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 ‘Category Two’ Category

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 »

The Definition of Math is …

Posted by mark schwartz on February 25, 2017


Do you have a definition of math that you’ve come to by virtue of your experience or have you adopted a particularly viable definition offered by an outstanding mathematician? In either case, does the definition help you understand math in a way which enables you to help others make sense of math?

The Story

I have a definition which has evolved over time and is likely to continue to evolve. I came to my current definition by virtue of discussing math as a concept with students. When a student offers me “when am I ever going to use this stuff in real life?”, I offer my definition. I do this because I can link my definition to actual daily behaviors. I’ll give you an example later.

My current definition is: math is a set of tools we use to identify, connect and summarize quantifiable relationships. The ‘set of tools’ part usually drives everyone nuts because it’s vague but if you think about it, this set of tools is built into our brain. One of the things we do automatically and in nano-seconds is to make judgements. Some of these judgements are quantifiable. A simple example is to toss an object to someone and ask them to catch it. Then ask: is there any math here? This discussion leads students to become aware of the quantifiable judgements made in order to catch the object – judging velocity, trajectory, position of object, etc.

If you are willing to accept that we have this set of tools, even without a clear delineation of what they are, what part of the brain is operational at that moment and how they function, then this allows for the rest of my definition to come into play.

If we can identify a quantifiable ‘event’, and thus have a bunch of these quantifiable events, we can then identify quantifiable relationships. Ever engage in a conversation where the topic was define love? Not exactly quantifiable. This is likely quite different from talking about an equation defining the relationship between x and y.

So basically, I’ve addressed not only identifying a quantifiable event but also connecting these events. I hope I need not make a somewhat exhaustive list to help you understand what I’ve stated. Rather, I’m hoping that your experiences can generate a list of examples.

Summarizing quantifiable relationships is nothing more than a formula such as y = mx + b!

If this definition seems too simplistic and doesn’t accommodate the kind of math you consider, then – since this is my current evolving definition – how about having a little discussion about it? Write to me at and let me know why this doesn’t work for you and let me know of your current definition, if you have one. I realize it’s not necessary to define math in order to learn or teach it well but by having one, I can talk to students about math in a context which seems to make sense to them and … it has helped many of my remedial students believe they can master math.

Posted in basic math operations, math instruction, mathematics, remedial/developmental math | Tagged: , , | 2 Comments »

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

Posted in basic math operations, Historical Math, math instruction, mathematics, percent | Tagged: , , | Leave a Comment »

Recognize x∧2 – x – 1 = 0?

Posted by mark schwartz on February 12, 2017


If you recognize the equation in the title then you are already familiar with the golden ratio of 1.618. There are a number of books that explore the occurrence of the golden ratio in nature, architecture, the pyramids, art, aesthetics and a variety of other suspected places. I’d like to explore something about 1.618 that I’ve not seen in any books; just something to add to the mythology of it. If you aren’t that familiar with it, just Google it and have fun.

The Story

When I was teaching a pre-service teacher’s class about math concepts, I introduced the golden ratio because it was a nice vehicle for demonstrating how a pattern can show up in a lot of places. I talked a lot about teaching students to explore for patterns because math is saturated with patterns and relationships.

I asked if anyone in the class was involved in mysticism, the occult and if they would feel comfortable talking to us about it. There was one person who spoke up. I directed the conversation to focus on the pentagram (a five pointed star inscribed in a circle) because I knew that it was a significant symbol. I asked her to talk about the significance of the pentagram and of all the stuff she talked about, the golden ratio wasn’t mentioned. I mention it because it’s a somewhat invisible yet significant characteristic of the pentagram. Let me start with the star.

From here on, you will be asked to do a lot of ‘construction’ but it doesn’t have to be elegant or perfect and hopefully my directions can be easily followed. I just believe that your participating – as I had my students do – will be entertaining and educational, not drudgery. There is some geometry and trigonometry involved so if you don’t recall stuff, I’ll provide the information needed to do the work.

