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 markdotmath@gmail.com. I’m not an antiquarian; the books I have are dusty, musty, brown-paged scribbled-in texts written by authors with insights into how math works. Unfortunately, most of their procedures have vanished. They’ve been overcome by more traditional perspectives, but you have to realize that at that time, they were teaching the traditional methods.

Archive for the ‘remedial/developmental math’ Category

The Definition of Math is …

Posted by mark schwartz on February 25, 2017

Introduction

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 markdotmath@gmail.com 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.

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A Short, Short Discourse on Digit Sum

Posted by mark schwartz on January 16, 2017

Introduction

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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Openings

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 …

Introduction

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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Yet Another Subtraction Algorithm!

Posted by mark schwartz on November 4, 2016

Introduction

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

7234
–567

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:

7234
– 567
6667

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.

Posted in basic math operations, Historical Math, math instruction, mathematics, remedial/developmental math, subtraction | Tagged: , , , | 1 Comment »

A 1st Day Handout to Students

Posted by mark schwartz on October 17, 2016

 

Author’s Note: The following is literally a handout given to students the first day of class. I give them time to read it and then we talk about it. The discussion set the tone for their learning and the idea of freedom was a surprising but satisfactory idea, although scary to some who expected this class to be like all previous math classes. What follows is the handout.

In the 1960s, a book titled “Freedom, not License” hit the bookstores. Briefly, it’s a story of the core philosophy of a school named Summerhill in England. The title refers to a subtle distinction between two conditions: freedom – being able to determine your own behaviors, live with the consequences, be self-determining, guided by your own internal discipline and control; and license – interpreting the circumstances in which you are allowed, permitted and “controlled” by an external authority. Actually, it’s misinterpreting the freedom as license, whereby the misinterpretation leads one to rely on external events, rather than understand the freedom to govern one’s own behavior and actions. License also is interfering with other’s freedom.

I give you freedom to succeed but it has to be your success, not driven by external rewards and punishments. I will teach well and you have to learn to learn well. Don’t rely on me to chase you down the hall demanding that you get assignments done on time. That’s your responsibility. Don’t rely on me to threaten you with loss of grade if you don’t attend class. Attendance is your responsibility. Don’t rely on me to control the classroom as is done in elementary school; hushing the noisy, punishing the “unruly”. It’s your responsibility to respect the classroom environment and not disrupt my teaching or the learning of others.

Freedom is a little scary if you’ve never experienced it in a classroom. But consider it a responsibility just like driving. You’re responsible for your car – for its maintenance and performance; for driving responsibly within the wide legal constraints of the speed limit, parking areas, passing, not drinking while driving, etc.

According to the Oxford English Dictionary, “education” is derived from its Latin root, “educare”.  Educare means “to rear or to bring up”.  Educare itself can be traced to the Latin root words, “e” and “ducere”.  Together, “e-ducere” means to “pull out” or “to lead forth”.  Hence we use the word “educare” to communicate the teaching method through which children and adults are encouraged to “think” and “draw out” information from within.

Notice the last three words: “information from within”. It is within you to learn well and to learn any subject well. I can help you draw it out, but the “you” is the important word in that sentence. You have to attend class, do the assignments, and act respectfully toward yourself and all others in the classroom.

Let me repeat – freedom is scary if you’ve never experienced it in the classroom. I will not check your classwork to see if you’ve done it and it is correct; answers are in the text. I will work with you if your answers are incorrect. You’re responsible for that and it will be hard for you to accept that responsibility because it will be tempting to leave class early and not do it because math makes you uncomfortable and anxious. But I can help you address the lack of math skills that lead you to feel that way.

My teaching doesn’t automatically lead to your learning. But take the freedom offered and use it; don’t let it become license that interferes with your learning.

