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Instructional Practices of Teachers in Integrating Critical Thinking and Problem Solving Skills in Mathematics Instruction

This qualitative research aimed to document the instructional practices, problems encountered, and suggestions offered by mathematics teachers in integrating critical thinking through problem-solving. To obtain the data, series of Focus Group Discussions (FGD) were conducted with seven (7) mathematics teachers and sixty (60) students in the Senior High School. Through the thematic analysis, four key themes were derived to present teachers’ teaching practices, problems encountered, and suggestions to intensify critical thinking promotion through problem-solving tasks. A Proposed Enhancement Plan for Mathematics Teachers was developed to hasten students’ exposure to activities that promote critical thinking and problem-solving. These activities involve enrichment sessions for mathematics teachers the problem-solving and refinement of instructional plans, particularly on strategies, assessment techniques, and instructional materials.


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Published by SCHOLINK INC. Polya (1940) presents a four-way linear iterative process in solving problems, which include the See, Plan, Do, and Check phases. Figure 1 presents the paradigm in which the study was anchored.

INPUT
PROCESS OUTPUT

Figure 1. The Paradigm of the Study
As its inputs, the study considered the 21 st Century Framework, K-12 Mathematics Framework and Curriculum, relevant theories on critical thinking and problem-solving, and participants' responses to the focus group discussions. The frameworks and theories served as supporting literature to analyze the participants' responses to the focus group discussions. The study outputs are the baseline information on the teachers' teaching practices, problems encountered, and suggestions for promoting critical thinking and problem-solving skills. The study further endeavored to develop an intervention plan to enhance students' skills as defined in the mathematics framework.

Statement of the Problem
This study aimed to document instructional practices as well as the problems encountered, and suggestions offered by teachers and students to promote the acquisition of critical thinking and problem-solving skills. Results were used as bases for a proposed mathematics teaching-learning intervention plan.
More specifically, the study aimed to answer the following sub-problems: 1.
What are the instructional practices of mathematics teachers as regards the promotion of critical thinking and problem-solving skills?

2.
What are the problems encountered by the teachers and students in the integration of critical thinking and problem-solving skills?

3.
What do students and teachers suggest to enhance the integration of critical thinking and problem-solving skills?

4.
What proposed teaching-learning intervention plan can be implemented for mathematics instruction to strengthen students' critical thinking skills through problem-solving?

Research Design
This qualitative research utilized the descriptive research design. It documented the instructional practices, difficulties encountered, and teachers' suggestions to integrate critical thinking skills and problem-solving in mathematics instruction.

Participants of the Study
The study participants include the population of mathematics teachers and a cluster sample of students (n=60) at St. Paul University Philippines handling mathematics subjects for the first and second semesters of 2017-2018. There were seven (7) teachers who were involved in the study. Of the 20 sections of G-12 students, there were 3 students taken for each section, thus, the study covered 60 students for the FGD. One FGD session was conducted for teachers and 6 FGD sessions for the students. Each FGD cluster covered 10 members.

Instrumentation
The study utilized the Focus Group Discussion (FGD) Guide to elicit information on the teachers' teaching practices, difficulties encountered, and suggestions to facilitate the integration of critical thinking and problem-solving skills in mathematics instruction.

Data Gathering Procedure
To obtain the data needed for the investigation, the following procedures were undertaken by researchers: 1. The researcher sought permission and endorsed by the principal of the Basic Education Unit (BEU).
2. Upon the principal's approval, the researcher met with the participants, sought their consent, and conducted the Focus Group Discussion (FGD).

Data Analysis Tools
The thematic analysis was used to organize participants' responses in the FGD, particularly on the teaching practices, the difficulties encountered, and suggestions on integrating critical thinking and problem-solving skills in mathematics instruction.

Instructional Practices in the Integration of Critical Thinking and Problem-Solving in Mathematics
Instruction

Students' Suggestions on the Integration of Critical Thinking and Problem-Solving Skills in
Mathematics Instruction dealing with mathematical problems, sustain the use of integrative approach and discovery method, maintain the practice of promoting students' understanding and mastery of concepts, and use appropriate instructional materials.
Furthermore, the students suggest that teachers should establish a meaningful connection among mathematics concepts, consider individual assessments for group work, use innovative strategies to enhance students' level of critical thinking, use appropriate instructional materials for visualization purposes and model the logical process in solving problems.

Proposed Teaching-Learning Plan to Promote Critical Thinking through Problem Solving
A proposed teaching-learning plan was conceptualized to hasten students' exposure to instructional activities that promote critical thinking and problem-solving. This proposed plan was based on the extent of students' exposure to content, strategies, assessment, and instructional materials that promote critical thinking; students' level of critical thinking based on their solution to problems; teaching practices, difficulties encountered, and suggestions offered by students and teachers to intensify the promotion of critical thinking through a problem-solving approach.

