## The Psychology of Number and Its Applications to Methods of Teaching Arithmetic |

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Halaman v

While the mere processes

deepening insight continually increase its power to grasp details in more

extensive combinations . Methods must be chosen and justified , if they can be

justified at all ...

While the mere processes

**become**mechanical , the mind should by ever -deepening insight continually increase its power to grasp details in more

extensive combinations . Methods must be chosen and justified , if they can be

justified at all ...

Halaman vi

This twofoldness of number

wherein one number is the unit and the other expresses the multiplicity — the

times the unit is taken . Fractions form a more adequate expression of this ratio ,

and ...

This twofoldness of number

**becomes**explicit in multiplication and division ,wherein one number is the unit and the other expresses the multiplicity — the

times the unit is taken . Fractions form a more adequate expression of this ratio ,

and ...

Halaman vii

... are trained to readily execute such exercises the pupil can take up a sonata of

Beethoven or a fugue of Bach , and soon

the old lessons in counterpoint , the pupil found himself helpless before such a ...

... are trained to readily execute such exercises the pupil can take up a sonata of

Beethoven or a fugue of Bach , and soon

**become**familiar with it . On the plan ofthe old lessons in counterpoint , the pupil found himself helpless before such a ...

Halaman viii

The constituent unit

measured and being made the measure ... the idea of individuality , which is still

a qualitative idea , and does not

PREFACE .

The constituent unit

**becomes**the including unit , and vice versa , through beingmeasured and being made the measure ... the idea of individuality , which is still

a qualitative idea , and does not

**become**quantitative until viii EDITOR'SPREFACE .

Halaman ix

a qualitative idea , and does not

composite and made up of constituent units homogeneous with itself . The true

psychological theory of number is the panacea for that exaggeration of the

importance ...

a qualitative idea , and does not

**become**quantitative until it is conceived ascomposite and made up of constituent units homogeneous with itself . The true

psychological theory of number is the panacea for that exaggeration of the

importance ...

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abstract activity actual addition already amount applied arithmetic attention become begin carry cent child clear common complete conception conscious constructive cost counting decimal defined definite denotes digits direct divided division divisor dollar educational equal exact example exercises expressed factor facts feet figures five foot four fractions fundamental given gives groups hundred idea inches interest involved kind learned least less magnitude means meas measured quantity measuring unit mental method mind minor multiplicand multiplication nature necessary objects once operations original practice present primary principle problem psychological pupil rational reason recognition reference relation remainder repeated represents result root seen sense separate simply square stage subtraction taken teacher teaching tens things third tion true unit of measure unity ured vague whole yard