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.

Archive for the ‘percent’ Category

One 1873 View of Percent

Posted by mark schwartz on February 18, 2017


I’m writing this short piece to give you the flavor of what students had to do when they studied percent using Rev. York’s 1873 The Man of Business and Railroad Calculator. Today’s texts typically present one formula for percent and then discuss variations on it, like percent increase, decrease, percent proportion or finding values given a percent. Also, typically, the problems are very similar to the examples given and rarely, if ever, include fractional values. Rev. York presents a much more demanding idea.

The Story

In his book he discusses percent across 11 pages, making 13 different conditions (like ‘given x, find y’) and ends the discussion with a presentation of 8 formulae. In essence, these 8 formulae are simple variations on ‘percent = part/whole’ but his presentation gives the appearance that these 8 formulae are to be used depending on the nature of the problem. In addition to the typical presentation of percent in today’s text, you can see my concept of percent proportion in this blog (see Percent Proportion). At no point does he state the basic, simple relationship algebraically.

The best way to show what students had to do is to list the kinds of problems he presented. I’ve included the answers as well. I’m not going to list his 8 statements. Let me remind you that students in the 1870s had no calculators and that the work Rev. York presented suggests the importance of mastering fractions. At that time, units of measurement weren’t as standardized and a lot of conversion between systems involved fractional relationship.

The problems as he presented them are below; the answers are at the bottom, in the event you want to play with the problems.

  1. What percent is 1/4 of 2/5?
  2. If a merchant sell calico at 12 1/2 cents per yard and makes 12 1/2 percent. ; what did it cost per yard?
  3. If I sell an article for $250, and make 125 percent; what did it cost me?
  4. One of the stockholders of a rail road company owns 19 shares of $50 each; the dividend is declared to be 7 1/2 per cent premium; what ought he to receive?
  5. If I sell 4/7 of an article for as much as I paid for 2/3 of it; what percent did I make?




  1. 62 1/2 percent
  2. 11 1/2 cents
  3. $111 1/9
  4. $67.50
  5. 14 2/7 percent

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An 8th Grade Final Exam: Salina , KS – 1895

Posted by mark schwartz on August 16, 2016


In the story below is an 8th Grade Final Exam given in Salina, KS in 1895. After you look at the problems, I’ve posed some questions and commented on a few things.

The Story

Here are the problems.

Name and define the Fundamental Rules of Arithmetic.

  1.  Name and define the Fundamental Rules of Arithmetic.
  2. A wagon box is 2 ft. Deep, 10 feet long, and 3 ft. Wide. How many bushels of wheat will it hold?
  3.  If a load of wheat weighs 3,942 lbs., what is it worth at 50cts/bushel, deducting 1,050 lbs. for tare?
  4. District No 33 has a valuation of $35,000.. What is the necessary levy to carry on a school seven months at $50 per month, and have $104 for incidentals
  5. Find the cost of 6,720 lbs. Coal at $6.00 per ton.
  6. Find the interest of $512.60 for 8 months and 18 days at 7 percent.
  7. What is the cost of 40 boards 12 inches wide and 16 ft. long at $20 per metre?
  8. Find bank discount on $300 for 90 days (no grace) at 10 percent.
  9. What is the cost of a square farm at $15 per acre, the distance of which is 640 rods

That’s it; just 9 questions. I guess the belief was that a student can demonstrate what they know in 9 questions, rather than 20 or so as we seem to do in today’s examination mode. In essence, either they learned it or they didn’t.

Let’s consider these problems one at a time in the context of what an 8th grader had to know. It is most likely that the information they had to know to answer these problems was, at some point, presented and discussed in class. Basically, it’s a set of memorized information. Think about this in the pedagogy at that time compared to what students today are expected to know. What they needed to know then is in the context of their daily lives and the business of the day.

In question 1 students had to know the Fundamental Rules of Arithmetic. Do we teach this today? Would it be of value for students to know? What are they? They are the four basic operations – addition, subtraction, multiplication, division which we teach but don’t identify as the fundamental rules.

