<p>Why the arms race? The schools want to pretend that they are teaching something. The problem is that while it is hard to ruin a good student, it is pretty easy to ruin less than stellar ones. Math education has become much more abstract and has less practice than it used to. This harms all but the most gifted students.</p>
<p>Kids’ brains wire up at different rates. Pushing some kinds of math too early discourages later developers. Another problem is that the acceleration has reduced the amount of practice kids get. This creates problems for the kids who would be good but need more time and practice to get to a give level.</p>
<p>Don’t get me started on how badly the AB/BC calculus course teaching and structure are in my child’s school. Or how stupid it is to teach matrices in 8th grade.</p>
<p>It could be worse: you could be here in NJ, where the math standards (soon to be replaced :)) specify a strand for “functions” that begins in second grade. And yet, I see many 11th graders struggling with the most basic function questions. NJ (and probably many other states) tend to teach broadly and repetitively in math (via the “loop back around” method), not deeply to mastery and automaticity. This situation, along with the stats I posted before, lend me to associate irony with the OP’s question.</p>
<p>Yeah, the schools don’t give credit because they found that more kids with AP 4s and 5s were flunking when they skipped the introductory course.</p>
<p>^I can’t reconcile your first sentence with your second. If early introduction to functions and function concepts doesn’t lead to mastery of functions later on, then the value of this approach is … ?</p>
<p>You said that students in the 11th grade have problems with functions. They aren’t currently being taught functions early - you said that it is a proposal or a consideration.</p>
<p>No, I said that the NJ math standards specify functions as a strand in the 2nd grade. The Common Core Standards, which first introduce the concept of functions in 8th grade, have been adopted here in NJ, to be phased in over the next 2-3 years.</p>
<p>Ummm…my son learned matrices in 5th grade (addition, subtraction and multiplication). He found it useful at several points in HS as well as for computer science classes.</p>
<p>Functions without a thorough grounding in algebra is very tough. Our experience that poor, watered-down Alg I instruction led to problems in Alg II/pre-calc (where functions is taught) because topics were omitted and/or not given enough reinforcement to truly sink in. Sometimes it also take a while for that abstract thinking part of the brain to kick in with this kind of stuff.</p>
<p>Don’t hate on math. I’m so happy I was able to jump ahead. The biggest benefit is being in a class where 97% of the kids actually LIKE math, or want to do well in it. </p>
<p>The worst part about any non-honors, not advanced course is the section of the class that slows it down. Either they horse around, need extra attention, or whatever.</p>
How young? The second grade? A lot of kids don’t really know fractions, decimals, exponents, any of this stuff in the second grade. I think that’s when I learned my multiplication tables. To me all that stuff is far more important than some cursory tangential knowledge about functions. I don’t see a problem teacing functions in a rudimentary way to young kids, but I also don’t really see much value.</p>
<p>How would you teach functions to a typical seven year old (especially since you don’t want to include algebra as part of the teaching)? Something like, “Each mommy birdie has three chicks. So if you ask a baby birdie who its mommy is they only have one answer, but if you ask a mommy birdie who her baby is she has more than one answer?” Really, I’m interested how knowing about functions has any real value before you learn any algebra.</p>
<p>IMO functions should be taught early in the Algebra curriculum, but before that? I suspect very few kids will link the concept to anything concrete before they have any Algebra. It reminds me of all the kiddie set theory I learned in the early grades. I’m sure it might have been beneficial to a few kids, but I bet it merely confused most of them. I remember having a hard time relating it much of anything else I was learning at the time.</p>
<p>and I better clarify - I’m talkng about the typical 7 year old. Not somebody’s mini Liebniz…</p>
<p>Wasn’t there a program, later abandoned, called “New Math” a couple of decades ago that left young students more confused about the basics than ever before? I think a big problem in the lower grades in math is the textbooks get away from concrete real-world story problems too soon and enter conceptional math where you copy a formula and plug in numbers, yet seldom truly understand WHY you’re doing it. After awhile, most kids become slaves to precise method-following, rather than discovering that they are in a magnificent tinker-toy/Lego playland where they can construct their own solutions with the parts they were given.</p>
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Yea, I missed you, too, Bovertine. Thanks for thinking of me. ;)</p>
<p>I had new math back in 1964 4th grade. I learned a lot about set theory. I don’t think it hurt me. </p>
