ISSN 1881−6134

鳥 取 大 学 数学 教育 研究
Tottori Journal for Research in Mathematics Education

http://www.rs.tottori-u.ac.jp/mathedu

Japanese Elementary School Students
Math Performance:
A Case of Tottori Prefecture regional “Math Diagnostic Test”

Tatsuya Mizoguchi & Yusuke Shinno

vol.18, no.1
Sep. 2015

鳥取大学 数学教育学研究室

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Japanese Elementary School Students Math Performance:
A Case of Tottori Prefecture regional “Math Diagnostic Test” *1
Tatsuya Mizoguchi Tottori University, Japan
Yusuke Shinno Osaka Kyoiku University, Japan
1. Introduction
The purpose of teaching mathematics to students is to enhance “students’ mathematicalpower” (NCTM, 1995), that is to extend their mathematical performance to a higher level.
Especially in Japan, we set up to develop students’ “mathematical way of thinking” as thepurpose of mathematics education traditionally (cf. Japanese Course of Study). This is theexpression including not only the aspects of students’ mathematical knowledge or skills but also
their attitudes toward mathematics.
On the other hand, the Japanese Course of Study andthe textbooks as its realization should be referred to as
the “Intended Curriculum” according to the TIMSS
curriculum model (Mullis & Martin, 2013), and then thelessons are practiced as the “Implemented Curriculum”
the next. Based on these contexts, we need to pursue theevaluation and the assessment of students’ performanceas part of curriculum research, i.e., the “AttainedCurriculum.”
In Japan, “Students Achievement Surveys” of thenational level had been implemented intermittently by
the initiative of the Ministry of Education in the past.
Since 2007, the National Assessment Test has been
Fig.1 TIMSS Curriculum Model
carried out every year continuously. The purpose ofNational Assessment Test is to clarify the actual condition of national student, and to improveeducational policies and lessons. That is, it is a verification of achievement of the “IntendedCurriculum” which is the holistic trends rather than individual performances, and then to infer
the global guiding principle for the curriculum development or teaching and learning.
Unlike nationwide survey, each region in Japan has been carried out their own achievement
test (the names are different for each region) continuously for many years. Some are theprefecture-scale, other are the city-scale, for depending on the population size. These tests look
the same way as the national survey with respect to the surface practice, however thoseinitiatives and purposes differ greatly. Although the scales differ, regional tests may be similar
to the national survey in terms of aiming to grasp the holistic trends. However, the intention ofregional tests mainly attempts to obtain a direct suggestion for the improvement of“Implemented Curriculum,” i.e., the lesson, which differs from the national survey including theverification of curriculum. In addition, most of regional tests are deployed by teachers in theregion (belonging to the math subcommittee in the case of elementary schools) under thesupervision of university researchers to develop problems and analyze results, instead thenational survey is done by the selected expert committee (researchers, policymakers, teachers).
Especially, when developing problems, it is the high tendency to develop problems conforming
to textbooks that are adopted in the region.
In this paper we verify elementary school students’ math performance through Tottori
Prefecture regional “Math Diagnostic Test” as a case of regional tests in Japan, in particular, wefocus on “problems for examining the thinking process” in this Test using a few years data.
2. Tottori Elementary Math Diagnostic Test
2.1 Organization

1!!!!!

Tottori Prefecture is located in the Chugoku region, western Japan.
It is the least populous prefecture in Japan (area is not the least).
However, the effort poured into education field is considerable suchthat top expense item of the prefectural budget is education (although
the amount tends to reduce in recent years).
This area has been organized in three broad teachers grouporganization (eastern, central, western). Developing problems andanalysing results of the “Math Diagnostic Test” are deployed by eachgroup in two years replacement. Although all teachers belong to themath subcommittee in their schools, their major were notmathematics necessarily. What subject subcommittee a teacher
belongs to is dependent on his/her school condition. Therefore, ateacher whose major is mathematics doesn’t belong to mathsubcommittee, and vice versa.
It is a suitable opportunity for teachers to study mathematical andpedagogical content knowledge and recognize what areemphasis/important contents through the curriculum:What kinds of problems are essential/appropriate to verify
students’ math performance?
What kind of performance do students show to such problems?

Fig.2 Tottori Pref.

