critical thinking activities for math

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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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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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Math Starters: 10 Simple Strategies

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Ignite student learning with math starters— quick activities used at the beginning of math class to engage your kids’ critical thinking and problem solving skills. This post shares 10 math starters that will jump-start student learning.

On my first day as a middle school teacher, my principal said the strangest thing.

He said, “Your students cannot leave during the first five or last five minutes of class.” 

Why? I thought to myself.

Maybe he didn’t want students to get caught in a tardy sweep?

I had no clue. 

“Because,” as he said, “they are the most important minutes of the class period.”

I’d never heard that before, but it made me stop and think about the message behind it. 

As I thought more about this expectation, I finally understood.

I needed to grab my students’ attention during the first five minutes of class– which meant I should program the first five minutes with a fun and engaging math starter. 

What are Math Starters?

Math starters, critical thinking activities , designed to get students thinking about math, provide opportunities to “sneak” in grade-level content and skills in a fun and engaging way. Intended to take no more than 5 -10 minutes of instructional time, starters can include a wide variety of tasks.

Over the years, I have used a number of starters, including number of the day and word wall activities; however, my favorite activities are challenges that allow me to reinforce a variety of critical thinking and problem-solving strategies. 

Math Starter Ideas

Here’s a list of my favorite ways to jump-start my math class. 

Emphasize Algebraic Thinking

1. Ask students to find five ways to represent a number, such as 1000, or have students represent the day of the month or the ____ day of school in a variety of ways.

2. Encourage algebraic thinking with number logic. To complete the task, students must determine the value of one or more symbols, such as snowflake + snowflake = 12 and snowflake – snowman = 2.

3. Challenge students with a logic puzzle . To create this puzzle, fill a 3 x 3 grid with three or four different objects, such as some fun seasonal erasers (you will need multiples of some objects). Then assign each object a value (keep it secret). Record the sum of each row or column on the outside of the table and challenge the students to find the value of each object. (See the picture above for an example.)

Encourage Problem Solving

4. Give students a quick problem-solving task to complete, such as “During the winter snow season, the Snowy Mountain Coffee Spot sells hot chocolate for 47 cents. If Katrina paid for a cup of hot chocolate with exactly five coins (quarters, nickels, dimes, or pennies), which five coins did she use?” Note: Consider opening the task by removing the condition of “five” to say “which coins did she use?” (See the image above.) 

5. Challenge students with an Open Middle task. These tasks end with the same solution but have multiple solution paths to allow students at various levels to achieve success. 

Use Clues to Eliminate Options

6. Give students clues to determine the mystery number. See the example below.

The number is:

• not an odd number,
• more than the value of a penny,
• less than the number that is two more than five,
• more than the number of sides on a square.

7. Using a set of pattern blocks, give students clues, and have them sequence the pattern blocks within a horizontal set of four numbered boxes. For example, students can order the hexagon, the blue rhombus, the trapezoid, and the triangle with the following clues.

• There is a quadrilateral in the even-numbered boxes.
• The shape with the least number of sides is not on the right side.
• The shape with one pair of parallel sides is not next to the triangle.

Analyze Relationships

8. Review math vocabulary with “What’s the Relationship?” To use this activity, write 5 – 7 related math words on the board. Then have students determine how the words are related to each other. Add a twist by including a non-related word and ask students to determine which word doesn’t belong.

9. Create a Venn Diagram with pattern blocks and then ask students to determine the common attribute for each set of shapes.

Build Number Sense

10. Tiling tasks require students to use a set of numbered tiles, 0-9, to complete a set of problems. Typically, each tile fits in one, and only one, solution. Use tiling tasks, like the one above, to challenge students to find the value of each symbol. (Use the form below to grab this freebie and challenge your students!)

Engage ‘Em from the Start!

Let’s jump-start the new semester with a bang! Start your next math class with one of the activities above. Then tell us all about it in the comments below.

Give one of the ideas above a try. Then tell us about your experience in the comments below.

Shametria Routt Banks

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Math Activities

Five fun math board games to motivate your child while practicing math.

When faced with the task of practicing math through things like math homework, a child’s typical response would be of exasperation and hesitance. In fact, 84% of children find taking out the trash or going to the dentist more appealing than completing their math homework. If your child exhibits the same attitude towards math homework, then what else would help them practice their math skills? The answer is simple: math games.

Math games do a lot more than simply raising a child’s entertainment levels. These games encourage critical thinking, logic, enhance computational fluency, and deepen children’s understanding of math concepts.

While also giving children and parents an opportunity to further bond, math games also help build the perspective that math is applied everywhere, not just in school. Below are some examples of enjoyable math games to help your child see the fun in math:

1. Proof! Math Game

As a Teacher’s Choice Gold Award Winner, Proof! is a fast-paced yet exciting game that engages children, families and even teachers with its mental math olympics. The game starts with a selection of number cards where players have to make an equation among these cards. Players have to shout the result of the equations they find and then show proof with the cards presented, gathering these cards once proven. The player with most cards wins the game!

This game is an amazing opportunity to practice one’s math skills regardless of their level. The rules can be easily customized for young children or played normally, as gifted students will also enjoy its challenges with each play. Engaging and easily differentiated for all types of students, Proof! is an excellent math game to practice computational fluency.

2. Math War (Addition and Subtraction Cards)

Math War is a classic math game that brings children delight and math fun every time they play! As a multiplayer game, children flip over cards and solve the equations they present. The child with the highest card result wins the round, keeping both cards.

