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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Archive for the ‘fractions’ Category

In 1877, Mr. Ray Reasons with Fractions

Posted by mark schwartz on September 8, 2016


In Mr. Ray’s 1877 Ray’s New Intellectual Arithmetic, an elementary school text, he presents some of the problems with their solution. A sample of these are worth looking at because in every case he shows a solution method which is based on fractions and knowing how to handle a sequence of fractions. But it’s not only the sequence of fraction operations but also the logic of these fraction operations that elementary school children had to follow. This required them to think about the relationships in the problem. I’d like to further note that this method of solution for all 7 problems presented here is seen in many of the texts of that era. It really required students to understand fractions! I’m not proposing that we use this “fractional” method in lieu of solving them by either proportions – the first 4 problems – or simple equations, the last 3 problems.

The Story

All these problems are from his text. Read the solutions slowly to really enjoy the subtlety of the method.

  1. A yard of cloth costs $6, what would 2/3 of a yard cost?  (Pg. 48, # 3)

Solution: 1/3 of a yard would cost 1/3 of $6, which is $2; then, 2/3 of a yard would cost 2 times $2, which are $4.

  1. If 3 oranges are worth 15 cents, what are 2 oranges worth?  (Pg. 49, #19)

Solution: 1 orange is worth 1/3 of 15, or 5 cents; then 2 oranges are worth 2 times 5 cents, which are 10 cents.

  1. At $2/3 a yard, how much cloth can be purchased for $3/4?  (Pg. 75, # 5)

Solution: For $1/3, 1/2 a yard can be purchased, and for $1, 3/2 of a yard; then, for $1/4, 1/4 of 3/2, or 5/8 of a yard can be purchased, and for $3/4, 9/8 = 1 and 1/8.

  1. If 2/3 of a yard o cloth costs $5, what will 3/4 of a yard cost?  (Pg. 101, # 2)

Solution: The cost of 1/3 of a yard will be 1/2 of $5 = $5/2; and a yard will cost 3 times $5/2 = $15/2; then, 1/4 of a yard will cost 1/4 of $15/2 = $15/8; and 3/4 of a yard will cost 3 times $15/8 = $5 and 5/8.

Note that these 4 problems lend themselves well to being solved using proportions. What follows now are 3 more problems, which if presented in today’s texts would likely be solved with simple equations, but again Mr. Ray’s solutions are a sequence of fraction operations.

  1. If you have 8 cents and 3/4 of your money equals 2/3 of mine, how many cents have I? (Pg. 52, #17)

Solution: ¾ of 8 cents = 6 cents; then 2/3 of my money = 6 cents, 1/3 of my money is 1/2 of 6 cents = 3 cents, and all my money is 3 times 3 cents = 9 cents.

  1. Divide 15 into two parts, so that the less part may be 2/3 of the greater.  (Pg. 106, #1)

Solution: 3/3 + 2/3 = 5/3; 5/3 of the greater part = 15; then, 1/3 of the greater part is 1/5 of 15 = 3, and the greater part is 3 times 3 = 9; the less part is 15 ̶ 9 = 6.

  1. A and B mow a field in 4 days; B can mow it alone in 12 days: in what time can A mow it?  (Pg. 110, #14)

Solution: A can mow 1/4 ̶ 1/12 = 1/6 of the field in 1 day; then he can mow the whole field in 6 days.

I hope you appreciate what elementary school students had to do at that time. Since it was elementary school, they weren’t taught proportions and simple equations but they were “exercised” with fractions in a way that I believe could benefit today’s students understanding of fractions.


