against human nature.rnUse of manipulatives. These are basicallyrntoys of various kinds—counters,rncubes, sticks, marbles, grids, and so on.rnEven the NCTM admits in its Standardsrnthat manipulation of manipulatives is incapablernof proving any mathematicalrnproposition, yet it advocates such arnhands-on approach well into the fifthrngrade.rnThis approach is closely related to verbalization/rnvisualization in that it is innumerate,rnantitheoretical, anticonceptual,rnand antiabstract. Remember the multiplicationrnprinciple, or basic set theory?rnTo determine the total possible numberrnof sets of certain variables, one multipliedrnthe quantity of the first group ofrnvariables and the quantity of each otherrngroup. But not any more. Now studentsrnare instructed to make an “organizedrnlist.” This means writing down, line afterrnline, all the possible combinations ofrnvariables, after which you add them allrnup. This chore may then be followed byrnthe manipulative exercise of coloring andrncutting and pasting little paper representationsrnof your sets, which are not ofrncourse called “sets.” Math is now a lot ofrndull busywork.rnWilliam G. Quirk, a Connecticutrnsoftware consultant and former universityrnmath teacher, has battled the math establishmentrnfor years on these issues. Hernhas posted what he calls “The TruthrnAbout the NCTM Standards” on thernInternet (http://www.webcom.eom/~rnwgquirk/welcome.html) and has this tornsay about manipulatives: “Prolonged reliancernon concrete ‘pacifiers’ interferesrnwith the most important social reasonrnfor studying math, the development ofrnthe average citizen’s ability to think abstractly.”rnIt does indeed. Are we beginningrnto get the idea that the average citizen’srnability to think abstractly is notrnsomething ardently desired by those inrnpositions of authority?rnRepetition. The same ground is coveredrnin grade after grade, especially in elementaryrnschool. The “organized list”rnbusiness, for instance, has been handedrnmy children in both second and fourthrngrade, without any conceptual differencernbetween the presentations. Myrnfourth-grader, by no means a math whizrn(possibly thanks to the new mathlessrnmath), was instantly able, however, torngrasp real set theory, remember it, andrnapply it; so the rationale that “kids aren’trnready for it” is not supportable.rnGuess and check. Otherwise known asrntrial and error, this “strategy” says brightlyrn(actual example): “Sometimes yourncan solve a problem by identifying twornconditions. You can guess at an answerrnthat satisfies Condition 1. Then you canrncheck to see whether your guess also satisfiesrnCondition 2.” Sometimes studentsrncan stumble upon the solution, too, ifrnthey recognize it as such. How many diagonalsrncan be drawn inside a four-sidedrnfigure? Guess. Four? Okay, draw them.rnNo, only two. What about a pentagon?rnFive? Three? Yes, five. A hexagon? Six!rnNo, nine. And so on. Guess, then drawrnthe tiny lines. The teacher told myrnfourth-grader, “There’s a formula forrnthis, but I don’t remember what it is, andrnyou don’t need to know it.”rn”Everyday problems” or “real-world”rnmath. The New New Math argues thatrnmath instruction must be related to everydayrnlife and practical solutions. Theyrncall this a switch from “skills orientation”rnto “meaning orientation.” Arithmeticrndoes of course help children tell time,rnmake change, count their Halloweenrncandy, and so forth. But as WilliamrnQuirk points out, “Most math has norn’everyday’ application.” Mathematics isrnan abstraction that exists solely as a resultrnof human mentation. “Who will buildrnthose bridges in the 21st century?” asksrnQuirk. “Right now, it looks like thernAsians.”rnTeamwork vs. individual effort. Usedrnsparingly, the team approach can be anrnexhilarating change of pace within thernschool day as well as a valid means of arrivingrnat new knowledge. In the contextrnof the New New Math, though, it is justrnanother means of devaluing the conceptrnof a teacher teaching objective facts andrnskills to students who need to pay attentionrnand learn, memorize and practicernthem in order to get—yes!—the correctrnanswer. The theory behind havingrnteams of ignorant students wrack theirrnempty little brains to arrive at estimatedrnapproximate solutions is called “constructivism”rnor “discovery” learning.rnNeedless to say, since the teacher is nornlonger conveying a body of knowledgernbut presiding over a vague “discoveryrnprocess,” the testing of these “discoveries”rnis necessarily problematic and to bernavoided.rnPerhaps the most disturbing statementrnin the NCTM’s entire document isrnthis: “Students might like mathematicsrnbut not display the kinds of attitudes andrnthoughts identified by this standard. Forrnexample, students might like mathematicsrnyet believe that problem solving is alwaysrnfinding one correct answer usingrnthe right way. . . . Although such studentsrnhave a positive attitude towardrnmathematics, they are not exhibiting thernessential aspects of what we have termedrnmathematical disposition.”rnMost students who “like mathematics”rndo so precisely because it offers arn”right way” to arrive elegantly at thern”correct answer.” The NCTM seems tornbe implying that such students are inrnneed of “disposition” modification. Andrnit seems to be imposing just that uponrnour hapless young.rnForget memorization. The NCTMrnStandards call for a curriculum focusedrn”on the development of understanding,rnnot on the rote memorization of formulas.”rnHere is proof positive these folk dornnot know what thinking is. The contentrnof human memory is what “thought”rnoperates upon: no content, no operation.rnIt’s like trying to open a computerrnfile without an application program.rnWithout stored knowledge of facts,rnsense cannot be made of past or currentrnexperience: there is literally nothing forrn”understanding” to build with. Torndownplay memory is to disable the brainrnitself. According to Jack Youngblood, arnteacher who advocates the Kumonrnmethod to undo the damage of publicrnschool math, “You can’t teach ‘concepts’rnwithout teaching math.”rnUse of calculators. Astonishingly, thernNCTM Standards declare, “There is nornevidence to suggest that the availabilityrnof calculators makes students dependentrnon them for simple calculations.” Dornmath educators never witness the agoniesrnof young cashiers when the registerrngoes down? The Council promotes calculatorsrnin the classroom and proclaimsrnthey have transformed the way math isrnunderstood; they have freed humanityrnfrom the need to compute; we can nowrnforget about number facts and concentraternon “meaning.” Meanwhile, evenrnwith calculators, American students trailrnthe industrialized world in math, andrnthey not only can’t do sums but don’trnknow what they “mean,” either.rnCD. Chakerian and Kurt Kreith haverncompared the “constructivist,” “meaning-rnoriented,” “discovery” approach tornthe Pythagorean theorem with the traditional,rn”skills-oriented” approach andrnhave sorrowfully observed: “Experimentsrnperformed under the tutelage of unskilledrnguides can lead students into arnchaotic jungle, one in which their mindsrn48/CHRONICLESrnrnrn