Learning is rapid. By age 11, some children beat their elders at chess, play music that adults pay to hear, publish poems, or win trophies for spelling or sports or some other learned skill. Others survive on the streets or kill in wars, mastering lessons that no child should have to study. How do they learn so quickly?

Piaget called the cognition of middle childhood concrete operational thought, characterized by new concepts that enable children to use logic. Operational comes from the Latin word operare, meaning “to work; to produce.” By calling this period operational, Piaget emphasized productive thinking.

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The school-age child, no longer limited by egocentrism, performs logical operations. Children apply their new reasoning skills to concrete situations, which are situations grounded in actual experience, like the concrete of a cement sidewalk. Concrete thinking arises from what is visible, tangible, and real, not abstract and theoretical (as at the next stage, formal operational thought). A shift from preoperational to concrete operational thinking occurs between ages 5 and 7: Children become more systematic, objective, scientific—and educable.

One logical operation is classification, the organization of things into groups (or categories or classes) according to some characteristic that they share. For example, family includes parents, siblings, and cousins. Other common classes are animals, toys, and food. Each class includes some elements and excludes others; each is part of a hierarchy.

Food, for instance, is an overarching category, with the next-lower level of the hierarchy being meat, grains, fruits, and so on. Most subclasses can be further divided: Meat includes poultry, beef, and pork, each of which can be divided again. Adults realize that items at the bottom of a classification hierarchy belong to every higher level: Bacon is always pork, meat, and food, but most food, meat, and pork are not bacon. However, the mental operation of moving up and down the hierarchy is beyond preoperational children.

Piaget devised many classification experiments. For example, a child is shown a bunch of nine flowers—seven yellow daisies and two white roses. Then the child is asked, “Are there more daisies or more flowers?”

Until about age 7, most children answer, “More daisies.” The youngest children offer no justification, but some 6-year-olds explain that “there are more yellow ones than white ones” or “because daisies are daisies, they aren’t flowers” (Piaget et al., 2001). By age 8, most children can classify: “More flowers than daisies,” they say.

Several logical concepts were already discussed in Chapter 9—in the explanation of ideas that are beyond preoperational children, such as conservation and reversibility.

Another example of concrete logic is seriation, the knowledge that things can be arranged in a logical series. Seriation is crucial for using (not merely memorizing) the alphabet or the number sequence. By age 5, most children can count up to 100, but because they do not yet grasp seriation, they cannot correctly estimate where any particular two-digit number would be placed on a line that starts at zero and ends at 100 (Meadows, 2006).

Concrete operational thought correlates with primary school math achievement, although many other factors contribute (Desoete et al., 2009). For example, logic helps with arithmetic: Children at the stage of concrete operational thought eventually understand that 12 + 3 = 3 + 12 and that 15 is always 15 (conservation), that all the numbers from 20 to 29 are in the 20s (classification), that 134 is less than 143 (seriation), and that if 5 × 3 = 15, then 15 ÷ 5 is 3 (reversibility). (Developmental Link: These concepts are explained in Chapter 9 and detailed in a recently reissued classic, Inhelder & Piaget, 1964/2013a.)

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Logic connects to math ability, as just shown. However, researchers find more continuity than discontinuity as children master number skills. Thus, Piaget’s stage idea was mistaken: There is no sudden shift in logic between preoperational and concrete operational intelligence.

Nonetheless, Piaget’s experiments revealed something important. School-age children can use mental categories and subcategories more flexibly, inductively, and simultaneously than younger children can. They are cognitively advanced, capable of thinking in ways that younger children are not.

Like Piaget, Vygotsky felt that educators should consider children’s thought processes, not just the outcomes. He appreciated the fact that children are curious, creative learners. For that reason, Vygotsky believed that an educational system based on rote memorization rendered the child “helpless in the face of any sensible attempt to apply any of this acquired knowledge” (Vygotsky, 1994a, pp. 356–357).

Unlike Piaget, Vygotsky stressed the centrality of instruction. Vygotsky believed school could be crucial for cognitive growth. He thought that peers and teachers provide the bridge between developmental potential and needed skills via guided participation and scaffolding, in the zone of proximal development. (Developmental Link: Vygotsky’s theory is discussed in Chapter 2 and Chapter 9.)

Confirmation of the role of social interaction and instruction comes from a U.S. study of children who, because of their school’s entry-date cutoff, are either relatively old kindergartners or quite young first-graders. At the end of the school year, achievement scores of the 6-year-old first-graders far exceeded those of kindergarten 6-year-olds who were only one month younger (Lincove & Painter, 2006). Obviously, they had learned a great deal from their year of first grade instruction.

