I agree that such lists of objectives that you have proposed here are good for students to pursue. Too many of my own students in online classes lack many of these abilities, at least in the classroom. Their discussions are shallow because they stick to the obvious. I would guess that, outside the classroom or at least the math classroom, they can think better than they actually do. As I continue to see more and more shallow discussions, I continue to wonder if many of the students are really just faking their efforts to participate.
I will have to try tracking down more articles, but I recall seeing some earlier this year, one a few days ago and another one this past spring that suggest students' abilities to think are actually better than what "school testing" indicates. One of them is a preprint of one chapter of a book by Terezinha Nunes and Peter Bryant, which I had found in a files section of a Yahoo group on math education. Whether the book is published or what the title of the book is, I will have to check. Since it is a preprint, I will also have to check if it is acceptable to share this document with others. I do not believe there is a web reference for this document; Google searches have turned up nothing.
This preprint is the one I had seen this past spring. The main idea in this document is that not only does proportional reasoning develop without formal education, formal education may actually significantly hinder that development in many students.
The other article I had seen several days ago is one written by Ralph A. Raimi: "Basketball Logic and the NAEP." The link to the article is found at
The following conclusion he reaches is quoted below:
"I believe, therefore, that the public is being given the wrong information, and is led to believe things that are simply not so. The NAEP scores affect to tell us that children fail to understand certain things touching on mathematics as it is presented to us in the real world, whereas what these children are exhibiting ignorance of is something quite different: the skill to read an examination paper and to imagine the real-world situation those words think they are describing to him. Very often, as happens repeatedly in other examinations where parents protest and exam publishers apologize, those words don?t really do their job. But even when the writing is plain, and good, should we label this NAEP ?story-problem? skill ?Mathematics??
"Can we really say that half of our high school students seriously believe that because all basketball players are tall every tall person is not only a basketball player but one of our own team? Yet we do solemnly say that. NAEP itself says so. They have measured it."
On 10/15/2010 at 3:48 am, GS Chandy wrote:
> Shimon Zinbovsky posted Oct 12, 2010 6:34 AM: > > > > High schools should prepare their students > foremost > > for college acceptance. We should begin with that. > > Gaining the necessary skills to succeed in college > > and eventually graduate is a function of both > > successful high school preparation and college > > performance. We shouldn't be so naive to assume > that > > a student's college success can be attributed to > his > > high school preparation. > > > I'd suggest that high schools should pursue various > objectives, amongst which may be things like** > [** Objectives listed below appear in the order they > came to mind - the numbers preceding each element on > lists represent only that and nothing more - many of > the objectives are not necessarily specific to high > school level learning only]: > > 1. To help the student become all that he/she can be > 2. To ensure the student graduates with the kind of > knowledge and understanding of various disciplines > that would help him/her to deal effectively with the > working world he/she may encounter > 3. To provide the student with needed skills and > abilities to learn further and deeper in subjects of > interest > 4. To prepare the student for college acceptance > 5. To promote the student's ability to think > analytically and critically on issues > 6. To ensure the student effectively read, write and > do arithmetic effectively enough to be able to cope > with the real world around after he/she graduates > from high school > 7. To enable the student use a library effectively > > 8. To enable the student to use a dictionary > effectively > 9. To enable the student to use an encyclopedia > effectively > 10. To convince the student that all this 'stuff' > that he/she has to learn is really worth something > for him/her in his/her later life > 11. To ensure the student develops the ability to > think analytically and critically on issues for > himself/herself > 12. Etc,etc, etc, ETC! > > Some of those objectives above are things that should > have started from Day 1 of his/her school career, and > should continue right through life. > > I trust the list above may help indicate that the > *tasks* a school is entrusted with are more than > preparing the student for college. > > More useful (and much more important) than the plain > and simple list is the 'structure' attached herewith: > this is an Interpretive Structural Model (ISM) > showing my perceptions of how each of the above > objectives may "CONTRIBUTE TO" the others - no > special validity is claimed for my perceptions > indicated in the model - but I do hope it helps > convince that the objective of "school > teaching+learning" is a whole lot more complex than > may be widely recognized. > > (Regrets to Robert Hansen: The structure illustrating > the point alas contains the 'boxes' that you hate to > see - sorry about that). > > [Some background about the concepts and 'systems > science' underlying this kind of structure is > provided along with my posting at > http://mathforum.org/kb/message.jspa?messageID=7234262 > &tstart=0 ]. > > GSC > > > Message was edited by: GS Chandy