The arithmetic operations from elementary school: add, subtract, multiply, divide – these are usually conceived as two-input, one-output operations.  There are some more you learn in secondary school: exponentiation, and perhaps the min and max operators we looked at in the previous post.  Interestingly, most of the new operations you learn in secondary school are single-input types: absolute value, square, square root; and later on sin, cos, tan and other trigonometric functions, and log.  Occasionally, secondary school students run into situations where there are more than two inputs (e.g. to find a slope of a line going through two given points) but these rarely rise to the level of standardized notation.

Even things like the quadratic formula, which looks for the value of for which , and where the numbers , and could be regarded as the inputs, is rarely seen or treated as an operator.  Instead, it is treated as a series of one-offs: here’s an equation, solve it.  Here’s another equation, solve it too.

There is a vocabulary to distinguish these kinds of operators and functions, though the vocabulary is more firmly established in computer science than it is in mathematics.  The one-in, one-out kind are called unary operators and unary functions; the two-in, one-out kind are called binary operators and binary functions.  Functions with three values coming in and one coming out would be called ternary functions, and so on.  Sometimes you see the notation “n-ary function” for a function that has n values coming in (for some unspecified n) and one value coming out.

Some mathematicians might scoff at all this, and bring up that in some sense, all operators and all functions are of the one-in, one-out type.  What comes in, no matter what it is, is called the domain, and what comes out, whatever it is, is called the range.  Neither domain and range have to be numbers.  They can be whatever they need to be.

For example, subtraction can be seen as a single-in, single-out function where the domain (what comes in) consists of pairs of numbers.  The range (what comes out) consists of single numbers.  The image below compares the way we normally think of subtraction, on the left, and subtraction as an operator on a pair of numbers, on the right.

Though the view on the right may seem artificial, it seems less so if you see what we have to do to the diagram on the left to keep straight what input is what.

As another example, sort-by-due-date can be seen as a single-in, single-out operator where the domain consists of a bag of bills and the range consists of a linear arrangement  of these same bills – a special arrangement where the bills that need to be paid soon are up front and the bills that need to be paid later are in the back.

In prior posts in this series, we’ve looked at state machines, where the operators operate on the state of the machine.   We’ve played with stopwatches and calculators as examples of such machines.  Sometimes the state of these machines consisted of a single number, but more commonly, the state consists of multiple numbers and other things.  For example, the state of the tab in your browser window includes a history of sites, so that the “back” operator has a way to get you back to the site you were before you clicked on a link.

In the next post, we’ll play with some operators that have interesting domains and interesting ranges.

To recap, a black box is something where we have access to the behavior but not the internal organization.  We can propose a model for a black box, and we can talk about the equivalence between the model and the black box.  We call the black box and the model equivalent if they show the same behavior, regardless whether the model accurately matches the internal organization of the black box.  In the figure, two different models are shown, on the right, and I suggested in the previous post that those two boxes at the right have identical behavior, and hence would be equivalent to each other, even though they have a different internal organization.

Since the claim about equivalence is a claim about behavior, it might be tempting to just test the two boxes with many input values, and watch the output values to see if they are indeed the same.

Yet how many inputs would you have to try, and in what order, to gain sufficient confidence that the two models behave the same?  Would a thousand be enough, or would you give up after five or six and consider it done?  Or, alternatively, you could just take my word for it, and save yourself even the minimal amount of work of checking five or six behaviors?

When we only look at some particular in-out behaviors, or grant authority to somebody and accept their take on what is going on, we are engaged in something other than mathematical reasoning.  We’ve given up on making sense of a situation for ourselves.  In this situation, we would be ignoring the information available to us, ignoring that we’ve been offered a glance inside of the box.  This is akin to playing poker without looking at your own cards – you might still luck out and win, but your odds of winning have gone way way down.

If we have the luxury of knowing what is inside of the box in model 1, we can use that info to see how the insides drive its behavior.  There are many different ways to approach this.  We might use the common language of algebra to describe it, and say that if “x” goes in, then “2x+1″ comes out.  Or we might simply note that whenever we increase the value of what goes in by one, the value coming out of the “double” box will have gone up by two, and therefore the value coming out of the “add one” box will also have gone up by two.  The upshot of all of this is that if “in” goes up by one, then “out” goes up by two.  If we can agree that this is true for both boxes, and if we can further agree that both boxes have identical outputs for one identical input, then they must always behave the same, or would they?

