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5 Effective Thinking Exercises from the Elective Everyone Loves

What will students remember from your course in 20 years? Dr. Edward Burger offers a few exercises in effective thinking that can be used for a lifetime.

What will students remember from your course in 20 years? Dr. Edward Burger offers a few exercises in effective thinking that can be used for a lifetime.

Edward B. Burger, PhD


Southwestern University, Georgetown, TX

PhD in Mathematics, BA in Mathematics

Dr. Edward Burger’s mind is constantly in motion. As a leading teacher of effective thinking, innovation, and creativity, it is not just his nature: It is his life’s work to make people better. Well, his latest life’s work.

Burger first rose to fame for being the Best Math Teacher in America (as named by Reader’s Digest in 2006)—largely thanks to the unique sense of humor and exuberance he brings to the thousands of math videos he has created. Millions of students have viewed his YouTube video Calculus in 20 Minutes. He has authored more than 70 scholarly papers and books, including Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas (coauthored with Michael Starbird), which has been translated into seven languages, and he served as mathematics advisor for the TV series NUMB3RS for three years.

Interestingly, though, the title of Burger’s latest book, Making Up Your Own Mind: Thinking Effectively through Creative Puzzle-Solving (Princeton, 2019), is conspicuously missing the word math, though he says the foundations of that subject form the backbone of his “effective thinking” approach. “It’s basically trying to take how mathematicians think, and realize that anyone can think that way about anything,” he says.

He has proven that theory over and over by engaging students from all majors in his wildly popular course Effective Thinking through Creative Puzzle-Solving, which he created and has taught for four years in his current role as president of Southwestern University in Georgetown, Texas. Below, he shares a quick look at the “5 Elements of Effective Thinking” from his book and course, plus a quick way to put each method to work in any classroom.


“The way I put it at Southwestern University is: We’re not just thinking about the material, which ends at the material, but we’re thinking through the material. So you’re learning modes and practices of thought that allow you to not only probe the depths of that subject but also apply those mindsets to think about everything else.”
-Edward B. Burger, PhD

Course: University Studies 232: Effective Thinking through Creative Puzzle-Solving

Description: This two-credit, mini-semester experimental course will sharpen your problem-solving skills, your ability to create novel approaches, see issues from a variety of perspectives, and develop ways of understanding at a deeper level by engaging with different ideas across the curriculum and through logic puzzles and mind-benders. An intentional component of this course is to connect these mindfulness practices to your other classes and to the rest of your life. In addition, weekly special guests having a wide variety of intellectual interests will visit, share their ways of thinking, their life stories, and engage in thought-provoking conversations.

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An overview of the five elements of effective thinking

“Effective thinking” is not the same as “critical thinking,” which tends to be associated with faultfinding, says Burger. “Effective thinking has those critical aspects but also involves a more creative, empathetic, intuitive kind of emotional intelligence.”

The 5 Elements of Effective Thinking (see also the book by that same name also coauthored with Michael Starbird) are simple in concept and wording, making them easy to remember (if a bit challenging to apply):

  1. Understand deeply.
  2. Fail effectively.
  3. Create questions.
  4. Go with the flow of ideas.
  5. Be open to change.

To put these into practice, Burger has students apply this thinking on a series of three puzzles each week, ranging from straightforward to very challenging.

The process of becoming a more effective thinker, he says, is not linear. “In fact, understanding deeply and change go together, because if you understand deeper, then you’ve changed. The other three—fail effectively, create questions, go with the flow of ideas—are some of the practices that will lead you to deeper understanding. Then, therefore, you will change.”

5 classroom exercises to enhance effective thinking

Before teaching effective thinking strategies in the classroom, Burger notes that the educator has to practice them actively. “It’s not enough to say, ‘Oh, I love this idea about failing effectively or deep understanding, I’ll have my students do it,’” he says. “Educators have to embrace those mindsets for themselves, then model that thinking for the students, and create little, micro episodes in which the student gets to practice that thinking on a very small scale.”

Here are practical exercises Burger says lend themselves well to any subject matter or course:

To help them understand deeply:
Ask them to add the adjective

To help students dive below the surface of a particular topic, Burger asks them to “add the adjective.” He gives them 40 seconds to come up with as many descriptors as they can to label or explain the topic at hand.

For example, he introduces a puzzle using a chessboard that is missing some squares. Students might note that the squares have “different colors.” They may say it is “broken” or “incomplete.” Depending on which squares are missing, it could be “symmetrical” or “asymmetrical.”

“Then you have an engaged conversation about the descriptors,” he says. He asks students to consider, “What words did we use? What words didn’t we say? How do we see the issue differently?” This exercise works well because it is not intimidating, it is fun, and it is fast.

To show them how to fail effectively:
Tell them to make an intentional mistake

In math courses, Burger often gives students a challenge and says, “OK, I want you to create a solution that contains a deliberate mistake.” (The same could be done with a draft of an essay or an initial sketch of a figure, for instance.) Then he has students trade papers with a neighbor, who is tasked with finding the flaw.

