Mathematical Notation and Schools: The Series

Here’s a summary of the series (thus far) of posts on mathematical notation, with links. This allows you to read them in order, from oldest to most recent.

In this series, I’m investigating variations of standard mathematical notation, with an eye on its use in schools.

1 – In the first installment, I look at the raised minus sign you see in American schools, to indicate a negative number.  You see ⁻7 to indicate negative seven, distinct from the normal minus sign used in e.g. 10 – 7 to indicate subtraction.  How is that useful, and if so, why do people stop using it after a certain grade?

2 – In the second installment, I look at notations for multiplication, and the way that “×” tends to be phased out in secondary school.  You see “•” used in middle school, but both “×” and “•” are too easily confused with common symbols in middle school: “x”, the variable, and “.”, the decimal point.  The secondary school (and later) convention of simply juxtaposing things to indicate multiplication (where 4ac means 4 times a times c) works, but makes it necessary to write 3(4) to mean 3×4.  I suggest an alternative in using “*” as a variant for “×” from about fifth grade on.

3 – In the third installment, I look at notations for sequencing and nesting of operations, and suggest an alternative notation for parentheses which I call bags.  These bags are easy to draw, but hard to type on a keyboard.  On a keyboard, these bags naturally devolve into ordinary parentheses.  Bags are easy to nest, and they look just as natural for terms as they do for factors.

4 – In the fourth installment, I started a look at the many uses of the equals sign, confusing if not conflicting, and looked at a common non-symmetric use of it, the one you see on many calculators: “do this now”.  What alternative notations might help disambiguate this?

5 – In the fifth installment, the use of “=” for equivalence, e.g. x + 1 + 1 = x + 2 is highlighted.  The “=” is used here to assert that left side and right side will evaluate to the same value regardless of the value chosen for x.  Is this use of “=” worth differentiating from the use of “=” in equations, and if so, what notation would help student learning?

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Mathematical Notation and Schools – 1

Notations for negative numbers

This blog entry marks the start of a series about mathematical notation and its uses in school.  I intend to look in detail at aspects of the standard mathematical notations, and how these help or hinder understanding of math in the various grades.  I also intend to look at special notations introduced in school to aid in understanding key math concepts, and to propose additional such notations.

A simple and useful example is the use of two different “minus” signs in school (at least in the USA), the normal “-” sign for subtraction, and the raised minus sign for indicating a negative number, ⁻7 for negative seven.  This notation, typically in use up to 8th grade, allows the text books to write  8 – ⁻7 rather than having to write 8 – (-7), for instance.  Teachers typically think of this negative number notation as a kind of training wheels.  This is akin to the use of pointing in Hebrew to indicate vowels, which is primarily used in texts for children or beginning Hebrew students.  Indeed, most students are encouraged to drop this special notation as they get further along in school.

From my own observations with students up to 8th grade,  this use of a distinct minus sign to mark a negative number is both successful and non-intrusive.  By ‘non-intrusive’ I mean that, as far as I can tell, the notation can usefully coexist with the standard notation, and that if some 8th grader decided to keep using that notation for the rest of his or her life, the rest of the world would barely notice and would not be in the least inconvenienced.  This would be true even if the student ended up in an engineering school or became a math major.  The use of the raised minus sign (the official name of this symbol appears to be superscript minus) is a harmless variation out there in the world, and is demonstrably useful in the lower grades.  These may be precisely the conditions under which a variant notation can spread far and wide.

In this series, I will bring up other examples of variant notations and try them out, to see if they meet these same tests: harmless in the wider world, and useful in the context of math education.  In addition to looking at notation, I will also occasionally bring in examples of variant vocabulary and terminology.

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Operators, Functions, and Properties – part 43

In the last several posts in this series, we’ve looked at cyclical patterns of behavior of numbers and operators.  For example, in the previous post we saw that when counting up, the last digit of a number consistently repeats after ten counts.  We also saw that the last digit of a sum only depends on the last digit of the (positive whole) numbers we are adding.

This is shown in the table above, and  it is easy to check this on particular numbers, e.g. 28+2076.  The table predicts that the last digit of 28+2076 is the same as that of 8+6, which the table shows us as 4.  Since 28+2076=2104, its last digit indeed matches the 4 from the table.