As our starting point, I’d like us to imagine a small device with a single red button, and a screen that shows a single digit.  When you receive it, the device shows “3″ on the screen, and nothing further happens until you push the button.

The screen then shows “4″ on the screen, and the “4″ stays there until your patience runs out and you push the button again.  After playing with this device for several days, you’ve observed that the screen cycles through the single digits as follows, and it has kept up its repetitive behavior for that entire time:

Can we tell what the device does and how it works?  We have seen its outside, we’ve been able to shake it and rattle it, and listen for any signs of a clock or a rodent inside.  Probably more importantly, we’ve played with it and observed its behavior, which we characterized in the diagram above.

Just as important, but trickier: you probably have formed a model for what’s inside of the device.  The model may be detailed enough so that a device could be built from your model, or it may be a simple sketch or a simple mental image with lots of details lacking.  It is easier to predict that you have in fact formed some kind of a model of what the device is like than to predict what your model looks like.  There are many different models that fit the behavior observed.

For right now, I’d like to focus on the button and on what it does.  You might say that it makes a “4″ appear on the screen, but that is only true in certain situations.  Since the device was showing a “3″ when you got it, the first time you pushed the button, it did make a “4″ appear.  Also, later, when another “3″ is showing, when you pushed the button again a “4″ appeared.  Yet in other situations, e.g. when a “0″ is showing, pushing the button does not make a “4″ appear.

So the description of what pushing the button does is not as straightforward as saying it makes a “4″ appear – and yet clearly the button does something, regardless of what particular digit is currently showing, and there are different ways to characterize what it does.  We can talk about the button as doing something as part of the device, as an operator acting on the current state of the device.  How we talk about the operator, and what language we use for that, will depend on our model for the device.  In future posts, I will show three different models which yet all show the same outside behavior, each described in a different language.  And our view of the button as an operator will be colored accordingly.

We call two black boxes equivalent if they exhibit the very same behavior.  This doesn’t require them to be identical.  A simple example is the telephone.  Let’s say I’ve been calling you every week for years, and I do the same today.  I dial the number, and I expect you to pick it up and talk.  Unbeknownst to me, you got a new phone yesterday, one with new-fangled bells and whistles: it plays music, takes pictures, and files your taxes.  Yet those differences, so important to you, don’t affect my call to you at all.  You still pick up and we still talk, and unless you tell me about your new phone, I would be none the wiser.  From my end, your phone behaves exactly the same as it did before.  My phone can still “talk” to your phone, and I can still talk to you.  It wasn’t necessary for me to upgrade my phone at the same time you upgraded yours.  For a phone to properly connect to the telephone network, it needs to behave in a particular way.  But if it does, it can be yellow or green, play Bach or Beatles for its ring tone, take pictures, be wired or wireless, have a headset or a speaker phone, fit in your pocket or hang from the wall.  The telephone network itself, the network to which telephones connect, has changed gradually: it used to only respond to the clicks of a rotary dial, then for a long time would accept either rotary dials or push button tones, and now many regions no longer support rotary dials.  Rotary dials and push button tones are an example of phones that are not equivalent: they behave differently in ways that “count” as far as the telephone network is concerned.  Land lines and cell phones (mobile phones) are likewise not equivalent, yet as long as the cell phone is within the reception area of the towers, the person on the other side of the telephone might not necessarily notice the difference.

Just like we can compare two black boxes and see if they have the same behavior, we can compare a black box and a model.  Imagine we had a black box and tried to figure out what the organization inside is like.  In doing so we might come up with a possible model for what the box is like.  The model might explain the behavior of the black box or it might give us a way to replicate the box.  If the model’s specification is too vague, none if this will work, but if it is precise enough, we can look at the functional match between the black box and the model.

