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.

Posts Tagged ‘fractions’

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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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.

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Are We Adding Ratios (rates?) or Fractions?

Posted by mark schwartz on May 5, 2016


This writing is a discussion and speculation on why I think fractions are so hard. I say speculating, yet when this was presented in class, the speculation was transformed into reality based on what students said. The ideas in this writing were sometimes presented to a class at this level of detail but even if the discussion never reached this depth, the core idea was discussed with the class. After such a discussion, and when assignments were given, there was always at least one student who would ask “What kind of problem are you asking us to do?” You’ll understand why this question was asked after you’ve looked at this article.

The Story

One of the elements of this story is embedded in the title. For the most part, we tend to not pay attention to the difference between ratio and rate; they tend to blend together. Yet, texts will carefully distinguish between them. I bring this up now because some of my colleagues may be distressed by my using “ratio” where they would see what’s being described, technically and formally, as a ‘rate”. For purposes of this discussion, ratio will be the term used although it’s a fine distinction in the examples I use.

What seems to be somewhat confusing is the concept of adding ratios as opposed to the concept of adding fractions. But you contest, ratios are fractions. Yes, they are but adding ratios is a different procedure from adding fractions. At least it is as we present addition of fractions in the classroom.

I present this because I believe it carries the kernel of misconception about adding fractions. For example, for students who have struggled with addition of fractions, a question like 1/2 + 1/3 will commonly get the erroneous result 2/5. In essence, add the numerators and add the denominators. Most math instructors would say this is wrong. However, it is a correct answer depending on what question was asked!

For example, if there are 5 people in the room, 2 men and 3 women, and 1 of the men is wearing glasses and 1 of the women is wearing glasses, the ratio of men who wear glasses to all the men in the room is 1/2, while for women, the ratio of women who wear glasses to all the women in the room is 1/3. If asked “what is the ratio of those wearing glasses to all those in the room?” the answer is 2/5. This comes from simply doing a count or from adding 1/2 + 1/3.

So, the answer to the abstract question, “what is 1/2 + 1/3?” depends on what the 1/2 and 1/3 represent. If it represents 1/2 (1 of 2 equal parts of something) and 1/3 (1 of 3 equal parts of the same thing), then it represents a fraction and the fraction addition algorithm is to be used. If it represents 2 ratios — how many of the total has a certain characteristic and there are two of these ratios – then the ratio addition algorithm is to be used.

This difference between adding fractions and adding ratios is a critical and typically unvoiced point. It typically is unvoiced because the concept of “lowest common denominator” (LCD) pushes us past this consideration. Most LCD algorithms involve students’ first knowing prime numbers, prime factors, equivalent fractions and how to build the LCD. After this, the students then return to “addition”.

I would contend that students should ask (or be told) what circumstance is the context for the question. In most texts, the addition of ratios may be noted but not dwelled upon because the focus is on adding fractions. This, however isn’t explicitly stated, but it is the assumption on which “adding like fractions” is based. Yes, ratios are fractions but as noted, adding two different ratios is allowable but doesn’t use the same addition algorithm as the traditional algorithm for addition of fractions.

I propose that this issue be discussed with students as part of the introduction to fractions. I believe it clarifies the issue for students who erroneously want to add the numerators and add the denominators when adding fractions. In a sense, this is a more “natural” act than adding fractions as we teach students to do, and because it is more common in everyday situations, it seems the correct procedure to follow. For example, in a classroom of 20 students (9 females and 11 males), I could ask for 3 female and 4 male volunteers. The unvoiced part – as happens in many “selection” situations – is that 3/9 and 4/11, or 7/20 were selected. We simply don’t pay attention to the denominator!

If given the abstract question 1/2 + 1/3, I would expect students to ask “Am I adding ratios or fractions?” In my experience of allowing a discussion and demonstration of this point, students have come away with a better understanding of adding fractions. The traditional addition algorithm seemed to make more sense after discussing the difference between adding ratios vs. adding fractions. The discussion actually clarified both algorithms, as the students asked very incisive questions about the difference and how they can determine which algorithm to apply.

