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3 Simple Ways to Add Creativity in Math

In a study by the US Department of Education, 81% of 4th graders reported having a positive attitude towards mathematics but that number drops significantly to 35% for 8th graders. Somehow, in the span of four years, children lose their interest in the subject, and as a result their performance declines. Professor Eric Mann, believes that “keeping students interested and engaged in mathematics by recognizing and valuing their mathematical creativity may reverse this tendency.”

In fact, research has shown that creativity can actually help students acquire content knowledge. But, how can we encourage creativity in mathematics, a subject usually considered linear and inflexible? A few people have come up with ways to twist math problems to make them more creative and fun. So, here are three simple ways to add more creativity in mathematics.

Problem Finding

Problem finding, or problem posing, in any domain is considered to be an important and integral aspect of creativity. For this activity, students are asked to come up with as many different problems as they can with a given situation. One example that Harpen and colleagues used to evaluate creativity is:

In the picture above (Fig. a), there is a triangle and its inscribed circle. Make up as many problems as you can that are related to this picture.

Most students were able to come up with problems involving Lengths (“If the triangle is a right triangle, the hypotenuse is 2, another angle is 60o, find the radius of the circle”) or Areas (“Given the radius of the circle r, find out the minimum are of the triangle”). Creative students were able to pose questions from different fields of mathematics including less obvious ones like probability (“If you are to drop something to the circle, what is the probability of it falling into the triangle?”).

Divergent Thinking

Mathematical problems with many possible solutions can help build divergent thinking skills. Exploring multiple solutions forces students to look beyond the obvious and . One such problem posed by Haylock is:

Given a nine-dot centimeter-square draw as many shapes as possible with an area of 2cm2, by joining up the dots with straight lines.

Most children come up with the easiest solution of drawing a rectangle 2cm x 1cm, but finding other kinds of shapes gets harder.  One highly creative solution shown in the picture above (Fig. b) uses two new ideas –  connecting non-adjacent dots and using an internal angle of 315o.

Overcoming Fixation

A key aspect of creativity is to break free from routine patterns of thinking (flexible thinking). By forcing students to drop their established mindsets helps them in examining a problem from different perspectives and arriving at better solutions. In the same study, Haylock gave students a series of questions in which the students are asked to find two numbers given their sum and difference. The first few example use only positive integers which sets the student’s mindset to expect solutions that use only positive whole numbers. Then the students are asked:

Find two numbers where the sum is 9 and the difference is 2.

A surprisingly large number of students assert that this is not possible. The more creative students were able to remove the self-imposed constraint of using only whole numbers and get to the right solution (5.5 and 3.5).

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Connecting the Dots

“Creativity is just connecting things. When you ask creative people how they did something, they feel a little guilty because they didn’t really do it, they just saw something. It seemed obvious to them after a while. That’s because they were able to connect experiences they’ve had and synthesize new things.” ~ Steve Jobs

The ability to connect unrelated things, or associational thinking, is a fundamental process that underlies creative thinking. Professor Mednick, who created the Remote Associates Test for creativity, defines the creative thinking process as “the forming of associative elements into new combinations which either meet specified requirements or are in some way useful.” The more remote the elements, the more novel the solution. Consider how the structure of Benzene was discovered. August Kekule had a dream about a snake eating its own tail, which he couldn’t shake off. By connecting that image with his work on chemical structures he got the idea of the cyclic structure of Benzene, thereby making a significant contribution to the understanding of aromatic compounds. But is there a better way to increase creativity without having to wait for serendipity?

The answer is surprisingly simple – by actively looking for different associations! MacCrimmon and Wagner developed a software tool that can help find useful connections. They make a distinction between two kinds of connections – internal and external. As they describe it, “Internal connections are those between elements of the focal problem itself. External connections are those between the focal problem and external factors.” Internal connections can be discovered by combining various form and function attributes in different ways. Examples of external connections are finding connections with related problems or using random stimulus like poem fragments to trigger ideas. The 3-step creative problem solving approach encompasses the process of finding connections through dissection (internal connections), manipulation and association (external connections).

Playing games that help build associational thinking can improve creativity. For example, in  the Crime Scene Investigation (CSI) brainteasers at MindAntix, users have to find a plausible scenario for how a make-believe crime could have been committed. To do that, they have to incorporate some random pieces of evidence (like a feather or a belt) that were found at the crime scene. Similarly, in the Wacky Inventions category of brainteasers, users have to combine unrelated objects in interesting ways to create a new product. These games use random stimuli to trigger the brain to think in different directions.

