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## How To Encourage Critical Thinking in Math

By Mary Montero

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Critical thinking is more than just a buzzword… It’s an essential skill that helps students develop problem-solving abilities and make logical connections between different concepts. By encouraging critical thinking in math, students learn to approach problems more thoughtfully, they learn to analyze and evaluate math concepts, identify patterns and relationships, and explore different strategies for finding the solution. Critical thinking also involves a great deal of persistence. Those are critical life skills!

When you think about it, students are typically asked to solve math problems and find the answer. Showing their work is frequently stressed too, which is important, but not the end. Instead, students need to be able to look at math in different ways in order to truly grasp a complete understanding of math concepts. Mathematics requires logical reasoning, problem-solving, and abstract thinking.

## What Does Critical Thinking in Math Look Like?

When I think about critical thinking in math, I focus on:

- Solving problems through logical thinking . Students learn how to break down complex problems, analyze the different parts, and understand how they fit together logically.
- Identifying patterns and making connections. Students learn how to identify patterns across different math concepts, make connections between seemingly unrelated topics, and develop a more in-depth understanding of how math works.
- Evaluating and comparing solutions. Students learn to evaluate which solution is best for a given problem and identify any flaws in their reasoning or others’ reasoning when looking at different solutions

## Mathematician Posters

These FREE Marvelous Mathematician posters have been a staple in my classroom for the last 8+ years! I first started using a version from MissMathDork and adapted them for my classroom over the years.

I print, laminate, and add magnetic stickers on the back. At the beginning of the year, I only put one or two up at a time depending on our area of focus. Now, they are all hanging on my board, and I’ll pull out different ones depending on our area of focus. They are so empowering to my mathematicians and help them stay on track!

A Marvelous Mathematician:

- knows that quicker doesn’t mean better
- looks for patterns
- knows mistakes happen and keeps going
- makes sense of the most important details
- embraces challenges and works through frustrations
- uses proper math vocabulary to explain their thinking
- shows their work and models their thinking
- discusses solutions and evaluates reasonableness
- gives context by labeling answers
- applies mathematical knowledge to similar situations
- checks for errors (computational and conceptual)

## Critical Thinking Math Activities

Here are a few of my favorite critical thinking activities.

## Square Of Numbers

I love to incorporate challenge problems (use Nrich and Openmiddle to get started) because they teach my students so much more than how to solve a math problem. They learn important lessons in teamwork, persistence, resiliency, and growth mindset. We talk about strategies for tackling difficult problems and the importance of not giving up when things get hard.

This square of numbers challenge was a hit!

ALL kids need to feel and learn to embrace challenge. Oftentimes, kids I see have rarely faced an academic challenge. Things have just come easy to them, so when it doesn’t, they can lack strategies that will help them. In fact, they will often give up before they even get started.

I tell them it’s my job to make sure I’m helping them stretch and grow their brain by giving them challenges. They don’t love it at first, but they eventually do!

This domino challenge was another one from Nrich . I’m always on the hunt for problems like this!! How would you guide students toward an answer??

## Fifteen Cards

This is a well-loved math puzzle with my students, and it’s amazing for encouraging students to consider all options when solving a math problem.

We have number cards 1-15 (one of each number) and only seven are laid out. With the given clues, students need to figure out which seven cards should be put out and in what order. My students love these, and after they’ve done a few, they enjoy creating their own, too! Use products, differences, and quotients to increase the challenge.

This is also adapted from Nrich, which is an AMAZING resource for math enrichment!

This is one of my favorite fraction lessons that I’ve done for years! Huge shout out to Meg from The Teacher Studio for this one. I give each child a slip of paper with this figure and they have to silently write their answer and justification. Then I tally up the answers and have students take a side and DEBATE with their reasoning! It’s an AMAZING conversation, and I highly recommend trying it with your students.

Sometimes we leave it hanging overnight and work on visual models to make some proofs.

## Logic Puzzles

Logic puzzles are always a hit too! You can enrich and extend your math lessons with these ‘Math Mystery’ logic puzzles that are the perfect challenge for 4th, 5th, and 6th grades. The puzzles are skills-based, so they integrate well with almost ANY math lesson. You can use them to supplement instruction or challenge your fast-finishers and gifted students… all while encouraging critical thinking about important math skills!

Three levels are included, so they’re perfect to use for differentiation.

- Introductory logic puzzles are great for beginners (4th grade and up!)
- Advanced logic puzzles are great for students needing an extra challenge
- Extra Advanced logic puzzles are perfect for expert solvers… we dare you to figure these puzzles out!

Do you have a group of students who are ready for more of a fraction challenge? My well-loved fraction puzzlers are absolutely perfect for fraction enrichment. They’ll motivate your students to excel at even the most challenging tasks!

## Math Projects

Math projects are another way to differentiation while building critical thinking skills. Math projects hold so much learning power with their real-world connections, differentiation options, collaborative learning opportunities, and numerous avenues for cross curricular learning too.

If you’re new to math projects, I shared my best tips and tricks for using math projects in this blog post . They’re perfect for cumulative review, seasonal practice, centers, early finisher work, and more.

I use both concept-based math projects to focus on specific standards and seasonal math projects that integrate several skills.

## Error Analysis

Finally, error analysis is always a challenging way to encourage critical thinking. When we use error analysis, we encourage students to analyze their own mistakes to prevent making the same mistakes in the future.

For my gifted students, I use error analysis tasks as an assessment when they have shown mastery of a unit during other tasks. For students in the regular classroom needing enrichment, I usually have them complete the tasks in a center or with a partner.

For students needing extra support, we complete error analysis in small groups. We go step-by-step through the concept and they are always able to eventually identify what the error is. It is so empowering to students when they finally figure out the error AND it helps prevent them from making the same error in the future!

My FREE addition error analysis is a good place to start, no matter the grade level. I show them the process of walking through the problem and how best to complete an error analysis task.

When you’re ready for more, this bundle of error analysis tasks contains more than 240 tasks to engage and enrich your students in critical thinking practice.

If you want to dig even deeper, visit this conceptual vs computational error analysis post to learn more about using error analysis in the classroom.

## Related Critical Thinking Posts

- How to Increase Critical Thinking and Creativity in Your “Spare” Time
- More Tips to Increase Critical Thinking

Critical thinking is essential for students to develop a deeper understanding of math concepts, problem-solving skills, and a stronger ability to reason logically. When you learn how to encourage critical thinking in math, you’re setting your students up for success not only in more advanced math subjects they’ll encounter, but also in life.

How do you integrate critical thinking in your classroom? Come share your ideas with us in our FREE Inspired In Upper Elementary Facebook group .

## Mary Montero

I’m so glad you are here. I’m a current gifted and talented teacher in a small town in Colorado, and I’ve been in education since 2009. My passion (other than my family and cookies) is for making teachers’ lives easier and classrooms more engaging.

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## One Comment

Mary Thankyou for your inspirational activities. I have just read and loved the morning talk activities. I do have meetings with my students but usually at end of day. What time do you

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## 20 Math Critical Thinking Questions to Ask in Class Tomorrow

- November 20, 2023

The level of apathy towards math is only increasing as each year passes and it’s up to us as teachers to make math class more meaningful . This list of math critical thinking questions will give you a quick starting point for getting your students to think deeper about any concept or problem.

Since artificial intelligence has basically changed schooling as we once knew it, I’ve seen a lot of districts and teachers looking for ways to lean into AI rather than run from it.

The idea of memorizing formulas and regurgitating information for a test is becoming more obsolete. We can now teach our students how to use their resources to make educated decisions and solve more complex problems.

With that in mind, teachers have more opportunities to get their students thinking about the why rather than the how.

Table of Contents

## Looking for more about critical thinking skills? Check out these blog posts:

- Why You Need to Be Teaching Writing in Math Class Today
- How to Teach Problem Solving for Mathematics
- Turn the Bloom’s Taxonomy Verbs into Engaging Math Activities

## What skills do we actually want to teach our students?

As professionals, we talk a lot about transferable skills that can be valuable in multiple jobs, such as leadership, event planning, or effective communication. The same can be said for high school students.

It’s important to think about the skills that we want them to have before they are catapulted into the adult world.

Do you want them to be able to collaborate and communicate effectively with their peers? Maybe you would prefer that they can articulate their thoughts in a way that makes sense to someone who knows nothing about the topic.

Whatever you decide are the most essential skills your students should learn, make sure to add them into your lesson objectives.

## When should I ask these math critical thinking questions?

Critical thinking doesn’t have to be complex or fill an entire lesson. There are simple ways that you can start adding these types of questions into your lessons daily!

## Start small

Add specific math critical thinking questions to your warm up or exit ticket routine. This is a great way to start or end your class because your students will be able to quickly show you what they understand.

Asking deeper questions at the beginning of your class can end up leading to really great discussions and get your students talking about math.

## Add critical thinking questions to word problems

Word problems and real-life applications are the perfect place to add in critical thinking questions. Real-world applications offer a more choose-your-own-adventure style assignment where your students can expand on their thought processes.

