This is done with the purpose of teaching the association of a number cardinal with a set and preparing for mental calculation for addition and multiplication. Third, multiplication is a binary operation and the answer is given by repeated addition at the introduction of multiplication Fig. To get the answer in multiplication, the students have to know polynomial notation first before they can interpret the meaning of polynomial notation.
For indirect comparison, through the comparison of A , B and C , students make an order and visualize transitivity, clearly: if B is smaller than A and B is smaller than C , then C is smaller than A.
For direct and indirect comparisons, the differences are usually discussed. These are necessary to produce arbitrary units. The differences can produce a unit for measuring a Euclidean algorithm. In Japan, students learn how to produce arbitrary units in this way.
Those activities are the bases to understand that any object can be seen as a unit See Table 1. And this processes are prepared for students who are able to learn how to produce the necessary unit.
Those preceding four preparations are the bases for the introduction of multiplication in the first grade. In addition, there are other preparations. For example, multiplication is the base for proportionality. The number line is a key preparation for representing times and extending it to proportionality.
In the first grade, it is implicitly introduced as a line of numbers by using repetitions of the unit tape Fig. In particular, the Gakkotosyo textbooks enhance the rule of three by using arrows to show the pattern on the table see the four-column tables in Figs. Proportionality is also embedded in these tables.
When students discuss the mutual relationship of their ideas in Fig. Two meaning of division are shown in Fig. The well-memorized and proceduralized multiplication-table is adapted to the tables that embed proportionality by multiplications and division rules. For filling in the blanks in the table, division is treated as the inverse operation of multiplication, as shown in different contexts in Fig.
In the Japanese curriculum sequence, the definition of multiplication by measurement is consistent with the multiplication table, division, fractions, and ratio and proportion. This consistency is supported by the model representations, proportional number lines see Fig. This is the reason why Japanese distinguishes the multiplier and multiplicand at the introduction of multiplication.
Actually, if we do not distinguish both, we can not distinguish partitive and quotative division, then, can not distinguish dividing and operational measurement fraction. After this consistency and extended adaptation of multiplication, the formal definition of proportions is introduced. To introduce Proportion formally, Gakko Tosyo Textbooks evolve the proportional number line from tape diagrams on Fig.
It illustrates the sequence to develop sense making for using multiplicative reasoning and proportionality on the principle of extension and integration. This sequence and representations make possible to apply the definition of multiplication by measurement to different teaching content see Chap. With this meaning, Japanese can introduce multiplication as preparation for division, fractions, and proportions. In the Japanese approach, the teaching content and sequence are usually preparation for future learning.
It is not only just making sense for reasonable explaining at every moment but also to develop sense making for extending and integrating by and for themselves. There are a number of misconceptions between additive and multiplicative structures in relation to ratios, rates, and proportions, such as misusing addition in multiplicative or divisional situations.
An origin of this type of misconception is originated from the properties of the multiplication table see Fig. When students learn multiplication in the second grade, they find and use this additive property.
If students extend multiplications to fractions in the upper grades, they can realize the difference of these two properties, such as half of two in Fig. Thus, in the second grade, they cannot easily distinguish additive and multiplicative structures in the table like the one shown in Fig. In relation to this, the Japanese definition and notations of multiplication are also preferred because of the following consistencies: Consistency among situations, repeated addition, and the multiplication table Consistency with other content such as measurement, division, fractions, ratios, and proportions in relation to distinguishing the multiplier and multiplicand Expandability to decimals and fractions by using consistent representations such as dot—area diagrams, proportional number lines, and tables for the rule of three Consistency in proportionality.
Cajori, F. A History of Mathematical Notations, Vol. Google Scholar. Common Core State Standards Initiative. Freudenthal, H. Mathematics as an educational task. Dordrecht: Riedel. Revisiting mathematics education. China lectures. Dordrecht: Kluwer. Haylock, D. Mathematics explained for primary teachers 4th ed. Los Angeles: Sage. Hiebert, J. Conceptual and procedural knowledge: The case of mathematics. Hillsdale: Lawrence Erlbaum. Hitotsumatsu, S. Study with your friends: Mathematics for elementary school , 11 volumes.
