It is hard to appreciate how hard it is to teach complex concepts successfully. As adults, we have mastered quite a bit of math. We have learned to read critically. We have a basic understanding of the core sciences. And so it can be difficult to remember what it is like not to know these things.
In the first post in this series, I talked about how many of our concepts are rooted in the ways that we interact with the world. That is, our minds and bodies are deeply connected, so that even seemingly abstract concepts have their roots in the way that our bodies interact with the world.
The recognition that body and mind are connected has led to many proposals for methods for teaching complex concepts. An important lesson from these attempts, though, is that it is harder to connect mind and body than you might think.
Basic mathematics provides a good example that Usha Goswami discussed at length in her book Analogical Reasoning in Children.
In the first few years of school, children have to learn a lot about math. They need to learn that numbers represent how many things there are in a set. They need to learn that the number system has a place-value structure so that the second digit in a number represents bundles of ten items, and that the third digit represents bundles of one hundred. They need to learn that adding pairs of numbers means combining the sizes of two groups, and that subtracting involves taking away items from a larger group.
These basic math concepts are abstract, but they clearly have roots in the world that children encounter. For example, it is much easier to think about a group of five eggs than to understand what a negative number is.
And so a number of proposals have been made to try to teach math by using concrete objects. For example, in the 1980s, methods were created in which children spent a lot of time playing with blocks that would stack or connect together. To learn to add numbers, children would stack groups of blocks together to represent the sizes of the sets named by the numbers, and then they would combine pairs of stacks to add the numbers together. The blocks would fit together in sticks of ten. Groups of ten sticks could be combined into squares to represent hundreds. These groupings were meant to help children learn about place value.
These methods seemed like a brilliant way to bring together children's love of play and their need to root abstract concepts in the world. The only problem was that these methods were not effective. Children were great at learning to manipulate the blocks, but that did not seem to help them to develop basic math concepts more easily or more effectively. Even giving children lessons about how the blocks related to the numbers did not seem to help them much.
One problem with these methods is that playing with blocks is too specific. As adults, we know that the only thing that matters is the number of blocks. To a child just learning math, though, it is hard to know that only the number matters. What about the shape of the blocks, or the color, or the satisfying click the blocks make when they fit together?
More recently, methods have been developed that do get children's bodies and minds involved in learning about numbers, but use a slightly more abstract way of teaching about number. For example, Geetha Ramani and Robert Siegler describe their work with board games in a March, 2008 paper in Child Development.
They took five-year-old children who are just learning about how numbers relate to sizes of sets and had them play a simple board game. The game had a line of ten squares. The squares had the numbers 1 through 10 in them in order. On every turn, players spun a spinner that had either a one or a two on it. They then moved their piece along the board the number of spaces that the spinner showed. As they moved the piece, they had to name the number in the square. So, if their piece was on square 4, and they spun a "2," they would count "5, 6" as they moved their piece forward. A second group of children played a similar game, only the board had the colors red and blue on it, and the spinner also had those colors. On each turn in the color game, the students moved their piece to the next square of that color.
Children who played this number game for about an hour (spread out over a few days) were much better than those who played the color game at a variety of skills involving numbers including identifying the numbers, comparing the sizes of different numbers, and estimating where the numbers would be on a line. This difference was found both soon after the children played the game and also at a test session 9-weeks after playing the game.
These results suggest that giving children bodily experience with concepts can really help learning. But it is important that the experiences that students have are set up so that they minimize the number of potentially distracting elements. For example, the board game involves mostly numbers and the order of numbers. There are few other elements to the game that might get in the way of learning.
This work is important for parents to think about as they help their kids navigate through school. When trying to explain a complex concept, parents often try to draw on real-world examples. That is important to do, but it is also important to keep those examples focused on just those aspects that are important for the concept the child is trying to learn. Otherwise, it is easy for the message to get lost in the details.