They also allow your students to get creative and think outside of the box. These problem-solving skills play a critical role in helping your students develop critical thinking abilities.

## Keep reading for math critical thinking questions that can be applied to any subject or topic!

When you want your students to defend their answers.

- Explain the steps you took to solve this problem
- How do you know that your answer is correct?
- Draw a diagram to prove your solution.
- Is there a different way to solve this problem besides the one you used?
- How would you explain _______________ to a student in the grade below you?
- Why does this strategy work?
- Use evidence from the problem/data to defend your answer in complete sentences.

## When you want your students to justify their opinions

- What do you think will happen when ______?
- Do you agree/disagree with _______?
- What are the similarities and differences between ________ and __________?
- What suggestions would you give to this student?
- What is the most efficient way to solve this problem?
- How did you decide on your first step for solving this problem?

## When you want your students to think outside of the box

- How can ______________ be used in the real world?
- What might be a common error that a student could make when solving this problem?
- How is _____________ topic similar to _______________ (previous topic)?
- What examples can you think of that would not work with this problem solving method?
- What would happen if __________ changed?
- Create your own problem that would give a solution of ______________.
- What other math skills did you need to use to solve this problem?

## Let’s Recap:

- Rather than running from AI, help your students use it as a tool to expand their thinking.
- Identify a few transferable skills that you want your students to learn and make a goal for how you can help them develop these skills.
- Add critical thinking questions to your daily warm ups or exit tickets.
- Ask your students to explain their thinking when solving a word problem.
- Get a free sample of my Algebra 1 critical thinking questions ↓

## 8 thoughts on “20 Math Critical Thinking Questions to Ask in Class Tomorrow”

I would love to see your free math writing prompts, but there is no place for me to sign up. thank you

Ahh sorry about that! I just updated the button link!

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## Engaging Maths

Professor catherine attard, promoting creative and critical thinking in mathematics and numeracy.

- by cattard2017
- Posted on June 25, 2017

What is critical and creative thinking, and why is it so important in mathematics and numeracy education?

Numeracy is often defined as the ability to apply mathematics in the context of day to day life. However, the term ‘critical numeracy’ implies much more. One of the most basic reasons for learning mathematics is to be able to apply mathematical skills and knowledge to solve both simple and complex problems, and, more than just allowing us to navigate our lives through a mathematical lens, being numerate allows us to make our world a better place.

The mathematics curriculum in Australia provides teachers with the perfect opportunity to teach mathematics through critical and creative thinking. In fact, it’s mandated. Consider the core processes of the curriculum. The Australian Curriculum (ACARA, 2017), requires teachers to address four proficiencies : Problem Solving, Reasoning, Fluency, and Understanding. Problem solving and reasoning require critical and creative thinking (). This requirement is emphasised more heavily in New South wales, through the graphical representation of the mathematics syllabus content , which strategically places Working Mathematically (the proficiencies in NSW) and problem solving, at its core. Alongside the mathematics curriculum, we also have the General Capabilities , one of which is Critical and Creative Thinking – there’s no excuse!

Critical and creative thinking need to be embedded in every mathematics lesson . Why? When we embed critical and creative thinking, we transform learning from disjointed, memorisation of facts, to sense-making mathematics. Learning becomes more meaningful and purposeful for students.

How and when do we embed critical and creative thinking?

There are many tools and many methods of promoting thinking. Using a range of problem solving activities is a good place to start, but you might want to also use some shorter activities and some extended activities. Open-ended tasks are easy to implement, allow all learners the opportunity to achieve success, and allow for critical thinking and creativity. Tools such as Bloom’s Taxonomy and Thinkers Keys are also very worthwhile tasks. For good mathematical problems go to the nrich website . For more extended mathematical investigations and a wonderful array of rich tasks, my favourite resource is Maths300 (this is subscription based, but well worth the money). All of the above activities can be used in class and/or for homework, as lesson starters or within the body of a lesson.

Will critical and creative thinking take time away from teaching basic concepts?

No, we need to teach mathematics in a way that has meaning and relevance, rather than through isolated topics. Therefore, teaching through problem-solving rather than for problem-solving. A classroom that promotes and critical and creative thinking provides opportunities for:

- higher-level thinking within authentic and meaningful contexts;
- complex problem solving;
- open-ended responses; and
- substantive dialogue and interaction.

Who should be engaging in critical and creative thinking?

Is it just for students? No! There are lots of reasons that teachers should be engaged with critical and creative thinking. First, it’s important that we model this type of thinking for our students. Often students see mathematics as black or white, right or wrong. They need to learn to question, to be critical, and to be creative. They need to feel they have permission to engage in exploration and investigation. They need to move from consumers to producers of mathematics.

Secondly, teachers need to think critically and creatively about their practice as teachers of mathematics. We need to be reflective practitioners who constantly evaluate our work, questioning curriculum and practice, including assessment, student grouping, the use of technology, and our beliefs of how children best learn mathematics.

Critical and creative thinking is something we cannot ignore if we want our students to be prepared for a workforce and world that is constantly changing. Not only does it equip then for the future, it promotes higher levels of student engagement, and makes mathematics more relevant and meaningful.

How will you and your students engage in critical and creative thinking?

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- DOI: 10.4236/CE.2015.64045
- Corpus ID: 54507500

## Developing Critical Thinking Skills from Dispositions to Abilities: Mathematics Education from Early Childhood to High School

- Einav Aizikovitsh-Udi , Diana S. Cheng
- Published 24 March 2015
- Mathematics, Education
- Creative Education

## 150 Citations

Developing students critical thinking ability through lesson study.

- Highly Influenced

## Cognitive Growth Learning Model to Improve the Students’ Critical Thinking Skills

The effect of a problem centered learning on student’s mathematical critical thinking, the relationship between prospective middle school mathematics teachers’ critical thinking skills and reflective thinking skills, student understanding of derivative: mathematics education in the senior high school, supporting activities for critical thinking skills development based on students' perspective, critical thinking skills based on mathematical dispositions in problem-based learning, the students’ mathematical critical thinking skill ability in solving mathematical problems, an innovative model to promote secondary students' critical thinking skills in algebra learning, developing critical thinking skills of students in mathematics learning, 32 references, the change in mathematics teachers' perceptions of critical thinking after 15 years of educational reform in jordan, higher order thinking skills and low-achieving students: are they mutually exclusive, the disposition of eleventh-grade science students toward critical thinking, infusing the teaching of critical and creative thinking into content instruction: a lesson design handbook for the elementary grades, ' needed : research in critical thinking, teaching the language of thinking., “you're going to want to find out which and prove it”: collective argumentation in a mathematics classroom, education for critical thinking, practical strategies for the teaching of thinking, critical thinking and subject specificity: clarification and needed research, related papers.

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## How to Develop Critical Thinking Skills in Mathematics

## Introduction

In the realm of education, mathematics goes beyond memorizing formulas and solving equations. It is a discipline that fosters critical thinking, logical reasoning, and problem-solving abilities. Developing strong critical thinking skills in mathematics is not only essential for academic success but also for nurturing cognitive abilities that are applicable throughout life. In this article, we will explore the significance of critical thinking in math education and provide students with practical techniques to enhance their critical thinking prowess.

## The Significance of Critical Thinking in Math

Critical thinking is the ability to analyze, evaluate, and synthesize information to make well-reasoned decisions and solve problems effectively. In the context of mathematics, critical thinking is the driving force behind understanding concepts deeply, approaching problems from multiple angles, and devising innovative solutions. It encourages students to question assumptions, think logically, and develop a genuine understanding of mathematical principles rather than merely memorizing procedures.

## Analyzing Problems

Encourage students to approach mathematical problems with a critical eye. Emphasize the importance of reading problems carefully, identifying the given information, and understanding the context. Encourage them to break down complex problems into smaller, manageable parts, making connections between different components. This analytical approach lays the groundwork for a structured problem-solving process.

## Asking Questions

Incorporate a culture of curiosity in the math classroom. Encourage students to ask questions about concepts, procedures, and problem-solving approaches. Asking “why” and “how” questions fosters a deeper understanding of the subject matter and challenges students to think beyond the surface. Teachers can create an open and supportive environment where questioning is welcomed and encouraged.

## Making Logical Deductions

Students can enhance their critical thinking by making logical deductions from given information. Encourage them to draw inferences, make predictions, and explore relationships between mathematical concepts. Engage students in discussions that require them to justify their answers and provide evidence for their conclusions. This practice strengthens their ability to think logically and reason through mathematical challenges.

## Exploring Real-World Applications

Connect math concepts to real-world applications, illustrating how critical thinking is integral to problem-solving in everyday life. Show students how math is used in various professions, such as engineering, finance, and technology. Engaging with real-world scenarios helps students recognize the relevance of critical thinking in their future endeavors.