And the simplest of all associative thinking games? Spotting shapes in clouds! A similar technique that Leonardo da Vinci used often was to throw a paint filled sponge at a wall and try to make sense of the meaningless stains. His ability to make remote associations helped him in connecting unrelated systems leading to his numerous inventions. It’s always possible to find some way to connect unlikely elements, even if that leads to bizarre ideas occasionally. Like da Vinci himself said, “Realize that everything connects to everything else.”

 



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MindAntix Brainteaser: Make-it-Better

The Wright brothers, Orville and Wilbur, after experimenting with gliders for a couple years, built and tested their first powered plane in 1903. The flight lasted 59 seconds. The next year, after making some design improvements, the brothers managed to stay in air for more than 5 minutes. And finally in 1905, they broke all records by flying 24.5 miles in a little over 38 minutes and landing safely when the fuel ran out.

Interestingly, despite having witnesses and photographic evidence, people were skeptical that two bicycle repairmen, with no expertise in designing airplanes, would have beaten well-funded experts in the field who were actively building their own planes. In fact, a 1906 article on the Wright Brothers in the Paris edition of the Herald Tribune was captioned “FLYERS OR LIARS?”. It took another couple of years for people to finally accept that the Wright brothers had indeed managed to create a flying machine. So how did these two amateurs end up outthinking the experts?

To fully understand that, you have to look at what some educators believe our current education system lacks. Dr. Maureen Carroll, Director of Stanford University’s Research in Education & Design Laboratory, is an advocate for introducing Design Thinking into the K-12 classroom. Our educational focus, thus far, has been on building analytical thinking skills. But, as she explains, “While analytical thinking is critically important, design thinking blends in equally powerful creative thinking.” And, “It’s not that creative thinking is more important… a blend of both types of thinking are more productive for finding truly unique and transformative innovation.”

So, what does the design thinking process look like? As Dr. Carroll and her colleagues describe, the design thinking process has six key components – Understand, Observe, Point of View, Ideate, Prototype and Test. This is an iterative process, and not a linear one. Making prototypes and testing helps in understanding what works and what doesn’t, and in modifying the point of view.

How does this all relate to the Wright Brothers? Essentially, what made the Wright brothers succeed, was their exceptional design thinking skills. In an analysis of the Wright brothers’ thinking, Johnson Laird proposes that the brothers superior reasoning skills gave them the edge over others. Wilbur first spent three months reading up about aeronautical history and recognizing some of the gaps in the knowledge (understanding). They also developed their own, unique point of view on what factors would be most important in designing airplanes. For instance, while Wrights’ contemporaries believed building a light but powerful motors was key, the brothers believed that the ability to control the the plane was more important. They also used analogies from bicycles and nature to design specific parts of the plane (ideation, creativity). And of course, they spent years observing, iterating and building prototypes to test out their ideas.

Encouraging design thinking is the goal behind the Make-It-Better category of MindAntix Brainteasers. The goal is to look at everyday objects – understand how they evolved the way they did, observe how people use them, develop a point of view about what could be improved about them and then come up with ideas on how to make them even better. Design thinking, like other creative problems, helps build both critical and creative thinking. And because of the focus on users, it also helps build empathy. As Carroll and colleagues explain, “Empathy develops through a process of ‘needfinding’ in which one focuses on discovering peoples’ explicit and implicit needs.”

After my son had done a couple of these brainteasers, he identified his own problem –  he wanted to make stickers better. His problem was that stickers lose their stickiness quickly when you try to use them on different shirts (well, it was a problem for him). His solution was to attach the sticker to one magnet and use another magnet to hold it in place. Not bad for a six year old!

Developing basic design skills isn’t hard, even without having to prototype and test. There are hundreds of objects we interact with everyday that are waiting to be improved upon. All it needs is an inclination to pause, reflect and imagine.

 

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MindAntix Brainteaser: Twist-a-Story

In a study in the late 80s, researchers gave a group of 4-6 year olds the following information:

  • “All fishes live in trees.”
  • “Tot is a fish.”

They then posed a question to the children: “Does Tot live in the water?” This syllogism was presented in two different ways – as matter of fact or with a make-believe prompt like “let’s pretend I’m on another planet”. What the researchers found, turned a long held assumption upside down. More students in the make-believe prompt group answered correctly with a “No” compared to the matter of fact group, upending the belief that imaginative thinking constricts deductive reasoning.

