether you are asking a favor or giving a command, and whether you are talking to a child or to your boss. To interpret a spoken sentence, you must identify phonemes, segment them into words, interpret their semantic meaning, analyze the syntax, and activate pragmatic rules (Trout, 2003). Your brain is able to process all these components almost instantly in your native language.

7-4aDevelopment in Language

The typical preschooler speaks almost fluently—in fact, some parents will complain that they speak nonstop! However, one of the charming errors that preschoolers often make is  overregularization, which you learned in  Chapter 3 refers to applying the rules for regular word change to a word that changes irregularly (Marcus et al., 1992). For example, the regular rule for making nouns plural is to add an –s, but for words like  tooth, the plural is  teeth, not  tooths. The regular rule for making verbs past tense is to add –ed, but for verbs like  come, the past tense is  came, not  comed. Common overregularizations include  cutted, hitted, bringed, drived, deers, gooses, and  mouses. We recently heard a 4-year-old say, “He bringed it” and a 6-year-old say, “He wasn’t hurted.” Listen to this conversation:

A 5-year-old boy excitedly announces to his father that he won a race with his infant brother.

Boy (gleefully):

I raced baby Luke and I beat him!

Dad (amused):

You mean you defeated an immobile infant?

Boy (proudly):

Yep, I defought him.

Although his father’s humor was lost on the boy—in  Chapter 12 you’ll learn that children don’t understand sarcasm until about 4th grade—the boy’s attempt to apply what he has learned about irregular verbs is impressive, although a little misguided.

Overregularization usually begins about age 2 or 3 but is inconsistent; that is, children sometimes make the error and sometimes they do not. Ironically,  younger children often use irregular words  correctly, but then as they infer the rules of English they begin to overregularize. Later, in middle childhood, they become more correct in their usage. However, even adults sometimes commit overregularization errors. What can you do about children’s overregularization? Talk to them. Children learn correct structures more quickly if they are frequently exposed to them (Ambridge, Kidd, Rowland, & Theakston, 2015). However, evidence suggests that deliberately correcting them does little to hasten their progress in overcoming overregularization; they figure it out on their own as they hear correct usage.

7-4bIndividual Diversity in Language and Literacy

Some preschoolers have greater language ability than others. If you are concerned about a particular child, seek the help of a language specialist. The federal Individuals with Disabilities Education Act (IDEA) mandates that screening services be available to all children. Let’s discuss what factors are linked to typical variation in language ability.

What Do Individual Differences in Language and Literacy Predict?

Two components of language discussed above are critically important skills for learning to read: phonological and morphological awareness (Hulme & Snowling, 2013). In addition, both verbal and nonverbal language ability predict academic achievement and social competence at school.

Academic Achievement

In  Chapter 3, you learned that language problems have academic consequences. One aspect of verbal ability—vocabulary size—may be particularly important because it influences reading. Once they get to school, vocabulary size is related to how fast students can process information, measured in milliseconds (Marchman & Fernald, 2008), which is important for fluent reading. Learners with large vocabularies learn to read more easily, enjoy reading, and read a lot, which increases their vocabulary even more.

Verbal ability is also linked to math ability, perhaps because the same areas of the brain are involved in both (Gelman & Butterworth, 2005). They may also be linked because poor language ability predicts poor executive functioning (Kuhn et al., 2014) and slower processing speed, which in turn predict low academic achievement.

Poor nonverbal language ability also predicts low academic achievement. Learners who don’t read others’ nonverbal cues tend to have lower standardized test scores (Nowicki & Duke, 1992).

Social Competence

Both nonverbal and verbal ability predict social competence. Children who don’t read others’  nonverbal cues tend to be friendless and rejected by peers. Similarly, children who have problems expressing themselves  verbally tend to have behavior problems (e.g., Dionne, Boivin, Tremblay, Laplante, & Perusse, 2003).

What Predicts Individual Differences in Language and Literacy?

In  Chapter 3, you learned that social and verbal interaction predict verbal ability. One key source of such interaction in early childhood is play. Children who engage in frequent, age-appropriate sociodramatic play have better verbal ability. When young children with low verbal ability are trained to play more, they develop better verbal ability. Play provides motivation to use language with peers and opportunity to learn from them. Children learn language better when they attend preschool with peers who have high verbal ability (Mashburn, Justice, Downer, & Pianta, 2009).

In  Chapter 3, you also learned that shared reading (also called joint storybook reading) promotes language. Questioning is particularly important during shared reading. An experimental study found that when adults asked high-demand or low-demand questions about a story, 3-year-olds learned more vocabulary than when there were no questions (Blewitt, Rump, Shealy, & Cook, 2009). High-demand questions are questions that do not just ask what happened or who did what; rather, they require inferences and may ask about why something happened, how a character felt, what a word means, or how story events relate to the child’s experiences. In the study, a low-demand question was “What were they selling in the  pagoda?” and a high-demand question was “Do you think the ticket man lives in the  pagoda? Why (or why not)?” (p. 304).

Both play and questioning during shared reading also predict better literacy skills. That is, preschoolers who use literacy in their play—like reading to dolls, writing a shopping list, and putting letters in a mailbox—develop greater literacy. So do elementary students who make their own books, write a play for their puppets, write out rules for games they make up, or try to find letters in billboards during car trips. In addition, during shared reading, if adults start with low-demand questions and work up to high-demand questions, they scaffold children’s comprehension (Blewitt et al., 2009).

Electronic storybooks that can be read on computers, tablets, and smartphones can enhance children’s emergent literacy skills. However, some storybook features detract from learning. Whether they are helpful or distracting depends on whether the feature supports or detracts from the theme and plot of the story. Generally, animated pictures, music, and sound effects are beneficial, whereas inserted games and hotspots that take the reader away from the story are distracting (Bus, Takacs, & Kegel, 2015; Takacs, Swart, & Bus, 2015). For example, an image of a tree that sways while stating that the tree is swaying supports children learning the meaning of swaying. However, clicking on the word  swaying in order to hear a definition can distract the child from the story and interfere with comprehension. This partly depends on the child’s literacy. Struggling children are more easily distracted and are less likely to make connections from the feature to the text, whereas more fluent readers are more likely to be able to make sense of the feature in the context of the text.

