In this series, we’ve done a review of mathematical notation, with an eye on how each notation helps or hinders student learning. In the prior posts, we started to explore function notation, and played with variations. I’m not done with that topic, but will interrupt that sequence for a post on scientific notation.

Let’s start with the following picture:

Each line in the picture represents a number, and one line is different from the next only in where the decimal point is placed. Some of the digits are grayed out. I’m using the grayed-out digits as a short hand for indicating equivalent ways of writing the same value. The number in the top row, for example, is most commonly written as .03412 (at least in the United States; in some countries, writing the 0 before the decimal point is standard – there it would be written as 0.03412), but the common form is not the only one, the number .03412 can be written as .034120 or .0341200 or 0000.03412000 etc. For our purposes here in this post, it is useful to think of the version with all the grayed-out stuff present as the “real” or the “full” version of the number, and think of the version without the grayed-out stuff as the “common” or “abbreviated” version of that same number. It also works to think of the grayed-out digits as “invisible” digits of the number.

Looking at these “full” versions, it is accurate to say that one line differs from the next only in the placement of the decimal point. Looking at the “common” versions, it is not quite accurate to say that. From one line to the next, some of the grayed-out digits have to be made solid, or some of the invisible digits have to be made visible. In general, all digits between the left-most non-zero digit and the right-most non-zero digit need to be visible in the standard notation, as well as all the zeros between the decimal point and the rest of the number. In addition, in some countries, at least one digit needs to be shown to the left of the decimal point, and at least one digit needs to be shown to the right of the decimal point, though this does not hold true in the United States.

One way to introduce *scientific notation* is to think of a calculator with a narrow display window. For example, let’s assume, for example’s sake, that the window can only show 5 digits.

In these 5 digits, only the following rows of the picture above could be represented directly:

.03412

.3412

3.412

34.12

341.2

3412

34120

Calculators that can handle scientific notation are able to represent both larger and smaller numbers in the same 5-digit width, by using an additional symbol, usually “E”, and this “E” symbol is followed by one or more digits.

In the picture above, the numbers on the right are as before; on the left we’ve shown the corresponding number in scientific notation. Note that all the numbers on the left fit in the 5 digits our hypothetical window of the calculator is capable of showing us. Let’s look at the number on the left in the last row. Here is one way to read it: “The number I’m showing you (3.412) is not quite the real number, the real number is seven rows lower.” And if you start from the highlighted number on the right, 3.412, and go down seven rows, you get 34120000. Another way of saying this: “The number I’m showing you (3.412) is not quite the real number, the real number has the decimal point moved seven places to the right.” The “E7” construct tells you that the real decimal point is seven places to the right of where it is shown. What the calculator is relying on is that the user can make the necessary adjustment easily and fill in the zeros appropriately. If we now look at the number on the left in the top row, we can read this as follows: “The number I’m showing you (3.412) is not quite the real number, the real number has the decimal point moved 2 places to the left.” The “E-2” construct tells you that the real decimal point is two places to the left of where it is shown. Again, the calculator relies on the user to make these adjustments.

Almost all real-life calculators will show a number in scientific notation only if the more familiar form doesn’t fit in the window. This hybrid approach is more practically useful, but doesn’t look as regular:

**I propose** that in school settings, teachers freely use the scientific notation as introduced above, consistent with how almost all calculators show it. The essence of this notation is not even that it is able to show very large numbers and very small numbers in a limited space – though that is clearly the motivation for it – but that it shows a *number*. To see it as a number, all you need to do is accept the new symbol E as part of a number, just like the decimal point is part of a number, and just as commas can make a number more easily scanned, (as in 3,000,000 for three million) – it still is just a number.

In contrast, the way scientific notation is often introduced in school textbooks is as an expression: instead of the calculator’s way of showing 3.412E7, the textbook will show . It is true, of course, that when you evaluate this expression you will end up with the same result of 34120000 (or 34,120,000). For students seeing this for the first time, this is unnecessarily confusing. I’ve seen plenty of kids take to their calculator to turn this expression into a single number! (Which, of course, usually doesn’t work in the sense that the calculator will not show the usual form of the result, but give it back in the 3.412E7 format.) And the reverse, to take a number like 34,120,000 and be told to write it as makes even less sense for the students – they already got the answer, why would they want to turn it back into an expression that then needs to be calculated? Many of these students never get that the scientific notation is an alternative way to *write the number*, and that it was never intended to treat it as an expression to be calculated. In my experience, most of these same students don’t have the same confusion with the calculator format. Since the calculator format (also used in any number of computer languages) is in no way inferior to the standard textbook format, it can be used throughout the classroom. The teacher can simply note that there are people who were taught to write and that it is useful to be aware of this standard, but that it amounts to an old way of writing 3.412E7. This statement is one that students can check on their calculator, if they so choose.