For an example, let’s look at a particular black box with one input and one output.  Whenever you put “1″ on the input, you get “3″ on the output, whenever you put “5″ on the input, you get “11″ on the output, and the following table records the results of the experiment:

On the left, I show the black box, and next to it a table of its behavior, to the extent we tested it in our experiment.  To the right of the table is a model.  In it, we figure that if we have a box that double the number fed in to it, and then feed its output to another box that adds one, we get a box that matches the behavior in the table.  Is this model equivalent to the black box?  Truth is, we don’t know.  We haven’t done enough experiments to be confident that the pattern holds beyond the four values we established in the table.  The model represents a generalization of what has been observed, it represents a hypothesis.  It is a real question whether any amount of experimenting would ever establish the equivalence between the black box and the model – and this is a question coming back to at greater length.  Right now, I’m more interested in comparing this model – the one with the “double” box inside of it – with the model shown on the right.  This model, too, matches the behavior laid out in the table.  I am confident that I can make a much stronger claim than that – that the two models will show identical behavior no matter what values are put on the input.  I suggest that it is possible to show equivalence between the two models without relying on a large amount of experimentation.  To do so (and I won’t do this in this post) requires a kind of thinking and a kind of making sense that’s of a very different nature.  It is this kind of thinking that we usually call mathematical thinking.

The “coin purse” approach for representing number has its advantages and disadvantages.  The most obvious advantage is the ease of addition.  To add a number represented by a pile of coins to another number represented by a pile of coins, I simply join to the two piles of coins.  Done -  I am now left with a pile of coins representing the sum of the two numbers!  If this seems like cheating – it really isn’t.  In the world of the coin purse representation, the question “how much is in the coin purse?” is as nonsensical a question as “how much is 10?” in our normal way of representing number.  The answer to “how much is in the coin purse?” in the world of the coin purse representation is  simply showing the coin purse and saying “this much”.  However, this whole tale points to a disadvantage of the coin purse representation: the conversion to our normal representation takes some work, and takes some time.  However, the primary use for wallets and coin purses is when paying in stores, and there the question isn’t usually “how much money is in your coin purse?” but something more like “do you have two pennies?”.  Another common use for coin purses is when dealing with soda machines or the like.  Then the question is: “do you have enough quarters?”  All of these are easy to answer even without knowing the total make up of what’s in your coin purse.  Increasingly, the major use of my coin purse has become accepting coins in change – and not payment at all.  I pay with bills, and get coins back, which are put in the coin purse without any counting whatsoever.  When coin purse gets full, it is emptied into the coin jar, and once a year or so I find a way to exchange the stuff in the coin jar for bills, which then go into the wallet.
For the kinds of uses that wallets and coin purses are most often used for, that representation of number is quite successful.

The main theme here is that the suitability of a particular representation is very related to what the representation is being used for.  In rare cases, a representation can be so well suited for a particular use that we are completely willing to put up with the conversion into and out of that representation from the more familiar representations.  (A simple example:  if I ask you how much 34 x 1376 is, you may be completely willing to convert the “34″ from a representation on paper to one inside of a calculator by pressing the “3″ and the “4″ button, etc.  You would judge the conversion worth it because the calculator is so much better suited to the job of multiplying large numbers than the sheet of paper is.)

As a final example in this blog entry, let’s examine the representation of number used in the picture below.  Imagine that you have a friend who counts cars at a particular intersection, to collect evidence that a traffic light might be needed.  You relieve her at the agreed-upon time, and she hands you this, saying “sorry – got to run.”  You notice that she has been doing the counting differently than you do.  You would have used tally marks.  How did she do it instead?  How does she represent the number of cars on her piece of paper?

You figure that as cars approach the intersection, she wrote down the number of cars she saw.  You figure that when three cars approached the intersection at the same time, she simply wrote down “3″ instead of “|||”.  At the top, on the left, you see such a “3″, and it is followed by a “2″, so you figure she then saw two cars approach the intersection.  And then one car, and another, and then three.  Or was it eleven cars and then three?  You then notice that the top left 3 and 2 are crossed out, and you see a “5″ below.  You surmise that she must have used periods of relative calm to replace some groups of numbers by their total.  The two “1″s have also been crossed out, and there is a “2″ below, so you figure your guess that the two “1″s represented two single cars rather than a wave of eleven cars is correct.  Below the “3 1 2 4″, which are crossed out, you find a “1″ and a “0″, and since 3+1+2+4=10, you figure that this “1 0″ represents ten.

So while you have been tally marking the cars that arrived since you took over, you think you have figured out her system, and you decide that it is a workable system.  It occurs to you that not only is it a system for marking cars as they arrive at the intersection, it also gives a representation of number.  In this system, number is represented as a bunch of smaller numbers, those not crossed out.  Like the coin purse, what matters is the total.  Unlike the coin purse, the pieces that make up the total don’t come in standard denominations.   Yet unlike the coin purse, I never need to worry about making change.  In the coin purse, when I need a quarter, and there is no quarter in the coin purse, I need to find somebody to give my two dimes and five pennies to, in return for the quarter.  In the car count situation, if I have an “8″ and a “7″ on my paper, I am free to cross them both out and replace them with a “9″ and a “6″, or a “10″ and a “5″, or a “15″, all according to my choosing.

