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Tuesday, October 23, 2012

In defense of the number line: Reform methods for teaching negatives fail on decimals, fractions ... and negatives

By Laurie H. Rogers

Because every time you use the Charged Particles Method to teach negatives, a brain cell commits suicide.

It’s simple to teach mathematical positives and negatives to a child. It’s been done successfully with the number line around the world, in private schools, homes, tutoring businesses and online. Unfortunately, many schools in America no longer teach the number line, don’t teach it to mastery, or they cloud any fledgling understanding of it by emphasizing other, less-effective methods.

First, I’ll explain the number line. Then I’ll show you what’s being emphasized in its place.

Traditional Math Method Used to Teach Negatives

The Number Line
A number line is a straight line with a series of real numbers listed at intervals. Typically, "zero" is a point in the middle, negative numbers are listed to the left of zero, and positive numbers are listed to the right of zero. Arrowheads are placed at each end to show that the line and numbers continue indefinitely. Each point is assumed to correspond to a real number, and each real number corresponds to a point. Like this:
(Comment: A number line is flexible and comprehensive. It can be drawn with integers, fractions, decimals, irrational numbers or a combination thereof. It can illustrate zero, positive and negative numbers, and small or large numbers. It provides an effective visual for adding and subtracting positive and negative numbers. It can be used to articulate changes in measurements, temperature, money, time or quantities. It’s clear, efficient and functional. It’s easy to draw, explain and understand.)
Naturally, people who love reform math prefer to not use the number line. Various alternate models are used that are not efficient and not comprehensive. Here are a few.

Reform Math Methods Used to Teach … Uh

These models (with myriad variations) are common to reform curricula. The examples illustrated below are quoted (with permission) from materials originating from two public entities: Henrico County Public Schools and the Georgia Department of Education. The wording is theirs. Please notice that the Georgia materials are said to be based on the Common Core State Standards.
Story Model
In this model, children are given a story that supposedly illustrates a mathematical calculation or equation.
Example 1: “If you have spent money you don't have (-5) and paid off part of it (+3), you still have a negative balance (-2) as a debt, or (-5) + 3 = (-2).”
Example 2: “Getting rid of a negative is a positive. For example: Johnny used to cheat, fight and swear. Then he stopped cheating and fighting. Now he only has 1 negative trait so (3 negative traits) - (2 negative traits) = (1 negative trait) or (-3) - (-2) = (-1)”
(Comment: Stories and problems do help to apply math skills that have already been learned, but - as a teaching method - the Story Model provides no real mathematical understanding. The story in Example 2 also attributes adverse characteristics to negatives. In mathematics, a negative is simply a lessening of a quantity, not an adverse characteristic.)
Charged Particles Model
In this model, imaginary items are added to replace items that are already there: “When using charged particles to subtract, 3 – (-4) for example, you begin with a picture of 3 positive particles.”
“Since there are no negative values to ‘take away,’ you must use the Identity Property of Addition to rename positive 3 as 3 + 0. This is represented by 4 pairs of positive and negative particles that are equivalent to 4 zeros.”
“Now that there are negative particles, you can ‘take away’ 4 negative particles. The modeled problem shows that the result of subtracting 4 negative particles is actually like adding 4 positive particles. The result is 7 positive particles. This is a great way to show why 3 – (-4) = 3 + 4 = 7.”
(Comment: How does a child make the leap from 3 particles to 11 particles? Where is the explanation of what's actually happening? As I was typing in the Charged Particles Method, my brain felt like it was melting. Brain cells began to give up and die. My daughter had to rescue me with chocolate.)
The Stack Model and The Row Model
In the Stack Model, students draw boxes on top of each other in “stacks” and then count them. In the Row Model, students draw boxes in “rows” and then count them. Subtraction for the Stack and Row methods means creating pairs, as in the Charged Particles Method, then “adding zeroes,” crossing out items, and redrawing boxes over and over. This figure illustrates the solution to 3 - (-4) = ?
(Comment: The children will draw a lot of boxes, but they will not come to understand negatives.)
The Postman Model
In this model, a story is provided that supposedly leads students through understanding negatives. Children are to act out parts using manipulatives and props. The concept can be done with other scenarios, but this example from Henrico County School District is based on a mail carrier:
“A postman only brings financial mail. Sometimes she brings bad news, e.g., a bill for $5 = –5. Sometimes she brings good news - a check for $5 = +5. If she brings both you get two pieces of paper but zero dollars. You always start a zero with a cash drawer full of matching checks and bills that equal zero dollars. So if she brings me two checks for $5, no sweat, she helps me by $10, answer = +10. Similarly if she brings me 2 checks for $5 the result is 2 • –5 = –10. Now here is the tricky part: –2 • + 5 = ? Well the – sign means takes away from me. But if we start at zero how can she take anything away? This is where the cash drawer of matching checks and bills saves us. We just take away 2 checks and are left with 2 bills to pay. –2 • + 5 = –10. Similarly, if she takes away our bills, she helps us and the money we would have used to pay the bill can now be spent on bubblegum. –2 • + 5 = +10.”I think the second “2 checks for $5” is actually supposed to be two bills for $5. Please also remember that young students are the intended audience. The story suggests using a bag filled with Monopoly money and paperclips. Students pretend to deliver mail, cash checks at a bank and pay bills. At some point, however, the mail carrier makes a mistake, and the story goes on to say: “If it was a check, that would be subtracting a positive. To get the check out of the bank, you would have to pay the bank (which would make you lose money). If it was a bill (taking away a negative), you can keep the money attached to the bill and give the bill back to the mail carrier. This would show that taking away a negative would give you more money.”(Comment: There is so much wrong with this example, the explanation and the reasoning behind it, it's hard to know where to begin. The process is complicated; the financial philosophy is suspect. And what does that second paragraph even mean?)