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Revisiting Mr. Stoddard’s 1852 Subtraction

Posted by mark schwartz on September 29, 2016

Introduction

In this blog is a posting Mr. Stoddard Subtracts in 1852. If you haven’t read it, you don’t need to (but of course you can!). Mr. Stoddard presents an idea in subtraction which avoided the need for “borrowing”. For some reason, I was playing with a subtraction idea and after I had written out the entire algorithm, I realized that I basically had modified Mr. Stoddard’s; thus the title.

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.

Here’s why it works. In essence, it could be said that borrowing has happened but it’s hidden as well as not written!

In essence, 10  ̶  ( c ̶ b ) is borrowing, but it’s hidden. The ‘10’ in the 10 ̶ (8 ̶ 2) could be said to have been borrowed from the 10s column in the minuend. Given that, that ‘10’ can be said to have been subtracted from the10s column in the minuend. It’s known that if the same value is subtracted (or added) from both the minuend and subtrahend of a subtraction problem, the answer will be the same. Thus, adding a 1 to the next place value in the subtrahend adds a value which will be subtracted.

There it is. It’s a mild modification to Mr. Stoddard, but my ‘aha’ moment with 10 ̶  ( c ̶ b ) may well have been his idea incubating all this time. Try it with other problems – like 20801 ̶ 278 and after a while it becomes as automatic as doing the problem using borrowing.

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In 1877, Mr. Ray Reasons with Fractions

Posted by mark schwartz on September 8, 2016

Introduction

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 »

Math Stories

Posted by mark schwartz on August 30, 2016

We all believe were good teachers, yet also we all believe we can improve. Our training and experience, and occasional validation by our peers helps us feel good at what we do. Students also give us comforting feedback, directly through their improved performance and “thank you” and indirectly as we hear from other students, friends and colleagues. We continuously share our ideas with our peers, discuss successes and failures, ask for guidance and help with a pedagogical idea. We teach with inspiration and concern, and treat students with respect, care and a nurturing attitude.

So, what’s the problem? How come we are distressed at the 4-5 students in a class of 20-some who don’t seem motivated, don’t do the work, sometimes don’t even show up for class? Are we to parent them? Are we to act as shepherds of their lives as well as math instructors? Are we to be motivational speakers to grab their attention and rouse them to dynamic learning?

Perhaps we believe, at the core, that everyone is capable of learning math, if only the learning environment were appropriate. If only we were able to be more adept at unraveling the history of their prior experiences and build from the math rabble they possess. And, paying attention to the importance of learning styles, we tend to adopt a host of teaching methods until we find the one that’s just right for each student. Well, at least, we try.

So, what’s the problem?

Is it them or us? This short question is the problem. Once ‘them’ and ‘us’ are separate entities, once we see ourselves as separate, the problem begins. Really? After all, we are individuals and it is pure happenstance that brings us together in the same classroom.

And that’s the point. We are a unit and theoretically, I’m the instructor and they the learners. There is no point to my teaching if there is no audience and there is no point of their sitting in a classroom if no activity is to occur. So, we are a unit; an educational unit.

And what is the dynamic of this unit? One way to view this is to allow that the instructor is to tell stories to the students. The stories will be about relationships between quantities. Usually these collective and cumulative stories will be called math; there are prescribed relationships that have been identified through the ages as essential and core to mankind, and these relationships are bundled together in a text and it is called mathematics.

Some of these relationships – some of these stories – don’t seem to make sense to the students because they have heard these stories before and the stories don’t seem to make sense or don’t seem to have any meaning, or further, don’t seem to have any meaning to that person’s unique story. Is this a critical event? Must all math – or for that matter – any content area have immediate and purposeful meaning to a person before they feel engaged with it and feel impelled to learn it because it has a visible connection to the daily lives of the student?

What about history, English, biology, psychology? Do these disciplines have value to the learner? Can the learner see how these can be used, integrated into their daily lives and become valued learning? What about math?

