Mathematical Notation and Schools – 8

Subtraction: The Dot For “Borrowing” Can Also Denote Negative Numbers

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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Mathematical Notations and Schools – 7

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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Mathematical Notations and Schools – 6

The Overloaded Equals Sign: Solving Equations and Checking Solutions

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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Mathematical Notation and Schools – 5

The Overloaded Equals Sign: Equivalence

In this series about mathematical notation and its impact on the learning of mathematics, the previous post started to look at the overloading of the equals sign – so many different meanings and uses cramped in this single notation “=”.   Sometimes, like the “=” key on a calculator, the meaning is “do this now”, perform the calculation and get me a single number.

Other times, the “=” sign is used to indicate equivalence between two different ways of describing what is (in some sense) the same thing.  If you compare x + 1 + 1 with x + 2 , you may agree that they look different, but that they have something very important in common.  That is, by some measures they are different, by another measure they are the same.  This phenomenon is known as equivalence.  Equivalence is a very powerful and general idea.  It is different from equality, in that two equivalent things need not be the same.  They just need to match in the particular attribute that you care about.   In textbooks, as well as in actual practice, the thing you care about is most often left implicit.  When we say that x + 1 + 1 and x + 2 amount to the same thing, what we are focusing on is the values you get when the expression is evaluated for a particular choice of x, and then noticing that these values will be the same for both expressions regardless of what number we choose for x.  By focusing on the values, and on nothing else, we say that x + 1 + 1 and x +2 are the same.  And yet they are not the same in other aspects.   These other aspects may be considered trivial, but that doesn’t make them less real.  For example, many teachers would not accept x + 1 + 1 as a correct answer to a problem, saying that it should be simplified to x + 2 .  This is typical for the realm of equivalence: some things are more equal than others.

The particular kind of equivalence where we focus on the values when choosing a particular value for x, this kind of equivalence is often called identity.  The standard mathematical notation for both equivalence and identity is “≡”, though ‘standard’ doesn’t mean universal.  Most people and textbooks write x + 1 + 1 = x + 2 .  The idea is that both sides are equal, regardless of the value of x.  The part of “regardless of the value of x” is not made explicit.

The use of “=” for identity is often confused by students with the use of “=” for equations.  When we write x + 1 = 5 , we are not saying or suggesting that both sides are equal regardless of the value of x.  We’re saying almost exactly the opposite: we’re saying that by telling you that x + 1 = 5 , we’ve given you a powerful hint or clue as to what value x should have.  Students often rely on verbal clues like “simplify” or “expand” or  “evaluate” or “prove” or “solve” to figure out what they are supposed to do.   Teachers’ consistent use of “≡” for identity, even if the student is never asked to do the same, and even if the textbook doesn’t play along, is still a powerful tool for students to become aware of the different meanings for “=” they see in their textbooks.

Hence I propose that teachers use “≡” instead of “=” whenever the intended meaning is that the two sides are identical, that is, they evaluate to the same value whatever is chosen as the  value for x.  In my experience, both students and teachers find this helpful.

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Mathematical Notation and Schools – 4

The Overloaded Equals Sign: “Do This Now”

In this series about mathematical notation and its impact on the learning of mathematics, we’ve looked so far at the raised minus sign for negative numbers, the various ways of denoting multiplication, and various ways to denote the sequencing of operations.  In the next few posts, I’ll be looking at the different mathematical ideas that are expressed, in standard notation, through the equals sign “=”.

In this post, we’ll look at the most familiar use of the equals sign, the one learned first, and the one in common use on calculators.  In school, we all know what we are supposed to do with:

5 + 4 =

7 + 8 =

5 – 2 =

The job here, as we all learned, is to put a number to the right of the “=” sign, and this number is called “the answer”.  The number we put there is either the correct answer or the wrong answer, and it is the job of the teacher to decide which is which.  This is such an important part of the teacher’s job that the teacher even has a special edition of the textbook, different from the students’ version, and the teacher’s edition includes something called an “answer key”.  The answer key helps the teacher in this all-important task.  Like a traditional country doctor who is instantly recognizable by the white coat and the stethoscope, the teacher is distinguishable by the answer key.

Any well-trained kid will feel an urge, whenever she sees an equal sign with nothing to the right of it.  The same urge is felt, clearly, by a four-function calculator whenever the “=” key is pushed.  Something needs to be done!  Right now!  We need a number, and we need it stat!

This means that, at this stage, the equals sign has a clear direction to it, it is not symmetric in its use.  For a kid, seeing
5 + 4 = 9 would look normal: it would look like an addition problem that somebody else had already answered.  Yet
9 = 5 + 4 would look positively strange, and 5 + 4 = 6 + 3 would look positively wrong.  To adults these don’t look wrong, but that means we are used to the equals sign in different contexts, for other purposes.  In upcoming posts, I’ll investigate those other uses.

