<p>BTW, the math dh uses most doing cancer research is statistics and he's constantly frustrated that many of his colleagues don't really understand it or use them properly.</p>
<p>Science programs require math through calculus and stats, but there are many venues that do not go any further than that.</p>
<p>He may come into his own when he takes more advanced (possibly, more abstract) math classes in college. But these days one must absolutely, positively succeed in math to succeed in most scientific and technical subjects. A favorite quotation of mine, found in a very good book on math, is "The proper thing for a parent to say is, 'I did badly at mathematics, but I had a very bad teacher. I wish I had had a good one.'" W. W. Sawyer, Vision in Elementary Mathematics (1964), page 5. In other words, don't EVER suggest that it is part of your child's nature, or your nature, or ANYONE's nature to not be able to learn math. Instead, go looking for good instruction and then apply effort to learning it. </p>
<p>For homeschooling parents of children somewhat younger than your oldest, I prepared a FAQ </p>
<p><a href="http://www.artofproblemsolving.com/Forum/viewtopic.php?t=6606#33140%5B/url%5D">http://www.artofproblemsolving.com/Forum/viewtopic.php?t=6606#33140</a> </p>
<p>several years ago that includes much advice that has worked very well for my son. You mentioned IQ tests, and I will say, feeling very gratified as I say this, that my son once had a pairing of an IQ test and an achievement test in which his math scores were FAR above what would be predicted from his IQ. In other words, he has had some very good instruction in math and has done his part to do his homework and think about his lessons. (No comment about my son's IQ, but the bottom line is that whatever the predicted level of achievement, that level can be exceeded by the high-IQ child who gains especially good instruction in a subject he pursues avidly.) </p>
<p>Your son should be able to use his science interest to revisit his math abilities and figure out how to strengthen them. The key motivational point is to know that MANY students at beginning college age in the United States have </p>
<p>a) crummy math preparation, far below international standards, </p>
<p>but </p>
<p>b) time to learn and opportunity to learn and improve as math students. </p>
<p>I have another FAQ </p>
<p><a href="http://www.artofproblemsolving.com/Forum/viewtopic.php?t=11946%5B/url%5D">http://www.artofproblemsolving.com/Forum/viewtopic.php?t=11946</a> </p>
<p>that gives links to detailed suggestions about how to succeed as a college math student. College math courses are notorious "weed out" courses, limiting the major subjects that college students can pursue successfully, so knowing how to succeed as a college math student opens lots of doors. </p>
<p>One more suggestion, a specific helpful program which I have used as a customer (I have no affiliation with the company) is the online lessons from ALEKS </p>
<p><a href="http://www.aleks.com%5B/url%5D">http://www.aleks.com</a> </p>
<p>The great thing about ALEKS is that it offers UNLIMITED forty-eight-hour free trials, so you can see if it will be helpful in your situation. It is online, computer-based practice in math up through precalculus. </p>
<p>Best wishes for much success for your son.</p>
<p>the option to take is the life sciences.. because if at this point he could easily understand science.. probably later in college he won't as the proofs of physical or chemical theories become based on math..
medicine is a viable option....
there is also one more thing... does he consider himself not distinguished in math because of problems of understanding or just that the grades aren't high enough..</p>
<p>my daughter graduated with a bio degree despite tested math disability- she had to take not only chem, but ochem, calc and stats for her degree & she managed. Higher level math, she doesn't have as much difficulty with as simple computation, but there is usually tutoring available and programs to assist understanding</p>
<p>
[quote]
Bright kids can breeze through mathematical concepts, just instinctively "getting" them with little or not effort - up to a point.
