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.

Posts Tagged ‘remedial/developmental math’

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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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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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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Unequations Buzz

Posted by mark schwartz on August 11, 2016

Introduction

Had a thought. Simple one-variable 1st degree equations, by definition, state that there is a bunch of stuff “here” that equals a bunch of stuff “there”. For example, 2(3x ̶ 1) = 5(x + 1). What is meant by “equal”? Looking at this equation, obviously the two bunches of stuff are not equal! What this statement means is that if you can find the value of the variable “x”, replace the “x’ with that value in both sides of the equation and evaluate both sides, the value on both sides of the equation sign will be equal. Thus, that’s why one solves for the value of “x”.

The fundamental rule for solving equations is “whatever you do to one side of the equation, you do to the other side.” This, in essence, maintains the equality. My thought was that rather than start with an equality and burp out the rules, start with an unequation and have students play with it to find out how to make it an equation. However, we won’t use paper and pencil; we’ll use poker chips.

The Story

In order to solve an equation of this order, students need to know a lot of stuff – identification of terms, order of operations, distributive law, the four basic operations with signed numbers and to verify their answer, substitution of a value for the unknown and of course the basic rule of “whatever you do to one side of the equation, you do to the other side.”

Solving unequations is simpler and is a kinesthetic, visual way to have students play with all those things which, in my view, expands their conception of equations. In many instances, I’ve seen students who know all the elements but somehow can’t blend them together to solve equations. Here’s how unequations work.

Each group of students (2 or 3 to a group) gets a handful of white poker chips and each chip has a positive on one side and a negative on the other. You can use other markers if you choose.

I ask them to put 1 to 5 chips in each pile but the total value in each pile can’t be the same. Two questions that always comes up are (1) can we put positives in one pile and negatives in the other and (2) can we put positives and negatives in the same pile? So, right away, they’re thinking about this exercise; they’re engaged. We have a discussion about this and although they don’t yet know what to do with these 2 piles (although some guess they’re equations), I let them determine what is allowable. So again, right away they “own” this exercise because they have determined what’s allowable. By the way, the discussion about what is allowable has many branches and typically includes a lot of “what if” banter. I just listen.

Once this is resolved, I then ask them to label the pile on the left “A” and the pile on the right “B”. This also is fun because there typically is someone who stacks the piles vertically rather than horizontally, so I simply say the pile furthest from you is A and the pile closest is B.

When everyone is ready I then ask them to do something to their pile A such that the total value in both piles is equal. This is also a fun point in the exercise for classes that allow positives in one pile and negatives in the other, but overall the buzz within each group again is one of the goals of this exercise. When this is done, I ask them to return to their original piles and then I ask them to do something to their pile B such that the total value in both piles is equal.

In both cases, I ask them if there was only 1 way to make the piles equal. Buzz, buzz again and the consensus was yes.

The next question to them was do something to both piles at the same time such that the total value in both piles is equal. This really generates buzz and questions to me, which I say I’ll answer later. The reason I won’t answer is that I want them to explore how this works. What they discover is that there is an unlimited number of ways to do this. For example, if A = 2 and B = 4, add 5 to A and 3 to B and both piles equal 7. There usually is an “aha” moment when they realize that as long as the difference between the two numbers added to A and B is 2, the total value will always be equal. Some also discover that unequal amounts can be subtracted from both piles and further that two numbers differing by 2 can result in an equal value in both piles. And there’s another “aha” moment – the total value in both piles can be negative if both were positive at first! And what’s more, zero is a valid value!

So, we played with these 3 options for a while and there was discussion all along about not only what was allowable but also the range of answers under the different conditions. Then we moved to equal piles to begin the exercise.

I ask them to adjust their piles so that there is an equal number in both piles. This then brings up the issue of their rule allowing positives in one pile and negatives in the other, if they allowed this. They realize they have to rule it out. But I then ask if they can have an equal value in both piles while having positives and negatives in the same pile. Can the total in both piles be positive or negative? Buzz, buzz and the conclusion is that it’s ok but this comes after a lot of discussion and this really gets them going about signed numbers. For example, if they are to have 3 positives in both piles to begin, they could put 4 positives and 1 negative, or 6 positives and 3 negatives or … here it goes again with an unlimited number of both as long as the total is 3.