With vibrant and kid-friendly numbers and illustrations, this math game has children improving their math skills without them realizing it. Time limits can be set to test the players’ mental math skills, motivating them to find strategies for quick yet efficient problem-solving!

3. MOBI (Numerical Tile Game)

Build your child’s math confidence with MOBI ; a strategy game that reinforces children’s math skills, sequential thought and automaticity. As a simple yet fast-paced math game, each player draws tiles and uses them to create simple equations, forming a crossword-like pattern. Whoever uses all their tiles first has to yell “Mobi” to win!

With only these little tiles needed to play, MOBI is a travel-friendly math game that allows children to become familiar with math at their own pace. Children can practice mental math in short intervals to keep them engaged yet educated. Perfect for family fun and friendly competition, this easy-to-play math game accommodates up to 6 players.

4. Sum Swamp Game

Sum Swamp is an adorable math game that features lively illustrations and immersive math gameplay. Players navigate the swamp using dice, facing math challenges and comical swamp creatures as they race through this engaging adventure.

Children playing the Sum Swamp math game apply their knowledge of addition and subtraction, having their mental math skills help them to victory. Colorful game pieces, challenges like the ‘endless loop,’ and mischievous shortcuts delight children in this fun-filled math adventure.

5. Kanoodle

As an easy-to-understand yet difficult-to-solve math game, Kanoodle encourages children to explore math strategies using their critical thinking and math knowledge. Tactile and colourful, Kanoodle can be played using the easy levels of 2-D puzzles, in which players have to complete a 5 x 11 board with 12 pieces – very reminiscent of Tetris. When a player has successfully won all 2D levels, they can move onto solving the more challenging 3D puzzles requiring 5 tiers with the same puzzle pieces.

Kanoodle ’s devious game concept brings joy to those who solve the puzzles, reeling them in with the more difficult puzzles – even for adults. Children as young as 7 can play the game, developing spatial orientation and understanding geometry in an entertaining manner. Parents and children can tinker with puzzles, reinforcing math knowledge and quality time.

Check out what other parents have to say about us, here !

Proof math game – proofmathgame.com, math war addition & subtraction game cards – schoolzone.com, mobi fast fun numerical tile game – playmobi.com, learning resources sum swamp game – learningresources.com, educational insights kanoodle – educationalinsights.com, math games and math homework – mathgamesandactivities.com, why play math games – nctm.org.

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Engaging Maths

Dr 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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Increasing Critical Thinking Skills in Math

• Math , Planning

It’s important that we are building critical thinking skills in math. Too often these are overlooked or assumed that students do it because they have to problem solve sometimes. While that does help build the all-important critical thinking skills, we need to make sure we are also finding ways to purposely bring it into instruction.

One such way that I like to implement critical thinking skills in my math class is through a game called Puzzlers. Recently I discussed why you should use games in the classroom and this one is no exception. Games go beyond just having fun and “entertaining” students. They aren’t just fillers.

Building Critical Thinking Skills with the Puzzler Game

The puzzler game is a game that not only increases critical thinking skills, but it also practices both fact fluency and the order of operations!

In the puzzler game, students are given a target number. This happens by rolling a die or dice, but it can also be any chosen number between 1 and 36. For instance, I have randomly chosen the date before.

Next, students are provided with a 3×3 grid of the numbers 1 through 9 mixed up. (See the image below.)

Once students have their target number and a mixed up grid of the numbers 1-9, they are ready to begin. This is where the critical thinking skills will come in.

Now, students will need to come up with a way to use ONLY three numbers (in a row, diagonally, or in a column) to get that target number. They will do this by creating equations that total the target number. They can add, subtract, multiply, divide, or even come up with a combination of them. If needed, they can use parentheses. This is where knowing the order of operations is necessary!

For instance, let’s take the example above with the 9 numbers on the sticky notes. Let’s say that the target number was 18. The student could create these two equations to come up with the solution of the target number 18:

• (9 x 6) ÷ 3
• (9 + 8) – 1

Here’s an example of a puzzler card with multiple solutions:

What I love about this puzzler game is the variety of ways it can be used to help build critical thinking skills! For instance, students could list all of the equations, or solutions, to get the target number:

or go through multiple cards trying to list as many solutions as they can:

Or they could skip rolling the dice altogether and see how many solutions they can find for the target numbers one through ten. Why not even through in zero?!

Students love this game and it’s perfect for independent work, early finishers, small groups, and even enrichment. It’s differentiated and there are cards that are strictly for adding and subtracting for students who can’t multiply yet.

You don’t have to purchase my puzzler resource to play this critical thinking skills builder! You can easily create it in your classroom as a bulletin board and change out the numbers each day!

If you want to save some time, grab the extra differentiated materials, and the specifics, head to my store now to purchase it! It’s definitely worth it!

Click here to purchase this Puzzler Game.

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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 ↓

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How to Turn Your Math Classroom Into a ‘Thinking Classroom’

The researcher Peter Liljedahl evangelizes for practices that prioritize and stimulate more hard thinking in classrooms.

In traditional math classrooms—and in classrooms where challenging, unfamiliar work is often assigned, more generally—the progression " I do, we do, you do” often becomes the default approach to learning, according to researcher and professor of mathematics at Simon Fraser University, Peter Liljedahl.

It makes sense in many cases, particularly when difficult concepts need to be addressed in time windows that are compressed by bell schedules, holiday breaks, and summer vacation. But used too frequently, this approach, Liljedahl recently told the Cult of Pedagogy , inhibits higher order thinking and results in students who “mimic” teachers. Students who engage in too much rote work miss out on some of the challenging, sometimes confusing work that builds up their self esteem to face difficult thinking tasks in the future.