[/quote]
</p>
<p>It's always seemed to me that everyone has a built-in wall for math. I really understood statistics in college, but never quite "got" calculus. I <em>used</em> calculus, but didn't have an instinct for it. My friend the Caltech math major says that at some distant spot in math (far beyond basic calc) he hit a wall where he just couldn't visualize anything anymore. He didn't get past his wall, though, as he was lured away by journalism and transferred out after his junior year. WashDad Jr is in Calc BC and can't see his wall yet -- he started trying to reinvent calculus when he was in algebra. </p>
<p>It would be interesting to get more input from forumites about the "math wall." Has anyone experienced hitting the wall, and still achieved deep understanding after working through it? I mean, I learned to use differential calculus but I didn't "get it" the way Junior has. Just wondering...</p>
<p>I hit a wall in my high school precalculus course, a new math course labeled "analysis," and never took a math course again. But I knew I knew more math than what my high school record suggested, and indeed the next year I sometimes helped high school classmates with SOME problems in their calculus homework. Different minds resonate to different problems, so one learner may hit a wall when there is still an open path beyond that wall for the same learner. </p>
<p>Because my oldest son is fairly interested in math, I have invested a lot of my personal reading time into learning more about learning math. A hugely important, not to be missed book is Liping Ma's Knowing</a> and Teaching Elementary Mathematics, which explains how lousily math is taught in the United States. Following up on that book, I looked into math pedagogy in other countries, eventually collecting quite a few Chinese-language math textbooks and popular books about math. </p>
<p>This last summer I attended a course for schoolteachers on new ways of teaching math. I am not a schoolteacher by occupation. It was interesting to compare what I know about math--in some respects, much more about upper-division undergraduate topics than they do, as to book-learning--with what they know about math. They can do algebraic manipulation without getting flustered the way I do, and they can operate Texas Instruments graphing calculators with ease. What a person knows how to do in math has much to do with BACKGROUND: what you read, and what you practice. Today, at age forty-something, I am beyond where I was when I hit the wall in high school, and thus my advice for young people choosing college courses is to get help early and often, look for books other than those assigned for your course, and keep pressing at the wall until you break through it.</p>
<p>Being the math-challenged parent of a math-lover, I can only comment from the sidelines. Many young students who excel at arithmetics do not "get" algebra easily; and algebra is the foundation of calculus. Many students hit their walls in geometry, which is more proof-based than other types of k-12 math. Some are highly concrete learners and need "real-life" examples to understand the math.
Tokenadult is right, however, that math can be very badly taught in this country, hence the popularity of the singaporemath program and the constant search for new pedagogies.</p>
<p>I don't know if S can be said to have hit a wall. He just does not care for more applied math; and this lack of appreciation affects his performance, just as his love of abstract math earlier drove him to forge ahead.</p>
<p>EDIT: cross-posted with Tokenadult.</p>
<p>Wow! These are all such great responses! Already, this has been very helpful! Thanks to everyone!</p>
<p>I have a lot of responses to all the responses...will take me a bit of time to get them all done...here goes...</p>
<hr>
<p>“...She chose the "best but hardest grading " prof. She is working very hard and has yet to make an A on an assignment but..... her writing is improving dramatically...I'll wager that your kid will do the same. Of course, it has to be his choice. Kids are good at what kids value.”</p>
<p>So glad to hear that your daughter overcame the writing challenge. Yes, I think that is a very good idea. </p>
<p>“You said he can do it if he works at it, so what's the problem?</p>
<p>”...As well, it is an opportunity! That God that there is one thing that is not easy for him such that he can learn a little discipline and perseverance - both much more important to success than is a high IQ...”</p>