Come to think of it, my assets are represented as a bunch of smaller numbers to be totaled up: the cash in my pockets, the coins in my coin jar, the balance in my checking account, the balance in my brokerage account – and this is even before we look at the less easily unlocked value in my house, my car, my collection of CDs and so on.  When I withdraw $60 from the ATM, I’ve got $60 more in my wallet, and $60 less in my checking account.  It nets out.  From a bird’s eye view, from the view of the representation of my assets, nothing has really changed.  It wouldn’t make sense for me to look at my wallet and exclaim that I was now richer by $60.  Nor would it make sense for me to look at my bank account balance and exclaim that I was now $60 poorer.  Until I spend the cash, I’m neither richer nor poorer.  Representing number as the total of a bunch of smaller numbers might actually be quite common and quite useful, even if I never actually total up these numbers.

If I write , then people the world over will agree that this means the multiplication of three quantities: the quantity indicated by , the quantity indicated by and again the quantity indicated by .   Within the same general scheme, we can yet find alternative ways to render this mathematical expression: or .

Certainly, stands as the standard – some of you may even consider the alternatives I gave as incorrect.  After all, you might say, the constant must come before the variable, and exponents should be used rather than repeated multiplication.  True, as far as it goes – but I don’t think it goes very far.  Among several equal renderings , , , , some are “more equal than others”, in the wonderful phrasing of George Orwell.  By writing the formula in the standard way, , others will recognize it immediately, and remember it as the formula for the area of a circle.  The other variants require more effort to recognize as the area of a circle.

Yet the historical fact of the spread of a particular system of notation needn’t blind us to the virtues of other kinds of representations, even if we don’t typically encounter them in math class in secondary schools.  It isn’t actually hard to imagine an alternate history in which a whole different system of representation would have come down to us, a system we then would think of as the true one.  As recent as a hundred years ago, we would still encounter peoples in jungles or in remote places like the highlands of New Guinea who had lived in isolation, with strange languages and strange cultures and strange civilizations.  Star Trek also got us interested in boldly going to seek new civilizations.  Such new civilizations may well know how to get the area of a circle, but there is no reason to assume their way to represent this would be .

Let’s imagine a civilization where the formula for the area of a circle would routinely be represented as follows:

Let’s be clear: I’m not asking you to look at this as a picture for the formula, I’m asking you to consider a civilization in which this would be the formula.  In this civilization, the basic building blocks for formulas are boxes where something goes in and something comes out.  If what happens in a box is simple enough, you simply state it (“multiply”), if what happens is more involved, you draw the box with smaller boxes inside that indicate what happens to produce the quantity that comes out.  You could also imagine that when people in this civilization jot down something quickly on scratch paper, they would cut corners and streamline it some, perhaps coming up with something like this:

The representation with boxes, as well as the streamlined version with arrows, shows quantities being obtained from other quantities through some orderly process.  It’s the same “orderliness” that underlies the notation but note how it is expressed in a completely different way.

Here is yet another way to represent these ideas, this time in a more verbal way, but still quite precise.

The area of a circle =  pi times s,
where pi is a constant, which is often given as 3.14 or 22/7, though each is an approximation,
where s is the area of a square that has r as its side:   s = r times r,
where r is the radius of the circle,
where the radius of the circle is the distance from the edge of the circle to its center.

This style of representation, based on “where clauses”, may seem cumbersome at first glance, yet it contains a lot of information.  It also maps onto a picture of the situation really well:

We could spend a lot of time on each of these representations, and showing how come each is in fact a legitimate and precise way to render the idea of the area of the circle.  In addition, we could look at the advantages and disadvantages of each.  Each pushes something into the foreground, relegating something else into the background.  Some may be more suitable for learners, some may be more suitable for experienced folks.  Some may be more suitable on a piece of scratch paper or a blackboard, some may be more suitable using a standard keyboard.

Of all these representational approaches, quite different from each other, one approach is the one we happen to have inherited.  The result is that, for most of us, we think of as the true representation, better than all the other ones.