“By and large students spend most of their class time not thinking, at least not in ways we know they need to think in order to be successful in mathematics.” Liljedahl explains. “If they’re not thinking, they’re not learning.”

Liljedahl, the author of Building Thinking Classrooms in Mathematics, Grades K-12: 14 Teaching Practices for Enhancing Learning advocates for “Thinking Classrooms”, which offers a different take on how classroom work is organized, how tasks are assigned, and how students learn and work together. His conclusions are based on more than a decade of research, experimentation, and collaboration with over 400 K-12 teachers.

In a 2017 article for Edutopia , Liljedahl clarified that a “non-thinking” classroom is “predicated on an assumption that the students either could not or would not think.” When presented with difficult problems, students in these classrooms have a hard time pushing through to find their own solutions, he argued, and often wait for teachers to step in to do the heavy lifting for them. 

To stimulate independent thinking, Liljedahl says, reorganize some of your classroom approaches: Start with hard puzzles and problems that push kids to their limits; confront the fundamental passivity of classroom seating; and use highly structured group activities to promote discussion, peer review, and iterative thinking. 

Starting With ‘Thinking Tasks’

Instead of starting a lesson with direct instruction, give students novel “thinking tasks” they can work on, ideally in groups. Liljedahl describes these tasks as problem solving activities and mental puzzles that, early on in the school year, should be “highly engaging, non-curricular tasks” to motivate students and get them in the mindset of challenging themselves. As the school year progresses and students become more accustomed to this mode of working and thinking, the activities and challenges can be replaced with tasks directly related to the curriculum. 

Liljedahl points out that the tasks should be carefully sequenced so they get incrementally more challenging. “The goal of thinking classrooms is not to get students to think about engaging with non-curricular tasks day in and day out—that turns out to be rather easy,” he told Cult of Pedagogy . “Rather, the goal is to get more of your students thinking, and thinking for longer periods of time, within the context of curriculum, which leads to longer and deeper learning.” 

Liljedahl has a long list of sample “thinking tasks” to look through. They include challenging dice-related problems, such as: Imagine a typical 6-sided die, and notice that the sum of opposite faces is always seven. The one is across from the six, the two is across from the five, and so on. Now imagine that you were making your own six-sided die that did not have this restriction. How many different dice could be made?

Use Standing, Randomized Group Work  

Central to Liljedahl’s approach is student collaboration and group work. Instead of grouping students by ability, or allowing them to choose their own groups to work in, his research has shown him students work more effectively—and are more likely to contribute—in randomized groups. According to Liljedahl , interviews with students show that randomized groups “break down social barriers within the room, increase knowledge mobility, reduce stress, and increase enthusiasm for mathematics.” 

At Design39Campus , a K-8 school in San Diego, eighth grade math teacher Kyle Asmus puts Liljedahl’s approach into practice by using a random group generator (such as this one from Classtools or this one from Keamk ) to ensure that different kids work together across multiple exercises. 

To ensure that all students feel included, Asmus assigns group roles: The scribe writes down possible solutions; the speaker communicates the group’s thinking to the broader classroom; the inquirer asks the teacher questions; and the manager makes sure the rest of the group stays on track. 

Having students stand while they engage in this collaborative, messy thinking is yet another way to engage them, according to Liljedahl: It makes them much less likely to withdraw from the work, or assume that others will handle it. “It turns out that when students are sitting, they feel anonymous,” Liljedahl told Cult of Pedagogy . “And when students feel anonymous, they disengage.” 

Work on Non-Permanent, Vertical Surfaces  

In a thinking classroom, students put notebooks away and participate in group work while standing at vertical non-permanent surfaces such as whiteboards, blackboards, or windows—surfaces that Liljedahl believes promote more risk-taking. 

According to Liljedahl, his experimentation with students shows that when comparing a group working on a whiteboard versus a group working on flip chart paper, the group working on whiteboards start working within 20 seconds. 

“They’ll start making notations on the board. They’ll try anything and everything because they feel like they can just erase it if it’s wrong,” he told Cult of Pedagogy . Meanwhile, students working on chart paper take upwards of three minutes to make a single notation, because, Liljedahl said, they often wait until what they write is perfect—“and that hesitation leads to a lower form of thinking.”

Design39 students also practice Liljedahl’s “vertical surfaces” learning technique, which Asmus says promotes “high quality collaboration” and facilitates “questions and rich conversations” among students. According to Asmus, when students are presented with challenging tasks and work vertically, the time for students to get to a task is not only quicker, but the time they spend on a task is longer. 

The large surfaces spread out around the room also allow students to see the work their peers are doing in other groups, and build off of each other’s understanding. “It makes it easy for the whole class to see what everybody else is doing so we can be inspired by each other’s ideas,” said Iniyaa, a student in Asmus’ class. 

Answer the Right Questions

As students work vertically in groups, teachers can easily see how they’re progressing and bounce around the room. Questions will undoubtedly arise, but Liljedahl says teachers should avoid answering questions asked for the purpose of reducing student effort and getting to an answer more quickly—such as “is this right?” Instead, they should prioritize addressing student questions that will lead to further independent thinking. 

In her elementary school classroom in Brooklyn, teacher Tori Filler says that instead of rushing to provide hints and solutions to the hard parts of a lesson, she often asks them to evaluate what’s tough about it, and encourages them to sort it out on their own before stepping in to help. 

If most of the class is struggling, ask more questions and turn the struggle into a productive discussion for everyone. Asking questions like “What makes this hard?” or “What have we tried?” gets students to think metacognitively and develop the skills to push through challenging work on their own. 

