Divergent Thinking: A Better Reason for Teaching Math

Divergent thinking is why Gattegno’s textbooks feel strikingly different.  I credit his textbook to the positive transformation of how we homeschool and divergent problems are at the core of this. 

When you have creative children running around, basic math problems are a real bore. You can stick all sorts of cute clipart on the page, but after five problems, they’ve lost interest. The page begins to be filled with doodles.

Do you have a doodler? Doodling is known for setting the stage for divergent thinking, but what is divergent thinking?

What is divergent thinking?

divergent thinking
convergent thinking

Divergent thinking is a creative process of exploring many possible solutions to a problem. It is also looking at an idea from many viewpoints. 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. The problem is that most 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 standardize tests are convergent problems with only multiple choice answers.

What does a divergent problem look like?

Divergent problems differ in that there are many solutions to one problem. It is important to note that no solutions are offered to choose from. Instead, many solutions are created during the divergent exploration. Gattengo sets these types of problems up in the form of number studies.

Children explore each number finding all the ways to build it. Because multiplication and fractions are taught side by side with addition and subtraction, the diversity of solutions is endless.

For example, currently Sonya’s child at Arithmophobia No More is exploring the number six. When you check it out, you will see the depth of exploration in which the child expresses six in fractions, addition, multiplication and fractions.

Why are divergent problems important?

The problem is simple. “How many ways can you create six?” The solution are many. This problem permits for creativity. More importantly, the child reasons, from his own understanding and experience with the Cuisenaire rods, to create his solutions.

openness of divergent thinking

It forces a deeper understanding of ideas and how to manipulate them. That is important in today’s world.

We have solutions to many problems that are lacking. Because of the strictly convergent thinking, people tend to cling to these poor solutions. They are hesitant to go off the beaten path. This leaves us in a cycle of dealing with the same problems.

Even more so, as textbooks remove the journey of discovering a solution and hand children a black and white world, children grow up to have no compassion for the gray area where people often need to journey to find the right answer.

The Journey of Divergent Thinking to Convergent Thinking

Divergent thinking leads to convergent thinking

It maybe that the worse part of avoiding divergent problems is that there is no practice for true convergent thinking. While we have convergent problems in standardize test around the world, it doesn’t necessarily mean there is a thorough development of convergent thinking.

Convergent thinking is about synthesizing information. In the best problem-solving processes, you begin with divergent thinking. You approach the problem from many viewpoints. However, natural rules and constraints, force us to converge our divergent thinking into a clear, efficient solution.

Let’s look back at our example. How many ways can you create six? This open-ended question requires you to converge, that is synthesize all that the student has learned previously to create the diverse solutions.

The mathematical properties of Cuisenaire rods constrain the child to find the right solutions. It forces the child to converge their understanding of the rods and the language of math.

For example, a child sees a red (2) rod plus a purple (4) rod is the same length of dark green rod (6). The child knows that orange (10) is much longer than dark green. It constrains the child to think differently on how it uses the orange rod. Understanding differences, the child ponders the difference of the two rods to find a solution to represent dark green.

Divergent Thinking as Playground for Convergent Thinking

Divergent problems are not void of convergent thinking. There is a right answer. It's just that there are many right answers. Gattengo’s textbooks offer a playground for the development of both divergent and convergent thinking.

Without convergent thinking via rules and constraints, we may have lots of solutions but maybe no best process or best solution.  When we are stuck with convergent problems, we are limited in our practice of converging information. Divergent thinking opens the door for a more thorough development of convergent thinking.

convergence streamlines

Let's look at a larger expression, 56 x 23 = ?  There are lots of ways to solve this problem.  You maybe acquainted with "new math."  "New Math" is about teaching the child several different ways to solve a problem.  The problem is that children struggle to remember math facts.  Adding tons of processes is only making math more difficult for children.

Instead, through Gattegno's textbooks, students have been manipulating Cuisenaire rods long enough to know a few basic principles behind math to find a process to solve the problem.  At first, the processes they use are cumbersome and lengthy.  But through experience, the student begins to converge their understanding into one clear process.  

