During the course of normal brain development, what comes first:

- logical thinking or
- counting?

Counting, easily. It's a matter of rote and repetition, dominated by procedural learning. Critical thinking requires more advanced circuitry and learning. There is a lot of really rapid neural development still occurring in the first couple years and children are learning from society and parents and experience in the meantime. During this time, they can easily count and say the alphabet.

Logical thinking isn't apparent until around 3 or 4, though I suppose a robust discussion would require a formal definition of what constitutes logic.

For a real answer on this topic, you need two things: one is to understand your question, and the other is an understanding of Jean Piaget's masses of research into the development of intelligence (from birth to adulthood). Regarding understanding your own question, you must appreciate that decades of philosophy and psychology reveal that you cannot separate logic from mathematics. Even at the simplest of levels , young children around 4 years old will demonstrate semi-logical skills together with primitive counting skills. The reason for all this is that the level of abstraction above perception (i.e. in thinking) required for both logic and number is the same, and there is a hierarchy of such abstractions that can be traced throughout the development of intelligence common to every individual. Jean Piaget is the one you want to read, and epistemology (study of knowledge) is the domain you are asking about.

Logical Thinking is a very broad class of phenomena whereas counting is a very specific one. Logical Thinking dependes on the invidual having logical operations, which follow a course of development from early childhood, as proposed by Piaget. Counting itself has been a subject of research that shows that indeed young children can learn to count without understanding what number is. For example, a child may count two rows of items and count both as having "six", yet when one of such rows's elements are spaced, thereby removing the optical correspondence between the items, that same child will deny that both rows still have the same amount. A review of this research and other topics related to early mathematics learning can be found in http://www.nuffieldfoundation.org/sites/default/files/P2.pdf

## Piaget's Theory of Moral Development

Jean Piaget first published his theory of child development during the 1920's but his work did not become prominent until the mid-twentieth century. Piaget is perhaps best known for his theory of children's cognitive development, but he also proposed his own theory about children's moral development. Piaget recognized that cognitive development is closely tied to moral development and was particularly interested in the way children's thoughts about morality changed over time. In this article we limit our discussion of Piaget's theory to adolescent moral development. The Middle Childhood Development Article discusses Piaget's theory with respect to younger children.

According to Piaget, youth develop the morality of cooperation, at the age of 10 years or older. As youth develop a morality of cooperation they realize that in order to create a cooperative society people must work together to decide what is acceptable, and what is not. Piaget believed that youth at this age begin to understand that morals represent social agreements between people and are intended to promote the common good. Furthermore, they recognize people may differ in the way they understand and approach a moral situation or problem. They also begin to understand that the difference between right and wrong is not an absolute but instead must take into account changing variables such as context, motivation, abilities, and intentions. Contrast this to younger youth who believe rules and laws are created by indisputable, wise authorities and believe that rules established by these wise authorities ought never be challenged or changed. Moreover, Piaget believed youth at this age begin to understand that the morality of a decision does not rest solely on the outcome of that decision. For example, youth at this age realize that running a stop sign is wrong, regardless of whether or not a person receives a traffic ticket, or causes a traffic accident.

Furthermore, youth begin to understand the reciprocal benefit of moral decision-making i.e., a moral decision creates the optimal solution for everyone involved, even when only two people are affected. Youth begin to realize that when situations are handled in a manner that seems fair, reasonable, and/or beneficial to all parties, it becomes easier for people to accept and honor the decision. This concept of fairness is called reciprocity. Initially youths' understanding of reciprocity can be very literal and simplistic. For example, last week Terrell, age 11, lent his brand new video game to his good friend Randy. This week, it is Randy who has a new video game. Terrell is likely to insist that Randy should allow him borrow the new video game because from Terrell's perspective, "it's only fair" since he graciously allowed Randy to borrow his new game the week before. Terrell believes that fairness is simplistically determined by exact reciprocity.

By middle adolescence youth expand their understanding of fairness to include ideal reciprocity. Ideal reciprocity refers to a type of fairness beyond simple reciprocity and includes a consideration of another person's best interests. It is best described by the familiar adage, "Do unto others as you would have them do unto you" which many people know as the Golden Rule. Teens who have reached ideal reciprocity will imagine a problem from another person's perspective and try to place themselves in another person's "shoes," before making a moral decision. This concept is best illustrated by the following example:

Suppose Maria, age 14, was looking out the living room window one day and happens to see her older sister, Ava, backing the family's car out of the driveway. As she was watching, Maria saw Ava accidently bump into the mailbox just as she was pulling out into the street. Next, Maria saw Ava get out of the car and examine the damage to the car and mailbox. But, instead of coming back into the house to tell her parents, Ava just drove away.

At a younger age, Maria would have immediately run off to tell her parents about Ava's accident because she knows it is wrong for Ava to drive away without telling her parents what happened. Instead, if Maria has reached ideal reciprocity she will stop herself and imagine what the experience must have been like for Ava. She might realize that if she were in Ava's shoes, she might have done the same thing because she would be embarrassed and scared to tell her parents about the accident. Furthermore, she might decide that Ava would probably prefer to tell her parents about the accident herself, rather than having her little sister "tattle" on her. Therefore, Maria would wait until Ava comes home so she can talk to Ava. During this discussion Maria would encourage Ava to go to her parents with the truth in order to make things right. Thus, ideal reciprocity would enable Maria to examine the problem from her sister's perspective and to make a moral decision based upon the "Golden Rule."

According to Piaget, once ideal reciprocity has been reached moral development has been completed. However, we now know that many youth will continue to refine their moral decision-making process well into early adulthood. So although Piaget pioneered our initial understanding of moral development, research has not always been able to confirm certain portions of his theory. For instance, not only do youth continue refine their criteria for moral decisions into adulthood, but they also continue to improve their ability act according to these criteria. In other words, their moral compass operates to guide their choices and to direct their behavior. Piaget also under-estimated the age at which children are able to take into account another person's moral intention. Piaget believed that this ability did not develop until late childhood, or early adolescence. However, more recent research indicates that this ability develops sooner that Piaget once believed. Younger children are able to recognize the importance of someone's intentions when evaluating the morality of a decision but, younger children tend to be quite naïve in their belief that people's best intentions will dictate the actual choices people make. Despite these weaknesses, Piaget's contributions were very significant because they heavily influenced the later work of Lawrence Kohlberg who published his theory of moral development during the 1950's. Unlike Piaget's earlier theories, Kohlberg's theory of moral development has generally been supported by contemporary research. Kohlberg's theory is discussed in the next section.

## The Sensorimotor Stage: Birth to Age 2

The first stage is aptly named after how infants learn until age two. From birth, infants absorb information through their senses: by touching, looking, and listening. They are very orally fixated and tend to put everything in their mouths. Piaget believed that this stage was valuable to their development, and each consecutive step is built on the growth that occurs in this stage.

We can observe the thought processes of infants through their actions. From about 6 months on, children begin to organize ideas into firm concepts that do not change. An infant may first not make sense of a specific toy, but as they begin to look at it, feel it, and manipulate it often, they are able to represent the object in their minds. This is how we can begin to observe knowledge in babies, as they begin to show understanding of an object for what it is. For example, by playing continuously with a toy animal, an infant begins to understand what the object is and recall their experiences associated with that toy. Piaget labeled this understanding as *object permanence*, which indicates the knowledge of the toy even if it is out of sight. He considered this understanding to be a major milestone in the sensorimotor stage and believed that it demonstrated the differences in the thought processes of toddlers compared to young infants.

The sensorimotor stage is unique in that is occurs without the use of language. As infants cannot speak, Piaget developed a few creative experiments in an effort to understand what they were thinking. His experiments were able to demonstrate that infants do represent objects and understand that they are permanent. In one of his experiments, Piaget consistently hid a toy underneath a blanket. Toddlers, or children between the ages of 18 and 24 months, took initiative to look for the toy themselves, but infants less than 6 months of age did not. The older infants interpreted the hiding of the toy as a prompt to search for it, which is thought to support the idea of object permanency.

## Does logical thinking or counting develop first in children? - Psychology

Cognitive development is an adaptive effort of a child in response to various environmental influences. This is accomplished in the following two ways.

- Assimilation

I ncorporating new thoughts, behavior and objects into the existing structures.

### Cognition

Cognition is a scientific term for thought processing of information, applying knowledge, and changing preferences.

The term is derived from Latin word “cognoscere” which means “to conceptualize”. Animals’ cognitive skills development process is nearly similar to that of humans.

Children have sharp senses with keen observation and are active learners: Note the dynamic brain growth and development during the first 2 years of life. They continuously perceive information and store it as their memory in the specific areas of the brain. Good and bad experiences right from early age modulate human brain development.

Memory is the basis of children's cognitiveꃞvelopment.

Memories are classified into concepts so that thinking, reasoning, and understanding is made possible and used to solve problems.

### Factors that influence cognitive development

- Neurological maturation: It enables children to understand the new experiences and to apply more complex reasoning as they get older.

It permits the central nervous system to reorganize neuronal networks in response to environment’s positive and negative stimulation.

In face of injury or new learning, the body draws upon the reservoir of neurons bank created as a result of large number of neurons produced during the third trimester and early infancy. Each neuron develops about 15,000 synapses by 3 years of age.

The neuronal “pruning” leads to preservation of synapses of frequently used pathways and atrophy of those that are less-used. The strength of synapses increases or decreases as a direct effect of a child’s experiences that influence the synaptic activity.

for child to be able to effectively exhibit the developed cognitive skills.

### Nativism versus Empiricism

There has been a major debate on “nativism versus empiricism” in cognitive development theory.

Nativism implies that children ਊre born with certain skills native abilities, On the other hand, Empiricism believes in theory of 'blank slate' of mind at birth. Thereby meaning that the brain has inborn capabilities for learning from environmental experiences but does not contain any inherent skills.However, this dichotomy is now proved to be false.

Evidences from biological and behavioral sciences confirm that genetic potential interacts with experiences right from birth through entire childhood development process and cognitive development is no exception.

Children because of their individualized environmental influences and genetic make up differ in their temperament and skills development and elicit different stimuli from the same environment.

### Theory of cognitive development

In 1929, the Swiss psychologist, Jean Piaget detailed the most comprehensive cognitive development theory. His theory was based on a basic fact that *“to survive adaptation is a must”.*

This implies that the cognitive development is the result of interaction between an individual and the environment. The contribution of each is complex, as both are continuously changing over time.

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### Types of intelligence

**There are two types of intellectual structure:**

- Schemaਊre those that a child is born with.

They are the internal representation of some specific action, such as sucking and grasping. Most of the neonatal reflexes get included in this category.

### Stages of cognitive development

Piaget proposed that children move through following 4 stages of cognitive skills development:

1. Sensorimotor stage is from birth to 2years of age.

- During the early infancy , behaviour of an infant is dominated by inborn reflexes like sucking, swallowing, visually following an object and so on.

At this stage of development, a child performs actions in response to stimuli from the five senses. The behavior that leads to an interesting or pleasant result is repeated.

2. Pre-operational stage is from 2-7 years of age, and refers to

early childhood development.

- Language use begins and a child can now represent objects by image and words.

3. Concrete operations stage is between 7 and 11 years of age.

- Thinking at this age becomes more logical and less egocentric.

4. Formal operations stage starts from 12 years onwards,

*This represents the most complex mode of thinking. Please note that not everyone achieves this stage of thinking, even as an adult!*

- Children in this age group use logical operations in a systematic fashion.

*Piaget’s four stages of cognitive development theory has had tremendous impact on educational concepts and teaching that has put a new meaning to children’s acquisition of early literacy and numeracy skills.*

## Sorting

Being able to classify things is a key aspect of numeracy. Matching and sorting – for example by size, shape or colour – helps children make sense of the world and develop their logical thinking skills. Here are some ways in which you can support them:

**Set up sorting activities**– there are plenty of variations to explore (and lots of ideas if you search online), but for example you could get the children to sort crayons by colour, pasta by shape, or blocks by size.**Invest in sorting toys**– there are many educational toys designed to help develop sorting skills.**Incorporate sorting into other activities**– ask the children to sort things as they tidy away, or make it part of a nature walk (eg collect leaves and then sort them by shape, colour or size).

## Does logical thinking or counting develop first in children? - Psychology

Middle childhood development is the concrete operational stage, when the focus is on logical thinking, intelligence and psychosocial development. Contrary to the usual concept, middle childhood is as special a stage of childhood development as toddlers and teenagers.

The American Academy of Pediatrics defines middle childhood from onset of sixth year to completion of tenth year of life.

Development in middle childhood is the outcome of interaction between the milestones achieved and ongoing learning experiences.