Fig.3 Three Districtsin Tottori

2.2 Procedure
The group organization in charge advances the effort of “Math Diagnostic Test” as follows.
1) In response to the determination of the subcommittee in affiliation schools in April, themembers in charge of “Math Diagnostic Test” are appointed in math subcommittee ofthe district. (In math subcommittee, there are also efforts other than “Math DiagnosticTest”.)
2) Staffs in charge of each grade are determined among members. Based on the analysis ofpast implementation and results, they discuss about improvement of the problems
and/or development of the new problems of the year.
3) They improve such problems and develop new problems actually.
4) A problem list of each grade is made via work of 3).
5) All members in charge including editorial supervisors (university researchers and aSupervisor of Prefectural Board of Education) participate the workshop to discuss about
verification of problem(s) and balance in and between grades (adjustment of asystem/connection of contents, appropriateness of the degree of difficulty, andanticipated time for students solving problems).
6) The organization orders printing Test booklets for a printer.
7) Almost all schools in the Prefecture participate the Test (not mandatory).
8) Scoring the Tests is done by homeroom teachers of each school.
9) Each school reports results (scores) of the Test to a teacher in charge of the district.
10) Staffs aggregate and analyze the results. A report of the analysis will be completed inthe next June-July.
2.3 Examinee students
The number of examinee students of Math Diagnostic Test implemented in February 2014 isshown in Table 1.

2!!!!!

GradeTable 1. Number of students by each grade
1st
2nd
3rd
4th

Number of students

4908

4867

4968

5138

5th

6th5057

5241

For reference, the results of the National Assessment Test that the 6th grade students took
exam in that year (2013) are shown in Table 2. The score of students of Tottori Prefecture was
11th place of 47 prefectures in Japan.
Table 2. National Assessment Test 2013
Tottori Prefecture (%)

National Average(%)

Elementary Math A

78.1

77.3

Elementary Math B

60.2

58.6

3. Problems for examining the thinking process
3.1 Nature of the problem
Although each Test problem requires only the answer basically, “problems for examining thethinking process” require describing the thinking process by an indication: “Let's leave yourwriting/trajectory of thought.” It is the reason why we focus on “problems for examining thethinking process.” However, we cannot see students’ writings directly. The authors can only
verify the trends of students’ solving indirectly to be shown in the report of the analysis(procedure 10 above mentioned).
There are 1-3 “problems for examining the thinking process” in each grade. (In all grades, 25
problems are provided by the Test generally.) In the following, we verify the Test problems andthe trends for each grade.
3.2 Problem of 1st GradeProb.
ProblemNo.
(8)
When distributing 12 pencils
by one for one person, 5
pencils are left.
When 10 pencils, how many
pencils are left?

Ave. of correct answer (%)
2012
2013
2014

39

39

38

Errors(Rate to the whole errors)7 (29%)
5 (29%)
2 (14%)others (14%)
no answer (14%)

Intention of the Problem
To find the unknown number from the given condition;
To identify the number of pencils and persons;To represent a problem situation with a diagram or a picture.
Expected Thinking
Finding the distributed number of the pencils by subtracting the number of remainders fromthe number of the beginning. This number is the number of people. Then, subtracting thenumber of pencils to distribute to seven persons from ten: 12 − 5 = 7 , 10 − 7 = 3 .
Finding the difference of the number of pencils between the first and the second situation.
Since the two small pencil in the second situation, the number of remaining pencils also
become two small: 12 −10 = 2 , 5 − 2 = 3
Error Analysis
Some students can’t do two-step thinking, they finish problem solving by finding the numberof distributed pencils: 12 − 5 = 7 .

3!!!!!

Some students subtract five remainders from ten pencils: 10 − 5 = 5 .
Some students misread the problem as “how many pencils remain when distributing 10 from12?”: 12 −10 = 2 .
3.3 Problems of 2nd Grade
Ave. of correct answer (%)
Prob.
ErrorsProblem(Rate to the whole errors)
No.
2012
2013
2014
(9)
There are two numbers which2 (19%)
one is 3 larger than 1000 and9 (13%)
another is 5 less than 1000.
47
51
51
others (47%)
How many is the differenceno answer (21%)
between two numbers?
Intention of the Problem
Appreciating to be able to find the number by using variables/differences based on thereference number.
Using a number line.
Expected Thinking
Finding the number by using variables/differences based on the reference number: 3 + 5 = 8 .
Finding the number by using a number line (two cases):a)

995 996 997 998 999 1000 1001 1002 1003
53535+3 = 8b)

995 996 997 998 999 1000 1001 1002 1003
(start) 1

2345

678(counting)Finding the number by calculation after obtaining the two numbers:
1000 + 3 = 1003 , 1000 − 5 = 995 ,.
Error Analysis
Some students use the difference without recognizing number sequence: 5 − 3 = 2 .
Some students include also the reference number in counted number: 5 +1+ 3 = 9 .