If math is taught as it traditionally is, then the likely answer is no, there is no apparent real-life immediate utility. This, of course, is based on the assumption that if the utility were made visible, the students would then be more engaged in learning it. But let’s return to the idea of ‘traditional’ for a moment. Traditional means that sequence of information presented in texts which math curriculum developers and teachers believe to be important for students to learn.

In basic math the four basic operations are learned, and then the order of operations is learned because a person confronted with a problem in which all the operations occur must have a ‘rule’ for which operations to do first. Why? So that not only will they get consistent answers but also so that all the people doing to problem will arrive at the same answer.

Is there something sacred about the order of operations? Not particularly, but it has been made to be so. Violate the rules and you will err. For example, in class the other day the problem was (2/5)^3x(5/8)^2.

The solution in the text showed (2/5)(2/5)(2/5)(5/8)(5/8) and then showed some canceling of common factors, resulting in an answer of 1/40. However, notice that in order to do this, the order of operations has been ignored. If one were to literally apply the order of operations the exponents would have been the first operation to do, giving 8/125 x 25/64. The next step would have been to do the multiplication of the fractions, and finally the answer would be simplified completely. Somewhere in the text, by the way, the way this problem was worked was consistent with what was described as a ‘shortcut’, meaning when you have multiplication of fractions you should look at the fractions to see if they can first be reduced, and then do the multiplication. This is a valuable and time-saving strategy, but in the example as worked out in this case, reference to the ‘shortcut’ wasn’t made.

The issue here is the story of how to do this kind of a problem. Is there more than one story that can be told? Yes. Be literal and follow the order of operations or alternatively, use the ‘shortcut’. Which one is correct? Both are correct. Which one should a student use? Whichever story resonates with them. What? Isn’t math a little more linear and logical than that? Allowing students to craft their own method of solution?

Well, isn’t crafting an individual and unique method for problem solution what students do in everyday life? Why not in math class?

A lot of students have the perception that math is very linear, very logical, and has one and only one path to the answer. Starting with the very basic idea, for example, that 2 + 2 always and only equals 4, this concept scaffolds into other math operations, like the above noted order of operations. Until of course, one has to do algebra, where the numbers don’t exist and letters representing them exist.

For example, what happens to the order of operations in algebra vs. in arithmetic? In arithmetic, students are taught to do the work inside any grouping symbols first, if there is work to do. So, 2(5 + 6), is done by adding 5 and 6 = 11 and then multiplying by 2 = 22. What happens if a student knows the distributive law and uses it? Is that student to be corrected?

But back to algebra. In algebra a statement like 2(5 + 6) can be seen as a(b + c) and of course, the order of operations still applies. ‘Do the work inside the grouping symbol first’, so the student is to add ‘b’ and ‘c’., Well, algebraically, it can’t be done. All one can do is indicate it by leaving the expression as written, b + c. Then multiply this expression by ‘a’. Is this consistent with order of operations? Absolutely, so why does this bedevil some students?

There are some students who contend that b + c = bc. We now have to address issues of notation. In algebra, ‘bc’ means to multiply ‘b’ times ‘c’, once we have actually substituted numerical values for ‘b’ and ‘c’. But if we were to be very literal, if ‘b’ = 7 and ‘c’ = 8, then what would be written is 78. But this isn’t done. By the way, 78 means 7(10) + 8(1), a whole other notation issue, and certainly doesn’t mean 7 times 8.

But when we write 78 because of having substituted ‘b’ = 7 and ‘c’ = 8, different notation is employed. To keep the arithmetic consistent with the algebra; we use one of the available notations for multiplication such as (7)(8). Why not use (b)(c)? Because b + c doesn’t result in (b)(c).

I think you get the point. Math stories seem to have inconsistencies. We seem to lurch around and come to realize that some of the arithmetic rules work with algebra and some of them don’t. And, what’s more, we have to know both domains, because while doing algebra, the arithmetic rules still apply, but the converse isn’t necessarily true. So, although 2 + 2 = 4 is true in arithmetic and in algebra, students must understand that although the order of operations is the same in both cases, it ‘feels’ different.

