

































 

 

Problems Solvingand Thinking Critical Ability Student Through 
Realistic Approach Based on Problem Non-routine by Blended 
Learningin the Department Mathematics,Faculty Mathematics 

and Natural Science, Medan State University 
 

Hasratuddin 1) ; M. Amin Fauzi 2 ) ; Budi Halomoan Siregar 3) ; and, Dwi Novita Sari 4) . 

1) and 2) Lecturer in Mathematics at Medan State University, and 
3) and 4) Students of S3 Mathematics Education, State University of Medan 

Email 1) ; siregarhasratuddin@yahoo.com, 2) aminunimed29@gmail.com, 
3) budihalomoan@unimed.ac.id, 4) dwinovitasari31@gmail.com 

Abstract. This study aims to analyze students' problem-solving and critical thinking ability through 

realistic approach based on non-routine problem by blended learning in the Mathematics Department, 

Faculty of Mathematics and Natural Sciences, State University of Medan. This type of research is a 

quasi-experimental. The research population is all undergraduate students of the Mathematics Education 

Study Program (MESP) in 2021. The research sample is taken from 2 classes of the population, namely; 

1) MESP A 2021 as an experimental class that is given learning with a realistic approach based on non-

routine problems using blended learning, and 2) MESP B 2021 as a control class that is given regular 

online learning. The research instrument on problem solving and critical thinking skills is a test. Data on 

problem solving abilities and critical thinking were analyzed using ANOVA. The results showed that the 

problem-solving and critical thinking skills of students who were given geometry learning through a 

realistic approach based on non-routine problems using blended learning were better than students who 

were given regular online learning. Furthermore, from the results of the study it was found that there was 

no interaction between learning and students' initial abilities, both on problem skills and on students' 

critical thinking skills. This shows that students' problem-solving and critical thinking skills are always 

better using learning with a realistic approach based on non-routine problems using blended learning 

compared to students who are given regular online learning. Thus, what is suggested from the results of 

this study is that in improving students' problem-solving and critical thinking skills through a realistic 

approach based on non-routine problems using blended learning, it is not necessary to classify students 

into low, medium or high initial abilities. 

Keywords: realistic approach; ability; solving problem; critical thinking; blended learning; non- routine. 

Introduction 

Realistic Mathematics Education originally came from the Netherlands and has been 
developed since the 1970s. As for what inspired it was Freudenthal's view which said that 
mathematics is a human activity. So that mathematics should not be given to students in the 
form of ' finished results', but must construct / find their own mathematical concepts, 
principles or procedures through solving non- routine problems. There are three key 
principles in the realistic approach [1], namely: 1) Guided reinvention/progressive 
mathematizing, 2) Didactical phenomenology, and 3) Self-developed model. From these 
three principles, the mathematics learning process with a realistic approach is divided into 
five characteristics, namely: constructing and concretizing , level and models, reflection 
and special assignment, social context and interaction, structuring and intertwining [1], [2], 
[3]. Mathematics learning with a realistic approach has wide-ranging consequences for 
children's learning and thinking processes [4]. Meanwhile, the mathematization process is 
seen as an activity that is constructive, reflective and interactive. Learning through a 
realistic approach is an activity that is meaningful for them, so that it can improve their 
attitudes and higher levels thinking skills. 

One offactor that affect thinking ability level tall is a stimulus through challenge in 
problem non-routine [5]. Challenge in the form of problem non-routine could stimulate 

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somebody to fully understand the problem based on observations and investigations, 
explore and prepare tools, conduct experiments or investigations. Therefore, by starting 
from non-routine problem conflicts, students will not feel unfamiliar with the topics they 
are going to learn and will grow curiosity for themselves, and it is hoped that in the end it 
will eliminate the impression that learning mathematics is no longer something that is fun. 
difficult and scary. 

Associated with the learning process that does not determined of consequence covid 19 
situation, then most potential approach for held in learning is based blended learning with 
an online or offline system through a realistic approach that emphasizes student - centered 
processes as well as load element constructive, interactive and reflective. Given the 
importance of choosing proper learning _ in dominate field mathematics then necessary 
analyze ability solving problem and think critical mathematical student through based 
realistic approach problem non-routine by combined offline with online. 