7-4cGroup Differences in Language and Literacy

There are some group differences in language and literacy that may be relevant to your classroom. These include socioeconomic status and whether a child speaks a different dialect or language at home.

Socioeconomic Status

In  Chapter 3, you learned that low-SES children tend to have lower verbal ability and that one explanation is less opportunity to learn at home. They may have less opportunity to learn at school as well. At school, low-SES learners are segregated, beginning with preschool programs like Head Start, where they are exposed to peers with low verbal ability (Mashburn et al., 2009). In kindergarten, low-SES learners may receive less vocabulary instruction than their higher-income peers (Wright, 2012).

There are similar SES differences in literacy. Some families have dozens of children’s books, and others have none at all. Some preschoolers are read to several times a day, culminating in thousands of hours before entering school, whereas others are not read to. College-educated parents are more likely to read to their children, to talk with them during storybook reading, and to take them to the public library (Aikens & Barbarin, 2008; Raikes et al., 2006). For example, a national report found that 74% of college-educated mothers read to their preschoolers daily as compared with 40% of high school–educated mothers and mothers living in poverty (Federal Interagency Forum on Child and Family Statistics, 2009). Thus, high-SES children have more print exposure, which refers to activities related to print and reading. Print exposure predicts reading comprehension, spelling, and academic achievement (Mol & Bus, 2011). Differences in print exposure continue into adolescence, as high-SES students spend more non–school time reading than low-SES students (Larson & Verma, 1999). However, your low-SES learners will become literate if they like to read and have good instruction.

African American English

The predominant dialect used in classroom interaction, instruction, and textbooks is  Standard English (SE) , also called School English or Formal English.  Most learners experience mismatch between the way informal language is used in the home compared with the more formal usage at school. However, the mismatch is larger for learners who speak a dialect other than SE. One such dialect is  African American English (AAE) , sometimes called Ebonics or Black English. Teachers who are not used to AAE need to guard against forming negative expectations toward learners who use AAE (Pearson, Conner, & Jackson, 2013).

AAE is a full dialect that has its own rules. Many language forms that are incorrect in Standard English are accepted in AAE. There are different versions of AAE, but some commonalities include the following (Champion, 2003):

· Saying  ax, bidness, and  ’posed to for  ask, business, and  supposed to.

· Stressing the first syllable in words like  PO-lice and  DE-troit.

· Omitting possessive /s/ as in  That man hat is on the table. ( That man’s hat is on the table in SE.)

· Omitting final /ed/ as in  They talk yesterday. ( They talked yesterday in SE.)

· Omitting contractions as in  She done well. ( She’s done well in SE.)

· Omitting final consonants as in  las for  last.

· Unique use of  done as in  I done did her hair or  I done her hair. ( I did her hair in SE.)

· Unique use of  be as in  He be happy.

· Using  f for  th as in  toof for  tooth.

Thus, when AAE-using learners say, “My dog name Lady,” they are not using incorrect Standard English, but rather are using the home dialect correctly.

Should you force learners to use Standard English at school? You should respect learners’ right to maintain their heritage dialect. However, learners also need to learn Standard English because it allows them to participate fully in school, commerce, and society, and learning earlier is easier than later.

How should you teach Standard English? Simply correcting informal usage is not effective. Instead, teaching learners about  code-switching may help. This refers to using different language varieties for different situations, such as SE at school and AAE at home. Point out to learners that they code-switch as they move between settings, like from the classroom to the playground or sports field. Discuss different patterns of speech without labeling them as correct or incorrect. Invite learners to code-switch and teach you aspects of their language that differ from SE. This makes them codiscoverers with you of their own language patterns. When you work with children who are beginning to read and write, you can create charts that list the informal version of a phrase next to the SE version and discuss when to use each (see  Table 7.1).

Table 7.1

Samples for Teaching Code-Switching

	AAE	Formal SE
Showing possession	“Yes,” said Annie mom. My dog name is Caesar.	“Yes,” said Annie’s mom. My dog’s name is Caesar.
Subject–verb agreement	She work hard.	She works hard.
Using “be”	She be my best friend. Jesse be goin’ to the store.	She is my best friend. Jesse is going to the store.
Showing past time	Yesterday I turn on the TV.	Yesterday I turned on the TV

Source: Adapted from Wheeler and Swords (2010).

This approach can be applied to varied dialects and languages of immigrants. For example, an Appalachian speaker might say, “We have 20 mile to go” instead of “20 miles to go.” Learners benefit when they can code switch between the home and school language, depending on which is most appropriate.

Immigrant Students and Bilingualism

According to the 2010 U.S. Census, more than one in five children is an immigrant, meaning either the child or at least one parent was born outside the United States. Immigrant children are likely to speak a language other than English at home. Today, about half of immigrant children speak Spanish. The next largest group speaks Chinese. However, immigrant children in the United States have many different heritage languages, such as Russian or Arabic ( Photo 7.2).

Photo 7.2

Classes in some schools can have English language learners who speak many different native languages, such as Russian, Spanish, Vietnamese, or Chinese.

Most U.S. immigrant children are  bilingual , meaning they speak both English and their heritage language fluently. Some children are  English language learners (ELL) , meaning they are not yet proficient in English.     It takes an average of 3 to 7 years for ELL students to reach proficiency in the new language; younger learners take longer but achieve greater fluency (Dixon et al., 2012). In 2013, about 9% of public school students were ELL (U.S. Department of Education, 2015). California has roughly half of the ELL students in the nation. In some districts in California, 60 to 70% of students are ELL.

ELL learners are not all immigrants. One study found that 60% of 9th-grade ELL students were born in the United States, meaning they had spent many years in school without becoming fluent in English. Some reached an intermediate level of English competence but then stopped making progress.  Foreign-born ELL students tend to catch up with U.S.-born ELL students by high school (Slama, 2012).

Skill in English is important because it predicts academic achievement (Parker, O’Dwyer, & Irwin, 2014). A study in Boston and San Francisco found that highachieving ELL learners tended to have greater English proficiency than low-achieving ELL learners (Suárez-Orozco et al., 2010). There were five patterns of achievement (see  Figure 7.9). Two-thirds of learners had low or declining achievement across 5 years, and one-third had high or improving achievement. Some low achievers believed their undocumented status would prevent them from attending college, which undermined their attitudes toward school. A protective factor that nearly every improver had was a mentor, which is a role you could play for your immigrant learners.

