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? Several researchers have found ways to make math more creative, fun and engaging. Here are five simple ways to add more creativity in mathematics.
1. Make Problems Open-Ended
Giving students open-ended problems where multiple solutions are possible, as opposed to the traditional one right answer problems, allow students to experience the first stages of mathematical creativity. Traditional mathematical problems can be converted into open-ended problems relatively easily. Consider an example where students have to find the volume of an aquarium that is 12in wide by 14in long by 12in high. An open-ended version of the same problem was:
You have been asked to design an aquarium in the shape of a rectangular prism for the school visitor’s lounge. Because of the type of fish being purchased, the pet store recommends that the aquarium hold 24 cubic feet of water. Find as many different dimensions for the aquarium as possible. Then decide which aquarium you would recommend for the lounge and explain why you made that choice.
The open-ended nature of the problem allowed students to compute multiple options and use additional strategies in picking their final design. For instance, one group picked 8ft long by 1ft wide by 3 ft tall as the length allowed teachers and visitors to not crowd around the aquarium.
2. Have Students Create Their Own Problems
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. For example, consider the following situation:
5 boys and 5 girls are standing in a line.
A simple problem based on this situation, which uses just addition, is “what is the total number of children in the line?” However, many other problems could be posed using different areas of math including combinatorics (“if no child gets surrounded on both sides by children of the same gender, how many different ways could you form the line?”) Research studies have found that creative students are able to pose questions from different fields of mathematics including less obvious fields.
3. Build Divergent Thinking Skills
Mathematical problems that challenge students to think in different ways help build divergent thinking skills. An example of such a problem from the Creative Ability in Mathematics Test is:
Suppose that instead of paper or whiteboard, you could only draw geometry figures on a large ball or a globe. List all the possible things that would happen as a result of doing geometry on the ball. For example, if we started drawing a “straight” line on the ball, we will eventually end up where we started.
Some of the more common responses are that geometrical figures would be distorted and measurement of distance would be different. Less common responses that reflect higher originality include that Pythagorean theorem would change or that there would be a need to establish a new mathematical system.
4. Overcome 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 examples 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. To get the right solution (5.5 and 3.5), they need to remove the self-imposed constraint of using only whole numbers.
5. Encourage Analogical Thinking
While analogical thinking is often considered relevant in the scientific domain, it is a cognitive skill that underlies creative thinking and is equally relevant in mathematics and other fields.
George Polya, the renowned mathematician, recommends asking students questions as they solve a problem that taps into their analogical thinking. For example, “Can you know of a related problem?” or “Can you think of a simpler problem?” and similar questions can nudge students towards using analogical thinking in problem solving.
Consider the following problem by Polya:
The vertex of a pyramid opposite the base is called the apex. Let us call a pyramid “isosceles” if its apex is at the same distance from all vertices of the base. Adopting this definition, prove that the base of an isosceles pyramid is inscribed in a circle, the center of which is the foot of the pyramid’s altitude.
In this case, by using the analogous theorem for isosceles triangles – the foot of the altitude is the mid-point of the base in an isosceles triangle – it becomes much easier to solve for the pyramid case.
Further Reading
- Education Week – Ditch the Math Worksheets and Stop Killing Kids’ Curiosity
- The New York Times – Math Teachers Should Be More Like Football Coaches
- Raleigh News & Observer – As some teachers defend controversial math curriculum, critics say it’s causing anxiety, PTSD