Categorizing Students’ Mathematical Problem Posing: A Case on

Counting

Marhayati, Siti Faridah, Intan Nisfulaila, Imam Rofiki, Muhammad Islahul Mukmin, and Elly

Susanti

Universitas Islam Negeri Maulana Malik Ibrahim Malang, Jl. Gajayana No. 50 Malang, Indonesia

Keywords: Category, Mathematical Problem Posing, Counting Operation.

Abstract: This research aims to categorize mathematical problems proposed by students based on the concept of

counting operations. The counting operations which are used in this research are addition, subtraction,

multiplication, and division. This research intends to a descriptive study. Data were collected from the 7

grade students of Islamic Junior High School in Malang, Indonesia. The topic given to the students was

Social Arithmetic. Students were asked to pose problems based on the given situation without giving the

solution. The analysis of the results shows that there are four categories of students’ problem posing, they

are respectively the problems that involving one counting operation (the first category), two counting

operations (the second category), three counting operations (the third category), and four counting

operations (the fourth category). Even though the students did not solve the problem, but some of them had

predicted the solution of the problem. When posing mathematical problems, the students associated the

given situation to their mathematical abilities. Also, some students added some new quantitative information

and develop the mathematical problems.

1 INTRODUCTION

Learning in school has encouraged a lot of students

to pose and solve mathematical problems (Cai and

Hwang, 2002; Crespo and Sinclair, 2008; Silver,

1997, 1994; Singer et al., 2013; Xia et al., 2008).

The students are asked to find the solutions for the

given situation and information. This result was a

very great weapon for students to think

convergently. The students must be trained to think

divergently so that their mathematical abilities can

be developed. One of the ways for students to come

up with ideas and develop their mathematical

abilities is by assigning tasks to pose some

mathematical problems.

Many researchers associate the task of problem

posing with creativity (Haylock, 1997; Kontorovich

et al., 2011; Leung and Silver, 1997; Marhayati and

Sa’dijah, 2018; Pehkonen, 1997; Shuk-kwan, 1997;

Silver, 1997; Siswono, 2010; Van Harpen and

Sriraman, 2013). In mathematical problems, the

students are trained to think out of the box. Students

do not only solve the problems given by the teacher

but also formulate the problems. In formulating

problems, students use their mathematical abilities.

Using mathematical abilities, creative questions are

needed so they can formulate various questions.

A semi-structured problem posing situations can

be used to train students’ mathematical abilities.

According to Stoyanova and Ellerton (Stoyanova

and Ellerton, 1996), semi-structured problem posing

situations are the submission of problems based on

open-ended situations. The students are asked to

explore open-ended situations using knowledge,

skills, concepts, and relationships from their

previous mathematical experiences. Thus, from the

questions posed, students can show into their

knowledge, skills, concepts, and mathematical

experiences.

One of mathematical abilities that is needed in

the submission of mathematical problems is the

ability in counting operations. Basic counting

operations consist of addition, subtraction,

multiplication, and division. Counting operations are

the foundation in learning mathematics. If students

have controlled the concept of counting operations,

it will be very helpful to pose or solve the

mathematical problems. In addition, the

mathematical experience that the students have been

444

Marhayati, ., Faridah, S., Nisfulaila, I., Roﬁki, I., Mukmin, M. and Susanti, E.

Categorizing Students’ Mathematical Problem Posing: A Case on Counting.

DOI: 10.5220/0008523404440447

In Proceedings of the International Conference on Mathematics and Islam (ICMIs 2018), pages 444-447

ISBN: 978-989-758-407-7

INFORMATION

On Sunday Zaki and Ega went to the Keripik Eco store. The

food store sells tempe chips, banana chips, cassava chips,

corn chips, and potato chips. The price of tempe chips is

IDR 24000/kg. The price banana chips is IDR 20000/kg.

