Found this, if this is correct, the previous version of CMP didn't have ANY standard methods, in the teacher or student version.
Connected Mathematics Program: A CritiqueJune 22, 2004As high school teachers, we are painfully aware of the gaps in our students' skill sets, gaps of skills that should have been cemented in middle school, gaps which make learning high school content all the more difficult.Mathematical FoundationIn elementary school, it is assumed students learn how to add, subtract, multiply and divide, and learn a little about fractions, decimals, and percents (just to name a few core concepts). It is in middle school--around the 6th, 7th and 8th grades--that students go deeper with these same skills, such as adding, subtracting, multiplying and dividing fractions, decimals and percents, and the conversions between fractions, decimals, and percents (again, just to name a few core skills).Middle school students should also learn the basics of exponents and roots, and get a solid grounding in early algebra, such as using variables and manipulating equations. By the end of their 8th grade year, students should be doing bona fide algebra (albeit at a middle school level), so that upon entering high school, the student is ready for the high school math sequence.This sequence (for prepared students) typically is Algebra I, Geometry, Algebra II/Trigonometry, with Calculus in the senior year.The foundation for a successful 4-year high school math sequence lies in solid teaching and learning in middle school.With this in mind, let's revisit the University of Washington study (which we mentioned last time) which compared the middle school curriculum of the Connected Mathematics Program (CMP), Mathematics in Context (MIC), and Singapore Mathematics.NCTM bustersAs we've pointed out, this study is actually cited on the CMP website as a supportive reference, for the simple fact that it complies with NCTM standards. The U of W researchers (mathematicians, all) don't appear to have much faith in the NCTM's push for student-centered learning, constructivism, and the discovery process. (We liken reliance on the discovery method to reinventing the wheel.)So while the researchers rank the Singapore curriculum below CMP, they do so only using the NCTM measuring stick, which they clearly find inferior to, oh, the measuring stick the rest of the world uses. On the merits of what is really important to middle school mathematics (namely the teaching of middle school mathematics), the U of W rearchers rate the Singapore math program very highly.With a boost from the Kids Do Count website, we reprint here some juicy quotes from this study.Discovery LearningAn early casualty of New-New Math programs is our best and brightest:
Moreover, we are skeptical about the possibility of maintaining the interest of high-end students while progressing at the [slow] pace necessitated by the discovery process . . . Reinventing the wheel is more time-consuming than using an existing wheel:
A related comment is that discovery-based learning naturally takes more time than the traditional lecture-then-practice format.Which then detracts from how much math can be learned in the same amount of time:
Also, in order for students to effectively discover the mathematics, more time needs to be devoted to the lessons than in a traditional curriculum. The recommended minimum of 45 minute-long classes seems insufficient.As we've discussed before, there's nothing wrong with judicious use the discovery method, as long as it is immediately followed by concrete teaching, to make sure the lesson is learned, practiced, and remembered. Unfortunately, in CMP, we have this:
[E]xponents are discussed, but the exponential laws are not explicitly written down even after they are discovered. In one exercise students discover that 26 = (22)3, but they need more practice to reach the generalization that (an)m = anm.But practice is supposed to be such a drag, so that too is skipped.The Fractions NuisanceApparently, even though CMP was designed to NCTM standards, they fall short in one key area, namely that having to do with numbers:
CMP and MIC do not meet these new standards in the number strand, one of the most fundamental subjects in Mathematics. For example, division of fractions is not discussed at all even through 8th grade in CMP . . . This probably explains why our high school students don't know what to do when faced with "three-fourths x equals nine."
[In the 2000 NCTM Standard for fractions, which Connected Math will try to follow], it appears to suggest that division [of fractions] should be done by repeated subtraction . . . which is a flawed algorithm in our opinion and not generalizable to all fractions.Hmm . . . where do we remember the concept of "division by repeated subtraction"? Oh, that's right, second grade. Simply teaching kids that "dividing by a fraction is the same as multiplying by the reciprocal" is a horrifying thought to the CMP folks, for it involves direct instruction without the use of a handy toy or manipulative.So it is skipped altogether.Remember finding the lowest common denominator? Well with a calculator, you needn't bother!
