Another sum was set on the board by the teacher and worked orally 670,800 - 387,986 The answering here was equally satisfactory. Any inference from such satisfactory answering to arithmetical capacity would be insecure; yet I was of opinion, from the nature of the mental arithmetic and from the use of books containing large numbers, that arithmetic throughout this school was above the average, and so could hardly be regarded as typical. Berlin. Primary School. Class V. Fourth School Year. Forty-six children present, aged nine, ten, and eleven, with a few of twelve years. The director was kind enough to let me set a test in arithmetic which did not involve problems, but simply accuracy of computation. I turned to the arithmetic books and proposed to set some sums in time measure, but the teacher preferred a metric example, so that the following were decided upon and written on the blackboard :— The teacher of the class thought that the sums were easy, and that the children would do well. They began to work the sums on paper, and twelve minutes was decided on as a reasonable time for the test. I noticed the absence of that attentive stillness which, in our schools, is more marked when the children are doing an exercise themselves than when the teacher is teaching. In Germany the attention to the teaching is, on the average, closer than our own, but the pupil's attention to his own work is not so concentrated. There was, moreover, reference by some of the scholars to the work of others. The following were the results : Twenty-seven pupils worked three sums correctly. Fourteen Five " two The teacher seemed pleased with the result, and himself set the following exercise in time measure :— Year. Forty boys present. Official age, eleven.1 Boys were allowed to count on fingers for the first school year, and, in a few, the habit seemed to survive. 1 The boys certainly appeared older, and I noted on several occasions that, in Berlin, the children seemed older for their respective standards than in the smaller towns. There is some reason to suppose that a similar relation exists between the ages in London, and those in provincial towns. The lesson commenced as usual with Mental Arithmetic. The following sums were given : In all these examples the pupils were required to stand up and call out the answer thus: One-half plus three-fourths equals one and one-quarter; and so on. The mental arithmetic was good, but the children muttered very audibly. The next section of the lesson was a difficult addition worked by a boy on the blackboard. First the L.C.M. was obtained and placed above the second column. The next steps are obvious, but the arrangement struck me as new and well worth trying. Column 3 was added up, and the whole number placed under column 1. The teacher's questions were a constant stimulus to the boy who was working the problem; but whether this was necessary for him, or merely expository to the class, I am unable to say. Berlin.-Primary School. Class III. Sixth School Year. Forty-three present. Nineteen pupils were eleven years old, sixteen were twelve, and eight were thirteen. Through the courtesy of the director I was allowed to give a problem from a part of the arithmetic book recently studied. The problem was— 1 dozen articles cost 18 marks. What are the receipts of a merchant who sells The teacher said this class was very weak in arithmetic. I inquired what effect that would have on the annual promotion, and was told that, though not more than five would usually be kept back in a class of this size, yet here more would be kept down. The tabulated results in arithmetic certainly bore out the teacher's statement as to weakness, and the results were as follow Thirty-five worked a correctly. but many pupils had not worked d when the teacher called the answer out. The discipline during the working was not what we should call excellent; there was much relaxation of attention and some references on the part of the children to each other's work. The teacher then gave exercises in mental arithmetic, but these did not reach the usual high level of excellence. I do not think it was a typical class in this subject. Hamburg.—Primary School. Class III. a. Fifty-three boys present. Twenty-five boys were eleven years old, eleven were twelve, five were thirteen, and two were fourteen. The boys had been working in their books, but, on my entry, the teacher had them put away, and questioned in mental arithmetic. The discipline was excellent, and nearly every pupil raised his hand to answer every question. As was almost invariably the practice in Germany, the questions were entirely oral. The following are specimens of them 15 ÷ 3; 1214 ÷ 2; 13 ÷ 4. The usual accuracy and readiness were shown in answering. At my request the teacher was kind enough to set the children three sums from their arithmetic books to be worked by them on paper. The sums were— 76 ÷ 17= 14519= 154 ÷ 18= |