“Quite often, of course, there’s the intentional mistake, and about two other unintentional errors,” he says with a laugh. “Their colleague has to look at every single step and find them all.”

By “forcing” students to fail and then push past it and examine what led to the failure, Burger helps them realize that getting stuck is simply an opportunity to gain greater insights. “If your last step is failure, that’s not good, and I am not suggesting, ‘Hey, you failed your calculus final! Yay!’ Absolutely not,” he says. “The question you need to ask after a failed attempt is, ‘What do I learn from this misstep and, from that, where do I go next?’ That’s a way in which failure is a positive learning tool and becomes effective.”

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To help them create questions:
Name an official questioner

Burger advises against asking any group, “Are there any questions?” If they are engaged, there should always be questions, so he suggests prompting with, “What are your questions?”

In his courses, Burger encourages this strategy by naming an “official questioner” in each class. That person creates at least one question to ask before the end of the period. It can be a “meta-question” a la Socrates (What does this mean in the grand scheme of life?), a fundamental question (Why does the Pythagorean theorem work?), or a question that is tangentially related to the subject (I am learning about the Renaissance in another course—how was the art at the time impacted by concurrent mathematical discoveries?).

He recommends adapting it slightly for huge stadium-seating courses. “If you pick two or three questioners, the rest of the students will check out or simply go to sleep,” he says. For those classes, he recommends stopping the lecture at various points and having students work in teams of two or three to create two questions. “Then I just pick people from the audience,” he says. The questions that those students share with the entire class then engender an engaged discussion.

To go with the flow of ideas:
Ask students what to do next

Burger is decidedly anti-syllabus—if that syllabus delineates what will happen every day for every class and every night for homework for an entire semester. “I had a negative, visceral reaction to that format as a student. I looked at the syllabus and said, ‘Where am I in this picture? Suppose that we are struggling? It doesn’t make a difference because on Friday, we’re doing something new.’ I thought that was so ineffective, because the student was given no voice at all.”

Today, Burger has a different suggestion for educators. “As your class is about to end for the day, stop two minutes early and ask, ‘Given what we’re talking about, what should we do (discuss) at our next class?’ If it’s a math class, they could say, ‘Maybe we should look at more complicated equations. What happens if we have a lot of negative numbers? What happens if instead of squares there are cubes?’”

Burger adds, “Quite often, what they will suggest will not be what you had been planning to do.” By following up with what students want to know, Burger says, educators ensure a more attentive and engaged audience overall.

To help them be open to change:
Call attention to biases

Many students go into class—and, in fact, college—not expecting to or wanting to change, says Burger. “Students are not vested in seeing each and every course in which they enroll as opportunities that will change who they are,” he says. “That is interesting … and, I would argue, problematic. To have a high-impact educational experience that’s focusing on intellectual and personal growth, you must change.”

Interestingly, change is actually a by-product of the other principles of effective thinking, he adds. If you learn from failure, ask questions, and go with the flow to find new ideas, you will have already changed by developing a deeper understanding than you understood before. “It’s a natural consequence,” he says.

One way to help open up students to change is to first show them that they are not as open-minded as they imagine. For example, in one of his math classes, Burger asked the students to identify themselves if they did not believe the mathematical theorem that they had just proven as a class. “Despite creating three different mathematical arguments to establish the validity of the theorem, there were students who raised their hands.”

So Burger engaged them in a conversation, and one student shared why he thought the theorem was wrong—not that he didn’t understand or had a question or was confused, but it was just “wrong.” “Maybe we are actually less open-minded than we care to admit,” Burger then pointed out. “And maybe we should admit that reality, because that might help us to be more open-minded.” When the class ended, another student approached Burger and said, “Wow, that was the most powerful moment I’ve ever had in a class. You actually got to me.”

An educator can help students be more open to change by recognizing and acknowledging those aha! moments in which you can see the light bulb go on. By being clear that change is the goal of education, Burger hopes his students will be more likely to seek and embrace it.

Bonus tip: Have them use all the strategies—everywhere

In his “puzzle class,” Burger does not give exams. Instead, students submit solutions to puzzles, along with essay reflections in which they talk about the strategies they used to arrive at the solution. The solution itself counts for part of the grade, but not the whole thing.

In addition, he asks students to offer insight into how they used the elements of effective thinking on each of the puzzles. They must consider:

  • How did you use “understanding deeply” to think about this puzzle?
  • How did you learn from a misstep?
  • What question did you create that allowed you to get some kind of insight?
  • How did the flow of ideas take you someplace?
  • How do you see this puzzle differently now?

Burger also has them perform a similar exercise outside of his class that he calls PETs (Practices of Effective Thinking). They must choose a homework assignment from another course and highlight how they have used an effective thinking practice to amplify that work.

“It’s so easy to incorporate these things at the micro level, but it’s got to be a sustained practice,” he concludes. “You can’t just offer these mindsets on the first day of class and say, ‘OK, now you know how to think.’ It has to be on an ongoing or recurring basis to become a true mindful practice of living.”

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