English translation of Japanese textbook. Tokyo: Gakkotosyo. Hulbert, E. A focus on multiplication and division: Bringing resarch to the classroom. Isoda, M. Designing problem solving approach with cognitive conflict and appreciation. Iwamizawa: Hokkaido University of Education. Developing lesson on mathematics problem solving approach: Conflict and appreciation on the dialectic among conceptual and procedural knowledge.
Tokyo: Meijitosho. Elementary school teaching guide for the Japanese course of study: Arithmetic grade 1—6. English translation of the edition published by the Ministry of Education, Japan. Fraction for teachers: Knowing what before how to teach. Elementary school teaching guide for the Japanese course of study: Mathematics grade 1—6. English translation of Japanese curriculum standards, ; and guide, Introductory chapter: Problem solving approach to develop mathematical thinking.
Katagiri Eds. Hackensack: World Scientific. CrossRef Google Scholar. Mathematization for mathematics education: An extension of the theory of Hans Freudenthal applying the representation theory of Masami Isoda with demonstration of levels of function up to calculus. Tokyo: Kyoritsu in Japanese. Katagiri A. Jel-drez, Trans. Elementary school and lower secondary school teaching guide for the Japanese course of study: Arithmetic and mathematics grade 1—9.
Study with your friends: Mathematics for elementary school , 12 volumes. Study with your friends: Mathematics for elementary school , 10 volumes. Izak, A. Developing a coherent approach to multiplication and measurement. Educational Studies in Mathematics, , 83— Kupferman, R. Elementary school mathematics for parents and teachers Vol.
Singapore: World Scientific. Ministry of Education. Course of study for elementary school mathematics. Tokyo: Ministry of Education. Muroi, K. Tokyo: Kyoritsu. A focus on fractions: Bringing research to the classroom 2nd ed. Rasmussen, K. The intangible task—a revelatory case of teaching mathematical thinking in Japanese elementary schools.
Research in Mathematics Education, 21 1 , 43— Rey Pastor, J. Reys, R. Helping children learn mathematics 10th ed. Hoboken: Wiley. Sugimura, K. The guidebook of secondary school mathematics, cluster I and II. Tokyo: Secondary School Textbook Publisher. Szilagyi, J. Journal for Research in Mathematics Education, 44 3 , — Summative Assessment — Card 6 — Exit Ticket Analysis — Informal observations and asking questions will allow me to see if students are able to decontextualize real-life situations, and then apply this understanding into symbolic numbers, drawings representation.
This Standard Curriculum provides the children a chance to make predictions and solve the problems around them. Through this mathematics curriculum, the students will be able to solve their mathematical problems in their daily routines effectively and systematically.
Multiplicative thinkers see and use the connections between addition and multiplication; they are able to use multiplicative arrays to see the flexibility with numbers and multiplicative problems. Additionally, they are able to use the notion and language of addition and multiplication and learn to extend to multiplication and division to solve problems.
Hurst suggests three key elements for multiplicative thinking and development. Therefore, it is important for teachers to be aware of the. Students who used Sim Calc had a better understanding than students who study in standard class. By use of autograph, students could gain a better understanding of what gradient are, what gradient look like and how gradient formed. Use autograph to create interval between two points and form a right-angle triangle where interval as the hypotenuse of the triangle could l However, the activity link to the mathematical tool should be carefully designed to meet the needs of all students as well as the assessment for learning in the progress.
The auditory feedback helps the student check for mistakes and accuracy of the keys they have pressed to verify the answer before putting the work on paper.
The product is designed for anyone who is having problems with math, it also can be used to create more involvement in a math assignment. The ten blocks system, and tangrams comes with some additional material which includes workbooks which the pages can be copied and used as parts of the lessons.
The Calculator does not come with batteries and they will need to be supplied. In this way, the teacher is making sure that his students understand the basic so that he can build new knowledge for better and effective learning. Otherwise, the confusion from the start will lead to further misunderstanding and confusion.