You can ‘construct’ a five-pointed star, starting by sketching a circle. Don’t make it too small because some of the points and lines will need room to be labeled. When you get done, you will have constructed a pentagram.

Starting at 12 o’clock, place 5 equidistant points on the circle. Starting with the point at 12 o’clock, label that one A and then going clockwise label the others B, C, D, and E. Now, draw a straight line from A to C, then a line from B to D, and continue with C to E, D to A and E to B. You should have a somewhat respectable looking 5 point star. I want to highlight that this pentagram has a pentagon in the middle and each face of the pentagon has a triangle on it. Take the triangle at the top that has the vertex A and label the other two vertices in that triangle F and G. Still with me?

Now if your pentagram were a perfectly crafted one, the pentagon and each triangle would appear to be equilateral. The pentagon would be but not the triangles! What? – they look equilateral. How can we determine that they are not?

To do this we need to know something about the angles in triangle AFG. Allowing that AF and AG are equal to 1 (we could pick any arbitrary value and the outcome of this exercise wouldn’t change; it’s just that 1 simplifies the calculations). Geometry allows us to determine that angle A equals 36 degrees. Now drop a perpendicular from vertex A to side FG, labeling that point on FG as H. Why? Well, we just formed a right triangle AFH which allows us to use some trigonometry to determine the length of FH. The perpendicular also bisects angle A, so angle FAH is 18 degrees. The sine of this angle is FH over 1, or just FH. The sine of 18 degrees is .809 – the length of FH – so the length of FG is .618, thus demonstrating that this triangle and all the triangles in the pentagram are not equilateral.

Given this, the perimeter of triangle AFG is 1 + 1 + .618, or 2.618. If you’re not familiar with the golden ratio, let me note a few relationships. First, the square root of 2.618 is 1.618 and clearly 1.618-squared is 2.618. And, as trivial as it seems, you can see that 1 + 1.618 equals 2.618.

And now the point of this exercise! Recall that the title of this article is x2 – x – 1 = 0. If you solve this equation, you get 1.618. But substituting 2.618 for x2, 1.618 for x you now have 2.618 – 1.618 – 1 = 0, which is a true statement for the perimeter of the triangle. So, triangle AFG and all the other triangles in this pentagram contain the golden ratio. But, there’s more.

Look at the pentagon in the pentagram. I won’t ask you to do all the construction (you can do it if you like) but I’ll step through it to show that the pentagon also contains the golden ratio. Briefly, starting with point F and moving clockwise, the vertices of the pentagon are G, I, J, K. Drawing a straight line from K to G, forming triangle KFG and assigning KF and FG a value of 1, what is the value of KG? As with the star construction, geometry lets us know that angle KFG is 108 degrees, and a perpendicular dropped from F to KG (label this point M) bisects the angle, and results in a right triangle KFM with angle KFM equal to 54 degrees. Again, trigonometry lets us determine that the length of KM is .809, and thus the length of KG is 1.618.

So, what is true for each of the triangles is also true for the pentagon, although it’s not the perimeter of the pentagon, but a construct (a ‘chord’ of the circle too) within the pentagon.

And one more thing. All of this analysis came about because I was watching the National Geographic Channel and the show was all about creatures of the sea. One of the creatures shown was the star fish. If you connect the tips of the starfish arms you get a pentagon!

So, in addition to the pentagram having a powerful influence in the occult, perhaps reinforced by the presence of the golden ratio, nature again (using a little imagination) provides another possible example of the ‘hidden’ sway of 1.618.

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A Short, Short Discourse on Digit Sum

Posted by mark schwartz on January 16, 2017


My daughter and I play with numerology; it’s just play, nothing serious. We play with calendar dates, prime numbers, birthdays, etc., looking for patterns and such. Most recently she texted me that she added the digits in my wife’s birthday and continued to add them until there was a single digit, the result was 4. She wasn’t aware of digit sum nor casting-out-9s. So, I played back.