## Encouraging Collaborative Learning

Promote collaborative learning environments where students can engage in discussions, share ideas, and learn from one another. Collaborative activities challenge students to think critically, consider different perspectives, and defend their reasoning. Peer-to-peer interactions can foster a supportive learning community that encourages critical thinking and intellectual growth.

Critical thinking is a cornerstone of mathematics education, empowering students to become independent and proficient problem solvers. By analyzing problems, asking questions, and making logical deductions, students can develop their critical thinking skills and apply them to various academic and real-life scenarios. As educators, parents, and mentors, we play a pivotal role in nurturing these abilities in our students. By cultivating a culture of curiosity, fostering collaborative learning, and highlighting the relevance of critical thinking in everyday life, we can equip our students with invaluable skills that will serve them well beyond the boundaries of the math classroom.

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## Developing Critical Thinking Skills of Students in Mathematics Learning

2015, Journal of Education and Learning (EduLearn)

Critical thinking skills should be owned by students. Therefore, schools should be responsible to develop and evaluate critical thinking skills through teaching and learning process in schools. This study aims to identify the effects of mathematical learning modules based on problem-based learning to critical thinking skills at secondary school students in District of Bone. Assessment of critical thinking skills in mathematical problem solving non-routine includes three parts; the identification and interpretation of information, information analysis, and evaluate of evidence and arguments. This study involved a total of 68 students grade 12 science state secondary school (SMAN) in Bone District of South Sulawesi, Indonesia in academic year 2014-2015. The sample consists of 38 students in the city and 30 rural students. The design of the study was quasi experimental one group pretest-posttest. The data was analysed using the inferential t-test with SPSS 20.0 for windows. The study...

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The objective of this research is to analyze the twelfth graders' mathematics critical thinking skills using a mathematics learning model to stimulate fundamental critical thinking abilities of science courses in SMA Negeri, Pacitan Regency, East Java Province, Indonesia. This quasi-experimental research design was used in this study with one group posttest only design using multiple substantive posttests. The sample of 141 students from the total population of six public schools involving the twelfth graders of the natural sciences was selected through purposive sampling technique, data were taken through tests of students' critical thinking skills and interviews. The data analysis consists of five stages, namely an analysis of one sample t-test, an analysis of students' grades, an analysis of problem-solving stages, an analysis of critical thinking abilities indicators, and an analysis of mathematics critical thinking abilities indicators. The results showed (1) The re...

THE 2ND SCIENCE AND MATHEMATICS INTERNATIONAL CONFERENCE (SMIC 2020): Transforming Research and Education of Science and Mathematics in the Digital Age

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The ability to think critically is one of the goals of education in Indonesia by emphasizing students' ability to think and act uncritically for every element of life. This study uses content analysis of 28 articles published in the Scopus database from 2013 to 2023 regarding students' critical thinking skills in mathematics education as the main focus. This research reveals that 2020 saw a very significant increase in publications in Q2 and Q3. The Indonesian Education University is an institution that has a lot of cooperative relations with other institutions in this field. Among these publications, real material analysis is the most researched topic with RME as the most frequently used treatment. The dominant research design is quantitative. Tests and t-tests are the most commonly used data analysis instruments and techniques, respectively. There are various types of design, treatment, collection techniques and data analysis used, therefore future researchers are advised ...

JOHME: Journal of Holistic Mathematics Education

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Critical thinking is one of the most important issues in education. However, based on field observation results it is found that students have low critical thinking skills. One reason is that learning activities in the classroom do not foster students’ critical thinking skills. The purpose of this research study to enhance students’ critical skills by implementing the Problem-based Learning (PBL) method. The research subjects were 29 grade 7 students in a junior high school in Manado. The research method used was Classroom Action Research (CAR), conducted from September 12 to November 3, 2017. The instruments used were diagnostic tests, pre-tests and post-tests, observation sheets, student interviews, checklists by observers and students, and the researcher’s reflection journal. Data results were analyzed using the descriptive qualitative method. The results showed that the implementation of the PBL method was able to enhance students’ critical thinking skills in learning math with ...

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The synthesis of some empirical researches revealed that the implementation of problem-based learning (PBL) has a heterogeneous effect size on the mathematical critical thinking skills (MCTS) of primary school students in Indonesia. However, it seems that no researches were investigating the causative factors of the heterogeneity of primary students’ MCTS through PBL. Therefore, this meta-analysis research was employed to examine three potential factors that were PBL class capacity, PBL treatment duration, and mathematics topic predicted as the causative factors of the heterogeneity of primary students’ MCTS. The Q Cochrane test by using the Comprehensive Meta-Analysis (CMA) application was used to examine these potential factors. The synthesis of ten primary studies published in 2015 – 2020 and indexed by Scopus, Google Scholar, Sinta, and Web of Science, showed that PBL treatment duration and mathematics topic were the significant factors causing the heterogeneity of primary stude...

Dr. SANTHOSH AREEKKUZHIYIL

Mathematics has a prominent position in modern education. The objective of the study was to analyse the relationship between critical thinking and Mathematical problem solving, abstract reasoning and mathematical problem solving, critical thinking and abstract reasoning. The research design is cross sectional in nature. The study has been carried out on representative sample of 330 secondary school students included 128 boys and 202 girls in Kozhikode district. Survey Method has been employed to collect data and the sample was selected by stratified sampling technique. The result reveals that there is a significant relationship between critical thinking and mathematical problem solving, abstract reasoning and mathematical problem solving, critical thinking and abstract reasoning. And also found that there is a significant difference between boys and girls in their critical thinking, abstract reasoning and mathematical problem solving.

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This study aims to determine the effectiveness of the mathematics learning model to stimulate critical thinking on student mathematics learning outcomes compared to the direct learning model in class XII science students of senior high school in Pacitan. The type of research is quasi-experimental, with a population of all students of class XII IPA of SMA N 2 Pacitan. The sample consists of 28 students of class XII IPA-1 as an experimental class and class XII of IPA-2 of 28 students as a control class. The sampling technique was purposive random sampling. Data collection techniques using the test. The test is a student's initial ability test and a student's mathematics learning outcomes test. The analysis technique of the data used in this study is the t-test, while the analysis of prerequisite tests includes tests of normality and homogeneity tests. The results showed that mathematics learning using the mathematics learning model to stimulate critical thinking was more effec...

ANDREWS COBBINAH

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- John Kevin A. Artuz, Dennis B. Roble. Developing Students’ Critical Thinking Skills in Mathematics Using Online-Process Oriented Guided Inquiry Learning (O-POGIL). American Journal of Educational Research . Vol. 9, No. 7, 2021, pp 404-409. https://pubs.sciepub.com/education/9/7/2 ">Normal Style
- Artuz, John Kevin A., and Dennis B. Roble. 'Developing Students’ Critical Thinking Skills in Mathematics Using Online-Process Oriented Guided Inquiry Learning (O-POGIL).' American Journal of Educational Research 9.7 (2021): 404-409. ">MLA Style
- Artuz, J. K. A. , & Roble, D. B. (2021). Developing Students’ Critical Thinking Skills in Mathematics Using Online-Process Oriented Guided Inquiry Learning (O-POGIL). American Journal of Educational Research , 9 (7), 404-409. ">APA Style
- Artuz, John Kevin A., and Dennis B. Roble. 'Developing Students’ Critical Thinking Skills in Mathematics Using Online-Process Oriented Guided Inquiry Learning (O-POGIL).' American Journal of Educational Research 9, no. 7 (2021): 404-409. ">Chicago Style

## Developing Students’ Critical Thinking Skills in Mathematics Using Online-Process Oriented Guided Inquiry Learning (O-POGIL)

Mathematics is an essential subject in the Philippines Department of Education (DepEd) K-12 curriculum that requires critical thinking abilities because it is crucial in everyday life along with the growth of other sciences. However, the majority of Filipino students had difficulty understanding mathematical concepts that require higher-order cognitive skills. As a result, students' mathematical process abilities, such as critical thinking, must be strengthened. This study aimed to develop students' critical thinking in mathematics in an online-POGIL environment. The study was conducted to second-year Early Childhood Education students of Pangantucan Bukidnon Community College during the first term of the academic year 2020-2021. It utilized a quasi-experimental Pretest-Post Test Control Group research design. A validated 6-item teacher-made test for critical thinking skills with a reliability coefficient of 0.752 was used to measure students’ critical thinking performance with a rubric scale adapted from St. Petersburg College. Mean and standard deviation was used to evaluate students’ pretest and posttest scores in the critical thinking skills test. A one-way analysis of covariance (ANCOVA) was used to analyze the significant effect of O-POGIL on students’ critical thinking skills. The findings showed that O-POGIL motivates students to actively participate in activities and improves their learning. Furthermore, students exposed to O-POGIL showed a substantial improvement, demonstrated on communication, analysis, and problem-solving in mathematics. To improve students' critical thinking skills, it is advised to employ the O-POGIL teaching methodology and design resources based on the O-POGIL construct. Other 21st-century process skills that O-POGIL could build and enhance can be researched further.