Motor skills, language proficiency and cognitive capabilities attained in the previous years are further mastered during the formative years of middle childhood.

Children in their middle childhood years begin to understand the rules of society and its moral bindings.

*Middle childhood development helps children to cope with the increasing demands of the classroom curriculum. Learning disabilities and behavior problems surface at this stage that need medical attention.*

### Middle childhood development leads to logical thinking

Jean Piaget’s has described middle childhood development as the concrete operational stage, when the focus is on developing logical thinking.

The thinking process during middle childhood years gets more logical, flexible, and organized than it was during early childhood development. Children during this phase of development are “critical thinkers” with the ability to reason, remember, repeat, reorganize, relate, and reflect.

Enhanced logical thinking are easily noticed in their social relationships strive for scholastic, cultural and sports excellence. Its reflections are also seen in their creativity, which gets significantly refined during middle childhood.

### Aspects of middle childhood development

**Logic **Children learn to reason and use the acquired knowledge constructively. They evaluate the thoughts both that of the others and their own before putting them in to action.

Constructive application of the information gathered involves child’s newly developed capability to accommodate and assimilate the thoughts.

**Decentration **Thinking changes from perception bound to concrete logical during middle childhood years. Cognitive development during middle childhood development helps children overcome early childhood tendency to focus only on one aspect of a situation and enable the children in middle childhood to perceive the events/problems from different angles.

**Reversibility**

A ny process can be returned to its original state.

**Causality ** Principle of or relationship between cause and effect and that nothing can happen without being caused.

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### Intelligence development

Middle childhood development shows a sharp rise in child’s capacity to understand the world. Children in this phase begin to think rationally, and use resources effectively when faced with challenges and **that is intelligence.**

**The components of intelligence that i ncrease during middle childhood.**

**Fluid Intelligence:**

It gives the ability to remember, reason, and process information increases.

**Crystallized Intelligence:**

Increases in this type of intelligence gives children the ability to gather information, master skills, and develop strategies.

**Metacognition:**

Metacognition is the process of monitoring own thoughts and memory development.

**Development of information processing**

- Decline in time needed to process information and so increase in information processing capacity.

### Attention span

The attention span increases during middle childhood development as a result of more c ontrolled an adaptable thought process of children at this age.

Consequently, children develop the capabilities to -

- Plan the attention time they need to assign to a task.

**Inability to control attention span**

Inability to control attention is seen in children with ADHD.

These children face great difficulty staying on a specified task. They tend to act impulsively and usually are hyperactive.

### Memory strategies

Memory strategies are the deliberate mental activities used to store and retain information. Working memory plays an important role in learning during middle childhood.

General intelligence is highly correlated with working memory capacity.

Memory capacity shows a steady developmental increase during middle childhood years as:

- Capacity of short term memory improves significantly.

### Child's scholastic performance as related to memory span

Memory strategies require time and effort in order to show desired results.

Memory span improves from the age of

5 years onward and the adult levels of performance is reached by about 15 years of age.

During middle childhood, children arrange the vast amount of information in their memories into increasingly elaborate, hierarchically structured networks.

**Strategies to improve memory:**

- Rehearse

Rehearsal i nvolves repeating information to oneself over and over again.

### Communication capabilities

### Communicative skills development during middle childhood

- Children in their middle childhood can engage in

constructive conversation with a variety of relevant questions.

## Adding It Up: Helping Children Learn Mathematics (2001)

Whole numbers are the easiest numbers to understand and use. As we described in the previous chapter, most children learn to count at a young age and understand many of the principles of number on which counting is based. Even if children begin school with an unusually limited facility with number, intensive instructional activities can be designed to help them reach similar levels as their peers. 1 Children&rsquos facility with counting provides a basis for them to solve simple addition, subtraction, multiplication, and division problems with whole numbers. Although there still is much for them to work out during the first few years of school, children begin with substantial knowledge on which they can build.

In this chapter, we examine the development of proficiency with whole numbers. We show that students move from methods of solving numerical problems that are intuitive, concrete, and based on modeling the problem situation directly to methods that are more problem independent, mathematically sophisticated, and reliant on standard symbolic notation. Some form of this progression is seen in each operation for both single-digit and multidigit numbers.

We focus on computation with whole numbers because learning to compute can provide young children the opportunity to work through many number concepts and to integrate the five strands of mathematical proficiency. This learning can provide the foundation for their later mathematical development. Computation with whole numbers occupies much of the curriculum in the early grades, and appropriate learning experiences in these grades improve children&rsquos chances for later success.

Whole number computation also provides an instructive example of how routine-appearing procedural skills can be intertwined with the other strands of proficiency to increase the fluency with which the skills are used. For years, learning to compute has been viewed as a matter of following the teacher&rsquos directions and practicing until speedy execution is achieved. Changes in career demands and the tasks of daily life, as well as the availability of new computing tools, mean that more is now demanded from the study of computation. More than just a means to produce answers, computation is increasingly seen as a window on the deep structure of the number system. Fortunately, research is demonstrating that both skilled performance and conceptual understanding are generated by the same kinds of activities. No tradeoffs are needed. As we detail below, the activities that provide this powerful result are those that integrate the strands of proficiency.

### Operations with Single-Digit Whole Numbers

As students begin school, much of their number activity is designed to help them become proficient with single-digit arithmetic. By *single-digit arithmetic,* we mean the sums and products of single-digit numbers and their companion differences and quotients (e.g., 5+7=12, 12&ndash5=7, 12&ndash7=5 and 5×7=35, 35÷5=7, 35÷7=5). For most of a century, learning single-digit arithmetic has been characterized in the United States as &ldquolearning basic facts,&rdquo and the emphasis has been on memorizing those facts. We use the term *basic number combinations* to emphasize that the knowledge is relational and need not be memorized mechanically. Adults and &ldquoexpert&rdquo children use a variety of strategies, including automatic or semiautomatic rules and reasoning processes to efficiently produce the basic number combinations. 2 Relational knowledge, such as knowledge of commutativity, not only promotes learning the basic number combinations but also may underlie or affect the mental representation of this basic knowledge. 3

The domain of early number, including children&rsquos initial learning of single-digit arithmetic, is undoubtedly the most thoroughly investigated area of school mathematics. A large body of research now exists about how children in many countries actually learn single-digit operations with whole numbers. Although some educators once believed that children memorize their &ldquobasic facts&rdquo as conditioned responses, research shows that children do *not* move from knowing nothing about the sums and differences of numbers to having the basic number combinations memorized. Instead, they move through a series of progressively more advanced and abstract methods for working out the answers

to simple arithmetic problems. Furthermore, as children get older, they use the procedures more and more efficiently. 4 Recent evidence indicates children can use such procedures quite quickly. 5 Not all children follow the same path, but all children develop some intermediate and temporary procedures.

Most children continue to use those procedures occasionally and for some computations. Recall eventually becomes the predominant method for some children, but current research methods cannot adequately distinguish between answers produced by recall and those generated by fast (nonrecall) procedures. This chapter describes the complex processes by which children learn to compute with whole numbers. Because the research on whole numbers reveals how much can be understood about children&rsquos mathematical development through sustained and interdisciplinary inquiry, we give more details in this chapter than in subsequent chapters.

#### Word Problems: A Meaningful Context

One of the most meaningful contexts in which young children begin to develop proficiency with whole numbers is provided by so-called word problems. This assertion probably comes as a surprise to many, especially mathematics teachers in middle and secondary school whose students have special difficulties with such problems. But extensive research shows that if children can count, they can begin to use their counting skills to solve simple word problems. Furthermore, they can advance those counting skills as they solve more problems. 6 In fact, it is in solving word problems that young children have opportunities to display their most advanced levels of counting performance and to build a repertoire of procedures for computation.

Most children entering school can count to solve word problems that involve adding, subtracting, multiplying, and dividing. 7 Their performance increases if the problems are phrased simply, use small numbers, and are accompanied by physical counters for the children to use. The exact procedures children are likely to use have been well documented. Consider the following problems:

*Sally had 6 toy cars. She gave 4 to Bill. How many did she have left?*

*Sally had 4 toy cars. How many more does she need to have 6?*

Most young children solve the first problem by counting a set of 6, removing 4, and counting the remaining cars to find the answer. In contrast,

they solve the second problem by counting a set of 4, adding in more as they count &ldquofive, six,&rdquo and then counting those added in to find the answer.

Children solve these problems by &ldquoacting out&rdquo the situation&mdashthat is, by modeling it. They invent a procedure that mirrors the actions or relationships described in the problem. This simple but powerful approach keeps procedural fluency closely connected to conceptual understanding and strategic competence. Children initially solve only those problems that they understand, that they can represent or model using physical objects, and that involve numbers within their counting range. Although this approach limits the kinds of problems with which children are successful, it also enables them to solve a remarkable range of problems, including those that involve multiplying and dividing.

Since children intuitively solve word problems by modeling the actions and relations described in them, it is important to distinguish among the different types of problems that can be represented by adding or subtracting, and among those represented by multiplying or dividing. One useful way of classifying problems is to heed the children&rsquos approach and examine the actions and relations described. This examination produces a taxonomy of problem types distinguished by the solution method children use and provides a framework to explain the relative difficulty of problems.

Four basic classes of addition and subtraction problems can be identified: problems involving (a) joining, (b) separating, (c) part-part-whole relations, and (d) comparison relations. Problems within a class involve the same type of action or relation, but within each class several distinct types of problems can be identified depending on which quantity is the unknown (see Table 6&ndash1). Students&rsquo procedures for solving the entire array of addition and subtraction problems and the relative difficulty of the problems have been well documented. 8

For multiplication and division, the simplest kinds of problems are grouping situations that involve three components: the number of sets, the number in each set, and the total number. For example:

*Jose made 4 piles of marbles with 3 marbles in each pile. How many marbles did Jose have?*

In this problem, the number and size of the sets is known and the total is unknown. There are two types of corresponding division situations depending on whether one must find the number of sets or the number in each set. For example:

**Addition and Subtraction Problem Types**

Connie had 5 marbles. Juan gave her 8 more marbles. How many marbles does Connie have altogether?

Connie has 5 marbles. How many more marbles does she need to have 13 marbles altogether?

Connie had some marbles. Juan gave her 5 more. Now she has 13 marbles. How many marbles did Connie have to start with?

Connie had 13 marbles. She gave 5 to Juan. How many marbles does Connie have left?

Connie had 13 marbles. She gave some to Juan. Now she has 5 marbles left. How many marbles did Connie give to Juan?

Connie had some marbles. She gave 5 to Juan. Now she has 8 marbles left. How many marbles did Connie have to start with?

Connie has 5 red marbles and 8 blue marbles. How many marbles does she have altogether?

Connie has 13 marbles: 5 are red and the rest are blue. How many blue marbles does Connie have?

Connie has 13 marbles. Juan has 5 marbles. How many more marbles does Connie have than Juan?

Juan has 5 marbles. Connie has 8 more than Juan. How many marbles does Connie have?

Connie has 13 marbles. She has 5 more marbles than Juan. How many marbles does Juan have?

SOURCE: Carpenter, Fennema, Franke, Levi, and Empson, 1999, p. 12. Used by permission of Heinemann. All rights reserved.

*Jose has 12 marbles and puts them into piles of 3. How many piles does he have?*

*Jose has 12 marbles and divides them equally into 3 piles. How many marbles are in each pile?*

Additional types of multiplication and division problems are introduced later in the curriculum. These include rate problems, multiplicative comparison problems, array and area problems, and Cartesian products. 9

As with addition and subtraction problems, children initially solve multiplication and division problems by modeling directly the action and relations in the problems. 10 For the above multiplication problem with marbles, they form four piles with three in each and count the total to find the answer. For the first division problem, they make groups of the specified size of three and count the number of groups to find the answer. For the other problem, they make the three groups by dealing out (as in cards) and count the number in one of the groups. Although adults may recognize both problems as 12 divided by 3, children initially think of them in terms of the actions or relations portrayed. Over time, these direct modeling procedures are replaced by more efficient methods based on counting, repeated adding or subtracting, or deriving an answer from a known number combination. 11

The observation that children use different methods to solve problems that describe different situations has important implications. On the one hand, directly modeling the action in the problem is a highly sensible approach. On the other hand, as numbers in problems get larger, it becomes inefficient to carry out direct modeling procedures that involve counting all of the objects.

Children&rsquos proficiency gradually develops in two significant directions. One is from having a different solution method for each type of problem to developing a single general method that can be used for classes of problems with a similar mathematical structure. Another direction is toward more efficient calculation procedures. Direct-modeling procedures evolve into the more advanced counting procedures described in the next section. For word problems, these procedures are essentially abstractions of direct modeling that continue to reflect the actions and relations in the problems.

