I've inadvertently pressed "Reply" instead of "Reply All".
____________________________________ From: DCJLEE@aol.com To: email@example.com Sent: 12/4/2010 12:39:40 A.M. Eastern Standard Time Subj: Re: Bloom taxonomy in Math
Before addressing Alain's points, I must express surprise that this topic hasn't caught the attention of more people, given that myriad school districts advertise the fact that they use Bloom's taxonomy in curriculum design.
A) Your preoccupation with instructional material seems to preclude the meaningfulness of assessment independent of how knowledge was acquired.
B) There are strategic and tactical aspects to assessments. In principle, why are you against the possibility or usefulness of checking students' grasp of the material presented in each class meeting?
C) I wonder if other subscribers to this forum agree with your conclusion. I rather wait and see if someone cares to address this point.
D) Of course, you're right. But, as announced, it was supposed to be a simple example.
E) Again, you seem to conflate/identify knowledge/understanding with the process of acquiring it.
F) Mental arithmetic was just a device used here to preclude the use of calculator/paper&pencil. No time constraint was imposed. The formula was explained earlier in class.
G) Please refer to A) and E).
In a message dated 12/3/2010 12:22:03 P.M. Eastern Standard Time, firstname.lastname@example.org writes:
On Dec 2, 2010, at 12:25 AM, DCJLEE@aol.com wrote:
> 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,
A. Assessment instruments cannot really be designed independently of the materials: Mathematical entities are connected in many different manners and can be seen in many different ways. The materials have to be designed/written with some particular conceptual organization in mind, which implies a particular selection among all these relationships and thus a particular viewpoint. As a result, their conceptual understanding then necessarily depends, at least in part, on this organization. Independent assessment tends to be limited to the "mechanical" aspects which is not to say that the latter are of no importance but only that they do not really relate much to conceptual understanding.
> and may have implications for lesson planning.
B. One ought to plan at least a semester, if not a sequence; certainly not a lesson. Moreover, "lessons" invariably devolve to the students "taking notes" (just so as not to draw attention by appearing idle). In developmental mathematics, with appropriate materials read by the students before class, time is much better spent discussing whatever comes up from said reading.
> 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?
C. I do not see what impact the above "scenarios" can have other than resulting in more drilling.
D. The above"scenarios" may just be "still photographs" out of a whole continuum of possible behaviors.
E. The use of the term "level" would seem to imply a hierarchy which I do not see the scenarios to imply. Consider the following: faced with the same problem, my wife, whose field is on the analysis side of differential geometry will get her ideas from the look of the computations whereas I, whose field was somewhere between differential geometry and fluid dynamics, would get mine from (a considerable amount of) doodling. Are we still talking "levels"?
F. Mental arithmetic is a very special talent which many mathematicians do not have. On the other hand, given any two numbers, my father could give you what percentage of one number the other one was and he was not even remotely a mathematician,.
G. The differences in the scenarios can be explained in ways other than "levels of understanding of the formula". For instance, it may be that the explanation didn't really "explain" at all so that the formula became voodoo and ended up having to be memorized. Which in fact is what "student knows this formula" seems to point at.