Posted in algebra, basic math operations, fractions, Historical Math, math instruction, mathematics, proportion, Proportions, remedial/developmental math | Tagged: , , , , , , | Leave a Comment »

Walk the Clock: It’s Fractions

Posted by mark schwartz on August 3, 2016


For some reason, or perhaps reasons, fractions don’t make sense to many students. Despite the visual representations in text and/or the use of manipulatives such as Cuisenaire rods, fractions seem to remain a mystery to students. One day I asked all my basic math classes “What makes fractions so hard?” The overwhelming response focused on remembering the steps of the 4 basic operations. For them, operations with fractions seemed nothing more than trying to remember the steps to get the answer. Somehow, math instruction throughout elementary and secondary education led students to think not about what fractions mean and what they represent but rather to think about how to “solve the problem”. So, I played with an idea which seems to have provided a way for students to “see” fractions a little differently.

The Story

DON’T TELL STUDENTS THIS IS FRACTIONS! If someone asks if this is fractions, tell them it will be discussed after the activities are done. I’ve provided an idea on how to do this in the discussion section following the demonstration of the activity.

The students preferably will work in groups of 3 (or 2, depending on the size of the class). The minimum grouping is 2. Each group gets a magic marker and 12 paper plates. The students are to number the plates 1 to 12. The plates are to be placed on the ground as a clock face. This activity is best done outdoors but if not, move the desks and chairs to allow for each group to have enough space for one person to walk inside and one person to walk outside a clock face circle. If neither of these spaces are available, the plates can be cut down in size and placed on a table top. If it’s a rainy cold day and going outside is a bad idea, and if the curriculum allows and time allows, make it a “review” day and hope for sunny and warm tomorrow. This activity works indoors and on the desk top but outside is best; it’s more fun. If done on the desk top, 2 markers per group will be needed. These markers will be the “walkers” in the activity (this will be explained below).

Here’s how it works. Lay out the plates as a clock face. One of the people in the group will walk outside the circle (call this person the outsider); one person will walk inside the circle (call this person the insider); the third person will be the reader/recorder (call this person reader). Give each group a copy of the activities (a sample is below) which states what the insider, outsider and reader are to do. Once the groups have figured out who will do what, give a demonstration of what they are to do, using the 1st activity.

Using the first activity and using one group to demonstrate, note that both walkers will walk twice. Both walkers start at “12”. In each activity, the insider walks first and then the outsider. The first walk is done when the insider reaches “12”. The second walk for both starts where their first walk ended. The reader is to watch and verify that each walker takes the right number of steps (others in that group can help verify).

1st Activity: on the first walk, the insider walks 2 units while the outsider walks 1 unit. On the second walk, the insider walks 3 units while the outsider walks 1 unit. The reader will note “outsider.insider”. In this activity, the record should show 10.12.

If there is confusion about the walking and/or the recording, just repeat the first activity. When everyone’s ready, move on to the next activities.

2nd Activity: on the first walk, the “insider” walks 4 units while the “outsider” walks 1 unit. On the second walk, the “insider” walks 6 units while the “outsider” walks 1 unit. The reader should note 5.12.

3rd Activity: on the first walk, the “insider” walks 2 units while the “outsider” walks 1 unit. On the second walk, the “insider” walks 3 units while the “outsider” walks 2 unit. The reader should note 14.12. (There should be questions on how to record this. Show students “military” time.

It’s important that if more activities are to be done, don’t allow students to do it. The reason: activities provided by students may result in a very time consuming set of walks and more critically, present a new issue to handle. For example, although subtraction of fractions can be done this way, I suggest not doing it. You could get a negative answer and you might want to avoid this. Just stick with one concept at a time; adding fractions (although they may not realize it). Given this, you might want to prepare and walk through a bunch of activities and be careful that none of them take too much time, yet enough time for the students to play and enjoy it.

Again, do not say anything about fractions at this point, but what has happened is that the problem 1/2 + 1/3 has been done. The record “outsider.insider” is 10.12 , or in reduced fractional notation is 5/6. Most likely, someone has noticed that the insider always has a value of 12. You sort of have to weasel your way around this and don’t yet call it a common denominator.