The method children might use to solve a class of problems is not necessarily the method traditionally taught. For example, many children come to solve the &ldquosubtraction&rdquo problems described above by counting, adding up, or thinking of a related addition combination because any of these methods is easier and more accurate than counting backwards. The method traditionally presented in textbooks, however, is to solve both of these problems by

subtracting, which moves students toward the more difficult and error-prone procedure of counting down. Ultimately, most children begin to use recall or a rapid mental procedure to solve these problems, and they come to recognize that the same general method can be used to solve a variety of problems.

#### Single-Digit Addition

Children come to understand the meaning of addition in the context of word problems. As we noted in the previous section, children move from counting to more general methods to solve different classes of problems. As they do, they also develop greater fluency with each specific method. We call these specific counting methods *procedures*. Although educators have long recognized that children use a variety of procedures to solve single-digit addition problems, 12 substantial research from all over the world now indicates that children move through a progression of different procedures to find the sum of single-digit numbers. 13

This progression is depicted in Box 6&ndash1. First, children count out objects for the first addend, count out objects for the second addend, and count all of the objects (count all). This general counting-all procedure then becomes abbreviated, internalized, and abstracted as children become more experienced with it. Next, they notice that they do not have to count the objects for the first addend but can start with the number in the first or the larger addend and count on the objects in the other addend (count on). As children count

**Box 6&ndash1** **Learning Progression for Single-Digit Addition**

on with objects, they begin to use the counting words themselves as countable objects and keep track of how many words have been counted on by using fingers or auditory patterns. The counting list has become a representational tool. With time, children recompose numbers into other numbers (4 is recomposed into 3+1) and use thinking strategies in which they turn an addition combination they do not know into one they do know (3+4 becomes 3+3+1). In the United States, these strategies for *derived number combinations* often use a so-called double (2+2, 3+3, etc.). These doubles are learned very quickly.

As Box 6&ndash1 shows, throughout this learning progression, specific sums move into the category of being rapidly recalled rather than solved in one of the other ways described above. Children vary in the sums they first recall readily, though doubles, adding one (the sum is the next counting word), and small totals are the most readily recalled. Several procedures for single-digit addition typically coexist for several years they are used for different numbers and in different problem situations. Experience with figuring out the answer to addition problems provides the basis both for understanding what it means to say &ldquo5+3=8&rdquo and for eventually recalling that sum without the use of any conscious strategy.

Children in many countries often follow this progression of procedures, a natural progression of embedding and abbreviating. Some of these procedures can be taught, which accelerates their use, 14 although direct teaching of these strategies must be done conceptually rather than simply by using imitation and repetition. 15 In some countries, children learn a general procedure known as &ldquomake a 10&rdquo (see Box 6&ndash2). 16 In this procedure the solver makes a 10 out of one addend by taking a number from the other addend. Educators in some countries that use this approach believe this first instance of regrouping by making a 10 provides a crucial foundation for later multidigit arithmetic. In some Asian countries this procedure is presumably facilitated by the number words. 17 It has also been taught in some European countries in which the number names are more similar to those of English, suggesting that the procedure can be used with a variety of number-naming systems. The procedure is now beginning to appear in U.S. textbooks, 18 although so little space may be devoted to it that some children may not have adequate time and opportunity to understand and learn it well.

There is notable variation in the procedures children use to solve simple addition problems. 19 Confronted with that variation, teachers can take various steps to support children&rsquos movement toward more advanced procedures. One technique is to talk about slightly more advanced procedures *and* why

**Box 6&ndash2** **Make a Ten: B+6=?**

they work. 20 The teacher can stimulate class discussion about the procedures that various students are using. Students can be given opportunities to present their procedures and discuss them. Others can then be encouraged to try the procedure. Drawings or concrete materials can be used to reveal how the procedures work. The advantages and disadvantages of different procedures can also be examined. For a particular procedure, problems can be created for which it might work well or for which it is inefficient.

Other techniques that encourage students to use more efficient procedures are using large numbers in problems so that inefficient counting procedures cannot easily be used and hiding one of the sets to stimulate a new way of thinking about the problem. Intervention studies indicate that teaching counting-on procedures in a conceptual way makes all single-digit sums accessible to U.S. first graders, including children who are learning disabled and those who do not speak English as their first language. 21 Providing support for children to improve their own procedures does *not* mean, however, that every child is taught to use all the procedures that other children develop. Nor does it mean that the teacher needs to provide every child in a class with

support and justification for different procedures. Rather, the research provides evidence that, at any one time, most children use a small number of procedures and that teachers can learn to identify them and help children learn procedures that are conceptually more efficient (such as counting on from the larger addend rather than counting all). 22

Mathematical proficiency with respect to single-digit addition encompasses not only the fluent performance of the operation but also conceptual understanding and the ability to identify and accurately represent situations in which addition is required. Providing word problems as contexts for adding and discussing the advantages and disadvantages of different addition procedures are ways of facilitating students&rsquo adaptive reasoning and improving their understanding of addition processes.

#### Single-Digit Subtraction

Subtraction follows a progression that generally parallels that for addition (see Box 6&ndash3). Some U.S. children also invent counting-down methods that model the taking away of numbers by counting back from the total. But counting down and counting backward are difficult for many children. 23

**Box 6&ndash3** **Learning Progression for Single-Digit Subtraction**

A considerable number of children invent counting-up procedures for situations in which an unknown quantity is added to a known quantity. 24 Many of these children later count up in taking-away subtraction situations (13&ndash8=? becomes 8+?=13). When counting up is not introduced, many children may not invent it until the second or third grade, if at all. Intervention studies with U.S. first graders that helped them see subtraction situations as taking away the first *x* objects enabled them to learn and understand counting-up-to procedures for subtraction. Their subtraction accuracy became as high as that for addition. 25

Experiences that focus on part-part-whole relations have also been shown to help students develop more efficient thinking strategies, especially for subtraction. 26 Students examine a join or separate situation and identify which number represents the whole quantity and which numbers represent the parts. These experiences help students see how addition and subtraction are related and help them recognize when to add and when to subtract. For students in grades K to 2, learning to see the part-whole relations in addition and subtraction situations is one of their most important accomplishments in arithmetic. 27

For students in grades K to 2, learning to see the part-whole relations in addition and subtraction situations is one of their most important accomplish-ments in arithmetic.

Examining the relationships between addition and subtraction and seeing subtraction as involving a known and an unknown addend are examples of adaptive reasoning. By providing experiences for young students to develop adaptive reasoning in addition and subtraction situations, teachers are also anticipating algebra as students begin to appreciate the inverse relationships between the two operations. 28

#### Single-Digit Multiplication

Much less research is available on single-digit multiplication and division than on single-digit addition and subtraction. U.S. children progress through a sequence of multiplication procedures that are somewhat similar to those for addition. 29 They make equal groups and count them all. They learn skip-count lists for different multipliers (e.g., they count 4, 8, 12, 16, 20,&hellipto multiply by four). They then count on and count down these lists using their fingers to keep track of different products. They invent thinking strategies in which they derive related products from products they know.

As with addition and subtraction, children invent many of the procedures they use for multiplication. They find patterns and use skip counting (e.g., multiplying 4×3 by counting &ldquo3, 6, 9, 12&rdquo). Finding and using patterns and other thinking strategies greatly simplifies the task of learning multiplication tables (see Box 6&ndash4 for some examples). 30 Moreover, finding and describing

**Box 6&ndash4** **Thinking Strategies for Single-Digit Multiplication**

In single-digit arithmetic, there are 100 multiplication combinations that students must learn. Commutativity reduces that number by about half. Multiplication by 0 and by 1 may quickly be deduced from the meaning of multiplication. Multiplication by 2 consists of the &ldquodoubles&rdquo from addition. Single-digit multiplication by 9 is simplified by a pattern: in the product, the sum of the digits is 9. (For example, 9×7=63 and 6+3=9.) Multiplication by 5 may also be deduced through patterns or by first multiplying by 10 and then dividing by 2, since 5 is half of 10.

The remaining 15 multiplication combinations (and their commutative counterparts) may be computed by skip counting or by building on known combinations. For example, 3×6 must be 6 more than 2×6, which is 12. So 3×6 is 18. Similarly, 4×7 must be twice 2×7, which is 14. So 4×7 is 28. (Note that these strategies require proficiency with addition.) To compute multiples of 6, one can build on the multiples of 5. So, for example, 6×8 must be 8 more than 5×8, which is 40. So 6×8 is 48. If students are comfortable with such strategies for multiplication by 3, 4, and 6, only three multiplication combinations remain: 7×7, 7×8, and 8×8. These can be derived from known combinations in many creative ways.

patterns are a hallmark of mathematics. Thus, treating multiplication learning as pattern finding both simplifies the task and uses a core mathematical idea.

After children identify patterns, they still need much experience to produce skip-count lists and individual products rapidly. Little is known about how children acquire this fluency or what experiences might be of most help. A good deal of research remains to be done, in the United States and in other countries, to understand more about this process.

#### Single-Digit Division

Division arises from the two splitting situations described above. A collection is split into groups of a specified size or into a specified number of groups. Just as subtraction can be thought of using a part-part-whole relation, division can be thought of as splitting a number into two factors. Hence, divisions can also be approached as finding a missing factor in multiplication. For example, 72÷9=? can be thought of as 9÷?=72. But there is little

research concerning how best to introduce and use this relationship, or whether it is helpful to learn a division combination at the same time as the corresponding multiplication combination. Further, there is little research about how to help children learn and use easily all of the different symbols for division, such as 15÷3, and

#### Practicing Single-Digit Calculations

Practicing single-digit calculations is essential for developing fluency with them. This practice can occur in many different contexts, including solving word problems. 31 Drill alone does not develop mastery of single-digit combinations. 32 Practice that follows substantial initial experiences that support understanding and emphasize &ldquothinking strategies&rdquo has been shown to improve student achievement with single-digit calculations. 33 This approach allows computation and understanding to develop together and facilitate each other. Explaining how procedures work and examining their benefits, as part of instruction, support retention and yield higher levels of performance. 34 In this way, computation practice remains integrated with the other strands of proficiency such as strategic competence and adaptive reasoning.

Practicing single-digit calculations is essential for developing fluency with them.

It is helpful for some practice to be targeted at recent learning. After students discuss a new procedure, they can benefit from practicing it. For example, if they have just discussed the make-a-10 procedure (see Box 6&ndash2), solving problems involving 8 or 9 in which the procedure can easily be used provides beneficial practice. It also is helpful for some practice to be cumulative, occurring well after initial learning and reviewing the more advanced procedures that have been learned.

Many U.S. students have had the experience of taking a timed test that might be a page of mixed addition, subtraction, multiplication, and division problems. This scattershot form of practice is, in our opinion, rarely the best use of practice time. Early in learning it can be discouraging for students who have learned only primitive, inefficient procedures. The experience can adversely affect students&rsquo disposition toward mathematics, especially if the tests are used to compare their performance. 35 If appropriately delayed, timed tests can benefit some students, but targeted forms of practice, with particular combinations that have yet to be mastered or on which efficient procedures can be used, are usually more effective. 36

#### Summary of Findings an Learning Single-Digit Arithmetic

For addition and subtraction, there is a well-documented progression of procedures used worldwide 37 by many children that stems from the sequential nature of the list of number words. This list is first used as a counting tool then it becomes a representational tool in which the number words themselves are the objects that are counted. 38 Counting becomes abbreviated and rapid, and students begin to develop procedures that take advantage of properties of arithmetic to simplify computation. During this progression, individual children use a range of different procedures on different problems and even on the same problem encountered at different times. 39 Even adults have been found to use a range of different procedures for simple addition problems. 40 Further, it takes an extended period of time before new and better strategies replace previously used strategies. 41 Learning-disabled children and others having difficulty with mathematics do not use procedures that differ from this progression. They are just slower than others in moving through it. 42

Instruction can help students progress. 43 Counting on is accessible to first graders it makes possible the rapid and accurate addition of all single-digit numbers. Single-digit subtraction is usually more difficult than addition for U.S. children. If children understand the relationship between addition and subtraction, perhaps by thinking of the problem in terms of part-part-whole, then they recognize that counting up can be used to solve subtraction problems. This recognition makes subtraction more accessible. 44

The procedures of counting on for addition and counting up for subtraction can be learned with relative ease. Multiplication and division are somewhat more difficult. Even adults might not have quick ways of reconstructing the answers to problems like 6×8=? or if they have forgotten the answers. Learning these combinations seems to require much specific pattern-based knowledge that needs to be orchestrated into accessible and rapid-enough products and quotients. As with addition and subtraction, children derive some multiplication and division combinations from others for example, they recall that 6×6=36 and use that combination to conclude that 6×7=42. Research into ways to support such pattern finding, along with the necessary follow-up thinking and practice, is needed if all U.S. children are to acquire higher levels of proficiency in single-digit arithmetic.