The Story

I first took a look through some of my old math texts, dating from about 1850 to 1900. I looked there because in those days, having a way to check your work was important and digit sum was popular, but not noted in all texts. What I don’t know is whether instructors may have taught it although it wasn’t in the text. I did find a few (I have about 75 old texts) that actually demonstrated how addition can be checked using digit sums. Oddly though, none of those that presented how to check addition indicated that digit sum can be used to check subtraction, multiplication and division as well. Yes, it can.

But, let’s take a look at why digit sum works. It’s based on what is called casting-out-9s. In essence, given a number – 23 – if you cast out 9s (which can be done by subtracting 9 until you have a single digit), you get 23 – 9 = 14, then 14 – 9 = 5. Notice that if you simply added the 2 and 3 in the number 23, you also get 5, so what simplifies getting a digit sum is simply add the digits repeatedly until you get a single digit. For the number 268, at first you get 16, then 7.

Why does this work? Let’s get basic. Using 23 again, this is really 2(10) + 3(1). Rewriting this in what I call ‘slow motion’ math, it becomes 1(10) + 1(10) + 3(1), then 1(9) + 1(1) + 1(9) + 1(1) + 3(1). If the ‘1(9)s’ are now ‘cast out’, the result is 1(1) + 1(1) + 3(1), giving 5(1).

This of course is not a rigid proof but rather a demonstration of casting-out-9s.

For example, 235 + 568 = 803. The digit sum for 235 is 1; the digit sum for 568 is 1, and the sum of these is 2. This equals the digit sum of 803, so it checks. I realize that in today’s technical world, this procedure isn’t likely to be taught nor used but in olden days without calculators, it seemed reasonable to check your work.

Now back to what I sent back to my daughter. I generated the digit sum for the birthdays for all the members of our family and then generated the digit sum for the sum of them and lo and behold the result was 1! Of course, the family is unity!

Told you it was a short discourse, but couldn’t resist sharing it. Can’t wait to see what she comes up with next.


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Posted by mark schwartz on December 18, 2016

Let me first apologize for the long delay between the last posting and this one … there was just a heap of other stuff that needed attention …


The first day of class for a remedial/developmental at the community college level is a classroom loaded with math anxiety. These students, by definition, bring not only anxiety but also expectations about how the class will be conducted based on their previous experiences; at best, they hope to finally master some of the math that has been confounding them. Given this, rather than only the usual presentation of the course information (book, assignments, grading system, attendance, etc.), I found that an opening exercise of some kind eased their minds about how things would go in the course. Below are examples of ‘openings’ that engage the students, rather than immediately plowing into the course content.

The Story

The first thing that happens is that I assign them to groups, typically 3 in each group. I give them time to introduce themselves to each other and announce that they will be working with those in their group the whole term. Basically, share what you know and discuss within your group how to manage the material and do the work. In addition to individual work, there will be some group work. When they’re ready, they do an ‘opening’.

The where-are-you-in-math line. I draw a horizontal line on the board, marking the approximate center. On the left end, I note something like ‘math sucks’ or ‘I hate this stuff’ and on the right end I note something like ‘I get it’ or ‘math is no problem’. I then tell them that I’m going to leave the room and I want them to mark where they are on this math line … don’t use your name or initials, rather an ‘x’ or star or smiley/frowny face and when everyone is done, come and get me. Questions?

The typical picture is that there is a cluster of marks to the left of center, reflecting somewhat realistically why they are in this remedial/developmental class. I start the discussion by pointing to one of the marks and asking, “what do you suppose it would take for this person to go from here to closer to the right end?” It takes a while for the discussion to get going because they’re not quite sure what the question means, but individuals start offering things like “getting the fraction stuff”, “learning the rules for signed numbers”, “word problems”.