Staying with the “do this now” meaning of the equals sign, let’s take inventory of other ways to indicate this meaning, both in school and out.  From this, a reasonable proposal might emerge for notational schemes that teachers could use to help disambiguate the mathematical ideas for their students.

If you listen to how 5 +4 = 9 is pronounced in parts of the English speaking world: “five plus four makes nine”.  Though a minority pronunciation (I assume “five plus four equals nine” is considerably more common), the “makes” pronunciation connotes a production, an action with a direction and an intention.  As one very obvious way to indicate direction is by using an arrow, “→” (which we might pronounce as “makes”), let’s play with this notation and see how it looks.  In an earlier post, I showed
75 + 10 = _______  × 1.09 =  ______ + 4 = _____ as a way to show the sequence of operations in ((75+10)×1.09)+4, knowing that this use of the equals sign upsets many teachers.  So here is an opportunity to see if it looks better using the “→” instead of “=”:
75 + 10 → _______ × 1.09 → _____ + 4 → _____.   I do think it looks better, but that isn’t a strong enough argument for proposing the → symbol.  Other notations may look even better, and also look more natural.  Moreover, other notations may be better able to coexist with standard textbook conventions.   We might take inspiration from what computer languages have long used for assignment operators, where it was desirable to have a notation for assignment that was distinct from a notation for equality.  If you ever learned the Basic programming language, it may have looked strange that you saw constructs like X = X + 1 used there.  Many programming languages use the construct “x := x + 1” instead.  The symbol “:=” for assignment doesn’t deviate too far from the “=” sign but makes the directionality clear.  But for our purposes in disambiguating 8+4=, the assignment operator is oriented the wrong way around.

In light of this, I propose “=:” as an informal notation for “makes”:

7 + 5 =: 12

75 + 10 =: _______ × 1.09 =:  ______ + 4 =: _____

The “=:” is easy to use in a classroom, is easily rendered from a keyboard; the extra colon is relatively unobtrusive, and can gradually be dropped as students get older.  It is something the teacher can use to disambiguate the various meanings of “=” in the standard notation, and the teacher can do this without ever requiring the students to use this notation themselves.  The notation is easily explained to parents,  though little harm is done if such explanation never reaches the parents.

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Mathematical Notation and Schools – 3

Notations for Sequencing and Nesting (Parentheses)

In  this series, I’m exploring mathematical notation as it is used in schools, and how these notations help or hinder students’ understanding of mathematics.  In the first installment of this series, I spoke positively about the practice in US schools to write negative numbers with a raised minus sign.  In the second installment, I explored the use of an asterisk (*) as an  alternative to the multiplication sign starting in middle school or slightly before.

In this post, I want to explore the various ways that sequencing of operations is indicated in schools.  But to set the stage, let’s first take a look at how we indicate to normal adults what sequence of operations to perform.  Here is an example, from one of the first Google entries of the “order form” search:

In such an order form, certain information must be provided, like Item Number and Description, that indicates what kinds of things are being ordered.  Then there are the columns labeled Quantity, Price and Amount.  You will most likely recognize this as requiring the Amount, on a given row, to be determined from the Quantity and the Price by multiplication.  The row labeled Subtotal will then be obtained from the Amounts by adding them all up.  To this Subtotal, a Discount will be applied, presumably some fixed percentage of the Subtotal, or some sliding scale.  The row labeled Total will be obtained by subtracting the Discount amount from the Subtotal.  The row labeled Tax (CO) presumably is a known percentage of the Total; the row labeled Shipping may be a fixed amount or may be found from a table.  Last, the AMOUNT DUE is obtained by adding Total, Tax (CO) and Shipping amounts.  In this set up, a sequence of actions is implied, leading from Quantities to the AMOUNT DUE through the Prices of the Item Numbers.  It makes little sense to compute the TAX (CO) before the first Quantity has been multiplied by the first Price.  On the other hand, there is some latitude in what order the multiplications of Quantity and Price are done, as long as all or done before the Subtotal is attempted.

I suggest we’ve all seen variants of this order form, and thus are familiar with the implied sequencing of actions in it.  Note that this is communicated to us in a relatively standard way, and yet there is no parentheses nor My Dear Aunt Sally to be found.