So, why don’t we as math instructors, teach students the algebraic story of grouping symbols to bring consistency? Why not allow that if one sees 2(3 + 4), use the distributive law? Is it because of exponents? A student would have to see that 2(3 + 4) and 2(3 + 4)2 have to be handled differently. And, by the way, look at what happens – and it happens regularly with some students – when a(b+c) and a(b+c)2 are presented. Have you ever seen (3 + 4)2 = 9 + 16?

Part of the issue here is that sometimes, the rules that apply when the values or expressions have an exponent of 1 are different from the rules that apply when a value or an expression has an exponent greater than 1. Wow, what a story.

For example, when trying to make the point of the difference between (-4)2 and –42, one very direct way is to say ‘follow the order of operations’. In (-4)2, a student would do the exponent first, thus 16. If the student were to do –42 according to the order of operations, the ‘-‘ has to be seen as subtraction, not the sign of the number, and the order of operations says to do exponents before subtraction, so the outcome is –16. Is this too subtle for students to see? Should this understanding of it be presented? And, why-oh-why does math allow the same symbol to be used for two different things: ‘+’ can mean addition or be the sign of the number and the same for the symbol ‘-‘.

So, look at (-4)2 and –42 again. Using grouping symbols makes it clear that the student is to apply the exponent to everything inside the grouping symbol. This works readily. But what if students had been taught to read the ‘-‘ not as subtraction but as the sign of the number? Or, that the symbol ‘-‘ can mean either? Given this understanding of ‘-‘, then (-4)2 and –42 could mean the same thing. The symbol ‘-‘ is the sign of the number, therefore understand this to mean ‘negative four times negative four’.

Here again, however, the idea of an exponent of 1 versus an exponent greater than 1 can come into play. If a value has an exponent of 1, the ‘-‘ can be either the sign of the number or the operation of subtraction, but if the exponent is greater than 1, the ‘-‘ means subtraction and not the sign of the number. Given this, should it ever be taught that ‘-‘ can mean both, or more critically, knowing that this distinction between the forms (-4)2 and –42 forces different meanings, perhaps the point should be made about how the rules differ when exponents are equal to 1 or greater than 1. Or, ignoring all this, simply point out that the order of operations applies.

However, texts present the following: “When subtracting, add the opposite” and demonstrate this algebraically as a-b = a + (-b). So, in a problem such as 25 – 42, if a student were to follow the subtraction algorithm, the outcome would be 25 + (-4)2 and thus again showing that (-4)2 and –42 mean the same thing. And it is again true that if a student were to be literal about the order of operations, the 42 has to be done before the subtraction. But where is it ever designated that changing the notation form isn’t allowed? After all, changing the notation form is not doing the operation. But, is rewriting 25 – 42 as 25 + (-4)2 actually doing the operation of subtraction or simply changing notation? Again, if it’s understood to be doing the operation, then again the order of operations prevails, and thus a student couldn’t apply the subtraction algorithm before doing the exponentiation. But, how is a student to know this? Well, maybe again, this defaults to distinguishing between the operations if the exponent is 1 versus greater than 1. If it’s greater than one, suspend transforming subtraction to addition. So, it seems that raising numbers to exponents greater than one has a subtle but critical meaning which isn’t addressed in texts or by instructors.

Perhaps all the above is irrelevant since the rules and algorithms that we present are presented in a very direct, concise, logical way. We need not presume that any student would ever get caught up in the tangle of symbols, words and relationships as written above.

But these little inconsistencies, which never reach consciousness necessarily, can generate cognitive dissonance. It may well be that some part of the students ‘math brain’ is saying, “Huh…this doesn’t make sense. We were taught that 6-2 means 6 + (-2) is true in one circumstance, but not true in another circumstance, 6-22 ≠ 6 + (-2)2.”