The point of the discussion is to identify not everything that needs to happen but rather that it may be that one (or maybe two) fundamental operations or rules can make a significant difference. I point out and emphasize that it’s not ‘math’ that they don’t get but rather some specific relationship that might be messing with their entire mastery. A good example is always operations with signed numbers. In the discussion, I make a point of doing the following: I ask that those who can finish the phrase I say, please do so out loud and I say “ a negative and a negative is a …” The response is of course “positive” but the I ask “when?” and I get some baffled looks and responses. They know the mantra but not what it really signifies. I ask for volunteers to come to the board and show me examples of when that mantra applies. Without correcting any of the statements – some of which are accurate – I simply point out that some are right and some are not and that rather than memorizing the rules, we will spend time talking about how the rules come about and how they really work.

I end this first day class at this point, unless collectively, they want to explore more about other math issues they may have. I won’t address the classic “when am I ever going to use this stuff?” but typically there are a few other issues we talk about, like “isn’t there an easier way to do fractions?”

Using an opening rather than diving right into the math content sets a different tone for the class; they realize that the class is more a dialogue than lecture; they feel comfortable asking questions; they like the idea of working in groups; they perceive math differently and this I note from questions at the beginning of the next class; they clearly have been thinking about what happened the first day and thinking about math is a very positive outcome.

Another first day opening I use once all the groups have settled down; is to ask if there are any ball players in the class – baseball, softball, basketball – and typically there are some. I ask one of them to stand and announce that I’m going to toss them an eraser and they are to catch it and throw it back. Once this is done, I ask “Was there any math done here?” This gets answers from “no” to “what do you mean?” I ask again if in the tossing and catching if any math was done and this typically gets things going. What gets focused on in being able to make judgements about trajectory, speed, acceleration, location and other quantifiable judgements which make it so that when you’re catching the eraser, you know how to place your hand to intercept the eraser in its flight and catch it. When it comes to quantifying the toss, it’s a matter of distance, energy, direction, flight path, etc. so that it makes it possible for the person to catch it.

The point of this opening and the discussion is to point out that we all do math all the time and if you ask “when am I ever going to use this stuff?”, the answer is “all the time”. I ask if anyone has any other examples of this kind of quantitative judgement. A typical response is “when I’m driving”. One student once proposed that walking up or down a set of stairs takes a lot of quantitative judgement.

The essence of this opening is that you already do math a lot and it’s a matter of realizing that a lot of stuff you will see this term are slowed-down algorithms that your brain does automatically and rapidly. This edges up to the philosophical question of “is math out there as a universal or man-made” and this sometimes comes up in discussion but the point is that it gets people – again – thinking about math. It again creates a different tone for the class and that this classroom will be different from their previous classes.

This next opening usually generates a lot of noise. First, I write an equation on the board twice, something like 2 + 3(2x ─ 1) + x = 3(x + 4). I put this equation on the left side of the board and on the right side of the board. I tell the class not to panic – they don’t have to solve it. But, what I do say is “where’s the math”? After we talk about this for a while, I make the following statement “what if I told you that numbers have nothing to do with math?” (sometimes, this question has popped up in the discussion, but if not, I state it). This really gets people going and after we talk about it for a while, I use the equations on the board to demonstrate what I mean.

I take the equation on the left side of the board and I write it without any numbers and I take the equation on the right side and write it with only the numbers.

The left side is     +   ( x ─ ) + x = ( x + )

The right side is       2 3 2 1   3 4

The question is “which statement makes the most sense?” That may not be the precise question to ask but the point is that when you compare the right side to the left side, there is an obvious difference. The left side has notation and the right side only has numbers. When we discuss this, it usually occurs that someone will say “the left side tells me things to do and I have no idea what to do with the numbers”.

This highlights the point of this opening. One can get a sense of the relationships and operations that are expressed in the equation by looking only at the notation; you get nothing by looking only at the numbers.