In secondary school, we may encounter formulas that look like  ((75+10) × 1.09 ) + 4, and to make sense of this, we must understand parentheses and “order of operations”.  In elementary grades, it is not very common to see multiple operations combined in a single formula.  It wouldn’t be uncommon to see it represented in elementary grades as follows:

75 + 10 = _______  × 1.09 =  ______ + 4 = _____ ; and even though many teachers would balk at this (mis)use of the equals sign, students don’t seem to have any problem with the sequencing aspect.  They seem to get they are supposed to proceed from left to right:  75 + 10 = 175, then 175 × 1.09 = 190.75 and finally, 190.75 + 4 = 194.75.

Like the order form we showed, there are ways to indicate the sequence of arithmetic operations that don’t look at all like mathematical  formulas with parentheses.  In an earlier post, I worked with representations like the following:

which can be used equivalently to ( 12 x + 8 ) / 4 .  Note that each box works on its own input and produces an output in return.  The sequence of arithmetic operations is obvious from the way the boxes are connected.

None of these examples is intended to suggest that parentheses are bad, merely that there are many other ways to indicate in what sequence operations should be performed.  When you learned the quadratic formula, did you ever notice that it doesn’t have parentheses?  If you’ve forgotten: one of the solutions of the equation a x^2 + bx + c = 0 is \frac{-b + \sqrt {b^2 - 4ac}}{2a} .  Here, the length of the horizontal line of the division is used to indicate what belongs to the numerator; similarly, the length of the horizontal part of the square root sign is used to indicate what belongs ‘inside’ of the square root.

Now let’s look at the kinds of formulas where traditionally we might use parentheses:

In standard notation, the formula on the top would be written as x^2 + 2x ; the bottom one would be written as x (x+2) .  In the picture, parts of the formula have been encapsulated in bags – like crude ellipses.  What the picture on top tries to suggest is that the formula x^2 + 2x is basically a sum of two parts.  One part is x^2 , the other part is 2x .  Similarly, the picture on the bottom tries to suggest that the formula x (x+2) is basically a product of two parts.  One part is x, the other part is x + 2 .

We can think of parentheses as an in-line version of the ellipses used in the picture: a rendering of the left part and the right part of the ellipses while leaving out the ceiling part and the floor part.  And, indeed, the formulas could be written as (x^2) + (2x) and (x)(x+2) , respectively.

From my observations with students in middle school, the notion of using parentheses is not that difficult to grasp.  More difficult for the students is getting facile with when these parentheses are left out.  In the standard notation, parentheses are only written “when needed”, by which is meant that parentheses are only written if leaving them out would alter the meaning of the formula.  In the standard system of mathematical notation, there is a fairly good reason for wanting to eliminate as many parentheses as possible from a formula: when you get too many parentheses, especially if they are nested, things become a bit confusing.  How easy is it to make sense of the following formula?

x = (-b + sqrt(b*b-4*a*c))/(2*a)

This is another rendition of a solution to the quadratic equation.  It has three left-parentheses: “(“, and three right-parenthesis: “)” – and each left-parenthesis is matched with a particular right-parentheses.  The matching is relatively easy to do with a pencil if the thing is printed on paper, but not quite so easy to do just in reading it.

The adult’s insistence on dropping all unneeded pairs of parentheses has an unexpected consequence for students: students start to think of parentheses as having something to do with multiplication.  They see (x+1)(x+3) and are asked to simplify it and they expand it to x^2 + 4x + 3 , but then if you ask them to simplify (x+1) + (x+3) , they likewise treat it as a multiplication instead of an addition.  The sight of the parentheses seems to make them think that multiplication is in order – after all, that’s where they have seen parentheses the most: in factors.  Sometimes teachers explicitly encourage students to think of the parentheses as indicating multiplication: they’ll tell their students that (x+1) + (x+3) has a hidden multiplication by 1 in it, so that the formula really means 1(x+1) +1(x+3) , which is then subject to the distributive property and expanded to 1*x+1*1 + 1*x +1*3 , which then becomes 1x + 1 + 1x + 3 and finally 2x + 4 .

My proposal is to introduce the bags early and frequently as ways to annotate a formula and indicate the order in which things happen.  Bags can be nested, meaning you can have bags inside of bags.  This use of bags, as in the picture below, can start in elementary grades as multiple operations are introduced.

Soon after, ‘real’ parentheses can be introduced as an abbreviation for drawing the bags, as follows:

After a while, kids can imagine the whole bags, just from seeing the parentheses.  The teacher can draw bags or parentheses interchangeably, while noting that the textbook only uses parentheses.

In middle school, I propose teachers use bags and parentheses whether they are “needed” or not, whenever their use can clarify the structure of a formula.  Teachers should use parentheses both for multiplication and addition, so students don’t automatically associate parentheses with multiplication as they tend to do today.

At the end of middle school, students can be eased into the standard notation, through being encouraged to write parentheses only when needed, like the textbooks already do.