For this topic, the teacher is also using the brainstorming to help the students to recall the information they learnt previously on solids and volumes. All numbers on the number line— even those that are not rational—can be approximated by finite decimals. For example, the number is approximately 1.
Expanding the rational number system to include all numbers on the number line brings you to the real number system. Finite decimals give you access to arbitrarily accurate approximate arithmetic for all real numbers. That is one reason for their ubiquitous use in calculators. The potential of the number line does not stop at providing a simple way to picture all rational numbers geometrically. It also lets you form geometric models for the operations of arithmetic.
These models are at the same time more visual and more sophisticated than most interpretations. Consider addition. We have already mentioned that one way to interpret addition of whole numbers is in terms of joining line segments. Now you can refine that interpretation by taking a standard segment of a given positive length to be the segment of that length with its left endpoint at the origin. Then the right.
To encompass negative numbers, you must give your segments more structure. You must provide them with an orientation —a beginning and an end, a head and a tail. These oriented segments may be represented as arrows. The positive numbers are then represented by arrows that begin at the origin and end at the positive number that gives their length. Negative numbers are represented by arrows that begin at the origin and end at the negative number. That way, 4 and —4, for example, have the same length but opposite orientation.
Note: For clarity, arrows are shown above rather than on the number line. It is difficult to add the arrows when they both begin at the origin:. But the arrows may be moved left or right, as needed, as long as they maintain the same length and orientation. To add the arrows, I move the second arrow so that it begins at the end of the first arrow.
The result of the addition is an arrow that extends from the beginning of the first arrow to the end of the second arrow. This geometric approach is quite general: It works for negative integers and rational numbers, although in the latter case it is hard to interpret the answer in simple form without dividing the intervals according to a common denominator.
Another method see below for illustrating addition on the number line is simpler because it uses only one arrow. The method is more subtle, however, because it requires that some numbers be interpreted as points and others as arrows.
Interpret the first number as a point and the second number as an arrow. Position the beginning of the arrow at the point. The result of the addition is given by the point at the end of the arrow. Numbers on the number line have a dual nature: They are simultaneously points and oriented segments which we represent as arrows.
A deep understanding of number and operations on the number line requires flexibility in using each interpretation. A principal advantage to this shorthand method for addition is that it supports the idea that adding 3, for example, amounts to moving the line translating three units to the right.
By similar reasoning, adding —5 amounts to translating five units to the left. In general, adding any number may be interpreted as a translation of the line. The size of the translation depends on the size of the number, and the direction of the translation depends on its sign i. Multiplication on the number line is subtler than addition. Multiplication by whole numbers, however, may be interpreted as repeated addition:. In what way does multiplication transform the line? Multiplication by 4, for example, stretches the line so that all points are four times as far from the origin as they previously were, given a constant unit.
Division by 4 or multiplication by reverses this process, thereby shrinking the line. Then multiplication by for example, may be interpreted as stretching by a factor of 3 and then shrinking by a factor of 5. Multiplication by —1 takes positive numbers to their negative counterparts and vice versa, which amounts to flipping the line about the origin. These geometric interpretations of addition and multiplication as transformations of the line are quite sophisticated despite their pictorial nature.
Nonetheless, these interpretations are important because they provide a way to picture the differences between addition and multiplication. Furthermore, the interpretations provide links between number, algebra, geometry, and higher mathematics. While the number line gives a faithful geometric picture of the real number system, it does not make it easy to see geometrically the expansion of the number systems from whole numbers to integers to rationals, with each system contained in the next.
The schematic picture in Box 3—7 illustrates how the number systems are related as sets. In the center is zero, surrounded on the right by the positive whole numbers and on the left by their negative counterparts.
Together they form the integers. In the next larger circle are the rationals, which include the integers as a subset. In elementary school, children begin with the right half of the innermost circle the whole numbers and then learn about the right half of the next larger circle nonnegative rationals.