## 1. Introduction

Mathematics is an indispensable subject of the Department of Education (DepEd) K-12 curriculum of the Philippines which necessitates the used of critical thinking skills and perceived as important in daily living as well as in the development of other sciences. However, studies in the context of Filipino student’s, reveals that majority of students excel only in knowledge acquisition but considerably low in understanding concepts which requires the use of their higher-order thinking skills (HOTS). This poor mathematics performance is evident in local, regional, national, and even international comparisons such as the National Achievement Test (NAT), Third International Mathematics and Science Study (TIMSS) and even the recent Programme for International Student Assessment (PISA) results and among others. These comparisons showed that Filipino students are underachievers in Mathematics. Braza and Supapo claimed that the shortcomings that can affect students’ achievements in Mathematics could be their lack of mastery of the basic concepts and skills, lack of problem solving and critical thinking skills, diverse behavior of students, and inappropriate teaching skills and approaches of teachers in dealing the students in the class of mathematics 1 .

Critical thinking is applied and use daily, specifically in decision making, systematic reasoning, analyzing information, and even in communicating ideas. Strengthening one's independent learning ability is paramount to this process skill. Proficiency with this skill is vital in enhancing fundamental knowledge in different learning areas like mathematics. Thus, this skill is important to be developed in order to successfully learn mathematics. Developing students’ critical thinking and problem-solving skills is an educational goal common to perhaps every academic program or discipline 2 . However, the above propositions and results show that students’ critical thinking skill is low and/or poor due to several factors. Irwanto, Prodjosantoso, Rohaete agree state that the low level of student’s critical thinking skills was caused by lecturer-centered learning 3 . Moreover, students tended to be trained to respond to problems by memorizing 4 which is a traditional practice in teaching mathematics. As a result, students have difficulty in solving problems that need reasoning and analyzing which are generally a component of critical thinking. With these, it is undeniable that there should be a modification as to how teachers should teach mathematics in order to enhance student’s critical thinking skills. Teachers in mathematics learning should facilitate students in developing critical thinking processes 5 . Developing critical thinking skills impacts educational fields, with education, not only providing information to the students but aiming to raise individuals who can think, examine and solve problems 6 .

One of the breakthrough methods in teaching mathematics today is the inquiry-based learning approach. Inquiry-based learning is a teaching method that facilitates asking questions, seeking information and finding new ideas related to an event 7 . One model of inquiry-based learning is the Process Oriented Guided Inquiry Learning (POGIL). It is a student-centered instructional strategy developed by chemical educators 8 for use in chemistry class 9 . POGIL is a collaborative learning technique that employs guided inquiry within a cyclic system of exploration, concept invention, and application 10 . This means that students under the POGIL method develop conceptual understanding collaboratively 8 . Additionally, in POGIL, students will conduct the activity themselves and teachers act as a guide to help them if needed 11 . POGIL shows a promising result on student’s critical thinking skills. As a strategy, it would enhance students’ problem-solving skills and critical thinking skills as ultimately evidenced by students’ improved performance 2 . However, previous researches show that there remains a gap as to how the POGIL method is not used to promote critical thinking skills; and although these skills have been developed in the previous decades, nevertheless the students’ performance needs to be improved satisfying level continuously 3 . Another teaching methodology which mathematics teachers can utilize to develop students’ critical thinking skill effectively is POLYA’s Problem Solving Method. As one of the fundamental goals of teaching mathematics, problem-solving has been pushed to be the central focus of the school mathematics curriculum. Solving a problem can be used to stimulate the skills of high-order thinking skills 12 . Thus, the application of problem-solving necessitates one to think critically and creatively and it is a systematic process 13 . In addition, problem-solving is about how to learn independently 14 . Polya’s Problem Solving Method follows four stages; Understanding the Problem, Devising a Plan, Carrying out the Plan, and Looking back 15 . This process helps the students to easily define the problem and create a bigger picture of the problem leading them to formulate and apply appropriate methods to solve the problem and check the solution. This methodology along with its positive effect on students’ critical thinking skills makes it an ideal approach in teaching mathematics. However, in reality, many students are still struggling to solve mathematical problems 16 . The effect of this is that students were found to be deficient in cognitive and critical thinking skills when they are faced with situations where they are expected to apply what they have learned to solve the specific problems 17 .

Today, technological breakthroughs change many aspects of human endeavor. As the world experience a rapid shift to a “new normal” due to COVID-19, this global pandemic opted many educational institutions and researchers both local and international to utilize technological innovation to maintain student learning and progress. This paved way for educators to analyze innovative approaches and implement appropriate methods that will actively engage students towards learning and acquiring 21st-century skills even amidst the threat of COVID-19. Concomitant to this, during this “new normal”, mathematics teachers teach students in order for the latter to acquire essential information and develop academically while actively strengthening their critical thinking and problem-solving skills. Hence, this scenario also prompted the researcher to shift to investigate the effect of POGIL on students’ critical thinking in mathematics in an online setting.

## 2. Literature Review

Studies and researches that support the current study are considered as regards to guided-inquiry learning, POGIL, O-POGIL, critical thinking skills in mathematics of students.

Teachers nowadays, resort to the advantages of digital learning or online learning method to improve students’ motivation. In the study conducted by Ichinose Bonsangue on their study on students’ mathematical self-related beliefs in an online mathematics course, they confirmed the link between success in an online learning environment and students’ beliefs in and motivation to achieve in an online environment for some, but not all, students 18 . Through instruction and content support, they suggested that online teachers and instructional designers must continue to create and implement experiences that will foster student beliefs and motivation that can accommodate students’ collective as well as individual experiences. With the use of 21st-century learning technologies, college instructors can create settings that promote challenging mathematics in a safe online learning environment 19 .

Online learning considerably supports access and provides opportunities not contained in the four corners of the classroom. The study of Ahn & Edwin introduced a mathematical e-learning model suitable for the modern digital era based on the learning theories of social constructivism, social realism, and connectivity. They proposed a mathematical e-learning model MCIEC (motivation, context, interactivity, evaluation, and connectivity), for making mathematical learning more interesting, meaningful, and applicable to the learners beyond the classroom knowledge. They argued that the teaching of mathematics beyond the primary level in most developing countries like the Philippines mainly emphasizes preparing students for high-stake national exams rather than linking the content to real-life problem-solving skills. To overcome these challenges, the MCIEC model emphasizes a flexible approach to teaching mathematics in which motivation, context, and dynamic evaluation are the backbone of any content design or delivery. The model places greater responsibility on the teachers to be more innovative and create materials that suit the learners’ abilities and environment. It is easier for the student to put in much effort to understand the mathematics in the content once the interest, motivation, and context have been attained 20 .

Researches on the effectiveness of Inquiry-Based Learning have been widely conducted as a factor in improving students’ learning performance. Ali Abdi studied the effect of the Inquiry-Based Learning method (specifically the 5E learning method) on students’ academic performance in Science courses. It was a quasi-experimental study with non-equivalent groups, which includes a pretest and posttest design with the control group. The study was conducted with 20 experimental and 20 control group girl students which utilized am Academic Achievement Test to both groups. The study concluded that there is a significant difference between the achievement of the students who have been educated by the inquiry-based instruction-supported 5E learning method and the students who have been educated by the traditional teaching methods. It then follows that the students who have been educated by the inquiry-based instruction supported 5E learning method have become more successful than the students who have been educated by the traditional teaching method 21 . Moreover, Stender, Schwichow, Zimmerman, & Härtig findings provide evidence that students can indeed learn new content knowledge by using inquiry skills to answer research questions 22 .

Rosadi, Sunarno & Maridi investigates the effectiveness of Process Oriented Guided Inquiry Learning to improve students’ analytical thinking skills on excretory topics. The subjects of the research were 60 11th grade students that were chosen by Simple Random Sampling. Moreover, the students’ analytical thinking skills were measured by a test instrument consisting of 30 multiple-choice questions. The study concluded that the developed POGIL method is effective to be used in teaching and learning to improve the students’ analytical thinking skills 23 .

The innovative teaching strategy is also a major factor in the study of Andrianni et al., in a study about the effect of the POGIL model on students’ logical thinking ability in mathematics. T-Test was used to investigate the effect of the independent variable towards the dependent variable which results stated that the students’ logical thinking ability who obtained POGIL is better than students’ logical thinking ability who obtained conventional learning model 24 . In addition, Sen stated that POGIL has a positive effect on students’ alternative conception. Thus, the students taught in the POGIL method had less misconception than those students who were taught using the conventional learning model 25 .

Another literature stated a positive impact of POGIL in the context of education. Soltis et al. conducted a study about enhancing students’ higher-level thinking skills in a pharmaceutical sciences course through the implementation of POGIL. It founds out that the use of the POGIL strategy had an overall positive effect on student learning and the classroom environment. Furthermore, it has been stated that students’ critical thinking skills and problem solving were improved with the use of the POGIL strategy 2 .