As we discuss this, the class reflects the importance of the notation and that the essence of math is not the numbers but – as one student said – how the numbers are connected. As in previous openings, this one again gets students thinking about math a little differently from what they had previously thought.

This last opening (I have more but 4 examples are enough for this posting) has several hidden messages; one is “read slowly and carefully” and the other is order of operations, although I don’t label this so in class. This is set up for a room that has a white board and uses markers but it could also be done with the standard chalkboard and chalk.

I give each group an envelope, in which there are brief statements, each statement on a separate piece of paper. I tell them that they are to put the statements in order and once they’ve done that, do exactly what it says to do – no more, no less. Once the instructions are clear, I watch each group sort through the statements, agree that they have the correct order and then do what it says to do. The statements, not in order are:

Walk to your seat

Write your name on the board

Cap the marker

Stand up

Uncap the marker

Sit down

Pick up a marker

Walk to the board

There is also one statement which says “choose one member of your group to do the following”. I need to note that I make sure that there are only two markers in the tray at the board because this is the core of this opening. You’ll see why in a moment.

In every class so far, every group fails the first time! When I announce this, I ask them to try it again. Sometimes someone gets it right on the second try but mostly people believe that there is the “trick” statement “write your name on the board”, so they correct themselves by writing that phrase rather than their name. No trick here.

Given that there are typically 6 or 7 groups in the class, it only takes two of them to get the exercise correct to bring out the point of the exercise. Note that in the statements, it does not tell the student to replace the marker in the tray. According to the statements, the correct thing to do is to take the marker with you back to your seat! So, once two groups get it right, the next groups can’t finish.

When this opening is done, I point out the importance of reading slowly and carefully and also of verifying what’s going on with members of your group. We talk about this when reading a text for information of when reading problems to solve.

As I said before, I believe it’s important to set a tone in these classes which signals students that this math class will be a little different from ones they’ve previously experiences. Let me conclude by quoting myself about what I consider to be the importance of an opening ….

“Using an opening rather than diving right into the math content sets a different tone for the class; they realize that the class is more a dialogue than lecture; they feel comfortable asking questions; they like the idea of working in groups; they perceive math differently and this I note from questions at the beginning of the next class; they clearly have been thinking about what happened the first day and thinking about math is a very positive outcome”

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Ted’s Question: Can I Graph a Decimal Slope?

Posted by mark schwartz on November 16, 2016


We were working on graphing lines using the slope-intercept method.

The equation to graph was y = 4/3x + 2. Traditionally, plot the point (0, 2) first – the y-intercept and from this point, move up 4 units (positive 4 on the y-axis) while moving 3 units to the right (positive 3 on the x-axis). This finds the second point at (3, 6). This process gives an accurate line between these two points.

Ted asked “If I use my calculator to find the value for the slope, I get 1.33 … can I use 1.33 as the slope to graph the line”? Having never heard this question before, I said I wasn’t sure but let’s look at it.

The Story

As it turns out Ted is correct … 1.33 can be used but it’s important to understand how to use it.

It goes back to a basic fraction relationship. In order to preserve the relationship between the numerator and denominator, it is allowable to multiply or divide both the numerator and denominator by the same value. This is what is done when searching to either find an equivalent fraction when reducing a fraction to lowest terms or finding an equivalent fraction for adding or subtracting fractions.

Given this, it’s not that the fraction is converted to a decimal by dividing 4 by 3. Rather the mathematical operation is to divide both the numerator and denominator by 3, giving the fraction 1.33/1. When we do this conversion, we typically don’t note the denominator of 1; it simply is ignored as if it weren’t there.

So, back to plotting the equation. Again starting at (0, 2), we would move up 1.33 (move positive 1.33 on the y-axis) while moving right 1 (move positive 1 on the x-axis). This is valid and falls on the line plotted when using slope = 4/3.