So, I ask them to consider there beginning equal value in both piles and typically they make it simple – either all positive or all negative and they do this partly – they tell me – because they don’t know what I’m going to ask them to do. At this point, the equation question arises and I have to admit that we’re headed in that direction. After playing with this for a while, the class concludes (again) that there is an unlimited number of values that can be added or subtracted to maintain the inequality.

The next step is to give each group a few blue chips. What the group is asked to do is have one person look away of shut their eyes while the others in the group do two things: (1) set up two piles with an equal number of chips in both and (2) remove a certain number of chips from one of the piles and place a blue chip in that pile. In essence, create a simple equation. When they are done setting it up, the closed-eye person is to look at what they’ve done and answer the question: what must you replace the blue chip with in order to make the piles have equal value?

Do each of these exercises until the class seems comfortable with all the ideas that got buzzed about.

At this point, if you’d like to extend this 2-pile concept to work with introducing work with equations, see Chipping Away at Equations in this blog. It links up with this posting and together it gives students a different view of equations.

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Walk the Clock: It’s Fractions

Posted by mark schwartz on August 3, 2016

Introduction

For some reason, or perhaps reasons, fractions don’t make sense to many students. Despite the visual representations in text and/or the use of manipulatives such as Cuisenaire rods, fractions seem to remain a mystery to students. One day I asked all my basic math classes “What makes fractions so hard?” The overwhelming response focused on remembering the steps of the 4 basic operations. For them, operations with fractions seemed nothing more than trying to remember the steps to get the answer. Somehow, math instruction throughout elementary and secondary education led students to think not about what fractions mean and what they represent but rather to think about how to “solve the problem”. So, I played with an idea which seems to have provided a way for students to “see” fractions a little differently.

The Story

DON’T TELL STUDENTS THIS IS FRACTIONS! If someone asks if this is fractions, tell them it will be discussed after the activities are done. I’ve provided an idea on how to do this in the discussion section following the demonstration of the activity.

The students preferably will work in groups of 3 (or 2, depending on the size of the class). The minimum grouping is 2. Each group gets a magic marker and 12 paper plates. The students are to number the plates 1 to 12. The plates are to be placed on the ground as a clock face. This activity is best done outdoors but if not, move the desks and chairs to allow for each group to have enough space for one person to walk inside and one person to walk outside a clock face circle. If neither of these spaces are available, the plates can be cut down in size and placed on a table top. If it’s a rainy cold day and going outside is a bad idea, and if the curriculum allows and time allows, make it a “review” day and hope for sunny and warm tomorrow. This activity works indoors and on the desk top but outside is best; it’s more fun. If done on the desk top, 2 markers per group will be needed. These markers will be the “walkers” in the activity (this will be explained below).

Here’s how it works. Lay out the plates as a clock face. One of the people in the group will walk outside the circle (call this person the outsider); one person will walk inside the circle (call this person the insider); the third person will be the reader/recorder (call this person reader). Give each group a copy of the activities (a sample is below) which states what the insider, outsider and reader are to do. Once the groups have figured out who will do what, give a demonstration of what they are to do, using the 1st activity.

Using the first activity and using one group to demonstrate, note that both walkers will walk twice. Both walkers start at “12”. In each activity, the insider walks first and then the outsider. The first walk is done when the insider reaches “12”. The second walk for both starts where their first walk ended. The reader is to watch and verify that each walker takes the right number of steps (others in that group can help verify).

1st Activity: on the first walk, the insider walks 2 units while the outsider walks 1 unit. On the second walk, the insider walks 3 units while the outsider walks 1 unit. The reader will note “outsider.insider”. In this activity, the record should show 10.12.

If there is confusion about the walking and/or the recording, just repeat the first activity. When everyone’s ready, move on to the next activities.

