Jean Piaget was one of the foundational researchers in cognitive development, particularly in 'constructivist' circles. Constructivism is the theory that learning is an active psychological process, and is contrasted with behaviourism (often associated with B.F. Skinner), in which learning is a passive process. Constructivism is popular amongst school teachers and behaviourism is unpopular.
Piaget's theories are very much rooted in learning about the physical world, and are thus more popular amongst teachers in the 'STEM' disciplines. The other foundational researcher in constructivism was Lev Vygostky, who worked in the Soviet Union under Stalin, and his theories reflect the political pressures of that environment; his earlier work remains influential especially amongst humanities teachers, whereas his later work is excessively ideological.
Piaget proposed a stage-based model of development, in which children progress through four different stages of cognition. The last two stages he called the concrete operational stage and the formal operational stage. In the concrete operational stage, children become capable of one layer of abstraction or symbolic indirection; it is not until they reach the formal operational stage that they are capable of multiple layers of abstraction or symbolic indirection.
It is interesting to note that Piaget suggested that the transition between these two stages usually occurs around the age of eleven, which is when British children transition from primary to secondary school.
So, to contextualise the linked article, the studied children developed an applied understanding of mathematics that fit their concrete operational cognition: the mental operations were readily made concrete by manipulating coins and banknotes.
However, when children study mathematics in school, it is very much done as formal operations: the numbers are not intended to represent anything physical at all. Consequently, many children learn mathematics as a set of arcane rituals for manipulating symbols on paper, because they can't yet understand the abstract meanings of those symbols.
"many children learn mathematics as a set of arcane rituals for manipulating symbols on paper, because they can't yet understand the abstract meanings of those symbols."
In the US?
I had pretty typical US public school education. Word problems and application were ubiquituous. Perhaps my experience is non-representative, or you like the study authors are speaking of other educational contexts (Asia).
I grew up in Australia and taught in Australia and the UK; and yes, the word problems are ubiquitous in those countries too. But those word problems are always deeply inauthentic, and extracting the relevant information from them becomes just another arcane activity.
One that comes to mind involves a farmer with a given length of fencing, and the student has to find the area of the largest rectangular field the farmer can surround with that fencing. It's a good mathematical puzzle, but the actual real-world problem is how much fencing the farmer needs to surround a given field.
Coming up with genuine, real-world applications for every mathematics lesson is extremely time-consuming, and maths teachers simply don't have the time.
With ChatGPT et al... it should start to be pretty easy. I envision a question answer game between humans and the AI. The AI would set up the scenario and the kids would have to ask the right questions. Teachers could supervise and evaluate
Maybe there's an element of both time and skill? How many folks that pursue math can point to a great early teacher as being influential. Not all, but I want to believe it's common and maybe true for most.
Some children read word problems, understand the question and apply math. Most children look for key words, use those key words to guess what operation to apply, then apply it to the numbers in the question (e.g. there were 3 numbers and the word "total" so I'll sum the numbers).
Piaget's theories are very much rooted in learning about the physical world, and are thus more popular amongst teachers in the 'STEM' disciplines. The other foundational researcher in constructivism was Lev Vygostky, who worked in the Soviet Union under Stalin, and his theories reflect the political pressures of that environment; his earlier work remains influential especially amongst humanities teachers, whereas his later work is excessively ideological.
Piaget proposed a stage-based model of development, in which children progress through four different stages of cognition. The last two stages he called the concrete operational stage and the formal operational stage. In the concrete operational stage, children become capable of one layer of abstraction or symbolic indirection; it is not until they reach the formal operational stage that they are capable of multiple layers of abstraction or symbolic indirection.
It is interesting to note that Piaget suggested that the transition between these two stages usually occurs around the age of eleven, which is when British children transition from primary to secondary school.
So, to contextualise the linked article, the studied children developed an applied understanding of mathematics that fit their concrete operational cognition: the mental operations were readily made concrete by manipulating coins and banknotes.
However, when children study mathematics in school, it is very much done as formal operations: the numbers are not intended to represent anything physical at all. Consequently, many children learn mathematics as a set of arcane rituals for manipulating symbols on paper, because they can't yet understand the abstract meanings of those symbols.
Piaget's theory on Wikipedia: https://en.wikipedia.org/wiki/Jean_Piaget#Theory
Paul Lockhart writes something parallel to this argument in A Mathematician's Lament: https://archive.org/details/AMathematiciansLament/mode/2up