``...U.S. curricula [that] generally review and extend at successive grade levels many (if not most) topics already presented at earlier grade levels, while the top-performing countries are more likely to expect closure after exposure, development, and refinement of a particular topic. These critical differences distinguish a spiral curriculum (common in many subjects in U.S. curricula) from one built on developing proficiency—a curriculum that expects proficiency in the topics that are presented before more complex or difficult topics are introduced.''Every math teacher knows the concept of spiraling, the revisiting of old problems to ``reinforce'' concepts lest students should forget. The NMAP in no uncertain terms clearly rejected this approach, but what does Common Core do? It introduces addition and subtraction in kindergarten, ``students should see addition and subtraction equations,'' yet CCSSI is still covering addition and subtraction of whole numbers into the fourth grade:
K.OA.5: fluently add and subtract within 5 (but also add and subtract within 10, see K.OA.2)
1.OA.6: demonstrat[e] fluency for addition and subtraction within 10, add and subtract within 20, (and also add within 100; see 1.NBT.4)
2.NBT.5: fluently add and subtract within 100 (but also add and subtract within 1000, see 2.NBT.7)
3.NBT.2: fluently add and subtract within 1000
4.NBT.4: fluently add and subtract multi-digit whole numbers
This seemingly endless list doesn't even cover every K–4 standard that focuses on addition and subtraction. What's with Common Core's obsession over concepts that very few students fail to grasp?
There are only two critical mathematical skills in addition and subtraction, carrying and borrowing, and they can be learned in a matter of days or weeks. (Memorizing the addition tables from 0-9 is essential, of course, and CCSSI lists that as a skill to be attained by the end of Grade 2, see 2.OA.2, but more on that below.)
Inexplicably, the words ``carrying'' and ``borrowing'' never appear in Common Core, but are obliquely referred to in Grade 4 as ``the standard algorithm''. Even worse are the terms ``compose a ten'' (1.NBT.4) and ``compose or decompose tens or hundreds'' (2.NBT.7), which describe one possible stepping stone in learning to add and subtract, but should never become a permanent part of a student's mathematical skills repertoire. They are not alternate ``strategies'' once carrying and borrowing are learned.
Under CCSSI, it seems carrying would have to be introduced in Grade 1 since students need to ``add and subtract within 20'' and ``add within 100...and sometimes it is necessary to compose a ten'', but what about carrying twice? Or a subtraction example like 101 – 23 which often causes confusion because of the double borrowing? That would mean carrying and borrowing have to be taught a minimum of twice, in both the 1st and 2nd grades (and then reviewed in Grades 3 and 4). Common Core has a lousy plan to fix something that isn't broken.
Grade 1 already has enough age-appropriate topics, such as counting up to 100 and beyond, and 1st graders should learn to solve 6 + 9 and 12 – 5 without knowing how to carry or borrow, but by counting and grouping. This is the only place in the learning sequence where ``composing a ten'' fits in.
Grade 2 learning goals should be distinct from what came before and what will come later. To attain the necessary skills in addition and subtraction, a better sequence—consistent with the NMAP recommendations—would be to begin the school year with double digit addition, such as 26 + 13, then problems requiring carrying, then 48 – 25, and follow that with borrowing. Once the skills are learned, there's no reason the skill cannot be extended to three or four digit numbers. In later years, when students start using really big numbers and decimals, the essential skills will already be learned.
How about addition and subtraction in Grade 3? Multiplication and fractions will be taking center stage, so why would anyone want to go over addition and subtraction again?
Composing a ten, such as 6 + 9 becomes 6 + (4 + 5) becomes (6 + 4) + 5 becomes 10 + 5 = 15 is a strategy for understanding addition, but should never become a lifelong practice. Such techniques certainly shouldn't be foisted upon a student who's fluent with sums. Any student who can tell you without hesitation that 6 + 9 = 15 shouldn't have to explain the strategy that was used to arrive at an answer that will always be the same.
Even for weaker students who may rely on ``composing a ten'' longer than others, once the concept of addition beyond ten is understood, there's no need to rehash the strategy. It's time to learn the addition tables, and that should begin toward the end of the first grade and memorized (drilled [yes, drilled] by parents with flash cards) over the summer between first and second grades. Then when second grade begins, students won't resort to ``composing tens'' or counting on their fingers.