Evaluate What You Value

To succeed in a thinking classroom, students need to develop skills like perseverance, academic courage, collaboration, and curiosity, among others. But according to Liljedahl, if we want students to develop these competencies, then we should find ways to evaluate them on it. 

“What we choose to evaluate tells students what we value, and, in turn, students begin to value it as well,” he writes. 

He argues for a mix of both formative and summative assessment in math classrooms that focus less on end products and student ranking, and more on the work leading up to those end products and collaboration between groups to get there. 

The educators at Bite Sized Learning , for example, suggest evaluating students on the work they produce, but also on how well they persevere and make an effort in response to challenges, how well they set individual goals and monitor progress toward achieving them, and how well they share information and resources with group members to solve problems and make decisions. 

Liljedahl’s own formative assessments focus on informing students “about where they are and where they’re going in their learning.” This can take the form of observations, check-for-understanding questions, or even unmarked quizzes. Summative assessments, meanwhile, “should focus more on the processes of learning than on the products, and should include the evaluation of both group and individual work,” Liljedahl writes. 

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Home » Blog Posts » Math & Technology » Math Riddles to Promote Critical Thinking Skills in Your Classroom

Math Riddles to Promote Critical Thinking Skills in Your Classroom

Have you ever watched your students’ faces light up with a “Eureka!” moment? There’s something incredibly rewarding about seeing young minds solve problems that challenge them. But how can we, as educators, create more of these moments in a way that’s both fun and educational? The answer might just lie in math riddles ! 🤔

Math riddles are not just entertaining puzzles; they are powerful tools for promoting critical thinking and problem-solving skills . In this post, we’ll explore how these clever questions can turn your classroom into a think tank of eager detectives. Plus, I’m excited to share that you can get my Math Riddle QR Code Task Cards for FREE —a resource guaranteed to engage your students and transform your classroom dynamics. So, are you ready to dive into the world of math riddles and unlock the potential of your students? Let’s get started!

This post may contain affiliate links to Amazon for your convenience. As an Amazon Associate, I earn from qualifying purchases, which do not cost any extra for you. Please see the   full disclosure here .

What is Critical Thinking, and Why Do Students Need It?

Critical thinking is the ability to analyze facts to form a judgment. In the classroom, it’s about going beyond memorization and recall to apply, analyze, and evaluate information. But why is this important?

In today’s fast-paced world, the ability to think critically is more valuable than ever. Students equipped with these skills can navigate complex problems, make informed decisions, and approach challenges with creativity and resilience.

Critical thinking in math, especially, teaches students to question assumptions, consider multiple solutions, and rigorously test their ideas. This not only enhances their mathematical abilities but also prepares them for real-world challenges where answers are not always clear-cut.

Getting Students to Think More Deeply

Thinking is the hardest work there is, which is probably the reason so few engage in it.      Henry Ford

One of my favorite things to do as a teacher is pushing my students to think more deeply.

So instead of just giving my former 2nd graders straight math drills to memorize, I would give them math riddles instead.

Promoting critical thinking through math riddles pushes students to engage in several different learning skills at once.

Students must read carefully to comprehend the riddle. Then, they have to complete the math computation in order to solve the riddle.

I love this because it combines ELA and Math content for my students.

Math Riddle Example:

This image below gives an example of a math riddle. Students engage in problem-solving skills as they work through each problem.

Math Riddles help promote critical thinking and place value practice. I absolutely LOVE being able to address and giving meaningful practice with more than one learning standard.

Ways to Use Math Riddles in Your Classroom

Incorporating math riddles into your classroom can be both fun and incredibly beneficial for student learning. Here are some creative ways to do it:

• Math Starter : Kick off your class with a math riddle to get those brains buzzing.
• Morning Work : Let students start their day with a puzzle, setting a positive tone for the day.
• Early Finishers : Keep those fast workers engaged with challenging riddles instead of busy work.
• Brain Breaks : Inject a quick, fun break into long periods of work with a stimulating riddle.
• Team Challenges : Foster teamwork and communication by having students work together to solve riddles.
• Homework Assignments : Include riddles as part of homework to make it more engaging.
• Math Centers : Create a riddle station for students to visit during math center time.
• Bulletin Boards : Post riddles on your bulletin board for an interactive display.
• Exit Tickets : Use a riddle as an exit ticket to assess understanding in a fun way.
• Weekly Challenges : Offer a riddle of the week for extra credit or just for fun.
• Parent Involvement : Share riddles with parents to engage them in their child’s learning.
• Interactive Notebooks : Have students record and solve riddles in their math notebooks.

Check Out These Related Posts!

• The BEST Math Read Alouds Ever
• Teaching Critical Thinking with Task Cards

Infusing Technology with Math Riddles

In addition to pushing my students beyond just rote memorization math practice, these QR Codes also give them practice with technology. My students absolutely love QR Codes anyway. Honestly, I love them too! They give students a chance to see their answers immediately. That is less grading for me and it gives students instant feedback.

Related Post: How I Save Time Grading with ZipGrade

When a student gets a problem incorrect, they go back and rework the math riddle–either independently or with a partner.

I’ve noticed how this also pushes their critical thinking because finding their own math mistake involves deeper thinking. Plus, it is a great exercise in showing grit and perseverance; something my kids definitely need!