Internationally as well, children who begin first grade earlier tend to be ahead in academic achievement compared to those who enter later, an effect noted even at age 15. The author of this study noted that Vygotsky’s theory is not the only one that explains this finding and that these results were not found in every nation (Sprietsma, 2010). However, no matter what explanation is correct, children’s academic achievement seems influenced by social context.

Vygotsky would certainly agree with that, and he would explain those national differences by noting that education before first grade in some nations is far better than in others. Remember that Vygotsky believed education occurs everywhere, not only in school. Children learn as they play with peers, watch television, eat with their families, walk down the street. Every experience, from birth on, teaches them something, with some contexts much more educational than others.

For instance, a study of the reading and math achievement of more than one thousand third- and fifth-grade children from ten U.S. cities found that high-scoring primary school children were likely to have had extensive cognitive stimulation. There were three main sources of intellectual activity:

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In this study, most children from families of low socioeconomic status did not experience all three sources of stimulation, but those who did showed more cognitive advances by fifth grade than the average high-SES child (Crosnoe et al., 2010).

Generally, poverty reduces children’s achievement because they are less likely to have these three sources of stimulation. However, for low-SES children especially, maternal education makes a notable difference in academic achievement—presumably because educated mothers read, listen, and talk to their children more. Also, an educated mother is more likely to place her child in a preschool with a curriculum that encourages learning.

Vygotsky’s emphasis on sociocultural contexts contrasts with Piaget’s maturational, self-discovery approach. Vygotsky believed that cultures (tools, customs, and mentors) are powerful educators. For example, if a child is surrounded by reading adults, by full bookcases, by daily newspapers, and by street signs, that child will read better than a child who has had little exposure to print, even if both are in the same classroom.

The same applies to math. If children learn math in school, they are proficient at school math; if they learn math out of school, they are adept at solving mathematical problems in situations similar to the context in which they learned (Abreu, 2008). Ideally, though, children learn math both in and out of school.

Context affects more than academic learning. A stunning example of knowledge acquired from the social context comes from Varanasi, a city in northeast India. Many Varanasi children have an extraordinary sense of spatial orientation: They know whether they are facing north or south, even when they are inside a room with no windows. In one experiment, children were blindfolded, spun around, and led to a second room, yet many still knew which way they were now facing (Mishra et al., 2009). How did they know? Perhaps social context.

In Varanasi, everyone refers to the spatial orientation to locate objects. (The English equivalent might be, not that the dog is sleeping by the door, but that the dog is sleeping southeast.) From their early days, children learn north/south/east/west in order to communicate. By middle childhood, they have an internal sense of direction.

Culture affects how children learn, not just what they learn. This was evident in a two-session study in California of 80 Mexican American children, each with a sibling (Silva et al., 2010). Half of the sibling pairs were from indigenous Mexican families, in which children learn by watching, guided by other children. The other half were from families acculturated to U.S. learning, via direct instruction, not observation. Those acculturated children had learned to work independently, sitting at desks, not collaboratively, crowding around a teacher.

In the first session of this study, a Spanish-speaking “toy lady” showed each child how to make a toy while his or her sibling sat nearby. First, the younger sibling waited while the older sibling made a toy mouse, and then the older sibling waited while the younger sibling made a toy frog. Each waiting child’s behavior was videotaped and coded every 5 seconds as sustained attention (alert and focused on the activity), glancing (sporadic interest, but primary focus elsewhere), or not attending (looking away).

A week later, each child individually was told there was some extra material to make the toy that his or her sibling had made the week before, and was encouraged to make the mouse or the frog (whichever one that child had not already made). In this second session, the toy lady did not give the children step-by-step instructions as she had for the sibling a week earlier, but she had a long list of hints she could give if the child needed them.

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The purpose of this experiment was to see how much the children had learned by observation the week before. The children from indigenous backgrounds scored higher, needing fewer hints, because they had been more attentive when their siblings made the toy (Silva et al., 2010) (see Figure 12.1).

The same conclusions have been found in other research. For example, in another study, children born and raised in the United States who are accustomed to learning by observation (as in some Native American cultures) were more proficient at remembering an overheard folktale (Tsethlikai & Rogoff, 2013).

Today’s educators and psychologists regard both Piaget and Vygotsky as insightful. International research confirms the merits of their theories. Piaget described universal changes; Vygotsky noted cultural impact.

