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 ‘signed number operations’

Concrete to Abstract

Posted by mark schwartz on April 16, 2017


When presenting operations with signed numbers, an instructor must deal with the issue of notation as well, to allow for the plus and minus having two different meanings; this has to be addressed. I spent a long time playing with this until I found a way for students to ‘see’ the difference. Given a number line which I call a road and a car which they can drive on this number-line-road, the car can be put into drive or reverse, So the direction of moving is with reference to the car. When the car is put on the number-line-road, it can face forward (toward the positive) or back (toward the negative), so the facing is with reference to the number line. Students twiddle with this a little but eventually get it. It seems clear that facing (positive or negative) is different from moving (positive or negative).

I wrote a piece in this blog (Driving the Integer Road), a somewhat long detailed almost lesson plan which describes how all this works but I never present the traditional notation until I’ve gone through a bunch of exercises which I call facing and movement. What’s interesting about it is that students seem to appreciate and understand the differences now between plus and minus in terms of the operation or the sign of the number. Although other instructors may use something like this I haven’t seen it presented anywhere. I’d never seen it in my collection of 1800s texts either … until recently. Here’s how Durell and Robbins, in a very similar way had people walk a number line. They omitted some detailed explanation but I will discuss this.

The Story

In Durell and Robbins 1898 School Algebra Complete (pgs. 20-21) they have the students visualizing walking on a number line. The students don’t actually walk the line but only visualize it. Before this ‘exercise’, the authors point out “ … the signs + and – are employed for two purposes – first, to express positive and negative quantity; and second, to indicate the operations of addition and subtraction.” This prompts the students to pay attention to the notation in a problem. I’m now going to paraphrase what the authors did to show students the relationship between walking on a number line and the traditional +/- notation.

On a number line with A at zero, B at + and C at – , a person walking from A toward B a distance of 5 units and then walking back toward A a distance of 3 units, has in total walked a distance of 2 positive units from A, or zero. In notation, the authors write “+ 5 + (- 3)”, essentially the sum of a positive and negative quantity. They point out that this is symbolically what was done on the number line. They discuss this in terms of positive and negative distance and demonstrate that “Hence, we see that adding negative quantity is the same in effect as subtracting positive quantity; therefore in the expression 5 – 3 the minus sign used may be considered either a sign of the quantity of 3, or as a sign of operation to be performed on 3.” That’s a very powerful statement and hopefully when instructors used this text, they emphasized this point because what this really does is show that + 5 + (– 3) = +5 (+ 3) = 5 – 3. The authors don’t detail this expression; they just state it … but it can be shown by walking on their number lines

Given that the authors point out “ … the signs + and – are employed for two purpose…” when they wrote the activity as + 5 + (– 3), were consistent with their own schema by designating the – as the sign of the number and the + as the sign of the operation. That’s interesting because the ‘walker’ is facing in the negative direction with reference to the number line, while walking forward with reference to his own movement and the authors don’t mention this. Their expression of the walking activity could have been written as + 5 –  ( + 3), meaning that the walker walked in a negative direction with reference to the number line, while facing in a positive direction with reference to his own movement. This is a subtle difference but consistent with their schema.

Look at the expression + 5 – (+ 3). With the ‘walker’ starting at A and taking 5 steps toward B, this is +5. If the walker does not change the direction he’s facing, then he could – with reference to his own movement – step backwards 3 units. This is the operation and thus the – outside the parentheses. So, facing forward with reference to the number line is the sign of the number, thus +3.

“With reference to” becomes a critical phrase in parsing one’s way through this demonstration of what the authors have done. Perhaps it’s too subtle for a class discussion but from my experience, this subtlety seems to make an appearance when students talked about the fuzziness in all the operations with signed numbers.

In summary, Durell and Robbins in 1898 captured the core elements of my “Driving the Integer Road” but didn’t explore the subtleties of the notation when talking about the sign of the number and the sign of the operation. I would urge instructors to explore ways of making concrete the ‘abstract’ use of – and + for this as well as other math relationships.

Posted in algebra, basic math operations, Historical Math, mathematics, signed number operations | Tagged: , , , , | Leave a Comment »

Exploring an 1864 Demonstration that ( – )( – ) = +

Posted by mark schwartz on May 1, 2016


When introducing (or reviewing) signed numbers to a remedial/developmental community college class, I try to identify how many in the class already have a mantra for some of the operations with signed numbers. My favorite way to check is to ask them to complete the sentence, out loud, that I’m going to say.  I make sure they’re ready and then say “a minus and a minus is a _____”. For those who voice the answer of “plus”, I pause and then ask them when. Here, I get muddled responses although that was the right answer. I also get a clue about the class because usually there are about only 5 to 10 who answer correctly. Students then take to the board to teach each other how this mantra works and this provides some more insight into how strongly this and other memorized shards of math interfere with learning and applying some core basic math resulting in blurry spots in their math experience.


