For the past 100 years, many brilliant men and women have been researching and thinking about how we learn number concepts and what are the best ways to teach math to children and adults. Combining their findings, we compiled a list containing 10 Rules of learning math.

##### the full story

The following 10 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.

##### 1. 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.

##### 2. 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.

##### 3. 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 “eight”. 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:

##### 4. construct order

When four-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. Once children learn to order objects in their heads, they can put them in relationships.

##### 5. 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 eighth place in the order. In other words, eight always includes 1,2,3,4,5,6,7. The idea of 8 is therefore a hierarchical relationship between the eighth 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.

##### 6. break relationships apart again

After building them, children need to learn to break relationships apart again.

We can see how hard this is when we present a five-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 “than 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 five year old children precisely can’t do. Only by age seven most children can see the whole and keep its abstraction in their heads and still divide the sum in its parts.

##### 7. experiences precede 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 five we can build a simple row of eight, 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.

##### 8. math can be expressed in different languages

100,000 years ago, we used objects to express our mathematical thinking. Later we used images. Around one thousand years ago we began to reduce images to Arabic numerals symbols. In the 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.

##### 9. 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.

##### 10. 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 and 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.

##### improve your math

The ideas presented in this video are based on the work of Jean Piaget, Constance Kamii , Keith Devlin, Georgia DeClark 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 description below for more details and links for the research.

## Sources

- Kamii, C., & Housman, L. B. (2000).
*Young children reinvent arithmetic: Implications of Piaget’s theory*. Early Childhood Education Seri. - Keith Devlin – Wikipedia.org
- Constance Kamii – Wikipedia.org
- Jerome Bruner – Wikipedia.org
- Jean Piaget – Wikipedia.org
- Nunes, Terezinha & Schliemann, Analucia & Carraher, David. (1993).
*Street Mathematics and School Mathematics*

## Dig deeper!

- Take this free online course : Introduction to Mathematical Thinking – Coursera.org
- Watch this Youtube video on What is Math Thinking and How Can Games Help Teach Them.
- Play Dragonbox Algebra
- Play Brain Quake
- Play Wuzzit Trouble

## Classroom activity

Check out this website for suggested activities on Bringing out the mathematician in every child. – Mathathome.org

## Collaborators

- Script: Jonas Koblin
- Artist: Pascal Gaggelli
- Voice: Matt Abbott
- Coloring: Nalin
- Editing: Peera Lertsukittipongsa
- Creative Director: Selina Bador
- Production Assistant: Bianka
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