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Counting in the absence of perceivable objects is the culmination of a rather intricate developmental process. The process includes the progressive development of an ability to create unit items to be counted, first on the basis of conscious perception of external objects and then on the basis of internal representations. One-to-one: there must be a one-to-one relation between counting words and objects;.
Stable order of the counting words : these counting words must be recited in a consistent, reproducible order;. Cardinal: the last counting word spoken indicates how many objects are in the set as a whole rather than being a property of a particular object in the set ;. Abstraction: any kinds of objects can be collected together for purposes of a count; and. Order irrelevance for the objects counted : objects can be counted in any sequence without altering the outcome. The first three principles define rules for how one ought to go about counting; the last two define circumstances under which such counting procedures should apply.
The variability of their performance makes fundamentally ambiguous the task of inferring their knowledge of principles from their behavior. For example, asked to judge the accuracy of counting by a puppet who counted either correctly, incorrectly, or unconventionally e. Four-year-olds were better than 3-year-olds at rejecting true errors. Presented with a larger set of counting strategies to judge, children in a later study did not perform quite as well.
Several studies have found that children 3 years and younger have a great deal of difficulty in using counting to produce sets of a given numerosity, even when that numerosity is well within their counting range. The ability of young preschool children to follow counting principles in their own counting and to focus on them in evaluating the counting of others is also quite vulnerable to situational variations. Nevertheless, two points are clear. First, both aspects of counting are important developmental acquisitions. Second, by the time they enter kindergarten, most U.
Procedural fluency refers to the ability to perform procedures flexibly, accurately, and efficiently. As we noted in Chapter 4 , procedural fluency makes it possible for children to use mathematics reliably to solve problems and generate examples to test their mathematical ideas. In the case of counting, the difficulties young children have in fluently performing the complex activities required to count a set of objects accurately are a major obstacle to their mathematical development. Thus, effort and concentration are important aspects of accurate counting.
The difficulty preschoolers have in coordinating the process of keeping track of objects and counting them seems to be a universal characteristic of learning to count, with children in different cultures showing comparable rates of recounting or skipping objects. One aspect of counting that preschool children find particularly difficult is learning the number names.
Learning a list of number names up to is a challenging task for young children. Furthermore, the structure of the number names in a language is a major influence on the difficulties children have in learning to count correctly. These difficulties have important implications for the initial learning of mathematics in elementary school. The number names used in a language provide children with a readymade representation for number.
Linguistic structure of number names. Names for numbers have been generated according to a bewildering variety of systems. These may be freely combined, with the place of a digit indicating the power of 10 that it represents. First, it is a widely used system for writing numbers. Second, it is as consistent and concise as a base system could be.
Box 5—1 shows how spoken names for numbers are formed in three languages: English, Spanish, and Chinese. All of these languages use a base system, but the languages differ in the clarity and consistency with which the base structure is reflected in the number names.
As the first section of the figure shows, representations for numbers from 1 to 9 consist of an unsystematically organized list. There is no way to predict that 5 or five or wu come after 4, four , and si , respectively, in the Arabic numeral, English, and Chinese systems. Names for numbers above 10 diverge in interesting ways among these different languages, as the second part of Box 5—1 demonstrates. The Chinese number-naming system maps directly onto the Hindu-Arabic number system used to write numerals. For example, a word-for-word translation of shi qi 17 into English produces ten-seven.
English has unpredictable names for 11 and 12 that bear only a historical relation to one and two. The English names for numbers in the teens beyond 12 do have an internal structure, but it is obscured by phonetic modifications of many of the elements used in the first 10 numbers e. Furthermore, the order of word formation reverses the place value, unlike the Hindu-Arabic and Chinese systems and the English system above 20 , naming the smaller value before the larger value. Spanish follows the same basic pattern for English to begin the teens, although there may be a clearer parallel between uno, dos, tres and once, doce, trece than between one, two, three and eleven, twelve, thirteen.
The biggest difference between Spanish and English is that after 15 the number names in Spanish abruptly take on a different structure. Thus the name for 16 in Spanish, diez y seis literally ten and six , follows the same basic structure as Arabic numerals and Chinese number names starting with the tens value and then naming the ones value , rather than the structures of the number names in English from 13 to 19 and the names in Spanish from 11 to 15 starting with the ones value and then naming the tens value.