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Mathematical Notation and Schools – 2

Notations For Multiplication

In the first part of this series, I looked at the practice in American schools to use a raised minus sign for negative numbers: e.g. ⁻7 for negative seven, in contrast to the “-” in 10 – 7.

Now I want to look at different notations for multiplication, and suggest one that is not currently widely used in schools but has wide currency elsewhere.

In elementary schools, by far the standard way to denote multiplication is by using the “×” sign.  We write 3 × 4 = 12, and we typically pronounce that as “three times four makes twelve”.  The main reason that “×” is so closely associated with elementary school is that this symbol is so easily confused with the letter “x”, and “x” is so much the domain of secondary school mathematics, as the primary symbol for an unknown amount.  At least, this is the standard explanation passed on from the distant past as to how come we switch in middle school to a different notation for multiplication.  This new notation is “•”, a dot.  So “a•b” is the new way to write “a times b”.  Unfortunately, this dot is not suitable as a universal symbol for multiplication either, since “3•4” (three times four) is very easily confused with “3.4” (three and four tenths).  In other words, the dot looks a lot like a decimal point, and neither your typical kids’ handwriting nor a typical laptop keyboard will let you easily distinguish the two.  So, the middle school textbooks tend to carefully use the “•” for multiplying letters (a•b), but also combinations like 2•a, but then switch to the old “×” to render “3×4”.

At roughly the same time, or later, the textbook will also introduce the idea that multiplication need not be written at all:
“ab” is read as “a times b”.   This very useful convention, which has been standard for centuries now, allows us to write
2x^2 + 6x + 4 to mean 2 \times x \times x + 6 \times x + 4 , eliminating any confusion between × and x.

However, this notational standard of writing nothing for multiplication (more precisely, to indicate multiplication by juxtaposition) comes at a cost.  This cost is quite considerable.  Yet because we adults are so used to this standard, we are barely aware of the cost.  To become aware of part of this cost, you can watch kids when they are first learning this convention.  Since juxtaposition doesn’t work for 12 times 34: nobody would see 1234 and be able to tell whether it meant 1234 or 123×4 or 12×34 or 1×234, a different convention for multiplying numbers is needed.  Teachers will tell kids to write 12(34) to mean multiplication.  The multiplication is still expressed through juxtaposition, and the parentheses are now used to make clear what the pieces are that are being juxtaposed.  For kids, this is not real obvious, so here is one cost.  A bigger cost, in my mind, is that it pretty much forces variables to be single letters.  This is not something that kids seem to have trouble with, and not something that teachers typically see as a limitation.  In the standard notation, “cost” means “c×o×s×t”, and can’t be a single variable that indicates how much something costs.  And indeed, if you look at college-level math texts, you will see how many alphabets mathematicians have appropriated (Greek, Hebrew, German) in order to have enough different symbols.  Even so, the convention is not used consistently.  When students see sin(45) in secondary school, they know somehow that this means the sine of an angle of 45 degrees, and not s×i×n×45.

When communicating mathematical formulas to computers, whether when writing computer software or entering formulas in spreadsheets like Excel, entirely different conventions are used, born from the limitations of typical keyboards.  The computer-based conventions resemble \sin(x)   a lot more than \sum_{i=1}^{10} t_i .
Since computers require very precise communication, a lot of work has been done to come up with clear and unambiguous languages for formulas.  Though there isn’t a single standard for these, there are commonalities in the conventions that have been developed.  In almost all of them, a variable is not restricted to a single letter, it can be a construct like total_cost_before_taxes.  In almost all of them, juxtaposition is avoided and all operations have to be explicit.  Many of these computer-based conventions rely on the “*” sign for multiplication.  This asterisk is found on all keyboards, and looks somewhat like a cross, a “×”.  Most importantly, it is not easily confused with x.

Here, then, is my proposal: introduce the asterisk “*” fairly early on in school as a variant of “×”, right around the same time that decimal numbers come into play.  Don’t make a big deal of it, don’t require its use, just get students used to seeing it as meaning the same thing as “×”.  This is not a new thing to students: variant notations for division are very common: “÷”, “⁄”, “—”.  So now multiplication has a variant, and this variant can gradually replace “×” and instead become a variant of the juxtaposition learned in middle school.  So a×b moves through a*b to ab, whereas 3×4 moves to 3*4 and never need become 3•4 or 3(4).

Pros: no confusion, no ambiguity, easy to use on a keyboard.  Kids learn this easily ( I’ve used this for years, and never had to explain it to kids beyond “this is just another way of writing ×”).   Secondary pros: easy transition to Excel or C# or a number of other computer languages that use this same convention for multiplication.

Cons: teachers are generally unfamiliar with this convention, and especially elementary grade teachers.



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