Mathematics

From learning basic numbers to memorizing multiplication tables to solving algebraic equations, children progress through an amazing range of mathematical abilities as they grow. In this section we will explore how mathematical reasoning develops through childhood.

Mathematical Skills in Infancy

Amazing but true: Researchers have found evidence that even newborns have rudimentary mathematical skills. Humans seem to be "born with a fundamental sense of quantity" (Geary, 1994, p. 1). For example, researchers showed newborn infants less than one week old!) a card with two black dots (Antell & Keating, 1983). The newborns looked at the dots for a bit, then started looking away. Looking away signals boredom.  When researchers then switched to a card that had three dots, the newborns regained interest in looking at the dots—they dishabituated. Newborns also showed dishabituation when the set size was changed from three dots to two. These patterns of habituation and dishabituation show that newborns can see the difference between two and three dots.
In the first months after birth, infants can already distinguish among small numbers of objects (e.g., among one, two, and three objects), whether the objects are similar or different, moving or still, or presented at the same time or in sequence. They can even match the number of objects they see with the number of sounds they hear. For example, when infants hear a sound track of two drumbeats, they prefer to look at a photo of two household objects rather than a photo of three objects. When they hear three drumbeats, however, their preference switches to the photo of three objects (Starkey, Spelke, & Gelman, 1983,1990). Impressive as these skills are, however, they apply only to very small sets. If researchers increase the number of objects in each set to five or more, then children don't show evidence that they recognize the quantities until they are about 3 or 4 years old (Canfield & Smith, 1996; Simon, Hespos, & Rochat, 1995; Starkey & Cooper, 1980; Strauss & Curtis, 1981; van Loosbroek & Smitsman, 1990; Wynn, 1992, 1995).
How are infants able to show such abilities? They clearly cannot count objects. They have no experience with a number system, and they don't have the language skills they would need to say the words that go with the numbers. Researchers propose that infants enumerate small sets by subitizing, a perceptual process that we all use to quickly and easily determine the basic quantity in a small set of objects. To see how subitizing works, try the following experiment. Have a friend toss three or four pennies onto a table while you have your eyes closed. Now open your eyes and, as quickly as you can, look to see how many pennies there are. Most people can "see" that there are three pennies, or four pennies, without needing to actually count each penny. There is something about the visual arrangement of the pennies that lets you know immediately how many there are. Of course we can't be sure that infants are subitizing object sets exactly the way adults do, but from the experimental evidence it does seem that they use a similar process. Somehow, without actually counting, they can determine that one set of objects has more items than another set, and they can match the number of things they see with the number of sounds they hear. Quite remarkable math skills for such a young age!

Mathematical Skills in Early Childhood

One of preschoolers' most obvious accomplishments is learning to count. Starting at about the age of 2, children begin to associate the counting words used in their language with the correct number of objects. They quickly become quite accurate in their counting and learn to count more and more things.
The structure of number words in a child's native language has an impact on some early mathematical competencies. For example, notice how we count in English. We count up through 10 ("one, two, ... ten"), but then we call the next numbers "eleven" and "twelve." Next come seven numbers with "teen" added to the end ("thirteen, fourteen, ... nineteen"). After the "teens:' we count up to 100 by naming the tens place followed by the ones place (as in "twenty-one, twenty-two"). Therefore, as they learn to count to 20 in English and many other Western languages, young children may not recognize the tens-ones system (called the base-10 system) that is the foundation of mathematics. In contrast, many Asian languages follow a much simpler rule (Miller, Smith, Zhu, & Zhang, 1995). After reaching 10 they go straight into naming the tens place followed by the ones place (essentially counting "ten-one, ten-two:' and so forth). In these languages children don't need to remember the inconsistent rules of giving special names to 11 and 12, then adding "teen" to the end of a few numbers before finally starting to count with the tens-ones system. As you can see in the figure below, it takes English-speaking children longer to learn to count than it does for their Chinese counterparts. By mastering the counting system at an earlier age, Chinese children get a head start on solving mathematical calculations and problems.
Language differences also affect young children's understanding of place value, and this understanding has a bearing on the strategies children use to solve arithmetic problems. For example, a common strategy for simple addition and subtraction problems is "decomposition." In a problem such as 6 + 7, a child might decompose the 7 into 4 + 3, then solve the problem by first adding 6 + 4 to get 10, then adding the remaining 3. Young Korean and Chinese children often use this strategy. But skill in decomposition depends on a solid understanding of the base-l 0 system, which Asian languages support better than English does (Fuson & Kwon, 1992; Geary, Bow-Thomas, Liu, & Siegler, 1996; Miller et al., 1995). Language differences also influence how quickly people can pronounce number names, which in turn affects how quickly children memorize basic math facts (Geary, Bow-Thomas, Fan, & Siegler, 1993; Geary et al., 1996). Similar language effects on learning fractions have also been found (Paik & Mix, 2003). So these seemingly simple differences in number words can contribute to more long-lasting differences in the development of math skills (Beaton et al., 1996; Geary, Liu, Chen, Saults, & Hoard, 1999; Stevenson, Chen, & Lee, 1993).
At about 4 years of age, children combine their developing counting skills with their knowledge of addition and subtraction. At this point they begin to use counting as a tool to solve simple arithmetic problems instead of relying on subitizing. This represents an important advance in their mathematical skills because a child can use counting with sets of any size and in the absence of concrete objects, whereas subitizing works only with small sets and visible objects. Children quickly begin to use several different counting strategies, approaches to solving problems that involve counting of the quantities. For example: A child figures out that 2 + 2 = 4 by first counting to 2 and then counting on (two more steps) to 3 and then 4. Preschoolers learn to solve simple problems whether the change in number is visible to them, screened from their view, or even described verbally in the absence of any concrete objects. Gradually, over the later preschool and elementary school years, children increase the complexity and sophistication of their counting strategies (Aunola, Leskinen, Lerkkanen, & Nurmi, 2004; Geary, 2006).