In the instance of addressing multiplying two negatives to obtain a positive, many math instructors have created a variety of approaches – physical, tactile, and melodic. Of the non-physical, non-tactile, and non-melodic approaches I’ve used, Mr. Greenleaf in his 1864 New Higher Algebra (pg.25) provides a simple and direct demonstration that seems to make sense to students.

Here are his statements:

“When the quantity to be subtracted is partly or wholly, negative. Let it be required to take b  ̶  c from a.”

“Operation.  a  ̶  (b  ̶   c) = a   ̶   b + c.”

“If we take b from a, we have a   ̶   b. But, in doing this, we take away c too much, consequently the true difference will be a   ̶   b increased by c, or a   ̶   b + c.”

“Hence, the ALGEBRAIC DIFFERENCE between two quantities may be numerically greater than either quantity.”

I took license with Mr. Greenleaf’s algebraic demonstration and put it in a numerical demonstration. I didn’t go through his algebraic presentation with the class.

First, I presented to the class the statement 8  ̶   (5  ̶  3) and asked them to carefully and in slow-motion do the order of operations to get the answer. Almost everyone knew to do the work in the parenthesis first, getting 8  ̶   2, and subtracting gives 6. After everyone was satisfied that this was correct, I left this on the board and I wrote the original problem on the board again.

Now, as I started to step through this differently, the class pointed out to me that I was doing it incorrectly; that I was cheating, that I had to do the work inside the parenthesis first, etc.  Technically, they were correct but I proposed that I would do it a piece at a time in slow motion. After much interesting discussion, they granted me permission to continue. Based on Mr. Greenleaf’s method, I first did 8  ̶  5,  noting to the class that this would be the first operation to do. They accepted this, so the result at this point is 8  ̶  5 = 3.  Are we done, I asked? Several sharp-eyed students pointed out that if the problem were done the “correct” way, the answer would be 6, not 3. Another student then pointed out that we hadn’t finished the problem! We hadn’t yet done the 8  ̶   (  ̶  3) part.

So, it was determined that when we subtracted 5 from 8, we were subtracting too much (because the real problem was to subtract less because 5  ̶  3 is less than 5), we had to account for the 3. So, how do we do this?

Again, it caught the attention of several students that the difference between the real answer of 6 and the answer here was 3, so can’t we just add the 3? A lot of back and forth about this until there seemed to a consensus that it worked.

So, after doing 8  ̶  5 and getting 3 – which was too low since we took too much away – we had to add 3 back and this +3 came from  8  ̶   (  ̶ 3).  So, can we conclude that 8  ̶   (   ̶ 3) = 8 + 3?

Again, more fun discussion and finally acceptance! A lot of the students expressed thanks for showing them why a minus and a minus is a plus and I was careful to remind them that they need to modify their mantra to say a minus times a minus is a plus. The class was quite engaged in seeing the “rule” emerge rather than simply being told it’s true. They owned it!

Posted in Category One, Historical Math, math instrution | Tagged: , , , , | Leave a Comment »

Driving the Integer Road

Posted by mark schwartz on April 27, 2016



A lot of texts these days use colored chips or other manipulatives to introduce students to the operations with signed numbers. This article goes one step further. I was using a “chip” concept but then realized that even the chip idea is somewhat abstract. So I visualized something that turned out to be effective … and fun.

The Story

Operations with signed numbers seem to escape a lot of students. This is particularly so when they are confronted with all the rules for the four basic operations. The one they seem to remember is “a minus and a minus is a plus” but they tend not to remember what operation is involved. This rule tends to get applied whenever they see two negatives in proximity to each other in a problem.

.Rather than presenting the rules to the class, they were provided a visual and kinesthetic activity which “generated” the rules. They were shown one way in which the rules were embedded in a daily activity. In essence, any rule or formula is a statement of a relationship between quantifiable activities and in our daily lives, one was found that could be used.

So looking at activities and playing with them might lead to the realization that there are quantifiable components, which ultimately may lead to or actually be the rule. It’s a matter of observation and a matter of determining which symbolizing system to use.

For example, in the following demonstration, letters and numbers will be used to indicate an action. The action will be precisely prescribed. Students will be asked to do a specified series of actions. These actions are based on something that they (well, at least most of them) do.

Here we Go …

The activity that most students do is drive. Their path is a road, but in this activity their path will be a number line. Certain conditions must be set: (1)  a car on the number line (the road) can face forward ( FF, toward the positive end of the number line) or face back ( FB, toward the negative end of the number line) and (2)  the car can move forward (MF, put it in drive) or move back (MB, put it in reverse).

A car (I use Tonka toys: small cars and trucks) can be put on the number line either facing forward or facing back. Once it’s on the number line¸ it can move either forward or back. At this point, it has to be clarified and emphasized that facing is with reference to the number line and moving is with reference to the car. I believe this to be a critical distinction because in our math symbolizing system, we use “ – “ for the action of subtraction but also the indicator for the sign of a number (negative). We use “ + “ to indicate the action of addition, but also the indicator of the sign of the number (positive). This is a confusing element for a lot of students. The notation needs to be discussed, examined, and understood. The exercise with the car makes this point very apparent, by having students begin with the FF, FB, MF, MB symbolizing system and ultimately showing how it is comparable to the use of “ + “,“ – “, and parentheses.