2nd Activity: on the first walk, the “insider” walks 4 units while the “outsider” walks 1 unit. On the second walk, the “insider” walks 6 units while the “outsider” walks 1 unit. The reader should note 5.12.

3rd Activity: on the first walk, the “insider” walks 2 units while the “outsider” walks 1 unit. On the second walk, the “insider” walks 3 units while the “outsider” walks 2 unit. The reader should note 14.12. (There should be questions on how to record this. Show students “military” time.

It’s important that if more activities are to be done, don’t allow students to do it. The reason: activities provided by students may result in a very time consuming set of walks and more critically, present a new issue to handle. For example, although subtraction of fractions can be done this way, I suggest not doing it. You could get a negative answer and you might want to avoid this. Just stick with one concept at a time; adding fractions (although they may not realize it). Given this, you might want to prepare and walk through a bunch of activities and be careful that none of them take too much time, yet enough time for the students to play and enjoy it.

Again, do not say anything about fractions at this point, but what has happened is that the problem 1/2 + 1/3 has been done. The record “outsider.insider” is 10.12 , or in reduced fractional notation is 5/6. Most likely, someone has noticed that the insider always has a value of 12. You sort of have to weasel your way around this and don’t yet call it a common denominator.

A Little Discussion. After these activities, you can transition to presenting fractions as you usually do. But, here’s one idea to consider in talking to students about how this activity demonstrates addition of fractions. What is seen and used but not referenced is the common denominator of 12. This explains why the insider’s walking the line twice isn’t counted twice. In the problem 2/3 + 3/4 , the denominator could be any multiple of 12 but in this case since it is 12 and you know it, don’t count it twice. Students may balk at this idea but it can be explained further. The insider always walks the line twice but always restarts the 2nd walk at “12”, while the outsider restarts the 2nd walk where the first walk ended so the insider’s walks aren’t added, rather they simply repeat.

Also not seen is the addition, but it occurs in the outside walk when the second walk starts where the first walk ended. The outside walker’s position at the end of the first walk is added to the beginning of the second walk. Please note that using this method for a problem such as 1/2 + 1/3 would give the answer 10/12, not 5/6, so clearly reducing fractions has to be addressed before this activity. Further, you might question how to get from this activity to the “rules” for addition and subtraction, but that’s not the point, although it can be seen because both fractions in this example, were converted to equivalent fractions with a denominator of 12, although in this case and others, it wouldn’t necessarily be the lowest common denominator. This again, could create a teaching moment, discussing the issue of common denominator versus lowest common denominator.

I suggest that different sets of students get a chance to walk the line. In fact, teams of students could do it; two walk and the others verify that their walking is accurate. Further, the point at which the transition from this activity to the traditional fraction work is to be made is a matter of how the class is collectively responding. In some instances, students caught on and realized that this was adding fractions. But even if they caught on, I still had them walk through all the activities. In several classes, students wanted more exercises. I think it was because it was a nice warm day. It’s a judgment call.

One more thing. Recall that the purpose of this activity is to give a visual and kinesthetic sense to the “rules” and it does seem to have a positive effect on students. When we got to the traditional rules and procedures, I heard students talking about how it “matched up” with what they were doing outside. Play with it.

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Metric to Metric Conversion: Ultimately, it’s a Proportion!

Posted by mark schwartz on July 27, 2016

Introduction

This discussion came about because one student in one class simply asked “Why does this work”? He was referring to the procedure for converting metric units to other metric units, for example, “how many centimeters are there in 10 kilometers?” He could see that the “moving the decimal point” procedure worked but he kept insisting that there must be more to it; that somehow someone had figured this out and he wanted to know how it had been figured out. I had no clear answer to this and told him (and the class) that I would think about it. What I came up with isn’t necessarily the reality of the derivation of that procedure, but it did start with something that we had already discussed in class – proportions – and he was willing to accept this as a demonstration of why it works but wasn’t about to consider it a true explanation. Loved this guy!

The Story

In Colaw and Ellwood’s 1900 School Arithmetic: Advanced Book (page 252) is a discussion of the metric system. Among other interesting things, they note that kilo hecta and deka are Greek, while deci, centi, milli are Latin. In reading through their discussion, I got to thinking about how they, as we do today, convert one metric unit to another: a 7-point scale and simply “move the decimal point” … but some of their commentary made me think about how and why this 7-point scale works.