Scientists have wondered if our ability to tell stories, or narrative intelligence, evolved to cope with the increasingly complex social dynamics.  Prof. Kerstin Dautenhahn, who proposed the Narrative Intelligence Hypothesis, explains, “narratives play a crucial role in how young human primates become socially skilled individuals” But like the study above suggests, narratives don’t just help with social learning – they also build logical thinking. Prof. Sarah Worth believes, “we learn to reason through the reasoning provided to us through hearing and telling stories. By engaging with narratives, we practice using our narrative reason.”

But what defines a narrative? A narrative is a story with the typical structure of exposition, rising action, climax, falling action and denouement, and focuses on unusual rather than stereotypical events. Or in other words, “narratives are about ‘unusual events’, ‘things worth telling’.” This focus on the unusual is one reason creativity and storytelling are so intricately linked. But more than being intertwined, storytelling can provide a great medium for practicing creative and critical thinking.

The Twist-a-Story brainteaser is a playground to build on creativity and narrative reasoning. These brainteasers use a familiar story but add an unexpected twist which users have to use to complete the story. A story, just like other creative problems, can be dissected into its various elements – plots, characters, events etc. which can be manipulated in different ways to make new creative stories.

These brainteasers help build the different kinds of creativity that Margaret Boden, author of  “The Creative Mind”, describes – exploratory (exploring a given space of concepts), combinatorial (combining existing concepts into new concepts) and transformative (changing the rules that delimit conceptual space). Examples of combinatorial creativity, posted by users, include the third Little Pig using Karate to fend off the fox when he couldn’t finish his brick house, or Snow White using incinerating liquid to defeat the evil queen (combining new elements, like karate and incinerating liquid, into the solution).

Stories are a great way to nurture creative thinking and reasoning skills, even when you don’t start from a blank page. You can always use existing stories to grow your creative and narrative thinking. The next time you read a story, try to change something and see how a new narrative emerges. As Ralph Waldo Emerson once quipped, “There is creative reading as well as creative writing.”

 

 

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An often overlooked but important aspect of Creativity

One of the earliest people to recognize that posing questions and finding problems can be an invaluable tool in learning was Socrates. Almost 2,500 years ago, Socrates developed an approach of asking questions (elenchi) to reach a state of contradiction (aporia) to help discover new insights for the concept under study. Even though he was eventually found guilty of “corrupting the minds of the youth” and sentenced to death by drinking poison hemlock, his ideas survived and influenced the present-day scientific method.

Jacob Getzels and Mihaly Csikszentmihalyi, leading figures in the field of creativity, have explored the role of problem discovery in creativity. In a landmark experiment, they brought in art students who were given the task of drawing still life from a selection of objects. They found that students displayed one of two behaviors – problem-solving students spent less time choosing and manipulating an object they painted, while problem-finding students spent considerably longer examining and manipulating their objects. What they learned next was quite interesting.

The problem-finding artists generated paintings that were judged to be more original by a panel of independent experts. What was even more fascinating was how these artists fared in the long run. Getzels and Csikszentmihalyi measured the success of these students seven years after the experiment and again after another eleven years. They found that problem-finding students were the most successful in their careers as artists compared to problem-solving students, many of whom had abandoned art altogether!

Problem posing isn’t just relevant in the art domain – it extends to even mathematics, a field conventionally not considered creative. In a study conducted on creativity and mathematical problem posing, researchers asked high school students in US and China to come up with as many mathematical problems in different tasks. An example task was a figure of a triangle with an inscribed circle where the participants had to make up problems related to the figure. Researchers then evaluated the responses on the fluency, flexibility and originality – key dimensions of creativity. They found that the more mathematically advanced students were also more creative in posing problems compared to their peers. Professors Singer, Ellerton and Cai, who study mathematical education in the different parts of the world, summarized as follows: “Problem posing improves students’ problem-solving skills, attitudes, and confidence in mathematics, and contributes to a broader understanding of mathematical concepts and the development of mathematical thinking”.

Creativity flourishes when problem finding meets problem solving. Professor Edward Silver, who conducts research related to teaching and learning of mathematics, observes, “The connection to creativity lies not so much in problem posing itself, but rather in the interplay between problem posing and problem solving. It is this interplay of formulating, attempting to solve, reformulating, and eventually solving a problem that one sees creative activity”.

Problem finding is at the core of MindAntix – users not only solve creative problems but are encouraged to find new problems that they have observed or discovered in the process. Problem finding, while often overlooked, is a meta-skill applicable to many different domains and is an indicator of both creativity and excellence.