Treating Children as Mathematicians

The concern that the United States is falling behind in mathematics compared to other countries such as Japan led the directors of TIMSS to look at aspects of mathematics teaching and learning outside of achievement scores. They wanted to know what other factors were behind this seeming disparity between U.S. students and those of other countries. Toward this end, the study included an intensive videotape survey of 231 eighth-grade mathematics lessons in the United States, Japan, and Germany. TIMSS was the first attempt to collect a nationally representative sample of videotaped observations of American classroom instruction. According to TIMSS, “The purpose of gathering this information was to understand better the process of classroom instruction in different cultures to improve student learning in our schools” (Third International Mathematics and Science Study, 2004a, 2004b; Greene, Herman, Haury, & ERIC Clearinghouse for Science, 2000).

By using the videotape study and other TIMSS data, we can begin to examine and compare instructional methods in the United States with those of Japan. One conclusion drawn from the videotape study is that the Japanese do a much better job of treating their students as mathematicians. In U.S. classrooms, however, the tradition is to teach mathematics through memorization and practice (Elkind & Piaget, 1979; Kamii, 1984, 1990; Wenglinsky, 2004; Wood, Nelson, & Warfield, 2001). Ultimately there is little difference in the teaching philosophies that inform how primary, middle grade, and high-school students are taught in the United States (Greene et al., 2000). The study found that the main goal in the U.S. classroom was “teaching children how to solve a problem and obtain a correct answer.”

Mr. Gerhig said to his class, “I am going to show you how to figure out the number of degrees in any figure. First you take the number of sides, then you subtract two and multiply by 180. Juan, how many degrees would a square have?” Juan answers, “360?” as more of a question that an answer. “Right,” says Mr. Gerhig. “All you have to do is remember this formula and you can compute the answer.”

You may ask, “Is it not the same in Japan?” In Japan, students are given the tools to solve problems; the emphasis is on concepts rather than answers. The study found that in Japan the goal is to support conceptual understanding—in other words, less memorizing of formulas and more thinking about concepts.

Mr. Okawa draws a polygon on the board and says, “Using what we know about the area of a triangle, can you change this 4-sided figure into a 3-sided figure without changing its area?” Students then work in groups and present their solutions to the class. Mr. Okawa asks one student, “Can you tell me how you know the area is the same?” The student replies, “If the height and the base are the same it must be the same area.”

When we begin to think of children as competent mathematicians who, while working on age-appropriate problems, are using the same thought processes as advanced mathematicians, it changes the way we think about curriculum development. To do this, we must know what mathematicians do when they are presented with a problem. We can then apply these principles to design mathematics curricula for young children. Throughout this discussion, we will step inside the Japanese and American classrooms of the TIMSS study to examine how these instructional methods can be applied.