Well, not exactly. Using 1.33 isn’t quite as accurate as using 4/3, simply because, in this case, it is a repeating decimal. But, even without a repeating decimal, there still is the possibility of a loss of accuracy. Of course, for classroom purposes this might be acceptable After all, we’re not designing a spacecraft that needs quite accurate calculations for design and flight.

Using this decimal idea with y = 3/5x + 2, we would have y = .6x + 2. The plot again begins at (0, 2). The issue now is the scale on the x and y axes. If these axes are laid out in .1 increments, then .6 can readily be used with the same accuracy as 3/5, but if the scale is in whole units, the .6 is an ‘eyeball’ estimate and may not be as accurate. As a reminder, in this case, when moving up .6 on the y-axis, move a corresponding 1 on the x-axis. When using a decimal, the denominator (change on the x-axis) is always 1.

However, the question was wonderful and exploring it was interesting and … well, educational.

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Yet Another Subtraction Algorithm!

Posted by mark schwartz on November 4, 2016


I recently posted Revisiting Mr. Stoddard’s 1852 Subtraction. In that posting I modified Mr. Stoddard’s idea by introducing a procedure which allows for subtraction without borrowing. This posting modifies that modification.

The Story

I’ll use a simple subtraction example to demonstrate the procedure, but I have examined much more sophisticated problems such as 20801 ̶ 278 and the procedure is still good.

Basically, treating ‘ab’ as a 2-digit number and ‘c’ as a single digit number, in the problem “ab ̶ c”, if c > b, the answer to ‘b ̶ c’ is 10 ̶ ( c ̶ b ) and then add 1 to the 10s place value in the subtrahend. For example, 12 ̶ 8 gives 10 ̶ (8 ̶ 2), or 4, then add 1 to the 10s place value in the subtrahend, giving 1 ̶ 1 or 0, which isn’t written.

What I didn’t note clearly are two things. First, if in that example, b > c, then write down that value as the answer. Do not add 1 to the next place value in the subtrahend. However, if c > b, then the algorithm as noted is to be used. And here’s the modification – continue with this algorithm!

Here’s an example in slow-motion math. Using the problem 7234 ̶ 567 as a traditional ‘vertical’ problem, we hav


In the 1s column, 7 is greater than 4, so the answer is 10 ̶ ( 7 ̶ 4) which is 7. Add 1 to the 6 in the subtrahend 10s column. Then in the tens column, 7 is greater than 3, so the answer is 10 ̶ ( 7 ̶ 3), which is 6. Add 1 to the 5 in the subtrahend 100s column. Then in the 100s column, 6 is greater than 2, so the answer is 10 ̶ ( 6 ̶ 2), which is 6. Add 1 to the zero in the subtrahend 1000s column. Then in the 1000s column, 7 is greater than 1, so the answer is simply the difference of 6. The solution looks like this:

– 567

There are many subtraction algorithms posted in this blog and most of them focus on avoiding the need to borrow, so if you feel like trolling through the entire blog and compiling them, you might find one you like.

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Heron’s Area of a Triangle

Posted by mark schwartz on October 24, 2016


This is a brief story about a fun event that almost always happens when discussing the area of a triangle. The formula for the area is A = 1/2 bh, where ‘b’ is the base and ‘h’ is the height.

The Story

Simple enough if it’s a right triangle and the base and height can readily be seen or calculated from a2 + b2 = c2. But what happens when it’s not a right triangle? Well, one has to wiggle around a bit and do a few more calculations to determine the height, but it can be found.

But being an instructor that likes to stretch students thinking and imaginations, I draw a very scalene triangle with sides of 4, 8, and 10 and present them the task of finding the area. As I roam the room watching them work and listening to their grumblings, I ultimately have them stop and present to them Heron’s formula. I don’t bother with the derivation and for those that are interested in knowing it, I recommend Googling it.