Acquiring proficiency with single-digit computations involves much more than rote memorization. This domain of number demonstrates how the different strands of proficiency contribute to each other. At this early point in

development, many of the linkages among strands result from children&rsquos natural inclination to make sense of things and to engage in actions that they understand. Children begin with conceptual understanding of number and the meanings of the operations. They develop increasingly sophisticated representations of the operations such as counting-on or counting-up procedures as they gain greater fluency. They also lean heavily on reasoning to use known answers such as doubles to generate unknown answers. Even in the early grades, students choose adaptively among different procedures and methods depending on the numbers involved or the context. 45 As long as the focus in the classroom is on sense making, they rarely make nonsensical errors, such as adding to find the answer when they should subtract. Proficiency comes from making progress within each strand and building connections among the strands. A productive disposition is generated by and supports this kind of learning because students recognize their competence at making sense of quantitative situations and solving arithmetic problems.

### Multidigit Whole Number Calculations

Step-by-step procedures for adding, subtracting, multiplying, or dividing numbers are called algorithms. For example, the first step in one algorithm for multiplying a three-digit number by a two-digit number is to write the three-digit number above the two-digit number and to begin by multiplying the one&rsquos digit in the top number by the one&rsquos digit in the bottom number (see Box 6&ndash5).

In the past, algorithms different from those taught today for addition, subtraction, multiplication, and division have been taught in U.S. schools. Also, algorithms different from those taught in the United States today are currently being taught in other countries. 46 Each algorithm has advantages

**Box 6&ndash5** **Beginning a multiplication algorithm**

and disadvantages. Therefore, it is important to think about which algorithms are taught and the reasons for teaching them.

Learning to use algorithms for computation with multidigit numbers is an important part of developing proficiency with numbers. Algorithms are procedures that can be executed in the same way to solve a variety of problems arising from different situations and involving different numbers. This feature has three important implications. First, it means that algorithms are useful tools&mdashdifferent procedures do not need to be invented for each problem. Second, algorithms illustrate a significant feature of mathematics: The structure of problems can be abstracted from their immediate context and compared to see whether different-looking problems can be solved in similar ways. Finally, the process of developing fluency with arithmetic algorithms in elementary school can contribute to progress in developing the other strands of proficiency if time is spent examining why algorithms work and comparing their advantages and disadvantages. Such analyses can boost conceptual understanding by revealing much about the structure of the number system itself and can facilitate understanding of place-value representations.

Research findings about learning algorithms for whole numbers can be summarized with seven important observations. First, the linkages among the strands of mathematical proficiency that are possible when children develop proficiency with single-digit arithmetic can be continued with multidigit arithmetic. For example, there can be a close connection between understanding and fluency. Conceptual knowledge that comes with understanding is important for the development of procedural fluency, while fluent procedural knowledge supports the development of further understanding and learning. When students fail to grasp the concepts that underlie procedures or cannot connect the concepts to the procedures, they frequently generate flawed procedures that result in systematic patterns of errors. 47 These so-called buggy algorithms are signs that the strands are not well connected. 48 When the initial computational procedures that students use to solve multidigit problems reflect their understanding of numbers, understanding and fluency develop together.

A second observation is that understanding and fluency are related. For multidigit addition and subtraction, given conventional instruction that emphasizes practicing procedures, a substantial percentage of children gain understanding of multidigit concepts before using a correct procedure, but another substantial minority do the opposite. 49 In contrast, instructional programs that emphasize understanding algorithms before using them have been shown to lead to increases in both conceptual and procedural knowledge. 50

So there is some evidence that understanding is the basis for developing procedural fluency. 51

A third observation is that proficiency with multidigit computation is more heavily influenced by instruction than single-digit computation is. Many features of multidigit procedures (e.g., the base-10 elements and how they are represented by place-value notation) are not part of children&rsquos everyday experience and need to be learned in the classroom. In fact, many students are likely to need help learning efficient forms of multidigit procedures. This means that students in different classrooms and receiving different instruction might follow different learning progressions use different procedures. 52 For single-digit addition and subtraction, the same learning progression occurs for many children in many countries regardless of the nature and extent of instruction. 53 But multidigit procedures, even those for addition and subtraction, depend much more on what is taught.

A fourth observation is that children can and do devise or invent algorithms for carrying out multidigit computations. 54 Opportunities to construct their own procedures provide students with opportunities to make connections between the strands of proficiency. Procedural fluency is built directly on their understanding. The invention itself is a kind of problem solving, and they must use reasoning to justify their invented procedure. Students who have invented their own correct procedures also approach mathematics with confidence rather than fear and hesitation. 55 Students invent many different computational procedures for solving problems with large numbers. For addition, they eventually develop a procedure that is consistent with the thinking that is used with standard algorithms. That thinking enables them to make sense of the algorithm as a record on paper of what they have already been thinking. For subtraction, many students can develop adding-up procedures and, if using concrete materials like base-10 blocks, can also develop ways of thinking that parallel algorithms usually taught today. 56 Some students need help to develop efficient algorithms, however, especially for multiplication and division. Consequently, for these students the process of learning algorithms involves listening to someone else explain an algorithm and trying it out, all the while trying to make sense of it. Research suggests that students are capable of listening to their peers and to the teacher and of making sense of an algorithm if it is explained and if the students have diagrams or concrete materials that support their understanding of the quantities involved. 57

Fifth, research has shown that students can learn well from a variety of different instructional approaches, including those that use physical materials to represent hundreds, tens, and ones, those that emphasize special counting

activities (e.g., count by tens beginning with any number), and those that focus on developing mental computation methods. 58 Although the data do not point to a single preferred instructional approach, they do suggest that effective approaches share some key features: The multidigit procedures that students use are easily understood students are encouraged to use algorithms that they understand instructional supports (classroom discussions, physical materials, etc.) are available to focus students&rsquo attention on the base-10 structure of the number system and on how that structure is used in the algorithm and students are helped to progress to using reasonably efficient but still comprehensible algorithms. 59

Sixth, research on symbolic learning argues that, to be helpful, manipulatives or other physical models used in teaching must be represented by a learner both as the objects that they are and as symbols that stand for something else. 60 The physical characteristics of these materials can be initially distracting to children, and it takes time for them to develop mathematical meaning for any kind of physical model and to use it effectively. These findings suggest that sustained experience with any physical models that students are expected to use may be more effective than limited experience with a variety of different models. 61

In view of the attention given to the use of concrete models in U.S. school mathematics classes, we offer a special note regarding their effective use in multidigit arithmetic. Research indicates that students&rsquo experiences using physical models to represent hundreds, tens, and ones can be effective *if* the materials help them think about how to combine quantities and, eventually, how these processes connect with written procedures. The models, however, are not automatically meaningful for students the meaning must be constructed as they work with the materials. Given time to develop meaning for a model and connect it with the written procedure, students have shown high levels of performance using the written procedure and the ability to give good explanations for how they got their answers. 62 In order to support understanding, however, the physical models need to show tens to be collections of ten ones and to show hundreds to be simultaneously 10 tens and 100 ones. For example, base-10 blocks have that quality, but chips all of the same size but with different colors for hundreds, tens, and ones do not.

A seventh and final observation is that the English number words and the Hindu-Arabic base-10 place-value system for writing numbers complicate the teaching and learning of multidigit algorithms in much the same way, as discussed in Chapter 5, that they complicate the learning of early number concepts. 63 Closely related to the difficulties posed by the irregu-

larities with number words are difficulties posed by the complexity of the system for writing numbers. As we said in chapter 3, the base-10 place-value system is very efficient. It allows one to write very large numbers using only 10 symbols, the digits 0 through 9. The same digit has a different meaning depending on its place in the numeral. Although this system is familiar and seems obvious to adults, its intricacies are not so obvious to children. These intricacies are important because research has shown that it is difficult to develop procedural fluency with multidigit arithmetic without an understanding of the base-10 system. 64 If such understanding is missing, students make many different errors in multidigit computations. 65

This conclusion does not imply that students must master place value before they can begin computing with multidigit numbers. In fact, the evidence shows that students can develop an understanding of both the base-10 system and computation procedures when they have opportunities to explore how and why the procedures work. 66 That should not be surprising it simply confirms the thesis of this report and the claim we made near the beginning of this chapter. Proficiency develops as the strands connect and interact.

The six observations can be illustrated and supported by examining briefly each of the arithmetic operations. As is the case for single-digit operations, research provides a more complete picture for addition and subtraction than for multiplication and division.

##### Addition Algorithms

The progression followed by students who construct their own procedures is similar in some ways to the progression that can be used to help students learn a standard algorithm with understanding. To illustrate the nature of these progressions, it is useful to examine some specific procedures in detail.

The episode in Box 6&ndash6 from a third-grade class illustrates both how physical materials can support the development of thinking strategies about multidigit algorithms and one type of procedure commonly invented by children. 67 The episode comes from a discussion of students&rsquo solutions to a word problem involving the sum 54+48.

The episode suggests that students&rsquo invented procedures can be constructed through progressive abstraction of their modeling strategies with blocks. First, the objects in the problem were represented directly with the blocks. Then, the quantity representing the first set was abstracted, and only the blocks representing the second set were counted. Finally, the counting words were themselves counted by keeping track of the counts on fingers.

## Score Information

**Credit Granting Score for Human Growth and Development**

ACE Recommended Score*: 50

Semester Hours: 3

Each institution reserves the right to set its own credit-granting policy, which may differ from that of ACE. Contact your college as soon as possible to find out the score it requires to grant credit, the number of credit hours granted, and the course(s) that can be bypassed with a satisfactory score.

*The American Council on Education’s College Credit Recommendation Service (ACE CREDIT) has evaluated CLEP processes and procedures for developing, administering, and scoring the exams. The score listed above is equivalent to a grade of C in the corresponding course. The American Council on Education, the major coordinating body for all the nation’s higher education institutions, seeks to provide leadership and a unifying voice on key higher education issues and to influence public policy through advocacy, research, and program initiatives. Visit the ACE CREDIT website for more information.

## Freak-Outs About Dogs (the Dark, Bugs, Automatic-Flushing Toilets)

**Your Take:** You’ve demonstrated over and over to your 3-year-old that the neighbor’s poodle won’t bite or that there really aren’t any monsters under the bed. But she still goes into hysterics when faced with these irrational fears.

**Your Child&aposs Take:** She can’t fathom why you’re so calm when that dog is so big and so loud. Dr. Damour puts it in perspective: “I𠆝 be anxious if anytime I walked outside I might encounter a horse that wanted to put its nose and teeth near my face!” Fears grow along with your child’s imagination. Now she can imagine what might be under the bed or in the toilet. But she doesn’t know enough about the world to understand why, say, an alligator is real but a monster isn’t, or be sure that neither of them will be hiding in a dark room.

**The Fix:** Avoid conveying your own anxiety. “If you pick your child up whenever you see a dog, you’re letting her know you’re worried too,” says Lansbury. “Just stand next to her so you can block the dog with your body.” For automatic-flushing toilets, put your hand or a Post-it over the sensor.

But don’t dismiss her fear or try to talk her out of it. “She’ll feel like she needs to keep an even tighter grip on her fear because you aren’t getting it,” says Dr. Damour. Acknowledge it as matter-of-factly as possible: “You’re afraid of the dark? Well, let’s figure out what to do because it gets dark every night.” Follow up with increasingly playful questions: “Is the monster furry? What sound does it make?” You won’t make her fear more real by validating it you’ll let her know it’s OK to vent her feelings and may transition the fear from scary to silly. Then build her confidence in baby steps. If you’ve been lying down with her until she falls asleep, sit in a chair for a few nights, then outside the door. 𠇍o it gradually, but stay the course,” says Dr. Kennedy-Moore. This lets her know that you believe she can cope with the challenge, even if it’s tough at first.

*This article originally appeared in Parents magazine&aposs May 2020 issue as “Think Like Your Kid.” Want more from the magazine? Sign up for a monthly print subscription here.*

## What is Moral Development?

******Morality is a code of conduct that guides our actions and thoughts based on our background, culture, philosophy, or religious beliefs. Moral development is a gradual change in the understanding of morality.**

Children’s ability to tell the difference between right and wrong is a part of their moral development process. As their understanding and behavior toward others evolve over time, they apply their knowledge to make the right decisions even when it’s inconvenient for them to do so.