Mathematical Operations

In the preschool years, children begin counting by reciting the number words in sequence (e.g., one, two, three, four, five). This rote counting helps them develop number sense. They learn that numbers come before or after each other and that numbers are smaller or larger than one another. They learn that one counting word corresponds with each object they count (i.e., one-to-one correspondence), and they also develop a sense of sequence and pattern (Franke, 2003). Counting, sequencing, and detecting patterns are often considered the basis of the elementary school curriculum. Comprehension of these concepts is necessary to support students in developing the skills and understanding necessary to operate on numbers and solve a variety of mathematical problems that are introduced in the early elementary school grades, such as addition and subtraction.

Multiplication, division, fractions, decimals, ratios, and geometry are some of the mathematical concepts and operations typically introduced during middle childhood. A number of developmental changes described in the previous sections may help explain why children in middle childhood might be cognitively “ready” to address these mathematical skills and concepts. For example, Piaget researched extensively how children develop the concept of number. A developmental pattern he observed was that as children exhibit decentration they can begin to generate multiple ways to solve a mathematical problem. For example, 9 × 2 = 18, but so does 3 × 6; and you can solve the addition problem 9 + 4 by adding 4 units to 9 or knowing that 10 + 4 is 14 and taking away 1.

Children who showed an understanding of seriation demonstrated a more sophisticated understanding of mathematical concepts. They were able to assimilate not only that one number can be larger than another but that it can also, and at the same time, be smaller than another number. Seriation allows for a comprehension of number lines as well as greater- and/or less than problems. Understanding reversibility allows a child to appreciate the relationship between addition and subtraction. Taking two apples out of a basket after adding two reverses or “undoes” the first operation. The same understanding of reversibility can be applied to multiplication and division.

Perhaps the most significant cognitive advancement toward understanding mathematical concepts is the ability to conserve number. If children begin with two equivalent quantities (i.e., numbers) and can comprehend that the quantity of 5 does not change when one group is spread out or bunched together, then they have begun to understand the invariance of numbers in the environment. With such knowledge, school-age children can better predict relationships between numbers and quantities.

Learning more demanding mathematical concepts also requires some memorization. Even though math educators argue that understanding is more important than rote memorization, students still need to memorize rules that can be applied to a variety of problems. Usually, school-age children either intentionally or unintentionally memorize their multiplication tables, as well as other mathematical “facts” or “rules.” Over time and with practice, children also tend to invent their own strategies to facilitate the learning of mathematical concepts and operations (Ernst, 2000). Once certain operations become automatized, attentional capacity becomes available to apply to the remainder of the problem. Learning how to identify the greatest common factor in the numerator and denominator of a fraction, for example, would be much easier if you could focus your attention on what those common factors are, rather than struggle to determine which factors go into either the numerator or the denominator.

Finally, mathematical concepts are learned best if they can be linked to something that is familiar or known to the child. In other words, if the concepts can be linked to previously learned math concepts or embedded in practical problems that make sense to children, they will be able to assimilate the information more easily. Using other terms, one’s knowledge base can and will facilitate the decoding and storage of this new information.

While the developmental progressions described in this chapter are relatively universal, there are individual differences in learning rates and styles among school-aged children. During the past decade in particular, researchers have determined that if the task is made more familiar, children will typically exhibit cognitive abilities, such as decentration, multiple-perspective taking, strategy generation, and even mathematical skills, at a much earlier age than Piaget predicted (Siegler, 1996). Despite the appearance of these skills in highly supportive environments, however, there is a tendency for children to approach the world in the patterns outlined earlier.

The patterns of skill development portrayed in this chapter describe “average” learners who share a common cultural background with their teacher, class, school, and curriculum material. Much of how children learn and what they learn can also be traced to their parents’ educational values, the pedagogical practices of their teachers, their classroom environments, and the support they receive from their schools and communities.