4!!!!!

Prob.
No.
(18)

Ave. of correct answer (%)
2012
2013
2014

Problem
There is a pattern arranged byright triangles as shown in thefigure.
How many triangles both large
and small are in this pattern?

42

43

47Errors(Rate to the whole errors)3 (63%)2 (10%)others (23%)
no answer (4%)

Intention of the Problem
Recognizing geometric figures by composing and decomposing.
Expected Thinking

Error Analysis
Some students count three small triangles only.
Some students answer that there are two kinds of triangles, big and small.
3.4 Problems of 3rd GradeProb.
ProblemNo.
(14)
Balls are packed in a box
accurately as shown in thefigure. How long (cm) is theradius of a ball?

Ave. of correct answer (%)
2012
2013
2014

54

54

57

Errors(Rate to the whole errors)6 cm (46%)
9 cm (13%)
2 cm (13%)
others (23%)
no answer (5%)

18cm
18cm

Intention of the Problem
Understanding spherical diameter and radius in relation to the length of vertical andhorizontal of the box.
Expected ThinkingFinding the radius after obtaining the diameter: 18 ÷ 3 = 6 , 6 ÷ 2 = 3 3 cm.
Find the radius on the basis of the number of radii (per side): 2 × 3 = 6 , 18 ÷ 6 = 3 3 cm.
Error Analysis
Some students answer the diameter: 18 ÷ 3 = 6 , 6 cm.
Some students recognize incorrectly the length of one side of the box as a diameter:
18 ÷ 2 = 9 (cm).
Some students divide the length of one side of the box by nine balls: 18 ÷ 9 = 2 (cm).

5!!!!!

Prob.
No.
(18)

Ave. of correct answer (%)
2012
2013
2014

Problem

There are pencils, notebooks,and paints.
A pencil is 60 yen.
A notebook is twice the priceof the pencil.
57
57
55
A paint is four times the priceof the notebook.
How many times the paint is
the price of the pencil?
Intention of the Problem
Solving a problem by using multiplication operators.
Expected Thinking
Finding the number by calculating operators: 2 × 4 = 8 8 times.

Errors(Rate to the whole errors)6 times (44%)
480 times (10%)
2 times (10%)
4 times (5%)
others (26%)
no answer (5%)×8pencil (60 yen)×2

notebook

×4paint

Comparing the prices of pencil and paint after finding the prices of notebook and paint inturn: 60 × 2 = 120 , 120 × 4 = 480 , 60 × □ = 480 , 8 is appropriate for □; or 480 ÷ 60 = 8 .
Error Analysis
Some students add the multiplication operators: 2 + 4 = 6 (times).
Some students obtain the price of paint (their thinking process are correct).
Some students answer either of operators.

Prob.
No.
(21)

Ave. of correct answer (%)
2012
2013
2014

Problem
Trees are lined in a row away by
2m. Trees of both ends are away
14m.
How many trees are lined?

3539

43Errors(Rate to the whole errors)7 (76%)
28 (7%)
16 (3%)
others (8%)

Intention of the Problem
Comprehending the problem situation accurately, and coping with it correctly.
Expected Thinking
Finding that the number of trees is always one greater than the number of interval of trees,
sometimes by representing the problem situation into a figure: 14 ÷ 2 = 7 , 7 +1 = 8

14m

2m

2m

2m

Error Analysis
Some students are regarded “the number of interval of trees” just as the "number of trees”.
Some students calculate inaccurately using the numerical values in the problem sentence:
2 ×14 = 28 , 2 +14 = 16 .

6

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3.5 Problems of 4th Grade
Prob.
ProblemNo.
(23)
I bought five apples. I paid 600
yen because the price was cutby 50 yen.
How much was an apple?

