I maintain that it wasn’t that unusual, but I’ll PM you the details.
Dfbdfb, I thank you for your replies regarding my list. I am afraid we might to disagree about the relevance of my list to the actual problems. I do believe to understand the outcomes of comparative tests such as PISA and TIMMS rather well, and also understand the various efforts to dismiss those results based on falsehoods such as different sampling. Again, this has been debated ad nauseam around here and there is little to gain from rehashing similar arguments. My conclusion is still that our system spends too much for the results it delivers. The fact that our system happens to deliver results comparable to the ones of countries that were solidly anchored in the Middle Ages not too long ago and recently joined the group of industrial nations should be reason to be concerned.
But again, we could spend ten pages of CC posts and not get past our differences. I am solidly in the camp that believes our system is abysmal, poorly structured, and has made no improvement in terms of learning. Our progress, if there is one, is one anchored in the new technology that has marked our past 20-30 years.
Lastly, regarding this “What percentage of 6th-graders would have been able to answer “What’s seven times nine?” (better: “What’s fourteen times twenty-one?”) in less than n seconds fifty years ago, compared to today?” my answer would be that my father who started middle school in 1965 would have solved it in a few seconds. I know this because by the time I was in 6th grade, he had taught me to solve it easily.
Fwiw, the approach would have included the following MENTAL steps
14 x 21
2x3x7x7
6 x 49
6 X (50-1)
300 - 6 = 294
Time to solve? How about 6 seconds? Oh, make that 10! Of course, this comes from a guy who has been telling students to drop their calculator when taking the SAT. A biased point if there was ever one.
Not how the above aligns itself with the Common Core, but I can tell you that my cousins in Europe would have approached AND solved the problem in just a few seconds. I also think that this would appear as voodoo in your typical US public middle school, or perhaps to juniors in high school.
On a last note, regarding the dumb and dumber tests, check what Harvard’s entrance exam in the 1930s looked like and check the evolution (and recalibration) of the SAT in the past decades.
Again, our perspectives might be different, but I believe that there is ample factual evidence to support a number of my bullet points.
I have been a fan of your posts for a long time, and don’t doubt that. But, I clearly said, “most” elementary teachers…
same question as hunt. I attended a fair-to-middlin’ public school system back in the dark ages and we had nothing close to a course in “logic”.
I Pm’ed you, too.
I did not have any courses specific to logic in (public) K-12, but high school math (particularly geometry with proofs) included some instruction in logic.
However, some students may eventually (in college) get more instruction and practice in logic than others. Perhaps it is not a surprise that math and philosophy majors tend to do well on the LSAT, which has a logic puzzle section.
Post #64 reminds me that my geometry instruction was classic, old-school, which demanded logic. (This is not the way geometry is taught today to my students.) I sailed through it with an A and loved it. Only math class I ever truly loved.
A quick web search for on line high school geometry textbooks reveals that they still do include instruction on logic and proofs today. So it is not like logic and proofs have disappeared from high school geometry.
Of course, whether students today or a generation ago pay as much attention as they should, remember the material learned, and notice its applicability to other things besides geometry, is another question entirely.
Wrong. I cannot find such instruction in any of their textbooks, with which I am well acquainted. Nor are they asked to use any kind of logical processes in reaching proofs. I’ve looked at several geometry books, and geometry is the most frequently requested subject for math tutoring, which is absurd. Most of my students aren’t even taught theorems any longer, or how to use theorems. This has been true for at least 8 years now, maybe 9.
No surprise to me, since Geom requires reasoning skills, which are in short supply in today’s students. (Reasoning skills are also tested by the SAT, and that is why many don’t like the test – their little kiddos don’t do well.)
fwiw: Geom in our highly-ranked public high school is tested using scantron. How one can complete an actual ‘proof’ with multiple choice is beyond my comprehension.
Perhaps the logic and proofs disappeared from high school geometry courses in your state or local area, but not all.
@epiphany, your BLANKET STATEMENTS make it hard to take your posts seriously. You may be skilled in “analysis and reasoning”, but your delivery is not persuasive and may actually have the opposite effect.
SOME high schooler students read a tremendous amount. SOME high school geometry classes are proof-based.
If you learn to qualify some of your statements, your arguments/points would seem much more reasonable.
Here are the 1997-2010 (pre Common Core) and 2010-current (Common Core based) California math standards:
http://www.cde.ca.gov/be/st/ss/documents/mathstandards.pdf (geometry on page 42)
http://www.cde.ca.gov/be/st/ss/documents/ccssmathstandardaug2013.pdf (geometry on page 133)
While the older one appears to have a relatively greater emphasis on proofs (which require instruction in logical reasoning as a prerequisite), the newer one still has significant proof content. So if a student is not learning anything about logic and proofs, it is a deficiency in either the teaching (by the teacher) or the learning (by the student), not a deficiency in what the course is supposed to contain.
ucb:
Again, I’ve examined thoroughly some of the current textbooks, but an additional problem, yes, may be the teaching. Much is being left up to the student to just “figure out” on his own because that’s supposedly “the new way.” Very little logic is employed and scant use of theorems.
As to the screaming/shouting in Post #70, I am speaking about a marked trend away from logic and toward an “experimental” way of teaching. Maybe it’s not true in your State or region; it is in mine, or maybe you’d like me to capitalize my words, since after all you’re so much more “reasonable” (not) than my statements (not “reasoning”) about my actual and recent experience, experience which has persisted over 8+ years. There’s nothing extreme about the statements. They reflect experience and acquaintance with the actual materials.
Have a nice evening. Or try to.