Ability solving problem is something very important in learning mathematics, because 
1) makes somebody Becomes skilled select and analyze something later information _ 
study the result, 2) makes something satisfaction intellectual arising _ from in self 
someone, 3) increase potency intellectual someone, and 4) someone will could finder 
through the discovery process that alone. Solution problem is very important thing in 
learning math, because with increase ability solving problem non-routine expected student 
will more analytical and capable in resolve problem and at the same time could prepare self 
in face changing situations in life _ later [6]. 

In everyday life, we cannot be separated from something called a problem, so problem 
solving is the main thing in learning mathematics. Most mathematics education experts 
state that the problem is a question that must be answered or responded to by students. By 
Specifically, math problems consist of routine problems and non-routine problems.A 
routine problem is a problem that is merely an exercise that can be solved using some 
commands or algorithms [7], [8] . Meanwhile, non-routine problems are more challenging 
and require creative abilities from problem solvers. Non - routine problems arise when 
problem solvers have problem characteristics that do not immediately known how to solve 
them [9]. Theproblems included _ inthing this is problems that contain challenges that are 
not immediately can be solved by known routine procedures. "For a question to be a 
problem, it must present achallenge that cannot be resolved by some routine procedure 
known to the student [10].” So, ability solving problem is ability think mathematical 
somebody in evaluate, connect and develop something theory. Problem solving ability is an 
essential competency in learning mathematics, so it is recommended to be trained and 
raised since children at the elementary school level up to college [11]. This means that 
mathematical problem solving skills need to be trained at every level of education. 

Problem solving skills require reflective thinking, including critical thinking and 
creative thinking skills. In other words, learning mathematics in the classroom needs to 
train critical and creative thinking skills that are carried out intentionally and 
planned.IDEAL is something term problem solving model or heuristic found by Brandsford 
and Stein [12] . This model consists of five stages of problem solving, namely Identifying 
potential problems, Defining and representing the problem, and Exploring possible 
strategies. Acting on those strategies, Looking back and evaluating the effects of those 
activities. Polya (1973) developed a problem solving model, procedure, or heuristic 
consisting of stages of problem solving, namely (1) understanding the problem; (2) make a 
problem-solving plan; (3) implementing a problem-solving plan; and (4) review [13] . Thus 
, in study In this case , the activities included in problem solving activities include: 
identifying elements that are known, asked about, and the adequacy of the elements 
needed, formulating problems from everyday situation with mathematics; implement 

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strategy for solve various problems inside or outside mathematics; explain and interpret the 
results according to the problem; develop mathematical models and solve them according 
to real problems and use mathematics in a meaningful way [14]. Strategies for solving 
problems mathematics depends on the problem to be solved. Problem solving strategies in 
general have four steps, namely ; 1) m understand the problem, inthis activity conducted 
stages or steps: what (data) is known, what is not known (asked), whether the information 
is sufficient, what conditions (conditions) must be met, restating the problem in detail 
operational , b) planning a solution, namely To do the activity of trying , looking for or 
remembering problems that have been solved that have similarities to the problems to be 
solved, looking for patterns or rules, making conjectures and compiling settlement 
procedures , c) solving problems according to plan, namely To do activities with doing 
according to the procedure that was made in the previous step to get a solution , d) re-
examine the procedure and the results of the settlement, namely To do the activity of 
analyzing and evaluating whether the procedures applied and the results obtained are 
correct and appropriate , whether there are other procedures that are more effective, 
whether the procedures made can be used to solve similar problems, or whether the 
procedures can be generalized . 

Critical thinking is one of the higher-order thinking processes that can be used in the 
formation of students' conceptual systems. Critical thinking is a sensible or reason-based 
reflective way of thinking that is focused on determining what to believe and do [15]. 
There are two main signs of critical thinking [16]. The first is that critical thinking is proper 
thinking that leads to deductive thinking and decision making rational.The second is that 
critical thinking is reflective thinking that shows complete awareness of the steps of 
thinking logical.Critical thinking must meet the characteristics of thinking activities which 
include: analysis, synthesis, problem recognition and solutions, conclusions and 
assessments [17]. Although mathematics is related to logical theory, critical thinking skills 
will not develop if in mathematics learning students are only trained to memorize 
(mechanistic) formulas, find formulas without knowing the relationship between draft with 
others (structuralistic), or solve problems routinely (empirical), without involving thinking 
skills. 

Critical thinking in mathematics includes the process of testing, questioning, 
connecting, evaluating all aspects that exist in a situation or a problem [19]. Actually 
critical thinking is a thinking process that occurs in a person and aims to make reasonable 
decisions about something that can be believed to be true and which will be done later. 
There are six basic elements that need to be considered in critical thinking, abbreviated as 
FRISCO, namely: focus , reason , infrent , situation , clear .and overview [18] . 