Figure 7.9Patterns of Academic Achievement among Immigrant Adolescents

The students were between 9 and 14 years old at Year 1. How many groups of immigrant children show declining grades over 5 years?

Source: Suárez-Orozco, Gaytán, Bang, Pakes, O’Connor, and Rhodes (2010).

When first immersed in a second language, it is common for children to have a temporary silent period that lasts from weeks to months as they focus on listening and comprehending. It is also common for children to be slightly delayed in language when mastering two languages simultaneously. Many ELL learners exclusively speak their home language until age 3, when they enter preschool. Often their progress in the home language slows, but their growth in the new language is rapid. Do not be concerned about temporary delays—in the long term, bilingual learners often have better language abilities than their monolingual peers.

Bilingualism has cognitive benefits. Bilingual learners are better at tests of executive functions, working memory, inhibitory control, math skills, and theory of mind than monolingual learners. This has been found across several languages and as early as age 3 (e.g., Adesope, Lavin, Thompson, & Ungerleider, 2010; Bialystok, 2015). They are also less egocentric and better able to understand a speaker’s meaning and perspective (Fan, Liberman, Keysar, & Kinzler, 2015). It appears that switching between languages may improve information-processing skills, although not all research supports this claim (de Bruin, Treccani, & Della Sala, 2014).

7-4dClassroom Implications of Language

In a kindergarten classroom, the teacher asked Jared, the meteorologist for the day, to report on the weather. Jared said it was rather brisk. When an observer asked him why he chose such an unusual word to describe the weather, he replied, “Well, it’s colder than cool, but it’s a long way from frigid.” (Adapted from Lane & Allen, 2010, p. 363)

Many high school teachers wish their students had the vocabulary of this 5-year-old. How did he come to be so verbally precocious? His teacher, Ms. Barker, deliberately taught verbal ability in her class. She selected novel words that she scaffolded for her learners, building on familiar class routines. Ms. Barker’s learners “distributed” rather than passed out paper, they lined up “adjacent” rather than next to the wall, and they “provided nutritional sustenance to our rodent friends” rather than fed the hamsters. Despite teaching in a high-poverty school, Ms. Barker demanded extraordinary vocabulary from her learners and provided them with the support to use it competently. This is important because strong language skills are linked to academic achievement and appropriate classroom behavior for learners of all ages (Forget-Dubois et al., 2009).

Promote Students’ Verbal Ability

To help your learners develop better language ability, follow the guidelines presented in  Chapter 3 as well as these guidelines:

1. Explicitly teach vocabulary. For preschoolers, this can mean modeling language and explaining what words mean. For literate students it can mean memorizing vocabulary. Help learners use mnemonic strategies to memorize vocabulary, which are discussed in  Chapter 11.

2. Help learners use new words in multiple ways. Give the definition of a new word, use it in a context that makes its meaning clear, and then have learners use it. Learners need repeated exposure to words, as well as the opportunity to use them in context.

3. Use academic language. Most learners, including ELL learners, will pick up day-to-day language. They struggle much more with technical vocabulary and complex language use, such as making inferences and hypotheses, summarizing, using analogies, and comparing/contrasting (DiCerbo, Anstrom, Baker, & Rivera, 2014).

4. Encourage children to use Standard English while respecting their heritage language or dialect. Ask questions that require more than a yes/no response. Pause to let learners contribute to classroom talk. To help learners speak SE, one high-poverty high school that has narrowed the achievement gap requires learners to give all answers in full sentences without slang. The teens initially resisted, but their ability to use SE has improved rapidly.

Let’s take a peek at a teacher using some of these guidelines to teach vocabulary to preschoolers.

Mr. Myers:

When it ( pointing to butterfly) was inside, its wings were together, but once it got out, it could splay, or spread out, its wings.

Aquala:

Ya!

Mr. Myers:

Splay means to spread out.

Aquala:

Yeah, like peanut butter. Like spread with a knife.

Mr. Myers:

Yes, but the peanut butter doesn’t really get splayed because it doesn’t have parts. Splay means to spread something that has parts. You have body parts that you can splay. You can splay your arms, legs. And spread out all over like this ( gestures).

Aquala:

( spreading arms apart) This splay?

Mr. Myers:

Yes, you are splaying your arms.

Aquala:

( to another child) And you are splaying your whole body. (Collins, 2012, p. 68).

Tactics like explaining word meanings or elaborating on what learners say may seem easy to do. Yet studies in which teachers are trained to follow these guidelines find that many do not implement the training (Dickinson, 2011; Justice, Mashburn, Hamre, & Pianta, 2008; Piasta, Justice, McGinty, & Kaderavek, 2012). One study found that kindergarten teachers averaged only eight episodes of vocabulary instruction per day, and teachers of low-SES learners did less of it, which would tend to perpetuate the vocabulary disparity between high- and low-SES children (Wright, 2012). Thus, you may need to deliberately practice promoting language skills in your classroom.

Use Nonverbal Language

Parents who use more gestures with their preschoolers tend to have children who develop better vocabulary, which predicts school achievement (Rowe & Goldin-Meadow, 2009). Nonverbal language is also important in your classroom in two ways. First, gestures help children learn and teachers teach. Second, teachers’ expectations can be unconsciously conveyed through nonverbal language.

Gestures and Instruction

Gestures include movements like inclining one’s arm to indicate different slopes of lines, pointing at a series of objects to indicate counting, or pointing with both hands at numbers on either side of an equal sign. Gestures help children learn, understand, and problem-solve (Goldin-Meadow & Alibali, 2013). See  Figure 7.10. Perhaps gestures foster learning because they lighten the cognitive load as learners talk about concepts they cannot fully articulate yet. Gestures serve as a bridge between concrete experiences and abstract concepts. As learners become more expert, there is less need for gesture when they talk about abstract ideas.

Figure 7.10Gesture Promotes Learning

Gesture helps this child understand the equivalency concept in mathematics.

Gestures can tell you about learners’ readiness for instruction. They are particularly ready to learn when gestures suggest accurate thinking despite inaccurate spoken language. For example, a boy explained his incorrect solution to the equivalence problem 7 + 6 + 5 =+ 5 as follows:

“I added 13 plus 10 equals 23” (an incorrect add-all-numbers strategy) while holding his whole hand under the 7 and the 6, pointing at the blank, and then pointing at the 7 and 6 (a correct grouping strategy).

