The 3rd paragraph Elementary Mathematics

**Chapter 1 Main Aim**

To make students master basic knowledges and skills through mathematics activities with number and quantity and diagram. And moreover, to foster students’ ability to think logically, foreseeing about daily phenomenon. And last, to foster students’ attitude to utilize it in their lives, noticing the apprreciations of pleasure of activity and mathematically management.

**Chapter 2 Aims and contents of each grades**

[The 1st grade]

**1 Aim**

(1) To enrich students’ sense with number through the activities using concrete materials and so on. To be able to understand not only the meaning of number and how to express it, but also the meanings of addition and subtraction. And to be able to think how to calculate them and make use of them.

(2) To repeat basic experiences to understand quantity and its measurement, and to enrich the sense with size of quantity.

(3) To repeat basic experiences to understand diagram through the activities using concrete materials and to enrich the sense with diagram.

**2 Contents**

**A Number and Calculation**

(1) To make students understand the meaning of number through students’ activities counting things and to able to use number.

*comparing the numbers by mapping operation

*counting and expressing the number and the order accurately

*making the number’s series and expressing it on the straight-line by thinking the size of numbers and order

*seeing the numbers, which might be related with other numbers, for example, to see one number as addion or subtraction against another number

*understanding the meaning and how to express it with numbers to 100 (2) To make students understand the meanings of addition and subtraction and use them. *Knowing the cases used addition and subtraction, expressing them as formula and reading them. *Thinking addition between one place numbers and subtraction as the reverse, and moreover, being able to calculate them surely. (3) To count concrete things collectively abd divide them equally, and moreover, to express them arranging orderly.

**B Quantity and Measurement**

(1) To enrich students’ experiences on the base of understanding of quantity and measurement through students’ activities comparing the length of materials. *Comparing the length directly. *Thinking one familiar material’s length as a unit and comparing the lengths by how times of it.

**C Diagram **

(1) To enrich students’ experiences on the base of understanding through students’ activities of observation and construction with familiar solids. *Recognizing the shape of a material and seizing the point of shape. *Using the terms related to direction and position accuratelly, e.g. before and behind, right and left, above and below, and moreover, expressing the posittion of materials linguistically.

[Terminology and Symbol]

one place two place + – =

[The 2nd grade]

**1 Aim**

(1) To enrich students’ sense with number through the activities using concrete materials and so on. To be able to understand not only the meaning of number and how to express it, but also the meanings of addition and subtraction. And understand the meaning of multiplication, after that, to be able to think how to calculate them and make use of them.

(2) To make students understand the unit of length and measurement, and to enrich the sense with size of quantity.

(3) To repeat basic experiences more and more to understand diagram through the activities using concrete materials and to enrich the sense with diagram.

**2 Contents**

**A Number and Calculation**

(1) To make students understand the meaning of number and how to express it, and to develop the ability to use number.

*counting things collectively as the same size and counting by grouping them.

*understanding how to express numbers (to 4 place numbers)on decimal notation and the size of numbers and the order.

*undrstanding relative size of number by considering a number as a unit of 10 or a unit of 100. *arranging and grouping simple things, expressing them with numbers and tables or graphs.

(2) To make students understand the meanings of addition and subtraction and to develop the ability to use them.

*understanding the relationship between addition and subtraction. *thinking addition of numbers to 2 place and how to calculate subtraction as the reverse. And moreover, understanding that these calculations can be done on the base of basic calculations of one place numbers and calculating them surelly. *researching properties that addition and subtraction consist of, and making use of them toward thinking how to calculate them or ascertaining them.

(3) To make students understand the meaning of multiplication and use them.

*knowing the cases used multiplication, and express them as formula or read the formula. *researching simple properties that multiplication consists of, and making use of constructing multuplication table and ascertaining the calculations.

*knowing multiplication table and being able to calculate the multiplication between one place number and one place number.

**B Quantity and Measurement**

(1) To understand the length and to able to measure length on the simple cases.

*understanding the meaning of a unit and measurement of length.

*knowing the units of length, milimeter(mm), centimeter(cm), and meter(m).

*being able to read the time in daily life.

