> This may be the last response since it is really unlikely that we will > convince each other except that we have fundamental differences that > cannot be reconciled. I enjoy discussing with you different issues on > which we have very different perspectives, but I'd rather discussion be > done publicly. Although some people on nctm-l have criticzed us (and > others) for dominating the list, I would really like to get multiple > perspectives - which the list was designed to do. Doing this discussion > in private just is not too satisfactory for me since I'm pretty sure I > will never convince you to change your perspective and you will not > change mine. We may be engaged in a very futile activity in that case > (with all due respect to you and your perspectives).
OK, you get what you want.
> > Math problems claim to assess themselves.
I do not understand this phrase. I never wrote it.
> > If a student cannot solve quadratic equations, > > he cannot solve quadratic equations. Period.
> I accept this point if the students are simply solving problem like > 2x-5=7, which is context-free. However, as a mathematician, I'm sure you > value students' ability to solve "problems" that exist within contexts, > don't you? A test with 50 equations measures something but not too many > facets of mathematical understanding.
Oh, yes. I attach enormous importance to word problems. But in a sense which is different from the common naive one. Many people think that a problem, say, about cars moving from two cities agaist each other is useful because it prepares students to manage cars. I think it is ridiculous.
> > Please, give an example. > > Only not with a student who does not know English. > > A very simple minded example: Suppose you have a word problem involving > dimensions of a garage. Many inner-city kids do not have garages at > their homes. Now, we can always argue that what type of building > involved makes no difference mathematically and that's true. And, that's > the exactly the reason we want to make sure, if a student get this > problem wrong, it was not because he was puzzled with this structure > called "garage" but because he did not understand mathematics involved. > You will probably think this is non-sense, but younger children (and many > adults, too) often get side-tracked with these items that are irrelevant > (from mathematical perspectives). Since we are assessing students' > mathematical understanding, what the (assessment) standards is saying is > that we need to make sure that we are indeed assessing students' mathematics.
From my perspective it means only that authors of problems should choose the most well-known and unambiguous real-life objects. Coins are among the best in this respect. And Standards are stupid enough to put coins first on their black list ! It also is related to why I attach so much importance to word problems. Because they teach something more important than mathematics. They teach an ability to formalize real objects, that is to operate with their abstract counterparts. For example, abstract cars (at the middle school level) always travel with a constant speed and never break down. This prepares students to manage MODELS, which is essential for any application of mathematics.
> Also, I'm not sure why you have to read "students' background" as > "students' skin colors".
Just to visualize the problem. We can speak about length of nose instead. The Standard explicitly mention `ethnic, cultural and social backgrounds' (p.15) among those which must be considered when assessing a student's work. Thus, to assess a student's solution of a problem I cannot simply say: `your solution is wrong because you think that volume of a garage equals the sum of its dimensions'. I must take the student's genealogic tree into consideration. And education of his parents. And nobody knows what else. Andrei Toom