Earlier I wrote about the importance of supporting the development of mathematical communication in a child. I thought break my thoughts up into chunks. Today, I'd like to talk about listening to understand a visual thinker.
One of the difficulties in communicating with a visual-spatial person, especially a child, about how a solution is reached is that they often understand things in pictures. For one thing, there isn't really a sequence of steps they took to arrive at "an answer." Rather, everything is there all at once in the picture. For another, since we converse using language, ask children to explain themselves using language, and often model using language to explain, children expect to explain themselves using language. As children get on in Math, this might evolve into a habit of trying to explain steps in a process using numbers or equations. We need to break out of these two boxes.
Continued from an earlier post...
Now where was I? (That's the trouble with taking quiet time in the morning, rather than at night: there's no way to steal from your sleep to get a blog post finished.) Right, communication.
What brought on these thoughts about helping children communicate their math? It was a combination of things that grabbed my attention this year. One contribution was the oft-repeated assertion that visual-spatial thinkers "just know" solutions to problems. The other was the similarity in approaches used by teachers in a variety of subject areas: In The Writer's Jungle, Julie Bogart emphasizes the importance of helping children express themselves by first listening to them and scribing for them, then offering your own words and structures to clarify or help them to better communicate their ideas. Grammar, spelling, punctuation are all there to support communication; they aren't ends in themselves. When my daughter brought her music composition to show her teacher, her teacher didn't critique her very unusual timing (In fact, she commented that she found it interesting.) but rather helped her by showing her how to add the bar lines and time signatures that would allow readers to understand the sounds she wanted to create. Visual art, too, has been described as sharing with the world what you see the way you see it. Learning to communicate in math needs the same type of support as learning to communicate in visual, linguistic, and musical mediums.
In North American society today, we tend to think of Math and Sciences as going hand-in-hand, perhaps because the Sciences so regularly use Math to communicate and understand their own ideas. However, Math is not itself a Science. There are parts of Math that are irrefutable and reproducible; those are the parts that seek truth in the way Sciences do. Math is also about beauty and creation; those parts are like the Arts. To validate only the mathematical ideas of a child that agree with a textbook is akin to accepting only the drawings of a child that are exactly like the given sample. To ignore or correct a child's own mathematical ideas is to dampen the mathematical spirit in him.
How then do we support the development of mathematical communication?
To be continued in...
Lately, I've been hearing a lot about kids who "just know" an answer to a problem and don't know how they got it. I must say, at first I felt a bit perplexed. In all my years of teaching and tutoring, I have never come across a single student who "just knew" an answer consistently and couldn't show work. Even if they began by saying that they didn't know how they knew, we could always tease out the line of thinking that led to their answer. What I believe, then, is that it's not okay to become complacent with these very intuitive kids. Showing work for marks is not necessarily a goal for everyone, but being able to communicate thoughts is an important step in learning. Yes, communication is necessary for the sharing of ideas that leads to piggybacking and synergy. More importantly, though, successful communication ensures that a child will understand that their mathematical ideas are valuable, that their logic is valid, that they are capable of "doing math."
Continued...