**C Diagram **

(1) To enrich students’ experiences more and more on the base of understanding diagram through students’ activities of observation about the shapes of things and construction of them .

*making various shapes and breaking them apart.

*knowing triangle and square and drawing them or making them.

[Terminology and Symbol]

unit straight-line *

**[The 3rd grade]**

**1 Aim**

(1) To make students use addtion and subtraction appropriately, and moreover, deepen understanding of multiplication and use it appropriately. And to make students understand the meaning of division, and think how to calculate it and use it.

(2) To make students uderstand units of quantity, weight, time,so on, and measurement.

(3) Aiming at elements constructing diagram, to understand basic diagrams.

(4) To make students express data as arranged tables or graphs and use them, and understand these usefullness.

**2 Contents**

A Number and Calculation

(1) To deepen students’ understandings the ways of expressing numbers, and to develop students’ ability to use numbers.

*knowing the unit of 10000

*knowing the sizes of 10 times number and 100 times number or number divided by 10 and how to express them.

*deepening understanding of relative size of number.

(2) To make students calculate addition and subtraction surely, and to develop the ability to use them appropriately.

*thinking how to calculate addition and subtraction of 3 place numbers and understanding that these calculations are based on basic ones of 2 place numbers.

*calculating addition and subtraction surely, and using them approproately.

*searching properties that addition and subtraction consist of, and making use of thinking how to calculate them or ascertaining calculations.

(3) To deepen students’ understandings of multiplication, and moreover, to calculate surely and develop the ability to use them appropriately.

*thinking how to calculate multiplication, 2 place number or 3 place number multiplied 1 place number, 2 place number multiplied 2 place number, and understanding that these calculations are based on the calculations with multiplication table.

*calculating multiplication surely and using them appropriately.

*searching the properties that multiplication consists of and making use of thinking how to calculate them and ascertaining calculations.

(4) To make students understand the meaning of division and use them.

*knowing the cases used division and expressing them or reading them. And understanding the meaning of remainder.

*understanding the relation between division and multiplication or subtraction.

*calculating division with both 1 place divisor and 1 place quotient surely.

(5) To make students know how to express numbers in an abacus and calculate simple addition and subtraction by using an abacus.

*knowing how to express numbers in an abacus.

*knowing how to calculate addition and subtraction.

**B Quantity and Measurement**

(1) To make students understand length, quantity, weight, and on the simple cases, be able to measure them.

*knowing the unit of length, kilo-meter(km).

*understanding the meanings of units and measurement about quantity and weight.

*knowing the unit of quantity, liter(l).

*konowing the unit of weight, gram(g).

(2) In length, to make students estimate it approximately, and to make students measure it selecting a unit and a gauge according to purposes.

(3) To make students understand time.

*knowing day, hour, minute, second and the relation among them.

*In the simple cases, finding necessary o’clock or time.

**C Diagram**

(1) Through students’activities observing the shapes of materials and constructing them, to make them understand basic diagrams.

*Through observing the box-shapes and constructing them, knowing the elements constructing diagrams.

*aiming the elments constructing diagrams, and knowing square, rectangle, right triangle, and moreover, drawing and making ,spreading them on the plane.

**D Numerical and quantitative relation **(appears above 3rd grade)

(1) To make students express data in tables or graphs simply, and read them.

*grouping data from the view point of date, place, etc, and representing them as table after arranging them.

*knowing how to read and draw a bar graph.

[Terminology/Symbol]

sign of equality, right angle, / (division symbol)

**[The 4th Grade]**

**1 Aim**

(1)To deepen students’ understanding of division and make them use it approproately.And, as well as understanding of the meaning and representaion of both decimal number and fraction, to make them think how to calculate and use them.

(2) To make students understand the meaning of area and be able to measure the area of simple plane diagrams. And moreover, to make them understand the meaning of the size of angle.

(3) Aiming at the elements constructing diagrams, to deepen students’ understanding of basic diagrams.

(4) To make students represent or consider of number and quantity, or their relation by using formula and graph. And moreover, to make them search the dependant-relationship according to the purpose and group or arrange them.

**2 Contents**

**A Number and Calculation**

(1) To deepen students’ understanding of represent integer with decimal notation.

