September 1997

Math for an Information Age

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Kids who lack a basic number concept when they start school don’t do well in math. Can we teach them the concept?


By Rosemarie Bahr

Five Grade 1 students are playing a game. They’re delivering letters in a neighbourhood that consists of a line of 40 cardboard rowhouses. The houses all have numbers on their doors, from 1 to 40. The first 10 houses have the same colour doors. The second 10 doors are a different colour. The even-numbered roofs are flat. The odd-numbered roofs are pointed. The kids are finding different ways to get to house 34 from house 15. They’re having fun, and they’re learning math.

Robbie Case

This is part of Rightstart, developed during the research carried out by Robbie Case, a fellow in the Human Development Program of the Canadian Institute for Advanced Research and professor at the University of Toronto.

Our kids don’t do as well as they should

International comparisons show that North American children don’t do as well in math as their Asian peers. Young Brazilian street vendors, children with no formal schooling, also do quite well compared to Canadian children.

Other studies reveal that North American children have a systematic misunderstanding of some math concepts they have been taught. For Robbie Case, these problems indicate a need to reshape the teaching of math.

Case and his colleagues first sought to discover how children learn math. They found that children develop a "central conceptual structure," which is the basis for further learning. Children who do not have this number central conceptual structure, or basic number concept, by the time they get to school, do not do well in math.

These students do not develop a good number sense, which would let them move between the real world of quantities and the mathematical world of numbers and symbols. Students who have good number sense can invent their own procedures and recognize and use number patterns. They can recognize gross errors, and they can represent a single number in many ways.

For example, suppose you have 19 marbles and someone gives you 15 more. How many would you have all together?

Many children, and adults for that matter, line up the 19 and 15 in their minds, the way they were taught. They add the nine and the five to get 14 and carry the one. Then they add that one to the one and one to get three, ending up with 34.

People with good number sense, however, intuitively invent an easier procedure on the spot. It might be to add 15 and 20 to get 35, then subtract one to get 34. Another method would be to add 15 and 15 to get 30 and then add four to get 34.

Both methods are based on an understanding of the base 10 nature of our number system. In the second procedure, 19 is shown as 20 minus one. This change displays a confidence that their own procedure is valid and shows an attitude that the students have somehow made mathematics their own.

Can basic number concept and good number sense be learned?

Case sees a connection between numerical and spatial knowledge. Mathematicians often say their intuitions have a strong spatial component. Adults with brain injuries that impede spatial understanding often also have difficulty with numbers. Children who are good at reading but poor at math also often show weaker spatial abilities.

If there is a connection, it should be possible to create a learning experience in which numbers are linked to space and children can use their spatial intuitions to learn about numbers.

Students enjoy inventing their own equations as they
deliver the mail with Rightstart.

Case and his colleague Sharon Griffin developed a program, Rightstart, to create that learning experience. They first tried it in Toronto in the late ’80s with a group of children from three kindergarten classes serving middle to low-income families who had immigrated from rural Portugal. The second study took place in kindergartens in three schools in Massachusetts. These schools had the largest proportion of visible minority students in the city and generally drew from low to middle-low-income families.

These and later trials showed that children who went through the program demonstrated increased number sense compared to a control group. The improved number sense also showed up in related contexts such as science, telling time and dealing with money.

When the children were followed up at the end of Grade 1, they were doing much better at arithmetic than the control group. When the program expanded and was added to the first and second grade curriculum, it improved the responses in an entire population at risk for school failure. The Rightstart group became the equivalent of a more advantaged population in most respects, even superior in some.

Over the years, Case and Griffin worked on refining the games and components of Rightstart. The program is being translated into French for use in some downtown Montreal schools. Griffin, who is now at Clark University in Massachusetts, has been working on workshops for teachers to help them feel comfortable with the materials.

How does Rightstart work?

One of the keys to Rightstart is the spatial embodiment of numbers and number systems. The numbers stay fixed, as they are on the rowhouses in the neighbourhood.