A demonstration of how the system works is given by drawing a number line on the board and using a picture of a car to face and move. I drew one on a piece of cardboard so that facing forward or facing back can be on opposite sides of the cardboard. The number line need only go to plus and minus five. The exercise always begins at zero. The direction to follow is stated as FFMF2. This means “face forward, move forward two”.  But combinations of facing and movement can be strung together as a “command line”. For example, FFMF2, FBMF3, FBMB1, FFMB3 would be done in sequence, beginning at zero with the second, third, and fourth command done from the previous end point.

Using an arrow to indicate where the car starts and which way the car is facing, and using a ● to indicate where the car would finish, the sequence FFMF2, FBMF3, FBMB1, FFMB3 would look like this.


-5     -4     -3     -2     -1     0     1     2     3     4     5

FFMF2                                →           


-5     -4     -3     -2     -1     0     1     2     3     4     5

FBMF3                        ●                  


-5     -4     -3     -2     -1     0     1     2     3     4     5

FBMB1                         ←   


-5     -4     -3     -2     -1     0     1     2     3     4     5

FFMB3        ●                      

After having students practice a bunch of these, a shift to conventional symbols can be made. But don’t hurry the class through this because it takes a while for them to get comfortable with the idea that facing is with respect to the line and moving is with respect to the car.

For example, FF and MF are +; FB and MB are −, and here’s a substitution to show how it works (BUT THIS IS NOT TO BE SHOWN TO STUDENTS UNTIL LATER).

FFMF2,    FBMF3,    FBMB1,    FFMB3

+ ( +2 )  – ( +3)   – ( –1 )  + ( –3 )

This standard notation can be further simplified but this relationship between facing and moving and the standard notation needs to be discussed, examined and understood before proceeding.

There are a few side benefits with this. Doing the facing and movement slows them down, engages cognition, sight, hand movement, decision making, all of which might be put in the context of problem solving. Further, when they are working in groups (in all my classes, students work in groups), they learn how to accept and give questioning and correction from peers, evaluate others’ actions as well as their own, and be deliberate. It also provides a kinesthetic and visual representation that aids the transition from positive-number-only operations to operations with signed numbers.

The key outcome, however, is that the “code” of facing and moving has a parallel pattern with the traditional notation. The pattern can be generated by asking the following for each command: Ask the class: for FFMFn (“n” represents any number), in what direction do you move? For FFMFn, you move in a “forward” direction. Asking the question for FFMB, FBMF and FBMB; for FFMBn, you move in a “back” direction; for FBMFn, you move in a “back” direction and for FBMBn, you move in a “forward” direction. As the answer to each of these questions is given, write down the following pattern:

Command       direction moved

FFMFn            forward

FFMBn           back

FBMFn           back

FBMBn           forward

This pattern contains most of the rules for operations with signed numbers, but this “code” has to be translated to the traditional notation. This is done by talking to the students about the slight difference in notation, including the use of parentheses as an indicator of multiplication. We earlier established (and it’s still on the number line on the board) that forward is “+” and back is “─”. Also remind students that facing and moving are two separate events.

The traditional notation can now be substituted in the above “coded” pattern. The “command” code list is the traditional notation “action’ list and the “direction moved” code is the traditional “outcome” list.

……  code   ……                                   …  traditional notation …

Command       direction moved             action           outcome

FFMFn                forward                       + (+n)              + n

FFMBn               back                             + (─n)              ─ n

FBMFn                back                            ─ (+n)              ─ n

FBMBn                forward                       ─ (─n)             + n

For example, take the problem 2 + ( ─ 3 ) + 4 ─ ( ─ 5 ) ─ ( +2 ). The first step is to look at the problem and see if any substitutions can be done based on the patterns. In talking about substitution, stress to students that this is, in fact, multiplication.

Look at the substitutions that can be made:  2 + (─ 3) + 4 ─ (─ 5) ─ (+2):

+ (─ 3) becomes ─ 3 ….. ─ (─ 5) becomes + 5 ….. ─ (+ 2) becomes ─ 2

So, the problem now is 2 ─ 3 + 4 + 5 ─ 2.

I recommend that at this point, students practice only the substitution idea with at least 10 more problems. I spend a lot of time at this point and don’t talk about what to do with this string of numbers. Finishing this problem after the substitution can be done a variety of ways. I prefer to have this string of numbers seen as a string of signed numbers to be added, employing the commutative and associate properties as needed. Again, there are multiple approaches but the focus of this article is to get to the string of numbers, not how to handle them, so the way to handle the numbers should come from experiences which you have found successful.

Posted in algebra, basic math operations, Category One, Category Two, math instrution, remedial/developmental math, signed number operations | Tagged: | Leave a Comment »