The Balloon Model
In the Balloon Model, things are moved up (added) and down (subtracted). This concept can be done with anything that moves up and down (airplane, elevator, swimmer, etc.), but in the Balloon Model, sand bags represent negatives, and air bags represent positives. This illustration shows -3 + 4 = 1.
(Comment: The Balloon Model is a vertical number line, but it labors to be more relevant by using a balloon (or elevator or airplane or the sea). Thus, the concepts don't match up with what happens in "real life." The biggest problem: "Up" and "down" is not equivalent to "more" and "less."

Balloon Model: In real life, subtracting a sandbag lessens weight, while adding a sandbag adds weight. In the model, however, subtracting a sandbag somehow makes a quantity larger, while adding a sandbag makes it smaller. In the Building and Airplane models, nothing is added or subtracted to make a quantity larger or smaller; the movement up and down is mechanically driven. In the Sea Model, you could say that blowing air out of one's lungs (subtracting air) makes a person sink, although the change in weight would be negligible. But how does one add air under water?

Where in these models is infinity? Where are fractions and decimals? Where is zero located? At ground level? If so, are negatives below the ground? Since when have balloons and airplanes flown, elevators descended, and swimmers paddled below the ground?)

The (Hey, let's call them) "Net Changes" model
In reform curriculum Investigations in Number, Data, and Space (aka TERC), students are taught to think of “net changes,” rather than addition and subtraction. Students use manipulatives to act out the "changes." They count things and keep tallies.

If you search the term “negatives” on the TERC Web site, you'll have a difficult time finding actual negatives, but games about thinking about negatives are explored. Actually, take a few minutes and skim through the 1st Edition units for TERC. This is the approach to K-6 math that has helped to kill off math proficiency in the United States.

(Comment: Looking through the TERC Web site, you can see the end of mathematics ... and possibly all of life as we know it. The school district in my city still uses this curriculum, despite my best efforts to change that.)

