Apples and oranges and the number line

You can add apples to apples and oranges to oranges, but you cannot add apples to oranges.

One of my teachers used this old chestnut to explain that we had to convert to the same units before adding similar quantities. It doesn’t make sense to add 2 and 6 to find the combined volume of 2 quarts and 6 gallons — before adding, you have to convert everything to quarts, or everything to gallons (or, perhaps, everything to litres).

Apple + apples = ?

Of course there are circumstances where it makes perfect sense to add apples to apples, but there are also a lot of situations where it doesn't.

What do you get when you add my PIN and my wife’s PIN? Or my brother's telephone number and my sister's telephone number,  or the grocery product numbers for tomatoes and cucumbers? It seldom makes sense to add numbers that are used as identification labels.

But even when the numbers are more than just ID numbers, it may still not make much sense to add them. Would you perform the following additions?

• Hours to travel by bus + hours to travel by car from Edmonton to Regina.
• Average pay in Alberta + average pay in Ontario.
• Johnnie's  Algebra grade + Tina's Algebra grade. 
• Life expectancy for a person born in 1990 + life expectancy for a person born in 2015.
• Current Montreal time + current Vancouver time. 
• Cruising altitude for flight 107 + cruising altitude for flight 108. 
• Sunrise time + sunset time for today in Saskatoon. 
• Number of voters in the 2011 election + number of voters in 2015 election. 

Apples − apples = oranges ?

The odd thing is that in each of the above cases, although it makes little sense to add the numbers, it does make sense to subtract them. Here's another example.

I’m at the corner of 105 avenue and 34 street. If I walk along the street from 105 avenue to 112 avenue, how far will I have to walk?  

You would probably answer "seven blocks." Here you are subtracting an avenue number from an avenue number and getting a distance expressed in blocks.

Even when the result of a subtraction uses the same language to express the units of the answer, the units are sometimes measuring something different. Consider the following:

The temperature in Calgary at 10 AM today.
The temperature in Calgary at 4 PM today.
The difference between these.

Although the three measurements are expressed as degrees Celsius, the first two have a different meaning than the last.

Apples + oranges = apples ?

And there are case where you can add apples to oranges. (There’s the useful observation that adding 2 apples to 3 oranges makes sense because it gives 5 things, but I’m not thinking of that.)

Dinner Time 

Q: What time is it?
A: 5 O'clock
Q: How long will it take to cook the roast?
A: 2 hours. 

Did you just add hours to O'clocks?

Number Lines

I imagine that most people, when asked what 5 + 2 means, conjure up what I would call an aggregate model, like a collection of 5 tokens together with a collection of 2 tokens.

We tend to think of numbers as describing an aggregate or assemblage — the number of marbles in a box, or the total weight of your luggage.

In the Dinner Time example, above, the numbers are not being used in this way. Instead, one number describes a position (time of day) and the other describes a movement (a duration of hours). If you were asked to explain to a child why 5 O'clock + 2 hours is 7 O'clock, perhaps you might draw something like one of these:

These are variations of the familiar number line¹, where numbers are depicted in the two different ways that numbers are often used:

1. as a position on the line, with positive numbers to the right of zero, negative numbers to the left, and
2. as a directed distance or movement (a vector), drawn as an arrow pointing left or right. The length of the arrow represents the magnitude of the number and the direction indicates the sign of the number (right = positive, left = negative ).

Addition

On the number line, addition is thought of as

current (or old) position + movement = new position,

as, for example, in

current bank balance + deposit = new balance.

With a number line, addition of 5 + 2 and 3 + (−7) is represented as:

For addition, the movement arrow is drawn with its tail at the old position. Its head indicates the new position.

Subtraction

By far the most common application of subtraction is to describe the movement that separates one position from another:

new position − old position = movement,

as in

Best before date − current date = days left before milk goes sour.

With a number line, here's one way to represent 2 − (−4) and 8 − 5:

As with addition, the arrow is drawn with its tail at the old position and its head at the new position.


Children typically learn subtraction as taking-away, which is demonstrated with an aggregate model by removing a certain number of tokens from a larger group and counting what's left. In fact, so strong is this interpretation of subtraction that the language often persists throughout the rest of one's life.

The take-away model of subtraction, can be described as

old position − movement = new position.

It is also possible to use a number line to model this interpretation of subtraction. Here's what 8 − 5 would look like (8 = old position, 5 = movement):

In this case, subtracting means to travelling backwards along the movement arrow to reach a new position.

The arrow is drawn with its head at the old position while its tail indicates the new position.

This is almost tragically different from the previous illustration for 8 − 5. Apart from my own kids, I've never taught arithmetic to children, but I cannot help but think that the difference will confuse a student. Moreover, with the position-minus-movement interpretation, the overwhelming temptation is to reverse the direction of the movement arrow, thereby conflating 8 − 5 and 8 + (−5), which may lead to further confusion.

So what?

