using only the topic-method inasmuch as all the problems they employ are taken from the specific subject being taught. For example, the algebra teacher who motivates the fundamental operations, or factoring, or fractions, etc., through their use or necessity in mastering the equation, is employing not the problem but the topic type of procedure. All texts, for class use, that aim by the organization of their material to assist in any motivation at all, are arranged on the topic-not the problem idea. Naturally this topic plan reveals itself in class-room practice, for have not Rugg and Clark in their monograph on Reconstruction of Ninth Grade Mathematics shown conclusively that the real course of study in algebra is the adopted text, no matter what else may be pointed to as the course of study? This is probably even more the case with high school geometry than with algebra, though the evidence has not been so convincingly marshalled for geometry as Rugg and Clark's monograph has done it for algebra. As a matter of fact the topic type of class exercise is to be seen everywhere and critical norms for judging it will be more frequently needed than for any other type of developmental lesson in high school mathematics.

The writer submits the following as critical norms for the topic lesson:

1. Was the class exercise really of the topic type? Did it purport to be of the problem type? Illustrate.

2. Was the topic merely the next in the text, or was it more inclusive, involving the thing being taught as a necessary detail?

3. Did the topic furnish a real motive and reason for the thing being taught?

4.

Was the class attentive and interested, or inattentive and languid? 5. Was there any life interest to pupils in general in the topic as used? 6. Was the topic kept before pupils as the impelling motive by frequent reference to it?

7. Was there any deviation from the adopted text made in the use of the topic?

8. Did the topic embrace the details of study in a manner to synoptize and unify the detail into an organized unit of thought?

9. Did the work move forward expeditiously? Illustrate.

10. Did pupils actually contribute suggestions of value? Illustrate. 11. Were pupils' suggestions attended to and critically evaluated? 12. What per cent of the class contributed suggestions? (Percentage of participation.)

13. What per cent of the class contributed suggestions of value? (Percentage of efficiency.)

14. Give the per cents of (a) participation, and (b) efficiency for the class.

THE APPLICATION TYPE AND CRITICAL NORMS.

It has long been recognized by good teachers that the proof of knowledge of a mathematical fact, precept, or principle, is neither the ability to recite it correctly nor to demonstrate it,

but is the ability to recognize and use it in an appropriate situation. No teaching technic is good that does not include the necessary element of teaching the application of the thing taught. The aim of mathematical pedagogy today is not knowledge alone, but usable knowledge. Perhaps one of the most acutely painful needs of current practical pedagogics in mathematics is a definite technic for teaching the application of mathematical knowledge. The long period, let us hope now happily closing, during which the main reliance of mathematical teaching has been upon mental discipline as an objective, has led us to see with Minnick1 that practical skill, modes of effective technique, can be intelligently, non-mechanically used only when intelligence has played a part in their acquisition, and furthermore, that only those things, which have been learned in useful relationships, will surely be useful. Certainly, we are ready to admit that the best way to get from a subject the largest measure of whatever disciplinary value it has is to get from it the largest possible measure of its practical value. A condition precedent to success in teaching the application of knowledge, mathematical, as well as other sorts, is that the learner be moved in some way to form within himself the intent to master application. Teaching effort must continue to go awry until we learn how to work the will-change within the learner of getting him definitely to determine to master what he has hitherto not cared for because he felt no benefit of any consequence came of it.

It would certainly seem that whatever technic we do ultimately set up for teaching the application of mathematical knowledge must include such study of subject-matter as will enable the pupil to learn it in its useful surroundings. Our besetting sin is teaching subject-matter isolated from its natural settings and afterwards attempting to furnish through a few sample applications the much-vaunted knowledge and skill to apply what has been improperly taught in the first instance. Such teaching procedure can never lead to ability to apply knowledge. The environment of knowledge in the learning act must be more adequately attended to in our teaching technic if we are to succeed in teaching the application of knowledge.

Furthermore, the environment of the learning must contain a variety, a multiplicity of applications. To touch and deal with utility in as many phases and as continuously as possible is the only sure way in which the learner can be led to focus attention and to hold it, upon the problem of mastering application. Using

the ideas during, not after, the learning of the ideas is the only way out of the present "Slough of Despond" of mathematics teaching.

The following critical norms will aid in judging this type of class exercise:

1. Was the old time spirit of "let applications take care of themselves" manifest in the teaching?

2. Did pupils seem to care about applying what they were being taught?

3. Was there a manifest attempt to teach varied applications of knowledge?

4.

5.

Was subject-matter taught first and application afterwards?
What sort of environment was used in the teaching?

6. Was this environment within the comprehension of pupils?

7. Did pupils suggest any uses, new to them, of the knowledge gained? 8. Were the applications valuable and vital, or only perfunctory?

9. Did pupils seem to feel they were learning something worth while or was the attitude of 'What's the use?' manifest?

10. Were the applications to present-day situations?

11. Suggest any applications not used in the class that you would have used.

12. Were the applications tedious, or complicated?

13. Were any specific instructions given as to how to apply the knowledge? Any needed?

14. Was the subject capable of much application by the pupils studying it? 15. How many different kinds of applications were shown?