Price of cassava chips is IDR 16000/kg. Price of corn chips

is IDR 10000/kg. The price of potato chips is IDR

36000/kg. After buying chips they went home.

experienced also supports the activity of posing

questions to provide varied results.

The mathematical experiences that are closely

related to everyday life such as shopping, discounts,

shopping trucks, price lists, and others will be more

meaningful in relating mathematics and everyday

life (Bonotto, 2011). Social arithmetic is closely

related to everyday life. Silver and Cai (Silver and

Cai, 1996) conducted study on the posing of

arithmetic problems. Social arithmetic learns about

the use of algebraic concepts to solve simple

economic problems. Social arithmetic material

discusses about profit, loss, discounts, interest rate,

percentage, buying price, and selling price. It is

often encountered by students in everyday life.

In this study, students were given the task of

posing math problems. The purpose of the study was

to categorize the mathematical problems proposed

by students in the terms of counting operations.

Teachers can use the result of the study for

developing students’ creativity related to counting

operations.

2 METHODS

The type of this study is descriptive research. Data

were collected from the 7

grade students of MTs

Surya Buana in Malang, Indonesia. The material that

has been used was Social Arithmetic. The task of

posing mathematical problems given to students is

the submission of problems without completion.

Students are asked to pose as many as possible

problems based on the information provided.

The information provided in the problem posing

is a word problem accompanied by an illustration.

The provision of illustration is intended to clarify the

given short story and to attract students' attention

and to give the students the freedom to choose the

questions from the story provided or from the

images provided. The story in the

situation/information provided is in the form of an

open situation. From the given situation, it is

expected that students can issue their creative ideas

in asking questions and developing them outside the

given situation. Students are given the freedom to

change or to add information to the given situation.

Information provided to students is presented in

Figure 1.

The research procedures carried out are as

follows: giving assignment to the 7th grade students.

Then, analyzing the results of the problem

submission. The problems raised by students were

selected in the form of mathematical problems that

could be solved. Although in this study students

were not asked to complete it. Furthermore, the

categorization of questions submitted by students is

carried out. Question categories are presented in

Table I.

Figure 1: Information given to the students in posing

mathematical problems.

Table I: Categories of questions raised by students.

Problem

Categor

Description

One counting

operation

The resulting problem involves

only one count operation, which

is it can be addition or

subtraction or multiplication or

division

Two counting

operations

The resulting problem involves

two counting operations

Three counting

operations

The resulting problem involved

three counting operations

Four counting

operations

The resulting problem involves

four counting operations

3 RESULTS AND DISCUSSION

Category of One Counting Operation

The problem that fall into the category one

counting operation are presented in Figure 2.

Categorizing Students’ Mathematical Problem Posing: A Case on Counting

445

The price of 1 kg of corn chips is IDR 10000

and the price of 1 kg of tempe chips is IDR

24000. Determine the difference!

Zaki bought 2 kg of banana chips, 1 kg tempe

chips, and 3 kg of corn chips. How much Zaki

should pay?

Ega bought 2 kg of potato chips and apparently

got a 10% discount. How much money must be

paid?

Ega bought 4 kg tempe chips, 3 kg corn

chips, 2 kg potato chips, 1 kg cassava chips.

Because Ega buy a lot of things, finally Ega get

a 25% discount. Hence, how much Ega should

pay?

Figure 2: Problems containing one counting

operation.

Figure 2 shows that it is known that the problem

posed by students contain one counting operation. If

the problem posed by the student is gained, the

solution will involve only one counting operation,

that is subtraction. The price difference between

corn chips and tempe chips is determined by

reducing the price of tempe chips with corn chips

(the price difference = 24000 − 10000 = 14000).

The students used quantitative information in a

given situation without adding some new

quantitative information.

Problem that was posed by students involve

addition or subtraction or multiplication or division.

This shows that in making problems, the students

only focus on the information provided and try to

ask questions with a low level of difficulty.

