Thoughts on Subtraction

Est. reading time: 20 min

Prologue

My first advice on teaching, when I first became a TA in my Master's program, was a 2-page list of Do's and Don'ts. I don't remember most of them, or even how many they were but I do remember the first three.

1. Know more than your students
2. Know a lot more than your students
3. Never tell your students everything you know.

This advice comes back to me approximately once a month, and its meaning always has subtle differences from the last time. I think it's extremely important advice. It's probably my most used benchmark for deciding how to participate in all sorts of discussions. The first point is a little "duh," but the second point makes you stop and meditate a little on the importance of the first. For anyone who has taught for more than 10 years, you get a sense of why 1 and 2 are important. But then, there's the importance of 3.  I thought 3 was a little silly and perhaps just a tad elitist for the first few years of teaching. But then, little by little, the power of 3 has crept up. I feel like, for me at least, there is a lot of wisdom in 3. Especially when it follows 1 and 2.

Here's an example.

Subtraction has been bothering me for some time now, and I've only recently started paying attention to this bother. I have been evolving the ways I've guided conversations around subtraction for the past few years, and I generally feel good about how the conversations go. However, I am always keenly aware of what parts of the conversation I'm leaving out (à la 3),  and I'm wondering what parts I should start adding in (yes, I note the pun of this sentence).

Pictures of Addition

Addition seems to have roughly only one meaning, in the context of, say, positive integers. Putting together two quantities. Take 2 + 7. Start with 2 apples and add 7 apples. Increase your apple quantity by 7 more apples. You can also start with 7 apples and add 2, or you can start with nothing and simultaneously grab a bag of 2 and a bag of 7. All of these concepts feel the same, largely because addition is commutative.

Here's a typical table of keywords [source]

The Addition words don't seem very different in concept. 

A typical picture (with apples replaced by frogs, because they're cuter) might look like

and nobody (I think) is going to argue that this is conceptually a different picture than the following.

A typical number line picture is

or you can just start at 7 and bump over 2

or you can start with 2 and then add 7. In all of these cases, though, whether you draw them or not, the implied "answer" is those 9 spaces you count from 0 to the tip of the last arrow.

Pictures of Subtraction

Subtraction, though, is different. Let's look at 9 – 2. For lack of more meaningful order, I'll just take keywords from the table above. I'll give scenarios and pictures. Each of these questions is answered by the operation 9 – 2.

Change

I started with 2 frogs, and then I ended up with 9 frogs. By how much did my frog quantity change?

or on the number line

Dissecting this, it seems that the underlying concept is a fill-in-the-blank addition equation.

Decreased by

I started with 9 frogs, and two of them hopped away. How many do I have left?

and the number line

This one doesn't seem conceptually connected to addition. If you twist your perspective slightly, you can see a related addition problem, but this is really an operation of subtraction.

Difference

I have 9 green frogs. My friend Jarra has 2 blue frogs. What's the difference in our frog populations?

The number line picture, in contrast to the previous ones, doesn't have a direction.

Now, I know that text books usually have some order specified, where "the difference of a and b" is always supposed to be translated as a – b, but I personally think that is a construct that was invented by math textbooks to make publication easier. When we use the word difference in a real sense, it either is assumed to be the positive, or it is accompanied by clarification of which thing is subtracted from the other. The word "difference" is not inherently signed. It's like a quantity-distance.

This is the beginning of the main conceptual difference (ahem) between addition and subtraction. Addition can always (always? somebody tell me I'm wrong here) be viewed as a changing of quantities, putting two things together; whereas subtraction can be viewed as a changing of a quantity (as in remove so much), or as a comparison of two quantities. This is important.

As for equations, this one might be either of 

 

or 

or, perhaps here it's not too awkward to relate to the fill-in-the-blank addition problem

 I think, maybe, the equations related to comparisons are not as rigid as those related to changing of quantities.

Fewer

I have 9 green frogs, and my friend Jen has 2 fewer frogs than I have (Jen's frogs are blue). How many frogs does Jen have?

While this looks the same number-line-wise as Decreased by, it is conceptually quite different. Largely because Decreased by involves a quantity-changing action, and Fewer is a comparison of two quantities. 