Verbally, this boy is communicating that he does not understand that the equal sign separates the two halves of the equation. However, his gestures communicate that at some level he does understand. He is ready for instruction because he is on the verge of change. His teacher built on his gestures, forcing him to notice that there was a five on each side:

“I am going to cover this up (while covering up the 7 and 6 with her hand). Now what do you see on both sides? Five and five, right?” (Goldin-Meadow & Singer, 2003, p. 516)

Teachers convey information through gesture. For example, 7- to 10-year-olds were taught, with or without gestures, about equivalence across an equal sign (e.g., 8 + 6 + 2 =+ 2). The group whose teacher used gestures learned more (Cook, Duffy, & Fenn, 2013). See  Figure 7.11. Learners are more likely to learn concepts when instruction includes  both speech and gesture—from teaching preschoolers how to count to teaching high school students physics concepts (Ping & Goldin-Meadow, 2008). For example, in teaching the preceding equivalency problem, you might point at the 7 and 6, then flick away at the 5s.

Figure 7.11Gesture Facilitates Learning

Proportion of problems correct for 2nd- and 3rd-graders who were taught math equivalence with speech alone compared with those taught with speech plus gesture.

Source: Cook, Duffy, and Fenn (2013).

Two important implications for your classroom from research on nonverbal language are:

1. Encourage learners to use gestures during explanations and problem-solving tasks. After activities, encourage learners to describe their experience using gestures rather than just asking them to “write it up.” This promotes deeper understanding and gives you a chance to clarify misunderstandings.

2. Use gestures in your instruction. You can convey problem-solving strategies to learners by combining speech and gestures as the teacher described earlier did.

Pygmalion in the Classroom

When teachers hold high expectations for their learners, they tend to learn more. This is called the  Pygmalion effect (named after a Greek myth) or the  teacher expectation effect (Rosenthal & Jacobson, 1966). This effect has been found in many studies.  

How might teacher expectations affect learner’s achievement? One explanation is that teachers communicate their expectations to learners nonverbally without being aware of it. Even when teachers try to hide low expectations and are good at emotional dissemblance, their real expectations leak out in body language and behavior (Porter & ten Brinke, 2008). Teachers tend to do the following at higher rates for learners toward whom they have high expectations: express warmth, smile at, call on, teach, wait for answers, and give informative feedback (Rosenthal, 2003). In contrast, when teachers have low expectations of learners, they ask easy questions, place learners in low-ability groups, use competitive motivation and grading strategies so losers are obvious, and make negative comments about certain learners (Bohlmann & Weinstein, 2013).

An important implication for your classroom is to  convey high expectations to every child. Show warmth, demand strong answers, and provide plenty of feedback to all learners, particularly low-SES and minority boys who may be most vulnerable to low teacher expectations (Hinnant, O’Brien, & Ghazarian, 2009). In one experiment, teachers recorded a lesson to analyze how their own nonverbal behavior communicated expectations of learners. Over a year, learners’ math achievement increased for the students in their classes as compared with a control group (Rubie-Davies, Peterson, Sibley, & Rosenthal, 2015).

Promote Literacy

Typically, in preschool through 2nd grade, schools emphasize phonological awareness, decoding, and beginning writing. From 3rd grade through high school, the emphasis shifts to fluency, comprehension, and advanced writing. This is because phonological awareness is the foundation of reading ability (Dickinson et al., 2003). You promote your learners’ phonological awareness when you:

1. Directly teach learners the names of letters and their corresponding sounds (Hulme, Bowyer-Crane, Carroll, Duff, & Snowling, 2012). Decoding  skills are the result of phonological awareness and knowledge of letters and their sounds (Melby-Lervåg & Lervåg, 2014). Remember to give more time to letters whose names do not match their sounds. Around age 3 to 4, children are particularly interested in the first letter of their name. Children whose parents talk about the first letter in their name tend to be better readers at the end of kindergarten than children whose parents talk less or not at all about the first initial (Treiman et al., 2015).

2. Sensitize learners to phonemes through games and nursery rhymes. For example, in one game, learners take turns saying a word that begins with the same sound that the previous child’s word ended in:  Apple—Lion—Nap—Pepper. Learning nursery rhymes is linked to phonological awareness, probably because nursery rhymes play with sounds (Snowling, Gallagher, & Frith, 2003; Williams & Rask, 2003).

3. Give learners spelling lists. Young learners who learn to spell words correctly learn to decode words better (Conrad, 2008).

4. Directly teach vocabulary, especially with low-income learners before age 6 (Hadley, Dickinson, Hirsh-Pasek, Golinkoff, & Nesbitt, 2016; Marulis & Neuman, 2013; Quinn, Wagner, Petscher, & Lopez, 2015). This can include using unusual words during instruction and then defining them. Poverty is a risk factor that strongly predicts poor vocabulary knowledge.

5. Read to learners. Reading may be most effective if you read in small groups (three to four learners); if you read rather than tell the story and when you read about four to five times a week (Adams, Treiman, & Pressley, 1998). Other aspects of reading are important as well:

· Talk about the book. As you learned above, use some high-demand questions. Simple yes/no questions and pointing to pictures are appropriate with toddlers. Ask older learners to predict events or analyze characters before and after reading, but not during the story, such as, “How do you think he feels?” Talk about new vocabulary words.

· Read a balance of familiar and unfamiliar books. When you read the same story a few times (there are diminishing effects after the second time), learners comment more on the story, particularly low-ability learners. However, new books expose learners to new vocabulary.

· Read information books, like books on weather or animals. Expository books tend to elicit more child talk than storybooks (Fletcher & Reese, 2005).

Each of the classroom implications discussed thus far pertains to all learners, whether native English speakers or not. However, teaching learners who are ELL brings added considerations.

