As I’ve been reading more about the Common Core Standards adopted by PA (see 7/3/10 post), it appears that they do not match the best in the country (CA, MA, IN, MN) but may be an improvement for some states. James Milgram’s critique caught my eye and is food for thought. An excerpt:
“Compare the first grade California Green dot standard
2.1 Know the addition facts (sums to 20) and the corresponding subtraction facts and commit them to memory.
with the much weaker standard in Core Standards:
1-OA(6) Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).
This standard misuses “Fluent,” I believe. One does not want students to develop fluent command of special tricks for doing arithmetic, as this could very well result in severe difficulties un-learning these methods in later grades where Core Standards asks for at least some degree of proficiency with standard algorithms.
Compare 5-NBT(5) Fluently add, subtract, and multiply multi-digit whole numbers using the standard algorithm for each operation.” (Milgram, 2010, p. 14)
As Wu points out in the Fall 2009 American Educator, the elegant simplicity of the standard algorithms is that “all whole-number computations are nothing but a sequence of single-digit computations artfully put together” (p. 6)
Happily, with the adoption of the Common Core Standards, teaching the standard algorithms now will be required in Pennsylvania and students will need to be able to fluently apply them. Three cheers for that!
The strategies mentioned in the Core Standard are not “tricks” but represent deep, foundational connections within mathematics, such as the very important idea of inverse operations. I have not read the entirety of either standards list, but the difference between the CA Standard and the Core Standard listed here is almost identical to the difference between the American teachers and the Chinese teachers in Liping Ma’s book Knowing and Teaching Elementary Mathematics.
The Core Standard is weaker not because of the “tricks” but because of its focus on fluency “within 10″ — a more appropriate goal for preschool or kindergarten than for first grade! But the emphasis on understanding rather than on memory is good. Memory of facts is useful, but deep understanding and flexibility in manipulating numbers is more important as a foundation for long-term learning.
And a student who is fluent in mental math by using these “trick” strategies should never have to “unlearn” them! The standard algorithms are elegantly simple, as Wu says. But why would a knowledge of inverse operations, for instance, make it harder to learn the standard algorithm? Gee, I wonder how all those Singapore students, who spend first grade mastering this sort of mental math (within 100!), manage to master the standard algorithm equally well in second grade…
The vital, unanswered question is how the standards are to be implemented in the classroom. Having a teacher who thoroughly understands the connections that underlie mathematics and having a curriculum that supports that sort of understanding (as the Singapore Primary Math program does) are both more important than the wording of whatever standard a state chooses.