<p>I liked the way my kid’s elementary school taught math - though they changed the program officially every couple of years. They ended up with one that people like to hate, but I thought worked pretty well when coupled with some practice on the side with math facts. My kids are both much better at doing math in their heads than I am. They learned early on (2nd or 3rd grade) to make “98+5” turn into the easier math problem “100 +3” without blinking.</p>
<p>Which means they were probably going to be advanced math students all along, since they could grasp conceptual math from an early age. I had two sons doing extremely advanced math before kindergarten, but I’m not going to pretend that the calculation shortcuts I taught them would work for the vast majority of kids. For one thing, both almost instantly understood that multiplication was just addition taken up a level and that exponents were just multiplication taken up a level. This is an ah! moment that cannot easily be taught.</p>
<p>Most young kids need something tangible and real-world to observe before they can get to this realization. Richard Feynman talks about this in one of his biographies; as a Noble-Prize winner, he once served on a committee that reviewed textbooks for acceptability. Often, even the word problems made no real-world sense: “In the sky there are 3 red dwarves at 2000 degrees, 4 yellow stars at 5500 degrees, and 2 white stars at 7500 degrees. What is the total temperature of these stars?”</p>
<p>I don’t doubt that the New Math had some value to some segments of the class. I only recall that enough students struggled with it that it was pretty much abandoned by the mainstream.</p>
<p>I don’t think it hurt me either. It also didn’t help me much at the time, as far as I can recall. And to the extent it took up time I could have used learning something of value, I guess you could say it hurt me.</p>
<p>Maybe I subliminally got something out of all those Venn diagrams, but I think the time I spent learning how to convert fractions into decimals was of much more value to me.</p>
<p>You could use something as simple as the number of legs for common
animals or the colors of items in a picture. You could also pick
sets of things and create arbitrary mappings from a set to itself.</p>
<p>If the student has facility with arithmetic, then you could introduce
the ideas in the first chapter of Spivak or perhaps The Philosophy
of Mathematics by Russell.</p>
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<p>Doing it earlier would allow you to describe arithmetic operations as
binary operators on sets of numbers. Kids would get the abstract notion
along with the concrete.</p>
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<p>There was a math professor at Purdue that used to have long discussions
about the new math on the misc.education newsgroup. His take was that
the teachers weren’t able to understand the New Math which resulted
in confused kids, parents and teachers.</p>
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<p>Here’s an article on the New Math from Patrick Suppes back in 1965 in
PTA Magazine.</p>
Yeah, that’s the problem. A lot of students, even those graduating from high school, never acquire facility with arithmetic.</p>
<p>I’m not a mathematician but I managed to learn a significant amount of math in my lifetime without learning about functions until I took Algebra. It wasn’t until that point where knowing whether something was a relation or a function, continuous or not, etc. really made much difference to me. I suspect if I asked any of my colleagues they would say the same thing.</p>
<p>Did you learn about functions in the 2nd grade? If not, did that fact somehow hamper your abilitiy to grasp mathematics? I’m assuming you’re some sort of mathematician or math educator.</p>
No, it means they practiced doing math in their heads at school more than I ever did. The entire class could do math in their head better than I can. I’m sure with practice I could do it too.</p>
<p>I admit, my older son was precocious in math. He looked at a clock and said five 12’s make up sixty minutes at age 4. He skipped ahead into 3rd grade math in 1st grade. But my younger son was a pretty ordinary math student - and seriously hampered by being below average at memorizing math facts. He’s still a bit shaky on his multiplication tables. His pre-calc teacher loved him because while he couldn’t remember formulas he could generally figure things out from first principles. </p>
<p>I have seen that abstract mathematical thinking does seem to be somewhat age dependent. My older son got the computer programming bug really early (2nd grade). In fifth grade his gifted class had a project where each kid tried to teach something to the other kids. My son tried to teach kids a very simple BASIC computer program to add two numbers. Parents had been invited to observe the class that day. My son might not be the best teacher, but the kids in his class just didn’t seem to be ready for the kind of abstract thinking about variables that was required.</p>
<p>My mother talks about learning algebra from her grandfather who did one of those tricks where you pick a number and add and subtract various numbers from it and the person tells you what number you started with. For her algebra was a parlor game.</p>
<p>At any rate the original question was are too many kids skipping ahead in math. My feeling is that while I’ve seen that plenty of fifth graders are less ready for abstract thinking than I would have thought, I do think that there are more kids ready for algebra by 7th grade than one used to think and it’s a good thing the schools are accommodating them.</p>