Development of Cognitive Structures Related to Mathematics

As children develop cognitively from pre-lingual and pre-symbolic stages to the use of language and symbols to manipulate concepts, their abilities related to later mathematics learning are also developing. Some of the most critical cognitive abilities for mathematics learning are memory, language skills, and the ability to make mental representations of number and space.
Young children begin using their memory abilities as they interact with the environment and recall those experiences. Infants will respond to familiar faces and music. Children enjoy retelling stories and singing songs over and over again. As they begin noticing environmental print, children begin to understand the role of letters and numbers as abstract representations for familiar things. Names of streets, stores, candy, and numbers on houses and roads begin to take on meaning. Children ask to be taught to write their names and memorize the markings. Some children are so delighted with their new skills that they make markings everywhere—on books, walls, and under furniture.
With formal schooling, memory tasks become more challenging. Children must recall the written character for letters, numbers, and other symbols used in writing and mathematics. They are required to remember math facts and the sequences for performing operations with numbers. In problem solving, children are encouraged to recall a similar problem type or situation. Memory tasks are more successful when children learn through concept understanding rather than by rote memorization. New concepts should be connected to real-life experiences of children so that cognitive structures are formed in long-term memory.
Language development is critically integrated with mathematics development. Children use language to express relationships, assign labels, manipulate concepts, and communicate understandings with others. Language becomes the mediation tool for performing more difficult mathematics tasks, as can be seen in native language comparison studies (see, for example, Miura & Okamoto, 2003). Teachers use carefully selected language to ask questions and explain new concepts.
Children with language delays may have corresponding delays in mathematics development. Cognitive abilities of symbolic thinking, temporal–sequential organization, verbal memory, and rate of language processing are language abilities directly related to mathematics tasks (Jordan, Levine, & Huttenlocher, 1995). Nonverbal mathematics tasks, such as the manipulation of objects without verbal requirements, are generally developed earlier than verbal tasks (story problems) and are less sensitive to socioeconomic differences. Jordan, Levine, and Huttenlocher examined the calculation abilities of young children (kindergarteners and first graders) and found that children with language impairments performed as well as their peers on nonverbal calculations but significantly worse on story problems (presented orally) than the non-impaired group.

Components of Effective Mathematics Instruction

Less is known about the components of effective mathematics instruction than about the components of effective reading instruction, because research in math is less extensive than in reading. However, conclusions still can be drawn from some very good studies that do exist, as well as from typical grade level expectations in math. As is true for reading, there is no single "best" program for teaching mathematics. Rather, certain key abilities involved in learning math need to be addressed in instruction, with the importance of different abilities shifting somewhat across the elementary and secondary grades.

Effective Kindergarten through Grade Four Instruction

At these grade levels, general education instruction in mathematics should include development of the following math-related abilities: concepts and reasoning (e.g., basic number concepts, meaning of operations such as addition, geometric concepts); automatic recall of number facts (e.g., memorization of basic addition facts such as 3 + 4 so that children know answers instantly instead of having to count); computational algorithms (the written procedure or series of steps for solving more complex types of calculation, e.g., for two-digit addition with regrouping, calculation starts in the right-hand column and tens are "carried" from the ones to the tens column); functional math (e.g., practical applications such as time and money); and verbal problem-solving (e.g., solving word problems).
Because progress in math builds heavily upon previously learned skills, it is important for instruction to be clear, unambiguous, and systematic, with key prerequisite skills taught in advance. For instance, children should not be expected to develop automatic recall of addition facts if they do not understand the basic concept of addition or the meaning of the addition sign. It is also essential for children to have sufficient practice to acquire new skills. For example, although manipulatives such as cubes or rods can be very helpful in developing basic concepts, many children will not spontaneously progress from accurately solving facts with manipulatives to automatic recall of facts. Instead, most children benefit from practice activities focused specifically on helping them to memorize facts. Similarly, learning computational algorithms such as those used in long division or two-digit multiplication often requires considerable practice.
Scientific investigators interested in learning disabilities have identified several patterns that may be found in youngsters with math disabilities. Some of these children have difficulties that revolve primarily around automatic recall of facts, coupled with good conceptual abilities in mathematics; this pattern characterizes some children with reading disabilities. Another common pattern involves difficulties with computational algorithms; yet a third pattern involves visual-spatial difficulties, such as difficulty lining up columns or with learning spatial aspects of math, such as geometry. Although effective general education instruction can help to prevent low math achievement in many children, some youngsters with genuine math disabilities will require more intensive, long-term instruction in order to be successful.