<p>"“...Has your son had any calculus? Or is he stuck at the precalc level? In my experience, a lot of HS kids (including mine) have more trouble with precalc than with calculus. Also, it sounds to me like he (maybe with help from you?) needs to do some backfilling in math. NOT NOT NOT to knock your educational methods here, but some kids who teach themselves almost everything else just fine really need the discipline of a structured series of math classes (which they will probably hate) to get themselves to the point where they can learn scientific math on their own. Depending on what is available to you, you might want to look for an engineering math class, IF you can find one which doesn't require facility in calculus / stat / diffeq.”</p>
<p>We actually did hire a tutor for him in 11th grade, who got him thru Algebra Ii and Trig. At that time, he was struggling with some math anxiety, so I thought that maybe the best approach was for him to get comfortable with math, hoping that it would click for him. And, there was a time constraint because the teacher was not going to continue tutoring the following year, so I wanted to get it all covered. So, I told the teacher to de-emphasize homework and focus on getting the material covered. He did some homework, but not nearly enough to make it stick, I now realize. He had a major anxiety at first - I mean on a phobia level - but I really do believe that was emotional in nature. He'd had a bad experience at the time he was starting algebra and I think that is really the root of the problem. It's an association thing, perhaps. When he was 4, he was reading 4th grade books AND doing multiplication for fun. I had no intention of homeschooling him at that time. I enrolled him at the best, most prestigious preschool & kindergarten in town (even though we really couldn't afford it) and they assured me that they would work with him on his level. Well, they didn't. They had him doing the alphabet and doing pages of worksheets adding 2+2 when he was way beyond that. I spoke to them about it and they started letting him go off to read by himself while the other kids were doing the alphabet, so his love of reading remained intact, but they made no provisions for his math ability or learning style. THAT is when we decided to homeschool, so that we could accommodate him. But, somehow, it seems that ever since then, he's had it in his mind that math=horrible. I know this probably sounds ridiculous - he was only 5 so how could a 1-year experience still affect him? But this is a kid who remembered being in the hospital when he was born - no kidding! (That's another story.) So all thru arithmetic, it was a struggle. But he did ok and even started algebra in 5th grade. But then, there were some family traumas that coincided with the algebra, and I think there is a negative association there.</p>
<p>I should clarify here that we DID do structured math with him. While science was self-taught and we employed an 'interest-led' approach to nearly everything else, math was the one thing that we believed NEEDED to be structured. Else, how would he ever learn it? That's why I don't agree with a complete 'unschooling' approach - that might work to 'just get by' but what kid is ever going to teach himself algebra?</p>
<p>Anyway, the math tutor was great: very patient, calm, and understanding. She helped him overcome the anxiety and one day, everything just started clicking, finally, and he zipped thru Algebra II and Trig in 3 months. We were ecstatic! But, I now realize that the lack of lots of repetition/homework (he just did the minimum) was a mistake, because it didn't stick in his long-term memory well enough to be able to perform the problems quickly and easily on tests. I mostly take the blame for that because that was my decision, not the tutor's, although, several times I did ask her if she was sure he was getting it without doing much homework, and she assured me that he was. In fact, she said he was one of the easiest kids to teach, because he understood it so quickly. She said some of her other students struggled for weeks on some things and he got it in 1 day. So, I trusted her assessment and figured that maybe he didn't need as much drilling.</p>
<p>"He may also need to take a step back. I took Calculus at 16 and felt that true understanding was just beyond my grasp. The 2 I got on the AP probably reflected my shaky understanding. Three years later I took it again at Harvard and I wondered what my problem had been. I got an A and finished the final exam before the halfway mark. He may just need a little time for his brain to grow into the math he's learned.”</p>
<p>He's had a bit of Calculus (read a book on the concepts and watched part of an instructional video from The Teaching Company) but seems to be stuck on the college algebra. He actually seemed to enjoy geometry and trig and found them easy. So I guess it's really the algebra that is the problem. That's good to know that some students obviously take calc in college (and even algebra!) because S has it in his head that he should not take anything below calc in college. Yet, obviously many students do!</p>
