What science knows about learning maths
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.
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.
This becomes clear when we look at how children learn to understand a number say eight, not the symbol eight 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.
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 and then 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.
As children construct order, they count the objects as follows one two three four five six seven, and eight as they do that the number eight represents the eighth place in the order, in other words, eight always includes one two three four five six seven the idea of eight 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.
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 and later form eight squares then eight roots only once we have constructed number concepts inside our heads, can we effectively learn how to express them with images symbols and language.
A hundred thousand 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 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.
We know for example that eleven-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.
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.
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.