Gifted Issues Discussion homepage Forums Identification, Testing & Assessment Common Core Mathematics: background

I'll speak to economics. All that's really needed to cover graduate material is the first year university series of calculus and linear algebra for math majors, with some second year stats to cover econometrics. To get published in AER or Econometrica, it's helpfully to have more econometrics.

I studied in Canada. My profs who had taught at Harvard and Princeton used their graduate textbooks for my upper year classes because, apparently, those universities' students didn't have the requisite math background to do the graduate work yet.

My husband stopped his studies in physics and math in high school. Had he had more rigorous treatment of math before university, he says would have gone into physics and not law/finance.

My DD is no Maths prodigy but she had all arithmetic operations on vulgar fraction 'down' by 7 just from talking about it with me from the back seat while we drove to places.

Why should she have to wait for everyone else?

I had no intention of 'accelerating' her Maths but just kept giving her stuff and she just seemed to effortlessly get it so we kept going.

I was reading about the uptick in hip displasia in infants being attributed to an uptick in the practice of swaddling. Apparently, the joint needs full mobility in the legs to develop properly.

What would we be doing to our kids' minds in 'swaddling' them by unnaturally restricting their natural learning abilities?

Wu's point is that students are made to memorize an algorithm, rather than being taught why it works and how it fits into the bigger picture. He's right about that. EPGY may very well be different, but the vast majority of today's textbooks present algorithms with minimal or no explanation about the ideas behind them.

Regarding "correct," he means that a lot of teachers don't understand this stuff, either, and so they don't see how standard algorithms fit into the bigger picture of mathematics. This problem makes them susceptible to the flawed reasoning behind goofy algorithms like partial products and guess-and-check.

This stuff is the bigger picture of the problem with the Dolciani algebra textbooks blog post that you posted.

Well, speaking to chemistry there, it was of NO use to come into chemistry (as a major) having already gone through a calculus sequence in high school, because most major's tracks have you taking general physics as a sophomore undergraduate, anyway, and you only need VERY good algebraic skills and the rigor taught by the study of geometric proofs (IMO and my DH's) in order to really learn gen-chem well.

HOWEVER-- those students who do NOT have good geometry backgrounds often find themselves at sea when it comes to molecular geometry (and all that comes with it), and truthfully, that "sense" of spatial intuition comes only with a great mastery of the subject. It's part of what makes chemists-- chemists, and not biologists, statisticians or pharmacologists. This makes organic chemistry in particular a NIGHTMARE for those students, many of whom change majors to either biology or physics at that point. (It also makes instrumental analysis, molecular spectroscopy, and inorganic chemistry far, far more difficult than they should be.)

So while I respect that most people feel that they have "never used" the geometry that they learned so arduously, I disagree. That deeply embedded understanding, and the diligent practice of logic applied to proofs is unmistakably preparation of the very best sort for some STEM fields.

Also true that as a chemist, having advanced skills in mathematics isn't really very useful (beyond calculus) until you get into distinctly graduate level topics. Stat Mech, instrumentation design, etc. Better to have had Statistics early and well, IMO, than calculus in high school (where it is often taught very BADLY-- by people who don't really understand it well conceptually).

So I think that what the university professors in mathematics are really saying is two-fold, but the second part of it is mumbled a bit so as not to offend those that they hope will go along with CCSSM:

a) students are NOT really "advantaged" significantly by taking very high-level mathematics topics earlier and earlier, and some of them may be "rushed" if this is encouraged; and

b) most high school teachers have no business teaching anything past trigonometry anyway, so they should leave the advanced math topics to those who DO understand them.

ITA with Val's post, by the way.

Honestly, the "allows study of more advanced topics" thing makes no sense to me unless one assumes that there is a finite shelf-life to learning math and science topics. As far as I can tell, there isn't.

Acceleration allows students to reach advanced topics EARLIER. The question is whether or not that is a good goal. I'd say that as a side-effect of natural learning and meeting an individual's needs and interests, it's fine-- but probably not a good thing to have as a primary GOAL, so much.

But that's me.

EMPHATICALLY agree.

My opinion is that this only highlights a major oversight in education, though, because except for Geometry proofs, and some conversations about the scientific method, high school students are not introduced to the basics of logic. Even college students mostly manage to avoid the topic, since it's mostly presented in a philosophy course which is elective for most students, and considered a to-be-avoided one at that. As a result, most people think they know what "logic" means, and then go about proving how they don't.

Logic is only the basic building block of scientific, mathematical, and philosophical thought, so how important could it be?

Yes, exactly. It all fits together in one big meshy intertwined whole.

... or 'mushy intertwined hole.'



As the case may be, I mean.



DH and I were discussing the learning process for him, me, and our DD this morning...

for him and her both, it's as though their "mastery" is an enormous floating dock of boxes... and "learning" is a process via which there is "capture" and then there is "placement" within that larger scaffold. Simply TYING a captured idea (or seagull) to the scaffold and letting it continue to fly in circles isn't mastery to either of them-- they have to be able to reel it in and place it within the larger scaffold in order to "own" it (mastery).

It BUGS them when it is assumed that just tethering more seagulls = "learning" because to them, it seems pointless and arbitrary. But without an expert teacher to respond to questions about where things fit within the scaffold, and whether there are ties to other parts of it... well, it just doesn't STICK for either one of them.

Me, I'm a trivia Goddess, so I can roll with the tethering of more and more seagulls. To a point.

But for learners like my DH and DD, having actual subject expertise in a teacher isn't just a nice bonus-- they in all probability CANNOT really learn much or very deeply from anyone else. DH is a pretty impressive autodidact now (as am I), but this was definitely not true at our DD's age. We both suspect that learners like us often HAVE to have a certain maturity and level of mastery (broadly) before we CAN function effectively as autodidacts. Our position of learning is dependent upon the strength and extent of the seagull-stuffed scaffold under us.

Like a Borg ship. Metaphorically.

Definitely a point of view I've had. Along with a range of meta- skills from research design, skepticism, logic (and fallacies), and set theory to heuristic development, mnemonics, mind-mapping, introspection, learning theory, and more... I'd have all that taught directly when kids are ready for it. Not accidentally and unmeasurably fluffed under other topics like literature or speech or geometry, etc.

I see the need for acceleration as being most important for polymaths who have multiple future avenues of study. Per Bostonian's point, if a student is potentially interested in a field that requires higher level math to access more than a conceptually basic understanding of the subject (e.g. finance, economics) then I do think math acceleration is warranted. Granted, most of our children will see the need for the math in their field(s) of interest and be intrinsically motivated to learn it, too, so the acceleration will be needs-driven.

Really, in many fields, you have to be at least at the senior undergraduate level in the topic before you have a realistic understanding of your interest in, and willingness to continue in, studying the topic.

I think all students would benefit from the topics Dude and Zen Scanner mention. Ideally, elementary math courses would incorporate proofs and logic as early as possible. To be an effective critical thinker requires a deep meta knowledge, and I think that is largely lacking in modern math instruction. (It's also why I'm keen on the idea of AOPS.)

And I agree with your critical seagull scaffold hypothesis for autodidacts to take flight. (Terrible pun!)