THE TEST TYPE AND CRITICAL NORMS.

Lessons of a test character and of the sort preparatory to the giving of standardized tests are coming to be a common occurrence. The tests may be given for diagnosis, for remedial effects, for measuring and comparing ability or attainment in a subject, or for practice effects. Recent texts for ninth grade mathematics and for first courses in algebra are beginning to include more or less test material, and to make the work on this material any great improvement on the old-fashioned written recitation and drill exercise certain conditions must be met.

In the first place, the pupil's state of mind regarding the test must be attended to. He must be made to feel that the test is more than anything else to let him see for himself how much he knows of the subject and how well he knows it. He must feel that he is merely showing himself to himself by the aid of standards that are as reliable and impersonal as they can be made. He must appreciate that if he works normally and vigorously he may, in a standardized test, get a reliable comparison of his achievement with that of many others of his own age and stage of advancement. He may thus have reliable standards of what

J. H. MINNICK: The Recitation in Mathematics, Mathematics Teacher, March, 1921; pp. 119-123.

he is and ought to be expected to meet. He will thus come to the attack upon advance work with more courage and zest than he otherwise would.

The rightly conducted test lesson enables the pupil also to locate his weak points that he may concentrate on the precise things that he sees he needs to strengthen. This assistance to him in economizing his efforts to get on through the setting up before him of independent standards of attainment, the defining of clear objectives for him and the converging of study effort precisely where it is most needed are appreciated by the learner as real helps, provided the tests are well devised and rightly conducted. They reduce dependence on the teacher and furnish the pupil his own gauges of excellence and habits of self-criticism and self-guidance..

Criteria for judging the test type of class exercise are as follows: 1. Were the pupils properly prepared and equipped for the test? 2. Were they willing to undergo the test?

3.

4.

Were the initial conditions made natural by the giver of the test? Did pupils show evidence of over-strain at the outset or subsequently?

6.

5. Did the test seem a real test of what it purported to measure? Was it a standardized test? For diagnosis, ability, achievement or practice?

7.

Was the timing properly administered?

8. Did all the pupils work all the time?

9.

Were pupils curious to know the results of the test?

10. Did pupils seem to accept the test as fair and legitimate?

11. How were the tests scored and by whom-teacher, pupil, or faculty?

12. Suggest any improvement on (1) the character of the test itself, or (2) the method of administering it.

13.

Would you undertake anything similar in your classes?

14. Have you ever employed such tests and, if so, with what results?

THE RESEARCH TYPE AND CRITICAL NORMS.

Good teaching must always seek to produce or to enhance two types of ability, viz.: ability to reproduce knowledge after a longer or shorter period of partial disuse, and ability to produce knowledge, or ability to arrive at new knowledge on the basis of old knowledge. The second type is research ability. The form it may take in high school classes of mathematics is the ability to do work analogous to what has been learned, but involving new elements and phases. Many teachers rate the sort of class work that leads to productive ability very highly, and plan much of their teaching definitely to attain it. Some teachers would make a course in high school algebra or geometry a continuous training course in problem attack and solution, or theorem attack and proof. They would exalt "the original" in algebra and geometry

to the rank of central importance. They consider their real business to be to discover and foster the research attitude. This type of teacher, despite the heavy handicap he is now laboring under, is increasing and is destined still more to increase, as educators and school people generally come to believe in the significance of research.

Critical norms for a class exercise of the research type are the following:

1. Was there a real problem before the class?

2. Did the class really do the work, the teacher only guiding?

3.

Was anything really new found out by the class?

4.

Did the class, as a whole, seem to enjoy participation?

6.

What per cent of the class participated in the original thinking?

7. Did any enthusiasm of discovery appear during the class exercise? 8. What was the teacher's method of keeping the attention focussed on the problem of the day?

9. Was the plan capable of being used by others, i. e., not too closely associated with the personality of the teacher?

10. Was a considerable part of the work real research for the class? 11. Suggest any ways you would vary the observed practice. 12. Give the per cents of participation and of efficiency of the class.

THE APPRECIATION TYPE AND CRITICAL NORMS.

In most school subjects excepting mathematics occasional class exercises are given in whole or in considerable part to getting pupils into an intelligent appreciation of the subject being or about to be studied. Why should this practice not be followed also for the mathematical subjects? May we not hope to see this type of class exercise become more frequent for mathematics as teachers learn the great value for high school work in mathematics of historical, industrial, and social material, and become more expert in the humanizing of mathematics? Appreciation of the value of mathematical attainment calls for a kind of ability that is much more widely distributed in the race than is the ability to become mathematically expert. To appreciate the contribution that mathematical skill and industry have made and are making to civilization is not only an inspiring preliminary to intensive mathematical study, it is also an element in sound culture in its own right. Many of the world's greatest thinkers have found stimulus and motive for high endeavor in this sort of study of mathematics. Surely we ought to give more systematic attention to mathematical appreciation in the teaching scheme than is customary. May it not be suspected that writers and speakers who voice their irritation at having the names of mathematicians attached to important theorems have not had much help