Perhaps this gives too much importance to the thinking students bring to math classes. Have they ever been bedazzled by other such seeming anomalies in math notation and math meaning? Have they every sensed something about the rules not always being applicable and no instructor has ever pointed this out? Should I, a student, ask about this if I see it?

Fundamentally, I believe teaching math is telling stories and listening to students’ math stories, the stories that they carry around in their heads, perhaps fantasies, hardly realities of math.

Getting people to reflect on what they know – actually, reflect on the fragments that they carry around that they call math – is important to the story telling. Again, these are the student’s stories. For example, writing on the board that 2/4 = and asking them what the answer is usually gets a collective ½. If it is then asked “What math operations did you do?” the answers will vary but be things like, cancelled, or divided, or reduced, simplified, got rid of. The point is that students know how to do, metaphorically, math. The words they use aren’t math operations but symbolically represent several math operations. In order to get 2/4 to 1/2 involves several math operations. These are sequential events, algorithmically organized, compartmentalized, and further trivialized by summarizing and compressing them into a word like ‘cancel’.

Getting students to reflect on what they know gives them a starting point from which understanding, not just doing, math can occur. Puzzling them, disabling their usual routes to answers, and making them appear as deer in the headlights. But not doing it in a threatening or punitive environment.

For example, we were talking about dividing by powers of 10, and used the example of 300 divided by 40. I then wrote it as a fraction, and brought it back to the same notation form as 2/4. I got a wow response, which was nice but also something which I could get only if students had been misperceiving what was really being expressed. How did they generate this misperception? They’ve been in math classes before mine and have come ‘armed’ with these fragments and pieces of math-like words.

I do believe that if they can gain an understanding of the basics, then they can always reconstruct in their own way with their own logic a pathway from the problem given to the set of possible solutions available.

Keeping it basic and visible is helpful. For example, most textbooks give one page to exponents and square root. It is written in ‘mathese’, dry presentation of what the notation is and what it ‘means’. However, what it means can also be demonstrated.

Put on the board the following: (2)(2)(2)(2)(2)(2)(2)(2)(2)

(3)(3)(3)(3)(3)

(5)(5)(5)

(7)(7)(7)(7)(7)(7)(7)

Then talk about notation; no mystery here, simply a notation issue.

Try this. Put on the board a cluster of 9 “postits” and a separate cluster of 16 “postits”. Ask for a volunteer to come to the board and arrange these in a square. Once done, they will be able to see that the square made by 9 is 3 on a side, and 4 on a side for the 16. Then ask for a volunteer to put them together in a square, and this gives a square of 5 on a side for the total of 25. This demonstrates square root.

Students have been trained to believe that if you do math well, you must also do it fast. This I find to be one of the perceptions that students bring which interferes with their being able to learn math. The part that isn’t seen is the thinking, the reflection, the connecting the ‘new’ stuff with stuff they already know. Their stories of math typically aren’t complete. Rather than identifying what they do as wrong, I try to identify what they do as incomplete. Sometimes I prompt them with the idea that if they have a strategy that works, they need to know if it works all the time. If it doesn’t, let’s talk about it and find out how to make it work all the time.

To be continued … meaning there is always more stories to tell – a lot of stories by a great number of teacher and students. I would hope that framing teaching math as a way that teachers and students can share math stories will reframe some of the classroom conversation.

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Is it ̶ 3 or is it ̶ 3?

Posted by mark schwartz on August 27, 2016

Introduction

I know. The title “Is it -3 or is it -3?” looks weird but it’s not a typographical error. It’s a way to bring attention to algebraic notation. The question is: how did you read -3? Did you say “minus 3” or did you say “negative 3”? Does it make a difference?