*knowing the units, ten million(Oku), hundred billion(Tyou),and understanding the method of decimal number.

(2) To make students understand round numbers and use them according to the purpose.

*knowing the cases used round numbers.

*undrstanding of rounding.

(3) To deepen students’ understanding of integer’s division and to make students calculate them surely. And moreover, to develop the ability to use them appropriately.

*thinking about how to calculate division that has the divisor as 1 place number or 2 place number and the dividend as 2 place number or 3 place number. And, understanding that these calculations are based on the basic calculations.

*calculating division surely and using it appropriately.

*searching the relation among dividend,divisor,quotient, and remainder, gathering them the following formula,

(dividend)=(divisor)*(quotient)+(remainder)

*searching the properties that division consists of, and thinking how to calculate it, and utilizing it for ascertaining calculations

(4) To make students understand the meaning of decimal number and how to represent it , and moreover, understand the meaning of addition and subtraction of decimal number and use them.

*using decimal number when we represent the size of remained part. And ,knowing how to represent decimal number and 1/10 place.

*not only knowing the fact that decimal number is represented as the same system as integer, but also deepening students’s understanding of teh relative-size of numbers.

*thinking how to calculate addition and subtraction of decimal number to 1/10 place.

(5) To make students understand the meaning of fraction and how to represent it.

*using fraction when we represent the size of remained part or equally divided part. And knowing how to represent fraction.

*knowing that fraction can be represented some times by a unit fraction.

**B Quantity and Measuremen**

(19 To make students understand the meaning of area, on the simple cases, measure area.

*understanding unit and the meaning of measurement about area.

*knowing the unit of area,(cm^{2}).

*thinking how to measure the area of square and rectangle and using them.

(2) To make students understand the size of angle, and measure it.

*catching the size of angle as one of rounding, and understanding its unit and the meaning of measurement.

*knowing the unit of angle,(ß ).

**C Diagram**

(1) Through students’ activities of observation of diagrams and construction of them, to deepen their understanding of basic diagrams.

*aiming at the elements constructing diagrams, knowing isosceles triangle, regular triangle, and moreover, drawing them and making, spreading them on the plane.

*knowing angle related to basic diagrams.

*knowing center,diameter, radius about circle, and drawing circle or making one. And knowing about cube related to circle.

**D Numerical and Quantitative Relationship**

(1) About two numbers or quantities which are changing the same, to make students represent their relation and search them.

*on the simple cases, thinking numbers or quantities mapped and search the relation by representing the set of values as a table.

*representing the phase of change, as graph of broken line, and reading the feature of change.

(2) To make students represent the relation of numbers or quantities as simple formula, and reading them.

*understanding mixed formula and formula used four operations and calculating accurately.

*understanding the idea of formula and using it.

(3) To make students collect data according to the purpose, and group or arrange them, and search the feature.

*searching the cases occurred on two matters.

*searching lack of data and double counting of data.

*searching data as graph of broken line, and searching the feature and tendency by seeing the graph.

[Terminology and Symbol]

sum, difference, product, quotient, integer, number line, decimal point, denominator, numerator, mixed fraction, proper fraction, improper fraction

**[ The 5th grade]**

**1 Aim **

(1) To make students understand the meaning of decimal and fraction, and how to represent them. And to make them understand the meaning of multiplication and division of decimal, moreover, think how to calculate them, and use them appropriately. And further, to make them understand the meaning of addition and subtraction of fraction, and think how to calculate them, being able to use them.

(2) To deepen students’ understanding of measurement of area, and make them be able to measure the area of basic diagrams.

(3) Considering and aiming at the elements constructing diagrams and their position-relationship, by doing so, to make students understand more and more basic diagrams.

(4) To make students consider data statistically,e.g. using percent or circle graph, and moreover, represent numerical and quantitative relationship as a formula, and search the relationship by themselves.

**2 Contents**

**A Number and Calculation**

(1) To make students understand the properties of integer.

*knowing that integer can be classified into even numbers and odd numbers when our view of point is fixed.

(2) To deepen students’ understanding of integer and decimal through numerical sysytem, and to make them use it effectively in calculation.

