Normat 60:4, 171–191 (2012) 171

On comparison and change

Leif Önneﬂod

Stora Hammar 301

370 42 Torhamn

leifskriv@telia.com

On Comparison and Change

Memory is short. The developers of modern mathematics once told us how to

reason about mathematical operations. Now important parts of this are forgotten

or misinterpreted. During the late 19

century eﬀorts were made to bring the

theoretical base for arithmetic to a closure. It ended up in a theory for logical

manipulation of symbols. But the want for “pure abstraction” went alongside of

a growing dissociation from reality. In the end we got proper theories for symbol

expressions, but poor descriptions of how to lend structure to real-world mathe-

matics. Accordingly a main issue in education is the diﬃculties with interpreting

between math as experienced in everyday life and math as expressed by symbol

expressions. They don’t agree in a simple and straightforward way.

In this essay I will lay forward some informative models for central mathematical

ideas and concepts. Those models take advantage from doing s ome deviations from

the conventional way of writing and interpreting expressions. Nevertheless do those

deviations agree better with modern mathematical theories, and concurrently with

practical thinking, as well as the reasoning of the ancient Greeks. The way to per-

ceive expressions will mutually support the perception of mathematical structures

in real life.

What a comparison tells

We set oﬀ with the concept of comparison. Seemingly a trivial concept, and still an

obstacle for pupils, when they meet word problems where comparisons are involved.

Nowhere (in any textbook) I can ﬁnd an in-depth analyse of the meaning and the

consequences of comparison. The main road oﬀered, to an understanding, is paved

by myriads of examples.

The ﬁrst thing to notice is that the answer to a (numerical) comparison comprises

both a number and an operation. The latter is more or less “hidden” in the way

we formulate the answer. Maybe we can accept that ‘times’ in the statement “One

foot is twelve times bigger than one inch”, is an operation. But what about the

statement: “One foot compared to one inch is eleven inch more”? Or: “Three pounds

are two pounds less than ﬁve pounds”. Some acquaintance with Latin helps to ﬁnd

the operations in disguise. ‘More’ says ‘plus’ in Latin and ‘less’ says ‘minus’! For

172 Leif Önneﬂod Normat 4/2012

convenience I introduce a symbol for ‘compared to’ – I suggest ‘

◊

’ – and write

the last two comparisons as “1 foot

◊

1 inch = +11 inch” and “3 pound

◊

5 pound =

= ≠2 pound”. Here the plus and minus sign informs whether ‘more’ or ‘less’ applies.

(Hence less need for words.)

In the same manner we can write “1 foot

◊

1 inch = ·12” for the ﬁrst compara-

tive statement above. Apparently comparisons can be made with diﬀerent methods:

an “additive” and a “multiplicative” one. To stress the diﬀerence, you may use

the symbol ‘

•

’ for the latter comparison and write “1 foot

•

1 inch = ·12”. T he

converse comparison is “1 inch

•

1 foot = /12”. Here ‘/12’ can be regarded as an

‘unit fraction’ (in Swedish: stambråk).

If this were all that there was to it, this would have been of minor use. However,

such elements as ‘+11’, ‘≠2’, ‘·12’ and ‘/12’, have a meaning on their own. They

are more but mere abbreviations for words. For a while, we leave comparisons and

pay our attention at elements of this kind.

Introduction of bound signs in integrated elements

It is a common notion, that an express ion like 6 ≠ 2+4 can be rewritten as

(+6) + (≠2) + (+4) (or as 6

≠

), where the parentheses enclose positive

and negative numbers. The problem is that those “numbers” have ambiguous inter-

pretations to real-world situations.

•

The purpose of using parentheses, is to rule

out subtraction and make the commutative and associative law apply.

There’s another possibility of interpreting (+6) and (≠2), but as positive and

negative numbers. That is as an “increase by six”, and a “reduction by two”, re-

spectively. In stead of ‘increase’ I will use the term ‘addition’, not in the conﬁned

mathematical sense, as it is known to us in Scandinavia, but in the more eve-

ryday sense, well-known to native English-speaking people. That is to say: here

‘addition’ © ‘increase’.