<p>"“...I find that it helps me to think of math as a language - I need to "speak" it well enough in order to dig the meaning out of an article...”</p>
<p>It's funny, he thinks of nearly everything as a language...he has experience with several computer programming languages that he is fluent in, and he even thinks of psychology as 'decoding the mind' and claims that he can decode pretty much anyone's m.o. (I suggested maybe he could become a criminal profiler since he's so good at it, but he nixed that.)</p>
<p>So it really does seem that he should 'get' this language!</p>
<p>He's even considering majoring in linguistics, and is currently looking into computational linguistics, since it is a blend of linguistics and computer science (speech recognition software, artificial intelligence, etc., all of which totally fascinate him).</p>
<p>This interest in linguistics started with being fortunate to have a really awesome German prof at the cc who used to teach at Berkeley. This guy is so incredible and I am so thankful that he has succeeded in inspiring my son and drawing out his strength. In fact, my son actually hit the wall in his 3rd semester of German, to the point of considering dropping the class, but the prof encouraged him to dig deep and work hard, and it would start to click. S stuck with it and is now one of the top couple of students in the class, the prof said he is on his way to 'getting the pattern' and 'breaking the code' so that it comes naturally. The prof also said that he is teaching the exact same class he taught at Berkeley (I wonder how he ended up teaching at the cc) and it's very hard to get even a 'B' in the class so making an 'A' is really an accomplishment. S is working hard for that A but he loves it. I wish the same thing would happen with math.</p>
<p>Well, this prof also teaches remedial math. I told him about the math issue and he recommended that he take lots of math, at a lower level, to gain mastery.</p>
<p>I even considered having S take his remedial math courses, but S scoffed at that. If he did that, he wouldn't get to calc until 3rd year of college! So maybe it's not necessary to go quite THAT far back. And, he will not be at the cc this spring, so he can't take the class with that prof.</p>
<p>So I thought maybe a compromise - there is an 'Algebra for Scientists and Engineers' class at the local 4-year university he is transferring to this spring (he will attend just part-time as a transition to his college of choice in the fall) which I thought would be good to strengthen his algebra. I even thought that maybe he should take it anyway, even if he scores high enough on the math placement test to get into Precalc. </p>
<p>He took the math placement last week. He took the Accuplacer first, and aced it. But the math placement, the goal of which was to place him calculus, he didn't do too well on. He said he knew all the material, but just didn't have enough time to finish. So here is the timed testing issue again. Still, I think if he knew it well enough, he could do the problems in the time allotted. </p>
<p>He will retake the math placement this week. He is stressing about it because he is concerned that if he doesn't do well the 2nd time, he'll have to take college algebra (or maybe that algebra for scientists class) and it would look bad on his college application.</p>
<p>Well, that is part of the problem - he wants to apply to the College of Natural Sciences at UT Austin, and although calculus is not REQUIRED, most of the applicants have had it in high school. S feels that he is 'behind.'</p>
<p>The rest of his app. is very strong - 3.8 college GPA, 32 college credits, Phi Theta Kappa (the cc honor society of top 3%), lots of computer stuff, webmaster in 9th grade, etc. So do you think being behind on the math will hurt his chances of getting accepted into the sciences college? The transfer applicants are assessed primarily on GPA and the admissions officer told me he is a very good applicant as a transfer and that they do not consider calculus in making their decision. However, someone at the sciences college told me otherwise - that since nearly all the engineering applicants have calc & physics, the ones that don't get into engineering go into the pool of applicants for sciences. But, she was referring to freshmen and S is not applying as a freshman.</p>
<p>I think the most important thing is that he get STRONG, even if it means getting in to the UT liberal arts college and transferring to sciences later.</p>
<p>”...I would guess, being homeschooled, he didn't have a peer group of math geeks. These kids do various timed tests in Math Club, some of which culminate in State and Natl. competitions. It's fun! Really!”</p>