Other Ways to Promote Critical Thinking Skills

Beyond math riddles, there are numerous strategies to encourage critical thinking in the classroom. Here are five additional ideas:

• Socratic Seminars : Engage students in discussions based on asking and answering questions to stimulate critical thinking.
• Problem-Based Learning : Present real-world problems for students to solve collaboratively.
• Debates : Organize debates on various topics to develop reasoning and argumentation skills.
• Reflection Journals : Encourage students to reflect on their learning processes and outcomes.
• Creative Projects : Assign projects that require innovative thinking and problem-solving.

How To Get Your FREE Download of these Math Riddles:

If you are not on my email list, be sure to subscribe to get this FREEBIE! The free products in my TpT store aren’t always the same as the ones I give to my subscribers.

This resource has additions that you can only get by signing up here.

If you are using a school email address, this process may not work for you. Many districts now block outside emails!

So use a personal email address that you check often, and you should be ready to roll!

I send all new FREEBIES to current subscribers automatically. If you haven’t already gotten yours, please check your SPAM or be sure that you confirmed your email with me.

Now that the “techy” talk has been covered, we can all go push our students to think more critically with some fun math riddles!

By integrating math riddles and these strategies into your teaching toolkit, you’re not just teaching math; you’re nurturing curious, independent thinkers ready to tackle the challenges of tomorrow.

Remember, fostering a classroom environment where questions are welcomed and challenges are embraced is key to developing critical thinking skills. And, don’t forget to grab your FREE Math Riddle QR Code Task Cards to get started on this exciting journey! 🎉

Happy Teaching,

The Butterfly Teacher

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5 Critical Thinking Activities That Get Students Up and Moving

More movement means better learning.

It’s easy to resort to having kids be seated during most of the school day. But learning can (and should) be an active process. Incorporating movement into your instruction has incredible benefits—from deepening student understanding to improving concentration to enhancing performance. Check out these critical thinking activities, adapted from Critical Thinking in the Classroom , a book with over 100 practical tools and strategies for teaching critical thinking in K-12 classrooms.

Four Corners

In this activity, students move to a corner of the classroom based on their responses to a question with four answer choices. Once they’ve moved, they can break into smaller groups to explain their choices. Call on students to share to the entire group. If students are persuaded to a different answer, they can switch corners and further discuss. 

Question ideas:

• Which president was most influential: George Washington, Thomas Jefferson, John Adams, or Abraham Lincoln?
• Is Holden Caulfield a hero: Strongly Agree, Agree, Disagree, or Strongly Disagree?

Gallery Walk

This strategy encourages students to move around the classroom in groups to respond to questions, documents, images, or situations posted on chart paper. Each group gets a different colored marker to record their responses and a set amount of time at each station. When groups move, they can add their own ideas and/or respond to what prior groups have written.

Gallery ideas:

• Political cartoons

Stations are a great way to chunk instruction and present information to the class without a “sit and get.” Group desks around the room or create centers, each with a different concept and task. There should be enough stations for three to five students to work for a set time before rotating.

Station ideas:

• Types of rocks
• Story elements
• Literary genres

Silent Sticky-Note Storm

In this brainstorming activity, students gather in groups of three to five. Each group has a piece of chart paper with a question at the top and a stack of sticky notes. Working in silence, students record as many ideas or answers as possible, one answer per sticky note. When time is up, they post the sticky notes on the paper and then silently categorize them.

• How can you exercise your First Amendment rights?
• What are all the ways you can divide a square into eighths?

Mingle, Pair, Share

Take your Think, Pair, Share to the next level. Instead of having students turn and talk, invite them to stand and interact. Play music while they’re moving around the classroom. When the music stops, each student finds a partner. Pose a question and invite students to silently think about their answer. Then, partners take turns sharing their thoughts.

• How do organisms modify their environments?
• What is the theme of Romeo and Juliet ?

Looking for more critical thinking activities and ideas?

Critical Thinking in the Classroom is a practitioner’s guide that shares the why and the how for building critical thinking skills in K-12 classrooms. It includes over 100 practical tools and strategies that you can try in your classroom tomorrow!

Get Your Copy of Critical Thinking in the Classroom

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11 Activities That Promote Critical Thinking In The Class

52 Critical Thinking Flashcards for Problem Solving

Critical thinking activities encourage individuals to analyze, evaluate, and synthesize information to develop informed opinions and make reasoned decisions. Engaging in such exercises cultivates intellectual agility, fostering a deeper understanding of complex issues and honing problem-solving skills for navigating an increasingly intricate world. Through critical thinking, individuals empower themselves to challenge assumptions, uncover biases, and constructively contribute to discourse, thereby enriching both personal growth and societal progress.

Critical thinking serves as the cornerstone of effective problem-solving, enabling individuals to dissect challenges, explore diverse perspectives, and devise innovative solutions grounded in logic and evidence. For engaging problem solving activities, read our article problem solving activities that enhance student’s interest.

What is Critical Thinking?

Critical thinking is a 21st-century skill that enables a person to think rationally and logically in order to reach a plausible conclusion. A critical thinker assesses facts and figures and data objectively and determines what to believe and what not to believe. Critical thinking skills empower a person to decipher complex problems and make impartial and better decisions based on effective information.

More Articles from Educationise

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• The Future of Education: 8 Predictions for the Next Decade
• The Latest in EdTech: 5 Innovative Tools and Technologies for the Classroom
• 8 Free Math Problem Solving Websites and Applications

Critical thinking skills cultivate habits of mind such as strategic thinking, skepticism, discerning fallacy from the facts, asking good questions and probing deep into the issues to find the truth.

