What Kids Learn in Arithmetic, Grade by Grade: Japan's Elementary Curriculum

Arithmetic in Japanese elementary schools follows the national Course of Study, published in 2017 and fully in effect since the 2020 school year. What children learn in each grade is set out in the “Numbers and Calculations” strand for that grade (Ministry of Education, Elementary School Course of Study, 2017, in Japanese).

Other countries place some topics a little earlier or later. In the United States, for example, the Common Core asks that by the end of grade 3 children “know from memory all products of two one-digit numbers” (Common Core, Grade 3, Operations and Algebraic Thinking). Japan finishes the times tables in grade 2. Still, the order in which the ideas build on each other is much the same everywhere, so I hope the list below is useful outside Japan too.

I read the “Numbers and Calculations” sections and sorted out the parts about calculation, grade by grade. Geometry and measurement are left out. The Course of Study fixes what is taught in each grade. The order within a year depends on the textbook and the school.

Grade 1: groups of ten, and single-digit addition and subtraction

Counting, two-digit numbers, and three-digit numbers in simple cases (the official commentary says up to about 120)
Seeing numbers in units of ten
Seeing one number as the sum or difference of others (7 is 3 and 4)
Adding two single-digit numbers, and the subtraction that undoes it, “reliably”. This includes sums like 8 + 5 that pass ten and subtractions like 13 − 8
Two-digit addition and subtraction in simple cases

Almost all of grade 1 arithmetic is about ten as a group. It is the foundation for the next six years.

Grade 2: column arithmetic and the times tables

Numbers up to four digits (10,000 is also covered)
Two-digit addition and subtraction, and how to write them in columns
Three-digit addition and subtraction in simple cases
The commutative and associative laws for addition, and how addition and subtraction relate
The meaning of multiplication and the times tables. Multiplying two single-digit numbers “reliably”
Two-digit by one-digit multiplication in simple cases
Simple fractions such as 1/2 and 1/3

Parentheses and boxes for unknowns may be used where needed. The times tables are the big hill of grade 2.

Grade 3: division, decimals and fractions begin

Ten thousands (up to 100 million)
Three- and four-digit addition and subtraction in columns
Multiplying two- and three-digit numbers by one- and two-digit numbers, in columns. Multiplying by zero
The commutative, associative and distributive laws for multiplication
The meaning of division, and remainders. Division with a one-digit divisor and a one-digit quotient, “reliably”
One-digit divisors with two-digit quotients, in simple cases
Decimals to the tenths place, and adding and subtracting them
The meaning of fractions, fractions as so many unit fractions, and simple addition and subtraction of fractions
Expressions with a box for an unknown
The abacus (soroban)

Grade 3 starts more new things than any other year. Division, decimals and fractions all arrive at once. The notes for this grade also ask that simple calculations be done mentally.

Grade 4: long division and the rules of calculation

Hundred millions and trillions
Rounding and estimating results
Division with one- or two-digit divisors and two- or three-digit dividends, in columns
The relationship (dividend) = (divisor) × (quotient) + (remainder)
Adding and subtracting decimals, extended to hundredths
Decimals multiplied or divided by whole numbers, including whole-number division that gives a decimal quotient
Adding and subtracting fractions with the same denominator, and equivalent fractions
Expressions mixing all four operations, and expressions with parentheses
Formulas, and expressions with boxes and triangles for unknowns
The commutative, associative and distributive laws

The notes for grade 4 division say it should consolidate “the ability to calculate with whole numbers shown from grade 1 to grade 4”. Whole-number arithmetic is treated as complete here. Long division uses the times tables, column multiplication and column subtraction all at once, so any shaky spot from earlier grades tends to show up here.

Grade 5: multiplying and dividing decimals, and common denominators

Even and odd numbers, factors and multiples
Making 10, 100, 1/10 and 1/100 times a number by moving the decimal point
Decimal times decimal, decimal divided by decimal, and the size of the remainder
Writing whole numbers and decimals as fractions, and fractions as decimals
The result of any whole-number division can be written as one number using a fraction
Multiplying or dividing numerator and denominator by the same number keeps the value (the basis of reducing and finding common denominators)
Adding and subtracting fractions with different denominators

Grade 6: multiplying and dividing fractions, and letters in expressions

Fraction times fraction and fraction divided by fraction, seeing division as multiplication by the reciprocal
Bringing whole-number and decimal multiplication and division together as fraction calculations
Expressions using letters such as a and x

The notes for grade 6 ask that decimal and fraction calculation from grades 3 to 6 be consolidated. Junior high mathematics then builds negative numbers and algebra on top of this.

Is it fine to go back a grade?

I think so. The Course of Study itself says that the content of each grade should continue to be taught “in later grades as needed”.

Once you lay the six years out, you can see that each grade sits directly on the one before. When a child struggles with grade 4 long division, the cause is often the times tables or grade 2 column subtraction. In that case, a few days of grade 2 column work is a shorter way forward than repeating grade 4 problems. Going back is not falling behind. It is checking the foundation.

What MathCal can do

MathCal is an arithmetic drill app I make, where children write their answers by hand on the screen. There are 22 drills, and they line up with the grade-by-grade arithmetic in this post. For example, the Column Arithmetic Drill for addition and subtraction suits grades 2 to 3, Column Arithmetic Drill (Multiplication) grades 3 to 4, Column Arithmetic Drill (Division) grade 4, the Fraction Drill grades 5 to 6, and the Find the x Drill grade 6.

If you enter a birth year and month under Birthday (adjusts questions) in the app settings, Today’s Drill on the home screen is chosen each day from the drills that suit the child’s grade. Grades follow the Japanese school year, which starts in April, so children in other countries may find this off by a year. Drills from earlier grades may also come up as review. When Today’s Drill is the Basic Calculation Drill, multiplication appears from grade 2 and division from grade 3. If no birthday is set, Today’s Drill is either the Basic Calculation Drill or the Fill-in-the-Blank Drill.

Today’s Drill, Today’s Challenge and Today’s Lesson are free every day. To pick any drill or lesson from the full lists, for example to go back to an earlier grade, you need the one-time unlock purchase. There are no ads.

For the big hills of grades 1 and 2, see How to Teach Carrying and Borrowing (Regrouping) and How to Learn Times Tables: Say Them, Write Them, Use Them. Making ten, the base of grade 1, is covered in How to Teach the Making Ten Strategy.

Laid out over six years, arithmetic has surprisingly few truly new ideas. Most of it combines what came in the grade before. When this year’s problems look hard, it helps to look one step back down the list.