<p>Very true. It's not like he was sitting at home all the time - he attended homeschool co-ops twice weekly from 6th - 11th grade. Co-ops are sort of 'the best of both worlds' because they give the kid a taste of a classroom environment, but the parent still has control of the education. The kid can take pretty much anything he wants, with no regard to grades. So younger kids can take high school courses if they're ready. He took a lot of great classes he wouldn't have gotten at home: painting, sculpting, speech, drama, journalism, chess, etc. And he did take a wonderfully hard 2-semester chemistry class complete with lab and very demanding lab reports, taught by a chemical engineer now homeschooling her own children. But alas, no math club! I thought we had pretty much everything covered, getting him in plays, robotics competitions, etc. but never realized this deficiency, but now see that a math club would have been just the thing he needed.</p>
<p>"I also suspect that he'll conquer the math without any problem that's required for his major, because it will MAKE SENSE and help him master the concepts.”</p>
<p>Yes, that is the way he tends to learn best. That's why I am seriously wondering if he'd do better at a college like Western Washington or Evergreen, both of which integrate math classes as part of an interdisciplinary approach, as opposed to just taking those classes as separate, stand-alone classes.</p>
<p>“...don't EVER suggest that it is part of your child's nature, or your nature, or ANYONE's nature to not be able to learn math. Instead, go looking for good instruction and then apply effort to learning it.”</p>
<p>Oops, I already made that mistake. I tried encouraging him, but at one point it occurred to me that, since he is so gifted in nearly everything else, maybe he needed to accept the fact that he might not be good at EVERYTHING.</p>
<p>"does he consider himself not distinguished in math because of problems of understanding or just that the grades aren't high enough..”</p>
<p>Well, last week he really though he was 'getting it' when he was preparing for the math placement test. He does seem to understand it, but just hasn't had enough drilling, I think. He finds it very hard to focus on math. I suspect he may have a touch of undiagnosed ADD, but it only seems to manifest when it's something he's not interested in. Funny how that works. Anyway his self-esteem is being challenged right now because he ran out of time on the math placement test. I wonder if there is some sort of provision for students to take more time? But, he won't have more time in an actual class, so I thought he needs to learn to do it quickly. He feels frustrated when he sees other students finishing their test in half the time. He can get it if he spends enough time on it, but that's just not gonna cut it - he must have a certain # of them done in the allotted time.</p>
<p>“The key motivational point is to know that MANY students at beginning college age in the United States have </p>
<p>a) crummy math preparation, far below international standards, </p>
<p>but </p>
<p>b) time to learn and opportunity to learn and improve as math students.”</p>
<p>I know this must be true, but it sure doesn't sound that way from reading the posts on this forum. Here it looks like ALL college applicants have AP Calculus!</p>
<p>He's interested in physics and computer science, maybe astronomy, but not biology or medicine.</p>
<p>Lealdragon:</p>
<p>Unfortunately, all the science fields your S is interested in (and that also applies to linguistics!) are very math-heavy.
Below are the listed requirements for physics majors at Harvard. I've put comments in square brackets:
1. Required courses:
1. Physics 15a, 15b, 15c. Students who have demonstrated sufficiently strong preparation in physics and mathematics may take Physics 16 in place of Physics 15a (See item 5f). [this means a score of 5 on BC-Calc exam]
2. Physics 143a.[quantum mechanics)
3.** Mathematics at least through Mathematics 21a[multivariable calculus] and 21b [linear algebra];23a and 23b; Mathematics 25a and 25b;[Math 23 and 25 are higher level versions of Math 21] or Applied Mathematics 21a, 21b. **While not required, taking one or more additional mathematics courses is strongly recommended. Students should give special consideration to the courses listed in item 1c of the Requirements for Honors Eligibility.
4. Two additional half-courses in Physics.
5. Additional half-courses in Physics, or a related field, to complete the requirement of twelve half-courses (see item 5d).
2. Tutorial: None.
3. Thesis: None.
4. General Examination: None.
5. Other information:
1. Courses counted as Physics courses include:
1. Applied Mathematics 201, 202 [these are graduate level courses]
2. Applied Physics (all courses)
3. Astronomy 145, 150, 191
4. Chemistry 160, 161, 242
5. Engineering Sciences 120, 123, 125, 128, 151, 154, 173, 181, 190, and any 200 level course containing a significant amount of physics. See the Director or Assistant Director of Undergraduate Studies for approval.