If asked how many decimeters are in .04 kilometers, one has a variety of strategies to use. If the 7-point scale (kilo hecta, deka, unit, deci, centi, milli) is known, one can write .04 at the kilo point on the scale and then visualize moving from kilo to deci, which would give a move of 4 places to the right. If the decimal point is moved four places to the right, this shows that .04 kilos is 400 decis. Typically, students are accepting of this ‘shortcut’ because it is much more manageable than other systems. But, the question was “why does it work?”

I believe the underpinning for the move-the-decimal method is to do the problem by first converting all the units to the amount at each point on the scale that equals one unit. This by no means is a rigid mathematical derivation but rather a way of demonstrating the relationships using a previously studied math relationship, namely proportions.

The traditional 7-point scale looks like this:

Kilo                 hecta                deka                unit                  deci                 centi                milli

1000             100                  10                    1                      1/10                 1/100               1/1000

This scale shows the number of units in a named place-value. “Kilo” means 1000 units; “deci” means 1/10 of a unit, etc.. But let’s ask the question from the point of view of the unit: how many kilos would it take to make a unit? How many decis would it take to make a unit, etc.?

Here’s how the “unitized” 7-point scale would look:

Kilo                 hecta                deka                unit                  deci                 centi                milli

1/1000          1/100                 1/10                 1                      10                    100                  1000

It appears as though the scale has been reversed, and it has because we are viewing the information from the perspective of what it takes to make one unit. For example, it can be read as “1/1000 of a kilo equals 1 unit” or “10 decis equals 1 unit”, etc. The point of this is that all of the place-value names are now on the same scale and having them on the same scale permits one to establish proportions.

For example, on this scale 1/1000 of a kilo equals 10 decis because they both equal 1 unit. Another way of stating this relationship is to state that “1/1000 kilos is to 10 decis”, which is a phrase describing the first rate of two rates that would make up a proportion. What would be the second rate? The original problem was “how many decimeters are in .04 kilometers?”

In this case, being consistent with the idea in proportions that the numerators are all the same type of units and the denominators are all the same type of units, what is seen is the relationship of kilos to decis, is:

kilo  1/1000  =  .04  = 400 decis
deci    10        x

It is this proportional relationship which provides the basis for conversions from one metric unit to another, as long as the units used are those that “equate” them to 1. Students must be comfortable knowing how many ‘dekas’ it takes to make one unit (since a ‘deka’ is 10 units, it takes 1/10 of a ‘deka’ to equal a unit). This may seem counterintuitive since we typically say, for example, that a kilo is a thousand units, which is true but the focus here is with how many kilos it takes to make a unit.

Given this discussion of the two methods, it seems most likely that students would tend toward the ‘move the decimal point’ system. It doesn’t require any computation. But the point of presenting both of these it to bring out the reality that the ‘easier’ system is based on a proportional system. Just another example of the power of proportions based on an interested student’s insightful inquiry.

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Rephrase That Impossible Application Problem

Posted by mark schwartz on July 19, 2016

Introduction

As I was presenting a topic one day, a student said that what I was saying didn’t make any sense and could I please say it differently. My first reaction was to ask the class if it was true for them too; some agreed. It wasn’t a rude statement and I took the comment seriously and did rephrase what I said and asked if that made more sense and apparently I got it right. Then I got to thinking not only about that moment but other moments where what I was saying may not have made sense, but nobody bothered to stop me. As far as I can tell, it wasn’t the math content but the language I used to describe the content that bothered them. Thus the story that follows.

The Story

Question: Let’s say that the first city 4th of July fireworks I attended was in 2005. Since then, I attended the city fireworks every year including this year, 2016. How many fireworks have I attended?

Before considering the answer, consider if that question is the same as: how many years have I attended the city fireworks display on the fourth of July?

If your answer to the first question is 11, you’re wrong and as vague as the second question is, the answer is 11.

