Is math too hard in US schools?

Parents Forum
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<p>Just a couple of thoughts on the main topic and the “sub-discussion”</p>

<p>I don’t think it is possible to teach a good high-school course in chemistry or physics to students who do not understand algebra. Trigonometry has been listed as one of the more useless topics, but trigonometry is needed for a reasonably good high school physics course, as are vectors (obviously).</p>

<p>In my opinion, one of the advantages of mathematics courses (and good chemistry and physics courses) is that they illustrate to students that there are elements of reality that are beyond the reach of authority, charismatic persuasion, or slick debate. Ideally, the student understands enough to obtain answers for him/herself–to simple questions, true–and to know to stick with them, even when others may be arguing against them. Also, the courses offer one of the few instances in the high school curriculum where a student can encounter an idea that he/she just doesn’t get, pour effort into the subject, and then get it. This is one of the most desirable experiences for a student to have, in my opinion. </p>

<p>If a person thinks that algebra is about “learning algorithms,” then I suspect that the person does not understand algebra.</p>

<p>On the sub-topic: It is quite difficult to figure out what math a student is ready for, especially with the non-mathematical outcomes (e.g., admission to a top college) rather closely tied to mathematical placement.</p>

<p>I think it is possible that in the future, public schools may be able to offer individualized instruction in mathematics via the internet, which might allow for as many paces as there are students. This would be my preferred solution.</p>

<p>In the meantime, the idea that students may hit conceptual “walls” is very real, I think. Some schools deal with this by eliminating troublesome topics from the curriculum. For example, many high schools now teach geometry without proofs (though that seems like an oxymoron to me). Schools that include proofs in the geometry course may omit solid geometry or–even more likely–locus. As ucbalumnus notes, a number of schools spread the Calc AB/BC curriculum across two years. This makes no sense to me if the two-year version is compulsory for all. The delta-epsilon concept of a limit has been removed from the AP Calculus curriculum, even in the BC course, though it may be taught to fortunate students in some schools. My university has actually removed the delta-epsilon definition of limit from the first-year calculus courses. In a way, this makes a (twisted) kind of sense: The students who are taking introductory calculus at the university either did not have it in high school, or didn’t get it then. Students often struggle with the delta-epsilon ideas. Removing it leads to fewer student complaints. The students who took AP Calculus didn’t see it either (for the most part), so no one really knows that it is missing . . . until they hit real analysis, of course. I don’t think a student has taken real calculus unless they have worked with the delta-epsilon concept of limits. Actually, I’d like to see that students understand uniform continuity, and see proofs of methods such as the chain rule, for a real calculus course.</p>

<p>I’d like to see schools offer what I’d call “way station” mathematics courses, for students who will continue in math, but who genuinely aren’t ready for the leap of conceptual complexity that the next course requires. How to identify the students who really should spend a year on mathematical “pause,” how to make it acceptable to the striver-class to take the course, and how to keep it from being the slacker-default would all be challenges, to be sure. At the university level, the mathematical curriculum usually has so many offerings once a student gets beyond the introductory calculus/diff eq sequence that it is possible to tune the course selection to the stage of mathematical maturity better than in high school.</p>

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I would just note that if a subject isn’t hard for you, it’s sometimes difficult to understand how it could be difficult for others.</p>

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<p>Will Spivak’s Calculus be a better textbook for AP CalBC? I thought my kid got a solid training in math in HS, but surprised to learn that a repeat of a higher course in college is needed. Mile wide and inch deep.</p>

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<p>What is the title of Spivak’s calculus book? When I think of Spivak, I think of fiber bundles–which, I suspect, is not what you have in mind for intro calculus.</p>

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<p>However, in my parents’ youth in the ROC(Taiwan) of the '50s, one needed to complete calculus by the end of 8th grade to fulfill admission requirements for the college-track high schools. Those that didn’t would mostly be tracked to vocational high schools, apprenticeships, or expected to start working in some sort of job unless they were extremely lucky to attend a remedial high school like mom did where she completed calculus as a high school sophomore.</p>

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<p>Spivak can be used to teach a first-year theory-style calculus course but I wouldn’t recommend it as it is very light on applications and doesn’t provide the coverage that you’d expect for the AP exam or that universities would expect as part of typical engineering programs. AP is pretty specific on what coverage should be. Spivak provides something else which is good in its own right but it is different.</p>

<p>The first chapter of Spivak is really cool stuff and middle-school kids with a decent understanding of algebra and some exposure to proof may benefit from the material there.</p>

<p>(this is from memory; I haven’t look at Spivak for several years)</p>

<p><a href=“http://www.amazon.com/Calculus-Michael-Spivak/dp/0914098896[/url]”>http://www.amazon.com/Calculus-Michael-Spivak/dp/0914098896&lt;/a&gt;&lt;/p&gt;