In the middle grades, the two circles are completed with the introduction of integers and negative rationals. In the late middle grades or high school, rationals are augmented to form real numbers. The number systems that have emerged over the centuries can be seen as being built on one another, with each new system subsuming an old one. This remarkable consistency helps unify arithmetic. In school, however, each number system is introduced with distinct symbolic notations: negation signs, fractions, decimal points, radical signs, and so on.
These multiple representations can obscure the fact that the numbers used in grades pre-K through 8 all reside in a very coherent and unified mathematical structure—the number line. In this chapter we are concerned primarily with the physical representations for number, such as symbols, words, pictures, objects, and actions.
These representations also support the development of efficient algorithms for the basic operations. Mathematics requires representations. In fact, because of the abstract nature of mathematics, people have access to mathematical ideas only through the representations of those ideas.
Mathematical ideas are essentially metaphorical. Because many mathematical representations are suggestive of the corresponding metaphors, mathematical ideas are enhanced through multiple representations, which serve not merely as illustrations or pedagogical tricks but form a significant part of the mathematical content and serve as a source of mathematical reasoning.
Numbers may be represented as physical objects, schematic pictures, words, or abstract symbols. For example, the number five may be represented by collections of physical objects, such as five blocks or five beads, by means of schematic iconic pictures like or or by abstract symbols like 5 or V. Operations can also be represented. For division this distinction is sometimes made through different notations e. Whether the symbols represent the concrete objects or vice versa depends upon where the child starts.
Both symbols and objects, however, represent a mathematical idea that is independent of the particular representation used. The remainder of this section considers one particular representation system for numbers, the decimal place-value system, which is a significant human achievement. It should be emphasized, however, that representation systems arise out of human activity, and much mathematical insight can be gained by considering the genesis and development of the representation systems of the Egyptians, the Babylonians, the Mayans, or other cultures.
Our intent here is more modest: to describe issues of mathematical representation by focusing on the representation system that is the major focus of school mathematics. It should also be emphasized that a representation system discussed previously, the number line, also deserves significant attention.
In fact, the main unifying and synthesizing point of the previous section was that the number systems of school mathematics, which remain often fragmented and disjointed in the perceptions conveyed by school curricula, are in fact all subsystems of a single system, which has a geometric model that is the foundation of later analysis and geometry. To use numbers effectively, to speak about them, or to manipulate them requires that they have names. Modern societies use decimal place-value notation in daily life and commerce.
With just 10 symbols—0, 1, 2,…, 9— any number, no matter how big or small in magnitude, can be represented. For example, there are roughly ,, people in the United States. Or the diameter of the nucleus of an atom of gold is roughly 0. The decimal system is versatile and simple, although not necessarily obvious or easily learned.
The decimal place-value system is one of the most significant intellectual constructs of humankind, and it has played a decisive role in the development of mathematics and science. Over the centuries, various notational systems have been invented for naming numbers. To represent numbers symbolically, the ancient Hindus developed a numeration system that is based on the principles of grouping 19 and place value, and that forms the basis for our numeration system today.
In this system, objects are grouped by tens, then by tens of tens hundreds , and so on. Hence, this numeration system is a base or decimal system. These are nontrivial ideas that took humankind many centuries to invent and refine. Although Roman numerals use grouping by tens and the interpretation of a numeral depends to some extent on the placement of the symbols, 20 they do not at all constitute a place-value system.
Also, the system of Roman numerals is ad hoc, in the sense that each new grouping requires a new symbol, so it is strictly limited in extent. A crucial steppingstone in the development of place-value notation was the idea of using a separate symbol to denote zero, which could then be used as a placeholder when necessary. This invention allows the same symbols to be used over and over to describe larger and larger groups. Since the grouping is by tens, only 10 symbols, the digits 0 through 9, are needed to indicate how many groups there are of a particular size.
In a numeral the size of the group depends on the place that the digit appears in the numeral. Some pictorial and physical representations can be helpful in understanding the decimal place-value system. Special blocks, called base blocks, for example, can be used to develop and support an understanding of the importance of tens and hundreds and the meaning of the various digits.