Importance of Acquiring Critical Thinking Skills

Acquiring critical thinking skills was never as valuable as it is today because of the prevalence of the modern knowledge economy. Today, information and technology are the driving forces behind the global economy. To keep pace with ever-changing technology and new inventions, one has to be flexible enough to embrace changes swiftly.

Read our article: How to Foster Critical Thinking Skills in Students? Creative Strategies and Real-World Examples

Today critical thinking skills are one of the most sought-after skills by the companies. In fact, critical thinking skills are paramount not only for active learning and academic achievement but also for the professional career of the students. The lack of critical thinking skills catalyzes memorization of the topics without a deeper insight, egocentrism, closed-mindedness, reduced student interest in the classroom and not being able to make timely and better decisions.

Benefits of Critical Thinking Skills in Education

Certain strategies are more eloquent than others in teaching students how to think critically. Encouraging critical thinking in the class is indispensable for the learning and growth of the students. In this way, we can raise a generation of innovators and thinkers rather than followers. Some of the benefits offered by thinking critically in the classroom are given below:

• It allows a student to decipher problems and think through the situations in a disciplined and systematic manner
• Through a critical thinking ability, a student can comprehend the logical correlation between distinct ideas
• The student is able to rethink and re-justify his beliefs and ideas based on facts and figures
• Critical thinking skills make the students curious about things around them
• A student who is a critical thinker is creative and always strives to come up with out of the box solutions to intricate problems
• Critical thinking skills assist in the enhanced student learning experience in the classroom and prepares the students for lifelong learning and success
• The critical thinking process is the foundation of new discoveries and inventions in the world of science and technology
• The ability to think critically allows the students to think intellectually and enhances their presentation skills, hence they can convey their ideas and thoughts in a logical and convincing manner
• Critical thinking skills make students a terrific communicator because they have logical reasons behind their ideas

Critical Thinking Lessons and Activities

11 Activities that Promote Critical Thinking in the Class

We have compiled a list of 11 activities that will facilitate you to promote critical thinking abilities in the students. We have also covered problem solving activities that enhance student’s interest in our another article. Click here to read it.

1. Worst Case Scenario

Divide students into teams and introduce each team with a hypothetical challenging scenario. Allocate minimum resources and time to each team and ask them to reach a viable conclusion using those resources. The scenarios can include situations like stranded on an island or stuck in a forest. Students will come up with creative solutions to come out from the imaginary problematic situation they are encountering. Besides encouraging students to think critically, this activity will enhance teamwork, communication and problem-solving skills of the students.

Read our article: 10 Innovative Strategies for Promoting Critical Thinking in the Classroom

2. If You Build It

It is a very flexible game that allows students to think creatively. To start this activity, divide students into groups. Give each group a limited amount of resources such as pipe cleaners, blocks, and marshmallows etc. Every group is supposed to use these resources and construct a certain item such as building, tower or a bridge in a limited time. You can use a variety of materials in the classroom to challenge the students. This activity is helpful in promoting teamwork and creative skills among the students.

It is also one of the classics which can be used in the classroom to encourage critical thinking. Print pictures of objects, animals or concepts and start by telling a unique story about the printed picture. The next student is supposed to continue the story and pass the picture to the other student and so on.

4. Keeping it Real

In this activity, you can ask students to identify a real-world problem in their schools, community or city. After the problem is recognized, students should work in teams to come up with the best possible outcome of that problem.

5. Save the Egg

Make groups of three or four in the class. Ask them to drop an egg from a certain height and think of creative ideas to save the egg from breaking. Students can come up with diverse ideas to conserve the egg like a soft-landing material or any other device. Remember that this activity can get chaotic, so select the area in the school that can be cleaned easily afterward and where there are no chances of damaging the school property.

6. Start a Debate

In this activity, the teacher can act as a facilitator and spark an interesting conversation in the class on any given topic. Give a small introductory speech on an open-ended topic. The topic can be related to current affairs, technological development or a new discovery in the field of science. Encourage students to participate in the debate by expressing their views and ideas on the topic. Conclude the debate with a viable solution or fresh ideas generated during the activity through brainstorming.

7. Create and Invent

This project-based learning activity is best for teaching in the engineering class. Divide students into groups. Present a problem to the students and ask them to build a model or simulate a product using computer animations or graphics that will solve the problem. After students are done with building models, each group is supposed to explain their proposed product to the rest of the class. The primary objective of this activity is to promote creative thinking and problem-solving skills among the students.

8. Select from Alternatives

This activity can be used in computer science, engineering or any of the STEM (Science, Technology, Engineering, Mathematics) classes. Introduce a variety of alternatives such as different formulas for solving the same problem, different computer codes, product designs or distinct explanations of the same topic.

Form groups in the class and ask them to select the best alternative. Each group will then explain its chosen alternative to the rest of the class with reasonable justification of its preference. During the process, the rest of the class can participate by asking questions from the group. This activity is very helpful in nurturing logical thinking and analytical skills among the students.

9. Reading and Critiquing

Present an article from a journal related to any topic that you are teaching. Ask the students to read the article critically and evaluate strengths and weaknesses in the article. Students can write about what they think about the article, any misleading statement or biases of the author and critique it by using their own judgments.

In this way, students can challenge the fallacies and rationality of judgments in the article. Hence, they can use their own thinking to come up with novel ideas pertaining to the topic.

10. Think Pair Share

In this activity, students will come up with their own questions. Make pairs or groups in the class and ask the students to discuss the questions together. The activity will be useful if the teacher gives students a topic on which the question should be based.

For example, if the teacher is teaching biology, the questions of the students can be based on reverse osmosis, human heart, respiratory system and so on. This activity drives student engagement and supports higher-order thinking skills among students.