If we wrap a number line around a cylinder, kind of like a slinky, as we did in previous posts, we can get cyclical behavior that is not necessarily based on cycles of ten.  Cycles of ten just happen to be very easy to spot from looking at the digits of the number, as our standard way of writing numbers is based on cycles of ten.

To see cycles other than cycles of ten, we need to stop focusing on the last digit of the number, and instead focus on the remainder after division.  This actually works for cycles of ten also, so let’s look at that first.  What is the remainder of 7653 after division by 10?  Well, 7650 is the multiple of 10 that comes closest to 7653 (yet stays below it), and so there are 3 left.  The remainder after division by 10 of a whole positive number is simple its last digit.

The diagram above shows the same result we had before, but this time stated in terms of remainders.  Now we can pursue similar results using different cycles.  Below is one based on remainders after division by 9:

This result is the basis of the well-known procedure of “casting out nines” by which people used to check their arithmetic before the widespread availability of electronic calculators.  The “casting out nines” process was considered convenient in part because the remainder after division by 9 is relatively easy to determine.  Though not as easy as finding the remainder after division by 10, you can get the remainder after division by 9 without doing long division, or any division for that matter.  Instead, it is sufficient to add up all the digits of the number.  For the number 7653, adding up all the digits gets us 21.  The claim (not further pursued here) is that the remainder of 7653 after dividing by 9 is the same as the remainder of 21 after dividing by 9.  We can do this trick – adding all the digits – repeatedly until we arrive at a single digit.  For 7653, this takes two steps: 7653 → 21 → 3.  And indeed, 7650 is a multiple of 9 (9×850), so we would agree that the remainder for 7653 is 3.

For the rest of this post, I’d like to introduce some of the common mathematical notation for expressing the kind of relationships we’ve been exploring.  It gets too cumbersome to continue to say things like “the remainder after dividing by 9”.  Unfortunately, the standard notation, though compact, has some drawbacks as well.  But here we go:

7653 ≡ 21 (mod 9)

is the standard notation for: “both 7653 and 21 give us the same remainder after we divide each by 9.”

The standard pronunciation of this notation would be: “7653 is equivalent to 21 modul0 9″.

In textbooks you often see it defined as follows:

a ≡ b (mod m) if a−b is a multiple of m.

Usually, this notation is only used if m is a whole number, and positive.  Typically, a and b are whole numbers as well, though they can be negative.  For example,

−14  ≡  6 (mod 10)

is considered valid, as -14 and 6 are 20 apart, an exact multiple of 10.

Many people think of the modulo notation as having less to do with division than simply with repeated subtraction or addition.  You can get from -14 to 6 by repeated addition of 10, and you can get from 7653 to 21 by repeated subtraction of 9.

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Operators, Functions, and Properties – part 42

In this series, we’ve been looking at black boxes, where you put something in and something comes out.  From looking at its behavior, you can often propose a model for what is happening inside.  If you build the model, and it behaves the same way as the original black box, you’ve got something that’s equivalent to the original black box.  Equivalence doesn’t mean equality – there may be many ways to model what is going on in the black box, and there may be no easy way to choose one model over another.  Our exploration in this series has led us in many interesting directions.  Most recently we looked at what happens if you take a number line and bend it, like a slinky, into a cylindrical shape.  In this post, we bent the number line so that numbers 4 apart would line up one above the other, and each number on the number line could be associated with either north, east, south or west.  In the most recent post, we imagined the number line being wrapped around an old-fashioned clock so that each number would correspond to an hour from 1 to 12.  On these slinky number lines, we looked at the effect of one operation, addition, and found that whether the result of addition of two numbers points north, east, south or west depends only on the compass direction of the two numbers involved and on nothing else.

This repetitive nature of a number line bent around a slinky might also remind us of the scenario we looked at in the very first post of this series: there we looked at a mysterious machine with a “next” button and a window in which a number shows.  Each time we hit the button, another number shows in the window, in what seems to be a predictable repetition of the digits 3,4,5,6,7,8,9,0,1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6,…   With the perspective of the recent posts, we can now think of a number line bent like a slinky – completing a full turn every ten numbers.  The mysterious box may move us along the number line on the slinky and show on the display where in the cycle we are.