Instructional Practices in the Integration of Critical Thinking and Problem-Solving in Mathematics
Instruction

Content
With regards to content, teachers promote the establishment of connections between and among mathematics concepts in problem-solving situations. Emphasis on the connection between previous and current topics, current topics with other mathematics-related concepts, and current topics with other related disciplines are reflected in the problem-solving scenarios explored during instruction. In this manner, students can see a meaningful link among mathematics concepts. Moreover, the teachers present logically organized mathematics content. Lessons are carefully planned to reflect the logical order of content presentation such that prerequisite topics are rehearsed first before the presentation of new topics, problems to illustrate the application of the topics are presented in an easy-difficult or simple-complex continuum. Furthermore, teachers consider problem-solving as an integral part of the content. The integration of problem-solving in the content presentation is done by presenting problems that shall serve as a springboard in generating the lesson content.

Instructional Strategies
Notable teaching practice of teachers along instructional strategies is the use of Problem Solving as a strategy in promoting Critical Thinking. Teachers mainly apply problem-solving strategies in teaching mathematics. The problem-solving approach refers to the teaching technique where teachers use problems to provide students with rich experiences that allow them to see practical situations that embody the explored mathematics content. Using a problem-solving strategy exposes students in learning activities that would enable them to utilize their higher-order thinking skills. This strategy also promotes rigorous interaction in class, allowing students to brainstorm, share and consolidate their ideas, and evaluate other students' ideas or solutions to given problems. Along with students' exposure to problem-solving is the modeling of the Polya's Plan of Attack and various techniques in Problem Solving. Problem Solving strategies during the instructional process were either carried out individually or in groups.

Assessment
Among the teachers' assessment practices is the use of group and individual assessments. Group assessment was used to allow students to share and evaluate every member of the group's ideas. The individual assessment was used to train students for independent learning. Also, a feedbacking scheme was used in the assessment process. This includes an evaluation of students' solutions to problems.
This practice provides feedback to students regarding their answers to the problems. Moreover, the use of rubrics facilitates rating students' performance in problem-solving. Through rubrics, teachers can easily determine what students can do and cannot do. Rubrics generate data to show evidence of their students' mathematical thinking and how they use that evidence to monitor students' progress and guide instructional decision making (Thompson & Senk, 1998).
Problem posing activities were also undertaken during the instructional process. Through this, students can create or construct problems that allow them to provide problem situations that are real to them, making learning more meaningful. Students' critical thinking skills are acquired as they must think of problem scenarios that fit the given content, creating strategies for students to understand the value of and schemes for posing problems on their own (Silver, 1994;El Sayed, 2002).
Creative projects were also required for mathematics assessment. These projects train students to solve practical problems that allow them to gather pertinent data (i.e., perform actual measurements (Cherif & Gialamas, 2000;Huckstep & Rowland, 2000).

Instructional Materials
The teachers maximize the use of Mathematics Resources such as scholarly sources (i.e., books, journals and mathematics websites). These materials are rich sources of problems to illustrate varied mathematics content topics (Hohenwarter, Hohenwarter, Kreis, & Lavicza, 2008). Teachers also practiced the use of visualization materials (Makina, 2010). Manipulatives were used to help students visualize problems (Carbonneau, Marley, & Selig, 2013;Sarama & Clements, 2009;Clements & McMillen, 1996;Uttal, Scudder, & DeLoache, 1997). Mathematics software was also used for problems involving graphs (Kilicman, Hassan, & Husain, 2010;Zengin, 2017). The use of calculators is also a notable practice of teachers when giving problems involving complex and tedious computations. Allowing the use of calculators in the classroom eliminates students' burden of doing lengthy calculations. Using a calculator gives them adequate time to process their thoughts and ideas to come up with an accurate answer (Gómez, 1998;Wheatley, 1980;Thomas, Hong, Bosley, & Delos Santos, 2006). Teachers' use of worksheets and activity sheets were also commended. The worksheet is an excellent resource for practicing problem-solving as this provides ready-at-hand problems of varying contexts and difficulty levels (Toumasis, 1995;Zulyadaini, 2017).