### Piaget’s Theory of Cognitive Development

Swiss psychologist Jean Piaget (1896-1980) was among the first to identify that the way children think is inherently different from the way adults do. Unlike many of his predecessors, Piaget didn’t consider children to be less intelligent versions of adults. They simply have a different way of thinking.

Piaget was the first psychologist to undertake a systematic study of cognitive development. His stage theory of cognitive development explains that children’s mental abilities develop in four stages: sensorimotor, pre-operational, concrete operational, and formal operational. Only after having mastered each one of them, children can reach their full intellectual potential.

For Piaget, children’s moral development is closely related to their cognitive development. In other words, children are only capable of making advanced moral judgments once they become cognitively mature and see things from more than one perspective.

Piaget formulated the cognitive theory of moral development in The Moral Judgment of the Child in 1932. His theory of children’s moral development is an application of his ideas on cognitive development.

## Score Information

**Credit Granting Score for Human Growth and Development**

ACE Recommended Score*: 50

Semester Hours: 3

Each institution reserves the right to set its own credit-granting policy, which may differ from that of ACE. Contact your college as soon as possible to find out the score it requires to grant credit, the number of credit hours granted, and the course(s) that can be bypassed with a satisfactory score.

*The American Council on Education’s College Credit Recommendation Service (ACE CREDIT) has evaluated CLEP processes and procedures for developing, administering, and scoring the exams. The score listed above is equivalent to a grade of C in the corresponding course. The American Council on Education, the major coordinating body for all the nation’s higher education institutions, seeks to provide leadership and a unifying voice on key higher education issues and to influence public policy through advocacy, research, and program initiatives. Visit the ACE CREDIT website for more information.

## Piaget's Theory of Moral Development

Jean Piaget first published his theory of child development during the 1920's but his work did not become prominent until the mid-twentieth century. Piaget is perhaps best known for his theory of children's cognitive development, but he also proposed his own theory about children's moral development. Piaget recognized that cognitive development is closely tied to moral development and was particularly interested in the way children's thoughts about morality changed over time. In this article we limit our discussion of Piaget's theory to adolescent moral development. The Middle Childhood Development Article discusses Piaget's theory with respect to younger children.

According to Piaget, youth develop the morality of cooperation, at the age of 10 years or older. As youth develop a morality of cooperation they realize that in order to create a cooperative society people must work together to decide what is acceptable, and what is not. Piaget believed that youth at this age begin to understand that morals represent social agreements between people and are intended to promote the common good. Furthermore, they recognize people may differ in the way they understand and approach a moral situation or problem. They also begin to understand that the difference between right and wrong is not an absolute but instead must take into account changing variables such as context, motivation, abilities, and intentions. Contrast this to younger youth who believe rules and laws are created by indisputable, wise authorities and believe that rules established by these wise authorities ought never be challenged or changed. Moreover, Piaget believed youth at this age begin to understand that the morality of a decision does not rest solely on the outcome of that decision. For example, youth at this age realize that running a stop sign is wrong, regardless of whether or not a person receives a traffic ticket, or causes a traffic accident.

Furthermore, youth begin to understand the reciprocal benefit of moral decision-making i.e., a moral decision creates the optimal solution for everyone involved, even when only two people are affected. Youth begin to realize that when situations are handled in a manner that seems fair, reasonable, and/or beneficial to all parties, it becomes easier for people to accept and honor the decision. This concept of fairness is called reciprocity. Initially youths' understanding of reciprocity can be very literal and simplistic. For example, last week Terrell, age 11, lent his brand new video game to his good friend Randy. This week, it is Randy who has a new video game. Terrell is likely to insist that Randy should allow him borrow the new video game because from Terrell's perspective, "it's only fair" since he graciously allowed Randy to borrow his new game the week before. Terrell believes that fairness is simplistically determined by exact reciprocity.

By middle adolescence youth expand their understanding of fairness to include ideal reciprocity. Ideal reciprocity refers to a type of fairness beyond simple reciprocity and includes a consideration of another person's best interests. It is best described by the familiar adage, "Do unto others as you would have them do unto you" which many people know as the Golden Rule. Teens who have reached ideal reciprocity will imagine a problem from another person's perspective and try to place themselves in another person's "shoes," before making a moral decision. This concept is best illustrated by the following example:

Suppose Maria, age 14, was looking out the living room window one day and happens to see her older sister, Ava, backing the family's car out of the driveway. As she was watching, Maria saw Ava accidently bump into the mailbox just as she was pulling out into the street. Next, Maria saw Ava get out of the car and examine the damage to the car and mailbox. But, instead of coming back into the house to tell her parents, Ava just drove away.

At a younger age, Maria would have immediately run off to tell her parents about Ava's accident because she knows it is wrong for Ava to drive away without telling her parents what happened. Instead, if Maria has reached ideal reciprocity she will stop herself and imagine what the experience must have been like for Ava. She might realize that if she were in Ava's shoes, she might have done the same thing because she would be embarrassed and scared to tell her parents about the accident. Furthermore, she might decide that Ava would probably prefer to tell her parents about the accident herself, rather than having her little sister "tattle" on her. Therefore, Maria would wait until Ava comes home so she can talk to Ava. During this discussion Maria would encourage Ava to go to her parents with the truth in order to make things right. Thus, ideal reciprocity would enable Maria to examine the problem from her sister's perspective and to make a moral decision based upon the "Golden Rule."

According to Piaget, once ideal reciprocity has been reached moral development has been completed. However, we now know that many youth will continue to refine their moral decision-making process well into early adulthood. So although Piaget pioneered our initial understanding of moral development, research has not always been able to confirm certain portions of his theory. For instance, not only do youth continue refine their criteria for moral decisions into adulthood, but they also continue to improve their ability act according to these criteria. In other words, their moral compass operates to guide their choices and to direct their behavior. Piaget also under-estimated the age at which children are able to take into account another person's moral intention. Piaget believed that this ability did not develop until late childhood, or early adolescence. However, more recent research indicates that this ability develops sooner that Piaget once believed. Younger children are able to recognize the importance of someone's intentions when evaluating the morality of a decision but, younger children tend to be quite naïve in their belief that people's best intentions will dictate the actual choices people make. Despite these weaknesses, Piaget's contributions were very significant because they heavily influenced the later work of Lawrence Kohlberg who published his theory of moral development during the 1950's. Unlike Piaget's earlier theories, Kohlberg's theory of moral development has generally been supported by contemporary research. Kohlberg's theory is discussed in the next section.

## Does logical thinking or counting develop first in children? - Psychology

Cognitive development is an adaptive effort of a child in response to various environmental influences. This is accomplished in the following two ways.

- Assimilation

I ncorporating new thoughts, behavior and objects into the existing structures.

### Cognition

Cognition is a scientific term for thought processing of information, applying knowledge, and changing preferences.

The term is derived from Latin word “cognoscere” which means “to conceptualize”. Animals’ cognitive skills development process is nearly similar to that of humans.

Children have sharp senses with keen observation and are active learners: Note the dynamic brain growth and development during the first 2 years of life. They continuously perceive information and store it as their memory in the specific areas of the brain. Good and bad experiences right from early age modulate human brain development.

Memory is the basis of children's cognitiveꃞvelopment.

Memories are classified into concepts so that thinking, reasoning, and understanding is made possible and used to solve problems.

### Factors that influence cognitive development

- Neurological maturation: It enables children to understand the new experiences and to apply more complex reasoning as they get older.

It permits the central nervous system to reorganize neuronal networks in response to environment’s positive and negative stimulation.

In face of injury or new learning, the body draws upon the reservoir of neurons bank created as a result of large number of neurons produced during the third trimester and early infancy. Each neuron develops about 15,000 synapses by 3 years of age.

The neuronal “pruning” leads to preservation of synapses of frequently used pathways and atrophy of those that are less-used. The strength of synapses increases or decreases as a direct effect of a child’s experiences that influence the synaptic activity.

for child to be able to effectively exhibit the developed cognitive skills.

### Nativism versus Empiricism

There has been a major debate on “nativism versus empiricism” in cognitive development theory.

Nativism implies that children ਊre born with certain skills native abilities, On the other hand, Empiricism believes in theory of 'blank slate' of mind at birth. Thereby meaning that the brain has inborn capabilities for learning from environmental experiences but does not contain any inherent skills.However, this dichotomy is now proved to be false.

Evidences from biological and behavioral sciences confirm that genetic potential interacts with experiences right from birth through entire childhood development process and cognitive development is no exception.

Children because of their individualized environmental influences and genetic make up differ in their temperament and skills development and elicit different stimuli from the same environment.

### Theory of cognitive development

In 1929, the Swiss psychologist, Jean Piaget detailed the most comprehensive cognitive development theory. His theory was based on a basic fact that *“to survive adaptation is a must”.*

This implies that the cognitive development is the result of interaction between an individual and the environment. The contribution of each is complex, as both are continuously changing over time.

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### Types of intelligence

**There are two types of intellectual structure:**

- Schemaਊre those that a child is born with.

They are the internal representation of some specific action, such as sucking and grasping. Most of the neonatal reflexes get included in this category.

### Stages of cognitive development

Piaget proposed that children move through following 4 stages of cognitive skills development:

1. Sensorimotor stage is from birth to 2years of age.

- During the early infancy , behaviour of an infant is dominated by inborn reflexes like sucking, swallowing, visually following an object and so on.

At this stage of development, a child performs actions in response to stimuli from the five senses. The behavior that leads to an interesting or pleasant result is repeated.

2. Pre-operational stage is from 2-7 years of age, and refers to

early childhood development.

- Language use begins and a child can now represent objects by image and words.

3. Concrete operations stage is between 7 and 11 years of age.

- Thinking at this age becomes more logical and less egocentric.

4. Formal operations stage starts from 12 years onwards,

*This represents the most complex mode of thinking. Please note that not everyone achieves this stage of thinking, even as an adult!*

- Children in this age group use logical operations in a systematic fashion.

*Piaget’s four stages of cognitive development theory has had tremendous impact on educational concepts and teaching that has put a new meaning to children’s acquisition of early literacy and numeracy skills.*

## The Sensorimotor Stage: Birth to Age 2

The first stage is aptly named after how infants learn until age two. From birth, infants absorb information through their senses: by touching, looking, and listening. They are very orally fixated and tend to put everything in their mouths. Piaget believed that this stage was valuable to their development, and each consecutive step is built on the growth that occurs in this stage.

We can observe the thought processes of infants through their actions. From about 6 months on, children begin to organize ideas into firm concepts that do not change. An infant may first not make sense of a specific toy, but as they begin to look at it, feel it, and manipulate it often, they are able to represent the object in their minds. This is how we can begin to observe knowledge in babies, as they begin to show understanding of an object for what it is. For example, by playing continuously with a toy animal, an infant begins to understand what the object is and recall their experiences associated with that toy. Piaget labeled this understanding as *object permanence*, which indicates the knowledge of the toy even if it is out of sight. He considered this understanding to be a major milestone in the sensorimotor stage and believed that it demonstrated the differences in the thought processes of toddlers compared to young infants.

The sensorimotor stage is unique in that is occurs without the use of language. As infants cannot speak, Piaget developed a few creative experiments in an effort to understand what they were thinking. His experiments were able to demonstrate that infants do represent objects and understand that they are permanent. In one of his experiments, Piaget consistently hid a toy underneath a blanket. Toddlers, or children between the ages of 18 and 24 months, took initiative to look for the toy themselves, but infants less than 6 months of age did not. The older infants interpreted the hiding of the toy as a prompt to search for it, which is thought to support the idea of object permanency.

## Sorting

Being able to classify things is a key aspect of numeracy. Matching and sorting – for example by size, shape or colour – helps children make sense of the world and develop their logical thinking skills. Here are some ways in which you can support them:

**Set up sorting activities**– there are plenty of variations to explore (and lots of ideas if you search online), but for example you could get the children to sort crayons by colour, pasta by shape, or blocks by size.**Invest in sorting toys**– there are many educational toys designed to help develop sorting skills.**Incorporate sorting into other activities**– ask the children to sort things as they tidy away, or make it part of a nature walk (eg collect leaves and then sort them by shape, colour or size).

## Freak-Outs About Dogs (the Dark, Bugs, Automatic-Flushing Toilets)

**Your Take:** You’ve demonstrated over and over to your 3-year-old that the neighbor’s poodle won’t bite or that there really aren’t any monsters under the bed. But she still goes into hysterics when faced with these irrational fears.

