454 REPORTS OK The state Or'' SCIENCE. 



a good plan to tuake pupils arrive at rough answers to i^roblems mentally 

 before working out a calculation in detail, for in this way they are less 

 liable? to be content with absurd answers. Until rough mental results 

 can be arrived at, it is of little use to proceed with exact calculations. 

 Moreover, in actual life approximate values quickly obtained from given 

 numbers are constantly required. The value of mental work in class 

 cannot be too strongly emphasised : no teaching is satisfactory in which 

 these exercises do not occupy the most impoi-tant place. 



Co-ordination with Experimental Work. 



The cO-ordination of arithmetic with any work in elementary experi- 

 mental science carried on in schools is very desirable. Simple experiments 

 in mechanics, physics, chemistry, and meteorology provide results that 

 afford practice in working with numbers to which pupils can attach 

 definite meanings ; and calculations of this kind have the advantage of 

 giving confidence in the applications of arithmetic to problems of everyday 

 life. The arithmetical work thus becomes part of a student's mental 

 machinery, instead of being regarded as a set of disconnected rules, leading 

 to results which have no real meaning outside the classroom. In the 

 upper classes especially, measurements and quantitative work in elementary 

 science should provide a large part of the material for arithmetical 

 exercises. The work may deal with such subjects as averages, sti-engths 

 of mixtures, velocity, conversion of thermometer scales, approximation, 

 specific gravities ; many examples of the application of graphical methods 

 should be given. 



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Graphs and Tables, 



In working out arithmetical results of simple experimental work in 

 science and in solving problems, particular attention should be paid to 

 (1) the limits of accuracy and (2) the shortest method of arriving at a 

 correct result. Expressed in another way, pupils should understand the 

 degree of approximation of the data supplied and should employ the best 

 tools available for working out a solution. Where squared paper provides 

 the quickest or easiest means of getting a result of sufficient accuracy, it 

 should be adopted ; but there is little advantage in ushig this graphic 

 method in cases in which the solution can be obtained more satisfactorily 

 by simple arithmetic. Frequent comparison of results obtained by graphs 

 and by calculation will soon show the degree of approximation which can 

 be reached on squared paper and will indicate that in many problems 

 graphic methods are the best to employ. It should also be remembered 

 that, even though arithmetical operations may give a result more quickly 

 than graphs, the advantage of the graphic method is that it provides an 

 illustration of a principle or relationship which is more easily remembered 

 than a set of numbers. 



Advantage should be taken of numerical tables such as are used by 

 men in the office or workshop. Tables of shillings and pence as decimals 

 of \l., tables of compound interest, squares, cubes, square root and cube 

 root, reciprocals and other values in common use by practical men, can be 

 consulted as labour-saving devices. Tables of logarithms might also be 

 used, as they are not so difiicult to understand as much of the abstract 

 work at present done in schools. 



