<p>PayFor, thank you for the clarification. I don’t want to get off into too technical a discussion on this already-long thread, and if I do, everyone will dose off.</p>
<p>In a nutshell, methodology has always stressed both (teaching the objective answer – when there is one! - and teaching the means to get there). One is not supposed to be sacrificed for the other, and it certainly never was during my own training.</p>
<p>The math situation is both more straightforward and more complicated. Clearly, finite mathematics, and math facts, start from a base of objective facts, answers, skills. Without those, there will be no math mastery, let alone fluency. The problem in this country, as I alluded to earlier, is that – unlike in many other countries – there is no consistent, systematic, integrated approach to math. Not only do approaches and even terminology vary from state to state, district to district, classroom to classroom, but also year to year. Rather than correct this frantically disunified situation, the response has been to continue to experiment, often wildly & desperately, with ever-novel approaches that compound the fragmentation of math teaching. </p>
<p>Again, reducing the argument here, the Chicago quote is concentrating on the fluency (which has been the greatest problem in middle/high school math until recently), rather than concentrating on facts. It’s important to have the facts first, but to build fluency simultaneously, which is the point behind teaching the fact family of 7, 5, and 12. The only way to make students competent, particularly when one moves from arithmetical operations to the more symbolic algebra & beyond, is to combine the understanding of why, along with the facts. </p>
<p>Chicago is again correct that teaching many ways to get to 12 is extremely important; otherwise there is neither understanding nor fluency. (That has also been proven in research.) However – speaking of Chicago – the U of Chicago developed an absolutely horrid math program (text) recently, showing zero understanding of actual classroom learning and children’s minds. It was very theoretically based and unfortunately widely adopted in my local district, with disastrous results. Making the analogy with a spoken language, if you were never taught how to conjugate a particular French verb (where you would start with a regular verb), you would be lost. That’s the parallel to draw here. It’s all theory & no substance. Further, this particular math program came with no manuals, training materials, etc. So I’m having to re-teach & re-center all my kids. But this is not because there are no educators who know how to teach math. What’s wrong (as I alluded to earlier) is the way decisions are made, and who makes curriculum (& textbook adoption) decisions.</p>
<p>The Chicago program I refer to merely looked superficially at the way some Asian countries teach math, which is to introduce all the strands of math at every level of learning – every age. But they didn’t look at how such introduction was done, and how thoroughly that was taught, approrpriate to grade level. It is only one aspect of Asian superiority at math teaching that ‘all subjects are introduced at every age.’ One of the other major factors of success is mastery of math facts until you can do them in your sleep. Another significant factor is the professional collaboration among math teachers, which prevents the fragmentation I mentioned. They decide together what they will do and how they will do it, so that there is coherence from one grade level to the next.</p>
<p>Our pre-algebra students – not to mention algebra students – are struggling because they were never taught fractions properly, or practiced enough, so things like GCF and LCM are problematic, as is Prime Factorization. A significant percentage of these students do not have their times tables memorized. Again, this is a cave-in on the part of U.S. teachers, who assume that “children won’t learn” math facts, partly out of boredom with repetition (& an assumed low attention span), partly due to calculator reliance for eveyrday use. I keep telling my students – and their teachers – that unless you know cold that 48 divided by 6 = 8, you will not be able to solve the equation, no matter how much you understand Fundamental Properties.</p>
<p>But speaking of Properties, there has been an underemphasis on learning math terms, formulas, and rules. I hate the modern approach to geometry, and few of my students understand it, either. Theorems have apparently been dispensed with – or at least the need to memorize them. Though I consider myself, relatively speaking a “math tard,” Geometry was one of my easiest & most pleasurable subjects in h.school. I Earned what I considered an Easy A, because it was taught in the most logical, straightforward way ever. (And I’m a visual person, which helps. :D) You memorized your theorems, you set up your proof with the Givens, and you basically just filled out the missing pieces of the puzzle. Bingo!</p>
<p>But American math teaching keeps trying to reinvent the wheel, which has resulted in most of the wagons breaking down.</p>