With (+6) + (≠2) + (+4) interpreted as a collection of additions and reductions,

the perspective can change from having “numbers submitted to operations” to

having “operational elements that compose”. This change is crucial. As the former

“numbers” are substituted for ‘operating elements’ like additions and reductions,

the intermediate plus signs reduce in signiﬁcance. They turn from active operations

to passive signs for composition. As such, they can be omitted. This suggests we

convert (+6) + (≠2) + (+4) into +6 ≠2+4. Those new number elements, supple-

mented with what I denote ‘bound’ (appendant) leading signs, obey with ease the

commutative and associative laws. E.g. +6 ≠2=≠2+6, that is: no matter what

order additions and reductions may come.

How then, to handle parentheses in the associative law? In expressions, fully

equipped with parentheses as in ((+6) + (≠2)) + (+4) = (+6) + ((≠2) + (+4)),the

law works well, but what about “(+6 ≠2) +4 = ...” ? Neither +6(≠2+4), nor

+6 ≠(2 +4) will do the thing. As the purpose of using parentheses is grouping, and

this now is directed at irreducible elements of change – not at numbers to be com-

bined by operations yielding new numbers – we need another kind of parentheses

submitted to that new kind of grouping. For reasons to be explained later on, the

natural choice is ‘square brackets’. A grouping like [+6 ≠2] yields, not a number,

•

Some take this diversity for an advantage. Not so done by pupils. They often nurture an

aspiration for single interpretations, to make life easier.

Normat 4/2012 Leif Önneﬂod 173

but a new element of change: [+6 ≠2] = +4. By using such square brackets, the

associative law works and will easily be understood: [+6 ≠2] +4 = +6[≠2+4]

i.e. [+4] +4 = +6[+2], where the latter (redundant) brackets surrounding sing-

le ‘changes’ are used only to highlight their origin. In plain words: the order of

grouping (additive) changes doesn’t matter.

The abstract concept of negative numbers, with one unique interpretation, does

have some quite diverse interpretations into real world. Although this is not the

main issue for this essay, subsequent sections will elucidate on the matter. The

basic notion for “negative number” (or rather “minus term”) will, in this article,

be “reduction”.

Another issue: Operations are said to be binary. Numbers are seen as targets

for operations. So can binary operations really be bound to numbers as if they

were unary? Actually, it’s a matter of choice. (After all, ‘+’ and ‘≠’ are used as

binary as well as unary operators.)

In the es tablished Category Theory, there are “morphisms”, where sample ex-

pressions may be ‘+4’, ‘·4’, and ‘≠1’. Morphisms can readily be combined by

composition. Morphisms change (morph) ‘objects’,

•

for example points on a

number line. Moreover, a translation of a point with ‘+4’ (drawn as an arrow

above the number line) is an example of ‘group action’ – in another discipline of

modern m athematics. “Operating elements” are ubiquitous. There are, of course,

mappings that out of necessity must be deﬁned as binary. Most mappings can

be done so, but that does not make it the one and only option.

A binary operation of major importance is composition! That one will be used

here.

Now back to “changes”. It lies near at hand to name those additions and reductions

‘plus terms’ and ‘minus terms’ in conformity with the current designations. Those

‘term elements’ simplify more issues but the me re practise of the commutative and

associative laws. For a start we can have a look at . . .

Order of operations

For expressions, we have “priority rules” stating the order of operations. Without

those rules, we would have to tell the order by means of parentheses. Instead of

(3 · (5

)) + (4 ≠ (36/12)) we can write 3 · 5

+4≠ 36/12 applying the priority rules.

Besides using those rules, a prerequisite to omit parentheses is that the associa-

tive law holds. This is commonly neglected. Take an expression like 4 ≠ 3 ≠ 2+1.

