Twenty-five years ago Frank Smith wrote an article which appeared in the Harvard, Educational Review "Making Sense of Reading¨and of Reading Instruction" in which he argued children must have two fundamental insights before they can learn to read:
1. print is meaningful, and
2. written language is different from speech.
In the article, he explains the importance of these two crucial insights and the fact that young children have to figure them out by themselves in order to make sense of written language.
The Glass Wall: Why Mathematics Can Seem Difficult does the same for making sense of mathematics. The book is about understanding mathematics. As in "Making Sense of ReadingÓ" Smith similarly presents two crucial insights about mathematics:
1. the physical world and the world of mathematics are separate and independent, and although the words and phrases we use to describe the physical world and the world of mathematics are the same, the meanings are different.
2. the world of mathematics resides in our minds.
We must enter the world of mathematics itself, Smith argues, in order to understand the relationships that mathematics elucidates. In order to make sense of mathematics we have to understand it on its own terms. In fact, he contends that our everyday language can serve as a powerful barrier to understanding mathematics if we don't have these two fundamental insights: that mathematics is separate from the physical world and that the relationships it describes come from our minds.
A Story: Jane, age 3, is at the kitchen table having lunch. A child with cystic fibrosis, she has the usual a pile of pills in front of her. On this occasions she's playing with them, spreading them out in front of her, I point to one and say "one," then another and say "two." She joins in and points and says "three," "four," Ó. We count to eight¨the number of pills on the table. She pops one pill in her mouth and says "one," another and says "two," and swallows them. She turns back to the pills still on the table and counts, "three, four, fiveÓ" touching each one in turn.
Several things intrigue me about what has just happened. First, she knows the conventional counting order and is able to keep a number name in mind and count on from that. She also seems to understand the number names as something separate from the pills themselves. I think it interesting that she points to a different pill as she says each number name, an indication that she is aware that counting allocates a single number name to each object in a collection. What I'm not sure about is whether she has any sense of quantity as a relational idea separate from the objects themselves.
It's coming to understand quantity as a mental construct separate from objects themselves that constitutes the beginning of mathematical thinking. It's a mental construct which children must invent for themselves, just as humankind must have invented it once long ago.
In The Glass Wall, Smith opens with "What is Mathematics?" in which he points out that mathematics is a social, interactive phenomenon¨the product of human mental activity, reflecting a host of logical, computational relationships invented perhaps initially as people solved problems about the physical world, but which has become a self-contained world of exploration.
How did mathematics begin? We have no record. "It wasn't in the human brain that mathematics developed, but in the interaction of the brain and mathematics itself," Smith conjectures.
"Mathematics," Smith believes, "must have its roots in spoken language, but there is no evidence of how this happened, only conjecture." He points out that mathematical concepts are among the first words children say and understand¨words representing understanding of quantity, proportion and ratio, sequence, speed, change. His discussion of mathematics in language, which focuses on the intersection of language and mathematics, identifies confusions that arise when we try to overlay language on mathematical idea. He makes it clear that the mathematical terms "add", "minus", "multiply", "divide" and "equals" for example, have very different meaning from the use of these words in everyday language.
Smith devotes three chapters to numbers. He describes the meanings of numbers, the naming of numbers and their written forms. What becomes evident is that an understanding of number is an understanding of relationships numbers have with each other. The moment we refer to numbers and build with numbers we have entered the world of mathematics, he contends.
The invention of ways of writing numbers and systems for representing calculation took centuries to develop. One major invention was the positional notation of numbers¨assigning value based on position: 5 X 100, 4 X 10, 3 ¨> 543. The challenge for children is to discover why numbers are written the way they are.
The remainder of the book deals with labeling, ordering and quantifying, calculating and measuring, notation, numbers between numbers, and numbers in space. He deals with memorizing, calculating and looking up. The final chapter "Getting Beyond the Glass Wall" points out that entering the world of mathematics means making sense of the network of mathematical relationships, of seeing the numerical patterns that represent these relationships. The challenge is to be able to leave language behind, to discover the meanings inherent in mathematical ideas themselves.
Although Smith refrains from discussing what might be involved in teaching mathematics, the implications are evident. Attempting to embed the learning of mathematics within language is likely to make learning mathematics harder. There is a profusion of instructional programs based on the assumption that mathematics can be made more understandable if the procedures are "wrapped" in everyday language, if the problems are connected to the physical world. If, however, Smith is right, such programs may be responsible for creating the "glass wall"¨for misdirecting learners away from the task at hand: discovering what constitutes the relationships which make up the world of mathematics itself.
What The Glass Wall makes very clear is that mathematics is a separate domain of understanding.
While the mathematical world intersects with the physical world and the language used to describe it, mathematics, like music, is an independent creation. As Smith argues "mathematics must come from the mind of the learner." The crucial thing about teaching mathematics is to help children invent these relationships for themselves.
Another Story: Robbie, aged 3, was driving with me in the car one day. From his car seat in the back he was able to see the odometer. As I was driving down Connaught Avenue (a wide boulevard) I heard from the back seat: "Hey Jude, you're going too fast¨you're going five-ty and the sign says you're supposed to be going three-ty." (This incident took place before the introduction of the metric system; when the speed limit was 30 mph.)
Intriguing, huh? This kid had figured out a rule for representing place value in spoken language. He simply hadn't yet made the connection with the conventional words for the values.
What Robbie illustrates is the mathematical mind at work¨constructing relationships and finding ways of expressing them. At the age of three this child was a confident mathematician, comfortable exploring the domain of numbers, inventing relationships, seeing patterns, and inventing ways of conveying them to someone else. ˛
Dr. Judith M. Newman is former Professor of Education at Mount Saint Vincent University and Dean of Education at University of Manitoba. She is an education consultant living and working in Halifax, NS.