In both cases, which seems to be the same case, I suspect you got your answer by simple subtraction, 2016 – 2005 = 11. The thing to consider though is what exactly is being subtracted? Let’s bring this into a more manageable range, like 5 – 1. If you were to do this operation on a number line, you could put your finger on the five and move to the one, counting as you go and thus you would get 4. That four represents the number of movements from point 5 to point 1 on the number line. When you move from 5 to 4, you say “1”, in essence, “scaling” the distance between 5 and 4 as 1 unit, regardless of the actual distance. Given that it’s a number line, the distance between the points on the number line will all be the same. So, when we say 5 – 1, we are asking how many distances are there between 5 and 1. By the way, this distance analogy is similar to the idea of having 5 kittens and giving 1 away – how many kittens have you? In this case, it’s not distance, it’s kittens but conceptually it’s the same. We need not bother with scaling the number of kittens, because it’s a quantity not a distance, although some consider distance a quantity. As far as “how many years have I attended the fireworks?”, what’s being counted here is the number of years – an amount of time scaled rather than a distance. So, 2016 – 2015 is one, etc. as far as counting.

What’s the point? Remember the first question? To repeat: “Let’s say that the first city 4th of July fireworks I attended was in 2005. Since then, I attended the city fireworks every year including this year, 2016. How many fireworks have I attended?”

What is being counted here? Again, consider the number line. We’re not counting the distance between points on the number line rather were counting the number of points. The first question then has to be a subtraction plus one, which really is asking for the inclusive count.

You might say “so what?” to the difference between the first and second questions but looking at them as I did points out that there is a difference. The real issue here is the nature of asking questions in a math class. If we, as instructors, ask ambiguous questions, or questions which require students to reflect on the context of the information as well as the information in the question (and students don’t see the need to reflect on these issues) then we are, in a sense, misleading them and adding to their confusion about math. The context in this case is the words we instructors use.

I’ve seen this in questions in texts. We glibly accept the questions and answers at the end of the chapter and if some of those questions are questionable, we simply don’t assign them. But it’s not just the questions in texts. It’s how we state information, it’s how we use the language to structure questions and present concepts. The difference between the first and second question demonstrates this.

We should be attempting to be better at some precision in our questions and presentations because, like it or not, instructors are math role models for students. If we expect precision and accuracy from students, we should also expect that they can phrase good questions and it’s the instructor that establishes the idea of a well-phrased statement. It also seems it’s a critical component of being able to arrive at a correct answer to a problem. The caution to read “word” problems until you understand it is reasonable, but what if you never “understand” the problem? My thought is that students have to have license to and practice in rephrasing problems, without changing any of the relationships in the problem.

For example, when teaching percent using the percent proportion model (you can see how this is presented in the Percent Proportion posting in this blog), I point out to students that most percent problems can be rephrased. An example: A farmer sold 180 sheep, which represented 16% of all the sheep he had. How many sheep had he after the sale?

There are a lot of extra words in this problem, but only two numbers. I asked the class to rephrase this problem focusing on the relationship between the numbers. The students wrote all their attempts on the board, so that we could discuss the thinking that brought about their answer. What they produced, based on the percent proportion model that I presented to them, were two rephrased problem: First, “180 is 16% of what number of sheep?” to get the total number of sheep and then subtract the 180 he sold. The second was to see that if 180 sheep were subtracted from the total (written as x – 180), this would represent the number of sheep left and the percent left would be 84 (100 – 16). The rephrasing then would be “The number of sheep remaining is 84% of what number of sheep?” The result of this rephrased problem still is the total but again, simply subtract 180 sheep. If you want to play with it, the answer is 945. Even if the percent proportion model isn’t used, this rephrased problem is much more manageable.

But here’s how to set up both results using the percent proportion model.

180  16
 x   100      … solving, x = 1125 … total after sale = 945
x - 180  84
  x      100   … solving, x = 1125 … total after sale = 945

We practiced this rephrasing idea some more and I reminded them that they don’t have to rephrase every problem, but if the problem seems “impossible”, rephrase it.

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