Only the ﬁrst one of the ﬁve (separable) orders of calculation – illustrated by

((4 ≠ 3) ≠ 2) + 1, (4 ≠ (3 ≠ 2)) + 1, (4 ≠ 3) ≠ (2 + 1),

•

4 ≠ ((3 ≠ 2) + 1) and

4 ≠ (3 ≠ (2 + 1)) – leads to the intended answer. No wonder doing the sums “from

left to right” is a widespread recommendation.

•

However, two kinds of problems

soon surface:

• How to compute 4 ≠ x ≠ 2+1 “from left to right”?

•

I.e. not (literally) changing an obje ct into another, but rather changing from an object to

another. All in compliance with how ‘mappings’ are to be understood.

•

This one is actually two “cases”, depending on which expression within parentheses that is

evaluated ﬁrst.

•

N.B. it follows the ﬁrst sample of orders.

174 Leif Önneﬂod Normat 4/2012

• How to perform computation shortcuts, as when cancelling the threes in

10 ≠ 3+5+3, or the x’s in 10 ≠ x +5+x, if the only familiar order is

“from left to right”?

Well, by adding still more “rules” (to be memorized), any problem can be settled.

That’s a safe horse, but not the one to be used here.

Now compare this to using “changes”. No more do diﬀerent orders of composi-

tion make any diﬀerence: [[+4 ≠3] ≠2] +1 = [+4 [≠3 ≠2]] +1 = ...

In addition, as the elements are trivially commutable, why not regroup like in

[+4 +1][≠3 ≠2] = +5 ≠5 ? An added x does not raise the complexity level:

+4 ≠x ≠2+1=[+4≠2+1]≠x =+3≠x. An intervening x becomes no obstacle

to compose remaining changes. A single “3 more” equals the aﬀect of the group of

“4 more, 2 less and 1 more” and may replace it.

•

The leading sign in each element

tells how the number aﬀects the expression as a whole. This is a crucial

piece of understanding.

That, what makes the order ((4 ≠ 3) ≠ 2) + 1 of calculation work, and work the

same as [[+4 ≠3] ≠2] +1, is that no left parentheses break the bond between

leading sign and trailing number. This too speaks in favour of “bound signs”.

The ordinary use of parentheses for regrouping purposes interferes in a complex

way with the operational structure. (As in (7≠ 5)+4 = 7+(≠5+4).) Regrouping

of elements using square brackets maintain a stable element interpretation. The

operation used between elements is ‘composition’ that is associative per s e, N.B.

Real-wor ld interpretation of terms

Let us have a closer look at the real-world correspondences to terms in expressions.

The trivial interpretation is as additions and reductions. What has to be stressed

upon is that in abstract expressions, this is the only conception of terms – they

express additions and reductions in some c ommon abstract unit.

•

There is yet

another interpretation in real life. To make my point, I go back to antiquity.

The ancient Greeks observed that the only possible way to relate two quantities

(–ﬂÿ◊µoÎ – an amount of units) was to add or s ubtract them, provided they had

a unit in common. The Greeks only recognised numbers as showing the magnitude

of some phenomenon: a number of feet, barrels, hours et cetera. The magnitude

represented a quantity. Only magnitudes with the same unit could be related.

•

One

way of relating was, so to say, to “compose” quantities by adding or subtracting

them.

Another way was to compare diﬀerent pairs of (multiplicative) relations. They

observed that 12 inch relates to 3 inch the same way 20 inch relates to 5 inch. If

the two relations were “the same”, they were said to be analogous; we had an

analogy.

•

The fact that 20 is 4 times greater than 5, was expressed as (with u

for some ‘unit’) “20 uisto5 u as 4 u is to u”. On purpose, I did not write ‘1 u’,

as the Greeks didn’t consider ‘1’ a number – ‘1’ was the unit itself. One unit is

•

This simple kind of rea soni ng is not accessible for quite a many pupils. When they become

familiar to it, th ey show reactions like relief and inspiration. The reasoning can easily be practi-

cally experienced by learners via hands-on materials.

