We are full of self esteem about our imaginary knowledge and mastery of difficult subjects, namely math. This is one of the reasons that test reality is so daunting. It is a blow to self esteem. This is why tests have to be made harder and to cover conceptual knowledge.
Conceptual knowledge = Abstraction.
Conceptual Math Knowledge = New Math
The tests need to test the abstract new math to force the students drugged on self esteem to learn it.
Ages past did not teach self esteem because they knew that people were inflated with self esteem already. They knew that reality was knocking down self esteem all the time and that what students needed in school was not self esteem building but to learn their lessons while they had time so they could use them in a world with low wages and hard conditions. That is the world that government is bringing back as much as fast as it can.
Debt as the start in life is now standard. Even the 19th century did not think of that. Yet know, the universities deliver it. They even want the parents and even the grandparents to go into debt. End of life debt for paying for the start of life of the young. This is the idea of today.
What is not needed is more self esteem building. We have to face the reality that abstraction is difficult to learn and requires attention and focused effort from the start.
Teachers and textbook makers can’t teach abstraction in math if they don’t learn it themselves. So they will have to search for materials they can understand, buy them, and read them. This is a foreign thought to our self esteem drenched education system. They expect this will just be delivered to them by colleges in pre-service courses or by the textbook manufacturers in their lesson plans.
Well, while waiting around for these others and trusting to self esteem, high stakes testing has arrived on the scene. So not understanding abstraction in math, which means not understanding math is no longer as viable an option as it once was.
To teach math you have to be able to recognize the problem, which is typically a problem in understanding an abstraction. This means learning the abstractions in the first place.
The Lance Rips, Jennifer Asmuth, Susan Carey type psychology research papers requires you to know the Peano Axioms, and some axiomatic set theory, formal logic, and so on.
Start reading at 8.1 in this Susan Carey paper.
8.1. Bootstrapping representations of natural number
TOOC draws on Quinian bootstrapping to explain all the
developmental discontinuities described in the previous
section. In the case of the construction of the numeral
list representation of the integers, the memorized count
list is the placeholder structure. Its initial meaning is
exhausted by the relation among the external symbols:
They are stably ordered. “One, two, three, four. . .”
initially has no more meaning for the child than “a, b, c,
d. . ..” The details of the subset-knower period suggest
that the resources of parallel individuation, enriched by
the machinery of linguistic set-based quantification,
provide the partial meanings children assign to the placeholder
structures that get the bootstrapping process
started. The meaning of the word “one” could be subserved
by a mental model of a set of a single individual
(i), along with a procedure that determines that the
word “one” can be applied to any set that can be put in
one-to-one correspondence with this model. Similarly
“two” is mapped onto a long term memory model of a
set of two individuals (j k), along with a procedure that
determines that the word “two” can be applied to any
set that can be put in one-to-one correspondence with
this model. And so on for “three” and “four.” This proposal
requires no mental machinery not shown to be in the
repertoire of infants: parallel individuation, the capacity
to compare models on the basis of one-to-one correspondence,
and the set-based quantificational machinery that
underlies the singular/plural distinction and makes possible
the representation of dual and trial markers. The
work of the subset-knower period of numeral learning,
which extends in English-learners between ages 2.0 and
3.6 or thereabouts, is the creation of the long-term
memory models and computations for applying them
that constitute the meanings of the first numerals the
child assigns numerical meaning to.
So if you have been following the Peano Axioms, set theory constructions of ordinal numbers as lists, cardinal numbers as 1 to 1 correspondences and their interplay, you can read this paragraph and others. If you have not, you will find these papers heavy going. Note this is a psychologist’s paper about children learning number concepts not another Milgram or Wu paper.
So you had better check your self esteem that you know the concepts of number, when you don’t, and study the materials on Peano Axioms, set theory, von Neumann ordinal constructions of ordinal numbers as lists, and so on.