Support Bilingual Learners

Imagine taking a chemistry class in Twi (a language in Ghana) or another language you do not speak well. You would not learn as much as if the class were in English. In the United States,  bilingual education refers to the use of heritage language for instruction with ELL learners. There are many different models of bilingual education. One model is to provide all instruction in the heritage language and then transition to an all-English classroom. Another model is to teach academic content mainly in English, with occasional tutoring for the same content in the heritage language. Yet another model is to teach academic content only in English and then teach English as a second language (ESL). Still another, nonbilingual, model is to simply immerse learners abruptly in English-only classrooms. The first two models are not an option in small school districts where few teachers are fluent in other languages, but they are feasible in large districts with high concentrations of a single heritage language, such as Spanish in Arizona.

Which approach is best? The answer to this politically charged question is not completely clear. Some studies show an advantage for bilingual education over immersion, but some show no differences. Rarely do studies show an advantage for immersion. Research supports the following guidelines for working with ELL learners (Baker et al., 2014; Dixon et al., 2012):

1. Devote time to teaching English. This provides opportunities to practice speaking. Make sure learners use full sentences in their responses, although they may resist. It may surprise you, but in many ELL classrooms learners have little opportunity to speak English.

2. Integrate oral and written English language instruction into content-area teaching. Engage learners in academic discussions about the course content, such as language arts, science, or math. You could encourage speech through cooperative learning and viewing short video clips that are then discussed.

3. Directly teach vocabulary. This includes everyday words and academic words in content areas like math, science, or history (Jansen, 2008). For example, learners may struggle with comparative words like  probably, very likely, and  almost certain, or math words like  estimate.

4. Teach Standard English grammar and syntax explicitly. Do not expect learners to become proficient in English by chance as you teach math, social studies, or other content areas. Provide clear, frequent feedback on grammar and syntax.

5. Teach language skills as early as possible. Learners who enter kindergarten with adequate English skills tend to do as well as native speakers, but learners who do not may lag in achievement throughout elementary school (Kieffer, 2008).

6. Build strong skills in the heritage language. Having a strong foundation in their first language helps preschoolers develop skills in English (their second language) faster and transition to English-only classrooms faster (Proctor, August, Carlo, & Snow, 2006).

7. Encourage English use in informal settings at school. Learners who use English in the hallway, cafeteria, playground, and with friends are more successful (Carhill, Suárez-Orozco, & Paez, 2008).

8. Remember that Standard English lags behind conversational English. Learners who can converse in English with you and who seem English-proficient may struggle to follow academic content and score poorly on standardized tests. When possible, test learners in their dominant language. ELL learners may know more math, science, and history than they can demonstrate in English (Abedi, 2004).

9. Support  additive bilingualism . This is where learners maintain proficiency in their heritage language while becoming proficient in English.  Subtractive bilingualism occurs when learners learn a second, majority language in a context that does not value their heritage language, which they eventually lose. Learners who are fluent in  both English and their heritage language fare better academically and emotionally.

Learners who must cross cultural borders may be inclined to choose one culture over the other. You can encourage them to be comfortable in English U.S. culture without losing their heritage culture; they can learn to feel positive about both cultures. ELL learners tend to participate more in classes where they feel welcome and where they share aspects of their heritage cultures, compared to classes where they feel like outsiders or invisible (Yoon, 2008).

These issues also apply to nonimmigrant learners who have a mismatch between language at school and at home. You have already seen that dialects like AAE are different from Standard English. In addition, many learners have limited opportunity to learn school vocabulary. Furthermore, the ways that learners use language at home may differ from the way it is used at school (Hemphill & Snow, 1996). At home, family members scaffold, butt in, and clarify what a child is saying; children talk when they have something to say. In contrast, at school, the teacher controls who gets to talk and what the talk is about. Speaking in class often consists of one-word comments, with very little elaboration. Expository talk, rather than conversation, is used.  Expository talk is formal, precise, and used to display information—like when a child is asked to summarize a text. You may need to help your learners develop expository talk. This can be done through direct instruction and indirectly through exposure to reading and writing.

7-5Putting the Theories to Work: The Case of Mathematics

To this point, you have learned about four major theories of learning and cognition—behaviorism, information processing, Piaget’s cognitive development theory, and Vygotsky’s sociocultural theory. In this section, you find out how to apply what you have learned about these theories to your classroom, using the domain of mathematics as an example. You are also introduced to concepts that are discussed in further detail in  Chapters 11 and  15. Key concepts are italicized. First, we begin with an overview of early math development.

7-5aAge Trends in Early Mathematics

You may not think of infants or preschoolers as budding mathematicians, but they do have skills that may surprise you. Scientists have developed clever ways of investigating those skills.

Infant Math

A 5-month-old sits before a large box (see  Figure 7.12). He sees a hand holding a mouse doll enter the box from a hole in the side and place the doll in the box. The hand retreats, empty. A screen rotates up to hide the doll. The infant sees the hand enter from the side with a doll again. It presumably adds another doll behind the screen, because it also retreats empty. The screen then rotates down to reveal only one doll. (A doll was removed through a trap door.) The infant looks longer at this impossible event than he did in preceding trials when two dolls were revealed.

Figure 7.12Infant Number Sense

Infants look longer at the impossible event, suggesting rudimentary number sense.

Source: Wynn (1992).

We suspect the infant looks longer at the impossible event because he is surprised that there are not two dolls in the box. Chimpanzees have similar reactions (Beran & Beran, 2004). Does this mean that infants and chimpanzees can count? Can they add? At age 3, our daughter enthusiastically announced a revelation: “6! There are 6 people in our family: 3 girls and 3 boys!” If infants, 3-year-olds, and chimpanzees can do basic math, why do children have so much difficulty learning math in school?

Infants have an intuitive, imprecise sense of number (called the  approximate number system) that stays with them across their lifetime. For example, some 6-month-olds will gaze longer at a screen that shows dots that change in number than a screen that shows the same number of dots that change color (Starr, Libertus, & Brannon, 2013). This suggests they are able to detect changes in quantity. Because number sense is present as soon as infants are old enough to be testable, it may be a biologically determined, innate ability that does not depend on learning—or a  core domain (Feigenson, Libertus, & Halberda, 2013). However, infants can succeed in these types of tasks only with small quantities (Desrochers, 2008). Furthermore, infants’ innate number sense stands in stark contrast to other math concepts like propositions, percentages, and algebra that are acquired later with considerable effort and varied success.