**Your Child&aposs Take:** She can’t fathom why you’re so calm when that dog is so big and so loud. Dr. Damour puts it in perspective: “I𠆝 be anxious if anytime I walked outside I might encounter a horse that wanted to put its nose and teeth near my face!” Fears grow along with your child’s imagination. Now she can imagine what might be under the bed or in the toilet. But she doesn’t know enough about the world to understand why, say, an alligator is real but a monster isn’t, or be sure that neither of them will be hiding in a dark room.

**The Fix:** Avoid conveying your own anxiety. “If you pick your child up whenever you see a dog, you’re letting her know you’re worried too,” says Lansbury. “Just stand next to her so you can block the dog with your body.” For automatic-flushing toilets, put your hand or a Post-it over the sensor.

But don’t dismiss her fear or try to talk her out of it. “She’ll feel like she needs to keep an even tighter grip on her fear because you aren’t getting it,” says Dr. Damour. Acknowledge it as matter-of-factly as possible: “You’re afraid of the dark? Well, let’s figure out what to do because it gets dark every night.” Follow up with increasingly playful questions: “Is the monster furry? What sound does it make?” You won’t make her fear more real by validating it you’ll let her know it’s OK to vent her feelings and may transition the fear from scary to silly. Then build her confidence in baby steps. If you’ve been lying down with her until she falls asleep, sit in a chair for a few nights, then outside the door. 𠇍o it gradually, but stay the course,” says Dr. Kennedy-Moore. This lets her know that you believe she can cope with the challenge, even if it’s tough at first.

*This article originally appeared in Parents magazine&aposs May 2020 issue as “Think Like Your Kid.” Want more from the magazine? Sign up for a monthly print subscription here.*

## Does logical thinking or counting develop first in children? - Psychology

Middle childhood development is the concrete operational stage, when the focus is on logical thinking, intelligence and psychosocial development. Contrary to the usual concept, middle childhood is as special a stage of childhood development as toddlers and teenagers.

The American Academy of Pediatrics defines middle childhood from onset of sixth year to completion of tenth year of life.

Development in middle childhood is the outcome of interaction between the milestones achieved and ongoing learning experiences.

Motor skills, language proficiency and cognitive capabilities attained in the previous years are further mastered during the formative years of middle childhood.

Children in their middle childhood years begin to understand the rules of society and its moral bindings.

*Middle childhood development helps children to cope with the increasing demands of the classroom curriculum. Learning disabilities and behavior problems surface at this stage that need medical attention.*

### Middle childhood development leads to logical thinking

Jean Piaget’s has described middle childhood development as the concrete operational stage, when the focus is on developing logical thinking.

The thinking process during middle childhood years gets more logical, flexible, and organized than it was during early childhood development. Children during this phase of development are “critical thinkers” with the ability to reason, remember, repeat, reorganize, relate, and reflect.

Enhanced logical thinking are easily noticed in their social relationships strive for scholastic, cultural and sports excellence. Its reflections are also seen in their creativity, which gets significantly refined during middle childhood.

### Aspects of middle childhood development

**Logic **Children learn to reason and use the acquired knowledge constructively. They evaluate the thoughts both that of the others and their own before putting them in to action.

Constructive application of the information gathered involves child’s newly developed capability to accommodate and assimilate the thoughts.

**Decentration **Thinking changes from perception bound to concrete logical during middle childhood years. Cognitive development during middle childhood development helps children overcome early childhood tendency to focus only on one aspect of a situation and enable the children in middle childhood to perceive the events/problems from different angles.

**Reversibility**

A ny process can be returned to its original state.

**Causality ** Principle of or relationship between cause and effect and that nothing can happen without being caused.

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### Intelligence development

Middle childhood development shows a sharp rise in child’s capacity to understand the world. Children in this phase begin to think rationally, and use resources effectively when faced with challenges and **that is intelligence.**

**The components of intelligence that i ncrease during middle childhood.**

**Fluid Intelligence:**

It gives the ability to remember, reason, and process information increases.

**Crystallized Intelligence:**

Increases in this type of intelligence gives children the ability to gather information, master skills, and develop strategies.

**Metacognition:**

Metacognition is the process of monitoring own thoughts and memory development.

**Development of information processing**

- Decline in time needed to process information and so increase in information processing capacity.

### Attention span

The attention span increases during middle childhood development as a result of more c ontrolled an adaptable thought process of children at this age.

Consequently, children develop the capabilities to -

- Plan the attention time they need to assign to a task.

**Inability to control attention span**

Inability to control attention is seen in children with ADHD.

These children face great difficulty staying on a specified task. They tend to act impulsively and usually are hyperactive.

### Memory strategies

Memory strategies are the deliberate mental activities used to store and retain information. Working memory plays an important role in learning during middle childhood.

General intelligence is highly correlated with working memory capacity.

Memory capacity shows a steady developmental increase during middle childhood years as:

- Capacity of short term memory improves significantly.

### Child's scholastic performance as related to memory span

Memory strategies require time and effort in order to show desired results.

Memory span improves from the age of

5 years onward and the adult levels of performance is reached by about 15 years of age.

During middle childhood, children arrange the vast amount of information in their memories into increasingly elaborate, hierarchically structured networks.

**Strategies to improve memory:**

- Rehearse

Rehearsal i nvolves repeating information to oneself over and over again.

### Communication capabilities

### Communicative skills development during middle childhood

- Children in their middle childhood can engage in

constructive conversation with a variety of relevant questions.

## What is Moral Development?

******Morality is a code of conduct that guides our actions and thoughts based on our background, culture, philosophy, or religious beliefs. Moral development is a gradual change in the understanding of morality.**

Children’s ability to tell the difference between right and wrong is a part of their moral development process. As their understanding and behavior toward others evolve over time, they apply their knowledge to make the right decisions even when it’s inconvenient for them to do so.

### Piaget’s Theory of Cognitive Development

Swiss psychologist Jean Piaget (1896-1980) was among the first to identify that the way children think is inherently different from the way adults do. Unlike many of his predecessors, Piaget didn’t consider children to be less intelligent versions of adults. They simply have a different way of thinking.

Piaget was the first psychologist to undertake a systematic study of cognitive development. His stage theory of cognitive development explains that children’s mental abilities develop in four stages: sensorimotor, pre-operational, concrete operational, and formal operational. Only after having mastered each one of them, children can reach their full intellectual potential.

For Piaget, children’s moral development is closely related to their cognitive development. In other words, children are only capable of making advanced moral judgments once they become cognitively mature and see things from more than one perspective.

Piaget formulated the cognitive theory of moral development in The Moral Judgment of the Child in 1932. His theory of children’s moral development is an application of his ideas on cognitive development.

## Adding It Up: Helping Children Learn Mathematics (2001)

Whole numbers are the easiest numbers to understand and use. As we described in the previous chapter, most children learn to count at a young age and understand many of the principles of number on which counting is based. Even if children begin school with an unusually limited facility with number, intensive instructional activities can be designed to help them reach similar levels as their peers. 1 Children&rsquos facility with counting provides a basis for them to solve simple addition, subtraction, multiplication, and division problems with whole numbers. Although there still is much for them to work out during the first few years of school, children begin with substantial knowledge on which they can build.

In this chapter, we examine the development of proficiency with whole numbers. We show that students move from methods of solving numerical problems that are intuitive, concrete, and based on modeling the problem situation directly to methods that are more problem independent, mathematically sophisticated, and reliant on standard symbolic notation. Some form of this progression is seen in each operation for both single-digit and multidigit numbers.

We focus on computation with whole numbers because learning to compute can provide young children the opportunity to work through many number concepts and to integrate the five strands of mathematical proficiency. This learning can provide the foundation for their later mathematical development. Computation with whole numbers occupies much of the curriculum in the early grades, and appropriate learning experiences in these grades improve children&rsquos chances for later success.

Whole number computation also provides an instructive example of how routine-appearing procedural skills can be intertwined with the other strands of proficiency to increase the fluency with which the skills are used. For years, learning to compute has been viewed as a matter of following the teacher&rsquos directions and practicing until speedy execution is achieved. Changes in career demands and the tasks of daily life, as well as the availability of new computing tools, mean that more is now demanded from the study of computation. More than just a means to produce answers, computation is increasingly seen as a window on the deep structure of the number system. Fortunately, research is demonstrating that both skilled performance and conceptual understanding are generated by the same kinds of activities. No tradeoffs are needed. As we detail below, the activities that provide this powerful result are those that integrate the strands of proficiency.

### Operations with Single-Digit Whole Numbers

As students begin school, much of their number activity is designed to help them become proficient with single-digit arithmetic. By *single-digit arithmetic,* we mean the sums and products of single-digit numbers and their companion differences and quotients (e.g., 5+7=12, 12&ndash5=7, 12&ndash7=5 and 5×7=35, 35÷5=7, 35÷7=5). For most of a century, learning single-digit arithmetic has been characterized in the United States as &ldquolearning basic facts,&rdquo and the emphasis has been on memorizing those facts. We use the term *basic number combinations* to emphasize that the knowledge is relational and need not be memorized mechanically. Adults and &ldquoexpert&rdquo children use a variety of strategies, including automatic or semiautomatic rules and reasoning processes to efficiently produce the basic number combinations. 2 Relational knowledge, such as knowledge of commutativity, not only promotes learning the basic number combinations but also may underlie or affect the mental representation of this basic knowledge. 3

The domain of early number, including children&rsquos initial learning of single-digit arithmetic, is undoubtedly the most thoroughly investigated area of school mathematics. A large body of research now exists about how children in many countries actually learn single-digit operations with whole numbers. Although some educators once believed that children memorize their &ldquobasic facts&rdquo as conditioned responses, research shows that children do *not* move from knowing nothing about the sums and differences of numbers to having the basic number combinations memorized. Instead, they move through a series of progressively more advanced and abstract methods for working out the answers

to simple arithmetic problems. Furthermore, as children get older, they use the procedures more and more efficiently. 4 Recent evidence indicates children can use such procedures quite quickly. 5 Not all children follow the same path, but all children develop some intermediate and temporary procedures.

Most children continue to use those procedures occasionally and for some computations. Recall eventually becomes the predominant method for some children, but current research methods cannot adequately distinguish between answers produced by recall and those generated by fast (nonrecall) procedures. This chapter describes the complex processes by which children learn to compute with whole numbers. Because the research on whole numbers reveals how much can be understood about children&rsquos mathematical development through sustained and interdisciplinary inquiry, we give more details in this chapter than in subsequent chapters.

#### Word Problems: A Meaningful Context

One of the most meaningful contexts in which young children begin to develop proficiency with whole numbers is provided by so-called word problems. This assertion probably comes as a surprise to many, especially mathematics teachers in middle and secondary school whose students have special difficulties with such problems. But extensive research shows that if children can count, they can begin to use their counting skills to solve simple word problems. Furthermore, they can advance those counting skills as they solve more problems. 6 In fact, it is in solving word problems that young children have opportunities to display their most advanced levels of counting performance and to build a repertoire of procedures for computation.

Most children entering school can count to solve word problems that involve adding, subtracting, multiplying, and dividing. 7 Their performance increases if the problems are phrased simply, use small numbers, and are accompanied by physical counters for the children to use. The exact procedures children are likely to use have been well documented. Consider the following problems:

*Sally had 6 toy cars. She gave 4 to Bill. How many did she have left?*

*Sally had 4 toy cars. How many more does she need to have 6?*

Most young children solve the first problem by counting a set of 6, removing 4, and counting the remaining cars to find the answer. In contrast,

they solve the second problem by counting a set of 4, adding in more as they count &ldquofive, six,&rdquo and then counting those added in to find the answer.

Children solve these problems by &ldquoacting out&rdquo the situation&mdashthat is, by modeling it. They invent a procedure that mirrors the actions or relationships described in the problem. This simple but powerful approach keeps procedural fluency closely connected to conceptual understanding and strategic competence. Children initially solve only those problems that they understand, that they can represent or model using physical objects, and that involve numbers within their counting range. Although this approach limits the kinds of problems with which children are successful, it also enables them to solve a remarkable range of problems, including those that involve multiplying and dividing.

Since children intuitively solve word problems by modeling the actions and relations described in them, it is important to distinguish among the different types of problems that can be represented by adding or subtracting, and among those represented by multiplying or dividing. One useful way of classifying problems is to heed the children&rsquos approach and examine the actions and relations described. This examination produces a taxonomy of problem types distinguished by the solution method children use and provides a framework to explain the relative difficulty of problems.