The formula is A = 1/4 the square root of (a + b + c)( ̶ a + b + c)( a ̶ b + c)( a + b ̶ c).

This is a really nifty formula because of the pattern in it – add all the sides together, then in the next three parentheses, just negate each side, one at a time, in order. And, given that there are 4 parentheses, just divide the square root of this product by 4.

In class, I first apply the formula to a 3, 4, 5 right triangle to demonstrate how it works. Then we play with a few other right triangles to get comfortable with the formula. Then we return to that weird scalene triangle. Using Heron’s formula, we get an area of 15.1987 … and now the fun begins!

Someone typically asks “how do you know this is correct?” So, this sets us up to explore how to find the area if we didn’t have Heron’s formula; so we do all the Algebra necessary to demonstrate that indeed 15.1987 is correct. In one class, this whole presentation got applause and I take that as their having had fun with it.

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Must We Filter Students Through the Math Sieve?

Posted by mark schwartz on October 19, 2016

Deborah Blum in The Best American Science Writing, 2011 (page 184) cites a California Institute of Technology science historian as saying “K-12 science classes in the United States are essentially designed as a filtration system, separating those fit for what he called ‘the priesthood of science’ from the unfit rest of us.”

I believe the same can be said for math classes. Of course, I can assume that math was included in science, but to be very specific about it, math actually seems to be a more severe filter than general science. Today, many science classes involve students in exploration and experimentation and some of the valuable lessons of accurate measurement, recording and analysis. And, some of these activities include the necessity of math. But, when doing math in a vacuum, unrelated to an activity – in essence, the math part of the activity is secondary – the filtering action seems more apparent.

For example, in today’s texts there are typically sections on “applications”. There are even entire texts dedicated to applications and these applications show the students how math is in our everyday activity – sports, statistics, banking, calculating interest, taxes, consumption, measurements of all kinds. And this is fine. But, it’s still done in the context of filtering those who have an aptitude for it from those who don’t because …

Texts still tend to present formulae and algorithms and teachers say “this is how to do it”. In essence, teachers are saying “here’s how to do it” rather than asking “how do you imagine how this can be done?”   We don’t ask students to generate their own conception of how to solve the problem, most likely because we believe they can’t or don’t. However, many math researchers of early childhood “math” capability have found that even before entering elementary school, most children are already identifying quantitative relationships, imagining algorithms that help them understand the relationships, verifying that their conception will always work, and subsequently and repeatedly, altering their algorithm if their conceptions don’t work. It’s sort of a fundamental, built-in scientific approach to what’s going on around them. So, having created their own quantitative environment, what happens not only to the environment but also – more critically – their formulating such systems when the teacher, the text, and “schooling” provides the algorithms for them? Who needs to continue exploring the pieces of the puzzle when a solution methodology is already provided? Further, if a student in elementary school proposes a solution differing from the text, is the teacher prepared to explore that proposal to its end and see if indeed it may be worthy of consideration?

When math is taught, it in essence teaches students not to think about the relationships. The tendency – and the pedagogy – is to teach students how not to think about it because we proffer the historically valid rule, procedure, formula or algorithm which allows them to get to the answer in the most efficient way (“rule” will be used from now on to summarize procedures, formula, algorithms, etc.). Why mess around with inefficient or erroneous methods? Just give them the rule and have them practice it. Well, this does two things: first of all, practice doesn’t make perfect, rather perfect practice makes perfect and second, it suppresses what seems to be a natural urge to play with the information presented and explore the quantitative relationships that might be there.

Let’s address the practice concept for a moment. A common phrase touted by math instructors is “math is not a spectator sport” or “you don’t learn math by watching others do it.” There is some validity to this, but there is also the reality that as Yogi Berra commented “you can observe a lot just by watching.” But the question is, what is it that students should be observing? Watching a math instructor use a predetermined rule to solve a pre-established problem and then ask students to mimic this activity may actually work for some students. But, in a broader sense, what is it that we want students to learn when we teach math?

