Mathematics and Society - OS X Lion Server

DIDACTlCSTEACHING VIA PROBLEM SOLVING(The students) were accustomed to being taught a method and then practicingit; to have to discover procedures was not only quite foreignto them in arithmetic but also in most school subJ'ects. It seems tous that the methods of teaching to which they had been subJ'ected tendedto develop, not adaptive responses, but fixations, so that a childmight know methods and formulas and yet not know where to apply themor how to determine what method best suited a particular problem.Our schools maybe JonJentrating so muJh on having the Jhild masterthe habits, that the hahitsare mastering the Jhild. ILuchins andLuchins, 1950, p. 286, emphasis added]The Luchinses' research has indicated that it is very easy to get a "mental set" in aseries of problems, with the result that one almost blindly proceeds with each newproblem just as he did with the previous problem (see number 10 on page 17).TRAf\JSLATI NGWORDS ----Po. EQUA,\ONEQUATIOt-.)--•• WORDSUsingonly one "type" of word problem in an assignment can foster this uncritical approach,of course, and makes it easy for students to ignore any work with problem-solving strategies.A key question is, how "mixed up" should the lists be? Sumagaysay [1972] hasshed some first light; a sequence of assignments in which the number of types of problemsgradually increased from one type in the first assignment to three types in thelater assignments was superior to sequences in which either only one type was usedthroughout or three types were used throughout (caution: these results cannot beregarded as definitive since the students were likely accustomed to only single-typelists). Here is one final note on mental sets: tension apparently:increases thelikelihood of adopting a mental set.Hence, "'hatever we can do to lessen tension--lessemphasis on time, partial credit, reduced emphasis on grades--may enable students toresist the tendency to fix on one approach.A technique endorsed by many teachersfor word problems is to give lots ofattention to translating from the wordform to the equation (or inequation) formand vice versa.This might include oraldictation of phrases or statements to beexpressed more symbolically ("six more than eight" or "after Don found the five dollarsand twenty-eight cents, he had eight dollars and nineteen cents"),The reverse,giving an equation and asking the students to make up a word problem which fits theequation, can be a valuable diagnostic tool.For example, a student may write aproblem that does not fit a given sentence at all ("I had $11 in the bank at 5%.How much did I get?" for the equation .05 x n = 11), Another diagnostic tool is to31