The notion of things repeating after ten steps is of course central to our decimal number system.  It applies most literally to the right-most digit in a number, as we count up.  Let’s assume we start with the number 966839, as on the mechanical counter below, and then repeatedly count up by one.

We would produce the following sequence of numbers: 966839, 966840, 966841, 966842, 966843, 966844, 966845, 966846, 966847, 966848, 966849, 966850, 966851, 966852, 966853, …  When you only pay attention to the right-most digit, you see them repeat in a cycle of ten steps.  But the other digits in the number each cycle through a set of ten values in turn.  It just happens a lot slower.  If you take the same set of numbers, and look at the second digit from the right, you would see the sequence 3, 4…4, 5…5, etc., where each digit sticks around for ten steps and then changes to its next value.  In the mechanical counter, this is realized by having each digit position consist of an identical wheel (ten digit values in the same sequence), and kicking each wheel forward only when the wheel to the right of it transitions from a “9” to a “0”.  The effect of this arrangement is that the wheels to the left rotate at a much slower rate than those on the right.  By putting enough wheels together in this arrangement, we can achieve very large cycles; in the mechanical counter shown, the overall cycle is one million.  This means that one million kicks of the right-most wheel will bring the whole counter back to its previous state, after which the whole thing repeats.

The rightmost wheel corresponds to the rightmost digit of the number.  If I have two numbers, and I focus exclusively on the rightmost digits, I can predict what the rightmost digit of the sum will be.  Also what the rightmost digit of the difference will be, and the rightmost digit of the product.  E.g. if I have a number X7 and another Y8, where the X and Y represent the digits that are covered up, you can predict that the sum will look like  Z5, and that the difference of X7 – Y8 will havea rightmost digit of 9, and the product X7 × Y8 will end in a 6.  Unfortunately, you can’t predict with certainty what the rightmost digit of the quotient is, even if there is no remainder.  This turns out to have something to do with 10 not being a prime number.

If it seems obvious to you that the rightmost digits of the sum and product only depend on the rightmost digits of the original numbers, please note that you cannot predict the next digit over quite so easily.  The second digit from the right of the sum does not depend only on the second digits from the right of the addends, and the second digit from the right of the product does not depend only on the second digits from the right of the factors.  However, it turns out that you can predict the final two digits of the sum and product by knowing only the final two digits of the addends or factors.  If our gear machine had wheels that had a hundred entries on them instead of ten, marked from “00” to “99”, we could again say that the position of the final wheel of the product could be determined from the position of the final wheels of the factors.

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Operators, Functions, and Properties – part 41

In this series, we’ve pictured operators as black boxes with inputs and outputs.  In the previous post we looked at rather ordinary addition from a less ordinary perspective: we imagined the number line arranged as a cylindrical spiral, like a slinky.  We imagined that numbers four apart would end up vertically aligned, with one above the other.

We’ll be looking at a variation of that in the current post, and our vehicle for introducing this will be an old-fashioned alarm clock:

We may be getting close to an era where children  won’t have any idea how to read time on a clock like this, never even having seen one in grandma’s house.  The saying “Even a broken clock shows the correct time twice a day”  will not make any sense to them either.  If a digital clock somehow got stuck and showed a single static display, we’d still expect it to show whether it was AM or PM, we would expect it to show the date, we might expect it to show the day of the week.  Even without the day of the week, such a clock would show the correct time only once a year.

The hour pointer (“Mickey’s little hand” for those from the right country and the right era) in the picture is pointing at the “1”, and exactly one hour later, it will be pointing at the “2”.  Each hour it advances one number around the clock face, and it keeps doing so: one hour after it is pointing to the “12”, it will point at the “1” again.  So if we focus only on the hour hand, and ignore anything else about this clock, we would expect to see behavior that strictly repeats every 12 hours.

The picture above symbolizes the “advance one hour” or “one hour later” effect on the position of the hour hand.  So if the input of the box s “11”, the output will be “12”, if the input of the box is “12”, the output will be “1”.  This is shown below in table form:

Now please look at the diagram below:

The top row of this diagram shows the repeated application of the “one hour later” operator.  The input on the far left will be a clock time (whole hours only), and as we move rightwards through the boxes, we trace the time on the clock going forward one hour per box.  The bottom row is a simple counter, starting at zero and going up by one each time.  What’s different from what we’ve done before: the intention is that the two run together: at one point, both machines would be at A, and some time later, both would be at B – they stay in synch.  There are other ways to model the synchronization of different parts of a machine, and we’ve done so earlier in this series.