11. Big Paper – Silent Conversation

Silence is a great way to slow down thinking and promote deep reflection on any subject. Present a driving question to the students and divide them into groups. The students will discuss the question with their teammates and brainstorm their ideas on a big paper. After reflection and discussion, students can write their findings in silence. This is a great learning activity for students who are introverts and love to ruminate silently rather than thinking aloud.

Finally, for students with critical thinking, you can go to GS-JJ.co m to customize exclusive rewards, which not only enlivens the classroom, but also promotes the development and training of students for critical thinking.

Read our next article: 10 Innovative Strategies for Promoting Critical Thinking in the Classroom

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Thanks for the great article! Especially with the post-pandemic learning gap, these critical thinking skills are essential! It’s also important to teach them a growth mindset. If you are interested in that, please check out The Teachers’ Blog!

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• Math Analogies  Gr. K-9
• Math Detective  Gr. 3-8
• Math Games  Gr. 3-8
• Math Mind Benders  Gr. 5-12+
• Math Ties  Gr. 4-8
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• Practical Critical Thinking  Gr. 8-12+
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• Red Herrings Science Mysteries  Gr. 4-9
• Science Detective  Gr. 3-6
• Science Mind Benders  Gr. PreK-3
• Science Vocabulary Crossword Puzzles  Gr. 4-6
• Sciencewise  Gr. 4-12+
• Scratch Your Brain  Gr. 2-12+
• Sentence Diagramming  Gr. 3-12+
• Smarty Pants Puzzles  Gr. 3-12+
• Snailopolis  Gr. K-4
• Something's Fishy at Lake Iwannafisha  Gr. 5-9
• Teaching Technology  Gr. 3-12+
• Tell Me a Story  Gr. PreK-1
• Think Analogies  Gr. 3-12+
• Think and Write  Gr. 3-8
• Think-A-Grams  Gr. 4-12+
• Thinking About Time  Gr. 3-6
• Thinking Connections  Gr. 4-12+
• Thinking Directionally  Gr. 2-6
• Thinking Skills & Key Concepts  Gr. PreK-2
• Thinking Skills for Tests  Gr. PreK-5
• U.S. History Detective  Gr. 8-12+
• Understanding Fractions  Gr. 2-6
• Visual Perceptual Skill Building  Gr. PreK-3
• Vocabulary Riddles  Gr. 4-8
• Vocabulary Smarts  Gr. 2-5
• Vocabulary Virtuoso  Gr. 2-12+
• What Would You Do?  Gr. 2-12+
• Who Is This Kid? Colleges Want to Know!  Gr. 9-12+
• Word Explorer  Gr. 6-8
• Word Roots  Gr. 3-12+
• World History Detective  Gr. 6-12+
• Writing Detective  Gr. 3-6
• You Decide!  Gr. 6-12+

Mathematical Reasoning™

Bridging the gap between computation and math reasoning.

Grades: Toddler-11

Mathematics

Full curriculum

•  Multiple Award Winner

Forget boring math lessons and dreaded drill sheets.  These fun, colorful books use engaging lessons with easy-to-follow explanations, examples, and charts to make mathematical concepts easy to understand.  They can be used as textbooks or comprehensive workbooks with your textbooks to teach the math skills and concepts that students are expected to know in each grade—and several concepts normally taught in the next grade. Every lesson is followed with a variety of fun, colorful activities to ensure concept mastery.  The lessons and activities spiral slowly, allowing students to become comfortable with concepts, but also challenging them to continue building their problem-solving skills.  These books teach more than mathematical concepts; they teach mathematical reasoning, so students learn to devise different strategies to solve a wide variety of math problems.  All books are written to the standards of the National Council of Teachers of Mathematics.

Beginning 1 , Beginning 2 , and Level A Contents

Understanding Pre-Algebra This book teaches and develops the math concepts and critical thinking skills necessary for success in Algebra I and future mathematics courses at the high school level.  It was written with the premise that students cannot problem solve or take leaps of reasoning without understanding the concepts and elements that lead to discovery.  The author—with 35 years of experience teaching mathematics—is a firm believer that understanding leads to confidence and confidence gives students the resolve to succeed in higher level mathematics rather than fear it.   It is standards-based, but what makes it different from other pre-algebra books is that it organizes concepts in a logical fashion, stressing practice and critical thinking. It avoids the mistakes—found in many other math books—of trying to teach new concepts before students receive the prerequisite skills and practice necessary for success. The concepts are presented clearly and in connection to other concepts. Math vocabulary is very important to success in higher mathematics, so this book includes easy-to-follow explanations and a user-friendly glossary.

Free Detailed Solutions are available!

Understanding Pre-Algebra Contents

Understanding Geometry The successful completion of this colorful 272-page book will prepare middle schoolers for high school geometry. It covers more than 50% of the concepts taught in high school geometry using a step-by-step approach and teaches the reasoning behind the properties taught in geometry–instead of merely asking them to memorize them. Students are also taught the basics of geometric proofs and coordinate geometry in a way middle school students can understand. Students who struggle with high school geometry usually have lower standardized test scores because it is a fundamental subject in high school standardized testing. A glossary of terms that every student should master is included. This book can be used as a classroom textbook in Grades 7, 8, or 9 (usually over a two-year period) or as a reference for high school students. This book covers more than the National Math Standards for middle school mathematics.

Understanding Geometry Contents

NOTE:  It is our recommendation that students complete Understanding Pre-Algebra (see description above) before attempting Understanding Geometry .