Informal Math

Preschoolers’ understanding of basic math concepts is referred to as “informal math” because it is acquired without formal schooling. What kinds of math abilities do preschoolers have? They understand that adding to a set produces more and taking away produces less. They can tell which quantities are larger. For example, they know a row of 12 has more than a row of 8 objects, even before they can count. Typical children begin counting around 2 years. For example, 22-month-old Connor counted his blocks by pointing to each one in turn and saying, “nine, nine, nine, nine.” He had the concept of assigning a number to each object, but he did not yet know the number names except nine. Learning number names begins at 2 to 3 years. At 4 to 5 years, most children can count up to 20 or even 100, and may use a combination of finger and verbal counting. They can evenly divide treats ( Photo 7.3). They can also solve simple arithmetic problems: “If you had 4 candies and someone gave you 3 more, how many candies would you have?” (Engel, Claessens, & Finch, 2013; Huntley-Fenner & Cannon, 2000).

Photo 7.3

Use math talk with toddlers and preschoolers, like “How many candies do you have?”

Denise Hager/Catchlight Visual Services/Alamy Stock Photo

Although most children will acquire these informal math skills before entering school, low-SES preschoolers may not (Jordan, Kaplan, Olah, & Locuniak, 2006). This is a concern because, as you learned in  Chapter 1, a strong predictor of academic achievement—even stronger than reading ability—is math skills at entry into kindergarten (Duncan et al., 2007; Romano, Babchishin, Pagani, & Kohen, 2010). Given that other preschoolers are developmentally ready to reason about math, low-SES preschoolers’ meager math skills may be due to less opportunity to learn these concepts.

School-Age Math

As children enter kindergarten, most are skilled at counting and understand numbers up to 10 (Engel et al., 2013). They commonly progress through the following strategies when solving a simple problem like 2 + 7: counting all (1, 2, 3, 4, 5, 6, 7, 8, 9), to counting on from the first number (2, 3, 4, 5, 6, 7, 8, 9), to counting on from the largest number (7, 8, 9). Thus, children are  using more sophisticated strategies and becoming faster and more accurate at addition. They are beginning to understand place value, and most master it by 2nd grade (Mix, Prather, Smith, & Stockton, 2014). If they do not, they are likely to have long-term math difficulties because this is foundational to higher-level mathematics (Byrge, Smith, & Mix, 2014).

To master fractions and other mathematical concepts, children must transition from additive to multiplicative reasoning. Children can be taught multiplication/division as young as age 4, like: “If 4 dogs want 3 treats each, how many treats do you need?” Typical children develop these concepts without instruction by age 6 for small numbers, perhaps because they have experience with division each time they share with peers. However, understanding the concept does not mean they will do computations correctly or use efficient strategies. This takes instruction and practice. In U.S. schools, relevant instruction usually begins in 2nd grade. Children who are going to have serious difficulties with math are usually identified by about 3rd grade.

Children’s counting strategies are eventually replaced by  memorizing facts like  and . Doubles (5 + 5 and 6 × 6) are memorized especially rapidly. These facts are stored in long-term memory as the result of frequent  spaced practice. Retrieval is an efficient  problem-solving strategy, as it becomes more rapid and automatic. Children then progress to using known facts to  reason about unknown facts (, so 9 + 8 must be 17). Some general rules are learned easily, like to get 10 n just put a 0 on the right side of  n, or 1 n is always  n. Knowing such general rules frees the child from having to memorize multiples of 10 or 1. Thus, memorized facts facilitate reasoning (De Brauwer & Fias, 2009; Sophian & Madrid, 2003).

Estimation is another important skill that develops in school-age children. Estimation is used often in daily life, like estimating how much each team member will have to contribute to buy the coach a $50 gift, and is foundational to math ability. One way estimation is measured is by giving children a number line with only 0 and 100 indicated at each end. The child is asked where a number, like 29, would be on the line. Preschoolers and kindergarteners typically can place numbers accurately on a 0 to 10 number line; 2nd-graders on a 0 to 100 number line; 6th-graders on a 0 to 1,000 number line. For some adolescents, the number line extends backward to negative numbers and broadens to include fractions. Thus, the ability to use a number line improves with age (Siegler & Lortie-Forgues, 2014). However, there are individual differences within each grade. Children who are better at number-line estimation have higher math test scores (Schneider, Grabner, & Paetsch, 2009; Siegler & Booth, 2004). Number-line estimation is important because it helps children understand the meaning of numbers and facilitates learning math in school.

7-5bImplications for Teachers from Different Theories

This brief review raises three questions:

· (1)

What causes these age trends in math?

· (2)

What causes individual differences in math? Stable individual differences in numeracy are apparent in toddlers (Feigenson et al., 2013). Over time, some learners will master calculus, but others will fail to develop basic math competence.

· (3)

How should math be taught?

Apparently some number sense (e.g., adding 1 + 1) is a core domain and does not need to be taught. However, more advanced and precise mathematics (e.g., adding ) is not a core domain and develops slowly over many years. There is still much to be learned that requires formal schooling. The answer to these three questions varies with each of the four major theories you have learned about.

Behaviorism and Math

From a behaviorist perspective, learning (or conditioning) begins with simple stimulus–response connections and then progresses to the complex level of abstract reasoning. Students cannot solve advanced problems if they have not mastered the prerequisite low-level skills. Research supports this premise of behaviorism. That is, preschoolers with stronger informal math abilities will come to understand fractions better in elementary school (Feigenson et al., 2013; Zhang et al., 2014), which will lead to greater success later in algebra (Siegler et al., 2012; Watts et al., 2015), which is essential for still later success in higher mathematics and science.

Behaviorists tend to use  behavioral objectives—statements of specific behaviors students must manifest to show that they have learned—to structure lessons. These lessons are organized hierarchically, with basic skills mastered before attempting advanced skills. Behaviorists also tend to emphasize  direct instruction with drill and  practice to create strong basic connections. Some behaviorist teachers use the “mad minute”; students do as many math problems as possible in 60 seconds, several times a week, until the skill becomes automatic. Research supports the effectiveness of practice for building a rapid, automatic fact base (in addition to other instruction), suggesting that it may even help narrow the achievement gap for low-SES children (Gersten et al., 2015).