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In this series, we’ve done a review of mathematical notation, with an eye on how each notation helps or hinders student learning. In the prior posts, we started to explore function notation, which in middle school and high school shows up as , and examined its use. In this post, we will continue the thread of looking for notations that have the power of lambda calculus but are suited to middle school use. Our goal here is to come up with a notational scheme for functions that is suitable for composition and that can gradually morph into the standard notation students will encounter in text books in later grades. This exercise is motivated by the hunch that a light-weight notation for functions could usefully inform the way that expression, equations and variables are viewed and understood.

**Basic Functions**: functions of the kind shown below.

These correspond to the basic arithmetic operations from elementary school, in a special way. Adding a number, subtracting a number, multiplying by a certain number, and dividing by a particular number.

**Linear Functions**: The composition of one or more basic function results in a linear function. For example:

It may not be obvious that all of these are linear functions, though the third one should look the most familiar. It takes the number coming in, multiplies it by 2 and then adds 3. This one corresponds directly with the y= m x + b form of the linear function that students learn in middle school. Here, the slope is 2 (that is, positive two) and the y-intercept is 3 (positive 3). The second one, first subtracting 5 and then multiplying by 2, corresponds to the x-intercept form of the linear function – not always taught in middle school – it’s the line through (5,0) with a slope 0f 2. In contrast, the third one shows the line through (0,3) with a slope of 2. In the figure below, we show more linear functions:

On the left, we have a function that corresponds to the point-slop form of the line; here it is the line that goes through the point (5,3) with a slope of 2. In general, the composition of any two linear functions is itself linear. The second function (multiply by 2 and then add 3), composited with the third function (multiply by 5 and then add 1) gives rise to the fourth function (multiply by 2 and then add3, then multiply by 5 and then add 1), and this is indeed linear. It could be re-written equivalently as a multiplication by 10 followed by an addition of 16. (Note that I’m not slowing down this account by substantiating my claims here – I’m just playing with a notation and showing some of its power here.)

**Polynomials**: In standard notation, a polynomial in x may look as follows: .

This could be shown as:

Each of the vertical constructions that feed into the +-bar is one of the terms (nomials) of the polynomial. Each of these terms shares an input, and this input corresponds to the “x” in the standard notation. The boxes like ” ^ 5″ show exponentiation, the boxes like ” * 2 ” show multiplication.

There is another way to show polynomials, named after William George Horner, which sidesteps exponentiation altogether. Below is the picture for :

In this rendition, you see each of the coefficients used as one input into a section of two boxes. This section multiplies and then adds. Each section gets its other inputs from the previous section, and from the input to the entire box. Once you see how this works, you can see that it will work for polynomials of any degree and with any coefficients. There is a patterned-ness to this that, though strange at first, becomes easy and powerful after a while.

More on this in the next post.

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In this series, we’ve done a review of mathematical notation, with an eye on how each notation helps or hinders student learning. In the previous post, we started to explore function notation, which in middle school and high school shows up as , and examined its use. This post is a continuation of the previous post.

Let’s continue to look at functions as little boxes that take an input and produce an output, and play with that a bit. The most obvious way to play with it is to connect the output of one box to the input of another. An example is shown below:

Two simple function boxes, one which adds three to whatever comes in, and one which doubles whichever comes in. Together, they take a number that comes in at a and produce a number that comes out at c. We could hide this entire apparatus (i.e. the combined +3 box and the double box) in a single box, as follows:

and now present it to the world as a single function box with its input at a and its output at c. The new function box is obtained from the two other function boxes by *composition*.

Since we know what the smaller function boxes do, we can figure out specifically what the new function box does. We could try some different numbers at a, and follow them through the boxes at b and then at c. The value 1 at a will result in the value 8 at c. The value 10 at a will result in the value 26 at c. If we collect a whole set of value pairs like this, we can graph them. We would discover that the graph is linear.

When we don’t know much about the smaller boxes, we need a different approach. In the picture above, we have two function boxes, labeled and , respectively. Also notice that we’ve labeled the inputs and outputs again, this time as . With regard to the function box , we’d say that x is the input and y is the output. In the standard notation, we write . Similarly, we write . Combining both, we’d get . Yes, in the standard notation, the is shown before the .

We should note that in this standard notation, we need to give a name to the value coming in to the function box. The function , in standard usage, is pronounced “ef of ex” ( f of x) rather than plain “ef” (f). This appears to be because in standard notation, the name of the variable x is important; for example, if we say , the name of the variable x shows up again inside of the expression .