Students are not working with general fractions to compare them by finding common denominators. By the end of the 8th grade, we feel this is a skill students should have. Instead they use a calculator, which converts the fractions to an approximate decimal form. CMP and MIC were designed to the 1989 NCTM Standards, which had very low standards with regard to fluency and skills involving fractions.In summary, regarding fractions, decimals, and percents (arguably one of the main reasons kids take math in middle school):
Specifically we find that CMP students are not expected to compute fluently, flexibly and efficiently with fractions, decimals and percents as late as 8th grade. Standard algorithms for computation with fractions . . . are not used.Keep it Concrete (Death of Abstraction)One common aspect of NCTM-based programs is the obsession with reality-based problems and concrete examples. If they can't find a real-world example or find an easy visualization to express a concept, then the concept couldn't be very important, right?
CMP and MIC meet the 2000 NCTM algebra standard, although the mathematical level is much lower than that covered in the Singapore texts. Generalizations and abstractions of concepts discovered and learned, which could have been easily included in the curricula, are mostly absent in the American texts [Connected Math & MIC]. It appears that this may be done deliberately in the authors' attempt to offer easily visualizable problems . . . As for fractional exponents, you'll never run into one at the grocery store, so it's probably best to skip that as well:
There is no discussion of negative and fractional exponents except when students explore exponential functions using graphing calculators. As a result, students miss an oportunity to revisit square roots and cube roots...CMP [Connected Math] misses the opportunity to discuss the quadratic formula or the process of completing the square.Remember, the NCTM folks who designed these weak standards aren't scientists or engineers, thus they can see no need to actually multiply an exponent by an exponent:
However, multiplying polynomials that are higher than the first order [for example, x2] is not covered in the entire [Connected Math] curriculum. This could be because it is difficult to come up with a context for multiplying an area by an area, or it could be the result of a decision [by Connected Math's authors] that the topic is non-essential to a middle school student since it is not explicitly called for by the NCTM Standards. In either case, it is an omission which requires attention for students who wish to be on an accelerated track in high school . . . Again, our gifted kids are given the shaft.
Similarly, the division of a polynomial by another polynomial of lower order is not covered, probably because it would have required conceptual understanding of long division at a level not covered by the [Connected Math] curricula . . . Understanding long division? Oh no, not that waste of time from elementary school! Too bad the NCTM folks 'taught' all our elementary school students 'how' to do long division using a calculator.
The Algebra level in CMP and MIC appear to be almost two grade levels lower than in the Singapore materials. Division of one polynomial by another or multipling two polynomials of order higher than one is not taught even by the 8th grade in these American curricula.No polynomials in the grocery store, either.Basic Skills versus Conceptual DevelopmentDoes it have to be a battle?
We feel that CMP's overwhelming emphasis on conceptual development neglects standard computational methods and techniques. In our opinion, concepts and computations often positively reinforce one another . . . there is a danger here of producing students with conceptual understanding but limited computational skills.Emphasis ours. Hmm . . . understanding but few skills. Kind of like Howard Cosell giving blow-by-blow commentary on a Cassius Clay fight. Just don't ask Howard to step in the ring.And here's some more "sacrifice the student" rhetoric:
CMP admits that "because the curriculum does not emphasize arithmetic computations done by hand, some CMP students may not do as well on parts of the standardized tests assessing computational skills as students in classes that spend most of their time practicing such skills." This statement implies that we [as math teachers] have still not achieved a balance between teaching fundamental ideas and computational methods.That's a very popular myth, that we need NCTM and their ilk to help give us balance, when in reality we were doing quite nicely without them for ages. The 'need' for balance shouldn't outweigh the need to teach basic skills.Onward and Upward?How does CMP bode for the future study of math? Here's a jaw-dropper:
It is our prediction that students wishing to take calculus before the end of the 12th grade year [or college] are likely not to be on track to do so after completing 8th grade CMP . . . Need we mention that CMP isn't some special "slow-track" program which is intended for kids not college-bound (which nonetheless would be offensive, being that we're talking about middle school). No, CMP is designed to be used by all middle school kids.We sure hope none of these kids was hoping for a career in science or engineering, for college-freshman calculus is nearly impossible for those who haven't had it in high school.
As mathematicians and applied mathematicians, we feel that a major shortcoming of [CMP and MIC], ironically, is that they adhered to the 1989 NCTM Curriculum Standards too literally at the expense of the level of the mathematics taught and the mathematical proficiency of the students.Something is definitely wrong when a group of mathematicians calls it a shortcoming to have adhered too closely to NCTM standards.A Plan of ActionOur advice is simple: if you're interested in your students really learning math, then stay away from the NCTM and any math program derived from their standards. Rather, the best course of action would be to look up the schools currently doing quite well by their students, and ask them how they're doing it.Learn from their experience, not the NCTM's theories.
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