Difficulties Encountered by Teachers in the Integration of Critical Thinking and Problem-Solving in Mathematics Instruction
Among the students' difficulties encountered by the students were related to the nature of Mathematics lessons. These are their difficulty to visualize abstract mathematics, difficulty to recall formulas, properties, rules, and algorithms; difficulty in dealing with numerical and algebraic expressions; difficulty in establishing the connection between and among mathematics topics; trouble in understanding Mathematics language; and poor mastery of the basic mathematics. When dealing with problem tasks, students find difficulty understanding mathematics terms used in problem-solving tasks; and students struggle to deal with problems involving many mathematics concepts. In the assessment area, students find some problems as very difficult to tackle: students consider unsupervised group problem-solving assessment as difficult to accomplish; students find problem-solving difficult when working within a limited time allotment, and students find difficulty in dealing with too long problem sets.
For instructional aids, students are not skillful in using the calculator.
Teachers' difficulty is traced from the nature of students (i.e., handling students with poor basic mathematics foundation and conceptual understanding, with varying instructional exposure in their basic education, with poor comprehension skill, who were used to routine problems, and those who lack skills in using the calculator); nature of mathematics (i.e., mathematics concepts are abstract and hard to contextualize, broad coverage of mathematics content, and limited time for deepening and remediation activities); and instructional materials (i.e., shortage of instructional materials to concretize mathematics concepts, and little reading resources to substantiate lessons).

Mathematics Instruction
Among the students' suggestions in integrating critical thinking in mathematics are as follows: Teachers should strive to teach Mathematics with meaning and establish a meaningful connection between and among mathematics concepts, relate terms to the vernacular, and find time to activate prior knowledge. Students further suggest that group work must be evaluated both as a group and individually; and the use of oral and written assessment. As to instructional approaches and strategies, students suggest that: (1) Teachers should present problems from easy to hard tasks; (2) Problem-solving activities should be conducted to allow students to reason out; (3) Teachers should continue exploring more strategies to assist students in tackling difficult topics and problem tasks, and (4) Teachers must be critical in the selection of problems. Regarding the use of instructional materials, teachers should continue using appropriate materials to visualize concepts (Makina, 2010), provide interesting PowerPoint presentations, not just texts, and to continue modeling the logical processes and heuristics in problem-solving. Furthermore, students suggest for the increased time allotment in dealing with complicated problems, devote more time to process the lessons, promote student-engagement, accommodate students' questions or clarifications, promote student-teacher interactions, and conduct remediation activities.
Among the teachers' suggestions on the integration of critical thinking and problem-solving skills in mathematics instruction include the following: (1) Teachers should take initiatives in unlocking students' difficulties (i.e., promote advanced reading, activate prior knowledge, employ scaffolding activities); (2) Teachers must sustain the use of integrative teaching approach (e.g., relating topics with other topics, establish a connection among topics, and (3) Teachers must integrate interesting activities; maintain practices that promote students' understanding and mastery of the concepts (e.g., allowing students to explain their work, encouraging students to explore alternative solutions, giving sufficient examples/drill work for practice, promoting visualization strategies (Makina, 2010) and drawing problems from students' experiences (Kurniati, Kusumah,, Sabandar, & Herman, 2015); (4) Teachers must sustain the use of discovery method to train students to process their thinking; (5) Teachers should sustain the use of instructional materials to visualize mathematics concepts and substantiate lessons through online books and materials.

Proposed Teaching-Learning Plan to Promote Critical Thinking through Problem Solving
A proposed teaching-learning plan was conceptualized to hasten students' exposure to instructional activities that promote critical thinking and problem-solving. This proposed plan was based on the following: (1) extent of students' exposure to content, strategies, assessment, and instructional materials that promote critical thinking; (2) students' level of critical thinking based on their solution to problems; (3) teaching practices; and (4) the difficulties encountered, and suggestions offered by students and teachers to intensify the promotion of critical thinking through a problem-solving approach.

Conclusion and Recommendations
Based on the findings of the study, the following conclusion was drawn: Actively engaging students in problem-solving activities from the motivation phase to the assessment phase encourages students' critical thinking development. Constant exposure of students to problem-solving activities is an avenue for modeling the thinking process, which is considered an efficient way to promote the acquisition of critical thinking. Fostering critical thinking is made possible when students are engaged in meaningful and challenging problem-solving activities that require them to reflect, brainstorm, explore alternatives, evaluate, and make decisions.
From the results of the investigation, the following recommendations are generated: 1.
Mathematics organizations may consider the study's findings in conceptualizing seminars and conferences that will provide inputs on strategies, activities, and instructional tools that foster students' acquisition of critical thinking skills.

2.
Mathematics teachers at all levels must find time to model problem-solving heuristics and strategies in their mathematics classroom to acquaint the students with the process involved in problem-solving.

3.
Mathematics teachers handling G11 and G12 mathematics subjects through the principal's support will consider integrating the suggested activities in the proposed plan to promote critical thinking and problem solving among students.

4.
The proponent shall disseminate the results of the study to the identified beneficiaries for maximum utilization.

5.
Researchers may conduct more studies to explore more effective strategies in promoting critical thinking and problem-solving skills.