6. Summer School PHYS S-123ab, an eight-week course that counts as a halfcourse.
2. Related courses include:
1. Applied Mathematics
2. Applied Physics
3. Astronomy (except Astronomy 1 and 2)
4. Biophysics 164r
5. Chemistry
6. Computer Science
7. Earth and Planetary Sciences 108, 121, 131, 132, 133, 140, 161, 166, 167, 200, 201, 236, 243, 260, 263, 264
8. Engineering Sciences
9. Mathematics at the 100 or 200 level
10. Statistics (except Statistics 100, 101, 102, and 104)
3. Neither Physical Sciences 2 or 3, Physics 11a or 11b, nor any Core course may be counted for concentration.
4. Physics 90r and 91r can be used, together or individually, to satisfy at most two of the required courses.
5. Pass/Fail: Two half-courses may be taken Pass/Fail. These may not include Physics 15a, 15b, 15c or 16.
6. Students with exceptional preparation in physics may wish to discuss the possibility of substituting more advanced courses for some of these introductory courses. Written permission of the Director of Undergraduate Studies is required if this is done.</p>
<p>Yup, he's got to make peace with this, no doubt about it.</p>
<p>Thanks for the links, tokenadult. I wish I'd had that first one when he was little.</p>
<p>I guess instead of fretting about not having calculus on his college app, I should just convey to him that he should be glad he's applying as a transfer. Maybe it will matter less as a transfer than as a freshman, I'm hoping. The original plan had been for him to stay at the cc another year (which would normally be his 1st year of college had he graduated and started college at the 'normal' time) but he has definitely outgrown the cc so suddenly everything is moved up a year. I knew he'd need some time to catch up and I thought he'd have another year to do that, and now suddenly he doesn't have that year.</p>
<p>Good math instruction, sometimes hard to find, is probably all that is needed. You may want to see how he does with Daniel Kleppner and Norman Ramsey's self-instruction book "Quick Calculus." I have seen this book, which is entirely self-instructional, turn that light bulb on. An Amazon review of the book with which I concur:
[quote]
I used the 1st edition of this book to prepare myself to take courses in chemical thermodynamics, kinetics and electrochemistry in 1979 after I began my Ph.D. program in Geology at Michigan State University. I had taken one college course in calculus eight years prior and did not perform well. The book is well named, I was "quickly" up to a level where I had no problem with the math in physical chemistry, and I did quite well in these courses. I found myself wondering why calculus had been so "hard" as an undergraduate as it certainly was not presented in a difficult manner in "Quick Calculus". Now, many years later with 6 years in industry and more than 17 years experience teaching at the university level, I am of the opinion that most math faculty in universities simply are very poor teachers of mathematics. It is significant that the authors of this fine book are both physicists (one a Noble Prize winner). This is as it should be because the calculus was invented, more than 300 years ago, specifically to solve very applied problems in the physical sciences. I would not expect such a book as "Quick Calculus" from a pure mathematician. I have recommended the book to numerous students who needed a review of calculus, or who, like me, failed to learn it the first time in their university courses. In fact I just recommended it to a student today and was checking to see if the book was available at Amazon, and decided to write this review.
[/quote]
</p>
<p>You mentioned he is a very good writer, there is science reporting & writing, and there is the philosophy of science as well.</p>
<p>To OP, </p>
<p>While it is uncommon to be a physics major without having been exposed to calculus in high school, it is not a serious deficiency. Most strong LAC programs will be able to accomodate such students. It is more problematic for such students in large universities becuase most introductory calculus courses are populated by premeds who have had the subject and are simply there for the grade.</p>
<p>Explore your options directly with colleges your son is interested in. Most physics program chairs will sit down and discuss with him the options in their curriculum. </p>
<p>Good luck, from a science prof.</p>
<p>... namely, your son may think that he knows everything on the test, but he doesn't. </p>
<p>If your son is going to study science successfully, he (perhaps with help from you) needs to get up to snuff on math basics. NOW. I wouldn't be worried about getting calculus on the transcript for college. At this point, basic math backfill is a lot more important.</p>
<p>Some of this may sound hurtful, but I don't know how to get around that. I am trying to be helpful. If I look just at the FACTS you've reported about your son's formal math training and self-study, I would conclude that </p>
<p>1) Compared to others with comparable or lesser ability, your son is behind in math.