In my experience with sixth and seventh graders, the representation with the boxes appears natural for them.  You show it, they start using it on their own.  The boxes don’t seem to occur to them as something that’s hard to learn – it seems to occur to them as something that doesn’t require learning at all.  It is interesting to watch them play with these and discover notions like “nesting” (boxes within boxes) and never having to stop to talk about parentheses or order of operations.  Still, I am not suggesting we change which representations we teach, and in what grades.  Rather, I’m suggesting that freeing yourself up from the single standard representation yields important benefits when working with students.  It is good when math students show a fluidity and flexibility in moving in and out of various representations, and coming up with their own.  Sometimes, such non-standard use of representations is discouraged in the classroom rather than celebrated.

]]> http://unlearningmath.com/2009/08/23/representations-formulas-and-some-alternatives/feed/ 0 kweetal Area of Circle Area of Circle, streamlined Area of Circle "where clauses" http://unlearningmath.com/2009/07/27/key-math-ideas-not-taught-in-school-invariants/ http://unlearningmath.com/2009/07/27/key-math-ideas-not-taught-in-school-invariants/#comments Tue, 28 Jul 2009 06:14:51 +0000 Bert Speelpenning http://unlearningmath.com/?p=1194 ]]> “Counting” involves a number of different mathematical ideas

If you watch really young children count, you may notice that they aren’t all doing the same thing.  For some children, counting “one-two-three” is done in very much the same way as singing a song, or as reciting the Pledge of Allegiance.  One word comes after another, in a predictable and repeatable way.  The one-two-three sequence isn’t yet linked to a notion of quantity.

For other kids, counting “one-two-three” is done the same way as reciting “eeny meeny miny moe” while pointing at one thing after another, more or less in unison.  These kids show understanding of some idea of correspondence, that the counting words have some relation to the things being counted.

When kids start to be able to count piles of blocks or stacks of pennies – that is, they are clear that the number they come up with is a number that tells you something important about the pile of blocks – there are significant variations in the degree of repeatability, organization, and certainty in the process of counting.  Some kids carefully count the blocks moving from left to right, others count in what looks to the observer to be a random order but somehow seem to be able to keep track visually as to which blocks have already been visited and which blocks have not.  Other kids move the blocks as they count them, yet others move their fingers to help them separate the blocks that have been counted from the blocks that haven’t yet been counted.

Why am I looking at counting rather than at something more mathematically sophisticated?  I actually think there is plenty of sophistication in the conglomeration of ideas that make up “counting”.  Looking at what kids do with it is one way to really get the subtlety of it.  As adults, we are so familiar with counting that it is hard to see all the assumptions that go into it, hard to see all the stuff about counting that we take for granted.

One of the ways to talk about all the stuff we take for granted about counting is to come up with some scenarios in which we recoil from trying to count at all.  A simple one is counting the bees in a swarm of bees.  Me, if I needed to do such a thing, I’d be more inclined to trap them all in a big plastic bag, and then weigh the bag.  From the weight of the bag (strictly speaking, the difference in the weight before and after) I could make a pretty shrewd estimate of the number of bees in the bag.  But counting them?  Not likely.
A trickier example is counting the bubbles after you pour yourself a glass of Coke.  But here we might dismiss the example as one where maybe there isn’t a real count that the process of counting could reveal.  For we all know that bubbles come and go faster than we can count.

Counting takes time

One of the irreducible facts about counting is that it takes time.  We go through stages, steps.  And something is happening in these steps, something mathematical.  The mathematics doesn’t just magically show up when we’re done, it is there along the way.  There’s something we’ve got, all the way along the way, even before we’re all the way done counting, something worth standing still and paying attention to.  If our way of counting is intentional, we expect to be making progress, to get closer to a final result we call the count.  While we’re counting, we are proceeding from just beginning, to being somewhere along the way, to being almost done, to being done.  Clear, then, something is changing along the way.  It is not hard for us to pay attention to what it is that is changing, it seems a rather natural thing to do.  Not so natural, but just as important, is to look at what it is that is staying the same, all along the way.  Along the way, something is staying invariant, something important to the process of counting.