Again, the related equations could be any of the ones listed for Difference.

the rest of the keywords...

The rest in the list of keywords listed in the table don't (to me) offer any conceptual difference from the ones mentioned above. These are the main concepts that I see represented by subtraction, and plenty on their own to cause confusion.

I did a survey!

I think if someone were to stop me cold on the street and ask me, "What is subtracting?" I would say something like taking away. Like, if you have 9 frogs, and 2 of them hop away. I suspect that I'm among the majority on this, and I suspect that this is the notion that is most emphasized when we teach students about the concept of subtraction. While this is super-far from a random sample, I did get nearly 100 people to tell me whether they thought the pictures above depicted the calculation 9–2. (By the way, Thank You to all the people who took my survey and passed it along.) Let's see if they have confirmed my suspicions or not. It may or may not interest you to note that I wrote the entire portion above before looking at the results.

I asked the same question, "Does this picture show the operation 9 – 2?" for each picture. Here is a summary of the results. If you wish to see the full list of comments, you may do so here.

Image 1

Among the YES comments, many noted some version of "Originally there were 9 frogs, but 2 got crossed out." Unexpectedly and somewhat amusingly, a few people interpreted the X's in rather sinister ways. 

Both in the YES comments and in the KIND OF comments, people noted that students may need clarification on what the X's mean, including possible confusion because "the frogs are still there." There were also a few notes that the straight-line arrangement of the frogs made them difficult to count, and the picture might be improved by arranging them to aid subitizing.

I made a couple attempts to remedy these issues.

Despite my attempts for improving the image, I will note here that the context of this survey was a curiosity about what people are seeing as representing subtraction, and I would not attempt or advocate for just using these images to depict subtraction without some sort of supporting force (worksheet text, classroom conversation) to give it context. The need for context is present in this image, but it comes out much more forcefully in subsequent images.

Image 2

The YES comments generally mention comparison, and the most common explanation is "There are 2 fewer blue frogs than green frogs."

Among the NO comments, many said that it looks like addition, 9 + 7. There was also confusion about the fact that there were two different colored frogs. One unexpected result was that several people interpreted the left-facing frogs as "negative" frogs. Some made "zero pairs" by pairing up blue and green frogs, and used this to argue that it could be 9 – 7, but not 9 – 2. One person even commented, "If the blue frogs were red, I'd be inclined to say 'kind of.'" At this point, one of my thoughts is that the vast majority of the people taking this survey are math educators, and I wonder how much that colors their interpretation of the color and direction of the frogs. I wonder if a group of students would have similar responses. [Note: My intention in using left-facing frogs was simply to give a cue other than color that this was a different set of frogs, so these interpretations were quite a surprise to me.]

In the KIND OF comments, several people mentioned comparisons but felt that it was ambiguous whether it represented 9 – 2 or 9 – 7. This is not surprising to me, as I had noticed the potential for flexibility of representation in the Difference and Fewer keywords.

In all categories of comments, there was an echo of this image needing context. I couldn't agree more.

Image 3

The YES commenters mostly indicated that this was a comparison, or a difference, with some version of "There are 9 green frogs and 2 blue frogs; how many more green frogs are there than blue?" A couple people mentioned zero pairs here, again with the interpretation that the blue frogs are negative. One person even interpreted the blue frogs as dead frogs, but then wondered why the live frogs are still there. (I'm amused juxtaposing this comment with the zero pair comments.)

The most common comments in the NO and KIND OF categories were, it looks more like addition, why are there two different colors of frogs, and this needs context.

Image 4

The most common interpretation is that this is a "missing addend" problem. (This I noted in Change keyword.) Something that surprises me here is that people used this to explain their thoughts in Every category. Meaning, I got "YES, it's just 2 + ___ = 9, which is the same as 9 – 2" as well as "NO, this is 2 + ___ = 9, which is different from 9 – 2", and "KIND OF, it looks like 2 + ___ = 9, which is not exactly what you're asking."

And, of course, so many said "needs context." 