Drill-and-practice does not have to be rigid and boring; it can be play-like. In fact, structured block play (rather than free play), in which you ask preschoolers to imitate a model with blocks or Legos or some other building materials ( Figure 7.13), improves their mathematical skills (Verdine et al., 2014). The National Council of Teachers of Mathematics (NCTM) has called for instruction in spatial skills as a basic foundation for math in early childhood. There are several play-like math curricula such as  Big Math for Little Kids, Building Blocks, Number Worlds, and  Rightstart that are designed for 3- to 5-year-olds, based on developmental science (Clements & Sarama, 2008; Ginsburg, Lee, & Boyd, 2008). The  Building Blocks curriculum involves  asking children to explain their strategies (e.g., “How did you know?”) and  spacing practice over time. Some preschool teachers resist using math curricula, preferring that children learn through teachable moments in naturally emerging play. Unfortunately, many such moments are overlooked and those that are noticed may not provide enough opportunity to learn for low-SES children (Ginsburg et al., 2008).

Figure 7.13Visual–Spatial Play Builds Math Skills

Structured block play, in which you ask preschoolers to imitate a model you have made, requires children to follow a multistep process using spatial information, measurement, and multiple strategies. This kind of activity improves mathematical skills and elicits more “math talk” from teachers. Almost all children in a Head Start group could imitate the first sample, but fewer than 10% could imitate the second accurately.

Source: Adapted from Verdine et al. (2014).

According to a behaviorist perspective, some aspects of learning should not be substantially more difficult than others, yet they are. For example, young children understand that every number has a number after it, so the concept of infinity does not have to be taught. However, fractions have to be taught, and most children struggle to understand them. The fact that a fraction like  is larger than  does not readily map onto what children know about numbers—4 is bigger than 2, so  should be bigger than . Existing knowledge structures can  interfere with new learning. The cognitive developmental model explains this better than behaviorism.

Piaget’s Theory of Cognitive Development and Math

According to Piaget’s cognitive developmental theory, children construct knowledge. This means children reinvent number concepts on their own, based on experience. Ask a young child to write 642. You might get 600402. This misconception is an intelligent mistake. Children extract rules from the multidigit numbers that saturate their world (e.g., street addresses, labels on products). They construct their own knowledge of place value, such as numbers to the left are larger than numbers to the right; numbers with more digits are bigger; and zeros hold place value (Byrge et al., 2014). However, as learners  assimilate new experiences with what they already know, they can create  misconceptions that teachers may not have intended. For example, if 3rd- and 5th-graders are given an equation like , they typically will respond with 23 instead of 9. They make this mistake because they have had extensive practice in school with problems of the “” structure. They then infer false rules like “all equations have the form ‘operations = answer’” and “the equal sign means ‘the total’” (McNeil, 2014). From a constructivist perspective, errors are intelligent and a natural part of knowledge construction. Errors give you a window into the child’s thinking process.

Constructivism is probably the most popular approach in mathematics education and is clearly reflected in the NCTM standards. A constructivist teacher guides reinvention of math and emphasizes hands-on tools to illuminate concepts. This involves direct manipulation of materials relevant to math whenever possible and emphasis on student-initiated problem-solving activities. In popular constructivist curricula like  Investigations in Number, Data, and Space, algorithms are de-emphasized. Rather than mastering rules and procedures, learners are expected to develop a deep understanding of mathematics. They become flexible strategy users, with mental arithmetic playing a central role.

What does the research say about this approach? Research suggests that being taught number words promotes conceptual understanding of math concepts in toddlers, in contrast to Piaget’s view that counting is a rote skill and merely singsong to toddlers (Baroody, Li, & Lai, 2008). His view was that conceptual understanding depends on logical abilities, such as classifying and ordering. Research also suggests that although preschoolers come to understand place value to some extent without direct instruction, direction instruction helps (Mix et al., 2014). Furthermore, research suggests that accurate pictures, like diagrams or graphs, may promote math more than manipulatives or self-constructions. For example, in one study 1st-graders were either given accurate pictures of addition problems on a number line or they were asked to generate their picture, like 29 + 17 in  Figure 7.14. Children who were given the accurate picture learned addition better than those who generated their own (Booth & Siegler, 2008).

Figure 7.14Estimation on a Number Line

A 1st-grader was asked to first show where 29 would be on a number line in red ink. Then he was asked to show where 17 would be in blue ink. He was then asked where 29 + 17 would be in turquoise ink. His response was fairly accurate. Based on the research, is this boy likely to have high achievement in school? Try this with younger and older children, and 0 to 10 or 0 to 1,000 number lines.

Among older students, when direct instruction and invention are compared, students tend to perform better following direct instruction. For example, students will solve complex division problems (e.g., 736 ÷ 32) more accurately if they are taught to write it out in a standard algorithm (Hickendorff, van Putten, Verhelst, & Heiser, 2010). In one study, 3rd- and 5th-graders were taught equivalency (e.g., ). Some were directly instructed to “add 4, 9, and 6 together, then subtract the 4 that’s over here, and that amount goes in the blank.” Other students were left to invent their own strategy, with feedback about whether the answer was correct. Instructed students got more problems correct and were more likely to transfer the skill to new, different problems. More than a fourth of the children in the invention method never developed a correct procedure (Rittle-Johnson, 2007).

An important lesson from this, and other studies, is that  asking children to explain their strategy is important. Explainers learned more regardless of instructional approach (Rittle-Johnson, 2007). This suggests the instructional approach may not matter as much as getting children to actively process the strategy. Ironically, explainers were more likely to invent a second way to solve the problem following direct instruction. Thus,  direct instruction does not preclude invention, but it does help prevent the invention of incorrect strategies. Unfortunately, research suggests that even exemplary teachers seldom ask students to provide explanations, although they may use hands-on activities effectively (Silver, Mesa, Morris, Star, & Benken, 2009).

Piaget was right that children can reinvent math to a limited extent. Children do develop logic and number sense on their own. Thinking is a source of knowledge. However, children cannot by themselves efficiently reinvent the mathematical notation system or complex theorems, which is a cultural tool they need for strong math achievement.