A third approach to understanding cognition adds crucial insight. The information-processing perspective benefits from technology that allows much more detailed data and analysis than was possible for Piaget or Vygotsky. (Developmental Link: Information processing is introduced in Chapter 2.)

Thousands of researchers who study cognition can be said to use the information-processing approach. Not all of them would describe themselves as doing so because “information processing is not a single theory but, rather, a framework characterizing a large number of research programs” (Miller, 2011, p. 266).

The basic assumption of all such research programs is that, like computers, people can access large amounts of information. They then: (1) seek relevant units of information (as a search engine does), (2) analyze (as software programs do), and (3) express their conclusions so that another person can understand (as a networked computer or a printout might do). By tracing the paths and links of each of these functions, scientists better understand the learning process.

The brain’s gradual growth, now seen in neurological scans, confirms the usefulness of the information-processing perspective. So does data on children’s school achievement: Absences, vacations, new schools, and even new teachers may set back a child’s learning because learning each day builds on the learning of the previous day. Brain connections and pathways are forged from repeated experiences, allowing advances in processing. Without careful building and repetition of various skills, fragile connections between neurons break.

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One of the leaders of the information-processing perspective is Robert Siegler. He has studied the day-by-day details of children’s cognition in math (Siegler & Chen, 2008).

Apparently, children do not suddenly grasp the logic of the number system, as Piaget expected at the concrete operational stage. Instead, number understanding accrues gradually, with new and better strategies for calculation tried, ignored, half-used, abandoned, and finally adopted. Siegler compared the acquisition of knowledge to waves on an ocean beach when the tide is rising. There is substantial ebb and flow, although eventually a new level is reached.

One example is the ability to estimate where a number might fall on a line, such as where the number 53 would be placed on a line from zero to 100. This skill predicts later math achievement (Libertus et al., 2013). U.S. kindergartners are usually lost when asked to do this task; Chinese kindergartners are somewhat better (Siegler & Mu, 2008).

Everywhere, proficiency gradually builds from the first grade on, predicting later math skills (Feigenson et al., 2013). This has led many information-processing experts to advocate giving children practice with number lines in order to develop math ability, such as the ability to do multiplication and division.

Curiously, knowing how to count to high numbers seems less important for math mastery than being able to estimate magnitude (Thompson & Siegler, 2010). For example, understanding the size of fractions (e.g., that 3/16 is smaller than 1/4) is connected to a thorough grasp of the relationship between one number and another, a skill that predicts later math achievement internationally, according to a study of school children in China, Belgium, and the United States (Torbeyns et al., 2014). Overall, information processing guides teachers who want to know exactly which concepts and skills are crucial foundations for mastery, not only for math but for reading and writing as well.

Many scientists who study memory take an information-processing approach. They have learned that various methods of input, storage, and retrieval affect the increasing cognitive ability of the schoolchild. Each of the three major steps in the memory process—sensory memory, working memory, and long-term memory—is affected by both maturation and experience.

Sensory memory (also called the sensory register) is the first component of the human information-processing system. It stores incoming stimuli for a split second, with sounds retained slightly longer than sights. To use terms explained in Chapter 5, sensations are retained for a moment, and then some become perceptions. This first step of sensory awareness is already quite good in early childhood. Sensory memory improves slightly until about age 10 and remains adequate until late adulthood.

Once some sensations become perceptions, the brain selects the meaningful ones and transfers them to working memory for further analysis. It is in working memory (formerly called short-term memory) that current, conscious mental activity occurs. Processing, not mere exposure, is essential for getting information into working memory; for this reason, working memory improves markedly in middle childhood (Cowan & Alloway, 2009) (see Table 12.1).

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As Siegler’s waves metaphor suggests, memory strategies for processing information do not appear suddenly. Gradual improvement occurs from toddlerhood through adolescence (Schneider & Lockl, 2008). Children develop strategies to increase working memory (Camos & Barrouillet, 2011), and they use these strategies occasionally at first, then consistently.

Cultural differences are evident. For example, many Muslim children are taught to memorize all 80,000 words of the Quran, so they learn strategies to remember long passages. These strategies are unknown to non-Muslim children, and they help with other cognitive tasks (Hein et al., 2014). A very different example is the ability to draw a face, an ability admired by U.S. children. They learn strategies to improve their drawing, such as knowing where to put the eyes, mouth, and chin when drawing a face. (Few spontaneously draw the eyes mid-face, rather than at the top, but most learn to do so.)