•

Long since, already Aristotle pointed out that a unit can be abstract.

•

Their thinking alone, would demand for a lengthy essay to be fairly described. This is the

short-short version.

•

Greek logos = relation, ana- = the same. The ‘logos’ was the ‘ratio’ relation.

Normat 4/2012 Leif Önneﬂod 175

the unit, they reasoned. Nowadays we write “20 : 5 = 4 : 1”or“20 : 5 :: 4 : 1”.

All calculations corresponding to multiplication and division were performed via

“analogies”, but enough for that.

The s tateme nt “When relating (composing) quantities, the only operations to per-

form on them are addition and subtraction”, have a converse: “The only things

that can be added or subtracted are quantities”. (Alas, this cannot be inferred in a

few lines. The reader may accept this as an unproven postulate.) Well, how then to

relate this to additions and subtractions? The abstract terms are meant to reﬂec t

changes. There is a frequent misconception that expressions mirror some course of

events in real life. That is not so. The terms in an expression no more than mirror

how the quantities in reality contribute to compute the answer to a speciﬁc posed

question.

To make my point clear, let us imagine the scene of an empty bench. At 13:45

three people sit down. At 13:55 another two join. That makes them ﬁve. At 14:10

four of them leave. Now there is one left. At 14:15 the last one leaves.

Here are two samples of all possible questions to pose.

• How many more leaves at 14:10 than arrives at 13:55?

This is c omputed by 4 ≠ 2=2. Here we subtract the people that arrived

(= ‘more’) and those who left, corresponds to a “positive number”.

• Given that there are ﬁve people on the be nch at 14:00 and two arrived at

13:55, how many were they before 13:55?

This is computed by 5 ≠ 2=3. Here the 5 rightly is a positive number, but

once again we subtract with the arrivals.

Those examples might have shown my point. The idea of “contribution” is crucial

when to perform an interpretation from reality to an abstract expressions. It also

leads to the second interpretation of terms. The following reasoning may show that:

How do three people and two people contribute to the number of people on the

bench at 14:00? The answer is evident: they both add to the number of people!

Seen through the “glasses of the c ontribution idea”, all quantities (subject to plain

composition) correspond to plus terms. They add to the result. Thus, a quantity

cannot be negative.

•

However, the contribution to a computation c an be negative

as well! So if Kim loses weight from 120 pounds by 7 pounds, the computation of

the new weight can be written as +120 ≠7 = +113,where+120 is the quantity

“before”, ≠7 is the quantity change and +113 is the quantity “after”.

To conclude: positive terms can represent quantities or changes in quantity, whe-

reas negative terms only represent changes.

•

In its abstract general sense “change”

merely depicts relations. In real life terms are associated with quantities. Terms

are coherent (integrated) elements, that obey the associative and commutative laws

with ease. Grouping and regrouping turn trivial.

Reality structured with the RAY model

As the next step to close in on comparisons, we have a look at the RAY model.

The letters stand for Root or Reference (= origin value), Alteration (= change

•

This was why the resistance to negative numb e rs was so ﬁerce when they were introduced –

nothing in real life could have a negative size.

•

Ileaveoutthetopicof“directedquantities”.

176 Leif Önneﬂod Normat 4/2012

value) and Yield (= res ult value). The model simply states that a Root subject to

an Alteration results in a Yield.

For example: +5 kg

≠≠ ≠ ≠ ≠æ

+3 kg

+8 kg follows the pattern of R

≠≠æ

Here R =+5kg and Y =+8kg are quantities and A =+3kg a quantity change.

Writing the alteration over an arrow, emphasizes its role as an intermediate

•

between the “before” (R) and “after” (Y) component. Evidently, this models mathe-

matical events in real life. In most arithmetical (single-step) computations, the

values involved divide into the three categories of RAY. The need to perform a

computation arise when two values are known and the third is sought for. The

RAY model oﬀers a structure to the three constellations that the two known va-

lues and the sought value can form. The ﬁrst two are, built on the example above,

the following:

F. Forward computation R

≠≠æ

? e.g. +5

≠≠ ≠æ

Problem illustration: “A dog weighs 5 kg. After half a year, it weighs 3 kg

more. What is its weight then?”

