Proposed levels may not come as sharply distinctive, yet well defined levels, illustrated by lots of detailed relevant examples, seem to be needed as a practical guide for designing assessment instruments, and may have implications for lesson planning.
One simple example. Consider the formula for the square of a sum
>>> (a+b)^2 = a^2 + 2ab + b^2
I've regularly witnessed the following five distinct scenarios:
(1 ) Student knows this formula, and can use it to compute 103^2 mentally (without assistance). (1*) Student knows this formula, and can use it to compute 103^2 mentally (with the hint to use the formula).
(2 ) Student knows this formula, and can use it to compute (100+3)^2 mentally (without assistance). (2*) Student knows this formula, and can use it to compute (100+3)^2 mentally (with the hint to use the formula). (3 ) A student knows this formula, and cannot do any of the above (even with the hint to use the formula).
Haven't you experienced similar situations? Their daily occurence seems to indicate the existence of different "levels" of understanding of the formula, however "levels" are ultimately defined?
In a message dated 12/1/2010 1:45:59 P.M. Eastern Standard Time, firstname.lastname@example.org writes:
On Dec 1, 2010, at 8:59 AM, DCJLEE@aol.com wrote:
> From: DCJLEE@aol.com > To: email@example.com > Sent: 11/7/2010 7:34:40 A.M. Eastern Standard Time > Subj: Bloom taxonomy in Math > > Hi everyone, > > I'm interested in your views on the applicability of Bloom's taxonomy > (or a revised version of it) to math education. Also, I would be > grateful > for pointers to studies on this subject that go beyond repeating the > description of the various levels of achievement in the generic Bloom > taxonomy, and provide a specific, nontrivial example of those levels. > Thank you in advance for your help.
In order to respond, it would be useful, at least to me, to know a bit more precisely what the issue(s) actually are that you want to deal with.
While there obviously are "levels" at which mathematics is understood---my level of understanding is certainly not that of, say, Grothendieck or Thurston---I would argue that the notion of level doesn't concern our students just like, while there are levels that are meaningful for, say, sprinters or long distance runners, the concept seems a bit useless both for children learning to walk and for adults walking.
Beyond that, I would argue that while, for instance, Anderson and Krathwohl (2001)' pyramid
might look good, it is not really relevant: In those few fields in which I have some small competency---mathematics, building construction, political sciences, mechanics, ... ---all six verbs have all always been there and thus did not characterize levels and if the mix changed a bit over time, it certainly was not monotonically.