Finally, information from working memory may be transferred to long-term memory, to be stored for minutes, hours, days, months, or years. The capacity of long-term memory—how much can be crammed into one brain—is huge by the end of middle childhood. Together with sensory memory and working memory, long-term memory organizes ideas and reactions, fostering more effective learning over the years (Wendelken et al., 2011).

Crucial to long-term memory is not merely storage (how much material has been deposited) but also retrieval (how readily past learning can be brought into working memory). For everyone at every age, retrieval is easier for some memories (especially of vivid, emotional experiences) than for others. And for everyone, long-term memory is imperfect: We all forget and distort memories, with strategies needed for accurate recall.

Research on information processing finds that the more people already know, the more information they can learn. Having an extensive knowledge base, or a broad body of knowledge in a particular subject, makes it easier to remember and understand related new information. As children gain knowledge during the school years, they become better able to judge what is true or false, what is worth remembering, and what is insignificant (Woolley & Ghossainy, 2013).

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Three factors facilitate increases in the knowledge base: past experience, current opportunity, and personal motivation. The last item in this list explains why children’s knowledge base is not what their parents or teachers might prefer. Some schoolchildren memorize words and rhythms of hit songs, know plots and characters of television programs, or can recite the names and histories of basketball players, and yet they do not know whether World War I was in the nineteenth or twentieth century or whether Pakistan is in Asia or Africa.

Motivation provides a clue for teachers: New concepts are learned best if they are connected to personal and emotional experiences. Children who themselves are from South Asia, or who have classmates who are, learn the boundaries of Pakistan more readily.

The mechanisms that put memory, processing speed, and the knowledge base together are control processes they regulate the analysis and flow of information within the system. Control processes include emotional regulation and selective attention (explained in Chapter 10 and Chapter 11, respectively).

Equally important is metacognition, sometimes defined as “thinking about thinking,” understanding how to learn. Metacognition is the ultimate control process because it allows a person to evaluate a cognitive task, determine how to accomplish it, monitor performance, and then make adjustments. According to scholars of cognition, “Middle childhood may be crucial for the development of metacognitive monitoring and study of control processes” (Metcalfe & Finn, 2013, p. 19).

Control processes require the brain to organize, prioritize, and direct mental operations, much as the CEO (chief executive officer) of a business organizes, prioritizes, and directs business operations. For that reason, control processes are also called executive processes, and the ability to use them is called executive function (already mentioned in Chapter 9), which allows a person to step back from the specifics of learning and thinking and consider more general goals and strategies. Executive function is evident whenever people concentrate on the relevant parts of a task, using the knowledge base to comprehend new information and apply memory strategies.

Executive function ability is a foundation for learning in early and middle childhood, measurable already by age 5, and more evident among 10-year-olds than among 4- or 6-year-olds (Masten, 2014; Bjorklund et al., 2009). For example, fourth-grade students can listen to the teacher talk about the river Nile, ignoring classmates who are chewing gum or passing notes.

Deliberate selectivity is a control process at work. A child can decide to do homework before watching television or to review spelling words before breakfast, creating mnemonics to remember the tricky parts. All these signify executive function.

Both metacognition and control processes improve with age and experience. For instance, in one study, children took a fill-in-the-blanks test and indicated how confident they were of each answer. Then they were allowed to delete some questions, with the remaining ones counting more. Already by age 9, they were able to estimate correctness; by age 11, they were skilled at knowing what to delete (Roebers et al., 2009).

Sometimes, experience that is not directly related has an impact. This seems to be true for fluently bilingual children, who must learn to inhibit one language while using another. They are advanced in control processes, obviously in language but also in more abstract measures of control (Bialystok, 2010).

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Control processes develop spontaneously as the prefrontal cortex matures, but they can also be taught. Sometimes teaching is explicit, more so in some nations (e.g., Germany) than in others (e.g., the United States) (Bjorklund et al., 2009). Examples that may be familiar include spelling rules (“i before e except after c”) and ways to remember how to turn a light bulb (lefty-loosey, righty-tighty). Preschoolers ignore such rules or use them only on command; 7-year-olds begin to use them; 9-year-olds can create and master more complicated rules.

Many factors beyond specific instruction affect learning. For example, if children do not master emotional control in early childhood, their school achievement is likely to suffer for years (Bornstein et al., 2013).

Given the complexity of factors and goals, educators disagree as to what should be deliberately taught versus what is best discovered by the child. However, understanding the early steps that lead to later knowledge, as information processing seeks to do, may guide instruction and hence improve learning. Exactly how to do that is an important topic of current research, as you will see in A View from Science on page 391. But first, consider one specific domain of learning that advances during middle childhood, language.

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