Each of the reform math models illustrated here is missing one or more critical concepts, such as zero, negatives, fractions, decimals, large numbers, and/or infinity. They don’t adequately explain negatives, or show that the larger the negative number, the smaller the quantity. They don’t handle all scenarios. Perhaps some are “fun” for a very short while; others are probably no fun at all. Time spent on these models wastes time, builds frustration and creates misunderstandings. Instead of clearly cementing concepts about addition, subtraction and negatives, the children are filling paper bags or talking about “negative” traits.

Without an understanding of the principles behind the number line, students can’t add and subtract negative numbers with proficiency. Without that capability, algebra, geometry and calculus will be beyond them.

I don’t know why so many people in public education seem not to respect the need for efficiency, effectiveness, sufficiency or student proficiency. Math is a tool, used in the "real world" to get a job done. Time is important; efficiency is vital; correct answers are critical. Those who can use math properly will be hired over those who can’t. There is no time in the “real world” to discover methods, struggle with basic math, or constantly ask a “partner” for help. But math reformers seem to think they have all the time in the world. To them, math isn’t about efficiency and correct answers; it’s about struggle, failing, striving and playing games. Many reformers truly believe that if the teaching was efficient, the lesson failed. Thus, they’re motivated to not just prefer the inefficient models, but to actually eliminate efficient models, to mock them and to label them as counterproductive.

Help your children gain a solid grounding in math by teaching them the traditional methods, such as long division, vertical multiplication and the number line. Traditional methods were developed and honed over thousands of years by very clever adults so that they would be efficient. More-efficient models will be developed, no doubt, but the reform models I've described are not better models.

Don’t let anyone convince you that efficiency and effectiveness in math are unnecessary or counterproductive. People who actually use math (outside of a K-12 classroom) don't believe that.

Please note: The information in this post is copyrighted. The proper citation is:
Rogers, L. (October 2012). “In defense of the number line: Reform methods for teaching negatives fail on decimals, fractions ... and negatives." Retrieved (date) from the Betrayed Web site:  /.

This article was republished October 25, 2012, on Education News:


Bruce Deitrick Price said...

I'm more and more thinking of these "reform math" people as sociopaths.

What other description makes more sense?

R. Craigen said...

Some very good points, Laurie. As a professional mathematician I'll reinforce that, in terms of where mathematics goes in advanced studies, the number line is the principal model and should be the basis for numerical thinking. And, as you say it provides the epitome of clarity, conciseness an efficiency.

The quotations you provide are appalling. Even though the models individually may have some merit it is obscured by these poorly articulated lessons.

I see no reason anyone should prefer these other models although many of them could be peripherally used to supplement or illustrate the use of the number line in specific settings. But they are mere flesh to hang on the essential skeletal structure that things like the number line provide, giving solidity and superstructure to one's knowledge.

R. Craigen said...

Let me add that, in my mind, there are two particularly useful concepts of number associated with the number line, which I'll call "position" and "displacement". Each has its purpose and each illustrates the meaning of signed numbers. (Also, both provide important precursors for later ideas and skills -- something else that "reformers" appear clueless about!)

"Position" is a number which says, in absolute terms, where you are on the line. In this conception each point has an a priori "address" that allows us to locate it absolutely as if in a well-organized filing system in which "units" are uniformly spaced "steps".

"Displacement" is a number which tells one how to get from one point to another -- an instruction for relocating relative to your current location, if you will.

If your current position is 3 and your displacement is 5 then your destination is 3+5=8 (another position). It is an interesting (and elementary) theorem that one arrives at the same destination starting at position 5 and "adding" displacement 3. Since displacement always uses a referent direction (positive), going in the other direction automatically represents negative numbers.

R. Craigen said...

Subtraction naturally fits in this schema. One way is through questions like "If I'm here then how do I get to there?" Normally we orient the line so that the "positive" direction is to the right, so 3 "steps" to the left may be denoted by a displacement of -3. A displacement of -3 from position 5 gives destination 5+(-3) = 2. We may use the shorthand 5-3 for this as well -- in fact, this is precisely the mathematical concept of "subtraction" as we use it in very advanced math, and the earlier students understand it the better.