From the description above , the indicators of critical thinking skills in this study are; 
a) connect as well as apply draft by math ,b) explore , that is ability construct meaning  or 
meaning and investigate ideas mathematics , c) generalize , i.e. interesting conclusion or 
determine mathematical ideas _ _ inductive or deductive , d) clarify , that is ability evaluate 
and explain , determine the context of the idea mathematics , and e) solve problem , that is 
analyze problem so that find correct answer _ by logical . 

Research Method 
Type study this is quasi experiment. Where, researcher use design two class 

experiment and control. Class experiment given treatment Realistic learning based problem 
non-routine by blended learning, while class control given learning normal online. Class 
experiment and control are taken from study program student education mathematics year 
enter 2021 which takes eye studying geometry, that is MESP A 2021 class and MESP B 
2021 class. As many as 35 people are used for MESP A 2021 class as class experiment, 

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while MESP B class 2021 as many as 35 people are made as class the control. This 
research _ implemented in the Department of Mathematics Faculty Math and natural 
Science, State University of Medan. Study this use learning design based on the realistic 
approach. 

Study this use two class experiment-control design with pretest - posttest research 
instrument to ability solving problem and think critical is with use test essay form. The 
statistical analysis used in study this is with anova two path. 

Research Results 
Before conducted learning on both class experiment and control is done test start, use 

see is student on both class have same ability _ or different, at the same time for grouping 
ability beginning student. As for the results of p retest second found class _ in study this 
could seen in the table following. 

Table 1. Results of statistical calculations on ability beginning Solution Problem student. 

 Sum of Squares df Mean Square F Sig. 
Score * Class Between Groups (Combined) 

29,557 1 29,557 .201 .655 

Within Groups 
12655.523 86 147,157   

Total 
12685,080 87    

 

From table 1 above seen that score significance count is 0.201 and more big from level 
0.05 confidence, then set that H o : accepted. So that could concluded that ability beginning 
solution problem student class experiment with control class is same. 

Table 2. The results of the calculation of the ability statistics beginning Think Critical. 

 Sum of Squares df Mean Square F Sig. 
Score * Class Between Groups (Combined) 

127,682 1 127,682 .879 .351 

Within Groups 
12497.182 86 145,316   

Total 
12624,864 87    

 

From table 2 above seen that score significance count is 0.351 and more big from level 
confidence 0.05 then set that H o: accepted. So that could concluded that ability beginning 
think critical student class experiment with control class is same. 

In accordance with destination study this that is for analyze ability solving problem and 
think critical student who was given learning geometry with approach realistic based problem 
non-routine by blended learning, then following describe the process and results. 

The learning process carried out in accordance characteristics realistic approach that is 
in learning started with give problem non-routine to student, then give opportunity to student 
for finish it by independent or group as well as discuss result by classic, so expected student 
find draft or the knowledge contained in problem. 

Meeting first: Done online, with given problem _ is " observe " picture below, choose 
which one is _ side four, then mention what do you mean with side four”. 

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According to opinion I from the picture above which includes side four is build ABCD, 

EFGH and MNOP. While those who don't including side four is KILJ. 

is polygon with four side and four angle.

S2: 

 

According to opinion I from the picture above which includes side four is build ABCD, 

EFGH and MNOP. While those who don't including side four is KILJ. 

is polygon with four side and four angle. 

St

udent 

respon

ses to 

the 

proble

ms 

above; 

S1: 

 

According to opinion I from the picture above which includes side four is build ABCD, 

EFGH and MNOP. While those who don't including side four is KILJ. So, quadrilateral 

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S3: 

From the answers of the three respondents above, it can be concluded that students still 
have doubts about the definition of a quadrilateral. Thus, the non-routine problems above will 
require discussion in order to find answers rationally. Learning with problem non-routine will 
could increase ability solution critical problem [20]. In discussion class, lecturer provide 
scaffolding in the form of draw get up triangle on 3 lines, _ four over 4 lines, and so on. So, 
students could find knowledge or draft that get up side four is field flat closed by 4 lines. _ 
And, this mathematical process is referred to as a process of improving students' problem 
solving and critical thinking skills. 