Irwanto, Saputro, Rohaeti, & Prodjosantoso, conducted a study in promoting critical thinking and problem-solving skills of 48 pre-service elementary teachers through Process Oriented Guided Inquiry Learning. The research consists of the Critical Thinking Essay Test (CTET) consisting of 5 items and the Problem Solving Essay Test (PSET) consisting of 4 items. The research used non-parametric statistics for quantitative data analysis, descriptive were used to describe the characteristics of the participants, Mann-Whitney U-Test and Spearman’s rho correlation was performed to explore the correlation between the dependent variables. The study emphasized that the POGIL method is more effective in improving students’ critical thinking and problem-solving skills. Additionally, the researchers, claimed that the high score of the experimental group students is related to course activities designed to teach content and engage students in analyzing data, discussing ideas, making a conclusion, and building their own knowledge through teamwork in accordance with the inquiry approach principles 3 .

However, a study by Douglas & Chiu argued and identified that students did not recognize the benefits of working in groups, such as promoting critical thinking, learning cooperative skills, gaining a different perspective, and retaining content knowledge. This statement could further denote that POGIL, as a group-based strategy, may not promote critical thinking skills and problem-solving skills 26 .

The research aims to describe the level of students’ creative thinking skill in solving problems of mathematics Olympics based on the problem-solving Polya and Krulik-Rudnick model. This descriptive research employed a qualitative approach. The research participants were 27 students at State Junior High School (SMPN) on 2 Jember involving the guidance of Olympiad mathematics in the academic year 2017-2018. The data were analyzed by using the Miles and Huberman model. The findings indicated that the level of students’ thinking skills based on the stages of the problem-solving model of Polya was in the category “sufficient creative. In addition, the research results indicated that the problem-solving stage of the Polya model was not always suitable to be used to solve all types of mathematics Olympiad problems. Indeed, there are advantages and disadvantages to each of them. Problem-solving Polya models are well suited to the problems of the type matter of routine while solving the Krulik-Rudnick model was more suitable for problems whose type was non-routine, but when all the students were using the stages of solving the model Krulik-Rudnick, there were still shortcomings experienced by students, namely how to find the idealized early when faced with-math Olympiad problems 27 .

As per a review of the studies cited, the researcher decided to do further work on extending the POGIL approach in an online setting in improving critical thinking skills in mathematics as Ichinose Bonsangue & Ahn & Edwin suggested and emphasized to create setting that will encourage learning engagement and motivation. Ali Abdi, & Stender, Schwichow, Zimmerman, & Härtig iterated the learning success of students exposed to inquiry-based learning. Rosadi, Sunarno & Maridi, Andrianni et al., Sen Andrianni et al., & Soltis et al., stated the positive effect of POGIL in higher-order thinking abilities of students. Moreover, Irwanto, Saputro, Rohaeti, & Prodjosantoso claimed that POGIL is effective in improving students’ critical thinking skills. However, none of these researchers studied the effect of POGIL applied in an online setting. Douglas & Chiu also identified that POGIL may not promote critical thinking skills. This became an illuminating factor for the researcher to conduct a study on Online-POGIL and its effect on students’ critical thinking skills.

This study employed a quasi-experimental Pretest-Post Test Control Group design. Mean and standard deviation was used to evaluate the pretest and posttest scores of students in the critical thinking skill test and a one-way ANCOVA was used to account for the variation and significant effect of O-POGIL on students’ critical thinking skills.

The respondents were second-year Bachelor in Early Childhood Education students of Pangantucan Bukidnon Community College. A total of 64 students participated in the study. These students are taking Mathematics in the Modern World course for the 1st semester of the S.Y. 2020-2021. The respondents were randomly assigned as control and experimental group. The control group was exposed to Online Problem Solving Method while the latter was exposed to Online Process Oriented Guided Inquiry Learning (O-POGIL).

A 6-item teacher-made test for mathematics critical thinking questionnaire was utilized in this study. This 6-item critical thinking skills test aimed to assess students’ critical thinking skills which follow students’ ability to highlight the primary idea or problem and support it with several details using inductive and deductive reasoning that is organized logically and coherently. The instrument also measures students’ ability to identify key aspects of the problem using relevant evidence and logically supports viable solutions. In addition, the test was analyzed and undergone a reliability test for which it obtained a reliability coefficient of 0.752. The Assessment Rubric for Critical Thinking Skill designed by St. Petersburg College was adapted to rate students’ critical thinking skills. The said rubric features 3 levels of critical thinking constructs with 5 levels of mastery or rating scale were used to score students’ pretest and posttest.

During the first day of the conduct of the study, the researcher conducted a pretest for both groups. It was followed by an orientation about the functions and applications of features of the ZOOM Meeting App. Moreover, students in both the control and experimental group were divided into subgroups with 4 members each having a distinct role; leader, writer, timer, and presenter. Both groups had a twice a week 90-minutes online session via ZOOM. Each online session was divided into two parts; the group online exploration phase and the virtual reporting of the groups’ output.

The experimental group was exposed to Online Process Oriented Guided Inquiry Learning (O-POGIL). Students were oriented of their functions and roles to their respective groups and the teacher was the facilitator. There was a brief introduction of POGIL materials and activities which was followed by the designation of students to their respective breakout rooms. POGIL activities followed three learning cycles: exploration phase, concept invention phase, application phase. After the activity, all students rejoined the main meeting portal where each group presented their output. After the presentation, the researcher encouraged students from the other groups to ask questions to address and clarify any misconceptions. In addition, in order to maximize interaction, constructive feedback was also given by the researcher, and students were asked to give some insights into the things that they had learned and acquired after the presentation.

For the control group, Online Problem Solving Method was employed. At the start of every online session, students were oriented of their functions and roles to their respective groups and the teacher was the facilitator. There was an introduction to the topics and activities. Examples about the topics were given to each group for their references. It was then followed by the designation of students to their respective breakout rooms. The students were given worksheets and were tasked to work out more problems. The online Problem Solving Method strategy followed Polya’s Four-Step Problem Solving Procedure. The first step was to understand the problem, devise a plan, carry out the plan and lastly look back. It was then followed by a presentation of the group’s outputs. Students were also encouraged to ask questions to address and clarify any misconceptions. Constructive feedback was also given at the end. This modality was used throughout the 8-weeks duration of the conduct of the study. A posttest was administered at the end of the experimental period.

The performance of the students in the pretest and the posttest critical thinking skill test in Mathematics in the Modern World Course were evaluated and described in terms of mean and standard deviation. The variation, as well as the significant effect of O-POGIL in the critical thinking skills of students, were measured and determined by one-way analysis of covariance (ANCOVA). In testing the hypotheses, alpha is set at a 0.05 level of significance.

## 4. Results and Findings

The results of this study were presented in the following tables:

## Table 1. Mean and Standard Deviation of Students’ Critical Thinking Skill in Mathematics

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Table 1 shows the mean and standard deviation of the pretest and posttest results in their mathematics critical thinking abilities test for both the control and experimental groups. The overall pretest mean score for the Non-OPOGIL and O-POGIL groups was 15.452 and 14.788, respectively, indicating that their pretest results were at the “Did not meet expectation level”. It’s also worth noting that Non-OPOGIL scored 0.664 points higher than O-POGIL. This suggests that prior to the investigation, the degree of critical thinking skills among students in both groups was relative and so comparable. Furthermore, the standard deviation of O-POGIL is smaller than that of Non-OPOGIL, indicating that O-POGIL students' results are closer to the mean, whereas Non-OPOGIL students' scores have a larger dispersion.

On the posttest, both Non-OPOGIL and O-POGIL received “Did not reach expectation level.” Although both groups' scores were still at the “Did not meet expectation level,” there was a significant improvement, with O-POGIL scoring 3.72 points better than Non-OPOGIL. This implies that O-POGIL students demonstrated skill in identifying a concept or problem with some supporting details and using logic to make inferences. This is due to the various information-gathering techniques that O-POGIL provides during the exploration and concept invention stages. This allows students to explore and discuss several ways to solve the problem, as well as collaborate to arrive at a final solution.

## Table 2. One Way ANCOVA Summary of the Students’ Critical Thinking Skills

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Table 2 shows that there was a significant difference in the posttest scores in the critical thinking skills between Non-OPOGIL and O-POGIL as indicated by the F-ratio of 7.24 and p-value of 0.009 which led to the rejection of the null hypothesis. Based on the result, O-POGIL had helped students improved their critical thinking skills and subsequently increased their scores.

The problem-solving nature of O-POGIL helped students to develop independent process skills, giving them the opportunity to learn varied ways and even alternative methods of solving problems. It further allows students to deepen their conceptual understanding and strengthen their analytical skills. O-POGIL promotes a good impact on student's cognitive abilities, particularly in the areas of communicating and analyzing. Through controlled group learning, students were directed to use their previously learned mathematical understanding to develop their own knowledge through O-POGIL. It has also created an opportunity for debate as students work together to identify the core idea or problem and provide supporting facts from their conceptual understanding to support their problem-solving approach. This encourages students to think and speak mathematically leading them to draw precise conclusions. Furthermore, they were able to compare and contrast available solutions during the concept invention phase. With these, students were able to support their answers with coherent and constructive reasoning.