Understanding Algebra I This is a one-year Algebra I course for Grades 7-9. Students who have a solid algebra background will have no trouble with the algebra problems from SAT and even the GRE.  This 384-page book highlights vocabulary and notation, and has examples from the history of math. What makes this book unique and different from other algebra textbooks is that it is built from the experiences of an award-winning algebra teacher with more than 30 years of teaching experience. Many textbooks are written by a committee of authors, and many of those authors have little experience teaching beginning algebra students in middle school or high school. Understanding Algebra I presents the most essential concepts and skills needed to fully understand and gain confidence in algebra in a step-by-step fashion, teaching students that algebra is generalized arithmetic. It helps students see the connection between mathematics that they already know and algebra, so that learning algebra becomes easier and less abstract. This book provides students with real strategies to succeed in solving word problems by using charts and translating strategies that guarantee success.

Understanding Algebra I Contents

Essential Algebra for Advanced High School and SAT

Discover Essential Algebra for Advanced High School and SAT , a 241-page math book in the esteemed Mathematical Reasoning series written by award-winning author and teacher with 30 years of expertise in secondary mathematics. This powerful resource teaches the ‘essential’ connection of arithmetic and geometric concepts with algebraic concepts. Without this understanding, students tend to memorize Algebra I problem-solving steps—which is sufficient to pass Algebra I—but leaves them unprepared for math courses beyond Algebra I and the SATs. Algebra, the essential language of all advanced mathematics, lies at the core of this book's teachings. By delving into the generalized arithmetic that underpins algebra, students develop a solid foundation in the rules governing number and fraction operations, including factors and multiples. This vital knowledge empowers students to move beyond mere memorization of Algebra I problem-solving steps and confidently tackle the complexities of math courses beyond Algebra I. Without the knowledge and skills taught in this book, students often struggle or even fail in advanced mathematics courses and on the SATs. Imagine a good high school student who sees a problem like 3•x•y•4 and hesitates to write 12xy due to uncertainty about the rules governing multiplication. Or not understanding how to add 2x to 1/4y to combine it into a single fraction. Or why –6 2 is different than (–6) 2 . It is easy to see that not having a strong understanding of the foundational rules of algebra can stop even the smartest students from succeeding in advanced high school math courses. Essential Algebra for Advanced High School and SAT serves as a companion to an Algebra I course or aids in post-Algebra I readiness. To ensure students’ long-term success in advanced math beyond Algebra I, this book teaches the following 'essential' mathematics skills and concepts:

• Understanding Terms and Order of Operations
• Understanding the Family of Real Numbers
• Rationals and Irrationals
• Working with Terms and Polynomials
• Polynomial Division, Factoring, and Rational Expressions
• Solving Equations and Inequalities
• Ratio, Proportion, and Percent
• How Algebra is Used in Geometry
• Understanding Functions
• Working With Quadratic Equations and Functions

Mathematical Reasoning™ Supplements These supplemental books reinforce grade math concepts and skills by asking students to apply these skills and concepts to non-routine problems. Applying mathematical knowledge to new problems is the ultimate test of concept mastery and mathematical reasoning. These user-friendly, engaging books are made up of 50 theme-based collections of problems, conveniently grouped in self-contained, double-sided activity sheets that provide space for student work. Each collection contains relevant math facts at the end of the worksheet in case students need hints to solve the problems. Calculators are allowed on activity sets that have a calculator icon at the top of the front side of the set. Each activity set is accompanied by a single-sided answer sheet containing strategy tips and detailed solutions. Teachers and parents will appreciate the easy-to-understand, comprehensive solutions. These books are a wonderful enrichment tool, but also can be used to assess how well students have learned their grade level's math concepts.

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All products in this series.

    • Our eBooks digital, electronic versions of the book pages that you may print to any paper printer.     • You can open the PDF eBook from any device or computer that has a PDF reader such as Adobe® Reader®.     • Licensee can legally keep a copy of this eBook on three different devices. View our eBook license agreement details here .     • You can immediately download your eBook from "My Account" under the "My Downloadable Product" section after you place your order.

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Critical thinking definition

Critical thinking, as described by Oxford Languages, is the objective analysis and evaluation of an issue in order to form a judgement.

Active and skillful approach, evaluation, assessment, synthesis, and/or evaluation of information obtained from, or made by, observation, knowledge, reflection, acumen or conversation, as a guide to belief and action, requires the critical thinking process, which is why it's often used in education and academics.

Some even may view it as a backbone of modern thought.

However, it's a skill, and skills must be trained and encouraged to be used at its full potential.

People turn up to various approaches in improving their critical thinking, like:

• Developing technical and problem-solving skills
• Engaging in more active listening
• Actively questioning their assumptions and beliefs
• Seeking out more diversity of thought
• Opening up their curiosity in an intellectual way etc.

Is critical thinking useful in writing?

Critical thinking can help in planning your paper and making it more concise, but it's not obvious at first. We carefully pinpointed some the questions you should ask yourself when boosting critical thinking in writing:

• What information should be included?
• Which information resources should the author look to?
• What degree of technical knowledge should the report assume its audience has?
• What is the most effective way to show information?
• How should the report be organized?
• How should it be designed?
• What tone and level of language difficulty should the document have?

Usage of critical thinking comes down not only to the outline of your paper, it also begs the question: How can we use critical thinking solving problems in our writing's topic?

Let's say, you have a Powerpoint on how critical thinking can reduce poverty in the United States. You'll primarily have to define critical thinking for the viewers, as well as use a lot of critical thinking questions and synonyms to get them to be familiar with your methods and start the thinking process behind it.

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