Four basic classes of addition and subtraction problems can be identified: problems involving (a) joining, (b) separating, (c) part-part-whole relations, and (d) comparison relations. Problems within a class involve the same type of action or relation, but within each class several distinct types of problems can be identified depending on which quantity is the unknown (see Table 6&ndash1). Students&rsquo procedures for solving the entire array of addition and subtraction problems and the relative difficulty of the problems have been well documented. 8

For multiplication and division, the simplest kinds of problems are grouping situations that involve three components: the number of sets, the number in each set, and the total number. For example:

*Jose made 4 piles of marbles with 3 marbles in each pile. How many marbles did Jose have?*

In this problem, the number and size of the sets is known and the total is unknown. There are two types of corresponding division situations depending on whether one must find the number of sets or the number in each set. For example:

**Addition and Subtraction Problem Types**

Connie had 5 marbles. Juan gave her 8 more marbles. How many marbles does Connie have altogether?

Connie has 5 marbles. How many more marbles does she need to have 13 marbles altogether?

Connie had some marbles. Juan gave her 5 more. Now she has 13 marbles. How many marbles did Connie have to start with?

Connie had 13 marbles. She gave 5 to Juan. How many marbles does Connie have left?

Connie had 13 marbles. She gave some to Juan. Now she has 5 marbles left. How many marbles did Connie give to Juan?

Connie had some marbles. She gave 5 to Juan. Now she has 8 marbles left. How many marbles did Connie have to start with?

Connie has 5 red marbles and 8 blue marbles. How many marbles does she have altogether?

Connie has 13 marbles: 5 are red and the rest are blue. How many blue marbles does Connie have?

Connie has 13 marbles. Juan has 5 marbles. How many more marbles does Connie have than Juan?

Juan has 5 marbles. Connie has 8 more than Juan. How many marbles does Connie have?

Connie has 13 marbles. She has 5 more marbles than Juan. How many marbles does Juan have?

SOURCE: Carpenter, Fennema, Franke, Levi, and Empson, 1999, p. 12. Used by permission of Heinemann. All rights reserved.

*Jose has 12 marbles and puts them into piles of 3. How many piles does he have?*

*Jose has 12 marbles and divides them equally into 3 piles. How many marbles are in each pile?*

Additional types of multiplication and division problems are introduced later in the curriculum. These include rate problems, multiplicative comparison problems, array and area problems, and Cartesian products. 9

As with addition and subtraction problems, children initially solve multiplication and division problems by modeling directly the action and relations in the problems. 10 For the above multiplication problem with marbles, they form four piles with three in each and count the total to find the answer. For the first division problem, they make groups of the specified size of three and count the number of groups to find the answer. For the other problem, they make the three groups by dealing out (as in cards) and count the number in one of the groups. Although adults may recognize both problems as 12 divided by 3, children initially think of them in terms of the actions or relations portrayed. Over time, these direct modeling procedures are replaced by more efficient methods based on counting, repeated adding or subtracting, or deriving an answer from a known number combination. 11

The observation that children use different methods to solve problems that describe different situations has important implications. On the one hand, directly modeling the action in the problem is a highly sensible approach. On the other hand, as numbers in problems get larger, it becomes inefficient to carry out direct modeling procedures that involve counting all of the objects.

Children&rsquos proficiency gradually develops in two significant directions. One is from having a different solution method for each type of problem to developing a single general method that can be used for classes of problems with a similar mathematical structure. Another direction is toward more efficient calculation procedures. Direct-modeling procedures evolve into the more advanced counting procedures described in the next section. For word problems, these procedures are essentially abstractions of direct modeling that continue to reflect the actions and relations in the problems.

The method children might use to solve a class of problems is not necessarily the method traditionally taught. For example, many children come to solve the &ldquosubtraction&rdquo problems described above by counting, adding up, or thinking of a related addition combination because any of these methods is easier and more accurate than counting backwards. The method traditionally presented in textbooks, however, is to solve both of these problems by

subtracting, which moves students toward the more difficult and error-prone procedure of counting down. Ultimately, most children begin to use recall or a rapid mental procedure to solve these problems, and they come to recognize that the same general method can be used to solve a variety of problems.

#### Single-Digit Addition

Children come to understand the meaning of addition in the context of word problems. As we noted in the previous section, children move from counting to more general methods to solve different classes of problems. As they do, they also develop greater fluency with each specific method. We call these specific counting methods *procedures*. Although educators have long recognized that children use a variety of procedures to solve single-digit addition problems, 12 substantial research from all over the world now indicates that children move through a progression of different procedures to find the sum of single-digit numbers. 13

This progression is depicted in Box 6&ndash1. First, children count out objects for the first addend, count out objects for the second addend, and count all of the objects (count all). This general counting-all procedure then becomes abbreviated, internalized, and abstracted as children become more experienced with it. Next, they notice that they do not have to count the objects for the first addend but can start with the number in the first or the larger addend and count on the objects in the other addend (count on). As children count

**Box 6&ndash1** **Learning Progression for Single-Digit Addition**

on with objects, they begin to use the counting words themselves as countable objects and keep track of how many words have been counted on by using fingers or auditory patterns. The counting list has become a representational tool. With time, children recompose numbers into other numbers (4 is recomposed into 3+1) and use thinking strategies in which they turn an addition combination they do not know into one they do know (3+4 becomes 3+3+1). In the United States, these strategies for *derived number combinations* often use a so-called double (2+2, 3+3, etc.). These doubles are learned very quickly.

As Box 6&ndash1 shows, throughout this learning progression, specific sums move into the category of being rapidly recalled rather than solved in one of the other ways described above. Children vary in the sums they first recall readily, though doubles, adding one (the sum is the next counting word), and small totals are the most readily recalled. Several procedures for single-digit addition typically coexist for several years they are used for different numbers and in different problem situations. Experience with figuring out the answer to addition problems provides the basis both for understanding what it means to say &ldquo5+3=8&rdquo and for eventually recalling that sum without the use of any conscious strategy.

Children in many countries often follow this progression of procedures, a natural progression of embedding and abbreviating. Some of these procedures can be taught, which accelerates their use, 14 although direct teaching of these strategies must be done conceptually rather than simply by using imitation and repetition. 15 In some countries, children learn a general procedure known as &ldquomake a 10&rdquo (see Box 6&ndash2). 16 In this procedure the solver makes a 10 out of one addend by taking a number from the other addend. Educators in some countries that use this approach believe this first instance of regrouping by making a 10 provides a crucial foundation for later multidigit arithmetic. In some Asian countries this procedure is presumably facilitated by the number words. 17 It has also been taught in some European countries in which the number names are more similar to those of English, suggesting that the procedure can be used with a variety of number-naming systems. The procedure is now beginning to appear in U.S. textbooks, 18 although so little space may be devoted to it that some children may not have adequate time and opportunity to understand and learn it well.

There is notable variation in the procedures children use to solve simple addition problems. 19 Confronted with that variation, teachers can take various steps to support children&rsquos movement toward more advanced procedures. One technique is to talk about slightly more advanced procedures *and* why

**Box 6&ndash2** **Make a Ten: B+6=?**

they work. 20 The teacher can stimulate class discussion about the procedures that various students are using. Students can be given opportunities to present their procedures and discuss them. Others can then be encouraged to try the procedure. Drawings or concrete materials can be used to reveal how the procedures work. The advantages and disadvantages of different procedures can also be examined. For a particular procedure, problems can be created for which it might work well or for which it is inefficient.

Other techniques that encourage students to use more efficient procedures are using large numbers in problems so that inefficient counting procedures cannot easily be used and hiding one of the sets to stimulate a new way of thinking about the problem. Intervention studies indicate that teaching counting-on procedures in a conceptual way makes all single-digit sums accessible to U.S. first graders, including children who are learning disabled and those who do not speak English as their first language. 21 Providing support for children to improve their own procedures does *not* mean, however, that every child is taught to use all the procedures that other children develop. Nor does it mean that the teacher needs to provide every child in a class with

support and justification for different procedures. Rather, the research provides evidence that, at any one time, most children use a small number of procedures and that teachers can learn to identify them and help children learn procedures that are conceptually more efficient (such as counting on from the larger addend rather than counting all). 22

Mathematical proficiency with respect to single-digit addition encompasses not only the fluent performance of the operation but also conceptual understanding and the ability to identify and accurately represent situations in which addition is required. Providing word problems as contexts for adding and discussing the advantages and disadvantages of different addition procedures are ways of facilitating students&rsquo adaptive reasoning and improving their understanding of addition processes.

#### Single-Digit Subtraction

Subtraction follows a progression that generally parallels that for addition (see Box 6&ndash3). Some U.S. children also invent counting-down methods that model the taking away of numbers by counting back from the total. But counting down and counting backward are difficult for many children. 23

**Box 6&ndash3** **Learning Progression for Single-Digit Subtraction**

A considerable number of children invent counting-up procedures for situations in which an unknown quantity is added to a known quantity. 24 Many of these children later count up in taking-away subtraction situations (13&ndash8=? becomes 8+?=13). When counting up is not introduced, many children may not invent it until the second or third grade, if at all. Intervention studies with U.S. first graders that helped them see subtraction situations as taking away the first *x* objects enabled them to learn and understand counting-up-to procedures for subtraction. Their subtraction accuracy became as high as that for addition. 25

Experiences that focus on part-part-whole relations have also been shown to help students develop more efficient thinking strategies, especially for subtraction. 26 Students examine a join or separate situation and identify which number represents the whole quantity and which numbers represent the parts. These experiences help students see how addition and subtraction are related and help them recognize when to add and when to subtract. For students in grades K to 2, learning to see the part-whole relations in addition and subtraction situations is one of their most important accomplishments in arithmetic. 27

For students in grades K to 2, learning to see the part-whole relations in addition and subtraction situations is one of their most important accomplish-ments in arithmetic.

Examining the relationships between addition and subtraction and seeing subtraction as involving a known and an unknown addend are examples of adaptive reasoning. By providing experiences for young students to develop adaptive reasoning in addition and subtraction situations, teachers are also anticipating algebra as students begin to appreciate the inverse relationships between the two operations. 28

#### Single-Digit Multiplication

Much less research is available on single-digit multiplication and division than on single-digit addition and subtraction. U.S. children progress through a sequence of multiplication procedures that are somewhat similar to those for addition. 29 They make equal groups and count them all. They learn skip-count lists for different multipliers (e.g., they count 4, 8, 12, 16, 20,&hellipto multiply by four). They then count on and count down these lists using their fingers to keep track of different products. They invent thinking strategies in which they derive related products from products they know.

As with addition and subtraction, children invent many of the procedures they use for multiplication. They find patterns and use skip counting (e.g., multiplying 4×3 by counting &ldquo3, 6, 9, 12&rdquo). Finding and using patterns and other thinking strategies greatly simplifies the task of learning multiplication tables (see Box 6&ndash4 for some examples). 30 Moreover, finding and describing

**Box 6&ndash4** **Thinking Strategies for Single-Digit Multiplication**

In single-digit arithmetic, there are 100 multiplication combinations that students must learn. Commutativity reduces that number by about half. Multiplication by 0 and by 1 may quickly be deduced from the meaning of multiplication. Multiplication by 2 consists of the &ldquodoubles&rdquo from addition. Single-digit multiplication by 9 is simplified by a pattern: in the product, the sum of the digits is 9. (For example, 9×7=63 and 6+3=9.) Multiplication by 5 may also be deduced through patterns or by first multiplying by 10 and then dividing by 2, since 5 is half of 10.

The remaining 15 multiplication combinations (and their commutative counterparts) may be computed by skip counting or by building on known combinations. For example, 3×6 must be 6 more than 2×6, which is 12. So 3×6 is 18. Similarly, 4×7 must be twice 2×7, which is 14. So 4×7 is 28. (Note that these strategies require proficiency with addition.) To compute multiples of 6, one can build on the multiples of 5. So, for example, 6×8 must be 8 more than 5×8, which is 40. So 6×8 is 48. If students are comfortable with such strategies for multiplication by 3, 4, and 6, only three multiplication combinations remain: 7×7, 7×8, and 8×8. These can be derived from known combinations in many creative ways.

patterns are a hallmark of mathematics. Thus, treating multiplication learning as pattern finding both simplifies the task and uses a core mathematical idea.

After children identify patterns, they still need much experience to produce skip-count lists and individual products rapidly. Little is known about how children acquire this fluency or what experiences might be of most help. A good deal of research remains to be done, in the United States and in other countries, to understand more about this process.