Let’s look again at the function box labeled “+ 3” above. Notice that it doesn’t contain any variable. Though we might call the number coming in “x” (or “a” or anything else), the function box doesn’t use “x”. It just says “+ 3”. In contrast, in standard notation, we’d talk about the independent variable and the dependent variable , and would write . It seems like the price we pay for using a “normal” looking expression such as is that we have to commit to the use of a particular variable, here . And yet, a function defined as is the same function in all respects as the function defined as . The notion, so beloved in secondary school, that x is always the independent variable and y is always the dependent variable, this gets in the way completely once we look at functions as things that can be combined (composited) easily.

I’m by no means the first one to notice that the “x” in f(x) could just as easily be “y” or “z” or “t”. The development of lambda calculus in the 1930s gave us a careful and precise model for function definition and function invocation, complete with a system of notation. This system of notation, involving the Greek letter lambda (λ) has become standard in certain branches of mathematics and computer science. It makes a clear and precise distinction between *bound variables* and *free variables* and elucidate how substitution works, to enough precision so that computers can do it automatically.

All the same, I’d say that lambda calculus is overkill for secondary school, even if introduced only for the notation, e.g. . I think there are easier ways to make clear through notational means that the bound variable (also called *dummy variable*) doesn’t matter. In fact, we’ve already seen examples of it.

Above are shown five identical function boxes, but with different notation. Box (a) shows the action of the box as an expression, x + 3, and labels the input as x. The suggestion is that the “x” in the x + 3 expression matches the number on the input. Box (b) shows the action of the box as an equation, y = x + 3, and labels both input and output, with x and y, respectively. Box (c) shows the action of the box as a function using lambda notation, and the input is not labeled. Here, the suggestion is that the label on the input has no bearing on the notation of the function in the function box. Box (d) shows the function in typical high school notation, using the function label “f” (so the function now has a name, f, even if that name is not used anywhere else.) Box (e) simply says “+ 3”, suggesting that whatever number is on the input gets three added to it.

Though we might have esthetic preferences for one of the boxes above over the others, it is the use of composition that will really drive up the reasons to prefer one over the other. Remember that all these 6 boxes are identical inside, and differ only in the labels. They all add three to the number going in. I could connect two of these identical boxes, output to input, and achieve the net effect of adding 6 to the number going in:

In situation (a), we see that the two identical boxes need to be given different labels, since the number going into the bottom box is not x, is not the same number as the number going into the top box. In situation (b), we also need to use different labels for identical boxes, since neither the number going into the bottom box nor the number coming out of the bottom box is the same as those for the box above. In situation (c), we can indeed use the same labels, since the lambda notation doesn’t presume anything about the name or value of the number coming in. In situation (d), we could call both the functions f, but can’t consider both boxes defining instances of the function f. In situation (e), as in (c), we can use the same box with the same label in both places, and have the notation work consistently.

So, for my money, the notation used in situation (e) gives us all the power and grace of the lambda notation while being much simpler for use at the middle school level. None of the more traditional school notations for functions has the same power and grace once we start to use functions in composition.

My experience with students at the middle school level suggests that the boxes and situations (e) give no problems. However, we need to examine this approach with examples other than adding constants to get a good feel for how expressive this notation really is. This will be the subject of our next post.

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In this series, we’ve done a review of mathematical notation, with an eye on how each notation helps or hinders student learning. The focus here is modest: on what teachers can do, even if their textbook sticks to the standard notation, to help disambiguate the standard notation for students. In the last several posts, we’ve been looking at notations for expressions.

The subject of *functions* in K-12 is typically only broached very tentatively, gingerly. Usually, kids learn to associate functions with graphs. Typically, there may be a situation, which relates two quantities. For example, a square has a certain side length and a certain perimeter. As the side length changes, the perimeter of the square changes also. The perimeter and the side length are related. This relationship can be shown in a table, shown in a graph, and shown in an equation.

When the side length is 3 units, the perimeter is 12 units. This is seen in the table as the row 3 | 12, and it is seen in the graph as the point (3, 12).

Yet when the question is asked “what is the notation for a function” in middle school, that question seems strange. When later in middle school or high school kids learn the notation , that notation seems to have exactly zero relevance. Students don’t typically get why saying is in any way better than saying . And no wonder. They are told that the stands for function, and it may take years before they ever see others like or . What functions are not, for these students, are objects that can be held up to the light and examined, and compared to other functions or combined with other functions. And truly, if all you ever do with is write instead of then there is no point. Even if students later encounter , it doesn’t seem to relate in any way with the function notation they’ve seen in .

Middle school students and middle school teachers can be forgiven for totally missing out on what’s important about functions at that level, and that is *composition of functions*. Composition of functions is a very simple and powerful idea to get a new function from other functions. For this to make sense, we need to move away from the beautiful and carefully developed notion of a function taught at the college level, which is based in set theory, where a function is seen as a special kind of *relation*. Instead, a useful model of what a function is at the middle school (even high school) level is that of a little machine with an input and an output:

We imagine that whenever something (usually a number) is put on the input, something is produced on the output (also usually a number). It is important that the output can only depend on the input, and on nothing else. Another way of saying this is that they are state-less: there is no internal state in the box, no memory of what went before, nothing that changes permanently based on what came before.