Related to the context comment, many didn't know how to interpret the boxes. While this doesn't really surprise me, it caused me to pause and think about how easily people were able to manufacture a meaningful context for the first image, but railed against lack of context for the others. And I'm wondering, is there something more natural about the "take away" notion of subtraction for us, or is it an indication of how intensely we are trained to look for "take away" pictures of subtraction? I have a conjecture that it is the latter.

Image 5

Overwhelmingly, the interpretation among the comments is some version of "take away 2 from 9" or "move left 2 spaces from 9." Many seemed relieved to see the clarity and simplicity of a number line. Until...

Image 6

Ha! Number line is no longer so simple and clear!

The YES's (for the most part) interpreted the picture as distance or difference between 2 and 9.

The NO's and KIND OF's said it needed context; maybe it looked more like 9 – 7, but for that it'd be better to have arrows. Or, it looks more like a missing addend problem.

[Est. reading time to finish: 7 min]

What are the ways we use subtraction? What is important for students to understand about subtraction?

As is indicated in the keyword categorizations, we use subtraction for conceptually very distinct calculations.

I have long suspected, and the survey confirmed for me (albeit unscientifically), that people most easily see subtraction as Decrease, or "take-away." However, I would venture to guess that this is one of the least-used concepts among all (real-world) uses of subtraction. My guess would be that the most-often-used subtraction concepts are Change and Difference, and they are very likely the most important for us to help students master. Further, it seems to me that putting so much emphasis on the "take-away" concept of subtraction may hinder students' ability to see subtraction as a Difference or Change. As an example: a student who fully understands subtraction as giving a difference between two quantities (or a number-line distance) will have very little difficulty accepting that –5 – (–8) is one of 3 or –3, because –5 and –8 are 3 units apart on the number line. Every year, I do this song and dance about what it means to subtract a bigger number from a smaller number, or what it means to subtract a negative number. But maybe, if I could get students on board with me accepting subtraction as a difference, we could skip all of the adding and removing assets and debts analogies, and just look at number line distances. Maybe.

But here. I'm actually not so interested in dissecting the best ways to get students to perform subtraction problems at the basic arithmetic level. I'm concerned that students don't have sufficient conceptual grounding in subtraction in to be able to use it in contexts where it's actually used.

Mathematical uses of subtraction

To indicate change (i.e. Delta)

How has this quantity changed from the last time we measured it until now?

In my experience, many students in my algebra courses are unfamiliar with this notion, and it takes serious work to get them to accept it when we are, for example, calculating rate of change. Or, rather, they're cool with it when looking at a table of values (the temperature went down 9.5º from the first reading to the next), but they struggle to get a calculation for the "change in y-values" when looking at a graph.

Related: I've recently started teaching the point-slope form of linear equations as a concept of change.

In fact, this notion of change of input and output is prevalent in graph translations.

Aside: I used to avoid mentioning ∆ (Delta) in my beginning algebra classes, thinking it was a notion and a symbol from Calculus. But then once I decided to say the word in a way that was, like, "here's a word that we sometimes use," fully prepared to tell students they don't need to remember it. And a student piped up saying that she had used it in business, in the context "what are our deltas from previous quarter." That gave me the permission I needed to use it as an Actually Used English Word. So I do. In discussions leading up to slope, as we're noticing patterns of change, we use delta notation.

To indicate Distance

A huge notion of distance comes up in Calculus, Statistics, and all sorts of applied sciences. In Calculus, think of epsilon-delta stuff. "When x is within delta-distance of a, f(x) should be within epsilon-distance of f(a)." Out in the real world, there are all sorts of codes regulating how long or deep or wide something can be, and tolerances built into the code. These are measured as distance between two values, also known as the difference. More specifically the absolute value of the difference. In building codes, for example, the difference between the actual length of a thing and the regulation length of the thing must be less than the prescribed tolerance.

Here's an example from an Algebra class, where I think we should really be using the notion of distance, yet I believe it is largely (if not entirely) ignored. Absolute value compound inequalities.

Solve the inequality |3x – 12| < 4.