In Day’s 1853 An Introduction to Algebra, he writes 5 pages on the topic – yes, 5 whole pages of words discussing negative quantities. He wants to make sure that students understand that the 4 basic operations in arithmetic are different from the 4 basic operations in algebra because of the introduction of negative quantities in algebra. In lengthy discussions he cites how negative quantities appear in profits of trade, ascent and decent from earth, progress of a ship relative to a latitude, and of course money. Clearly he’s conveying what I would call the algebraic trip-wire – how to handle negative quantities. This kind of lengthy discussion isn’t presented in today’s texts but rather students are presented with diagrams and number lines and visual aids to help them understand the rules. An instructor can supplement the text with their own creative explanations and demonstrations. But Day’s emphasis on this point may well be what is needed in today’s texts – a core understanding of the rationale behind the rules.

The Story

So, back to “is it  − 3 or is it – 3?”

Day’s writing prompted me to recall a question from a student. We were working with operations with signed numbers. Typically I am very careful to reference any “ ̶ “ in a problem or an answer as a negative or as a minus, depending on its use in the problem. Knowing, for example, that + ( ̶ 3) gives the same result as ̶ (+3), in the former the “ ̶ “ is understood as negative 3 but in the latter it’s understood as minus 3. As noted, it ultimately makes no difference, but a student stopped me during a discussion and pointed out that in the same problem I had referred to a term as both and it didn’t seem right to him … and in a most technical sense, he was right. I asked if he were the only one bothered by this and other students felt as he did.

I admitted to my sloppy use of the terms and we got back to discussing operations with signed numbers and then again, this student stopped me. He asked “what about – and in his words – minus a minus 5” – how come it’s plus 5?” I wrote ̶ ( ̶ 5) the board and asked him if this is what he meant and he said yes. I asked him then what operation is being indicated and he said that it indicated to subtract a negative. So, the sign inside the parenthesis isn’t a minus, rather it’s a negative sign, a sign of the number. The class was muttering about this somewhat lengthy Socratic discussion – and they participated too – which really was a very positive result of the initial question … what some might call an unintended consequence … but a good one.

And of course, there was the question of “does it make a difference what I call it if I get the right answer?” So, we played language games with various examples until there was consensus that there was a difference between “minus” as the operation of subtraction and “negative” as the sign of the number. But, for most of the class, this difference didn’t make a difference as long as they understood what the notation in the problem was asking. So, I asked them to think about this:

Don’t do this problem yet but within your group, discuss the “ ̶ “ signs in the problem 4 ̶ 6 + 2 ̶ 3 ̶ 5 + 7. Signs of the number of signs of the operation? It was fun to roam the room and listen to the within-group discussions. As expected, there were disagreements, yet those that disagreed came to understand that both were correct! It was a matter of what procedure made each person feel most comfortable.

After allowing for discussions, I asked for volunteers to go to the board and demonstrate their solution. There were two primary solutions: first, just use the order of operations and do the indicated operations from left to right, although there was some stumbling to explain how to handle “2 ̶ 3 ̶ 5”. The language used in explaining the whole problem was interesting. For example, “4 ̶ 6” equals minus 2 (not negative 2) and minus 2 and plus 2 is zero (adding two operation not two values). Then zero minus 3 (the “ ̶ “ is the sign of the operation) gave “minus 3” and the next operation was expressed as “a minus 3 and a minus 5 equals negative 8”. Think about that. Technically, the 3 and the 5 were expressed as adding two subtractions (minus wasn’t seen as an operation) yet the answer of negative 8 was correct notation. But the real thing to notice is that the answer is correct independent of technically incorrect labelling of the values.

As much as I believe in the importance of carefully using either minus or negative correctly, it clearly seems that – at least for this student and his group – knowing how to handle the negative is more important.

The second solution was given with a preface. This student rewrote the problem as 4 + ( ̶ 6) + (+2) + ( ̶ 3) + ( ̶ 5) + ( +7). She pointed out that her group saw all the signs as signs of the numbers and therefore they just added them all together. Neat.