The result shows that the use of OPOGIL significantly improves students’ critical thinking skills is similar to the result of Rosadi, Sunarno & Maridi who concluded that the developed POGIL method is effective to be used in teaching and learning to improve the students’ analytical thinking skills 23 . Soltis et al. also stated that students’ critical thinking skills and problem solving were improved with the use of the POGIL strategy 2 . This is also similar to the results of Irwanto, Saputro, Rohaeti, & Prodjosantoso, who emphasized that the POGIL method is more effective in improving students’ critical thinking and problem-solving skills 3 .

## 5. Concluding Statements

Based on the analysis and findings of the study, O-POGIL actively engages students in activities that develop their critical thinking skills in mathematics. It enables students to develop their process skills by giving them the opportunity to construct their own understanding while learning and discussing varied ways to solve the problem leading them to provide coherent reasoning to support their answer. Improving critical thinking skills necessitates this process of communication, evaluation, and problem-solving. To improve students' critical thinking skills in mathematics, the researchers advocate using the O-POGIL teaching method and developing resources based on the O-POGIL construct. Analyzing O-POGIL in a more controlled online environment where the facilitator can observe students’ progress synchronously may also yield data to deeply understand O-POGIL’s effect on students thinking ability in mathematics. There may also be an illuminating area for further research through the utilization of e-learning materials and software that keeps students engaged and motivated.

## Acknowledgements

The researchers would like to express their heartfelt gratitude to the Pangantucan Bukidnon Community College (PBCC), led by Atty. Mc Ferlin Taola, for enabling them to conduct the research.The researchers are also grateful for the College of Science and Technology Education (CSTE) headed by Dean Dr. Laila S. Lomibao, the chair of the Department of Mathematics Education headed by Dr. Rosie G. Tan of the University of Science and Technology of Southern Philippines for their approval and support of the conduct of this study.

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Published with license by Science and Education Publishing, Copyright © 2021 John Kevin A. Artuz and Dennis B. Roble

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In light of the importance of developing critical thinking, and given the scarcity of research on critical thinking in mathematics education in the broader context of higher order thinking skills, we have carried out a research that examined how teaching strategies oriented towards developing higher-order thinking skills influenced students’ critical thinking abilities. The guiding rationale of the work was that such teaching can foster the students’ skills of and dispositions towards critical thinking. In this article, we discuss ways in which critical thinking can be incorporated in mathematics instruction. In particular, we highlight how content taught in the probability strand can intentionally be focused on the development of students’ critical thinking. We report results of a study demonstrating improvement in secondary mathematics students’ dispositions towards critical thinking and abilities to think critically.

Critical Thinking , Probability , Mathematics , High School , Early Childhood

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1. Introduction

Critical thinking (CT) is a capability essential to contemporary life. Furthermore, the benefits of critical thinking are lifelong, supporting students in the regulation of their study skills, and subsequently empowering individuals to contribute creatively to their chosen profession. In this paper, we argue that critical thinking is constituted through both dispositions and abilities. While the abilities may be developed through direct instruction, the dispositions are better thought of as “habits of mind” and their development requires long-term participation in learning environments conducive to reflection and argumentation. Our research into the promotion of critical thinking has been undertaken in high school mathematics classrooms through the teaching of probability, but we will argue that all mathematics topics can be taught with the underlying goal of helping students learn to think critically and become mathematically proficient citizens of the world. We provide examples of activities that can help foster students’ critical thinking, from early childhood through high school. Critical thinking can be thought of as both a capability and a “habit of mind” by which individuals interact with the world of experience by questioning it, rather than just accepting it. This is more than a taught skill to be employed selectively; it is an orientation to the world. To encounter the world critically, ideally an individual must be initiated into the practices of critical thinking from an early age. Routines such as “I see, I think, and I wonder” can be used with very young children to develop the three skills of careful observation, thoughtful interpretation, and imaginative speculation. The development of such questioning habits of mind in young people may be culturally confronting to some communities. In this chapter, we explore approaches to the lifelong development of critical thinking as an essential process by which we can regulate our experience of a world in which few things are constant, where reliance on routines will not maintain our quality of life, and emergent problems will be overcome with imagination rather than just persistence. Central to our argument is the distinction between abilities and dispositions. It is our contention that critical thinking must become a pervasive element of the educational experiences of all students from pre-school to upper high school and university settings and that a structured program in critical thinking should start with the promotion of appropriate dispositions and progressively move to the development of critical thinking abilities.

2. Critical Thinking

2.1. Critical Thinking: An Overview

Critical thinking was originally developed during the ancient times when the sophists (740-399 BC) and Socrates (5th century BC) pondered theories of ethics and societal governance. In order for Socrates to understand an opinion on a certain issue, he first clarified the definition of that issue and then evaluated whether that definition was true. He was a person who thought independently, and taught others to think for themselves. He also believed that thinking independently was a learned trait; that is, people could be educated in thinking critically, rather than being innately born with critical thinking capabilities ( Bryan, 1987 ; Regev, 1997 ).

Critical thinking is reflective and reasonable thinking that focuses on deciding what to do or believe ( Ennis, 1985 ). Critical thinking abilities include such things as: applying available information to new situations, analyzing causes or motives for situations, and evaluating opinions on subjects.

The consensus emerging amongst science and mathematics educators worldwide is that the acquisition of higher-order cognitive skills (HOCS) by our students should be a major instructional goal in education ( Zohar & Dori, 2003 ). HOCS include, among other skills, the ability to ask questions, solve problems, make decisions, and engage in critical thinking ( Ennis, 1989 ; Innabi & Sheikh, 2007 ; Perkins, 1992 ). Ideally, students will gradually build upon their learning of critical thinking as they progress through their schooling; as opposed to being asked to think critically one year but not the next. National curriculum initiatives around the world (Australia, China, Israel, Japan and Singapore, for example) all advocate the structured promotion of thinking skills that can be variously labeled “higher order”, “heuristic”, “metacognitive”, “imaginative”, “creative” or “critical”. We address some of the strategies by which such structured promotion of critical thinking might be effected at different stages of schooling, from pre-school onwards.

McPeck defined critical thinking as “skills and dispositions to appropriately use reflective skepticism” ( McPeck, 1981 ). Lipman claimed that critical thinking is “thinking which enables judgment, is based on criteria, corrects itself, and is context sensitive” ( Lipman, 1991 ). A third definition, by Ennis (1987a) , is the one on which we have based our research. Ennis (1985) defined critical thinking as “reasonable reflective thinking focused on deciding what to believe or do.”

In light of his definition, Ennis developed a critical thinking (CT) taxonomy that relates to skills that include an intellectual aspect as well as a behavioral aspect. In addition to skills, Ennis’ (1987a) taxonomy includes dispositions and abilities. Ennis claims that CT is a reflective and practical activity the purpose of which is to moderate action or belief. In the study that is the principal focus of this chapter, we address the challenge of promoting both students’ abilities and their dispositions.

2.2. Critical Thinking Activity Examples in Mathematics

In order to give readers a more concrete view of the kinds of grade activities suggested by researchers internationally, we focus on the mathematical strand of probability, which can be taught starting from preschool children through high school and beyond. We provide examples from only this one significant mathematical topic, in order to provide a coherent picture of the progression of critical thinking with respect to mathematical content introduced.

In early childhood, students are often asked to sort objects by shapes and characteristics. For instance, they could have a bucket of red cars, red trucks, blue cars, and blue trucks; and be asked to sort out all of the cars from all of the trucks, or all of the red vehicles from the blue vehicles. This kind of activity can help students think about the definition of each class of objects and the application of this definition amongst the current sample space available to them. Lead by instructors who have critical thinking in mind, students can build fundamentals of inductive reasoning by comparing newly introduced objects with characteristics of the existing objects. Should a yellow car belong in the group of cars? Students must consider whether the definition of car is dependent upon color. Students can also be asked to develop categories of their own and to articulate the defining characteristics of each category.

Another kind of activity involves presenting students with dilemmas, or situations in which two solutions can be equally valid. For instance, elementary school students can explore the dilemma, “is it okay for someone to take another person’s pencil without asking?” and discuss whether there are conditions where it is acceptable (e.g., a doctor needing to write out a prescription for another classmate) and not acceptable (e.g., greed). By evaluating moral situations relevant to students in their daily lives, students are led to recognize and challenge their existing beliefs and employ such reasoning strategies as considering exceptions to a rule.