The reason I believe it is important to allow this as a legitimate question (an instructor can always state “from now on, all abstract statements are to be understood to be adding fractions, not ratios … unless otherwise stated”) stems from the way most texts define ratio or ratio notation. Collectively, they tend to state something like “the ratio of a to b is given by the fraction notation a/b, where a is the numerator and b is the denominator, or by the colon notation a:b”. Most texts only cite the fraction notation. But again, most texts discuss and have examples and exercises for students showing how to determine and write a ratio. But, I have never seen a text present a problem of adding ratios. I contend this is done intentionally to avoid the need to identify the difference between adding ratios and adding fractions. Yet, it seems that this identification of the difference should be included, for the reasons previously stated.

And this, I propose, is one reason why students stumble through fractions. Addition of ratios is not addressed. Further, when addition of fractions is presented it is usually preceded by a lengthy discussion of factors, prime factors, greatest common factor, least common factor, like fractions, least common denominator, and finally, equivalent fractions. Wow! … all that just to add 1/2 and 1/3. I can see why students take the easy way out and apply the ratio addition algorithm!

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Historically Multiplying and Dividing Fractions by a Number

Posted by mark schwartz on April 27, 2016


I was trolling through some of my old texts (1850-1875) and I noticed some consistency in their explaining and demonstrating multiplying and dividing fractions by a number. This method is not in today’s text because, once you’ve played with the examples in this article, you’ll realize that today’s rules are easier to apply. However, in some cases, at least knowing what was done in the 1850-1875 era might prove useful … you’ll find out as you read.

The Story

In today’s texts, there are methods for teaching how to multiply and divide a fraction by a number, with perhaps some slight variation by instructor but the core “rule” is the same. Basically, write the whole number as a fraction with a denominator of 1. For multiplication, multiply the numerators together and multiply the denominators together. For division, multiply the reciprocal of the whole number by the other term.

In Loomis’s 1872 Treatise on Algebra (page 62) he states (italics and bold mine):

In order to multiply a fraction by any number, we just multiply its numerator or divide its denominator by that number”. For division: “In order to divide a fraction by any number, we must divide its numerator or multiply its denominator by that number”.

The following carefully crafted problem demonstrates how his rules apply. Take your time reading each step and refer to his statements above as needed.

Multiply:  (12/15)(3) = (12•3)/15 = 36/15 = 12/5      or      (12/15)(3) = 12/(15 ÷ 3) = 12/5.

Divide:     (12/15) ÷ 3 = (12/3)/15 = 4/15      or        (12/15) ÷ 3 = 12/(15•3) = 12/45 = 4/15

I chose the above problems to demonstrate his rules because, depending on the numbers in the problem, his rules can lead to fussy and messy solutions. Note the second example.

Division:    (3/4) ÷5 = (3/5) ÷ 4 = 3/(5•4) = 3/20         or        (3/4) ÷5 = 3/(4•5) = 3/20

Multiplication:  (3/4)(5) = (3•5)/4 = 15/4     or     (3/4)(5) = 3/(4/ 5) = ?

The question mark is there because … what is to be done next using his rules? We’re not dividing a fraction by a number. This is where Loomis’s idea gets complicated and why today’s rules are easier to apply, although I believe playing with his rules gives some background for today’s rules. If he ever ran into this he didn’t say so.

Now, about the problem which ended with a question mark. Here’s how it can be finished. Step one is to take the reciprocal of the problem, which changes the problem to dividing a fraction by a whole number, which allows the use of the Loomis rule. Taking the reciprocal is one of those sneaky math things we occasionally employ and later in the process, we take the reciprocal of the answer. It’s unusual but it works.

The reciprocal of 3/(4/5) is (4/5)/3 = 4/(5•3) = 4/15, the reciprocal of which is the correct answer of 15/4. I told you that his rule would give the correct answer but it’s a process requiring the reciprocal of the reciprocal!

In applying Loomis’s rules it becomes clear that using the division option when either multiplying or dividing creates problems that can be solved but, as you saw, in one case it’s a complicated “trick” (double reciprocal) that makes it work. It seems then that unless you can see that the division will give a whole number quotient, use only the multiplication option … which is exactly what the current rule does … “in order to divide, multiply by the reciprocal”.

Loomis, I believe, provided a combined rules. Their application sometimes depended on the numbers in the problem and further, as I discovered, sometimes depended on the double-reciprocal trick. And one more thing. I was wondering, as you might be, if his rule worked when both terms are fractions. I played with it and it doesn’t; his rules are only for a fraction and a whole number.

It was fun exploring his ideas but again, the current rules for multiplication and division are easier to apply … stick with them!

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