<p>It seems that the quote from Michael Paul Goldenberg is a response to someone who may have heard her say that it is easier to teach–hence his emphasis on learning. But I’d have to dig into the Math Forum archives to find the original claim.</p>
<p>Liping Ma’s book is a refutation of the common idea that teaching basic math is easy and can be handed over to teachers whose main task is to teach children how to read and write (Chinese students get specialist math teachers much earlier than American students according to her).</p>
<p>One statement I’ve seen repeated many times is that students need to really master fractions in order to tackle algebra. And fractions were the concept that US teachers had the hardest time teaching. </p>
<p>Of course, there are many ways of teaching. One of them is the cookbook/megaphone approach: just repeat what the book tells you to say. But that’s not how Ma thinks teaching involves or what students need.</p>
<p>That is an absolutely perfect summary of Liping Ma’s overall point about math teachers, which should heeded by pretty much the entire United States population. </p>
<p>As to the more particular algebra vs arithmetic question, it is clear from what you quoted that Liping Ma was speaking only of what is considered easier by students in China, i.e. that their prior mastery of arithmetic makes algebra easier to learn. There isn’t any implication about the relative difficulty of algebra vs arithmetic in general, just that the Chinese degree of emphasis on one renders the other easy. The source for her remarks is easily located and confirms that this was her meaning:</p>
<p>You are absolutely right. I’m sorry that I gave the impression that algebra is easier than arithmetics to learn or to teach. It is indeed the greater mastery of arithmetics that makes learning algebra easier. I’ll have to read her book again to see if math teachers cycle through grades that involve the teaching of algebra. It appears they don’t always teach the same grade.</p>
<p>These last interesting points relate to the lack of integration, unity, cohesion, within the U.S. elementary math programs in general. It appears that in China there is implicit if not explicit integration in that they anticipate the learning of algebra by the way (& thoroughness) in which they teach arithmetics. Here the tendency is to treat algebra as a separate subject, virtually not related to the preceding learning. Most often in elem./jr. high, an “algebra teacher” is brought in separately to teach that specifically, & often has no experience teaching younger math. If anything these teachers are often borrowed from the high school set of math teachers. And our credentialing system is bifurcated twice: multiple subject vs. single-subject, elementary level vs. secondary level. The single-subject is geared for the secondary level, not primary/middle. </p>
<p>Not very enlightened, in my view. It should be no surprise that our students tend to be giant leaps behind the Asian students in mathematics mastery.</p>
<p>(And sorry if I’m repeating any points that were brought up earlier.) I think this is a really important discussion & I hope people concerned about SAT’s, college placements, & math mastery in college are visiting the Cafe.</p>
<p>To add to epiphany’s point, I’ve found this excerpt from Liping Ma (p. 122):
</p>
<p>When I looked at the set of textbooks my S’ math teacher lent us (to see whether S had already covered the materials and should be accelerated), I was surprised to see the number of topics included in a single year. The many topics are connected to the spiral method, whereby students move on to a different topic, whether or not they have mastered it, with the idea that they will revisit it later on, and it will then have the opportunity to “sink in.” Talking to some of my S’s classmates, I had the impression that they were befuddled by this approach, feeling frustrated that they had not mastered one topic before being asked to move on to another one. Meanwhile, those who had gotten it first time were also frustrated by this revisit. </p>
<p>Tangential to the discussion at hand, we had specialist teachers beginning in the French equivalent of 6th grade and had algebra in 7th grade, along with geometry (the two were taught concurrently throughout 6-12th grades, with the addition of trigonometry in 9th grade, followed by calculus for those in the math track). So a student who was good at math in 7th grade in my lycee might not have had any trouble scoring high on the SAT-Math.</p>
<p>The point is, that even here in ol’ Ameica, there are kind who are so good at Math in 7th grade that they do well on the SAT (point of OP!).</p>
<p>They don’t need all this looping and relooping of material. And half the time (or more than half, in our experience), these kids have much more math intuition and inherent arithmetic fluency that their teachers, particularly through grade 5.</p>
<p>Concurrence and spiraling are very important in math and in science, and are largely missing during the arc of U.S. K-12 learning, in my career experience. And of course that is not the same thing as non-progressive repetition (the “reinforcement” attested to by some systems, such as parochial schools my children have attended). The latter reveals an inattention to cognitive <em>development</em>. (Key word)</p>
<p>It’s astounding that in our multicultural country we have not made use of the best of other countries in the educational realm.</p>
<p>Note: The Saxon Math program claims to use spiraling. It sometimes (often) gets criticized by homeschool families using this curriculum as boringly repetitious. How much of that dissatisfaction reflects American impatience & modern attention spans I don’t know. I have not been in the position to follow the high school end of the Saxon Math, but since plenty of my students have used it but not performed well, the spiraling may not mimic the Chinese approach cited, or (again!) may not be taught well. It’s the teaching, the teaching – as well as the program, of course.</p>
<p>But some kids don’t need all that spiraling and repetition. The problem when such a system becomes educational policy, is that all kids are subjected to this methodology, whether they benefit or not (and many don’t).</p>
<p>There is a really good article about what the top scorers on the SAT at middle school age do to develop their mathematical ability. You can find it at </p>
<p>Kolitch, E. & Brody, L. (1992). Mathematics Acceleration of Highly Talented Students: An Evaluation. Gifted Child Quarterly, 36(2), 78-86.</p>
<p>The article is by researchers affiliated with the Johns Hopkins University Center for Talented Youth. The authors note, </p>
<p>“These students were highly involved in mathematical activities outside the classroom. Only 2 of the 43 students did not report any involvement in mathematics competitions. To varying degrees, students participated in school math teams; state and regional math competitions; MathCounts; the American High School Mathematics Examination; the USA Mathematical Olympiad; and other tests, contests, and competitions. . . .