A Little Discussion. After these activities, you can transition to presenting fractions as you usually do. But, here’s one idea to consider in talking to students about how this activity demonstrates addition of fractions. What is seen and used but not referenced is the common denominator of 12. This explains why the insider’s walking the line twice isn’t counted twice. In the problem 2/3 + 3/4 , the denominator could be any multiple of 12 but in this case since it is 12 and you know it, don’t count it twice. Students may balk at this idea but it can be explained further. The insider always walks the line twice but always restarts the 2nd walk at “12”, while the outsider restarts the 2nd walk where the first walk ended so the insider’s walks aren’t added, rather they simply repeat.

Also not seen is the addition, but it occurs in the outside walk when the second walk starts where the first walk ended. The outside walker’s position at the end of the first walk is added to the beginning of the second walk. Please note that using this method for a problem such as 1/2 + 1/3 would give the answer 10/12, not 5/6, so clearly reducing fractions has to be addressed before this activity. Further, you might question how to get from this activity to the “rules” for addition and subtraction, but that’s not the point, although it can be seen because both fractions in this example, were converted to equivalent fractions with a denominator of 12, although in this case and others, it wouldn’t necessarily be the lowest common denominator. This again, could create a teaching moment, discussing the issue of common denominator versus lowest common denominator.

I suggest that different sets of students get a chance to walk the line. In fact, teams of students could do it; two walk and the others verify that their walking is accurate. Further, the point at which the transition from this activity to the traditional fraction work is to be made is a matter of how the class is collectively responding. In some instances, students caught on and realized that this was adding fractions. But even if they caught on, I still had them walk through all the activities. In several classes, students wanted more exercises. I think it was because it was a nice warm day. It’s a judgment call.

One more thing. Recall that the purpose of this activity is to give a visual and kinesthetic sense to the “rules” and it does seem to have a positive effect on students. When we got to the traditional rules and procedures, I heard students talking about how it “matched up” with what they were doing outside. Play with it.

Posted in basic math operations, fractions, math instruction, mathematics, remedial/developmental math | Tagged: , , , | Leave a Comment »

Percent Problems from 1868

Posted by mark schwartz on June 28, 2016


After presenting percent and having students work many problems from the text, I decided to give them a set of problems from 1868. I did this for several reasons: first, I gave this as an in-class quiz in which I allowed the entire class to discuss strategies, compare answers, and work with others. I roamed the room to watch things happen. Second, I gave these problems because they all had fractions in them as well as related information that they had to decide where it fit in the problem. Problems like these are rare in today’s texts. It was a real challenge for the class but some of the students actually said it was fun! The story begins with a copy of what I handed to them. I’d like to note that I taught a visual percent proportion method only (no formulae, no short cuts) – Percent Proportion – which you can see in this blog. It’s a slower procedure but gives students a better understanding of percent. They said so.

The Story

Here’s the assignment I gave the class …

The 5 problems below are from The Progressive Practical Arithmetic by Daniel Fish. (Ivison, Phinney, Blakeman & Co, Chicago, 1868, pg. 228). There are some minor punctuation and word changes.

Please show all your setups and solutions. Consider carefully how to handle the fractions in the problems. Consider that only non-repeating decimals will give you a precise and accurate answer, so you may have to use improper fractions in your calculations, or use the “fraction” key if your calculator has one. Feel free to work with anyone (or everyone) in the class. Be careful handling dollars, cents and decimal points, and fractional answers. ALL ANSWERS ARE MONEY – DOLLARS AND CENTS.

Before you start, any questions?

  1. A miller bought 500 bushels of wheat at $1.15 a bushel, and he sold the flour at % advance (profit) on the cost of the wheat. What was his gain?

2. A grocer bought 3 barrels of sugar, each containing 230 pounds, at cents a pound, and sold it at percent profit. What was his whole gain?

3. A sloop, freighted with 3840 bushels of corn, encountered a storm, when it was found necessary to throw percent of her cargo overboard. What was the loss, at cents a bushel?

