When I said that perhaps 50% show no comprehension of mathematics, I should have said "mathematics above grade 4 level". I have seen, and said, for years that the problem is really at the grade 4 level, maybe some at grades 3 and 5 depending on student and school. By the time the students reach grade 6, having understood almost nothing of things like long division, and absolutely nothing of fractions, the game is lost and huge numbers never recover. \ I think the critical lack of comprehension at around grade 4 is the lynch pin of the problem. Long division is the first time students have to memorize a complex algorithm for which they have no visualizable model. I think the criticality of this transition from what I might call "visual mathematics" to "formal mathematics" is the nuclear issue. I welcome comments on this from those who may have more experience than I with students in these age ranges, although I think the information and my experience, though limited, on these students makes it undeniable that there is a critical problem at this transition.
On 1/27/2012 3:35 AM, Ed Wall wrote: > > A few comments below > On Jan 27, 2012, at 1:25 AM, Jerome Epstein wrote: > > > I am happy to see interest in what seems to me to be quite interesting > > and somewhat surprising results. Well, surprising depending on your > > point of view. > > > > I have been laboring over results from the CCI for a few years, and the > > BSDT for many years. Both tests show similar things in the large, > > although at different levels, with large populations. Leaving aside for > > a moment the results from China, which raise major questions in > themselves, > > the large view conclusions are abundantly clear, but the interpretation > > of them, and the development of some program to improve things is far > > less clear. > > > > I have been at this basic understanding testing in mathematics now > for a > > very long time. I am only one person with no assistance, and I really > > need to think about developing a team to begin developing some > > publishable research out of this. I think there is a huge amount of > very > > important stuff implicit in the results of these tests. > > > > A few conclusions are clear and I think undeniable: > > > > 1. A stunning percentage of US High School and College students have no > > meaningful, usable comprehension of meaning in mathematics beyond the > > 4th grade. > > > (see below, it may not be this simple). I'm not completely disagreeing > with you, but people do tend to make meaning out of the situations > they are placed in and do so reasonably well although not as we would > necessarily wish. > > > 2. I would say that results on the tests show that something like 5% of > > US students are still among the best in the world. Maybe 10% are very > > good (including the first 5%), 20% are competent to graduate from high > > school or Freshman year in college with a basic comprehension and > > functionality in basic mathematics. Once you get below the top 20% you > > start running into disaster very quickly. > > > > This seems right > > > 3. Perhaps as many as 50% of students graduate from high school with no > > conceptual understanding of mathematics beyond the 4th grade. Some of > > them learn to compute with memorized rules which they can apply to rote > > memorized situations, but not otherwise. > > > > It is more complicated than this. You are, in my opinion, defining > mathematics a bit too narrowly here and in the above. If you said they > have no conceptual understanding of school mathematics, then I might > tend to agree somewhat. However, they do have conceptual > understandings of mathematics. Talk to, for instance, > preKindergartners, skilled plumbers, electricians, heavy equipment > operators. > > > 4. It would be extremely worthwhile in my opinion (though I doubt it > > will ever happen) to come to wide-spread agreement on an examination of > > minimal conceptual understanding that must be mastered to graduate from > > high school, and how to test for it in a way that cannot be gotten > > around by memorized rules. /This is very hard to do/. This is > enormously > > difficult, socially, politically, etc. because I have little doubt that > > right now many millions of students in high school could not pass > such a > > test. What then? > > > > Things vaguely like this have been tried. Students fail. > > > 5. A critical question is how many teachers, teaching in elementary > > schools in particular, could pass such a test. The percentage that > could > > not, I believe, is very high and this fact is commonly not known. > > > I teach, among other things, elementary school teachers. You are > correct and this has been documented again and again. A good many > people do know this. The teachers, themselves, do know this and many > of them care. > > > 6. No real progress can be made on the overall problem until the > reality > > of the conceptual understanding of elementary school teachers is > > seriously addressed. I see no possibility that this can happen in my > > lifetime (not so much left. . . . ). But this is the lynch pin on which > > everything else depends. > > > I happen to agree again. However, a good many people prefer to their > money on middle and high school teachers. If you look at special > programs for secondary mathematics teachers and grant money, this is > very clear. There is another very large problem and that is the > training of elementary school teachers - I'm talking about content > pedagogy