Above 20, all these number-naming systems converge on the Chinese structure of naming the larger value before the smaller one. Despite this convergence, the systems continue to differ in the clarity of the connection between the decade names and the corresponding unit values. Chinese numbers are consistent in forming decade names by combining a unit value and the base ten. Decade names in English and Spanish generally can be derived from the name for the corresponding unit value, with varying degrees of phonetic modification e.
How to find standard deviation - Algebra 1
Psychological consequences of number names. Although all the number-naming systems being reviewed are essentially base systems, they differ in the consistency and transparency with which that structure is reflected in the number names. Several studies comparing English-. The relative complexity of English number names has other cognitive consequences. Speakers of English and other European languages face a complex task in learning to write Arabic numerals, one that is more difficult than that faced by speakers of Chinese.
Speakers of languages whose number names are patterned after Chinese including Korean and Japanese are better able than speakers of English and other European languages to represent numbers using base blocks and to perform other place-value tasks. When learning to count, children must acquire a combination of conventional knowledge of number names, conceptual understanding of the mathematical principles that underlie counting, and ability to apply that knowledge in solving mathematical problems.
Language differences during preschool. In one study, for example, Chinese and American preschoolers did not differ in the extent to which they violated the previously discussed counting principles or in their ability to use counting to produce sets of a given size in the course of a game. Nevertheless, these effects have implications for learning Arabic numerals and thus for understanding the principal symbol system used in school mathematics. As with other aspects of mathematics, counting requires combining a conceptual understanding of the nature of number with a fluent mastery of procedures that allow one to determine how many objects there are.
When children can count consistently to figure out how many objects there are, they are ready to use counting to solve problems. It also helps support their learning of conventional arithmetic procedures, such as those involved in computation with whole numbers. Preschool children bring a variety of procedures to the task of learning simple arithmetic. Most of these procedures begin with strategic application of counting to arithmetic situations, and they are described in the next section. As with the distinction between conceptual understanding and procedural fluency, this categorization is somewhat arbitrary, but it provides a good example of how children can build on procedures such as counting in extending their mathematical competence to include new concepts and procedures.
Strategic competence refers to the ability to formulate mathematical problems, represent them, and solve them. An important feature of mathematical development is the way in which situations that involve extended problem solving at one point can later be handled fluently with known procedures. Simple arithmetic tasks provide a good example. Most preschoolers show that they can understand and perform simple addition and subtraction by at least 3 years of age, often by modeling with real objects or thinking about sets of objects.
In one study, children were presented with a set of objects of a given size that were then hidden in a box, followed by another set of objects that were also placed in the box. The majority of children around age 3 were able to solve such problems when they involved adding and subtracting a single item, although their performance decreased quickly as the size of the second set increased.
Preschool arithmetic: A wealth of strategies. Much research has described the diversity of strategies that children show in performing simple arithmetic, from preschool well into elementary school. Some children will model the problem using available object or fingers; others will do it verbally. These strategies are discussed in detail in Chapter 6. Kindergartners use all of these strategies, and second graders use all of them except for counting all.
When 5-year-olds were given four individual sessions over 11 weeks in which they solved more than addition problems, most of them discovered the counting-on-from-larger strategy, which saves effort by requiring them to do less counting. They then were most likely to apply it to problems e. The diversity of strategies that children show in early arithmetic is a feature of their later mathematical development as well.
In some circumstances the number of different strategies children show predicts their later learning. Solving word problems. Young children are able to make sense of the relationships between quantities and to come up with appropriate counting strategies when asked to solve simple word, or story, problems. Word problems are often thought to be more difficult than simple number sentences or equations. Young children, however, find them easier. If the problems pose simple relationships and are phrased clearly, preschool and kindergarten children can solve word problems involving addition, subtraction, multiplication, or division.
Will every bird get a worm? In addition to using counting to solve simple arithmetic problems, preschool children show understanding at an early age that written marks on paper can preserve and communicate information about quantity. But they are less able to represent changes in sets or relationships between sets, in part because they fail to realize that the order of their actions is not automatically preserved on paper.
