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.

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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