Ave. of correct answer (%)
2012
2013
2014
58

59

Errors(Rate to the whole errors)

110 yen (24%)
120 yen (15%)
others (49%)
no answer (12%)

57

Intention of the Problem
Thinking back to the order by representing the problem situation into a diagram.
Expected Thinking

0□

600 ■

(price)−50

1(numberof apples)

5

□

×5■

÷5−50

600

+50

(600 + 50) ÷ 5 = 130

130 yen600 + 50 = 650 , 650 ÷ 5 = 130
Error Analysis(600 − 50) ÷ 5 = 110 (yen)

130 yen600 ÷ 5 = 120
Prob.
No.
(24)

Ave. of correct answer (%)
2012
2013
2014

Problem

The monkey weighs 12 kg, andis three times the weight of therabbit. The weight of the rabbit
is twice the weight of the72
74
squirrel.
How many kg does the squirrel
weigh?
Intention of the Problem
Solving a problem by using operators with drawing diagrams.
Expected Thinking

monkeyrabbit

72Errors(Rate to the whole errors)

72 kg (25	 )4 kg (12	 )others (38%)
no answer (25%)

×(2×3)

12kg

□

■kg

squirrel □kg×2
÷2

■×3

12

÷3

÷(2×3)
12 ÷ 3 = 4 , 4 ÷ 2 = 2 2 kg
12 ÷ (2 × 3) = 2 2 kg
Error Analysis
Some students can’t represent the problem situation into a diagram or a figure, andconsequently they do incorrect calculation variously with using the numerical values of the
problem sentence: 12 × 3 × 2 = 72 , 12 ÷ 3 = 4 , 12 − 3 − 2 = 7 .

7!!!!!

3.6 Problems of 5th GradeProb.
ProblemNo.
(14)
There is a parallelogram-likegarden. Find the area of thewhole flower beds.
2m 1m

5m

pass
flower bed
6m

55

flower bedage

5m

Ave. of correct answer (%)
2012
2013
2014

5m

58

54

Errors(Rate to the whole errors)

35 m2 (37	 )
32 m2 (11	 )
40 m2 (3	 )others (31%)
no answer (18%)

4m

1m 1mIntention of the Problem
Finding a way of obtaining the area of flower beds by using quadrature formulas of aparallelogram and a trapezoid, and equivalency transformation.
Expected Thinking
Subtracting the area of passage part from the area of the whole garden:(6 +1+1) × 4 −1× 4 = 28 28 m2,(2 +1+ 5) × 4 −1× 4 = 28 28 m2.
Considering flower bed parts to be two trapezoids: (2 + 6) × 4 ÷ 2 + (5 +1) × 4 ÷ 2 = 28 (m2).
Finding the area as a parallelogram by moving a part of the flower bed:
(6 +1) × 4 = 28 (m2), (2 + 5) × 4 = 28 (m2).
Error Analysis
Some students calculate the area by using the "hypotenuse" of a parallelogram as the"height”: (5 + 2) × 5 = 35 .
Some students don’t remove the area of passage part: (6 +1+1) × 4 = 32 .
Some students consider that there are two same trapezoids: (2 + 6) × 4 ÷ 2 = 16 , 16 × 2 = 32 .
Some students calculate the area both by using the "hypotenuse" of a parallelogram as the"height” and by considering that there are two same trapezoids: (6 +1+1) × 5 = 40 .

Prob.
No.
(16)

ProblemTaro has taken fiveexaminations. The average ofexam up to fourth was 88. Thefifth score was 98. Find theaverage of all five exam.

Ave. of correct answer (%)
2012
2013
2014

40

4247

Errors(Rate to the whole errors)

93 (34	 )37.2 (6	 )
138 (6	 )others (33%)
no answer (21%)

Intention of the Problem
Finding overall average by using the temporary average.
Expected Thinking
Finding overall average by calculating the total from temporary average to the fourth andadding the fifth score:
88 × 4 = 352 , 352 + 98 = 450 , 450 ÷ 5 = 90 ; or (88 × 4 + 98) ÷ 5 = 90 .
Finding overall average by dividing the difference between the temporary average to thefourth and the fifth score into five equal parts:
98 − 88 = 10 , 10 ÷ 5 = 2 , 88 + 2 = 90 ; or (98 − 88) ÷ 5 = 2 , 88 + 2 = 90 .