In question 2 students have to be able to calculate volume but the critical thing they have to know – because it’s not given in the problem – is the volume of a bushel of wheat. Quite likely, since this is an example in Kansas, knowing the volume of a bushel is a very handy piece of agricultural information. As it turns out, there are two possibilities and we have to assume that the teacher at that time made it clear what was being asked in the question. A bushel of wheat has a volume of about 1.2445 cubic feet. There is also a heaped bushel, which is 27.8% (sometimes 25%) larger than a regular bushel. The regular bushel is also called struck measure to indicate that the bushels have been struck, or leveled, rather than heaped. And by the way, I didn’t know any of this and had to look it up. I didn’t grow up in an agricultural area. One more thing – given 4 decimal places in the volume and the 27.8% (sometimes 25%), we have to again assume that the teacher gave precise directions on how to handle these two values, most likely – my guess – by rounding 1.2445 to 1.

Question 3 looks like a straight forward calculation problem. The only possible issues are students (1) knowing how to handle the 1050 lbs. – do I subtract the 1050 before or after calculating the worth or (2) mishandling the decimal point in the 50cts.

Question 4 is again a straight forward calculation problem. What got my attention here is carrying on school for 7 months. It made me wonder if school was a 7 month period or if they simply used 7 because it fit the other numbers well for the calculation? It seems plausible that there could be a 7 month school year because it’s an agricultural area and families worked together to get the farm work done. Just a thought.

In question 5, students need to know how many pounds in a ton, which we assume was talked about in class at some point.

Question 6 seems to be a rather sophisticated problem for the 8th grade, but again in the context of life at that time, it seems reasonable that an 8th grader might be involved in the family’s business and would use this kind of calculation. I suspect it was a simple plug-these-values into the formula they learned. At that time, rote knowledge was highly prized. Note several things: how to use percent in decimal form and also realize that this was all paper and pencil calculations; no calculators then.

Question 7 has inches, feet, and metre as measurements so the student is being tested on measurement conversions. Once all the conversions are done, it again becomes a straight forward calculation. The issue, since 1 meter is equal to 3 feet and 3.37 inches, is what decimal value was used or were they taught to use just 39 inches. But again, it’s something that they presumably were presented and were expected to know. It’s sort of cute to use 12 inches, simplifying the calculation.

Question 8 is like question 6 in the sense that it’s sophisticated for the 8th grade, but again something very useful if a 14-15 year old was helping out with a family business. I again suspect this was a simple plug-these-values into the formula they learned. And as in question 6, students had to know how to write 10 percent as a decimal and further, again, there were no calculators then; all pencil and paper calculation. And what, exactly, is a bank discount?

Question 9 presents conversion issues between acre and rod – common, useful agricultural measures at that time which I again will presume were covered in class and students had to know. But they still had to do the calculation with paper and pencil. Technically and precisely both acre and rod had decimal values but I suspect it’s possible it was rounded off when talking about it and when doing calculation. The rod was a measure of 5 and a half feet, so the students had to again know how to handle decimal calculation if that was the value used. It’s interesting that they noted a “square” farm, since acre is a measure of area and the farm could be any shape. Perhaps it was just the teacher tinkering with the students, as some teachers – even today – are wont to do.

When you look across these 9 questions, it becomes apparent that they are all what we call today “application” or real life problems. These questions also involved sophisticated calculations based on formulae they clearly had to learn. It seems that the information that 8th graders needed to know was normal for that time, yet the pencil and paper calculations they needed to do were demanding. In this day and age, would there be an outcry if students had to do similar problems, having had to memorize the formulae and had to do the calculations without a calculator? I’m not going to venture a guess.

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What? That Much Percent Increase?

