Mathematics and Society - OS X Lion Server

DIDACTICSTEACHING VIA PROBLEM SOLVINGAccordingly, it may be better not to assign several problems to be solved within oneclass period. Also, in recognition of individual differences we could include optionsat different levels of difficulty (e.g., "By next Wednesday see whether you can solveone of Problems 1, 2 or 3. These take some real thinking, so don't give up too soon").PROBLEM SOLVING CAN BE THE STAR OF THE CURRICULUMDrill through problem solving.Exercises(in the sense above) are routinelyused to provide practice on skills, butproblem-solving situations can alsoprovide lots of practice--plus othersorts of thinking.If you just skimmedProblems 3 and 4 above, you might tryone of them.Problem 4 will likelyinvolve as much .computation as you wouldordinarily assign. As another example,the solution of Problem 5 to the rightinvolves work with exponents and division(it is not as formidable as itlooks and is discussed further on pages4 and 5). Word problems also supplypractice in solving mathematical sentencesand in computing....------------------,DRILLNEW IDEAS ORPROBLEM TYPES!?roblem 55 999 + 7 givesPROBLEM-SOLVINGPROCESSESPROBLEMSOLVINGwhat remainder?TRANSFERMOTIVATION(New ideas/problem types through problem solving.Suppose the students havedealt only with positive whole number exponents and have just finished a discoverylesson in which the problem was to find a sbortcut for simplifying expressions like57 7 12 10 4 79 71~. They would then be primed for exercises like --- = n, ~ = n, --yr = nand5 7 12 10 0 79 0 0the question, "What seems to be a reasonable meaning to give to 7, 10, 79 ?" andthereby be introduced to zero exponents.Trimble has commented that a "problem is• • . a means to an end; the end is student involvement in the creation or re-creationof the key ideas and techniques of mathematics." [1966, p. 8] Quite a challenge!(22