When One Correct Math Answer is Not Enough

Two Kinds of Thinking

Divergent thinking means looking for many correct solutions. The opposite of divergent thinking is convergent thinking. Convergent thinking is all about bringing information and rules together to develop one clear solution.

Both types of thinking are important.

Divergent math activities have many correct answers.

Math in school often focuses on math activities that encourages convergent thinking and most math textbooks offer only convergent thinking problems. Convergent problems are easy to recognize. There is only one right answer and usually, only one process to get to the answer. Many standardized tests are convergent problems with only multiple choice answers.

A typical math problem in first grade often looks something like this, 32 +5 =­­__.  Generally, the only answer that would considered correct would be 37. Once a first grader answers the problem correctly once, they can stop thinking about the problem and move on to something else. Setting up problems like that encourages convergent thinking.

Reframing the above problem to encourage divergent thinking might look something like this:

“Find the sum of 32 + 5 and then create more equations that have the same sum.”

Theoretically, someone could keep thinking about this problem forever and generate infinite correct answers.

Six Benefits to Math Activities Encouraging Divergent Thinking

1. Math activities that encourage divergent thinking help students develop the ability to examine situations creatively. The more kids learn to keep searching for solutions, the more they have to stretch their creativity and think about problems from different perspectives.
2. It is easier for groups, like families, to find activities that fit several levels of math abilities. Every child can work at their own level and still feel successful. You could even award points for creating equations no one else in the group thought of. That will encourage kids to work to push their limits of mathematical creativity. Take the previous example, “Find the sum of 32 + 5 and then create more equations that have the same answer.” A first grader may come up with equations like 31 + 6, a third grader may come up with 900 – 863, a fifth grader may come up with .9 + 36.1, and a seventh grader may come up with 9² – 44, and so on. Children who are at a lower math ability will get exposure to higher level mathematical thinking. Children at a higher math ability will look for ways to articulate their mathematical thinking more simply.
3. Math activities that encourage divergent thinking tend to be more like “real-life” math problems. Problems in real life are usually complex and we have to consider many possible solutions. An example of a real life math problem needing convergent thinking would be “Is $40 enough money to buy two adult and two child movie tickets to the matinee?" An example of a real life math problem needing divergent thinking might be, “How can we redecorate a bedroom with a budget of $200?” The answer will require looking at the price of different paints, flooring, and furniture and compare the pros and cons of each cost. There is no one right answer, but some solutions are better than others.
4. Math activities that encourage divergent thinking foster cooperation, innovation and diversity. Since a better solution is always possible, the more heads thinking about it the better. Diversity of thought and approaches will be valued for increasing the likelihood of finding the optimal solution. It won’t always be the same few kids that always get the “right” answer.
5. Kids feel less pressure to be fast. Instead of kids being told they having to complete 30 problems in 45 minutes, they are told they have 45 minutes to come up with the best solution they can.
6. Kids explore math ideas in-depth and learn how to work with multiple steps. This goes along with spending more time and energy on more complex problems. Even if a child uses a calculator to figure out some of the math, they will still be using mathematical reasoning.

Family Math Activities to Encourage Divergent Thinking

Ready to try some divergent math activities? When you do a divergent math activity it is best to set a time limit. Since there is no end to number of answers, a time limit will give a natural end to the activity. Here are a few activities to get you started.

• Create a budget for something, like a meal or a party, and have the family come up with possibilities.
• Create menus for individuals to meet their nutritional requirements.
• Figure out the best mode of transportation and route to a destination taking into account the cost, travel time and number of people.
• Make a game of creating equations for a quantity, such as in the example above.
• Play a game, like Monopoly, that requires math and a strategy.
• For more ideas read 10 Fun Ways to Incorporate Math During School Breaks

Mathnasium and Divergent Thinking

Some of our methods and curriculum encourage divergent thinking. We also play games and have “warm-ups” that encourage divergent thinking. Have more questions about our program? Give us a call today at 303-840-1184

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