Adaptive reasoning refers to the capacity to think logically about the relationships among concepts and situations and to justify and ultimately prove the correctness of a mathematical procedure or assertion. Adaptive reasoning also includes reasoning based on pattern, analogy, or metaphor. Research suggests that young children are able to display reasoning ability if they have a sufficient knowledge base, if the task is understandable and motivating, and if the context is familiar and comfortable. Situations that require preschoolers to use their mathematical concepts and procedures in unconventional ways often cause them difficulty.
For example, when preschool children are asked to count features of objects e. A major challenge of formal education is to build on the initial and often fragile understanding that children bring to school and to make it more reliable, flexible, and general. Most preschool children enter school with an initial understanding of procedures e. In addition to the concepts and skills that underlie mathematical proficiency, children who are successful in mathematics have a set of attitudes and beliefs that support their learning.
They see mathematics as a meaningful, interesting, and worthwhile activity; believe that they are capable of learning it; and are motivated to put in the effort required to learn. Reports on the attitudes of preschoolers toward learning in general and learning mathematics in particular suggest that most children enter school eager to become competent at mathematics. In a survey that examined a number of personality and motivational features relevant to success in mathematics, teachers and parents reported that kindergarteners have high levels of persistence and eagerness to learn although teachers differed in their perceptions of children from different ethnic groups, as we discuss below.
In one study, first graders rated their interest in mathematics on average at approximately 6 on a scale from 1 to 7 with 7 being the highest. One important factor in attaining a productive disposition toward mathematics and maintaining the motivation required to learn it is the extent to which children perceive achievement as the product of effort as opposed to fixed ability. Extensive research in the learning of mathematics and other domains has shown that children who attribute success to a relatively fixed ability are likely to approach new tasks with a performance rather than a learning orientation, which causes them to show less interest in putting themselves in challenging situations that result in them at least initially performing poorly.
Most preschoolers enter school interested in mathematics and motivated to learn it. The challenge to parents and educators is to help them maintain a productive disposition toward mathematics as they develop the other strands of their mathematical proficiency. In some circumstances, preschool children show impressive mathematical abilities that can provide the basis for their later learning of school mathematics.
These abilities are, however, limited in a number of important ways. Because the algorithms that preschoolers develop are based on counting and on their experience with sets of objects, they do not generalize to larger numbers. For example, preschool children can show a mastery of the concepts of addition and subtraction for very small numbers. This limitation is an important feature of preschool mathematical thinking and is an important way in which preschool mathematical proficiency differs from adult proficiency. As stated above, the way in which a word problem is phrased can be the difference between success and failure.
Furthermore, if children succeed, the strategy they use is a direct model of the story; they, in effect, act out the story to find the answer. Most U. These abilities include understanding the magnitudes of small numbers, being able to count and to use counting to solve simple mathematical problems, and understanding many of the basic concepts underlying measurement.
For example, a large survey of U. A number of children, however, particularly those from low socioeconomic groups, enter school with specific gaps in their mathematical proficiency. Several promising approaches have been developed to deal with this developmental immaturity in mathematical knowledge. For example, the Rightstart program consists of a set of games and number-line activities aimed at providing children needing remedial assistance with an understanding of the relative magnitudes of numbers. Another intervention is aimed at ensuring that Latino children understand the base structure of number names, something that, as noted above, U.
Taken together, these results suggest that relatively simple interventions may yield substantial payoffs in ensuring that all children enter or leave first grade ready to profit from mathematics instruction. The kindergarten survey cited above reported smaller ethnic differences in factors related to productive disposition persistence, eagerness to learn, and ability to pay attention than in mathematical knowledge.
There were, however, some noteworthy differences between the reports of teachers and parents for different ethnic groups. Parents reported high levels of eagerness to learn e. Teachers and parents are, of course, judging children against different comparison groups, but the data at least raise the possibility that kindergarten teachers may be underestimating the eagerness of their students to learn mathematics. For preschool children, the strands of mathematical proficiency are particularly closely intertwined.