At the meeting second online, problem given _ is " Known " there is plot paper shaped 
rectangle long and have circumference 24 cm. Questions; a) how much many rectangle that 
has circumference 24 cm? b) how much size rectangle length that has large area biggest? 

 

Response student on problem mentioned above, among others; 

S1: Answer: Rectangular ABCD, EFGH and MNOP are side four, except rectangular IJKL. 
 So, in terms of four is field flat closed line formed by 4 lines. 

Rectangular IJKL instead is side four, because if every line be extended no will shape 
side four but side three. Whereas if the line on the third get up other extended , shape 
get up fixed and not change . 

  

 

S2: 

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S3: 

 

Of several response student to problem non-routine the student not yet could complete 
problem by criticalthinking. That's it, lecturer provide scaffolding in the form of definition 
side four, parallelogram, trapezoid. Then, lower get up parallelogram Becomes rectangle long 
and rhombic. Then from rectangle, together student found get up rectangle that is rectangle 
length that has size the same side. So, students find knowledge that rectangle is a rectangle 
length that has the same side. 

 
Meeting third conducted offline. Problem non - routine given is  
 

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Response student to above problem: _ 

S1: 

 

S2: 

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After learning online and offline, each meeting is held 2 times in class experiments, and 
4 times online on learning normal, then given test formative, with results as following . 

Table 3. The results of the calculation of ability statistics solution problem student in class experiment 
and control. 
 

 
 
 
 
 
 
 
 

 
 
 
From table 3 above seen that score significance between class experimental and control 

is 0.001, and less than level confidence 0.05. This show that H o rejected. So that could 
concluded that there is difference ability solving problem student control class with class 

experiment. By graphics, can seen in the 
picture following. 

From the graph on the side, it can be 
seen that that class line experiment more 
tall from control class, then could 
concluded that ability solving problem 
mathematics student who was given 
Realistic learning based problem non-

Source Sum of Squares df Mean Square F Sig. 
Corrected Model 721,742 a 3 240,581 4,863 .004 

Intercept 577307.756 1 577307.756 11669,714 .000 

Class 601.136 1 601.136 12,151 .001 

Group 120,606 2 60.303 1,219 .301 

Error 4155.530 84 49,471   

Total 617100,000 88    

Corrected Total 4877,273 87    

a. R Squared = .148 (Adjusted R Squared = .118) 

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routine by blended learning more good from 
student who was given learning normal online. 

Next seen that score significance 
between group ability beginning math in class 
experimental and control is 0.301, and 
morebig from level confidence 0.05. This 
show that H o accepted. So that could 
concluded that no there is interaction Among 

learning with ability beginning to ability solution problem mathematical student. And, by 
chart could seen that ability solving problem student who was given learning with based 
realistic approach problem non-routine more than every group on learning normal online. 
This thing in accordance with results study Nasution showing _ that ability solution 
mathematical students who are given more realistic approach good from learning 
conventional, good in groups ability beginning mathematics low, medium nor high in SMP 
18 Medan [21]. 

For ability think critical geometry student could explain as following. 

Table 4. The results of the calculation of ability statistics think critical student . 

Source Sum of Squares df Mean Square F Sig. 
Corrected Model 1565,350 a 3 521,783 13,460 .000

Intercept 641565,465 1 641565,465 16550,217 .000

Class 1472,727 1 1472,727 37,991 .000

Group 92.622 2 46,311 1.195 .308

Error 3256,241 84 38,765  

Total 669850.000 88   

Corrected Total 4821,591 87   

a. R Squared = .325 (Adjusted R Squared = .301) 

 

From table 4 above seen that score significance between class experimental and control 
is 0.000, and more small from level confidence 0.05. This show that H o rejected. So that 
could concluded that there is difference ability think critical student Among control class 
with class experiment. By graphics, can seen in the picture following. 

From the graph on the side, it can be seen that that class line experiment more tall from 
control class, then could concluded that ability think critical mathematical student who was 
given Realistic learning based problem non-routine by blended learning more good from 
student who was given learning normal online. 
Next seen that score significance between group ability beginning math in class experimental 
and control is 0.308, and more big from level confidence 0.05. This show that H o accepted. 
So that could concluded that no there is interaction Among learning with ability beginning to 
ability think critical mathematical student. And, by chart could seen that ability think critical 
student who was given learning with based realistic approach problem non-routine more tall 
from every group on learning normal online. Finding this strengthen results study ability 
think critical mathematical student school advanced first, show that which is given learning 
mathematics with more realistic approach online _ good from learning what teachers usually 
do at school that. 