When students are sufficiently cognitively advanced as to be able to consider at least three choices within a context, they can be asked questions requiring them to evaluate whether a situation is “always, sometimes, or never” true. For instance, “if a person has a dream to become a sports star, does this person always, sometimes, or never accomplish this dream?” or “when you add an odd number and an even number, is the result always, sometimes, or never an odd number?” The obligation to justify an assertion with evidence should be emphasized as soon as students can make claims. This would lead to the development of principles of reasoning, such as, when a statement has a counterexample, it can never be always true; however, if a statement is supported by only one example, the statement may not necessarily be always true. This is an important logical distinction to make and can be introduced early in a child’s school experience.

Some simple situations and related propositions can be evaluated and even proposed by students from the early years of schooling through high school. Here are three simple examples:

1) Did you notice that Dani is very clever at maths and he has really big feet? Do you suppose that all students with big feet are good at maths? (Grade 4)

2) The population of a country town fell by 50% over five years. Suggest why this might have happened and some methods that you could use to test your hypothesis. (Grade 6)

3) A research study showed a negative correlation between disciplinary statements and maths achievement. That is, classes where the teacher made many disciplinary statements were low achieving and classes where the teacher made few disciplinary statements were high achieving. It is clear that students would be much more successful if teachers would just stop disciplining them. Please comment. (Grade 9)

In upper elementary school, students could learn about fractions simultaneously with their meanings from a probabilistic point of view. Fractions are one representation of the notion of chance. When there is a 7/8 chance that you will pull out a blue marble out of a bag of blue and red marbles, it is still possible 1/8 of the time to pull out a red marble. Students should become aware of the notion of taking calculated risks and the implications of making decisions based upon chances; often seen in the contexts of sports, financial decisions, etc. There are many ways in which decision-making based on likelihood can be simulated in the classroom. Games of chance can be adapted to simulate such situations. Once statistical situations require decision-making, we have conditions conducive to the development of critical thinking.

Students generally learn about measures of central tendency during their years in upper elementary and middle school. There are many opportunities for students to think critically about mean, median, and mode. Students can begin to calculate, “if I want to earn an A in this class, what grade do I need to earn on the final exam?” or “if the majority of my classmates did not enjoy visiting a national monument, what are the chances that I will enjoy visiting it?” Reasoning on the basis of measures of central tendency simultaneously invokes statistical and critical thinking. Situations that highlight the interplay between mean, median, and mode (e.g., a mode that is low but mean is high, a median that is low but mode is high, etc.) can also help students to re-evaluate their first reactions to a situation and understand the underlying mathematics before making an informed decision. In such situations, critical reflection on the relevance and relationships of statistical concepts can enhance both the students’ understanding of statistics and their capacity for critical thought. That is, such classroom activities bring the multiple benefits of simultaneously developing cognitive or argumentative skills at the same time as statistical knowledge, and the pragmatic utility of developing effective strategies for evidence-based decision-making.

In middle school, students are ready to learn how to use argumentation, that is, using warrants and backings to support any claims that they may make ( Forman et al., 1998 ). Skills of argumentation can be applied to any discussion of a mathematical or non-mathematical nature, and can be especially used in mathematics classes when students solve cognitively challenging tasks ( Aizikovitsh-Udi, 2012 ). An example of a cognitively demanding task is to evaluate the claim, “in larger and better equipped hospitals, the rates of mortality are higher than in smaller and less well equipped hospitals”. Students would need to compare and contrast the two different kinds of hospitals, and coordinate their comparison with the mortality rates in those hospitals. They would need to identify relevant variables and consider alternative scenarios, leading to the development and investigation of rival hypotheses. Such tasks can be expressed simply and create the conditions to optimize student reflection and critical thinking in instructional settings.

3. Method: A Study Supporting the Early Introduction of CT

There are two approaches to teaching critical thinking using content disciplines: a) The “infusion” approach, where critical thinking skills are taught explicitly using the discipline’s content; b) The “embedded” or general approach, where critical thinking skills are taught in indirect ways without spelling it out to students ( Swartz & Parks, 1994 ). The study by Aizikovitsh-Udi (2012) used the “infusion” approach that had also been used elsewhere ( Ennis, 1987b ; Swartz, 1992 ; Swartz & Parks, 1994 ). In the infusion approach, the skills are taught in the framework of a particular subject or topic, and thinking becomes an integral part of teaching specific subject content, while general principles and terminology of thinking are explicitly emphasized.

In the course of the study, students were challenged to consciously examine data supporting claims and determine the validity of any data to be interpreted. The course consisted of fifteen sessions of ninety minutes each. During this course, students were given recent news from the newspaper or from survey results and were asked to judge how true these claims could be. In order to properly justify students’ claims, they needed to learn conditional probability, Bayes Theorem, and additional probabilistic and statistical tools.

3.1. Research Questions

In this study, Aizikovitsh-Udi (2012) examined the following two research questions:

1) To what extent do students learn critical thinking skills (abilities and disposition) via a CT-infused presentation of probability?

2) What critical thinking processes are invoked as students learn probability via an infusion approach?

Data collection was purposefully triangulated: 1) Five students were interviewed at the end of a class period as well as one week after. These personal interviews were conducted in order to reveal a change in the students’ attitudes throughout the academic year; 2) Students’ exams, in-class handouts, and homework were collected; 3) All class sessions were documented and analyzed―the sessions were recorded and transcribed; 4) The participant group was tested twice, both Pre and Post the instructional program, using the two questionnaires: Dispositions of Critical Thinking (CCDTI) and Abilities of Critical Thinking (Cornell test). Each of the questionnaires (both the CCTDI and Cornell test) generates sub-scores and a total-score.

CCTDI was used in order to evaluate the students’ dispositions toward critical thinking. This tool was designed to measure the general dispositions profiles of the students. CCTDI is divided into seven sub-tests: truth- seeking (sub-test T), intellectual openness (sub-test O), analyticity (sub-test A), systematicity (sub-test S), self- confidence in critical thinking (sub-test C), inquisitiveness (sub-test I) and maturity (sub-test M), although thinking dispositions are far less numerous than thinking skills ( Facione & Facione, 1992 ; Facione, Facione, & Giancarlo, 1996 ).

The Cornell Test was used to check the development of the students’ critical thinking abilities according to Ennis’ taxonomy. This test was developed by Ennis and his colleagues (e.g., Ennis & Millman, 2005b ). The Cornell Critical Thinking Test, Level Z and X were chosen by the researchers as it was more suited to the advanced high-afool students in the group. The test includes general content with which most of the students would be familiar and it assesses various aspects of critical thinking. It is a multiple-choice test with three choices and one correct answer. Although the test is meant to be taken within a fifty-minute period, it was thought that the students in the group would be unable to complete it within that time limit. For this reason it was decided to give them eighty minutes in which to take the test. The test includes five sub-tests and evaluates different aspects of critical thinking, including induction, deduction, value judging, observation, credibility, assumptions, and meaning ( Ennis & Millman, 2005a ).

In order to answer the research questions and to evaluate the change in the students’ knowledge, pre and post questionnaires were used according to the Counter Balance method19. To achieve these goals, we have passed the following questionnaires: 1) the CCTDI ( Facione & Facione, 1992 ): a 75-item, Likert test of critical thinking; 2) the Cornell Critical Thinking Test version “Z” ( Ennis & Millman, 2005b ), a 52-item, multiple choice test of critical thinking abilities.

The teacher kept a reflection journal after every session, as well. Data was processed using qualitative (interpretive) methods, which allowed the researcher to follow the students’ patterns of thinking and interpretation with regards to the learned materiel in different contexts.

3.3. Population

Fifty-five children between the ages of fifteen and sixteen participated in an extra curriculum program aimed at students from different cultural backgrounds and socio-economical levels. The students all attended the tenth- grade and recently finished learning from the textbook, “Probability in daily life” ( Lieberman & Tversky, 1996; 2002 ).

4. Result Related to the Disposition of Critical Thinking

In order to develop and foster critical thinking, we need first to define it and to understand the mental processes it involves. Previous research has investigated the ways in which students acquire technical tools such as evaluation, verifying results, assessing problems, making comparisons and conclusions, choosing solution strategies etc. This study, by contrast, focused on the more general and universal aspects of critical thinking, investigating the ways in which students develop abilities, such as induction, deduction, value judging, observation, checking the sources’ reliability, identifying assumptions, and extracting meaning, and dispositions, such as truth-seeking, open-mindedness and inquisitiveness, according to the taxonomy of Ennis (1989) . The overarching purpose of this study was to determine whether critical thinking abilities and dispositions could be developed through the study of probability.

Two prefacing remarks summarize the key findings:

・ Learning did occur but gains in specific CT areas could have been higher;

・ Students in interview asserted that CT was important and valued, and suggested that there were multiple uses of CT in their lives; however they thought CT should appear earlier in the educational process.

The details behind these general findings fall into two specific categories: those results concerned with student disposition towards critical thinking and those concerned with student critical thinking abilities. Results related to both are reported below.