If we only had a single kind of function box (even if we have a thousand copies of it), the box doesn’t need any particular kind of label. Even a stupid label like would do fine. But if function boxes come in all different kinds, we need good labels to keep things straight. If we know what a particular box does, we should find a way to say so on the label; and if we don’t know (or care) what is in a particular function box, the label should say so as well.

Though we could label function boxes in different styles and according to different criteria, we’d like function boxes to be able to take their place inside of expressions as full-fledged components right next to the My Dear Aunt Sally stuff. For example, a solution to the quadratic equation, rendered in standard format as , might look as follows:

where we’ve used simple function boxes for negation(change sign), square (input times itself), double (input times two) and square root. If you are very familiar and fluent with the standard notation, my sketch above will not have impressed you. Yet the picture below, showing a simple composition of functions, displays some interesting patterns:

You subtract one from the number coming in, square the result, and then add 5. If you know the behavior of the square box in the middle – for example, you know the parabolic graph that corresponds to it – then the function box on top tells you to move that graph one over to the right, and the bottom function box tells you to move that graph up 5, resulting in:

In the next post we’ll explore more about composition of functions, how it is useful and powerful, and how notation can help highlight this.

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In this series, we’ve done a review of mathematical notation, with an eye on how each notation helps or hinders student learning. The focus here is modest: on what teachers can do, even if their textbook sticks to the standard notation, to help disambiguate the standard notation for students. In the last several posts, we’ve been looking at notations for expressions. In the previous post, I reviewed slight variations on the standard notation for expressions; in this post I’ll offer a more drastic variation. This drastic variation doesn’t exhaust the subject – we’ll return for more later.

Let’s start with an expression given in standard notation: , and remind ourselves that its meaning is derived from the sequence of operations that are indicated: some values are to be multiplied, others added, etc.

This same meaning is expressed above in the form of a *tree*, and this tree is a particular kind of notation for the expression. (Some useful vocabulary: the tree is said to have nodes and branches. One node is called the *root*, and is shown at the bottom. Other nodes are called *leaf nodes* and represent the starting points: e.g. 3, the value of a. The remaining nodes are called internal nodes, and represent operations.) Here, “*” represents multiplication of two numbers, and “+” represents addition of two numbers.

Somewhere along the way, students become familiar with the idea that the operations of addition and multiplication are special compared to subtraction, division and exponentiation: when you add a bunch of numbers, it doesn’t matter in which order the addition is done. If I need to add a and b and c, I can add a + b and then add c to the intermediate result; or I can add b+c first and then add a to it; or I could add c+a and then add b to it. In formal terms, addition is both commutative and associative. Multiplication is commutative and associative as well.

Recognizing the special role of multiplication and addition allows for a very useful re-writing of the tree shown above, as follows:

This notation can peacefully coexist with the standard notation in the textbooks. Students get the pictorial form without any trouble, and can manipulate the expression trees with ease. They tend to think of the internal nodes as little machines that are sitting there waiting for inputs, and then producing an output.

I intend to come back to this notational scheme and other variations in future posts – I want to take an excursion first to look at notation for functions: I think this will prove very fruitful.

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In this series, we’ve done a review of mathematical notation, with an eye on how each notation helps or hinders student learning. The focus here is on what teachers can do, even if their textbook sticks to the standard notation, to help disambiguate the standard notation for students. In the previous post, I reviewed the standard notation for expressions; in this post I’ll review common variations. In future posts, I’ll look at less familiar – but powerful – notations for expressions.

**Exponentiation**: one commonly seen alternative for the raised exponent shown in a smaller font is a version of the up-arrow.

In the top left, we see the standard notation of x raised to the nth power. In the top middle, we see the up-arrow (↑) used as a way to suggest the n should be raised. The up-arrow was available on specialized keyboards in the past, but keyboards for laptops etc don’t tend to have this character. People have used the caret character (^) as a keyboard substitute for the up-arrow, and by extension, for the exponentiation. Several computer languages do this very thing.

There is an obvious advantage for being able to put an expression as a linear sequence using a standard keyboard, but that advantage comes with its own disadvantage. The bottom row of examples indicates this. On the bottom left, we see x raised to the power n-1. No parentheses are needed to express this grouping: the entire n-1 is rendered in small font and raised. To get the same effect, the bottom middle expression puts the n-1 in parentheses, and so does the bottom right.