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<p>“I would just note that if a subject isn’t hard for you, it’s sometimes difficult to understand how it could be difficult for others.”</p>

<p>Thank you Hunt!!!</p>

<p>This conversation is somewhat moving away from the point of the original article. For some, Algebra is a gateway they find it impossible to get through. Most of you posting seem to be math people, with brilliant kids. You might be losing sight of the fact that half the population are below average, and they need to be educated too, and that does not mean trigonometry in sixth grade, and maybe not even in 11th grade. Just understanding Algebra well might be a fantastic goal. </p>

<p>As I said, I have two typical children who have no trouble with math, and one with a learning disability who is a year or two behind in math. I will be thrilled if she learns Algebra well enough to pass the Regents. She’s not dumb, she has a lot going for her, but she has trouble with her working memory. She is by far not the only student in this country like this. I feel like many of you are forgetting about a huge part of the population in this discussion. Or do you only care about the typical cc-type child?</p>

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<p>Hmm, the Spivak Calculus book looks interesting–I wasn’t familiar with that text. A quick perusal of the reviews suggests that it covers calculus in a deep and logical way. In fact, I wish I had known of this book before QMP took real analysis. The reviews on Amazon suggest that it would be a great bridge between AP Calc BC and higher-level math courses.</p>

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<p>Sorry, redpoint, cross-posted with you. I think that my suggestion for “way station” courses would help students like your third child. I don’t think that algebra should be required for graduation from high school. On the other hand, I do think it probably should be required for entrance to most universities, with the option at other colleges for students who have a real block to learning math, but who can handle other topics at college level.</p>

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<p>One might consider Spivak to be a more readable version of Apostol - with less coverage of course. Apostol I, II cover something like five or six semesters of undergraduate math.</p>

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<p>BCE got the right book. I only heard others said it’s a good book. I myself don’t know much math. I think real understanding is the key, but it’s rare to find a teacher who can teach in laymen’s terms to begin with.</p>

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<p>Normal sequence for math in the US is the following:</p>

<p>9th grade: algebra 1
10th grade: geometry
11th grade: algebra 2
12th grade: trigonometry and precalculus (generally only college bound students take this, and not all do, resulting in high remedial math course enrollment in college)</p>

<p>As far as the usefulness of math for those not going to college for a bachelor’s degree, consider the scenario where a carpenter building an A-frame with certain angles wants to calculate the correct lengths of wood to cut.</p>

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<p>Did the kid do poorly on the BC test or calculus placement exam at the college, or attend a college where “freshman calculus” is really real analysis?</p>

<p>Many students who get scores of 5 on BC do fine skipping freshman calculus in college and taking the sophomore level math courses.</p>

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<p>Answer to OP ----> no.</p>

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<p>UCB, I never made the argument that Algebra is not useful or should be discarded. Obviously there are uses for people who are not going to college. The question is how to track, and how to teach it so that students learn it properly, so that it is relevant to their lives and their aspirations.</p>

<p>It is not useful to make a particular subject an impediment.</p>

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Really?! :eek: That was my yearbook quote!</p>

<p>I think my textbook way back then was Thomas.</p>

<p>Even an elementary school kid can understand series and limits with a drawing like this: <a href=“http://www.sfecon.com/2_Theory/22_Hyperbolae/223_Why/images/NatN1.gif[/url]”>http://www.sfecon.com/2_Theory/22_Hyperbolae/223_Why/images/NatN1.gif&lt;/a&gt;&lt;/p&gt;

<p>Yes, Japan and Germany do more sorting of kids, that’s why the study looked at 8th grade.</p>

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<p>Yes, I know it’s horrifying, mathmom!</p>

<p>The statement about our university intro calc courses came from the chairman of our Department of Mathematics.</p>

<p>The delta-epsilon definition of limit was taught in QMP’s Calc BC course after the AP exam was over–definitely not needed for the AP exam.</p>

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<p>It’s in an appendix in Stewart.</p>

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<p>I’m surprised that Thomas is no longer one of the big texts today. I bought a copy in high-school to teach myself. It disappeared one day - I think that my sister swiped it when she went to college. I still don’t know what happened to it.</p>

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What does one have to do with the other? German kids get tracked after 4th grade.</p>

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<p>I’m a math person myself, and I think Hunt is spot-on. I think there is a high degree of arrogance in this thread - “well, <em>I</em> get math, so really, why shouldn’t anybody who is willing to put in the effort”. Gag. Oh really? People have different skills. How well do you “get” art, or foreign language, or acting?</p>