At A, the bottom machine shows an output of 1, and the top machine shows the time one hour later than the input time.  At B, the bottom machine shows an output of , and the top machine shows the time two hours later than the input time.  Similarly, at the very right of the diagram shown above, the output of the top row will show the time 14 hours later than the input time, and the output of the bottom row will simply show 14.

In the diagram above, we show the same idea, two machines running in synch, one tracking the hours on the clock face, the bottom one counting the hours gone by.  We no longer assume the bottom row starts with zero.  Note that this time we’ve shown exactly 12 stages.  For the bottom row, looking at the net effect of running 12 stages – of adding 1 in each stage – is easy.  The next effect is adding 12 to the incoming cont.  For the top row, the net effect of running 12 stages – of advancing the clock face one hour in each stage – isn’t that hard either: regardless of the position on the clock face when we start, after 12 stages we will have made a full cycle around the clock face, and we will have returned exactly to the point where we started.  We can show this as follows:

The top box is what we’ve called an empty box or an identity operator in earlier posts.  One of the implications of this is that if we a long train of the “one hour later” operators, we can take any sequence of 12 consecutive “one hour later” operators, and remove it from the train without altering the net effect of the train.

So if we had a machine with 38 stages, with each stage a pair of “one hour later” on top and an ordinary +1 on the bottom, we could find 12 consecutive stages and reduce them as above, and repeat this process till we were left with:

From the bottom row we can confirm that there were indeed 38 stages (the net effect of the bottom  row is +38); from the top row it is clear that the net effect is two stages’ worth of “one hour later” (i.e. two hours later).

In terms of their effect on the clock face, 38 stages and 2 stages are equivalent.  The standard language for that is that 38 and 2 are equivalent modulo 12.

We will look further into this in subsequent posts.

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Operators, Functions, and Properties – part 40

In this series, we’ve spent considerable space playing with operators, which we’ve pictured as black boxes with inputs and outputs, often as part of a machine with state.  In more recent posts, e.g. this one , we’ve looked at operators that we likened to moving around on a contour line, towards a representative value in an equivalence class.

In the prior post, we looked at a number line on a slinky, and picked an equivalence relationship simply based on which numbers are vertically aligned.  We’re going to take a closer look at slinky arithmetic,

mapping numbers onto the slinky as below – so that 5 ends up directly above 1, and 6 directly above 2, etc.:

It may be helpful to think as the “1” position as South, the “2” position as East, the “3” position as North and the “4” position as West.  As you move around the spiral, you keep coming back to South etc, but you’ll be a little higher up.

Now let’s look at arithmetic on the slinky, starting with addition.   We can take some well-known addition fact, like
5 + 10 = 15, and then note the compass positions of the numbers involved.  5 is South, 10 is East, and 15 is North.  The question I’m now curious about is: If I start with a number in the South and add a number in the East, will I always end up with a sum in the North? After all, the compass directions seem to have nothing to do with how high up we are.  On this number line, East comes after South, and North comes after East, and West comes after North, and after West comes South again.

In the diagram above, the top line shows the compass marks for the numbers from 0-18.  The diagram at the bottom left shows  a normal addition table.  In the table on the right, all the numbers from the left have been replaced by their compass markings.  As you can see, the pattern on the right is completely regular and repeats after 4 rows, after 4 columns and after 4 diagonals.

From that table on the right, we extract a four-by-four core piece, and show it below.  It is this piece that repeats over and over.  We can translate the compass marks back into numbers, as shown below on the right.  We used the numbers 0-3 as representatives of the equivalence classes – unlike the situation in the prior post, we now have at least some reason to pick this set of numbers over other numbers.

In some regards, the table on the right does look like an addition table: most of the entries match what we would expect.  It is only where we would normally expect a sum of 4 or greater that the table looks somewhat strange.  Instead of 4, we find 0; instead of 5, we find 1, and instead of 6, we find 2.  This is at least consistent with the equivalence relationship we’re looking at here: 4 is directly above 0, 5 is directly above 1, and 6 is directly above 2 in our slinky.