Here's a common solution:

Here is a proposed other solution:

One thing that I like about this approach is that it preserves the absolute value and allows for interpretation as a distance problem. The beginning problem says that the absolute value of something must be smaller than 4. This suggests that x must be in some finite interval. As we simplify, we see that x must be within 4/3 units of 4 in order for the original inequality to be satisfied. And look! That's exactly what you see in the graph! This, to me, seems like a relevant interpretation of the result, whereas the first solution feels just like a following of procedures, followed by an answer that has no meaning.

By the way, the topics that made me think of inequalities this way are both Calculus-based: epsilon-delta proofs, and radii of convergence of series. In both of these cases, both teaching the topics and remembering being a student, I've felt like I had to teach my students to see that |x – 4|<4/3 means that x is within 4/3 units of the number 4. And I feel like this notion is part of a working mathematician's or scientist's core vocabulary of difference. I think by the time that students get to calculating convergence of series, they should have a fluent mathematician's understanding of difference. I don't have any other examples on-hand, and this post is already waaay too long, but I am confident that this notion is used extensively in many other areas.

So, this is why I'm thinking about how to discuss this meaning with my Basic Math students (actually all of my students; Basic Math just happens to be what I'm teaching currently).

As an act of decrease

Given that most people seem to think of subtraction as an act of decrease ("take away"), I don't have much to say about it. I'm not at all worried that, if I start thinking about how to teach my students about subtraction as differences or as changes, they'll suddenly be lost if I ask them to take 15 lemons away from a pile of 22.

As an inverse operation to addition

This is the first I've mentioned this one, but it's pretty darn important. Solving equations. Undoing stuff. Seeing relationships between related equations, such as 2 + 7 = 9 and 9 – 2 = 7 and 9 – 7 = 2

and related but different:

Subtracting a number is equivalent to adding its opposite

This is, in fact, the mathematical definition of subtraction. Starting with Peano's Axioms that give us the semiring of Natural Numbers, and constructing the ring of Integers, then from there going to the field of Rational Numbers and completing that to the Reals and, if you wish, closing that up in Complex Numbers (and even extending to Quaternions). The formal algebraic definition of a – b is a + (–b). For most of my years of teaching, I have avoided telling my students this (à la teaching tenet 3) but I've recently started doing so. No, I don't tell them about Peano's Axioms and the formal construction of the Integers, and in fact these days I'm slightly averse to even discussing Real Numbers with them (but that's another conversation). But I've started using the definition of subtraction as adding the opposite as just a way to deal with getting past lots of stuff.

So, disclaimer, I teach adults. One of my basic assumptions is that all of my students have had lots of experience with addition and subtraction, and they already know subtraction as Decrease. They have also all made peace with negative numbers as a concept (I'm open to meeting an exception to this, but haven't yet). So I don't know what it's like teaching kids who are learning these concepts for the first time.

For the most part, it's a short step from talking about positive and negative numbers, to adding positive and negative numbers. And it's not at all challenging to convince them that 5 + (–3) is the same as 5 – 3, and a couple more examples, generally demonstrating on the number line. From there, once it feels like they're on-board with this, I will insert, "in fact, that is the formal mathematical definition of subtraction: Subtracting 3 is defined to be adding the opposite of 3." And that (not absent other analogies involving subtracting debt, mind you) is one of the tools I offer my students for approaching subtracting negative numbers. It doesn't come without confusion, but I think it's a useful tool nonetheless.

What's next?

Honestly, I don't know. I'm not accustomed to blogging, I think partly because my posts end up being This Long. I think I ramble too much, and it takes me HOURS to write a post, which means I can only afford to do this every 5 months. And, because I don't blog much, I don't get much practice in filtering my posts down. You see the cycle?  :) 

But it's good. These are things that have been simmering in my head for awhile now. The survey and this post were an effort to organize my thoughts and start making a plan for change, and that is in the works. I'd love to get feedback on any of these thoughts.

Right now I'm thinking I like the idea of continuing to use "subtracting is defined by adding the opposite," but I'd like to incorporate more examples that get students thinking about the difference thing, and the change thing.

Very very soon I'll be creating course materials that implement some of these changes. I'll be grateful for input on those as well.