Because the student worked through and discovered an efficient process of solving the problems and similar ones, the student's understanding is fluent and memorable.  

This sounds like "New Math"?

It seems the greatest error of "New Math" application is bypassing this divergent process. In bypassing the process, kids are given a whole toolbox of ideas to memorize. Student's are already struggling to memorize math facts.  How does this solve the problem?

The problem is its just too many tools at one time without any strong context.  The student needs time to tinker and find the right tool for them.  I am certain this was the original intent, but like all bureaucratic systems, it got lost in translation. I believe Gattegno provides the structure to accomplish what many hoped "new math" could bypass. 

generalizing comes from convergent thinking

I tried the traditional textbook for many years.  It led to shallow math understanding and a lot of frustrated boredom.  I saw another child going through "new math" completely frustrated by all there was too learn.  Gattegno has been that key in turning math around for us. 

My creative children love to tinker, and Gattegno gives them permission to explore and discover math for themselves. I have also become a better math teacher, too. 

I remember when tinkering with building small sequential cubes and squares using Cuisenaire rods led to my son's discovery of an algorithm to find larger sequential cubes and squares.   The divergent process that led to that convergence was messy and cumbersome.  It was a process that was his own, and the discovery had meaning to him.  I knew he understood it because I didn't teach it to him.  He taught it to me. 

Why Divergent Thinking is Scary

It is the messiness and cumbersomeness that is scary.  There is an imaginary timeline that looms over us, and it doesn't have messy and cumbersome on the agenda.  That takes a lot of time.

It is why children get left behind.  the system doesn't understand the most efficient form of genuine learning can appear cumbersome in the beginning.  

It also doesn't trust the child to search for efficiency.  The brain, however, is designed to look for efficiency.  The brain uses up massive amounts of energy. Because of this, it is always looking to streamline any process.  

In teaching, we too want to streamline the process for them.  We want to bypass this cumbersome divergent process and give the student the streamlined algorithms already discovered.  I would suggest the reason we default to bypassing divergent thinking and jumping right into algorithms lies behind our reason for teaching mathematics. 

Often we see mathematics only purpose is to know how to handle daily arithmetic like counting back change, paying bills, passing the SAT, etc.  Because of our inherent drive towards efficiency, we invented calculators to streamline such basic activities.  If our reason for teaching mathematics lies in performing basic calculations for everyday use, we should see the calculator robs us of this reason.  

Maybe we should teach mathematics because it is an excellent playground for developing thinking skills.   In this reasoning, we can slow down and dig deeper and tinker with math.  But is every child suited for this kind of mathematics?

Is my child a divergent thinker?

Studies have shown that children have a much larger capacity for divergent thinking. With progression to adulthood, this capacity seems to dissipate probably due to lack of practice in divergent thinking.

Chances are that if you have an imaginative child, you probably have a divergent thinker. Divergent thinkers tend to be open, creative and thoughtful, even a slow thinker. They may arrive at what appears to be wrong answers, but when asked, they can justify those answers in clear, logical reasoning.

Gattegno felt every child was capable of deep mathematical fluency.   He believed a child's ability to pick up language and walk demonstrated the algebraic thinking present in the child.  I would argue it is more than that.  It is also evidence of the divergent and convergent thinking processes being developed.

The messy divergent beginnings of knowledge is converging rapidly today.   The best solutions won't be attained without the willingness and ability to continue to approach problems from a divergent process.  Likewise, the best and most efficiency solutions won't be attained without convergent thinking.   I believe here is a better reason for teaching math, creating a generation of problem solvers. 

Next post, I will dive into divergent problems and questions we use daily and how last year’s Gattegno conference was an the experience I needed to make the best change for our homeschool. 

Lacy | Play Discover Learn 24/7

Knowing the best kind of learning comes from a highly motivated internal drive, Lacy Coker cultivates tools and resources that help to make learning for young children playful and self-directed.

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