Some. A minority. All kinds of articles have been published about the trend away from sustained reading among adolescents, beginning especially at middle school. All of my posts were relating to trends, not to universals, but the trends have persisted and are corroborated abundantly. Trends are relevant to this thread as they affect current and future standardized testing – the subject of this thread.
Thank you, blue. Your experience corroborates mine.
Again, ucb, I never said “all,” but the trend has been expanding, first of all (over a period of years and over a larger area). Second, it’s troubling that it would be anywhere. As bluebayou has explained, it’s contradictory – kind of like instructing students to solve algebraic equations without teaching them factoring. All in the name of being “new” and “experimental.”
What state are you in where there is no instruction on logic and proofs in high school geometry?
What textbooks were you looking at?
I have to say that, based my own experience and some (limited) reading I have done, I tend to agree with @epiphany when it comes to the trend in geometry instruction.
I think advanced math teaching at good US high schools has improved over the 35 years since I was in high school. However, the near removal of truly rigorous reasoning (mainly from the Euclidean geometry course) is one area in which it has gone backwards. Perhaps this is the flip side of allowing advanced math students to take calculus in 10th or 11th grade, since geometry then is taught in 7th or 8th grade when children’s brains aren’t as developed.
Here’s some information that I posted before http://talk.collegeconfidential.com/discussion/comment/18159780/#Comment_18159780. It includes links to a few articles that support this belief. See in particular the quotes from the study Perspectives on the Teaching of Geometry for the 21st Century, which seems to be a survey that is very much on point …
… I first formed my opinion after observing my own children’s curricula, and conversations that I’ve had with fellow grumpy old (former) professors in a university math department have mostly confirmed it. My kids attended much better schools than I attended (they had future math olympiad finalists / winners as schoolmates), but in their geometry class they proved only a minority of the important theorems and proofs weren’t the basis of their homework and tests. An example which this article uses
http://www.educationnews.org/k-12-schools/the-modern-day-high-school-geometry-course-a-lesson-in-illogic/
is that students are simply told the angles of a triangle add up to 180 degrees; it isn’t proven. In contrast, when I took geometry we pretty much proved everything building up from the basic postulates (at least until we got to solid geometry); we even had a few ruler and compass constructions on tests. And I went to a fine but not particularly wonderful high school.
I think geometry courses now emphasize knowing and applying the theorems (not proving them); “experiencing” geometry in the everyday world, the role of transformations and symmetry, and computation using coordinates. I’m not saying that’s all bad, but it’s not formal deductive reasoning.
Also, if I remember right, when I took BC Calculus epsilon-delta proofs were on the exam; they definitely aren’t nowadays.
Some quotes and articles I found
An article that discusses the Introduction of “Intuitive” geometry - http://www.jstor.org/stable/30173550
In nearly any discussion of K-12 (and, to a somewhat lesser extent, college-level) instruction, nostalgia will rear its hoary head. In addition, people will remember their own experiences and often overextend them to others.
Was I taught how to do rigorous (in its grade-appropriate meaning) geometric proofs in high school? Sure.* Were most students in my high school taught to do geometric proofs. Nope. The whole idea of college-prep courses in my high school back in the 80s was that they were open only to the higher-achieving students—and this was a very, very widespread way of sorting (perhaps better: pre-sorting) students by ability.
I strongly suspect it was the same at your high schools (unless it was a terrifically small one, or a magnet or somesuch, of course)—you may well have been taught logic and reasoning and proofs and so on, but I’m pretty sure that wasn’t the case for everyone there.
So if some students aren’t taught those things now, well, that may or may not be optimal, but either way, without evidence to the contrary it’s wrong to simply assert that this is a lowering of standards over the years.
- Just like one of my daughters was taught in her public high school, and another in her public **middle** school. But I digress.
warning: perhaps another long digression below.
I liked the “old” way of learning math. Perhaps my mathematical mind was just suited to it, but that is neither here nor there. What xiggi’s solution of 14x21 reminded me of was how I liked to solve math problems when I was young. I almost instinctively would see the types of short cuts that solution shows. However, not all students see these things and they need to be taught how to see the short cuts. In my own kid’s education they had what I recall to be called “mental math”, which was mean to show these types of short cuts or ways of looking at numbers. (Maybe it was called something else, my mind is turning to meatloaf so I can’t be sure.) She also (I guess fortunately) was taught geometry in the same type of classical way that I remember learning it. None of her education was in public school (although mine was all public.)
However, many students struggle with numbers starting even in the early grades. That is where I think all the push for “new” math came from: there and from Chicago, SIngapore, and who knows where else. Not all the new concepts of teaching are awful. For example actually using manipulative pieces in math may help. I never saw those as a kid. There is so little time for individualized help in public school for those who don’t “get it” fast. So in general, the parents don’t seek out extra help (no time, can’t afford it, don’t have a good education themselves), that these students fall further and further behind.
In truth my own opinion of education has two big negative views of the system:
- Good to excellent students who are not challenged enough: This exists in privates that I have seen as well as public. There can be limited places in "gifted and talented" classes, or AP or Honors. These are often admission by test, not desire to learn or ability or even teacher evaluation. So children actually are turned away from learning more. IMHO this is a disgrace. To the schools who actually let children who want to challenge themselves quite a bit do so (and I have seen these too) - thank you.
nb- some teachers let the class clown take over the so called Honors, Advanced, Excellerated or whatever you call it, class and let them drag everyone else down, also a failing.
- Poor students who are not taught to succeed.
Everyone throws around the term “love of learning” at the highest levels of secondary school. Without going into a whole thing of who has it and who doesn’t, what I would like to see is that all students are taught to have a love of learning.