**Square Root**: I’m showing common alternatives below:

On top are the square root symbols with the long horizontal bar, on the bottom you see a square root symbol without a horizontal bar (√) available on some keyboards. On the left, where it shows square root of n, both the symbol with the horizontal bar and the symbol without will do fine. In the middle, where we show square root of n-1, the symbol with the horizontal bar indicates the grouping clearly and effortlessly, whereas the √ symbol requires parentheses to make the scope of the square root clear. On the right, we’ve shown a variant that’s easy to produce on the keyboard but requires both parentheses and a dedicated name, “sqrt” for square root. The “sqrt” designation is common in computer languages and is seen elsewhere as well; obviously this notation doesn’t generalize easily for third roots etc., but then neither do the variants based on the √ symbol. I get the impression that there is little demand for special symbols for third roots, n-th roots, etc., since these roots can all be rewritten as exponentiations with fractional exponents.

**Division**: Some common alternatives are shown below:

The ÷ symbol for division, as shown on the left, is very common in the early grades, yet rarely used past elementary grades. This, in itself, is a pretty good indicator that it is quite possible to introduce notations for lower grades that are gradually phased out without in any way damaging a student’s eventual understanding or skill or fluency. However, though rarely used by adults, adults do almost universally recognize the ÷ symbol, and this makes it suitable for use on calculators. Most calculators, by far, use ÷ to mark their division key.

The symbol seen in the middle, the slash (/), is also universally recognized as a division; on the bottom we see the case where parentheses are necessary. On the right we see a horizontal line used as the dividing line between top and bottom (here known as numerator and denominator). In neither the top nor the bottom situation are parentheses needed. The horizontal line is often pronounced as “over”. So it is n over 2, or n over n-1. This horizontal line seems like it would be very confusing with the line used for fractions. In practice, it is not confusing at all. Many students insist on fractions and divisions being not at all alike, and they pride themselves on being able to tell them apart. For them, “two fifth” is one thing, and “2 0ver 5” is something else altogether. From my perspective, the more kids see a fraction as a division, the better off they’ll be. Here is an example, I think, of a notation that is trying very hard to suggest that fractions and division are very closely related – and failing!

Note: In this account of notations for division, I’m deliberately leaving out notations for divisibility, e.g. 5|30 (read: “5 divides into 30”, or “5 is a factor of 30”) and I’m also deliberately leaving out the symbol seen in the long division algorithm. I’m also leaving out complications having to do with division (in the lower grades) resulting in a quotient and a remainder.

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In this series, I’ve been exploring issues of mathematical notation and their impact on student learning in school. In the last installment I veered a bit from the straight path and showed a way to denote negative numbers that is very familiar in computer science and yet completely missing from school math.

In this post, I’ll get back to the center of mathematical notation in middle school: expressions and formulas. Before embarking on various alternative notations in subsequent posts, let me use this post to review the standard notation and all that it implies.

First, a clarification on terminology. In an equation like , we call an *expression*. And as far as I can tell, *formula* is just a synonym for expression; other people claim that a formula is the kind of thing like F = C*9/5 + 32 where you convert from degrees Centigrade to degrees Fahrenheit by applying a recipe written in mathematical language – I’ll stay neutral on the issue here. Beyond middle school, the whole equation might be seen as an expression, in this case one with a value of *true* or *false*. The use of “=” as an operator, with possible values true or false, is one I deliberately skipped in my earlier posts about the overloading of the equals sign.

In middle school, we know what expressions consist of, since we’ve all learned the Please Excuse My Dear Aunt Sally thing about the order of operations. By implication, we’re looking at the composition of exponentiation, multiplication, division, addition and subtraction, where composition is done by use of parentheses. And this pretty much matches what students see in middle school, with only the occasional absolute value |x|, square root √x or trigonometry functions sin(x), cos(x) and tan(x) adding a wrinkle.

Since in the standard order of operations the last action we do is add or subtract, we can coin a name for those things that we add or subtract. Actually, there is an already existing name that will do fine, it is called a *term*. So an expression is either a term or it is the sum or difference of terms. According to the order of operations, what comes before additions and subtractions is multiplication or division. This means that these terms, these things we are adding or subtracting, are themselves products or divisions. The things we are multiplying or dividing, the are commonly called *factors*. So a term, in turn, is either a *factor*, or a product or division of factors. A factor, in turn, is either a *primary*, or a primary raised to an expression. Finally, a primary is either a number, or a variable, or a parenthesized expression.