What I am arguing here is that there is no reason writing a-b as a+(-b) should be treated as overly "abstract" (in the derogatory sense of being too divorced from tangibles to be understood easily) -- it is a simple concept derived direction from everyday experience.

It should be noted that position can be thought of as "displacement from 0". Thus, if you begin with a marked point called "0" and the concept of displacement, you can recover the concept of position (I'm not a primary grade teacher so don't know how this should be articulated for young 'uns but at some point it should become integrated into their concept of the number line. Thus "-3" is the "position" that you arrive at by a "displacement" of -3; that is, subtract 3 from 0 to get to -3, or 0-3 = -3.

Students who have this particular precursor "down pat" have a huge advantage over others when it comes to elementary linear algebra and vector geometry. I've noticed students getting weaker and weaker in this course, which probably stems from failure to emphasize the number line in early education.

In the terms I describe here this is probably inappropriate to teach to young children; I am using "adult" language. This is something that bothers me about the reform (I call it the "fuzzy") approach: they develop technical language for teachers and then appear to expect young children to learn this language. No. We can discuss "position" and "displacement" in these abstract terms but let us talk to children in ways that are natural for them.

Again I'll leave that task to people who actually teach small children, but I would love to see early-years classrooms with built-in number lines, on the floor or wall, both horizontal and vertical. Lines students can walk along and even perform elemental arithmetic with their bodies in this way ("Start at 4. Now move 3 in the positive direction: 4 + 3. Where are you? Now move 5 in the negative direction. Where are you now? Write these as a number sentences on the board ..."). I think if a master teacher used the number line in creative ways it would inject a lot of fun in the classroom. (A lot more than these silly, confusing alternative models, I'll suggest!)

Here's an idea: an addition slide rule. Mark a fixed reference "bar" and a sliding "bar" in the same uniform "number line scale". You add 3+5 by placing the 0 of the sliding bar over the 3 of the fixed bar and reading the number under the 5 on the sliding bar. Use the concepts of position and displacement (in appropriate language) to support understanding of the toy. If you've marked negative numbers, it also subtracts (even if you haven't, it is easy to devise a procedure, but I'd rather the whole number line is marked). Do it with two meter sticks to add (or subtract) numbers up to a total of 100.

Niki Hayes said...

Laurie, Your words here are perfect:

"...math reformers seem to think they have all the time in the world. To them, math isn’t about efficiency and correct answers; it’s about struggle, failing, striving and playing games... they’re motivated to not just prefer the inefficient models, but to actually eliminate efficient models, to mock them and to label them as counterproductive.

Thanks for a great article.

LaraM said...

Great article. It is so hard for me to get a handle on WHY someone would advocate for the types of convoluted questions that you cite above. What sort of ideologues are we dealing with? What can explain their behavior? Niki, given your experience (I've read about it), I'd love to hear your thoughts.

Niki Hayes said...

I realize that anecdotal evidence isn't always the best explanation but somtimes experience, old age, and hard-gained wisdom is pretty good stuff to go on.

What I'm about to say is something that I realized, after reading your question, that may be part of the answer to today's disaster in education.

About 10 years ago, I was publishing numerous articles on the New Horizons website in Seattle. Through time, I learned that those in charge were great believers in the New Age religion (or philosophy) and I was often out of synch with them. They were actually quite kind to me, however, compared to the nastiness I experienced with those affiliated with the Univ. of Washington School of Education. (They likened me to a Neanderthal.)

I learned that the New Age folks really did believe they had a huge advantage over the "ordinary" person because they saw themselves as extraordinarily intellectual and far more introspective and ethereal than most folks, especially those in America. Their goal was to bring all young learners into their world or "realm" of thinking with its nuances, abstractions, metacognitive expansions, and subsequent brilliant creativity. It was as though I was watching a Star Trek program where wonderful individuals were living in the clouds, far removed from the rabble on the earth below.

