Marveling At The Historical

Math Oldies But Goodies

  • About This Blog

    This blog is mostly about math procedures in textbooks dated from about 1825-1900. I’m writing about them because some of the procedures are exquisite and much more powerful, and simpler, than some of the procedures in current text books. Really!

    I update this blog as frequently as possible ... every 2-3 days. And, if you are a lover of old texts and unique procedures, you might want to talk to me about them, at I’m not an antiquarian; the books I have are dusty, musty, brown-paged scribbled-in texts written by authors with insights into how math works. Unfortunately, most of their procedures have vanished. They’ve been overcome by more traditional perspectives, but you have to realize that at that time, they were teaching the traditional methods.

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Some Old Commentary on Today’s Common Core Math

Posted by mark schwartz on June 11, 2016


The quotes herein are extracted from prefaces and introductions of texts ranging in date from 1839 to 1911. I’ve done this because then as well as now there‘s concern that math instruction and learning can always be made better, that students can perform better. Then, as opposed to now, they didn’t have nation-wide and international standardized tests and data on which to make their assessment of the need for change. Rather, they talked to each other and were motivated to improve instruction based on how students were performing. Even then, hints of the rationale behind common core math existed and it can be seen in the language. I’ve underlined select words and phrases that seem like those used to support common core today, but don’t limit your reading to what I underlined … you may have underlined different things. Italics are theirs.

If you would like more information about any of the authors or any of the books, write to me at . Reference the author, date and ”#n” (after the author and date).

The Story.

The design has been, to present these in a brief, clear and scientific manner, so that the pupil should not be taught merely to perform a certain routine of exercises mechanically, but to understand the why and wherefore of every step. Joseph Ray, 1866. (#1)

It is about 20 years since the first publication of the Elementary Algebra. Within that time, great changes have taken place in the schools of the country. The systems of mathematical instruction have been developed, and these require corresponding modifications in the text-books. Charles Davies, 1874. (#5)

Explanations rather embarrass than aid the learner, because he is apt to trust too much to them, and neglect to employ his own powers; and because the explanation is frequently not made in the way, that would naturally suggest itself to him, if he were left to examine the subject by himself. The best mode, therefore, seems to be to give examples so simple as to require little or no explanation, and let the learner reason for himself, taking care to make them more difficult as he proceeds. This method, besides giving the learner confidence, by making him rely on his own powers, is much more interesting to him, because he seems to himself to be constantly making new discoveries… this mode has also the advantage of exercising the learner in reasoning, instead of making him a listener. Warren Colburn, 1839. (#7)

The aim of this treatise is to meet, more fully than has been done heretofore, the requirements of the highest standard of mathematical instruction in the best high schools and seminaries. To this end, great care has been taken to include all the more important parts of analysis, to treat each topic with as much conciseness as is consistent with clearness and elegance, to introduce valuable original processes, and to secure throughout an arrangement most conducive to a philosophical development of the science. Benjamin Greenleaf, 1864. (#10)

That useful mental discipline may be attained, the theory and principles of numbers have been clearly presented, and problems have been given requiring thought and discrimination. The inductive plan has been followed throughout, principles have been developed from methods, rules derived from analyses, and oral and written exercises combined in a rational manner.  Benjamin Greenleaf, 1881. (#12)

The transition from the traditional algebra of many of our secondary schools to the reconstructed algebra of the best American colleges is more abrupt than is necessary or creditable. This lack of articulation between the work of the schools and the colleges emphasizes the need of a fuller and more thorough course in elementary algebra than is furnished by the text-books now most commonly used. Charles Smith, 1911. (#13)

The inductive method is applied throughout the book. New topics are introduced by carefully prepared questions and suggestions designed to develop a correct understanding of the principles to be taught, and to give a clear insight into arithmetical processes and relations.  John Colaw and J. K. Ellwood, 1900. (#16)

Here the object is, first, to lead the learner to see clearly the distinction between the process of reasoning he is led to pursue in the solution of a question, and the numerical operations he I required to perform as the result of that process. In order to do this, he is required to perform numerous questions, retaining the operations as he proceeds, and leaving them all to be performed at last, when the reasoning process has reached its conclusion. In this way he is led to see that the reasoning process is precisely the same, and the operations to be performed precisely the same, for all questions which differ only in the particular number that are given, that thus, in fact, he has obtained a general solution of his question. William Smyth, 1859. (#19)

When the more systematic treatment of the science is presented, the pupil is led by natural, progressive, and logical steps to an understanding of the definitions, principles, processes, and rules, before he is required to state them; consequently, all definitions, principles, and rules are but the expressions of what he already knows. It is evident, therefore, that the plan pursued in the work will develop in the student the habit of investigating for himself any subject which may claim his attention, and this is an extremely important part of proper teaching. William Milne, 1893. (#20)

There are two general methods of presenting the elements of arithmetical science, the Synthetic and the Analytic … Analysis first generalizes a subject and then develops the particulars of which it consists; Synthesis first presents particulars, from which, by easy and progressive steps, the pupil is let to a general and comprehensive view of the subject … Synthesis constructs general principles from particular cases. Analysis appeals more to the reason, and cultivates the desire to search for first principles, and to understand the reason for every process rather than to know the rule. Horatio Robinson, 1863. (#23)

The science of Arithmetic, until somewhat recently, was much less useful as an educational agency than it should have been. Consisting mainly of rules and methods of operation, without presenting the reasons for them, it failed to give that high degree of mental discipline which, when properly taught, it is so well calculated to afford. But a great change has been wrought in this respect; a new era has dawned upon the science of numbers; a “royal road” to mathematics has been discovered, so graded and strewn with the flowers of reason and philosophy that the youthful learner can follow it with interest and pleasure …   Edward Brooks, 1873. (#27)

It is the purpose of the book to lead the young to comprehend and appreciate mathematical reasoning, as well as to solve problems. Edward Olney, 1874. (#30)

Very many people will prefer to have the student trained to be rapid and accurate in computations, and they will esteem a rapid accountant more competent in mathematics than the learned astronomers of our time; while others will prefer that training which cultivates the reasoning powers, even at the expense of practical expertness in the use of numbers. William Milne, 1892. (#31)

The plan of this work is, first, to give a course of reasoning leading to those conclusions from which the rules are drawn – and this is given in language free from perplexing technicalities, and easily to be understood. Secondly, to give in plain and comprehensive language, the rule drawn from such reasoning. Thirdly, to give examples for practice in application of the rule given. Fourthly, to introduce, at proper intervals, miscellaneous questions, involving the several rules which shall have been passed through. The explanations are so written as to throw the student into the place of the original reasoner, as plodding his way through, until he arrives at a conclusion from which he can draw the rule for himself. Cornell Morey, 1856. (#38)

The explanations of the written processes are not designed to serve as models for the pupil to memorize and repeat. They are intended to supplement the analysis. In some cases, a formal analysis is given; in others, a principle is deduced or demonstrated; and in others, the process is described or its principals stated. Neither teacher nor pupil is denied the privilege of determining his own explanations. E. E. White, 1870. (#45)


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