Although their conceptual understanding is limited, as their understanding of number emerges they become able to count and solve simple problems. It is only when they move beyond what they informally understand—to the base system for teens and larger numbers, for example—that their fluency and strategic competencies falter. Young children also show a remarkable ability to formulate, represent, and solve simple mathematical problems and to reason and explain their mathematical activities.
The desire to quantify the world around them seems to be a natural one for young children.
They are positively disposed to do and understand mathematics when they first encounter it. Preschool children generally show a much more sophisticated understanding of small numbers than they do of larger numbers. They also have a great deal of difficulty in moving from the number names in languages such as English and Spanish to understanding the base structure of number names and mastering the Arabic numerals used in school mathematics.
Furthermore, not all children enter school with the intuitive understanding of number described above and assumed by the elementary school curriculum. Recent research suggests that effective methods exist for providing this basic understanding of number. Copeland, , p. Similar suggestions have been made by Baroody, a, b; Fuson, , ; and Siegler, The so-called Hindu-Arabic numeration system is in some sense a misnomer because the Chinese numeration system has been a decimal one from the time of the earliest historical records.
Because of the frequent contact between the Chinese and the Indians since the time of antiquity, there has always been some question of whether the Indians got their decimal system from the Chinese. Language has to be the product of its culture. So the fact that the names for numbers in Chinese, especially for the teens, reflect a base system indicates that the decimal system has been in place in China all along. By contrast, the Hindu-Arabic system did not take root in the West until the sixteenth century, long after the names for numbers in the various Western languages had been set.
The irregularities in the English and Spanish number names may perhaps be understood better in this light. Siegler, Alibali and Goldin-Meadow, , showed that in learning to solve problems involving mathematical equivalence, students were most successful when they had passed through a stage of considering multiple solution strategies.
Alexander, White, and Daugherty, , propose these three conditions for reasoning in young children. There is reason to believe that the conditions apply more generally. Alexander, P. Analogical reasoning and early mathematics learning. English Ed. Mahwah, NJ: Erlbaum. Allardice, B.
The development of written representations for some mathematical concepts. Alibali, M. Cognitive Psychology , 25 , — Antell, S. Perception of numerical invariance in neonates. Child Development , 54 , — Baroody, A. New York: Teachers College Press. The development of counting strategies for single-digit addition. Journal for Research in Mathematics Education , 18 , — Journal for Research in Mathematics Education , 20 , — Meljac Eds.
Hillsdale, NJ: Erlbaum. Remedying common counting difficulties. The relationship between initial meaningful and mechanical knowledge of arithmetic. Hiebert Ed. Bowman, B. Eager to learn: Educating our preschoolers. Briars, D. Developmental Psychology , 20 , — Carpenter, T. Journal for Research in Mathematics Education , 24 , — The acquisition of addition and subtraction concepts in grades one through three.
Journal for Research in Mathematics Education , 15 3 , — Copeland, R. How children learn mathematics 4th ed. New York: Macmillan. Dweck, C. Self-theories: Their role in motivation, personality, and development. Frye, D. Child Development , 60 , — Fuson, K. New York: Springer-Verlag. Research on whole number addition and subtraction. Grouws Ed. Relationships children construct among English number words, multiunit base-ten blocks, and written multidigit addition.
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The nature and origins of mathematical skills pp. Amsterdam: North-Holland. The acquisition of early number work meanings. Ginsburg Ed. New York: Academic Press. The acquisition and elaboration of the number work sequence. Brainerd Ed. Matching, counting, and conservation of numerical equivalence. Make you own patterns using one Pythagorean triangle in a range of sizes to make a nice tiling pattern as in the previous investigation question. Send some to me at the email address at the foot of this page and I will include them here.
Can you find a rectangle that dissects into different non-similar Pythagorean triangles? The Calculator earlier on this page opens in a new window is useful for the following. Find the only two Pythagorean triangles with an area equal to their perimeter. Which are the only 3 numbers that cannot be the shortest side of any Pythagorean triangle?
Can you find four consecutive numbers which are hypotenuses? What about five? Of the primitive ones in your list, what is special about their m and n values? List the values H that are squared to make these hypotenuses: where have you seen this series before on this page? It seems there are two triples for each and every hypotenuse that is a square number H 2.