What I described in words in the previous paragraph, is usually expressed with what is called a generative grammar:

expression : term

expression: expression + term

expression: expression – term

term: factor

term: term × factor

term: term / factor

factor: primary

factor: primary ^{expression}

primary: number

primary: variable

primary: ( expression )

The little grammar above assumes that it is clear what a number looks like, and what a variable looks like. This grammar can be extended to deal with roots, absolute value and trig functions by adding rules for *primary*, e.g.

primary: | expression |

primary: trigfunction ( expression )

trigfunction: sin

trigfunction: tan

These rules together make for a language of expressions that can be uniquely parsed; under broad circumstances, the ×symbol can be omitted and expressions can still be uniquely parsed. This language is compact and terse; it is fairly easy to appreciate why this language won out among professionals and spread throughout the world. Yet just because it is compact and can be parsed uniquely doesn’t mean that it is easy to interpret by student learners. In earlier posts I’ve alluded to the confusion students have between parentheses and multiplication: students see parentheses, and are primed to expect that there is a multiplication involved. Sometimes there is, as in , and sometimes there is not, as in , and yet I’ve routinely seen even ‘good’ students treat as some kind of multiplication. The standard method of writing expressions is linear and monochrome. We can do better in a learning environment, and subsequent posts will explore various alternatives.

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In this series, I’m looking at the impact of mathematical notation on student understanding and student learning. In the previous installment, I looked at a notation for “long subtraction” different from the standard USA way of writing it. The current post will not make sense without reading the previous post first.

The previous post compared the standard USA way of writing subtraction:

with a way found in other parts of the world. In that approach, a fat dot indicates borrowing:

I will extend this idea to subtraction in the situation where you subtract more than you originally have. (School kids typically know that you are not supposed to do this, and if you do, you don’t do it by putting the two numbers one above the other. It just isn’t done, the standard USA algorithm will self-destruct when you try that. Yes, kids learn later in school that you can subtract a bigger number from a smaller number and get something called a negative number, but you don’t use the subtraction algorithm for that.)

With the fat dots indicating borrowing, I can extend the use of the algorithm as follows. Imagine I’m subtracting 45 from 28. That’s right: 28 – 45. Here’s what you’d get by using the fat dot algorithm:

(We subtracted 5 from8 getting 3; tried to subtract 4 from 2 and needed to borrow; we place the fat dot over the third column from the right and proceed by subtracting 4 from 12 getting 8. For the leftmost column, we can’t do better than pull the fat dot down to the result, giving us •83 – and how shall we interpret this result?

One way to interpret the •83 is to see it as 83 – 100, since we tried to borrow 100. The algorithm is indeed correct in that it showed that 28 – 45 = 83 – 100. Another way to interpret •83 is to see it as paying 100 and getting 83 back in change.

Yet another way of interpreting •83 is to see them as three digits in a three-digit number. Each digit has its place value. From right to left, the 3 is 3 ones, the 8 is 8 tens, and the • is some number of hundreds. How many hundreds? Well, the total result is supposed to be ⁻17, so working backwards, the • has to stand for negative one. So if we think of • as a digit, just like 0 or 1 or … 9, then the value of this digit is negative one, and the value of • in a number depends on the place value in the normal way.

Allowing • as a digit in a number has benefits. I’ll be brief on these in this post, as the series is mostly about notation as such.

One benefit for writing •83 for ⁻17 is that it helps in adding positive and negative numbers.

For example, instead of 33 + ⁻17 (which in school is usually converted to a subtraction problem), we would have 33 + •83 which we could do as:

We add in the standard way, starting in the ones column, giving 3 + 3 = 6. I the tens column, we add 3 + 8 = 11, writing down the 1, and carrying the ten tens as 1 hundreds. In the hundreds column, we have the 1 that is carried in, and add it to the fat dot, which treat as an injunction to subtract 1 – or alternatively, as an injunction to add a digit with value of negative one. The net effect is that there are no hundreds. We could have a written a zero, but of course we all know that leading zeros don’t affect the value of the number.

In fact, just like there are many ways to write the same positive number in the decimal system, there are many ways to write a negative number using the fat dot:

On the left, we have various ways of writing positive 16, on the right, we show various ways of writing negative 17. The first one, •83 represents negative 17 as 83-100; the second one, •983 represents negative 17 as 983-1000, etc. This can also be read as: paying 100 and getting 83 in change, or paying 1000 and getting 983 in change.

It is a nice feature of our common number system that it is easy to tell if different ways of writing the same number are equivalent. This feature is often under-appreciated. Typically, we don’t even think of 0016 as the “real” way to write the number sixteen. We tend to think of 16 as the real way to write sixteen, and 0016 as weak and rather useless variant. For our notation for negative numbers, we might likewise settle on •83 as the “real” way to write negative seventeen, and treat •983 or •9983 as less desirable variants.

The notations I’ve suggested in this post for negative numbers may have seemed completely strange and arbitrary to you, not to mention impractical. Given our backgrounds, •83 simply doesn’t *look* like negative seventeen, and why would one want to even consider •83 as the way to write negative seventeen? But •83 is *not* impractical: it is the standard way by which computers represent negative numbers. Whether your computer is a PC or a MAC or whether you are reading this post from your cell phone, all these have processors that represent negative numbers in a way consistent with what I have sketched here. In the binary number systems used in computers, this way of representing negative numbers is called “two’s complement” and described in this Wikipedia entry. The decimal version of it, as shown in this post, is known as “ten’s complement”, and more details can be found in this Wikipedia entry. The notion of using the same fat dot from the subtraction algorithm as a marker for negative numbers in ten’s complement notation is my own.