Of course there are more ways to handle this problem but these two examples show that as long as students understand the basic rules and relationships with signed numbers, the right answer will be found. We talked about these two solutions and how to handle the signs and operations.

I then asked if all the talk we had about the difference between negative 3 and minus 3 made a difference for them. The consensus was yes and that it showed up when they were talking about the problem in their group. Apparently, it provided a clearer understanding of the difference.

There was also the comment that allowing them to challenge me (I pointed out it wasn’t challenging me but rather challenging the math content) gave them a sense that the “rules” and labels weren’t arbitrary – that there really was sense to it.

Finally, I’d like to note that hearing a student’s question as a real interest in knowing rather than a hostile kind of “whatever”, opened the door to the discussion which further opened the door for their better understanding – again an unintended positive consequence. If you have time, try it.

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The Importance of a Clearly Stated Algorithm

Posted by mark schwartz on August 22, 2016

Introduction

I posted a piece earlier in this blog titled Sheldon’s Compound Proportions. It describes what Sheldon labels the “cause and effect” method for solving compound proportions, which as far as I can tell, aren’t in todays’ texts. His work was in 1886. You might want to take a look at his idea because this posting talks about other compound proportion procedures at that time and I did it to emphasize the importance of a clearly stated procedure for doing an operation.

The Story

I strolled through my collection of old texts and in quite a few of them found the same prescription for solving compound proportions not using cause and effect. I picked 5 which cover about a 20 year span from 1864 to 1883. They all have the same procedure and what I suspect is that it was the established and accepted solution method at that time. As in todays’ texts, it was just a simple matter of “borrowing” a basic algorithm from someone else’s work. There are other texts of that era which reference Sheldon’s cause and effect method and a few of them introduce it along with the procedure I’ll cite below.

The point is that his method is a much clearer statement of how to handle the information in a compound proportion problem. Further, what I’m suggesting is that we should carefully examine some of our current traditional algorithms to see if the reason students have trouble with them is because of the way they are worded and presented. For example, finding the lowest common denominator (LCD) in order to add/subtract fractions doesn’t require the extended way it’s been typically taught. In fact, I have seen some texts introducing a method which doesn’t require finding an LCD at all. Certain mixture problems can more readily be solved with an 1864 method Mixing it up with Alligation, posted earlier in this blog.

By the way, the 5 texts in which I found this procedure are all arithmetic texts, which indicates to me that this somewhat sophisticated idea of compound proportion was taught in elementary school. I’ll give you example problems from an old text to indicate that, in my view, it was a very handy procedure for the real world experience at that time. Today we call these “application” problems.

Here’s the rule as stated in Greenleaf’s 1881 The Complete Arithmetic, page 235 (the other 4 books are cited below and present the same rule).

Rule for Compound Proportions

“Make that number which is like the answer the third term. Form a ratio of each pair of the remaining numbers of the same kind according to the rule for simple proportion, as if the answer depended on them alone. Divide the product of the means by the product of the given extreme, and the quotient is the fourth term, or answer.”

Embedded in this is reference to “…the rule for simple proportion …” which Greenleaf provides on page 233 and it is:

Rule for Simple Proportions

“Make that number which is of the same kind as the answer the third term. If from the nature of the question the answer is to be larger than the third term, make the larger of the remaining numbers the second and the smaller the first term; but if the answer is to be smaller than the third term, make the second term smaller than the first. Divide the product of the means by the given extreme, and the quotient is the fourth term, or answer.”

Students had to be able to apply this latter rule for simple proportion before being presented compound proportion. There is no conflict between the two rules; in fact, there is some overlap. For simple proportions, the rule directs the student to understand “the nature of the question …” and use that to determine which values go in which of the 4 places in the proportion. The students had to be able to assess and estimate if the answer was going to be larger or smaller and place the correct terms in the first and second places. Wow! There is a lot of estimating and juggling of values and basically it seems that all of this effort is aimed at what we would say today as determining whether it’s a direct or inverse proportion. With problems with simple values, this is a somewhat manageable issue.

