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 1800s Fractions That Might Fracture Today’s Students

Posted by mark schwartz on May 31, 2016


I noticed something while looking through my collection of old texts. Something seemed quite different from today’s texts. The more I stared at the books and thumbed through them I realized what it was; in the 1800s problems with fractions weren’t limited to the chapter on fractions, which seems to be the model for today’s texts. Further, the values in the fractions as well as some of the values in the answers were considerably different from the values in today’s texts (I included some of the more interesting answers). Also, the problems in today’s texts are very simplistic compared to the problems in the 1800s.  Below are some of them.

The Story.

The essence of this story – before I show the examples – is that the operations with fractions today are pretty much the same as in the 1800s. A big difference, as I noted above, is the values in the problems. But more importantly, the biggest difference is that students then didn’t have calculators. Students today moan and groan when learning and doing fractions and in most remedial/developmental classes, calculators can’t be used. Despite this, most students seem to master the basic operations, and I propose that their grasp of these basics might be stronger if they had to learn operations with the 1800s fractions and without calculators, if students were allowed sufficient time.

So, here are some old problems, with references and a little annotation.

From S. Mecutchen. Graded Problems in Arithmetic and Mensuration, E. H. Butler Co., Philadelphia, 1880, pg. 39.

325 3/7 acres + 119 1/4 acres + 13 3/5 acres – how many acres?

From  Joseph Ray. Rays New Practical Arithmetic, Van Antwerp, Bragg and Company, Milford, NY, 1877, pg. 146.

Add 13/18 + 8/15 +11/20 + 18/30

Add 2/5 + 7/16 + 7/50 + 3/140 + 8/2800 (wouldn’t you guess the LCD as 2800?)

From Benjamin Greenleaf. The National Arithmetic, Robert S. Davis Co., Boston, 1858, pgs. 159, 169, 191.

What is the value of 4/9 of 7/11 of 11/25 + 25/31 of 7 3/4?

From a cask of molasses containing 84 3/8 gal., there were drawn at one time 4 3/7 gal., at another time 11 gallons; at a third time 26 1/2 gal. were drawn, and 1/2 of 7 1/2 gallons returned to the cask; and a fourth time 13 8/11 gallons were drawn, and 3 1/2 gal. of it returned to the cask. How much then remained in the cask?  Answer: 35 597/616 gal. (wow … no calculator therefore no decimal).

What cost 1670 7/13 pounds of coffee, at 12 3/4 cents per pound? Answer: $212.99 9/52.  (notice the mix of notation in the answer: decimal and fraction).

From J. Colaw and J. K. Ellwood. School Arithmetic, B. F. Johnson Publishing Co., Richmond, VA, 1900, pgs. 108, 133, 134.

What is the value of 7 barrels of sugar each containing 344 1/2 pounds at .04 3/4 a pound? Answer: $114.54625 (yes, they took it to five decimal places! Also notice the mixed notation in the cost per pound)

Find the LCD of 6/7, 3/4, 5/6 and 7/11.

Add 5/16 + 11/12 +17/20 + 7/18. (hum? … the LCD is …)

From James B. Eaton. Eaton’s Common School Arithmetic, Taggard and Thompson, Boston, 1864, pg. 118.

What is the sum of 3 4/15 + 6 9/16 + 4 5/12 + 24 3/8?

There are a lot more like these but these demonstrate that students then had to be able to handle larger numbers than are typically in today’s texts when finding LCDs, when adding and multiplying and when checking if the answer was correct. I also believe that since they didn’t have calculators, in order to assure their calculations were accurate, they had to have a mastery of the procedures and work slowly, very slowly. I, personally and professionally, believe that slowing down is an important feature for learning math. You might take a peek at the article “Staring” on this blog. It’s another aspect of getting people to slow down when doing math.


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