'Integrative Math'


by Cynthia McCallister

Summer 2016. To the memory of my friend and inspiration, Jerry Bruner, October 1916-June 2016.

A new instructional procedure developed by Cynthia McCallister, called Integrative Math©, combines elements of Cooperative Unison Reading®: and modes of representation Jerome Bruner presented in his book, Toward A Theory of Instruction (1974). Students use the rules of Unison to read story problems. Then, using manipulatives, crayons, markers, and stories, they integrate enactive, iconic, and symbolic modes of representation, strengthening their mathematical thinking.

Integrative math integrates the way our minds work to unite diverse systems of memory so that they can be deployed and used for problems that come up in real-life situations.

For example, Integrative Math is an example of 'bihemispheric integration,' because the procedures let students engage both sides of their brains as they do activities focused on solving the problem at hand. The context-sensitive, receptive right brain handles the context-dependent, face-to-face, non-verbal communicative aspects of the activity, while the left-brain, logical-sequential, assertive left brain does the work of describing and encoding the experience. 

After doing the activity for a half hour, one graduating senior from Urban Assembly High School for Green Careers said he didn't want to graduate now that the kids were getting to play with blocks again!

Ready to learn how to do the activity? Here’s how Integrative Math works:

  1. Use the rules of Cooperative Unison Reading® (a group of no more than 5 people reads a text together in sync, breaching with questions or comments, and being promotive). This is how my ideas about instruction come into play.
  2. The text to be read is a word problem.
  3. The special rule that applies in this situation is where Dr. ‘Jerry’ Bruner’s fascinating ideas come into play.

Before I explain the next step, I’ll take a minute to explain how Bruner’s ideas come into play.

Bruner developed an insightful way of thinking about the way human beings represent ideas to themselves. He reasoned that people have three ways of representing information that unfold, one after the other, but also work interactively to support complex thinking.

Enactive: The first form of representative thought is stored through action. It’s in our bodies and muscles. It’s what Bruner called the enactive Mode of representation. It’s dominant at ages 0-1 years of age.

Iconic. The next type is the iconic mode of representation, in which we represent images as mental images in our minds. Diagrams and pictures are a form of iconic representation. Children ages 1-6 begin to rely upon iconic forms of representation in their thinking.

Symbolic. The symbolic mode of representation is the final form of thought, and it’s stored in our memory as a code or symbol that is able to represent something else. This ability—for the code to be flexible enough to stand for something different than the code itself—allows human beings to free up their thinking from the constraints of the ‘here and now’ boundaries of situations. This form of thinking allowed humans to conjure up the ideas that created mathematics and science and enabled civilizations to compile their histories.

Bruner thought that kids of almost any age could tackle most conceptual problems if the material were arranged in the appropriate sequence. That’s where Integrative Math as a protocol comes in. Which brings me back to Step 4.

  1. In Unison, the group reads the first sentence, breaching mid-sentence if necessary to work out any problems. If at any point—mid sentence or at the end of the sentence, any concept that arises can be put into an Enactive Mode of representation, it has to be done! In simple terms, if you can put the idea into a form of action using manipulatives, do it! Everyone at the table has to do it personally. But interpersonal stealing of great ideas is totally allowed. Look around the table and see who has a good idea. Ask what they’re doing, and steal their idea. Everyone needs to put the idea into action—to create the idea as an Enacted Mode of representation through manipulatives.
  2. Now each person needs to explain what they did. Once everyone has had a chance to explain, the group collectively decides which idea was the best, and explains why it was the best. This step is a good thing to do for your thinking. It takes an experience that was in your ‘muscle memory’ or that was handled mostly by your ‘right hemisphere’ of the brain and uses the ‘left hemisphere’ of the brain to explain. In other words, it integrates the two systems through the activity. Cool, huh?
  3. Next, you got it, we move to the Iconic Mode. Now everyone at the table needs to put their action into an image. Take out the white boards and markers or scrap paper and crayons. Same drill. Be a thief of great ideas. Lots of kids think they’re bad at drawing. If that’s you, is a time to rob shamelessly!  Then everyone explains their drawing, and collectively the group decides which is best.
  4. Finally (oh, you’re so smart!), we move to the Symbolic Mode of representation. Take the picture and move it into symbols. Use the procedure you already know so well from Steps 4-6, above. Feel free to take ideas from others. But everyone has to write their own symbol sentence.
  5. Decide as a group which person had the best symbol sentence, and explain why.
  6. Now, follow the rules of Unison, reread the sentence in its entirety before you go onto the next sentence. 
  7. Every time you encounter a breach or a sentence to which enactive-iconic-symbolic modes of representation can be applied, you need to apply them using the procedures you just learned.

 Now you have the tools to become a math genius!

Note: 

Theories embedded in this method:
Cooperative nature of human thinking and creativity (Tomasello)
Mindsight and neural integration (Siegel)
Modes of representation (Bruner)
Socio-relational theories of instruction (McCallister)

© 2016     Cynthia McCallister    All Rights Reserved

Integrative Math©, Cooperative Unison Reading, and other Learning Cultures methods are featured in online courses this Fall on www.LearningCultures.net.

 




Cynthia McCallister
Cynthia McCallister

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