One is easily explained as it has a simple relationship with the triple with hypotenuse H : what is that relationship? For the second, look at its generators to find a proof that it always exists.
120 Awesome Word Problems to Engage Students
Can you guess how many triples there will be with a hypotenuse that is a fourth power: H 4? Following on from the previous Puzzle, what can you find out about triples with a hypotenuse of the form H 3? What about Pythagorean triples having a smallest side which is a cube? What is special about their m and n values?
How many Pythagorean triangles have a side of length 48? Find a number that can be the side of even more Pythagorean triangles. Hint: there are 5 answers less than What is the highest number of triples you can find with the same side in each? Which number less than occurs in 32 triples? What is the smallest number that is the hypotenuse of more than one triple? What is the greatest number of triples you can find with the same hypotenuse? Is there one that is not a multiple of 5? Find some numbers which are the odd sides of more than one primitive Pythagorean triangle.
The first two are which is a side in both 8 15 17 and 15 and which is a side of both 20 21 29 and 21 Can you find a property to describe the factorizations into primes of each number in this series? We can find a whole series of Pythagorean triples where all the numbers are palindromes : 3,4,5 33,44,55 ,, Also we have the triple ,, What is the series of factors that has been used to generate these from 3 4 5? Another is 66,88, if we include an initial 0 in front of the hypotenuse: 66,88, What about the Pythagorean triples ,, and ,,?
Can you find any more infinite series of palindromic Pythagorean triples? There was another one that year '05 - when was it? Assuming that the years are in this century and are just two digits long, when is the next Pythagorean Triple Date? How many other such days are there in this century? If the date is any set of 3 numbers that are a Pythagorean triple that is, the numbers need not be in order , how many dates are there in one century? How many are there in a whole day if we use a hour clock with hours from 0 to 23?
Here is another way to do this. Think of a series of numbers that are like those in the lists above, e. Plug these numbers into the Triples generator and see if any patterns emerge. Sequences need a length which is the number of non-zero numbers. Some special sums of squares What is special about the numbers 14, 29, 50, 77, if we look at them as a sum of squares?
Find a formula for these numbers. It applies to the sum of K consecutive squares for all K. To prove this, find the formula for the sum of the K squares starting at n, let's call this S n. We have not used the value of K anywhere in the proof so the recurrence relation applies no matter value K has.
Look at the remainders when the odd numbers are divided by 8. There is a related pattern the evens in the list. The numbers have a remainder of 7 when divided by 8 or are 4 times a number in the list. The evens are all powers of 4 times another factor. What are those factors? The only odd numbers are 1, 3, 5, 7, 11, 15 and The evens are 2 or 6 or 14 or a power of four times one of these.
See A for the complete list of the 31 numbers and a reference to a proof. In Sprague published a proof that there are only 31 such numbers: 2, 3, 6, 7, 8, 11, 12, 15, 18, 19, 22, 23, 24, 27, 28, 31, 32, 33, 43, 44, 47, 48, 60, 67, 72, 76, 92, 96, , , See A These nine squares can be made into a rectangular jigsaw. The area of the rectangle is Cut out the nine squares and and solve the jigsaw puzzle. What are the dimensions of the rectangle? A rectangular jigsaw puzzle has exactly nine square pieces, each of a different size, and an area of If the square pieces have sides 1, 4, 7, 8, 9, 10, 14, 15 and 18, what is the width and height of the rectangle and how do the nine pieces fit into it?
According to Beiler's Recreations in the Theory of Numbers see Link and References below this is the smallest rectangular jigsaw where all the pieces are square and of different sizes. Take any two fractions or whole numbers whose product is 2 Notice that the fractions do not have to be in their lowest form :. Give two fractions whose product is 2 : they do not have to be in lowest form. Find m,n for a: b: h:. A 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 2, 1,.. A 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, A 0, 0, 1, 1, 2, 0, 1, 1, 1, 0, 1, 2, 2, 0, 2, 1, Triad Primitive?
Perimeter 3, 4, 5. Perimeter 27, 36, Perimeter 24, 70, Area 3, 4, 5. Area 27, 36, Area 24, 70, Primitive Triple Perimeter Area r 3,4,5. Non-primitive Triple multiple of Perimeter Area r 6,8,