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**Subtraction: The Dot** **For “Borrowing”**

In this series, I’ve been exploring notational issues and their impact on student learning. So far, I’ve been hesitating dealing with the standard subtraction algorithm as taught in the USA for two reasons:

1. I’m not at all convinced that teaching standard algorithms for “long subtraction”, “long division”, etc. is all that useful. Yes, it is very important that kids learn what subtraction is and what division is. It is important that kids learn algorithms for subtraction and algorithms for division. But that doesn’t mean that the algorithm they learn needs to be the standard one, and it doesn’t mean that the algorithm is taught as a series of steps. I know no adult professional who uses long division if they have an alternative. In most cases, the alternative will be the use of a calculator, or a spreadsheet, or some related tool. Before calculators and computers, professionals used log tables or slide rules. The notion that teaching long division teaches kids a useful skill is suspect. The notion that teaching long division teaches kids something fundamental about math is also suspect. After all, most kids who do know long division nevertheless have no idea why it works, and kids learning different methods (e.g. the method known as “the big 7”) may well end up with a better sense of what division is.

Still, I know that long subtraction is still taught in many places. My purpose with this blog is not to add fuel to controversies; my interest is in adding light, not heat.

2. I was taught a way to do subtraction different from the one that is typically taught in the USA, and I’m generally reluctant to tout things I happened to be taught as being better. There are so many ways in which one can be prejudiced and not impartial when it comes to one’s own background. People say their own mother cooked better pancakes than anyone else in the world, or people say their home team is the best. Others learn to ignore this, or treat it with suspicion.

Still, I think it useful to show a way of doing multi-digit subtraction that is standard in other parts of the world, and introduce it to US teachers, and suggest how it avoids certain confusion and misunderstanding.

Let’s start by reviewing the USA standard for subtraction, with particular attention to borrowing. Here is a diagram from an exposition on how to subtract14 from 402.

It starts by writing the two numbers one below the other, aligned so that the ones are below the ones and the tens below the tens etc. The general flow of the algorithm is from right to left. So we attempt to subtract the 4 from the 2, and conclude that we need to *borrow*. We borrow one from the place value to the left, which gets us ten in the current place value. In the picture you can see the ten as the orangish 1 written just above the 2, making it 12. The place value to the left is showing a zero, so borrowing one from it is problematic, and we need to go further left. We can borrow 1 from the 4 (shown as crossing out the 4, and writing 3 above it) and that allows us to write the gray one just above the zero, to turn that place value into 10. But we needed to borrow one from it for the right-most place value, so the 10 is itself crossed out and changed into the red 9. In this example, this completes all the borrowing necessary, and we can now subtract 4 from 12 in the rightmost position, and subtract 1 from 9 in the middle position, and subtract nothing from the 3 in the leftmost position.

Since this is so familiar to most of us, I don’t know if you can appreciate how messy this is. The diagram shown is nicely color coded, and the numbers are spaced properly. Even so, it looks messy. When kids do this, you can often no longer distinguish what the number was that we started out with, making it hard to double check the work. In addition to it being messy, the algorithm also doesn’t proceed smoothly from right to left. To complete the process of borrowing, we often have to move much further to the left, crossing over all the zeros, to get to a place where we can find a one to subtract. This one, becoming 10 in the place value to the right, then becomes a 9 and a propagating 10 further to the right.

Here is how this subtraction looks in the standard algorithm that I was taught:

In this algorithm, subtraction does proceed strictly from right to left. There is no crossing out of numbers. When subtracting 4 from 2 in the rightmost position, we decide, as in the standard US algorithm, that we need to borrow. To indicate this, we put the fat dot above the column to the left. The fat dot represents the fact that there has been a borrowing, it represents the IOU. It also indicates that we can now read the 2 as a 12, and we subtract 4 from 12 getting 8. Note that at this point, we haven’t done anything yet with the leftmost position. When we look at the middle position, we see a 0 on top, a fat dot above it, and a 1 below it to be subtracted. The fat dot indicates that we have to subtract an additional one, to make good on what was borrowed before. We can’t deliver based on what we have in the middle column, and thus place a fat dot on top of the leftmost column, borrowing ten units for the middle column. These 10, minus the 1 for the fat dot in the middle column, and minus the 1 from the bottom number, gives us 8. Now we turn our attention to the left column. Here we have 4, with a fat dot, reducing it to 3, and we have nothing from the bottom number to subtract from it. So we are left with a 3. The digits 3, 8 and 8 tell us that the result of our subtraction is 388.