Category of Two Counting Operations

The problem that meets the category of two

counting operations are presented in Figure 3.

Figure 3: Problems containing two counting

operations.

Figure 3 shows that the problem posed by

students contains two counting operations. They are

multiplication and addition. There are three stages in

finding the solution. The first is determining the

purchase price of 2 kg of tempe chips, by

multiplying 2 kg with 24000 so the result is 48000.

The second is determining the purchase price of 3 kg

of corn chips, by multiplying 3 kg with 10000 so the

result is 30000. The third, finding the total money to

be paid, that is 48000 + 30000 = 70000. The

problems submitted contain new quantitative

information, namely 2 and 3. This shows that in

formulating problem, the students are not only focus

on the information provided, but also, they

associated with their experiences. When the problem

in Figure 5 is compared with the problem that fall

into the category of one counting operation, the

problem in this category is more complex.

Category of Three Counting Operations

The questions that fall into category of three

counting operations are presented in Figure 4.

Figure 4: Problems containing three counting operations.

Figure 4 shows that the problems posed by

students contain three counting operations, they are

multiplication, division, and subtraction. There are

three stages in solving the problems in Figure 4.

First, the price of 2 kg of potato chips is calculated

as 2 × 36000 = 72000. Second, the amount of the

discount is calculated, which is 72000 × 10% =

7200. Third, the amount of money that must be

paid after the discount is 72000-7200 = 64800. At

the time of determining the discount, there are two

counting operations involved namely multiplication

and division. There is new quantitative information

added which is 2 kg and 10% discount. This shows

that students think they are developing, not just

focusing on the information provided. The questions

raised are also more complex than the questions that

fall into the category of one and two counting

operations.

Category of Four Counting Operations

The questions that fall into the category of four

counting operations are presented in Figure 5.

Figure 5: Problems containing four counting operations.

Figure 5 shows that the problem posed by

students contains four counting operations, namely

multiplication, addition, division, and subtraction.

There are seven stages of problem solving. First, the

price of 4 kg of tempe chips is calculated which is

4 × 24000 = 96000. Second, the price of 3 kg of

corn chips is calculated, namely 3 × 10000 = 30000.

Third, calculated the price of 2 kg of potato chips,

namely 2 × 36000 = 72000, Fourth, calculating 1 kg

of corn chips which is 16000. Fifth, the total price of

chips purchased is 96000 + 30000 + 72000 + 16000

= 214000. Sixth, calculated a discount of 25% that is

214000 × 25% = 53000. Seventh, the amount of

money that must be paid after applying the discount

is 214000-53000 = 160500. New quantitative

information added are 4 kg, 3 kg, 2 kg, 1 kg, and

25% discount. New quantitative information added

more when compared to questions included in

categories one, two, and three counting operations.

This shows that students develop the form of

problems and not only focus on the information

ICMIs 2018 - International Conference on Mathematics and Islam

446

provided. The problem posed is more complex than

the three other problems.

4 CONCLUSIONS

There are four categories of problems posed by

students, which contain one counting operation, two

counting operations, three counting operations, and

four counting operations. The characteristics of

problems that fall into category one are involving

one count operation, the absence of new quantitative

information and the low complexity of the question.

This shows that the source of ideas in posing

problems comes from the situation/information

provided. The characteristics of the problems

included in the second category involved two

counting operations, new quantitative information

and at least two steps in solving the problem. The

characteristics of the problems included in the third

category contain three counting operations, new

quantitative information and at least 3 steps to solve

the problem. The number of quantitative information

in the problem of third category is greater than the

categories in the second categories shows that the

students begin to develop out of the given situation.

The characteristics of the problems in the fourth

category involve four counting operations, the

presence of new quantitative information, at least 4

steps to solve the problem, and the development of

the form of the problem that was posed out of the

given situation. The development of the posed

problems in the fourth category can be seen with

more new quantitative information added and the

problems posed are more complex than the first,

second, and third categories.

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