This is not a simple question and doesn’t have a simple answer. Most likely, the answer is to get students to be able to do the indicated calculation or solve the problem. But is that what is intended for them to learn? Should the lesson be about applying a rule or about exploring the quantitative relationship? Rather, it’s establishing a context in which the student can imagine alternative rules and test those rules for reliability and validity. And what are we, as instructors to do, if a student discovers a less efficient but comparably valid rule? Here’s where we run into the range of expectation of the instructor as well as the training and experience of the instructor.

Going back to the premise of math learning as a filter system, it seems reasonable to assume that all students, those who can attain the priesthood and those who can’t, could manage in a system that allows and prompts for exploration, rather than being given the rules. It would still act as a filter system, but the real key is that those not destined for the priesthood would gain a better grasp of quantitative and mathematical relationships. Basically, it is math learning by doing but the “doing” is now differently defined.

Here’s something that happened in class one day. We were just beginning to work with simple equations in an introductory Algebra class. The text approached setting up the equation by making a statement which could be directly translated to an equation. This has become a typical introductory approach. For example, the student is asked “if you take a number, double it and add 1, the result will be 5. What is the number?” The expectation is that the student will write “x”, then double it by writing “2x”, then add 1 by writing “2x + 1” and then showing that 2x + 1 will have a result of 5 by writing the equation 2x + 1 = 5.

As I moved around the room watching and helping students work through this translation, this is what I saw on one student’s paper:

P   P   P   P   X                   The answer is 2.

I asked her how she got 2 as an answer and it went something like this: I knew there were 5 pieces when I got done, so I wrote “P” five times. But since one was added, I had to take one away. So, one of the “Ps” became an “X”. Then, since the number was doubled, I had to take half of it, so half of the 4 “Ps” that are left gave me 2.”

This is perfect logic and a valid way to reason through to the answer. In essence, she saw that the process could be reversed and mapped it. It doesn’t, however, meet the intended goal of

having a student construct and then solve an equation. What is an instructor to do? Consider that in the future, this student might be asked to solve the equation 4 ─ 2(2x + 1) = 3x + 5. Can this equation be solved using this student’s strategy? Yes, but not as efficiently as the traditional equation solving strategy. What happens to this student’s sense of self, sense of algebra and equations, and sense of quantitative relationships if, as an instructor, I have to say “no, that’s not the way to do it.”?

And the issue isn’t only the student; it’s the pedagogy. It seems that the pedagogy is probably more the issue because it doesn’t allow students to try out various strategies and come to the realization that their strategy works for some equations but not all equations. They now have a choice. They can learn several strategies and tailor the strategy to the circumstance, or accept the traditional pedagogy which offers an efficient method for solving equations of all types. It may be contended that if the student builds a library of different strategies for different equations, that it may be a big library and there may be an equation not amenable to one of the strategies. I would reply that a strategy developed and employed by a student is likely to be better remembered, and modified as necessary, than one that is presented and never “owned”.

There are ways of approaching the solution of equations which allow for the type of visual representation that this student used. Further, equations can be solved using objects and images or both; no paper and pencil need be used – at least not at first. All students could be started with this student’s approach and as the equations become more sophisticated, it could be noted that an alternative strategy needs to be used for these more sophisticated types of equations. Starting with their conceptions may well result in their all coming to the conclusion that the most efficient strategy – the classic traditional strategy – is most favorable. However, consider that getting to this point would take more time, yet that time is valuable in establishing students’ capability to imagine alternative methods, compare and contrast them, and conclude which is best. Further consider that when students are taught, for example how to solve systems of equations, texts and instructors teach the substitution and the addition method, and sometimes even matrix and determinants. We bother to do this because, with some examination before plunging into the solution, it may be determined that one method is better than the other, under the circumstance. So, why not allow students to use their methods as well as they work their way through solving the problem?

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