Note that the original numbers are still neatly visible, none of it has been crossed out or overwritten. So we can double check our arithmetic, or check the subtraction in another way (e.g. adding 388 to 14 to see if we get 402). Note that each column was handled in the same way, one at a time. For each column, there was a top digit, a bottom digit, a possible fat dot on top to indicate that 1 had been borrowed. The work on that column then gives us a result digit, and a possible fat dot for the column to the left. We are then ready to turn our attention to the next column to the left. The diagram below shows the mechanism at play in each column:

The digit for the top number, indicated by A, and the digit for the bottom number, indicated by B, are inputs to this operation, as is the signal indicated as borrow-in. The borrow-in represents the presence or absence of the fat dot on top of the column. The output of the mechanism is (1) the result digit, indicated as C; and (2) the presence or absence of a fat dot for the column to the left, indicated as borrow-out. The result digit C is computed, if A is big enough, from A-B if the borrow-in fat dot is absent, and A-1-B if the borrow-in fat dot is present. If A is not big enough, there is a borrow-out, and C is computed as “1A”-B if the borrow-in fat dot is absent, and as “1A”-1-B if the borrow-in fat dot is present.

Almost all computer arithmetic, e.g. the processors in your PC or MAC, use subtraction hardware that matches this algorithm in all essential aspects.

You can think of the fat dot as playing a dual role: when we place a fat dot to the left or our current column, we can read it as a leading 1 for the number in the current column, e.g. turning the “2” into a “12”. When we advance to the next column, the column to the left, this same fat dot now looks like a “-1”, an extra one to subtract.

Please note that in many aspects, the algorithm I sketch here uses the same ideas as the standard US algorithm. It is based in similar ways on place value, it is based in similar ways on taking “1” from the left and getting “10” in the current position. The major difference is in notation, and the fat dot notation avoids the crossing out of the digits we start with. A nice side-effect of this notation is that we can truly look at it as a borrow, rather than as an exchange. The term “borrow” is often used when talking about the US algorithm, but we travel all the way to the left as needed to ensure that we pay before we “borrow”. In contrast, the fat dot can be seen as a pure IOU, or can be seen as a new digit that represents “-1”.

**I propose** that teachers use the fat dot as a notational alternative for indicating borrowing, and avoid crossing out numbers. They can do this even when holding on to the notion that you have to go left all the way past zeros till you ‘land’ on something solid. The fat dot can be used as a way to show multi-digit subtraction even if kids later learn and use the standard US algorithm; it is especially advantageous for kids who already write messily or don’t have fine motor control.

In the next post, I’ll show how this notation can be extended to represent negative numbers. This extension is not actually part of the algorithm as I was taught it, but the extension is both useful, clean, and has real applications in the way computers do arithmetic.

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In this series about mathematical notation and its impact on the learning of mathematics, the previous post looked at the use of the “=” sign to indicate *equivalence* or* identity*, and then looked at an alternative notation. In this post we will look at the “=” sign as used in equations.

In the picture below I show a standard way of solving a linear equation:

The use of the “=” sign to indicate that the left side 5x – 6 and the right side 9 are the same for some suitable value or values of x – and that we are interested in what that suitable value for x is – is distinct from the “do this now” use, and also different from the “the left side and the right side will always evaluate to the same value regardless of the chosen value for x” use.

In an equation, the equality of left side and right side is something that you receive as a hint or a clue; it is a useful fact with informational value. In the example above, the equation tells you something about what x must be, and in solving it, we find that it indeed allows us to nail down precisely what value x must take.

In the way that we solved this equation, we proceeded from 5x – 6 = 9 through 5 x = 15 to x = 3. All three of these are equations, and all three equations have the same solution. In all three of these, the “=” sign means the same thing. Except – most of us don’t relate to x=3 as being an equation. We relate to it as our answer, our solution. We rarely ask: “what value must x take for x=3 to be true?”, and perhaps this is because the answer to that question is so obvious: “well, x must be 3!”

Once an equation has been solved, teachers usually impress the importance of checking the solution in the original equation. The next picture shows the checking:

What you see in the top line is the left side 5x – 6 where “x” has been replaced by the solution that we found: 3. The left side should evaluate to 9, but we don’t know yet if this will pan out. By the time we come to the third line, we have 9 = 9, and this, of course, checks out. The notation shown, with the question marks over the “=” sign while we are still checking, and the exclamation point over the “=” sign once we have verified that the solution does indeed check out, this is the notation that I **propose** teachers use. It is a simple notation, like an annotated “=” sign, that makes it viscerally clear that we are not yet ready to assert that left side and right sides are equal, but that we are working towards that goal. And of course, if we had made an error along the way, we might expect that during checking we’d discover that left side and right side are *not* equal, and that the equality check does not pan out. In that case, our final check line would use the “≠” sign.

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