Invariants during counting

Once you decide to look for things that stay the same during the process of counting, you won’t have much difficulty finding some.  A key idea in counting is that there is a pile of stuff that’s already counted, and a pile of stuff that hasn’t been counted yet.  How the piles are organized, and how you can tell where one pile ends and the other one begins, that depends on the details on how you do your counting.   I’ve given some examples of how kids count above.  The pile of stuff that’s already counted has a number associated with it, and this is a number important to keep track of.  It is the number that you hear children say out loud when they count.  The number they say out loud is the number that matches the number of items in the pile already counted.  This is the fundamental invariant of the act of counting, no matter the details of how you do your counting.  You may count by ones or by twos, from left to right or from right to left, you may move the things you count or simply move your finger.  Regardless of the details, you are keeping track of a pile or group that you consider already counted, and you have a number to go with it.  As you make progress in your process of counting, the pile of hasn’t been counted yet will shrink.  When that pile is all gone, the other pile will have all the stuff in it, and the number you’ve been keeping track of can now be promoted from intermediary result to final result.
(There are other invariants during the process of counting – you may have thought of some while we discussed this.  One is the total count – even though we don’t know what this number is (until we’re done), we assume there is such a number, and that it doesn’t change while counting.  If the things we’re counting are dying or breeding while we count, we can come up with a count alright, but we wouldn’t put much stock in being able to say what exactly it was that we counted.  We end up with a number, but without a clear idea of the pile of stuff that the number corresponds to.)

Other invariants

I discussed the process of counting, since we’re all familiar with it, and with a number of variations of it.  It provides us with a useful starting point from which to start looking for invariants.  Invariants are a useful thing to look for whenever the process of getting our result takes more than a single step.  Below is a picture for the standard algorithm for multiplying 34 x 12.  The final result is 408.  The 408 is obtained by adding 48 and 360.  Yet many people (kids as well as adults) who use this method have nevertheless no idea what the 48 and the 360 represent.  That wasn’t considered important.

The 48 represents 4 x 12.  In this, 12 is the top number, and 4 is the rightmost digit of 34.  By putting down 48 we’re saying essentially that we’ve now dealt with the 4, and the 30 is still to be dealt with.  When we put down 360, which represents 30 x 12, we are saying that both the 4 and the 30 of the 34 has been dealt with (and if 34 had any hundreds or thousands in it, we would have added more lines till all of those would be accounted for as well.)  Below is the same multiplication, but this time annotated.

The annotation is just for our benefit – I’m not suggesting that anybody should write their multiplications this way.  We start with having to do 34 x 12.  After the first line, we’ve got 4 x 12 of the 34 x 12 accounted for, and we’ve got 30 x 12 left to do.  After the second line, we’ve got another 30 x 12 accounted for, and we’ve got 0 x 12 left to do.  When we add the numbers on the left, 48 and 360, what we’re adding is 4 x 12 and 30 x 12, so we can see that 408 matches 34 x 12.  What is invariant in each step is that we have numbers that represent multiples of 12, and that more and more digits of 34 have been accounted for.  For us to be more precise in our formulation it would help to bring in a three- or four-digit bottom number, but I’ll spare you that.

What about invariants?

Looking for invariants in a computational process is something that can gain us clarity.  It isn’t necessarily a new way of doing things, but rather a different way to look at what it is we’re doing already.  When we’re doing something with many steps, we must be keeping track of some number of things while we move from step to step.  The numbers we’re saying out loud, or the partial results we’re writing down, they tell us something; they represent a truth about some part of the process we’ve already dealt with.  Articulating what truth those partial results represent, and finding and describing the invariants, turns out to be a very useful practice.

In a recent TED talk, Arthur Benjamin does exactly this, by suggesting that the apex of the arc may want to be Statistics rather than Calculus.  If I understand him correctly, he isn’t merely talking about the courses you take in college, he is talking about the material taught at lower levels that prepare people properly for Calculus or for Statistics.  His argument, roughly, is that no matter how important Calculus is as a cultural heritage, and no matter how important it is for college students in the sciences and engineering, very few others will use Calculus in daily life, and many more would benefit from a grounding in statistics and probability, as a way to deal with risk and uncertainty.

I’m not interested, at least not here, in discussing the details of Benjamin’s idea further.   My first reaction is two-fold: the first one is that it is great that somebody is doing this kind of thinking, though I predict that even such a relatively mild shift in how mathematics is taught (and seen) will undoubtedly provoke a massive amount of reaction arguing that the way we currently do things must not be altered lest Western Civilization collapse.  The second one is that Benjamin is looking at relatively late stages in the curriculum – too late, I think – given that vast masses of children will be permanently turned off from any kind of enjoyment of mathematics well before any fork in the road appears where one way leads to Calculus and the other leads to Statistics.