Analysis of each capability indicator solving problem between group class experiment 
with class control. 

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T

able 

5. 

Resul

ts of 

statis

tical calculations on capability indicators solution problem. 

 
 

 
 
 

 

 

From table 5 above seen that for each indicator of understanding problem, planning 
completion and implementation plan solution ability problem _ solution problem 
mathematics have score significance more than level confidence 0.050, then could declared 
that H o rejected. This show that ability understanding problem, planning completion and 
implementation plan solution problem mathematics student class experiment different with 
student control class against theory geometry. Because of the graphics ability understanding 
problem , planning completion and implementation plan solution problem mathematics 
student in class experiment more tall from control class, then could said that ability 
understanding problem, planning completion and implementation plan solution problem 
mathematics student to field geometry student who was given learning with based realistic 
approach problem non-routine by blended learning more good from student who was given 
learning normal online . Meanwhile, for the looking back indicator, the value of the 
significance is 0.001 and more small from tarap significance 0.05 then H o accepted. This 
thing show that ability check return solution mathematical student to given geometry material 
_ _ learning with based realistic approach problem non-routine by same blended learning just 
with student who was given learning normal online. 

Analysis of each capability indicator think critical between group to second class 
experiment and control. 

Table 6. Results of statistical calculations on indicators of ability think critical . 

Indicator Sum of Squares df Mean Square F Sig. 
understanding 

28,409 1 28,409 3,713 .057

Planning 
4,545 1 4,545 .598 .441

Doing 
1,136 1 1,136 .052 .821

Looking back 501.136 1 501.136 11,670 .001

Indicator Sum of Squares df Mean Square F Sig. 
Connection  .284 1 .284 .094 .760 

Exploration 10,227 1 10,227 1.358 .247

Generalization .284 1 .284 .015 .904

Clarification 92.045 1 92.045 6,100 .015

Solution 955,682 1 955,682 24,157 .000

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From table 6 above seen that for each indicator think critical; connect, describe, attract 
conclusion and clarify student show that class experiment have score significance more tall 
from level confidence 0.050, then could declared that H o rejected. This show that ability 
connect, describe, attract conclusion and clarify solution student class experiment different 
from student control class against Theory geometry. Due to the average ability connect, 
describe, attract conclusion and clarify student in class experiment more than control class, 
then could said that ability connect, describe, attract conclusion and clarify student to field 
geometry student who was given learning with based realistic approach problem non-routine 
by blended learning more than from student who was given learning normal online. 
Meanwhile, for the indicator to complete score the significance is 0.000 and more than 
significance value 0.05 then H o accepted. This thing show that ability complete student to 
field given geometry _ learning with based realistic approach problem non-routine by same 
blended learning just with student who was given learning normal online. 

Conclusion 

1. There is difference ability solution problem field geometry among student who was 
given learning with based realistic approach problem non-routine by blended learning 
with student who was given learning normal online. By general could said that ability 
complete problem geometry student who was given based realistic approach problem 
non-routine by blended learning more good from student who was given learning normal 
online. This thing showed that 3 out of 4 indicators ability solution problem geometry 
student that is ability understand, plan and complete problem student who was given 
based realistic approach problem non-routine by blended learning more than student who 
was given learning normal online. Only indicator check return from ability solving 
problem same geometry _ among student who was given learning with based realistic 
approach problem non-routine by blended learning with the given learning normal 
online. 
 

2. There is difference ability think critical Among student who was given learning with 
based realistic approach problem non-routine by blended learning with student who was 
given learning normal online. By general could said that ability think critical geometry 
student who was given based realistic approach problem non-routine by blended learning 
more than student who was given learning normal online. This thing showed that 4 out of 
5 indicators ability think critical student that is ability connect, describe, attract 
conclusion and clarify solution student who was given based realistic approach problem 
non-routine by blended learning more than student who was given learning normal 
online. And, only indicator completes from ability think critical same geometric _ among 
student who was given learning with based realistic approach problem non-routine by 
blended learning with student who was given learning normal online. 

 

3. Not there is interaction Among learning with ability beginning to ability complete 
problem nor ability think critical student. With so, can concluded that in increase ability 
solution trouble geometry student nor think critical geometric student through based 
realistic approach problem non-routine by blended learning no need to do grouping on 
ability student low, medium nor high. 

 

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	Introduction