*making 10 times, 100 times, 1/10, 1/100/ of a number and searching the relationship among them.

(3) To make students understand the meaning of decimal multiplication and division, and use them appropriately.

*understanding the meaning of multiplication and division on the cases that a multiplier and a divisor are integers.

*on the base of thinking of calculation that a multiplier and a divisor are integers, undrstanding the meaning of multiplication and division on the cases that a multiplier and a divisor are decimals.

*thinking about how to calculate decimal multiplication and division, and being able to calculate them. And understanding the size of a remainder.

(4) Not only to deepen students’ understanding fraction, but also make them understand the meaning of the same denominator’s fraction addition and subtraction, and use them appropriately.

*aiming at the fact that there are the same size fractions on the simple cases.

*arranging a integer and a decimal as a fraction, and representing a fraction as a decimal.

*understanding that the result of integer division can be always represented as a fraction.

*thinking how to calculate fraction addition and subtraction with the same denominator, and being able to calculate them.

(5) To deepen students’ understanding of round numbers.

*being able to estimate sum and difference as round number according to the purpose.

**B Quantity and Measurement**

(1) To deepen students’understanding that the areas of basic plane diagrams can be calculated, and to make them calculate the areas.

*thinking how to measure the ares of a triangle and a parallelogram and using them.

*thinking how to measure the area of a circle and using it.

**C Diagram**

(1) Through students’ activities observing diagrams and construcing them etc, not only deepen students’ understanding of basic plane diagrams more, but also to make students be able to consider it by aiming at the elements construcing diagrams and the positional relarionship among them.

*undrstanding the relation of parallel and perpendicularity with straight lines.

*knowing parallelogram, trapezoid, rhombus, and drawing them, making them, and spreading them on the plane.

*understanding the meaning of circle ratio.

**D Numerical and Quantitative Relationship**

(1) To make students put the properties that four operations consist of.

*deepning students’ understanding of commutative law, associative law, and distributive law.

(2) To make students understand the meaning of percent, and use it.

(3) To make students group and arrange data according to the purpose, and be able to represent them by using circle graph and band graph.

(4) About the relation represented as a simple formula, by aiming at mapping or changing between 2 numbers and quantities, to deepen students’ understanding of how to view and search the numerical and quantitative relationship.

[Terminology and Symbol]

parallel, perpendicularity, diagonal, %

[The 6th Grade]

**1 Aim **

(1) Deepen students’ understanding of fraction addition and subtraction, and to make students use them appropriately, and further, to make them understand the meaning of fraction multiplication and division, and think how to calculate them, using them appropriately.

(2) To make students undrstand the meanign of volume and measure the volume of simple solid diagrams. And to make students understand the meaning of speed, and be able to calculate it.

(3)To make students consider diagrams by aiming at construting elements and positional relations, and deepen students’ understanding of basic solid diagrams.

(4) To make students understand the meaning of ratio and proportion, and be able to use the idea of function in considering numerical and quantitative relationship.

**2 Contents**

**A Number and Calculation**

(1) To deepen students’ understanding the properties of integer more and more.

*knowing measure and multiple.

(2) To deepen students’ understanging of fraction more, and more, understand the meaning of fraction addition and subtraction with different denominations, and make students use them appropriately.

*understanding that a fraction, that is multiplied or divided by the same number to a numerator and a denominator of a fraction, shows as the same size as the former fraction.

*thinking about equallity and size of fractions, and putting how to compare the size.

*thinking how to calculate faction addition and subtraction with different denominatiors, and being able to calculate them.

(3) To make students undrstand the meaning of fraction multiplication and division, and use them appropriately.

*understanding the meaning of multiplication and division on the cases that a multiplier and a divisor are integers.

*on the base of the idea of calculation, in which a multiplier and a divisor are decimals, understanding the meaning of multiplication and division on the cases that a multiplier and a divisor are fractions.

*thinking how to calculate fraction multiplication and divison, and being able to calculate them.

(4) To deepen students’ understanding of round number more and more.

*estimating product and quotient as round number according to the porpuse.