4.1. Result Related to the Disposition of Critical Thinking

The distinction between Abilities and Dispositions is a vital one, because the individual who possesses the appropriate thinking abilities may still lack the drive, the will or inclination to act upon them. According to Sternberg, there are “close to a thousand” thinking skills, and according to Lipmann, “the list is infinite” ( Sternberg, 1984 ; Lipman, 1991 ). This distinction is by no means clear-cut―thinking dispositions include and encourage disposition to thinking―but it has theoretical and practical justifications. A thinking disposition, as we define it, is a rational impulse toward a particular thinking pattern or thinking quality, which encourages becoming actively involved in the process of thinking, investing oneself in thinking. Unfortunately, school does not traditionally provide room for the type of thinking that involves intellectual awakening; it can even appear to oppose it. Baber describes the following dialogue between a teacher and his student: “Teacher: What are you doing? Student: I’m thinking. Teacher: Then stop thinking and start working!” ( Beyer, 1987 ). We suggest, as a contemporary imperative, that schools initiate a structured program to develop students’ critical thinking and purposefully promote students’ disposition to thinking. We further suggest that the promotion of critical thinking must start in the early years of schooling if we are to maximize the educational benefits of such a program.

A consistent effort to encourage higher-order thinking skills not only promotes student critical thinking during the current instructional period, but also has a long-term effect, by becoming an integral part of these students’ thinking habits. Statistical comparison of averages in CCTDI tests administered in this study focused on the relative rate of improvement. A series of t-tests showed that the students of the Experimental group considerably improved their critical thinking dispositions relative to the Control group. Thinking dispositions in a specific area do not necessarily extend to thinking patterns in general. For example, one may tend to deep and nuanced thinking in one’s area of specialization but exhibit superficial thinking on political matters. That is, thinking dispositions are context-dependent. Aizikovitsh-Udi & Amit (2008) demonstrated that teaching the “Probability in Daily Life” unit in the infusion approach greatly developed the students’ critical thinking. It is clear from this research that one of the fundamental elements of good critical thinking is the development of the dispositions already discussed. Specifically, with respect to the major focus of this paper, the idea of dispositions as an inclination to perceive and interact with experience in a certain way suggests that the initiation of individuals (children) into this way of thinking (that is, the development of these dispositions) should begin at the earliest age possible.

4.2. Results Related to Abilities of Critical Thinking

Both training and knowledge are necessary to promote critical thinking abilities in students. Ennis (1962; 1963; 2002) believes that critical thinking involves inquiry, asking questions, offering alternative answers, and questioning traditional and accepted beliefs. He suggests that because society does not welcome people who challenge authority, critical thinking is not encouraged. “Old” education, as Dewey referred to it, focused on passing on knowledge, where knowledge was conceived to be a body of facts and propositions that need only be memorized and understood.

As in the case of critical thinking dispositions, consistent effort to encourage high order thinking skills contributed to the development of the students’ ability to think critically (not only during the experimental learning period but also in the longer term, as these skills became an integral part of the students’ thinking habits). To conclude, studying the learning unit “Probability in Daily Life” in the infusion approach contributed to the development of the students’ thinking skills. It is our opinion that instruction in such high order thinking skills requires that the teacher be personally expert in the various aspects of critical thinking. Further, the relative sophistication of the abilities being developed suggests that programmatic development of critical thinking abilities should be structured to provide a well-established dispositional foundation before attempting to develop the more sophisticated critical thinking abilities.

5. Conclusion

The results of this research demonstrate that if teachers consistently and systematically encourage CT in their classes by applying mathematics to real-life problems, encouraging debate, and planning investigative lessons, students are likely to practice CT skills and develop the language of critical thinking. The research cited in this paper also shows how the infusion approach can work in the specific case of the teaching of probability related to real life problems. Students practiced critical thinking using probability, while the stimulus material presented constituted the basis for practicing critical thinking skills together with the subject of probability.

It was important to note that in addition to the CT skills mentioned above, in the course of the reported study, students also gained intellectual skills such as conceptual thinking and benefited from a classroom culture that fostered CT. But to practice CT skills under teachers’ guide and develop the language of critical thinking does not mean that the internalization of critical thinking skills has been achieved by students. Such internalization is essential if our ultimate goal is that students should be able to apply critical thinking in any and all settings and situations in which they find themselves. This internalization is deeply connected to the establishment of the relevant dispositions and will only come over time, and hence we strongly advocate the use of critical thinking instruction for students beginning in early childhood education. The general educational implications derived from this research can and should be used to encourage the intellectual development of students beyond the technical content of conventional mathematics courses, by creating learning environments that foster CT. This approach will, in turn, encourage students to interrogate the issue at hand, evaluate relevant information, and respond as critical thinkers.

From this research’s findings and discussion, there arise the following research recommendations. A more comprehensive examination of the processes of critical thinking: to what extent could the students describe, orally and in writing, the processes of thinking, activate them and apply the thinking skills they studied at the procedural and metacognitive level? Did they make an informed use of terms and strategies of higher order thinking, including critical thinking? To examine the former question, researchers should examine to what extent participants use the “language of thinking” ( Ben-Chaim et al., 2000 ; Costa, 1991 ; Costa & Marzano, 1991 ; Harpaz, 1996 ; Zohar, 1996 ). Developing such a language involves, on the part of teachers, such skills as using precise vocabulary, presenting critical questions, presenting data rather than answers, aspiring for exactness, giving directions, and developing metacognition. As has already been discussed, students can be assisted to talk about their thinking, even at a very young age, through routines such as “I see, I think, and I wonder”. Through such relatively simple approaches, we feel that an appropriate classroom climate can be developed for the promotion of students’ critical thinking dispositions. Once these dispositions are established in the early years of schooling, the instructional program can progressively increase the sophistication of the critical thinking skills being developed. The research cited here suggests that instructional approaches such as the infusion approach are suitable for the development of both critical thinking dispositions and abilities throughout all years of schooling. It is hoped that the findings of this study will contribute to our understanding of the nature of critical thinking and to the further development of instructional approaches relevant to its promotion.

Conflicts of Interest

The authors declare no conflicts of interest.

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## DEVELOPING REALISTIC MATHEMATICS EDUCATION-BASED WORKSHEETS FOR IMPROVING STUDENTS' CRITICAL THINKING SKILLS

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This work intends to emphasise critical thinking as a key twenty-first century competence through an immersion approach where pre-service teachers take the role of students in order to experience an integrated teaching approach with an emphasis in the promotion of critical thinking in mathematics and science lessons.

Critical thinking skills should be owned by students. Therefore, schools should be responsible to develop and evaluate critical thinking skills through teaching and learning process in schools. This study aims to identify the effects of mathematical

Mathematics is an essential subject in the Philippines Department of Education (DepEd) K-12 curriculum that requires critical thinking abilities because it is crucial in everyday life along with the growth of other sciences. However, the majority of Filipino students had difficulty understanding mathematical concepts that require higher-order cognitive skills. As a result, students ...

Critical thinking skills are important in solving mathematical problems because, with this skill, students can find the most effective solution to a problem by processing known information in the ...

In this paper, we attempted to operationalize the concept of critical thinking in the context of elementary mathematics education, through specific skills that are emphasized during work with mathematical content and to examine if a more purposeful selection of content can influence the development of critical thinking.

The importance of math in developing critical thinking skills is undeniable. Mathematics inherently involves critical thinking, where students analyze problems, interpret data, and arrive at correct solutions. Educators and parents play a key role in this process by fostering curiosity, encouraging group work, and utilizing resources like Math ...

The Role of Mathematics in Developing Critical Thinking Mathematics is often perceived as a subject of numbers and equations, but it's true value lies in its ability to develop logical reasoning ...

Discover how teaching strategies focused on higher-order thinking skills can enhance students' critical thinking abilities in mathematics education. Explore the incorporation of critical thinking in probability instruction and the positive impact on secondary mathematics students' critical thinking dispositions and abilities.

The following recommendations were drawn: there should be the emphasis on the development of the student's character and other affective domains in learning the subject to develop critical thinking skills in Mathematics, students should exert effort to learn Mathematics to improve their critical thinking skills in the subject, students should ...

The mathema tical critical thinking ability test is use d to meas ur e math ematica l c ritical thinking skills base d on the indicators c ontained in the components of mathematical critical ...

This study aims to develop teaching materials based on Realistic Mathematics Education (RME) to improve students' critical thinking skills. The development model used is ADDIE consisting of Analysis, Design, Development, Implementation, and Evaluation phases.

We also compose two combined critical thinking score indices to test the overall critical thinking skills development. H5: ICM design can improve students' overall critical thinking skills. 4. Methodology ... Examining Effective Instructional Approaches," International Journal of Science and Mathematics Education, 16, 1065-1089.

One effective way to nurture critical thinking in math is through Problem-Based Learning (PBL). ... By engaging with real-world problems, students develop the skills to analyze, evaluate, and apply mathematical concepts in meaningful ways. As educators, fostering these skills through PBL not only prepares students for academic success but ...

Mathematics in the Modern World is a college course that aims to develop critical thinking and problem-solving skills through fundamental mathematical concepts, emphasizing their real-world ...