Importance of Informal Mathematics Knowledge

Baroody and Ginsburg (1986) termed the knowledge that children develop in everyday settings prior to attending formal schooling “informal knowledge.” Most preschoolers arrive at school with important mathematical competencies, such as a sense for numbers and counting that are foundational for formal mathematics learning if understood by educators. Even with older children, the everyday, informal knowledge that is developed through experience can be tapped for enhancing formal mathematics learning.
Seo and Ginsburg conducted an interesting study of the types of informal mathematical activities in which four- and five-year-old children were engaged in natural settings (2004). The researchers classified observable activities by their mathematics characteristics:

• Classification activities involved sorting, grouping, or categorizing objects.
• Magnitude activities were statements made about global magnitude of objects, direct or side-by-side comparisons, or judgments without quantification.
• Enumeration activities involved saying number words, counting, subitizing, and even reading and writing numbers.
• Dynamics involved putting things together, taking them apart, or making other transformations such as turning and flipping.
• Pattern and shape activities included identifying or creating patterns or shapes and exploring the properties and relationships of shapes. (Seo & Ginsburg, 2004, pp. 93–94)

After coding 15-minute videotaped segments of ninety children, the researchers concluded that most children (88%) engaged in mathematical activities naturally and that about forty-three percent of the time observed was spent in math-like activities. Very significant in their findings was the conclusion that there were no income level or gender differences in these activity levels. In general, children engaged in pattern and shape activities the most, and classification the least, and were capable of achieving quite complex levels of performance. For example, some children demonstrated estimating the number in a set without counting and transforming a rhombus shape into a trapezoid.
Both quadrilaterals because of their four straight sides, a trapezoid is a quadrilateral with two parallel sides; a rhombus is a quadrilateral with both pairs of opposite sides parallel (also called a rhomb or diamond).
Informal mathematics knowledge is also important for older students. It may be harder to extract informal from formal learning and some misconceptions may be more rigidly held by older students, but teachers need to assess prior knowledge in whatever form for better connections to new mathematics learning. For example, Jamie is a student in Mrs. Banks’ third-grade class and was taught in previous grades that applying multiplication to whole numbers would result in larger numbers. This concept was confirmed with Jamie’s experiences grocery shopping with her mother. If one apple costs 20¢, then four apples cost 80¢. Buying a six-pack of soda for $2.50 was cheaper than buying six sodas at the individual price of 50¢. Mrs. Banks is attempting to explain why would result in a smaller number. She could begin with Jamie’s understanding that six sodas 50¢ would result in a $3.00 purchase. Shown another way, that would be of a dollar, or 3 dollars.
Misconceptions of older students are often caused by inadequate concept development through a wide enough range of examples during formal instruction or by limited informal experiences. For example, if a child had only cube-shaped blocks to play with, he could not compare other three-dimensional shapes and their properties. Viewing a diagram of a pyramid would be confusing for this child.

Making Math Not Suck

In the 2006 State of the Union Address, President George Bush used the word math five times. Why is this significant? Because it shows those in power are starting to acknowledge that the United States is losing its competitive edge in terms of math, science and technology in the 21^st Century. So, what's being done about it? Since 2006, the National Mathematics Advisory Panel, appointed by the President, has studied how to get our students up to speed in math. They focused their attention on how to properly prepare students for algebra—a fundamental step towards understanding high school math.
On March 13^th, 2008 the panel came back with several recommendations for our public school system, with the goal of getting every student fully prepared to take algebra in eighth grade. They said students in the elementary grades need to master:

1. Computational fluency, meaning the student can effortlessly recall number facts.
2. Computations involving fractions.
3. Particular aspects of geometry, specifically understanding the concept of the slope of a line.

The main current in their findings is that right now, the curriculum in U.S. schools covers many topics superficially, and as a result kids walk into algebra without the fundamentals. The panel recommends that the curriculum be reined in, so that these three key skills can be worked to maturity, starting in the younger grades.
“Students in countries with high scores on international assessments, such as Singapore, China and Finland, study a very core basic set of materials up through the middle grades,” says Larry Martinek, a Los Angeles-based teacher and founder of Mathnasium, a chain of after-school math learning centers. For 34 years, Martinek has based his curriculum around the idea that math must hang together as a united body of knowledge, with one basic core skill leading on to another, for it to make sense to kids. Now, the world is catching on, and it's about time, according to Martinek.
“This report will cause a change in the thinking from the president working down to the Department of Education, through the state level, down to text books and ultimately, what's being taught in the class,” he says.
But, Martinek says all this change will take time, and current parents will most likely see the new curriculum in the backpacks of their grandchildren. That's all well and good, but if you're wondering what can be done for this generation of kids, Martinek offers an important piece of advice: do math with your child just as much as you read with your child. For parents who need help to get the ball rolling, here's a list of exercises that can be started in kindergarten, first and second grade, but are appropriate for any student of any age who needs help with basic mathematics concepts and skills. The trick is to do these exercises both orally and visually, with little or no writing. Pictures can be used as visual aides. Real-world objects (coins, blocks, etc.) should be used as appropriate.