Anyone who's supervised an elementary math teacher, say in the process of teaching fractions or fractional operations, e.g.
½
+
¼
__
can't help but realize that most of the mathematics education imparted to these young teachers becomes absolutely superfluous in the classroom. Of course, in their classes at University X, Y or Z, the teachers-to -be would have encountered fractions in the course of their Abstract Algebra courses - generally one of the 7 or 8 subject requirements to obtain the B.Ed. degreee.
In that setting, one will be concerned with some integral domain Z, and a congruence class determined by (a, b) ( Z X Z*, denoted by (a/b) - which is referred to as a "fraction" or a quotient of elements of Z, with the set of all such fractions denoted by Q(Z). Whence also, we expect:
(a/b) ( Q(Z)
Of course, other lemmas, axioms and theorems then follow which inexorably push fractions into a transition to "commutative rings". For instance, one lemma asserts:
"If e is the multiplicative identity of Z and (a/b) ( Q(Z) then:
1- For any non-zero x ( Z, a/b = ax/bx
2- The equality a/b = 0/e holds IFF a = 0.
3- The equality b/b = e/e holds for any b ( Z*"
and a theorem, whence:
"If Z is an integral domain with identity element e, then the set Q(Z) together with the binary operation of addition and multiplication of congruence classes defined by:
a/b + c/d = (ad + bc)/ bd
and
(a/b) * ((c/d) = ac/ bd
is a field.
Moreover Q(Z) contains a subring Z'_e that is isomorphic to Z"
and so on and so forth, with the ultimate goal to provide reasons for the fractional operations and outcomes and effectively them out of the purely mechanical, rote domain. The problem is that all that advanced education is useless as a pedagogical aid because: 1) it will sail serenely over any students' neads and (2) be irrelevant to their ongoing mastery of basic maths. This despite the fact that some have argued this isn't so and is more a fucntion of imprecision and sloppy notation at the advanced level. Hardly! Even with the most meticulous care no elementary teacher will be able to impart to 4th or 5th graders a thorough understanding of why fractional operations are as they are (including the inadvisability of having 0 in the denominator).
True, with some ultra-bright students some headway may be made but generally not consistently.
Many puzzled math teachers who've tried to grasp this usually are unable to because they fail to appreciate one singular thing: the current educational system and its pedagogical practices (never mind all the computers, spreadsheets and what not) are still modeled after the philosophy of 19th century practice. Thus, fractions or decimals or whatever ....will still basically be taught that way, if only on ipads or computers.
Meanwhile, true professional mathematicians abandoned this early in the 20th century.
What all this shows is that more advanced mathematics for teachers (including non-Euclidean geometry, non-commutative algebra and geometry as well as complex analysis) is basically a waste of time. They'd do better to focus on the pure pedagogy and leave the arcane stuff to the math professionals who actually publish papers, such as in The Journal of the American Mathematical Society. (Fortunately, educators themselves seemed to have 'jumped the shark' and rejected contemporary advanced approaches as ridiculous for normal humans who may never need more than an ability to say, finding interest on a loan, or a discount on a store purchase. )
Meanwhile, perhaps in their retirement if not before, those same teachers may wish to at least examine some of the advanced math basis for what they taught their charges.
½
+
¼
__
can't help but realize that most of the mathematics education imparted to these young teachers becomes absolutely superfluous in the classroom. Of course, in their classes at University X, Y or Z, the teachers-to -be would have encountered fractions in the course of their Abstract Algebra courses - generally one of the 7 or 8 subject requirements to obtain the B.Ed. degreee.
In that setting, one will be concerned with some integral domain Z, and a congruence class determined by (a, b) ( Z X Z*, denoted by (a/b) - which is referred to as a "fraction" or a quotient of elements of Z, with the set of all such fractions denoted by Q(Z). Whence also, we expect:
(a/b) ( Q(Z)
Of course, other lemmas, axioms and theorems then follow which inexorably push fractions into a transition to "commutative rings". For instance, one lemma asserts:
"If e is the multiplicative identity of Z and (a/b) ( Q(Z) then:
1- For any non-zero x ( Z, a/b = ax/bx
2- The equality a/b = 0/e holds IFF a = 0.
3- The equality b/b = e/e holds for any b ( Z*"
and a theorem, whence:
"If Z is an integral domain with identity element e, then the set Q(Z) together with the binary operation of addition and multiplication of congruence classes defined by:
a/b + c/d = (ad + bc)/ bd
and
(a/b) * ((c/d) = ac/ bd
is a field.
Moreover Q(Z) contains a subring Z'_e that is isomorphic to Z"
and so on and so forth, with the ultimate goal to provide reasons for the fractional operations and outcomes and effectively them out of the purely mechanical, rote domain. The problem is that all that advanced education is useless as a pedagogical aid because: 1) it will sail serenely over any students' neads and (2) be irrelevant to their ongoing mastery of basic maths. This despite the fact that some have argued this isn't so and is more a fucntion of imprecision and sloppy notation at the advanced level. Hardly! Even with the most meticulous care no elementary teacher will be able to impart to 4th or 5th graders a thorough understanding of why fractional operations are as they are (including the inadvisability of having 0 in the denominator).
True, with some ultra-bright students some headway may be made but generally not consistently.
Many puzzled math teachers who've tried to grasp this usually are unable to because they fail to appreciate one singular thing: the current educational system and its pedagogical practices (never mind all the computers, spreadsheets and what not) are still modeled after the philosophy of 19th century practice. Thus, fractions or decimals or whatever ....will still basically be taught that way, if only on ipads or computers.
Meanwhile, true professional mathematicians abandoned this early in the 20th century.
What all this shows is that more advanced mathematics for teachers (including non-Euclidean geometry, non-commutative algebra and geometry as well as complex analysis) is basically a waste of time. They'd do better to focus on the pure pedagogy and leave the arcane stuff to the math professionals who actually publish papers, such as in The Journal of the American Mathematical Society. (Fortunately, educators themselves seemed to have 'jumped the shark' and rejected contemporary advanced approaches as ridiculous for normal humans who may never need more than an ability to say, finding interest on a loan, or a discount on a store purchase. )
Meanwhile, perhaps in their retirement if not before, those same teachers may wish to at least examine some of the advanced math basis for what they taught their charges.
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