 The following ten rules, which are based on decades of research, aim to demonstrate that the standard way to teach children math is counterproductive, because it focuses on symbols. To be able to think mathematically, however, means not to think in symbols, but to learn to think in relationships. Math can't be taught Just like we can't learn ping-pong from watching videos, children can't learn math from reading textbooks or listening to a teacher. Instead, they learn math by doing math, ideally with real objects, because only when they do math, relationships are constructed right where math happens, in their heads. It happens in the head Whatever is on paper is merely a representation of mathematical thinking that happens in the brain, just like musical notes. What is on paper is just a representation of music that actually happens when someone plays the piano. To be a good musician, it's not enough to be able to read the notes. We also need to practice a lot. The same is true for math, which is why practicing mental arithmetic is so important. Math needs years of practice This becomes clear when we look at how children learn to understand a number, say, 8. Not the symbol 8, but the idea of the quantity of 8. To internalize this seemingly simple idea, children need a lot of practice in two skills. First, they need to learn how to create order, and then, later on, how to create hierarchical relationships. Let's look at order first. When 4-year-olds learn to count, most have trouble ordering objects in their heads if the things they count are unevenly distributed. Sometimes they skip objects, then they count the same ones twice. To do it right, children have to learn how to construct order in their heads. This seems easy, but actually takes our brains a lot of practice. As children learn to order objects in their heads, they can put them in relationships. Hierarchical relationships As children construct order, they count the objects as follows. 1, 2, 3, 4, 5, 6, 7, and 8. As they do that, the number 8 represents the 8th place in the order. In other words, 8 always includes 1, 2, 3, 4, 5, 6, 7. The idea of 8 is there for a hierarchical relationship between the 8th object and all those preceding it. If we don't learn to do this sort of abstraction by doing lots of math in our heads, we won't be able to form a solid foundation for arithmetic. After building them, children need to learn to break relationships apart again. We can see how hard this is when we present a 5-year-old an image of 6 dogs and 2 cats and then ask, are there more dogs or more animals? While most adults who see the full picture find this question odd, a 5-year-old typically just answers, more dogs. When you ask further, more dogs than what, the child replies, then cats. In other words, if you ask, are there more dogs or more animals, the child hears, are there more dogs or more cats? At age 5, most kids didn't practice enough math to break hierarchical relationships apart while still remembering the whole. This happens because once the child has to cut the whole into parts, for them, at that moment, the whole no longer exists. They have not yet constructed the concept of 8 without thinking of it as a sum of its parts. So when they divide the animals into cats and dogs, all they can think of are two parts of which one looks larger. The idea of 8 is then forgotten. To also think about all animals would require two opposite mental actions. First, divide the whole and then put it back together, a mental process that most 5-year-old children precisely can't do. Only by age 7, most children can see the whole and keep its abstraction in their heads and still divide the sum in its parts. Experiences Proceeds Language As we demonstrated, it takes a child a lot of mental training and hands-on experiences to form the concept of a number. At the age of 5, we can build a simple row of 8, later form 8 square then 8 root. Only once we have constructed number concepts inside our heads can we effectively learn how to express them with images, symbols and language. Math can be expressed in different languages. 100,000 years ago, we used objects to express our mathematical thinking. Later, we used images. Around 1,000 years ago, we began to reduce images to Arabic numeral symbols. In future, we might replace symbols with bits or express math in graphic simulations or games. In other words, while math thinking always happens in our heads, the language that represents our thinking is evolving. Most people don't have math but language problems. We know, for example, that 11-year-old unschooled street vendors are often highly proficient in complex money transactions but incapable of doing paper and pencil arithmetic. This phenomenon, known as street mathematics, shows that when smart kids struggle in school, they often just can't express their thinking in symbols. Their brains can do math but have language problems. One way to solve this is to do it your way. Just like nobody ever learned to speak a language just by learning the rules of grammar, nobody learns math by memorizing the rules of how to arrange numbers in symbols in order to find the right answer to a problem. Whenever we do that, we stop constructing fundamental principles inside our heads. To get better and confident, children should be encouraged to find their own path and use their own language to express a solution. Which brings us back to Rule 1. Math can't be taught, it has to be constructed. If we want to learn math, we have to do math in our heads, ideally with real life experiences. Later, we replace the objects with abstractions, such as language, symbols, or whatever the future might bring. The ideas presented in this video are based on the work of Jean Piaget, Constance Camille, Keith Devlin, Georgia de Clark, and Jerome Bruner, who all contributed immensely to the body of work and research on how children and adults learn math. If you want to get better at math today, join Keith Devlin from Stanford University and over 100,000 students from all around the world in his free course on Thinking Mathematically. See the descriptions below for more details and links for the research. Sprouts videos are published under the Creative Commons license. That means our videos are free and anyone can download, edit, and play them for personal use. 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