2) He really doesn't know yet how to learn math.</p>
<p>If he's going to succeed at science, he needs to 1) catch up and 2) learn how to learn. </p>
<p>In support of (1), many high-performing public school students take Algebra II and Trigonometry when they are freshman, not juniors. When they're juniors, they're in calculus. When they're seniors, they have 2 years of dealing with Alg. II / Trig concepts under their belt. They apply concepts readily and solve problems quickly. These are the kids your son will encounter (and compete with) in a first-class engineering or arts and sciences college like UT. </p>
<p>To put a human face on it: my son is bright enough, but he certainly wasn't at the top of his class and will never set the mathematical world on fire. He did take Alg. II / Trig as a freshman. In the spring of his freshman year, he took the math placement test at our local community college (he wanted to take a summer computer science course and math was a prereq). He zipped through the test - finished it ahead of time - and placed right into calculus. </p>
<p>The above is not a brag about my son. The point is, he is not unusual - I estimate that there were ~20 kids in his class of 400 who were better at math than he was. I'm sure that your son can also get himself to the point where he could zip through a college math placement test - IF he learns how to learn math. Which brings me to point (2).</p>
<p>My experience was that I didn't learn math from reading textbooks - I learned it from doing the problem sets. I could read right through a textbook sections, skim through the sample problems, and then think I had absorbed all of the concepts covered. Then I would hit the problem set and find myself unable to solve any of them! I had to go back, re-read the textbook section carefully (sometimes several times), work through every step in the sample problems. THEN I could start to make some progress on the problem set. They were boring (because they were hard), and they took me a long time, and even though I basically enjoy math there were a lot of afternoons and nights when I wanted to do ANYTHING other than my problem sets.</p>
<p>It sounds to me like your son has been reading textbooks and then convincing himself that he "knows it" because he has "seen it" (here I'm going by your report that he said that he "knew everything" on the placement test, he just didn't have enough time to finish it). For math, "seeing" .ne. "knowing". Back to the math-as-language analogy: being able to recognize a phrase as French does not necessarily imply fluency in French. Fluency takes practice ==> back to those problem sets!</p>
<p>I would suggest that, regardless of how the next try at the math placement test works out, your son enroll in some kind of "regular" math class where he has to do plenty of problems. This should help get him on the road to learning how to learn math. </p>
<p>Good luck to your son - and to you!</p>
<p>I am more concerned about math dislike than lack of calculus. Calculus can be learned in college, but dislike of math needs to be overcome for the fields the student is interested in.</p>
<p>I agree with Waterdaughter that doing the problems is important. A strong student does not need to do as many of the exercises as others, but s/he needs to do some in order to acquire fluency. It is tempting to read the expositions of concepts and think that one gets the concepts. </p>
<p>I don't know about other schools--I assume it's the same elsewhere-- but S gets weekly problem sets; they take rather a long time to work through.</p>
<p>I'll paste in here a story about Donald Knuth. Knuth was interviewed in the book Mathematical People: Profiles and Interviews edited by Donald Albers and G. L. Alexanderson (Boston: Birkhäuser, 1985) and this is what he had to say about repetition of mathematics class exercises:</p>
<pre><code>At Case, I spent hours and hours studying the mathematics book we used--Calculus and Analytic Geometry by Thomas--and I worked every supplementary problem in the book. We were assigned only the even-numbered problems, but I did every single one together with the extras in the back of the book because I felt so scared. I thought I should do all of them. I found at first that it was very slow going, and I worked late at night to do it. I think the only reason I did this was because I was worried about passing. But then I found that after a few months I could all of the problems in the same amount of time that it took the other kids to do just the odd-numbered [sic] ones. I had learned enough about problem-solving by that time that I could gain speed, so it turned out to be very lucky that I crashed into it real hard at the beginning.
</code></pre>
<p>(pages 185-186)</p>
<p>By anyone's account Knuth is a gifted man, and by anyone's account Knuth has applied his gifts to produce expert performance in his chosen domain of computer science. Knuth is definitely not an underachiever. He applied what surely must be unusually good brainpower by engaging in unusually diligent effort, and how he is a legend in the computer industry. </p>
<p><a href="http://en.wikipedia.org/wiki/Donald_Knuth%5B/url%5D">http://en.wikipedia.org/wiki/Donald_Knuth</a> </p>
<p><a href="http://www.npr.org/templates/story/story.php?storyId=4532247%5B/url%5D">http://www.npr.org/templates/story/story.php?storyId=4532247</a> </p>