The number is pictured with base blocks below. The composition of shown above might be expressed symbolically as follows:. In this case, 2 is called the exponent, and 10 2 is 10 to the second power. Making the meaning of the digits explicit in a larger number requires the use of higher powers of A number in the decimal system is the sum of the products of each digit and an appropriate power of 10, where the power in question corresponds to the position of the digit.
The system is general enough to represent any whole number, no matter how large. This conciseness, however, presents a challenge to young learners as they try to understand this compact notational system. Extending the decimal system to the right of the decimal point is accomplished by analogy. As you move to the left, the value of the place is multiplied by , 10, , 1,, and so on.
As you move to the right, this sequence is reversed, so that the value is divided by Continuing past the units ones place and over the decimal point, you continue dividing by 10, to reach places for tenths, hundredths, thousandths, and so on.
A rational number such as therefore, is written as 0. The values of the digits are sometimes shown in a place-value chart, in which Because the reciprocals of powers of 10 become smaller in magnitude as their exponents get larger in absolute value, such decimal representations can describe quantities that are arbitrarily small.
Consequently, any positive number, no matter how small in magnitude, can be represented by a decimal. To represent numbers that are not whole numbers, one could choose a fractional rather than a decimal representation.
Representational choices are much broader, however, than whether to use decimals or fractions. In the previous section, for example, we used points and arrows on the number line to indicate fractions, integers, and operations on integers. Fractional values are often represented with pictures, and relationships between quantities are often represented with graphs or tables. Communicating about mathematical ideas, therefore, requires that one choose representations and translate among them.
Such choices depend on balancing such characteristics as the following:. How easily can the idea be seen through the representation? Base blocks, for example, are more transparent than a number line for understanding the decimal notation for whole numbers, whereas the decimal numerals themselves are not at all transparent. Does the representation support efficient communication and use? Is it concise? Symbolic representations are more efficient than base blocks.
Does the representation apply to broad classes of objects? Finger representations are not general. The number line is quite general, allowing the representation of counting numbers, integers, rationals, and reals.
If digits on both sides of the decimal point are included, the decimal place-value representation of numbers is completely general in the sense that any number may be so represented. Is the representation unambiguous and easy to use? Representations should be clear and unambiguous, but that is often established by convention—how the representation is commonly used. See Box 3—8. Box 3—8 Clarity of Representations. For simplicity of use, representations should be as clear and unambiguous as possible.
Much of that clarity is not inherent in the representation, however, but is established through convention. This ambiguity is solved in part by omitting multiplication signs, using parentheses or juxtaposition instead. Thus, xy means x times y, and 5 3 means 5 times 3. But that practice creates another ambiguity. In the notation for mixed numbers, means It does not mean Furthermore, juxtaposing symbols to indicate multiplication creates confusion in high school mathematics with the introduction of function notation, where f 4 looks like multiplication but instead means the output of the function f when the input value is 4.
The ambiguities of such standard notations can interfere with learning if they are not acknowledged, explained, developed, and understood. Try a few different calculators. How close is the representation to the exact value? Graphs are usually not very precise. With enough digits to the right of the decimal point, decimal representation can be as precise as desired. And one-half is the simplest fraction.
Much more is involved in understanding and translating among representations of or rational numbers more generally. See Box 3—9 for an example. Perhaps the deepest translation problem in pre-K to grade 8 mathematics concerns the translation between fractional and decimal representations of rational numbers.
Successful translation requires an understanding of rational numbers as well as decimal and fractional notation—each of which is a significant and multifaceted idea in its own right. In school, children learn a standard way of converting a fraction such as to a decimal by long division. The first written step of the long division is dividing 30 tenths by 8. After three divisions, the process stops because the remainder is zero.
The quotient obtained, 0. The remainder at the seventh step is 2, which is where the first step began. Thus, a repeating decimal, where the horizontal bar is used to indicate which digits repeat. The process of using long division to obtain the decimal representation of a fraction will always be like one of the above cases: Either the process will stop or it will cycle through some sequence of remainders. So the decimal representation of a rational number must be either a repeating or a terminating decimal.
