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.