I am starting a new series, with this post, to look at important mathematical ideas that currently have no place in the math curriculum.  They aren’t necessarily complicated ideas, and the ideas aren’t necessarily alien to the current way of doing it.  They are mostly ideas whose relevance to the math curriculum has been largely unexamined.  The ideas that I will highlight mostly have their origin in the field of computer science, a branch of thinking that has revolutionized the way  we look at computation and algorithms over the last 50 years, and which has helped shape the computer revolution that has utterly changed the face of the world, yet has left almost no trace in the mathematics curriculum (other than in the arguments whether to allow children to use calculators when doing their math homework)!

The ideas I want to highlight in this series are interesting and fascinating in their own right, I think, and I don’t want to leave the impression that I have a fully-formed curriculum in mind in which those ideas would fit.  You may find it more useful to read the entries as background ideas aimed at teachers rather than content for a curriculum for children.

In the next few entries in this series we will be looking at the notion of invariants and transactions.  Both are, at bottom, very simple ideas, which have become refined and well-articulated in the world of computer science.  A number of well-known mathematical ideas and practices will look very different once viewed through this other perspective.

The way we denote money amounts is one of those where there appears to be widespread confusion.  An amount like $2.69 is easily read and interpreted as 2 dollars and 69 cents.  The “$” symbol announces the dollars, and the “.” symbol announces the cents.  This is a reading that is consistent with many other constructs: the symbols that announce can come before, in the middle, or after the number involved.  In the construct 2’3″, which you are supposed to read as 2 feet and 3 inches, the symbol ‘ comes after.  In the date 06/13/2009, the slash separates the month from the day, telling us the 06 is the month and the 13 is the day (following USA conventions), and the second slash separates the day from the year.  In fact, the leading zero is completely unnecessary, and I can write or 7/4/2009 or 7/04/2009 or 07/04/2009 interchangeably without confusion.

So when a child writes “2 dollars and 9 cents” as follows: $2.9  we know that this is wrong because the “.” really denotes a decimal point.  But how is the child to know this?  Unlike the case with dates, suddenly the leading zero is critical.   But it is only critical for the cents, not for the dollar part.

In the American system for denoting money values, there is a subtle clue the child gets when hearing “2 dollars and 69 cents”.  The clue is that we are talking about value, and not about currency.  When hearing “2 dollars and 69 pennies” the child knows we are talking about currency: 2 dollar bills, and sixty-nine penny coins.  In contrast, “2 dollars and 69 cents” may come as 2 dollar bills, 2 quarters, 2 nickels, and 4 pennies – or in many other ways.  There is no standard notation for 2 dollars bills, 2 quarters, 2 nickels and 4 pennies.  What this suggests is that the value is considered important, but not the make-up in terms of particular coins and bills.  This matches the convention that any number of coins adding up to 2 dollars and 69 cents will be accepted by the store.  Yet it doesn’t match the convention that Coca Cola machines take quarters but not pennies, and that machines that take bills rarely accept bills over $5, and rarely accept 2-dollar bills at all.

When the store rings you up and announces the total amount you are to pay, they present you with the value interface.  $2.69 can be paid in many different ways, and you expect the store to accept it all.  You give them Susan B Anthony dollar coins, Kennedy 50-cent pieces, and expect them to accept it, though perhaps not with a smile.  You expect to be able to pay them with a $20 bill and get change back.  Where the model breaks down is when you present them with a $1,000 bill (I have to admit I have never even seen one, but I’m told they are in circulation).  Conversely, if you present them with a $20 bill, they are allowed to give you the change as a huge stack of pennies, even if they will rarely do so.

It is fun to think about classifications of different kinds of quantities, and there isn’t a single way to achieve such a classification.  Those of you familiar with the subject will have noticed that I deviated from a well-known approach suggested by Stanley Smith Stevens in 1946.  Many of the discussions about this subject have been focused on statistics – for example, if I use a survey where the answer “1″ means “very unsatisfactory”, up to “5″ meaning “very satisfactory”, does it make sense to compute the mean (average) score of a bunch of these surveys?