Vygotsky’s Sociocultural Theory and Math

According to sociocultural theory, social interaction is an important source of knowledge. You promote young children’s math ability by using math talk, such as “You can have three crackers” or “Count how many cups we need for snack.” Use math-related vocabulary often (e.g., minus, bigger, altogether, edge, short, middle). This helps children develop better number sense (e.g., Jordan, Glutting, Dyosn, Hassinger-Das, & Irwin, 2012). You could also play number-oriented board games. Playing games that involve counting pieces along a number line, like Chutes and Ladders™, or simple homemade games, helps preschoolers develop better math skills, particularly if you ask them to say the number they spun and count the spaces as they move along (Ramani, Siegler, & Hitti, 2012).

Cultural transmission is also an important source of knowledge. A child could draw a fishbowl with eight fish in it and cross out three (they died) to indicate that five are left. Or a child could write . The latter is more efficient, and it is transferable to a wide array of other situations. The math symbol system used in writing  is a cultural tool. Cultural tools transform thinking. Children must be taught these tools. Although infants and toddlers may intuitively get the difference between 1 and 2, the concept of 3 or larger numbers requires social interaction and  scaffolding (Baroody et al., 2008). When children enter school, we expect them to learn their culture’s notation system for representing precise numbers (e.g., ) and mathematical concepts (Feigenson et al., 2013).

School-based instruction, a cultural creation, accelerates mathematical development. This is illustrated by child street vendors in Brazil, who can, without schooling, perform simple arithmetic with 98% accuracy when buying and selling. However, if those same children are given problems in writing (e.g., ), they drop to 37% accuracy (Schliemann & Carraher, 2002). They understand arithmetic but have trouble with mathematical notation. Unschooled child street vendors and 2nd- and 3rd-graders who had learned multiplication in school were asked to solve two problems:

1. A boy wants to buy 3 chocolates that cost 50 cruzeiros each. How much money does he need?

2. Another boy wants to buy 50 chocolates that cost 3 cruzeiros each. How much money does he need?

Schooled children solved the first problem with multiplication and were able to solve the second problem without doing any computation because they understood that . In contrast, the street vendors used addition to solve both problems. Adding 3 cruzeiros 50 times was slow and resulted in errors.

Thus, the sociocultural view is that full development of mathematical ability requires social interaction—opportunities to use and observe strategies, and receive  scaffolding from experts. In school, children should hear and use  math talk when they solve problems because they learn by explaining their reasoning to others.  Cooperative learning in the classroom—which we discuss in  Chapter 13—is linked to greater math achievement (Slavin & Lake, 2008).

The Information-Processing Model and Math

The information-processing model focuses on how children remember and reason. It is complementary to the other theories, with the exception of some aspects of Piaget’s theory. Information-processing theorists accept Piaget’s idea that children construct their own understanding, but they tend to a more direct instructional approach akin to behaviorism and socioculturalism. In addition, rather than portraying children as thinking differently than adults, as Piaget does, they portray children as knowing less and having slower processing and more-limited working memory.

All aspects of information processing are involved in math. To solve the problem , a child must access  long-term memory to get , then maintain this in  working memory while again accessing long-term memory to get .  Long-term memory of addition facts helps them  reason about the problem. W orking memory allows children to compare previously solved problems with the current problem.  Executive functions keep them moving through the steps of problem solving and flexibly selecting the most appropriate strategy. For example, mathematically skilled students use a mental approach on easy items and an algorithm on more difficult problems. Instead of using an algorithm for the problem , they might  chunk the problem like this: , and ; therefore,  (Hickendorff et al., 2010).  Metacognition acts on feedback about whether the strategy was useful or not.

Research shows that both age-related growth in math ability and individual differences in math ability are linked to each component of the information-processing model. For example, learners of all ages who have better  working memory and  executive functions are faster and more accurate at addition, multiplication, algebra, and solving word problems. In contrast, children with slow processing speed and limited working memory are likely to have math disability.  Knowledge also makes a contribution. Learners who know their numbers and youth who can readily retrieve math facts or theorems from long-term memory have higher math achievement.  

This suggests that your role as a teacher is to help learners acquire better information-processing skills and more knowledge. You can help learners  memorize math facts and procedures through  spaced practice and  frequent tests. Instead of practicing one type of problem in a massed block and then moving on to the next type, space practice of each type across time. Having more knowledge will help your students with problem solving, because one of the most efficient problem-solving strategies is simple retrieval of the answer from memory. You can help learners chunk problems to  reduce memory load. You can teach other strategies through  direct instruction and  modeling, although you have to be careful not to overwhelm children’s working memory capacity with complex strategies (Swanson, 2014). Children will gradually shift toward more efficient strategies. You can facilitate this shift by providing  feedback to students. You can also make children aware of their strategies by having them  explain the strategy, which facilitates metacognition.

In summary, some number sense is innate. However, children still have much to learn about math. Although Piaget underestimated young children’s mathematical reasoning ability, he was correct that children construct their own understandings (and misconceptions) of math. Contrary to Piaget’s theory, children do not follow an orderly progression in moving from less-advanced to more-advanced strategies for solving problems, but rather use overlapping strategies. They also do not reinvent the numerical notation system; rather, as Vygotsky pointed out, they learn this cultural tool in informal interactions as well as in formal school settings. Behaviorists tend to focus on very specific behavioral objectives in math learning, with emphasis on skill and drill. Behaviorists and constructivists are often at odds with one another—educators tend to avoid one and support the other. The information-processing model, on the other hand, fits either perspective. Information-processing researchers have demonstrated that children learn by direct instruction, skill-and-drill practice, modeling from others who are more skilled, and constructing their own knowledge through reasoning and metacognition as they receive feedback about the success of their strategies.

How to teach math has been a polarizing issue, particularly when traditional methods (often based on behaviorism) are compared with reform methods (often based on constructivism). Yet, when implemented well, both methods may be effective. Different math curricula are based on different theories, yet there are only small effect sizes on student achievement when researchers compare one curriculum with another (0.10 at elementary, 0.03 at secondary school), probably because most curricula are well designed (e.g., IES, 2011). The What Works Clearinghouse ( Chapter 1) has reviews of many curricula to help your school make good decisions. It also has practice guides for early childhood, elementary, and secondary school math instruction (see resource list). Other aspects of teaching, such as using cooperative learning, promoting time on task, and motivating students, have larger effect sizes than curricula (Harwell et al., 2009; Slavin, Cheung, Groff, & Lake, 2008; Slavin & Lake, 2008). You learn about these topics in later chapters, so stay tuned.