**B Quantity and Measurement**

(1) To make students catch the approximate shape about near diagrams, and calculate round areas.

(2) To make students understand the meaning of volume and be able to calculate the volume on the simple cases.

*understanding the meaning of units and measurement.

*knowing the unit of volume,(cm^{3}).

*thinking how to measure the volume of cube and rectangular parallelepiped, and using them.

(3) About numbers or quantities that can be considered as a ration in 2 quantities with different kinds, to make students understand how to compare them and use them.

*using the idea of unit quantity.

*understanding the meaning of speed and how to express, and thinking how to calculate speed and do it.

**C Diagrams**

(1) Through students’ activities observing diagrams and constructing them, to deepen students’ understanding of basic solid diagrams and moreover, to make students be able to consider aiming at the constructing elements of diagrams and positional relation among them.

*understanding cube and rectangular parallelepiped.

*understanding the relationship between parallel and parpendicularity on straight lines and plane related to rectangular parallelpiped.

*knowing prisms of triangle prism, quadrangular prism, and circular cylinder.

**D Numerical and Quantitative Relationship**

(1) To make students understand the meaning of ratio on the simple cases.

(2) About 2 numers changing together, to develop students’ ability to consider the relationship between them.

*understanding the meaning of ratio. And on the simple cases, searching the feature of them by using tables and graphs.

(3) understanding the meaning of average and using them.

[ Terminology and Symbol ]

greatest common measure, least common multiple, abbreviation, reduction common denominator, plane, base, side face, :(ratio),

Û Making a curriculum and treating the contents through each grade

1 To consider the following matters in making a curriculum.

(1) To teach each grades’ contents continuously when necessary after next grade.

(2)To foster the students’abilities of logical thinking, intuition, and problem solving, attempting the relation to various phenomena in real life, to take in mathematical activities, which are operative and active ,aggressively.

(3) To attempt the relation among the contents in each grade, that are [A Number and Calculation],[B Quantity and Measurement], [C Diagram], [D Numerical an Quantitative] Relationship]. In that time, to take in mathematical activities aggressively that generalizes some contents.

(4)About basic skills of calculation and measurement, to teach them deliberately by having the opportunities to exercise appropriately, in order to acquire and maintain them.

2 To consider the following matters about treating the contents.

(1) To foster students’ rich sense about number,quantity, and diagram, and moreover, to catch approximate size and shape, and to judge appropriately on the base of them, and to make students think out how to treat it efficiently.

(2) In the teaching of each grade’s [A Number and Calculation], when they think how to calculate and ascertain calculation, to make use of the estimated result of calculation.

(3) In the teaching of each grade’s [B Quantity and Measurement], not to treat the learning of formal unit-exchanging.

(4)[Terminology and Symbol] that each grade’s contents consist of shows the extent or range of the contents treated in the concerned grade, and in theaching them, to take up them with the related to each grade’s contents and find the appreciation to represent and think using them.

(5) In the process of problem solving, to use an abacus and an electronic calculator appropriately when they treat the calculation with large place numbers or calculate complex ones since 4 grade. In that case, to set the situations appropriately estimating the result of calculation and ascetening calculations. And, in teaching lower grade’s [A Number and Calculation], to use teaching tools of abacus or concrete materials and to consider that students deepen their understanding of the meaning of number and calculation.

(6) By utilizing computer effectively, to foster students’ abundant sense of number and quantity and diagram, and improve their ability to express them using a table and a grapgh.

**[ Conclusion ]**

The contents of new curriculum are reduced largely, and it is afraid whether the learning ability of students will decrease more and more. Recently in Japan, it is said that the scholastic ability of college and university students becomes to decline very much, for example, many students can not calculate fraction, 1/4+1/6=,etc. Can the new curriculum foster students’ scholastic ability really? We teachers have to teach students mathematics whithin short lessons, fostering the ability of problem solving. I think that how to teach mathematics becomes more important than ever. Awakening students’ interests, we should create mathematics lessons that are pleasant and interesting for students. That is, not to teach mathematics contents, but to create mathematics with students, which will connect to curriculum contents and deepen them. I have been investigated such mathematics lessons for the long time, and have found effective teaching methods.