The style of addition in this last table has a fancy name: it is known as addition modulo 4.  The same phrase, modulo 4, is used to talk about equivalence classes for whole numbers: when any two numbers that are 4 apart are considered equivalent.  For example, people would say that 5 and 9 are equivalent modulo 4, and that 9 and 13 are equivalent modulo 4.  By extension, 5 and 13 are also equivalent modulo 4.  Some people would describe equivalence modulo 4 in terms of division: two numbers are equivalent modulo 4 if they have the same remainder after dividing by 4.  Yet another way to talk about it is in terms of multiples: two numbers are equivalent modulo 4 if they differ by a multiple of 4.  All these ways of talking about it amount to the same thing, but that may not be immediately obvious.  “Two numbers are equivalent modulo 4 if they point in the same compass direction on the slinky” also expresses the same idea.

So far, we’ve talked about modulo 4 and only looked at the operation of addition modulo 4.  We can extend in both directions: we could look at subtraction, multiplication, and division modulo 4.  We could also extend in a different way and look at equivalence modulo 5 or modulo 6 etc.  We will take this up in future posts.

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Operators, Functions, and Properties – part 39

In this series, we’ve been looking at operators in various settings, and looked for ways to use them to shine a different colored light on some aspects school math.  In the last four posts, we’ve explored the notions of equivalence  and equivalence classes and representatives of equivalence classes – and saw a number of interesting things.  One, we saw that equivalence classes can be likened to contour lines on a geologic survey map, the number on the contour line that indicates the altitude can be likened to the property that is the same for all the members of the equivalence class, and that one useful kind of operator is the operator that takes us from any point on the contour line towards the point (the representative member of the equivalence class) that carries the label (the number).  Two, that many computations involving two numbers can be split into two parts: the first part locates the two numbers as a point on the map, and the second part moves us from that point along the contour line to the representative point which carries the label.  And there is additional stuff that we’ve hinted at but not anywhere fully developed, like the idea of introducing new kinds of numbers through equivalence classes.

In this post I want to get started on playing with what’s known as modulo arithmetic, also called modular arithmetic, from our framework of operators and equivalence classes and properties.

As a starting point, let’s take another look at the number line.  In this older post, I suggest that this one thing, the number line, takes on a different character as students progress from using it for counting to – much later – locate numbers like π and √3 on it.  When we are just counting, the fact that the numbers on the number line are carefully spaced is not critical – but what is important is that the mark for 4 comes after the mark for 3 and the mark for 5 comes after the mark for 4, etc.  The number line is usually drawn as a straight line, but this may only be critical once we start to use number lines as axes on graphs.

The measuring tapes shown above, whether in inches or centimeters, are essentially number lines also, though they aren’t necessarily straight lines.  If you measure things with them, you probably want to stretch them out to be straight – but if you want to use them for many of the functions you use a number line for, it may be quite sufficient to unroll the tape enough so you can see the numbers you care about.

You could even take a slinky and turn it into a number line.

To do so, you would have to put markings along the rim.  Though the slinky would never be suitable for measuring like a measuring tape, it might serve quite well as a number line.

A sketch of one possible such number line is shown below:

The numbers 1-7 are shown; the slinky extends in both directions, as indicated by dotted lines.  Numbers are shown evenly spaced – in this particular example, precisely 4 whole numbers are placed along each circle.  For example, by the time we get to “5”, we are exactly where “1” was, just slightly above it.

In prior posts, we’d look at a particular operation, and then decide on equivalence classes based on equal values of that operation, this time we’ll turn things around, and decide on the equivalence classes first.   Can we do this and get away with it?  Sure, if we cast it in terms of: “with respect to what operation(s) will these equivalence classes in fact be equivalent?”

On the slinky number line just sketched, we are going to consider the equivalence classes based on vertical alignment, like this:

Another way of talking about this equivalence is that we consider equivalent all numbers that are 4 apart.  The representatives of these equivalence classes, just like in our earlier posts, are found at the end points of the blue arrows.  So 1, 2, 3 and 4 are the representatives.  There is nothing magical about this particular choice, you can make a very good case for using 0, 1, 2 and 3 as the representatives instead.  We could also have picked -2, -1, 0 and 1.  In a sense, it is too early to argue which make for the best representatives – we haven’t looked at relevant operations yet.

In the next post, we’ll apply this particular equivalence structure to the operations of addition and subtraction.

x

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