Critical thinking can be as much a part of a math class as learning concepts, computations, formulas, and theorems.

Activities that stimulate critical thinking will also encourage students to think and speak in mathematical terms.

## Critical Thinking in Math

**Critical thinking** is a key factor in separating those students who can ‘do’ math from those who truly understand what they’re doing. When students do math, they can perform computations and explain concepts because they’ve learned formulas and definitions through practice and rote memorization. They don’t necessarily know why the formula works, but they can use it. Likewise, they may not know how some ancient mathematician defined the concept, but they know the definition.On the other hand, students who’ve been taught to think critically in math can explain why a formula works, and they can trace the steps used to define a concept.

Not only can they solve a problem, they can explain the logic behind the process they used to reach a solution.Think of it this way: sooner or later, most math students learn the Pythagorean theorem and its related formula. Students can become adept at using the formula A squared + B squared equals C squared to find the length of the hypotenuse or one of the sides. They can solve problems involving right triangles using the formula, provided they are given enough information. Students who have been taught to think critically can explain why the Pythagorean theorem works.

More importantly, they know when and how to apply the Pythagorean theorem to solve a problem even if right triangles aren’t an obvious part of the solution.

## Real World Problem Solving

Math textbooks only go so far when it comes to presenting real world problems that require mathematical solutions. Texts are organized around concepts, making it easy for students to see what strategies they need to use to solve a problem. If the chapter is on the quadratic formula, students automatically know they will use that formula at some point to answer every question, even a complex word problem.

Critical thinking kicks in when students have a variety of options for solving a problem. Students apply critical thinking to find the best strategy out of many possible methods to reach a solution.Here’s a problem that requires mathematical critical thinking:*Based on current trends in rising or falling temperatures, predict the average high and low temperatures for five different places on Earth five years from now.*

To solve the problem, students will need to analyze data, determine the trends in each place, and select a method for predicting the future temperatures. They may need to use a variety of formulas and statistical tools to form their predictions. Teachers can take this a step further by asking students to explain and defend the methods they used.

## Asking Questions

To think critically is to follow a clear line of logical steps and reasoning. To solve critical thinking problems, math teachers should model the way they think when solving a problem. Students can internalize a set of questions to ask that will help them think their way to a solution.

These questions could include:

- What is the problem? What am I trying to figure out?
- What do I know? What is the given information?
- What do I need to know to solve the problem?
- What problems like this have I solved before?
- What solutions could work? What strategies will work best in this situation?

After students attempt a solution, they can further ask: Why did my solution work? Or they might try to understand why their solution didn’t work. Critical thinking activities present the perfect opportunity for students to collaborate and have meaningful conversations using mathematical vocabulary, which is a good sign they have developed a deep understanding of concepts.

## Bellringers

**Bellringers** are short activities students do at the beginning of class as a warm-up exercise. Normally, bellringer activities require students to practice or apply skills they have already learned. Bellringers can also stimulate critical thinking.

For example, a critical thinking math bellringer might ask students to evaluate a pattern and determine a missing piece. For example:*What number would come next in this sequence: 3, 6, 9, 15, 24, _____? *Pattern questions may consist of numbers, shapes, variables, or colors. Part of the bellringer should include asking students to explain how they found the pattern since that’s where the critical thinking occurs.

## Puzzles

Solving puzzles is an entertaining way to engage students in critical thinking. Sudoku is a popular style of puzzle that requires students to think critically and mathematically.

To solve a Sudoku puzzle students fill in columns, rows, and boxes with numbers from one to nine using each number only once in each area.Many students enjoy brainteasers with a mathematical twist. Here’s one that involves understanding the principles of multiplication by zero:*I will take the Philadelphia Phillies. You can have any other team in the major leagues.*

* I will add together the runs the Phillies score in every game this season. You can multiply all your team’s runs together from every game this season. At the end of the season, I will probably win. Can you explain why?*The answer is that I can add zero runs from a shutout to my total and continue to build on my score. However, as soon as you multiply by zero from one shutout, your result will be zero.

## Project-Based Learning

**Project-based learning** tasks require students to apply math concepts and strategies while thinking both critically and creatively. While completing the projects, students not only practice thinking and speaking in mathematical terms, they also learn to collaborate with their peers. They have to think critically to respond to each other’s ideas, and they have to frame their criticisms constructively.Imagine all the mathematical critical thinking involved in a project requiring students to design an original product, calculate the cost of manufacturing it, and create a method for marketing the product.

Each stage from design to marketing requires a different type of research and different forms of math. In each step, students must think critically to ensure their imaginary product becomes a successful project.

## Lesson Summary

When students use **critical thinking** in math, they not only know how to solve a problem, but they also understand why the solution works. Likewise, students use critical thinking when they determine the best strategy for solving a problem. **Bellringers**, puzzles, and **project-based learning** assignments will teach students to think critically and mathematically.

## Learning Outcomes

After studying the lesson, students are prepared to:

- Describe critical thinking in math
- Identify five strategies to develop critical thinking in math
- List examples of how to incorporate critical thinking strategies in math classes