#### Single-Digit Division

Division arises from the two splitting situations described above. A collection is split into groups of a specified size or into a specified number of groups. Just as subtraction can be thought of using a part-part-whole relation, division can be thought of as splitting a number into two factors. Hence, divisions can also be approached as finding a missing factor in multiplication. For example, 72÷9=? can be thought of as 9÷?=72. But there is little

research concerning how best to introduce and use this relationship, or whether it is helpful to learn a division combination at the same time as the corresponding multiplication combination. Further, there is little research about how to help children learn and use easily all of the different symbols for division, such as 15÷3, and

#### Practicing Single-Digit Calculations

Practicing single-digit calculations is essential for developing fluency with them. This practice can occur in many different contexts, including solving word problems. 31 Drill alone does not develop mastery of single-digit combinations. 32 Practice that follows substantial initial experiences that support understanding and emphasize &ldquothinking strategies&rdquo has been shown to improve student achievement with single-digit calculations. 33 This approach allows computation and understanding to develop together and facilitate each other. Explaining how procedures work and examining their benefits, as part of instruction, support retention and yield higher levels of performance. 34 In this way, computation practice remains integrated with the other strands of proficiency such as strategic competence and adaptive reasoning.

Practicing single-digit calculations is essential for developing fluency with them.

It is helpful for some practice to be targeted at recent learning. After students discuss a new procedure, they can benefit from practicing it. For example, if they have just discussed the make-a-10 procedure (see Box 6&ndash2), solving problems involving 8 or 9 in which the procedure can easily be used provides beneficial practice. It also is helpful for some practice to be cumulative, occurring well after initial learning and reviewing the more advanced procedures that have been learned.

Many U.S. students have had the experience of taking a timed test that might be a page of mixed addition, subtraction, multiplication, and division problems. This scattershot form of practice is, in our opinion, rarely the best use of practice time. Early in learning it can be discouraging for students who have learned only primitive, inefficient procedures. The experience can adversely affect students&rsquo disposition toward mathematics, especially if the tests are used to compare their performance. 35 If appropriately delayed, timed tests can benefit some students, but targeted forms of practice, with particular combinations that have yet to be mastered or on which efficient procedures can be used, are usually more effective. 36

#### Summary of Findings an Learning Single-Digit Arithmetic

For addition and subtraction, there is a well-documented progression of procedures used worldwide 37 by many children that stems from the sequential nature of the list of number words. This list is first used as a counting tool then it becomes a representational tool in which the number words themselves are the objects that are counted. 38 Counting becomes abbreviated and rapid, and students begin to develop procedures that take advantage of properties of arithmetic to simplify computation. During this progression, individual children use a range of different procedures on different problems and even on the same problem encountered at different times. 39 Even adults have been found to use a range of different procedures for simple addition problems. 40 Further, it takes an extended period of time before new and better strategies replace previously used strategies. 41 Learning-disabled children and others having difficulty with mathematics do not use procedures that differ from this progression. They are just slower than others in moving through it. 42

Instruction can help students progress. 43 Counting on is accessible to first graders it makes possible the rapid and accurate addition of all single-digit numbers. Single-digit subtraction is usually more difficult than addition for U.S. children. If children understand the relationship between addition and subtraction, perhaps by thinking of the problem in terms of part-part-whole, then they recognize that counting up can be used to solve subtraction problems. This recognition makes subtraction more accessible. 44

The procedures of counting on for addition and counting up for subtraction can be learned with relative ease. Multiplication and division are somewhat more difficult. Even adults might not have quick ways of reconstructing the answers to problems like 6×8=? or if they have forgotten the answers. Learning these combinations seems to require much specific pattern-based knowledge that needs to be orchestrated into accessible and rapid-enough products and quotients. As with addition and subtraction, children derive some multiplication and division combinations from others for example, they recall that 6×6=36 and use that combination to conclude that 6×7=42. Research into ways to support such pattern finding, along with the necessary follow-up thinking and practice, is needed if all U.S. children are to acquire higher levels of proficiency in single-digit arithmetic.

Acquiring proficiency with single-digit computations involves much more than rote memorization. This domain of number demonstrates how the different strands of proficiency contribute to each other. At this early point in

development, many of the linkages among strands result from children&rsquos natural inclination to make sense of things and to engage in actions that they understand. Children begin with conceptual understanding of number and the meanings of the operations. They develop increasingly sophisticated representations of the operations such as counting-on or counting-up procedures as they gain greater fluency. They also lean heavily on reasoning to use known answers such as doubles to generate unknown answers. Even in the early grades, students choose adaptively among different procedures and methods depending on the numbers involved or the context. 45 As long as the focus in the classroom is on sense making, they rarely make nonsensical errors, such as adding to find the answer when they should subtract. Proficiency comes from making progress within each strand and building connections among the strands. A productive disposition is generated by and supports this kind of learning because students recognize their competence at making sense of quantitative situations and solving arithmetic problems.

### Multidigit Whole Number Calculations

Step-by-step procedures for adding, subtracting, multiplying, or dividing numbers are called algorithms. For example, the first step in one algorithm for multiplying a three-digit number by a two-digit number is to write the three-digit number above the two-digit number and to begin by multiplying the one&rsquos digit in the top number by the one&rsquos digit in the bottom number (see Box 6&ndash5).

In the past, algorithms different from those taught today for addition, subtraction, multiplication, and division have been taught in U.S. schools. Also, algorithms different from those taught in the United States today are currently being taught in other countries. 46 Each algorithm has advantages

**Box 6&ndash5** **Beginning a multiplication algorithm**

and disadvantages. Therefore, it is important to think about which algorithms are taught and the reasons for teaching them.

Learning to use algorithms for computation with multidigit numbers is an important part of developing proficiency with numbers. Algorithms are procedures that can be executed in the same way to solve a variety of problems arising from different situations and involving different numbers. This feature has three important implications. First, it means that algorithms are useful tools&mdashdifferent procedures do not need to be invented for each problem. Second, algorithms illustrate a significant feature of mathematics: The structure of problems can be abstracted from their immediate context and compared to see whether different-looking problems can be solved in similar ways. Finally, the process of developing fluency with arithmetic algorithms in elementary school can contribute to progress in developing the other strands of proficiency if time is spent examining why algorithms work and comparing their advantages and disadvantages. Such analyses can boost conceptual understanding by revealing much about the structure of the number system itself and can facilitate understanding of place-value representations.

Research findings about learning algorithms for whole numbers can be summarized with seven important observations. First, the linkages among the strands of mathematical proficiency that are possible when children develop proficiency with single-digit arithmetic can be continued with multidigit arithmetic. For example, there can be a close connection between understanding and fluency. Conceptual knowledge that comes with understanding is important for the development of procedural fluency, while fluent procedural knowledge supports the development of further understanding and learning. When students fail to grasp the concepts that underlie procedures or cannot connect the concepts to the procedures, they frequently generate flawed procedures that result in systematic patterns of errors. 47 These so-called buggy algorithms are signs that the strands are not well connected. 48 When the initial computational procedures that students use to solve multidigit problems reflect their understanding of numbers, understanding and fluency develop together.

A second observation is that understanding and fluency are related. For multidigit addition and subtraction, given conventional instruction that emphasizes practicing procedures, a substantial percentage of children gain understanding of multidigit concepts before using a correct procedure, but another substantial minority do the opposite. 49 In contrast, instructional programs that emphasize understanding algorithms before using them have been shown to lead to increases in both conceptual and procedural knowledge. 50

So there is some evidence that understanding is the basis for developing procedural fluency. 51

A third observation is that proficiency with multidigit computation is more heavily influenced by instruction than single-digit computation is. Many features of multidigit procedures (e.g., the base-10 elements and how they are represented by place-value notation) are not part of children&rsquos everyday experience and need to be learned in the classroom. In fact, many students are likely to need help learning efficient forms of multidigit procedures. This means that students in different classrooms and receiving different instruction might follow different learning progressions use different procedures. 52 For single-digit addition and subtraction, the same learning progression occurs for many children in many countries regardless of the nature and extent of instruction. 53 But multidigit procedures, even those for addition and subtraction, depend much more on what is taught.

A fourth observation is that children can and do devise or invent algorithms for carrying out multidigit computations. 54 Opportunities to construct their own procedures provide students with opportunities to make connections between the strands of proficiency. Procedural fluency is built directly on their understanding. The invention itself is a kind of problem solving, and they must use reasoning to justify their invented procedure. Students who have invented their own correct procedures also approach mathematics with confidence rather than fear and hesitation. 55 Students invent many different computational procedures for solving problems with large numbers. For addition, they eventually develop a procedure that is consistent with the thinking that is used with standard algorithms. That thinking enables them to make sense of the algorithm as a record on paper of what they have already been thinking. For subtraction, many students can develop adding-up procedures and, if using concrete materials like base-10 blocks, can also develop ways of thinking that parallel algorithms usually taught today. 56 Some students need help to develop efficient algorithms, however, especially for multiplication and division. Consequently, for these students the process of learning algorithms involves listening to someone else explain an algorithm and trying it out, all the while trying to make sense of it. Research suggests that students are capable of listening to their peers and to the teacher and of making sense of an algorithm if it is explained and if the students have diagrams or concrete materials that support their understanding of the quantities involved. 57

Fifth, research has shown that students can learn well from a variety of different instructional approaches, including those that use physical materials to represent hundreds, tens, and ones, those that emphasize special counting

activities (e.g., count by tens beginning with any number), and those that focus on developing mental computation methods. 58 Although the data do not point to a single preferred instructional approach, they do suggest that effective approaches share some key features: The multidigit procedures that students use are easily understood students are encouraged to use algorithms that they understand instructional supports (classroom discussions, physical materials, etc.) are available to focus students&rsquo attention on the base-10 structure of the number system and on how that structure is used in the algorithm and students are helped to progress to using reasonably efficient but still comprehensible algorithms. 59

Sixth, research on symbolic learning argues that, to be helpful, manipulatives or other physical models used in teaching must be represented by a learner both as the objects that they are and as symbols that stand for something else. 60 The physical characteristics of these materials can be initially distracting to children, and it takes time for them to develop mathematical meaning for any kind of physical model and to use it effectively. These findings suggest that sustained experience with any physical models that students are expected to use may be more effective than limited experience with a variety of different models. 61

In view of the attention given to the use of concrete models in U.S. school mathematics classes, we offer a special note regarding their effective use in multidigit arithmetic. Research indicates that students&rsquo experiences using physical models to represent hundreds, tens, and ones can be effective *if* the materials help them think about how to combine quantities and, eventually, how these processes connect with written procedures. The models, however, are not automatically meaningful for students the meaning must be constructed as they work with the materials. Given time to develop meaning for a model and connect it with the written procedure, students have shown high levels of performance using the written procedure and the ability to give good explanations for how they got their answers. 62 In order to support understanding, however, the physical models need to show tens to be collections of ten ones and to show hundreds to be simultaneously 10 tens and 100 ones. For example, base-10 blocks have that quality, but chips all of the same size but with different colors for hundreds, tens, and ones do not.

A seventh and final observation is that the English number words and the Hindu-Arabic base-10 place-value system for writing numbers complicate the teaching and learning of multidigit algorithms in much the same way, as discussed in Chapter 5, that they complicate the learning of early number concepts. 63 Closely related to the difficulties posed by the irregu-

larities with number words are difficulties posed by the complexity of the system for writing numbers. As we said in chapter 3, the base-10 place-value system is very efficient. It allows one to write very large numbers using only 10 symbols, the digits 0 through 9. The same digit has a different meaning depending on its place in the numeral. Although this system is familiar and seems obvious to adults, its intricacies are not so obvious to children. These intricacies are important because research has shown that it is difficult to develop procedural fluency with multidigit arithmetic without an understanding of the base-10 system. 64 If such understanding is missing, students make many different errors in multidigit computations. 65

This conclusion does not imply that students must master place value before they can begin computing with multidigit numbers. In fact, the evidence shows that students can develop an understanding of both the base-10 system and computation procedures when they have opportunities to explore how and why the procedures work. 66 That should not be surprising it simply confirms the thesis of this report and the claim we made near the beginning of this chapter. Proficiency develops as the strands connect and interact.

The six observations can be illustrated and supported by examining briefly each of the arithmetic operations. As is the case for single-digit operations, research provides a more complete picture for addition and subtraction than for multiplication and division.

##### Addition Algorithms

The progression followed by students who construct their own procedures is similar in some ways to the progression that can be used to help students learn a standard algorithm with understanding. To illustrate the nature of these progressions, it is useful to examine some specific procedures in detail.

The episode in Box 6&ndash6 from a third-grade class illustrates both how physical materials can support the development of thinking strategies about multidigit algorithms and one type of procedure commonly invented by children. 67 The episode comes from a discussion of students&rsquo solutions to a word problem involving the sum 54+48.

The episode suggests that students&rsquo invented procedures can be constructed through progressive abstraction of their modeling strategies with blocks. First, the objects in the problem were represented directly with the blocks. Then, the quantity representing the first set was abstracted, and only the blocks representing the second set were counted. Finally, the counting words were themselves counted by keeping track of the counts on fingers.