8

!!!!!

Error Analysis
(88 + 98) ÷ 2 = 93 .
(88 + 98) ÷ 5 = 37.2
Some students consider the total of five times to be five times of 98 (average):
88 × 4 = 352 , 98 × 5 = 490 , 490 − 352 = 138 .

Prob.
No.
(21)

Ave. of correct answer (%)
2012
2013
2014

Problem
A figure surrounded with fivestraight lines is called apentagon.
How many degrees is the sumof interior angles of thefollowing pentagon ABCDE?
EErrors(Rate to the whole errors)

900° (18%)
720° (7%)
520° (7%)
others (27%)
no answer (41%)
6267

67

A
D

B

C

Intention of the Problem
Finding the sum of a polygonal interior angles based on “the sum of a triangle's interior
angles is 180°.”
Expected ThinkingEADB

180 × 3 = 540

CEAD



B

360 +180 = 540

CEADC B
180 × 5 − 360 = 540
Sum of interior angles of the n-gon is 180 × (n − 2) in general.
So, if n = 5 , 180 × (5 − 2) = 540 .
Error Analysis
180 × 5 = 900 .
180 + 540 = 720 , or 180 + 540 = 520 (miscalculation).

9!!!!!

3.7 Problem of 6th GradeProb.
ProblemNo.
(20)
T-shirt of 1200 yen has
become to 960 yen at thebargain sale. What percent
discounts?

Ave. of correct answer (%)
2012
2013
2014

404544Errors(Rate to the whole errors)

80 % (50	 )
1.25% (9	 )
125% (7	 )
5% (5	 )others (28%)
no answer (18%)

Intention of the Problem
Understanding the proportion.
Expected Thinking
1280 − 960 = 240 , 240 ÷1200 = 0.2 , 0.2 ×100 = 20 (%).
960 ÷1200 = 0.8 , 1− 0.8 = 0.2 , 0.2 ×100 = 20 (%).
Error Analysis
960 ÷1200 = 0.8 , 0.8 ×100 = 80 .
1200 ÷ 960 = 1.25 ; 1.25 ×100 = 125 .
1200 − 960 = 240 , 1200 ÷ 240 = 5 .
4. Concluding Remarks
In this paper we described and analyzed the Tottori Prefecture regional “Math DiagnosticTest”, especially focusing on “Problems for examining the thinking process.” These efforts byteachers groups contribute to their professional development, especially in terms of“mathematical content knowledge study (kyozai-kenkyu, in Japanese).” Furthermore, such a
Test is also effective for improvement of teaching in the classroom. In fact, a student’s responseof a problem shows his/her Attained Curriculum. It leads to find where of the teaching andlearning process his/her teacher has a problem by analyzing it.
On the other hand, the teachers groups have difficulty for "the development of new problems.”
If an international database of math problems is developed, it is hoped that it helps the solution
to such difficulties.
Notes*1 This paper was presented at the International Conference on the Open Environment for the Worldwide
Mathematical Education, 8-11, April, 2015, Moscow, Russia. This version is slightly modified.

References
Japan Ministry of Education, Culture, Sports, Science and Technology (2009). The Course ofStudy for Elementary School.
Mullis, I. V. S., and Martin, M. O. (Eds.) (2013). TIMSS 2015 Assessment Frameworks. TIMSS& PIRLS International Study Center, Lynch School of Education, Boston College, andInternational Association for the Evaluation of Educational Achievement (IEA).
National Council of Teachers of Mathematics (1995). Assessment Standards for School
Mathematics. Reston, VA.
Tottori Prefecture Regional Elementary Mathematics Subcommittee (Eds.) (2014). The 56th
Annual Report of Tottori Prefecture Regional “Math Diagnostic Test 2013”. Tottori
Association of Mathematics Education: Elementary School Committee. (in Japanese)
Acknowledgement
This study was supported by JSPS KAKENHI Grant Numbers 24243078.

10鳥取大学数学教育研究

ISSN 1881−6134

Site URL：http://www.rs.tottori-u.ac.jp/mathedu

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鳥取大学数学教育学研究室

tsyabe@rstu.jp

溝口達也

鳥取大学数学教育学研究室

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