I had been accepted, I think, because I was a math teacher and everyone knows that math teachers are really, really smart people (snort, snort). And, I even enjoyed writing which meant I was creative, right?

The air of superiority has grown exponentially in schools of education as the Progressive/New Age philosophy has seeped into all crevices of the institution of education for the past 120 years (even before John Dewey "codified" the efforts). That air now holds the institution together as solidly as one would see with any well-built pyramid.

Can it be brought down? I don't think so. Can it be replaced? Yes, stone by stone.

Vain Saints said...

What's particularly risible about this is that real numbers are *defined* as distinct points on the number line. The number line is not a heuristic device or a pedogogical tool. It is, in real theoretical mathematics, the *definitional basis* for real numbers and real number operations. To ignore it in favor of stupid story time models is disgusting.

Anthony said...

Here is a comment which I had shared with Laurie earlier:

Just the other day, I was working with a middle school algebra student (not local) who used a "box and coin" method for solving linear functions with variables on each side. The model worked something like this:

Problem: 3x + 2 = 2x + 5

Solution: Draw 3 boxes (for the x term) on the left and 2 coins (for the constant term. Do the same on the right except with 2 boxes and 5 coins. Once the drawing is completed cross out one box on each side until you cannot do so any more. Do the same for the coins. There is your solution!


[ ] [ ] [ ] O O = [ ] [ ] O O O O O

[ ] O O = O O O O O

[ ] = O O O

All the problems done in class worked in this manner. All terms were positive integers. All solutions were positive integers. Each difference between the x terms was 1 as to get 1 box in the end.

However, the students homework had problems such as this:

(1) -1.2 + x = 4x + 5
(2) 5x = 2 + x
(3) -10 - 3x = 1.5x + 2.5

In my work, I have seem many confused children. I am not sure I have ever seen one so confused and understandable so. Unfortunately, the student's confidence in his educational experience did not increase when I told him "Sorry, that method is not going to be useful here". This model is similar to some methods used by SPS though they use colored chips (like Connect 4 pieces) for positive and negative.

On another note, elevators can technically go below ground. We do have basements after all. I actually feel the vertical number line works just as well as the horizontal one (perhaps minus silly unrealistic scenarios). It also is necessary when kids get to algebra and have the xy-plane. The construction of that typically begins with the number line. Whether we start with the y-axis (up and down) or the x-axis (left to right) is not really relevant. Most of the other models that I see are seriously flawed though and make it clear why our country is falling behind in math/science.

Doug1943 said...

What do people think about this idea, which involves using the number line to teach about adding and subtracting negative numbers?

I show the negative side of the number line as the 'mirror image' of the right (positive) side. I encourage my tutees to actually use the phrase 'left side numbers', since 'negative' has a, well, negative connotation.

Now, when we add a positive number -- no matter where we're starting from, no matter which side of zero we're on -- we're moving -- displacing -- to the right, to where the positive numbers get bigger. When we add a negative number, we're displacing to the left, to where the negative numbers get bigger (in magnitude).

Subtraction is just the opposite.

It's also necessary to teach that the word 'larger' in the context of negative and positive numbers usually means 'to the right of'.

These are not ideas which can simply be told to a student with their comprehension immediately folowing.

They've got to MEMORIZE (gasp!) the idea that 'negative numbers go off to the left, positive numbers go off to the right', and 'adding a positive number means going off to the right, adding a negative number means going off to the left' and 'subtracting a positive number means going off to the left; subtracting a negative number means going off to the right'.

And they must memorize, know parrot-fashion, the phrase 'larger than usually means to the right of'.

You've got to get them to know these ideas by heart, before you start working problems.

All the things about balloons and debts and submarines, if necessary, must come LATER. In THIS case, we must move from the abstract and general to the concrete and particular.

The problem with starting with particulars is that not all real-world problems can be mapped onto the number line. A debt is not necessarily the same as 'negative money' because you have a choice about whether or not to pay it -- if I owe someone $20 and I am paid $30, it's not at all necessarily true that I now have only $10.