Posted by mark schwartz on July 8, 2016


I like coincidences. Particularly when they provide learning opportunities for my students. We had just spent time learning about percent and percent increase and decrease. The problems in the text were good but not really challenging. The coincidence was that I was reading John McPhee’s The Curve of Binding Energy (1974) and I’ll start the story with what he said on page 18.

The Story

“Thousands of miles of tubes, pipes, and other conduits were needed to create a network of flow wherein the gas could now go through a membrane, now return to try again, now go on to a new membrane, gradually advancing, in a process of separation and elimination, until what had begun as seven-tenths of one percent U-235 was more than ninety percent U-235 – fully enriched, weapons-grade uranium.”

I’d never heard this detailed an explanation of how weapons-grade uranium was made. But what really got my attention was that his statement could be a percent increase problem. I worked it out before I gave it to the class, rounding off the initial “more than ninety percent” to a manageable 90%.

Further, I decided that it would be an in-class extra credit exercise and allowed that the students had to first work within their assigned group, but once they had an answer they could discuss it with other groups.

I did this because the percent increase is 12,757% and this size percent increase would cause the groups to question what they did, even if they got that number. There were occasional answers to the problems in the text that resulted in percent increases of more than 100% but nothing quite like this. Once I gave them the problem and answered any preliminary questions and they got to work, I roamed the room listening to the strategies they came up with to do the problem.

The first issue was how to numerically express seven-tenths of one percent. One group asked if they could talk to another group to get help expressing it. So, I stopped the class and said that if they are willing to accept the following condition, they can work as a class to get the answer. The condition was that everyone in the class would get the same grade. They accepted. There was an eruption of conversation and as I roamed around, I was asked if what they got was right. I just referred them to other members of the class.

Once there was consensus on how to represent the initial percent, they simply continued with what they had learned about setting up percent increase problems. By the way, I taught a somewhat non-traditional method that doesn’t use a formula, rather it uses a somewhat modified percent proportion approach. You can look at it in this blog at Percent Proportion.

Several groups quietly called me over to show me their result, asking if they were right. Some were and some weren’t but I wouldn’t say yes or no, reminding them of the condition under which they were working. So, more talk, discussion and exchange of how to set up the problem.

It was interesting to watch the evolution of the shared work – people got up and moved around the room; some asked to and did use the white board; I heard a lot of “show me” and “why did you do that?” and “that doesn’t seem right”. But, ultimately there was class consensus on the right answer.

They did, however, insist that I walk through how I thought about it even though they got it. So, I put on what I call my “slow-motion-math” hat and gave them the following:

Ninety percent is 90/100, so the amount of increase is 90 – .7, or 89.3%. Seven-tenth of one percent is (7/10)(1/100) or 7/1000 (I did this because I saw a lot of questioning on how to express it). This in percent is .7/100 (or if you were sure, you could have just written .7 over 100). So the question can be put in a percent increase frame. First, the amount of increase is 89.3 and since it started at .7, the amount of increase relative to the beginning point can be expressed as 89.3/.7, or 893/7 (they questioned if doing this would give the same answer and we discussed this). Using the proportion statement 893/7 = x/100 gives a percent increase of 12,757%, rounded. So, doing the original problem led to some other related talk about fractions, decimals and rounding. Neat.

After all was said and done, I got questioned about this exercise because there was a sense that it was a trick question. I have noticed that when students feel uncertain about a math problem, the frequently asked question is just that. I then heard stories from the class about their prior math experiences where trick questions unfortunately were used to presumably teach them something about math, but the only learning was frustration because a lot of the tricks were beyond the bounds of what had been taught and in essence they quit. Given what I heard, I may have quit too. Somehow they concluded that math is just knowing the right tricks.

But once they were accepting that it was an interesting problem, I noted to them that as they read books, magazines, watch TV or come across “mathy” stuff, they might play with it as we did with this problem. And of course they noted to me to record the “A” for all of them.

Posted in basic math operations, Historical Math, math instruction, mathematics, percent, proportion, Proportions, 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 »