Baby Steps

On the topic of supporting the development of mathematical communication, today I want to address patience.  One of the greatest obstacles to learning is the fear of making mistakes.  In other areas of study, we know that mistakes are the path to learning.  In Math class, we get points taken away for every mistake we make, and if the teacher doesn't understand the steps you made to get to the wrong answer, we get no points at all.  But in fact, we don't want children to know what steps to show; we want them to understand how to show enough so that their audience can follow them. We, as guiding adults, need to exhibit patience with our children as they learn to communicate their mathematical ideas.

When children are learning to write in the early years, we encourage them to get their ideas on paper without fussing about perfect spelling and punctuation.  When they are learning to express themselves as toddlers, we give them the words we think they are trying to use.  We need to do the same with "showing steps."  Before we ask children to show all of their work on paper, we need to talk out their reasoning with them.  We have the advantage in that we can make educated guesses about how they came up with a solution to a problem or what steps they might take.  The back and forth of a conversation allows the child to make baby steps in learning to communicate by both watching and hearing the adult model, and making small attempts themselves and getting immediate feedback.  As we converse, we might write the parts on paper that we think would go into "showing work."  Eventually, the child will make their own written attempts.  

The process of learning what steps to show takes time and is unevenly paced.  A child not only needs to learn the language and vocabulary of math, but also takes years to understand that different audiences require different steps to be shown or different ways to show them.  We need to embrace that what goes on inside the head of a child is usually more than they can express well.  Rather than the patience of waiting -- waiting for the child to mature when one day they will magically "get it" --  what is required is the patience of building.  Celebrate each small success.  Laugh over misunderstandings.  Learn from mistakes.  It's all valuable.

Supporting Communication Through Understanding A Child's Process

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. 

Communication II

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... 

• Understanding a Child's Process
• Baby Steps