"When you’re playing a board game," says Case, "the numbers are all laid out and you move through them. Math in the early grades can include bundling things, like popsicle sticks, into groups of 10 or counting with counters. But that doesn’t give a sense of how each number relates to each other number, that the teens belong together and 23 is like 33 in certain ways."

According to Case, children should have one or two contexts in which to explore the math problems. In the neighbourhood of row houses, the context remains the same, but 15 or 20 different games are played there. As the children learn, more houses are attached, using velcro, until there are 100.

Fun and storytelling are crucial. Case reports that both the children and the teacher have fun. The activities are part of a story, like delivering letters in the neighbourhood or going down a line of boxes to put out the fire of the dragon who lives in a box at the end and is terrorizing the city.

Ownership and diversity are embedded in the program. One child may use her 10-step boots to hop from house 15 to house 25, then to house 35, then walk back one to house 34 so she can deliver her letter. Another child may use his boots to hop to house 25, then walk nine steps to house 34 to deliver his letter.

Both students will tell the group about their methods. They may discuss the advantages and disadvantages and decide there aren’t any.

By Grade 1 or 2, Case reports, these kids are inventing their own equations. Just as there is no best single composition or drawing in a class, there is no one best equation. And they can all go up on the wall or home to parents. The children feel ownership. They don’t feel there is only one right mathematical expression. In that way they’re more like mathematicians and less like adding machines.

Moving on to fractions

Rational numbers (fractions, decimals, per cents) in Grades 5 and 6 are often a roadblock for students at the middle and top end of the socio-economic scale. In a national multiple-choice test, the majority of American children in Grade 12, when asked to estimate the sum of 9/10 and 11/12, picked 19 or 21 as the answer instead of two. Students have some serious misconceptions about fractions.

With Joan Moss, a teacher and lecturer at the University of Toronto’s Institute for Child Study, Case has been working on another set of spatial embodiments that starts with per cents rather than fractions, which fits better with children’s developing intuitions about proportions. This has been tried several times with advantaged students in Grades 4 to 6.

"If I said, ‘what is 65 per cent of 160?’ most people would say, ‘Excuse me, you expect me to do that in my head?’ Most of the kids who’ve come through the program have no problem with that. It’s like asking them what’s nine and 32. It’s a bit challenging, but not intimidating."

If Robbie Case has his way, all our children will be included in the information age.

The Canadian Institute for Advanced Research

The Canadian Institute for Advanced Research is a research university without walls. Established in 1982 by Fraser Mustard, it links more than 170 of the best scholars and researchers from Canada and elsewhere. CIAR gives these scholars the chance to develop and apply their work beyond one discipline and to study the effects of their work in a broader context. The research, particularly in the social sciences, generally has considerable applied relevance.

"In many areas of health and competent functioning there are differences between countries. Those that have very sharp differences – steep gradients – between people at the top and bottom of their social status scales do worse overall than countries that have more shallow gradients," says Dan Keating, a professor at the University of Toronto and chair of the CIAR Human Development Program.

"These patterns show up in many ways, including educational achievement and health. Evidence seems to emerge that the early experiential differences that are related to social differences do appear to be significantly implicated in this pattern of outcomes. There are certain kinds of experiences that help sculpt or shape the brain in ways that make it easier or harder to do other things later on.

"In mathematics," Keating continues, "as in many other things, what we do early matters a great deal, probably because the brain is so plastic early on, and if we deal with them early, we wind up with much better circumstances later."

CIAR has eight programs: population health, economic growth and policy, human development, cosmology and gravity, evolutionary biology, earth system evolution, soft surfaces and interfaces, and superconductivity.

Robbie Case is a Fellow with the Canadian Institute for Advanced Research and professor at the University of Toronto. He attended McGill University and received his PhD in education at OISE (applied psychology). Currently on leave from Stanford University, he has also taught at OISE and Berkeley.

For a copy of the research, contact the CIAR, 179 John Street, Suite 701, Toronto, ON M5T 1X4. Working papers cost $5.

For a copy of the 78 lessons and materials, write to Sharon Griffin, Department of Education, Clark University, 950 Main Street, Worcester, Mass., 01610 for information on when they will be published.