Most people work with numbers in an abstract way: 8 − 5  has no meaning unless the 8 and 5 refer to something "real," yet we have no hesitation in saying that 8 − 5 is 3. We do this without reference to any "real" setting whatsoever, and we do not revert to using a model to do the computation. We have completely abstracted the operation of subtraction from real life.

However, it seems that we need to give meanings to numbers in order to learn even the most basic arithmetic facts. But when we do assign meanings to numbers, we encounter the strangeness described in this post, namely:

• Two numbers can measure exactly the same thing but it may be nonsensical to add them. 
• Two numbers can describe quite different concepts, and yet it may make sense to add or subtract them.  
• Two numbers that don’t make sense when added can make perfect sense when subtracted.

I can't recall being taught about this strangeness when I was a student, and I don't know if it is explicitly dealt with by today's teachers, but it is conceivable that it could be a source of confusion.

To some extent, the number line model deals with this strangeness, but I haven't seen it used to this end.

*****

¹ Some number line models use only arrows. Positions are replaced by arrows whose tails are at zero.

Marilyn Burns and the number 11.

Recently I read an interesting post by Marilyn Burns about a pleasant discovery she made. It niggled at my mind (in a good way). It was not the discovery itself that struck me, but rather the reservations she had about how she reported it. This is what she said:

What I’ve done is given an explanation that falls into the category of "teaching by telling," which I avoid in the classroom when I want students to "uncover" knowledge that’s based in understanding relationships.

She had written a book about the number 11 for her grandson, and in it she had mentioned that 11 could be expressed both as 6 + 5 and as 6² – 5².

Marilyn's Problem 

Sometime after completing the book, she became curious about whether the same thing worked for other pairs of numbers that differ by one, and she found that indeed it did: for example, 

7 + 6 = 13  and   7² – 6² = 13.

He question was about how to generalize this. She explained how she did it both graphically and algebraically, but she was uneasy about the "teaching by telling" approach that she used. This lingered in my mind for some days because it reminded me that some of us (math educators) have not yet sorted out the relationship between "exploring the math" and "direct instruction," or, if you wish, between "learning mathematical reasons" and "applying the math trick."

Moreover, her two approaches (plus another that I have added) reflect the increasing distance between math via “discovery” and math via “direct instruction.” I think of the former as not requiring an extensive math background, and the latter as depending on already acquired knowledge. Perhaps I am being naive about this, but let me explain:

Her graphical approach, via (1) above, is related to what I think of as being "discovery based." You don’t need a whole bunch of content knowledge to understand it. The 11 blue squares visually illustrate the difference in areas between a 6 x 6 square and a 5 x 5 square, and the process can be extended to show the difference between 7² and  6² and so on.

Her approach, via (2) above, is a very natural transposition of the problem to the algebraic domain: compare the sum of the two numbers with the difference of the squares of the two numbers.

A third approach, via (3) above, is also a transposition to the algebraic domain. It is clean, and it leads very quickly to a solution (because, for consecutive numbers a - b = 1),  But, to me, it is a lot less natural than (2), for it depends upon having memorized a particular algebraic fact and having it always at the ready. For this particular problem, using (3) covers up the thinking about the math involved. It verges on being a mathematical trick.

Or does it? Do I really believe that it is a trick?

Using (3) certainly bolsters the argument that content knowledge helps you do mathematics. No mathematician would ever deny that content knowledge is important. An awful lot of mathematics takes knowledge from one area and applies it elsewhere, and one hopes students learn how to do this. 

I’m not sure how students can acquire content knowledge and learn to apply it without some teaching by telling. Yet, "telling" may lead to mimicry rather than understanding. (That may explain why teaching problem solving is difficult! And that’s maybe why we get a zing when we do solve a problem.)

Now, I think (or rather, I hope) fewer and fewer people are still invested in the total primacy of teaching content over everything else. An episode from my past suggests that content knowledge alone is insufficient.

Jim's Problem

In high school, my friend Jim would occasionally bug me with puzzles that exposed my poor abilities with mental arithmetic. He was fond of asking me things like 

Without using a pencil and paper, what’s 48 squared?  or  What’s 61 squared? 

(BTW, I existed as an entity long before calculators did, so that’s the pencil and paper reference.) 

But before I could even get started he would tell me the answer. 

48 squared is 2304.   61 squared is 3721.

How could he get it so fast? When pressed, he explained how he did it:

48 squared has got to be close to 50 times 46.  (I just added and subtracted 2.) The algebra goes like this:
50 × 46 = (48 + 2)(48 - 2) = 48²  -  4.
And 50 × 46 is easily seen to be 2300, so 48 squared is  2304.

Somehow, I didn’t see the trick until he showed it to me, even though by that time in my life factoring a² – b² had become an automatic reflex. If you're reading this, I’ll bet it’s an automatic reflex for you as well.

I find it interesting that both Marilyn’s problem and Jim’s problem can be resolved use exactly the same content knowledge. For Marilyn’s problem, that knowledge led me immediately to an answer, yet for Jim’s problem I did not make the connection.

Are you like me, or did you immediately see the connection with Jim’s problem?