What makes Japanese math textbooks so powerful?

Tad WatanabeKennesaw State Universitytwatanab@kennesaw.edu

http://science.kennesaw.edu/~twatanab/

Textbooks Matter

• “American students and teachers are greatly disadvantaged by our country’s lack of a common, coherent curriculum and the texts, materials, and training that match it.”

• “Top achieving countries have clearly focused and coherent curricula that help students acquire fundamental mathematics knowledge each year with sophisticated topics built upon previously mastered knowledge through careful sequencing.”

American EducatorSummer 2002

American EducatorSummer 2002

American EducatorSummer 2002

Mathematics textbooks in the U.S. cover more topics than texts in other countries, and, as a result, are substantially larger. The photo above compares five eighth-grade texts commonly used in the U.S. (right) to the eighth-grade texts from five of the A+ countries, which often use two slim books per year (left).

Characteristics of JapaneseElementary Mathematics Textbooks

• Focused• Coherent• Rigorous• Problem-solving oriented• etc.

A focused curriculum• includes fewer number of topics.• emphasizes mathematically significant

topics.– Topics must:

• be mathematically important, both for further study in mathematics and for use in applications in and outside of school

• fit with what is known about learning mathematics

• connect logically with the mathematics in earlier and later grade levels

NCTM, Curriculum Focal Points

A cohesive curriculum• builds new ideas upon prior knowledge• organizes its contents in a purposeful

sequence

A rigorous curriculum• develops concepts in a logical sequence.• emphasizes students’ reasoning and

justification.

Tokyo Shoseki TextbookAbout 80% of grade 1 curriculum is on number and operations: • Numbers up to 10• Ordinal numbers• Two numbers together• Addition (1) and subtraction (1)• Numbers up to 20• Calculation of 3 numbers• Addition (2) and subtraction (2)• Large number (numbers up to 100 and a

little beyond)

• Addition and Subtraction (finding more and less)

About 70% of grade 2 curriculum is numbers and operations:

• Addition: 2-digit numbers• Subtraction: 2-digit numbers• Numbers up to 1000• Calculation with 3-digit numbers• Numbers up to 10000• Multiplication (about 25% of grade 2 curriculum)

Tokyo Shoseki Textbook

Numbers up to 10 10 lessonsTwo numbers together 6 lessonsAddition (1) 6 lessonsSubtraction (1) 8 lessonsNumbers to 20 6 lessonsAddition (2) 13 lessonsSubtraction (2) 15 lessons

Large Numbers 14 lessons

Solution to the problem

A Problem-solving Curriculum

• develops concepts and procedures through problem solving.

• problems are carefully developed -contexts, numbers, representations, etc.

• students are to learn mathematics through problem solving - going much beyond “show and tell.”

Area Formulas

• Intersection of 3 content strands: measurement, geometry, and algebra

• Students “develop, understand, and use formulas to find the area of rectangles and related triangles and parallelograms” (NCTM Principles and Standards for School Mathematics).

Find the area.

(A) (B)

An overview of area measurement in Japanese textbooks

• Grades 1 - 4: Measurement of length, capacity, weight, and angle

• Grades 1 - 4: Important measurement principles– Measurement as comparison– Units

• Grade 4: Area of rectangles and squares• Grade 5: Area of parallelograms,

triangles & trapezoids• Grade 6: Area of circles

Geometry & Algebra• Grades 1 - 6: Writing and reading

mathematical expressions• Grade 2: Triangles, rectangles & squares• Grade 3: Circles & spheres; Sorting

triangles according to the lengths of sides; Mathematical expressions with

• Grade 4: Sorting quadrilaterals; Consolidated mathematical expressions; Co-variations

• Grade 5: Circles & regular polygons; Mathematical expressions with variables

Grade 1

Grade 4: Introduction of area

Direct comparison

Measuring each using a unit square

Developing the formulas

Important understandings

• When you are given an unfamiliar shape, you may still be able to find its area by changing it into a familiar shape (or familiar shapes).– Partition it into parts that are familiar

shapes– Make it bigger, then subtract– Cut and re-arrange

Grade 5

From a NJ Grade 6 classroomAshley: Today I learned that you cantransform the shape or increase the area.Example: Evi and UmmeÕs method PriscillaÕs method

(6 Š 2 + 2) x 4 = 24 sq. units (4x2Ö2) +(4x4)+(4x2Ö2) =24 sq. units.

RoshelleÕs method VeronicaÕs method

(6 Š 1 + 1) x 4 = 24 sq. units (6 Š 4 + 4) x 4 = 24 sq. units

JoseÕs method

(8x4) Š (2x4) = 24 sq. units

Area of a parallelogram is equal to the area of the related rectangle.

The height of a parallelogram is the distance between the base and its opposite side (or the distance between the parallel lines containing those sides).

The height of a triangle is the height of the parallelogram you can form by doubling its area.

The area of triangle does not change as the third vertex moves along the line parallel to the base of the triangle because the height does not change.

Elementary School Teaching Guide for the Japanese Course of Study :

Arithmetic (Grades 1-6)

• The main objectives of this grade are for children to determine the area of plane figure by utilizing the methods of determining the area of rectangles and squares that they have previously learned, and to develop and use new formulas for determining the area of other shapes. (Grade 5)

When teaching how to find areas of triangles or parallelograms, it is important that children have a concrete idea of what is the height and what is the base, etc. That is, if a certain side is selected as a base, the height is automatically defined, and no matter which side becomes the base, the area stays the same.

When teaching how to find the areas of triangles, quadrilaterals, and other polygons, the objectives should be not only to memorize the formula for finding areas or to do calculations based on them. It is also very important to foster mathematical thinking through the process of finding a new formula based on the already learned ways of finding areas. Therefore, it is important to devise teaching approaches considering the problem contexts/settings and children’s diverse thinking.

Japanese math textbooks

“…it is not clear whether most curriculum is written with teacher learning as a goal. To what extent do they aim to help teachers learn mathematics through the materials they write? Moreover, if a key element of the kind of teaching which … is responsiveness to students’ ideas, can curriculum materials adequately anticipate students’ mathematics?”

Ball, 1996