Thus a nonrepeating decimal cannot be a rational number and there are many such numbers, such as p and. In the process of converting a fraction to a decimal, all remainders must be less than the denominator of the fraction. Because the list of possible remainders is finite, and because each subsequent step is always the same brings down a 0, etc.
Understanding a mathematical idea thoroughly requires that several possible representations be available to allow a choice of those most useful for solving a particular problem. And if children are to be able to use a multiplicity of representations, it is important that they be able to translate among them, such as between fractional and decimal notations or between symbolic representations and the number line or pictorial representations.
Addition is an idea—an abstraction from combining collections of objects or from joining lengths. Carrying out the addition of two numbers requires a strategy that will lead to the result. For single-digit numbers it is reasonable to use or imagine blocks or cookies, but for multidigit numbers you need something more efficient. You need algorithms. There are many such algorithms, as well as others that do not use pencil and paper.
Years ago many people knew algorithms for computation on fingers, slide rules, and abacuses. Today, calculators and computer algorithms are widely used for arithmetic. Indeed, a defining characteristic of a computational algorithm is that it be suitable for implementation on a computer. And in fact, most of algebra, calculus, and even more advanced mathematics may now be done with computer programs that perform calculations with symbols.
When confronted with a need for calculation, one must choose an algorithm that will give the correct result and that can be accomplished with the tools available. Algorithms depend upon representations. Note, for example, that algorithms for fractions are different from algorithms for decimals. And as was the case for representations, choosing an algorithm benefits from consideration of certain characteristics: transparency, efficiency, generality, and precision.
The more transparent an algorithm, the easier it is to understand, and a child who understands an algorithm can reconstruct it after months or even years of not using it. The need for efficiency depends, of course, on how often an algorithm is used.
An additional desired characteristic is simplicity because simple algorithms are easier to remember and easier to perform accurately.
Again, the key is finding an appropriate balance among these characteristics because, for example, algorithms that are sufficiently general and efficient are often not very transparent. It is worth noting that pushing buttons on a calculator is the epitome of a nontransparent algorithm, but it can be quite efficient.
In Box 3—10 , we show some examples of algorithms with various qualities. Algorithms are important in school mathematics because they can help students understand better the fundamental operations of arithmetic and important concepts such as place value and also because they pave the way for learning more advanced topics. For example, algorithms for the operations on multidigit whole numbers can be generalized with appropriate modifications to algorithms for corresponding operations on polynomials in algebra, although the resulting algorithms do not look quite like any typical multiplication algorithms but rather are based upon the idea behind such algorithms: computing and recording partial products and then adding.
The polynomial multiplication illustrated below, for example, is somewhat like multiplication. The expanded method below shows the relationship a bit more clearly. The decimal place-value system allows many different algorithms for the four main operations.
The following six algorithms for multiplication of two-digit numbers were produced by a class of prospective elementary school teachers. They were asked to show how they were taught to multiply 23 by Note that all of these algorithms produce the correct answer. There is no further difficult in adding two signed fractions , only the interpretation of the results differs.
For instance consider addition of two Q3 numbers shown compare to the example with two 4 bit signed numbers, above. If you look carefully at these examples, you'll see that the binary representation and calculations are the same as before , only the decimal representation has changed.
This is very useful because it means we can use the same circuitry for addition, regardless of the interpretation of the results. Even the generation of overflows resulting in error conditions remains unchanged again compare with above. Multiplying unsigned numbers in binary is quite easy.
Multiplication can be performed done exactly as with decimal numbers, except that you have only two digits 0 and 1. In this case the result was 7 bit, which can be extended to 8 bits by adding a 0 at the left.
When multiplying larger numbers, the result will be 8 bits, with the leftmost set to 1, as shown. There are many methods to multiply 2's complement numbers. The easiest is to simply find the magnitude of the two multiplicands, multiply these together, and then use the original sign bits to determine the sign of the result.
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