My intentions for engaging in this exercise are somewhat different.  One, I am interested in the different mathematical reasoning that goes with different kinds of quantities, reasoning that is quite distinct even if the numbers used (e.g. “two”) are not.  What you can do with a “two” depends a lot on whether we’re talking about a scalar number or a key.  In some ways, the numbers take their meaning from the operations rather than the other way around.  More precisely, the operations take their meaning from the actions they represent in the context in which they are used.  In fact, what to you and me may sound like a single operation, like “minus”, may have quite different meanings depending on the context in which it is used.  We addressed this in a post on unary minus, but the idea applies just as much to ordinary subtraction (“binary minus”).  Ordinary subtraction has different meanings (though identical numerical results) when thought of as “take away”, as “missing addend” or as “distance”, and which one we think of most naturally depends on the situation.  As grown-ups, we’re have become so familiar with all three giving the same results that we are hard pressed to look at them as different operations.  We call them all subtraction, and we expect a single key on the calculator to stand for each of them, undistinguished.  For a child, there is nothing obvious or inevitable about this – it must all be learned.

Two, I’ve been wondering how to bring in some of the fundamental thinking and some of the fundamental ideas from computer science into the whole discussion about math learning.  There is half a century of re-thinking of mathematics that has been taking place in the world of computers and software, and almost none of that thinking has found its way into the thinking about mathematics we do in school.  Depending on how much computer programming you’ve done, you may be familiar with one or more of these terms: interface, data types, data abstraction, data hiding, records, classes, function prototype, implementation, class instance, inheritance.
There is a way of thinking about data, and about working with data, and about representing data in different ways, that comes together in the notion of a class – or at least this is what it is called in some common programming languages, such as Java or C# – that I think could be very usefully imported into the world of school mathematics.  I don’t have an easy way to talk about it yet – I haven’t articulated it carefully in ways that will make sense to educators and teachers of mathematics.  I don’t quite know where to start.  You shouldn’t need a computer science degree to make sense of this, and yet there seems to be a limit of how much I can say about them without bringing in computer stuff left and right.  I will attempt to talk about it in settings that are not computer-like, but I assume that some precision will be lost.

The main idea of a class is that of a carefully designed abstraction, one that leaves room for different kinds of implementation.  Let’s say I design a bank building, where customers walk in and stand in line for a counter, and behind the counter is a person who is authorized to perform certain transactions on the customer’s behalf.  The counter can be thought of as the interface, the boundary, between the customer and the bank.  Certain actions can be performed across the interface.  The counters provide the face of the bank to the customer.  Lots of the work that goes on in the bank will not be seen by the customer.  If I, as the customer, come in the bank to deposit a $20 bill into my checking account, I won’t typically see what happens to the $20 bill after I hand it to the teller.  I do see my new balance, increased by $20, and that tends to be the limit of my interest in the matter.  For example, if I were to come in a week later, and withdraw $20 from my checking account, I would expect them to hand me a $20 bill, but I could care less if it was the same $20 bill that I deposited.   Those are the kinds of details that the bank is free to implement in a variety of ways,  I just want to know for sure that I can withdraw all my money from my account, meaning I get back the kind of money I can spend in a store, in the amount that corresponds to my bank balance.
Almost all business have a similar distinction between their front office and back office.  Certain interactions are public, exposed, accessible from the outside, and they form the basis of my view of what a bank is.  It represents a contract of sorts, that the bank has with me, whether explicit and enforced by law, or implicit.  An example of an implicit contract would be the Google interface.  Google never promised me explicitly what would happen if I type “unlearning math” into their search engine.  But I do expect a page back with ten entries, sorted in some way that I trust is helpful to me, each of the entries representing pages that contain the words “unlearning” and “math” closely together.  How Google manages to deliver this is not my primary concern, and I trust that the details on how their search engine works have changed considerably since I first started to use them years ago.  I assume they have replaced their disk drives for newer models many times over, I assume their web crawling software has been redone many times over, I assume their web page sorting algorithms have been refined many times over, all behind the covers.  But for all I know, Google works by having a pool of really smart and really fast Marians the Librarians responding to my queries.
One of the key ideas behind all this is that things can be done in more than one way, and that it is essential to maintain some flexibility in the means and separate it carefully from the ends.  For the bank customer, the ultimate item is the stack of $20 bills, for the bank itself, cash and currency are a nuisance, a bother, an expense, compared with the low cost of a megabyte of data on a disk drive and the low cost of sending a megabyte of data over a wire from one place of the world to another.

For a first post on the notion of a class, and on how it may be a useful concept in school math, this will have to do; more to follow.