In addition, several students captained math teams, and 3 students were responsible for organizing teams.” </p>
<p>In other words, the students made up for the inadequacy of the standard school curriculum, which "[is</a> not designed for the top students](<a href="http://www.artofproblemsolving.com/Resources/AoPS_R_A_Calculus.php]is">http://www.artofproblemsolving.com/Resources/AoPS_R_A_Calculus.php)" by going beyond the standard school curriculum and doing lots of challenging problems in an extracurricular context. Another possibility besides competition math for such students is to participate actively in a local mathematical circle (which I lack in my community, despite attempts to organize one) or study in an intensive summer program such as PROMYS, MathCamp, or Ross. School curricular programs in the United States, even for highly accelerated students, are too easy by international standards. The way to be up to world standard in math for young people here is to go beyond the school curriculum.</p>
<p>I don’t know if we’re using “spiraling” in the same way.</p>
<p>From what I’ve read of Saxon Math, each lesson consists of a large amount of review to which is added a new topic and a few exercises on this new topic. The boredom that epiphany’s students experience may be due to this large degree of repetition. If a child is homeschooled, the parents can reduce the number of exercises they assign. In many cases, however, parents or teachers insist that all exercises must be gone through. I have not seen this approach described as spiraling. In fact, it seems to be its antithesis.
For example, the 7th grade math text we read included about 8 different books, each covering a single topic (some geometry, some statistics, some arithmetics, and so forth). Some of these books clearly revisited concepts that had been taught before–in other words, they were spiraling, according to what was explained to me. But with 8 different topics, it was also clear that students would be expected to move on after a topic had been “covered.” One parent complained to me that his child had not mastered one topic before having to move on to another one. The idea was that the topic would be revisited later; but he complained that revisiting the topic did not make his child any more proficient than the first time around. By then, his child would have forgotten what s/he did not understand in the first place and would have to tackle the concept de novo.</p>
<p>When I was trying to persuade our school to let my son skip 6th grade math I borrowed a friend’s copy of the 7th grade math textbook. I was astounded to see that the material in them was almost identical. If fact, the only difference I could find was something called a box and whisker plot something I never heard of in my math education. I hate spiralling curriculums. If it’s September it must be graphing…</p>
<p>I was astounded to see that the material in them was almost identical.</p>
<p>I was thinking about this some time last spring. Six years out of HS, it seemed to me that as best as I could recall, that if I were to difference my math education from 5th grad through my jr. pre-calc course, I wasn’t left with much of anything. That is, there was virtually no new material each year. </p>
<p>How sad. I cannot imagine this in any other subject. Reading the same books, year after year after year in English? Studying *basic *American history and gov’t or biology for seven years? That would be ludicrous, and yet in math…</p>
<p>In my daughters’ parochial school they studied the same basic physical science for 9 years. (Yes, ludicrous.) Words slightly more advanced, every few years, but that was it. Never any new scientific concepts learned. In retrospect, the pathetic curriculum reinforced how little respect the Catholic Church has for science. (Not too dissimilar from the Middle Ages, or from 2007.) I can say that as a Catholic, so no one needs to flame.</p>
<p>The texts we looked at were for 8th grade, but the idea is the same. In 8th grade, lots of students were still struggling with fractions; but they were expected to cover some statistics. Our district does not offer AP-Stats.</p>
<p>Statistics is totally out of place at this level, as it cannibalizes time from other more important things that have a chance to be understood. I think the NCTM standards may be partly to blame for this sub-fiasco within the larger disaster.</p>
<h2>“I browsed through the names and photos of the kids who got a perfect score on the AMC8. Obviously, that’s an incredible accomplishment, and these kids must be really gifted in math. Is it politically incorrect of me to point out that over 90% of them are Asian? And is it politically incorrect of me to ask, Why is this the case?”</h2>