here - is often done by people with little mathematical > background. Look at the ads for colleges/universities looking for > faculty to teach content pedagogy (I realize this is a little too > jargonish, but I'm thinking about teaching mathematics with a strong > emphasis on mathematics. There are, I think, some other things that go > into this, but that goes to far afield). You will see that seldom is > anyone looking for somebody that has even minored in mathematics. > > > 7. The CCI data for calculus are clearly showing -- very > dramatically -- > > that teaching methodology makes a staggeringly large difference. There > > is no such clear study for earlier grades in math. Such a study of > > teaching methodology in the younger grades needs to be done. A team > > needs to seek grant funding for this. > > > Talk to Deborah Ball (I, for some reason, think you may know her). > Some similar things are being done at Michigan and elsewhere, but not > necessarily with your emphasis. They seem a little more indirect. > > > 8. Nearly all the data for the BSDT is from the original test which is > > not multiple choice. I do have a small data set on a multiple choice > > version of the test. Overall, the multiple choice version showed no > > difference in scores from the non-multiple choice version, though > > individual questions differed widely. The number of students was too > > small to conclude anything. Testing of the multiple choice version > needs > > to be done with many more students. > > > > 9. There is no equivalent population of students for the BSDT that > there > > is for the CCI -- students from clearly Interactive-Engagement > > methodology programs. The difference based on methodology on the CCI is > > so enormous that it becomes really important to know if this holds also > > for the BSDT. > > > > 10. I think this is long range the most important. The skills and > > comprehension tested on the BSDT in schools are not the program of a > > specific course or a specific school year. A project to test the effect > > of IE methodology is far more difficult to design than in calculus > > because it is not just one course (though it can be done in one > > 2-semester course -- My Integrated Lab Program (ILP) was designed to do > > exactly that). > > > > 11. As I am one lone person, there is no possibility that I will carry > > out the above program. I welcome anyone's thoughts on how some progress > > can be made and documented. But I repeat my overall point above: There > > will be no major overall progress until we can really reform the > > training of elementary school teachers in large numbers. The forces > > maintaining rigidity in the elementary schools are enormously strong, > > teacher's unions being one but not the only one, so that I despair of > > any real effective change in my lifetime. I wonder if it might be > > worthwhile to look into trying to do some initial work in private > > elementary schools. Or in schools that are experimental, charters, or > > whatever. > > > I happen to agree and, I assume in some sense, Deborah Ball would > also. There are a number of people working in this direction although > with various agendas. However, she probably knows most of those who > are doing work at the elementary level. I have been working for a > number of years to change the training of elementary school teachers > (isn't there TEDS at Michigan State, by the way) within my institution > and the resistance is enormous at the collegiate level (and not, for > the most part, from the mathematics department). We have met the enemy > and he is, sadly, us. And I'm not talking about the teacher's union. > > > I welcome comments from any and all, > > > > Jerry Epstein. . . > > > > On 1/27/2012 12:19 AM, Ed Wall wrote: > >> > >> John > >> > >> This seems reasonable. But is that gut feeling or intellectual > >> discernment (smile). Of course, we would all like to think it is the > >> latter (and that is gut feeling).Ed > >> > >> On Jan 26, 2012, at 11:45 PM, John Clement wrote: > >> > >>> Here is an interesting article about the psychology of accepting > >> something: > >>> > >>> http://www.sciencenewsline.com/psychology/2012012017430043.html > >>> > >>> John M. Clement > >>> Houston, TX > >>> > >>> > >>>> > >>>> I've been reading this conversation as it has progressed and > >>>> thought I'd make a comment or so and put forth a speculation. > >>>> > >>>> I think I read Bob Hansen as saying he'd like to see some > >>>> correlation of traditional 'success' (and while I believe he > >>>> has some candidates in mind for this term, they seem to be up > >>>> for debate) with the interesting results Jerry has noted. > >>>> However, I really am doubtful much of anything will > >>>> necessarily convince those who don't want to be convinced. > >>>> Further, I rather doubt that those who don't want to be > >>>> convince are going to take John up on reading and trying. > >>>> However, strangely enough I am starting to see sort of a > >>>> Kuhnian paradigm surfacing with regards a number of folks who > >>>> were randomly anti. There is still skepticism but it has > >>>> become grudgingly nuanced. > >>>> > >>> > >>> > >>> > >>> ------------------------------------ > >>> > >>> Yahoo! Groups Links > >>> > >>> > >>> > >>> > >>> > >> > >> > > > > > > [Non-text portions of this message have been removed] > > > > > > > > ------------------------------------ > > > > Yahoo! Groups Links > > > > > > > > > > > >
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