4. A gentleman bought a store and contents for $4720. He sold the same for percent less that he gave, and then lost 15 percent of the selling price in bad debts. What was his entire loss?

5. A man commenced business with $3000 capital. The first year he gained percent which he added to his capital. The second year he gained 30 percent on the whole sum, which gain he also put into his business. The third year he lost percent of his entire capital. How much did he make in the 3 years?


Roaming the room and hearing the discussions and debates was a lot of fun. I occasionally answered questions with questions on how to handle the values and the relationships. As it turned out, when the work was turned in, everybody got it! And, of course, the class wanted this format for all future quizzes, but I noted to them that I had other formats in mind. Someone pointed out how valuable this learning experience was and why would I not want to repeat it? Ah, logic, but as it turned out the other quiz formats turned out to be good learning experiences too.

Below are the answers if you want to play with these 5 problems.

  1.    $95.83 &1/3 (I accepted $95.83; our money system stops with pennies)
  2.    $10.35
  3.    $900
  4.    $1209.50
  5.    $981.25

Posted in fractions, Historical Math, math instruction, mathematics, percent, proportion, remedial/developmental math | Tagged: , , , , | Leave a Comment »

Math Fragments Perpetuate Fragmented Learning

Posted by mark schwartz on June 8, 2016


Carla (not her real name) indicated that she had missed a lot of education because in the religious school she attended, time was spent on theological not academic issues. She has very fragmented math information and it seems that she tries to take any new material and fit it into the fragments she knows. She seems to have a need to preserve what she knows, not realizing that it interferes with learning. It is likely not a conscious act.

The Story.

An example of this is in her work on a quiz. I gave a bonus question. Typically, they aren’t too hard but do incorporate some material that has been learned but also include something that if you are aware, makes the problem somewhat easy.

The problem was

.25 ─ 1/2  + .4 ─ 1/4  + .5 ─  2/5.

Her work and her answer was

51/20 + 17/20 + 27/20 = 95/20 = 19/20

These numbers were puzzling until I realized that she had seen the problem as

25 1/2 + 4 1/4 + 5 2/5

The first thing to notice is the blurring of the “—“ sign. It disappears, thus giving her 3 mixed numbers to add.  She said that she saw the minus sign as dashes separating the numbers from the fraction.

Somehow one of the fragments she recalled was that mixed numbers can be converted into improper fractions, so the weight of that fragment also drove the disappearance of the “─”.  It is also possible, since it had been noted in class, that the notation of a mixed number “A b/c” really means “A + b/c”. This fragment seems to have stayed, but the “+” got replaced with “─”.

The next fragment that emerges is that in order to add fractions, you have to have a common denominator. I think this explains why there is a “20” in each denominator, rather than the 2, 4, and 5.  There was an awareness of this and she found the correct lowest common denominator, but the fractions weren’t converted to 20 correctly.

The decimal points seem to have been totally ignored; when we talked about this, she said she had never seen them before and when she saw them, she decided it was a printing error.

Finally, there is a fragment about reducing fractions. It was demonstrated that “canceling” really is looking for a common factor in numerator and denominator. Somehow, only the part about the numerator was operating.

Carla really wanted to learn math; she was motivated and trying very hard and I applauded her effort but talked to her about fragments and how the fragments were dominating her learning. When we talked about this she kept clinging to the fragments she knew because she had seen this material before and was sure that what was going on now had to fit into what she already knew. After presenting her with visual and kinesthetic procedures which didn’t offer her the chance to use her fragments, she slowly but surely started to restructure what she knew, rather than trying to restructure the new information to fit her fragments.

Recognizing a student’s learning problem sometimes has nothing to do with what we see, but rather is an invisible impediment to learning … and not easy to find, so keep snooping around.

Posted in basic math operations, fractions, math instruction, mathematics, remedial/developmental math | Leave a Comment »

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.

Posted in basic math operations, fractions, Historical Math, math instruction, mathematics, remedial/developmental math | Leave a Comment »