For example, a problem from the text is “If a man travel 319 miles in 11 days, how far will he travel in 47 days?” Using the rule for simple proportion, the setup would be:

11/47 = 319/x    (the rule doesn’t use “x”, but I did for demonstration purposes)

The solution is (47×319) ÷ 11 = 1363

However, in today’s approach to simple proportion, the setup (in most cases) simply follows from the order of the information in the problem, giving:

319/11 = x/47

This gives the same answer but notice that the rule states “Divide the product of the means by the given extreme …” and that doesn’t apply here. So, the 1881 rule is quite constraining when it comes to writing the proportion, when indeed there are several ways to set up the proportion for the problem.

Again, there is nothing wrong about the simple or compound proportion rules as provided by Greenleaf. The issue is that the rules are somewhat convoluted and constraining. If a student doesn’t learn this algorithm and follow it precisely, the likelihood is that the correct answer won’t be found. There are a lot of words referring to the terms and judgements that a student must make about which terms go where in the proportion. Further, look at what happens with a compound proportion problem, again from Greenleaf (#67, page 236):

“If 12 men in 15 days can build a wall 30 feet long, 6 feet high, and 3 feet thick, working 12 hours a day, in what time will 30 men build a wall 300 feet long, 8 feet high, and 6 feet thick, working 8 hours a day?”

Now, where does a student begin sorting through all this information if they use the rule above for simple proportion? What’s the “nature of the question”? For example, the rule states “…make the larger of the remaining numbers …” and how is a student to know which number is to be selected? I can visualize the instructor explaining in excruciating terms how all this works. Again, it’s not impossible to apply the rules as stated in 1881 but I urge you to look at Sheldon’s Compound Proportions in this blog and see how much more direct the rule is by framing information as cause and effect.

Briefly, Sheldon’s 1886 statement of the procedure:

“The solution of every example in proportion proceeds on the assumption that effects are in the same ratio as the causes that produce them. Every proportion is the comparison of two causes and two effects. In the method known as Cause and Effect, the causes form one ratio, and the effects the other. The first cause and the first effect are antecedents; the second cause and second effect consequents.”

Notice the simplicity of identifying cause and effect and then the causes forming one ratio and the effects the other. The words” antecedents” and “consequents” could be updated to 1st and 3rd term and 2nd and 4th term, respectively.

Taking the above compound problem the 1st causes are 12 men, 15 days 12 hours a day and the 1st effect is to build the wall 30 feet long, 6 feet high, and 3 feet thick. The 2nd causes are 30 men working 8 hours a day and the 2nd effect is to build a wall 300 feet long, 8 feet high, and 6 feet thick. You are to find “…in what time…” which is a 2nd cause. There is a shortcut that can be used but let me show you – in what I call slow-motion-math – one way to make sure the terms get placed correctly. I typically use the labels and then replace it with the values (for a lot of different types of problems, not just compound proportions). The proportion following Sheldon’s procedure is:

Causes                     Effects

1st       men, days, hours         length, height, thickness

2nd       men, x, hours               length, height, thickness

I used “x” for days in the second cause. If the numbers are substituted, we have:

12•15•12 = 30•6•3
30•x•8     300•8•6

Cross-multiply and divide, solving for x and the answer is 240.

Again, a detailed description of the “cause and effect” is in Sheldon’s Compound Proportions in this blog.

The essence of this posting is to demonstrate the importance of a well thought-out procedure expressed in easily understood language. If you are an instructor, you likely have done this kind of “simplifying” of the algorithm because as stated in the text, it seemed too fussy for students to follow. Not every algorithm can be simplified but I believe it’s an instructor’s responsibility to make math more accessible to students by removing the fog of awkwardly phrased rules and algorithms. Give it a try.

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