B. Backward computation ?

≠≠æ

Y e.g. ?

≠≠ ≠æ

Problem illustration: “A dog weighs 8 kg. It weighs 3 kg more than half a

year earlier. What was its weight then?”

Evidently, the last problem is solved by +8 ≠3=+5, but why does the altera-

tion have to be reversed? The answer lies in the “forward alternative” ﬁgure. It

demonstrates (by means of an arrow), that the alteration is applied on a value – a

starting point – that is a root value R. In the B alternative, the only value known

to apply the alteration on, is the Y value. However, in order to apply A on Y,the

direction of the arrow has to be reversed. This is apparent from the ﬁgures above.

Reversing the direction of the operation is coupled with reversing the operation

itself. The RAY concept forwards the idea by help of the ﬁgure

≠≠ ≠ ≠ ≠æ

Ω≠ ≠ ≠ ≠≠

≠3

where a reversed operation (‘≠3’) accompanies the reversed lower arrow.

This mutual dependence between an operation and its direction, can easily be

demonstrated to pupils and trained. I believe the reader can imagine how this

can be achieved in practise.

How then, to transfer those alteration diagrams into ordinary expressions? Well,

by applying the principle that an alteration follows after the value to be altered.

For the “forward example”, we write +5 +3 = +8. For the backward example, we

write +8 ≠3=+5, as the ‘≠3’ alters ‘+8’, which is evident from the direction of

the arrow.

Many pupils experience that problems, with this kind of reverse reckoning, are

diﬃcult to grasp. In the absence of a model like RAY (and thereby explicit

instruction on how to identify the ‘A’ component that needs to be reversed)

the pupil is left to her intuition as the only means to manage the problem.

•

I.e. a mapping.

Normat 4/2012 Leif Önneﬂod 177

The choice to use the reverse operation becomes (unless equation techniques are

used) more of an immediate choice, based on intuition. The reasoning, if any, is

a reconstruction afterwards.

Separate inversion operator

The use of (operationally) integrated elements, with explicit operations, clears the

way for directing inversions towards operations instead of numbers.

•

As an inver-

sion of operation becomes an operation on its own, there is motive for introducing

a separate operation, reserved for this single purpose. I suggest ‘

’ to be a general

symbol for inverse operation. Using the RAY model we now can write . . .

forward computing as RA = Y

backward computing as R = Y

A (or Y

A = R)

When solving the problem above we can write:

?+3 =+8 =∆ ?=+8

+3 = +8 ≠3=+5.

If we want to reverse an op eration performed, in theory (abstract algebra), we

cannot do that directly. The reason is that in group theory, a group is about on e

operation acting on a set of elements. Accordingly, subtraction cannot be the one

operation, since it does not comply with the associative law. So how to reverse

the adding of 3 to 5? Well, in theory, by adding the inverse of 3 with respect to

addition. The converse to 5+3=8 will be 8+(≠3) = 5. Here the number is

inverted. However, as no one can comprehend how a group of eight people can

be reduced by adding “three negative people” (however they might look), this is

explained to be the same as subtracting, i.e. reducing, by three people. (It can be

questioned whether something impossible can be “the same” as something quite

natural.)

All the line, reversing (inverting) operations is simpler to comprehend. Com-

pare using

/3=· 3 instead of (3

≠1

)

≠1

=3 or 1/

3 =3. Parallel the use of

≠5=+5 to that of ≠(≠5) = 5. Complex derivations, of how to create and

calculate inverses, become superﬂuous. Explicit knowledge of opposite operation

pairs will do.

Another point is that using subtraction to create “additive inverses”, and divi-

sion to create “multiplicative inverses”, obfuscates the real-world structure of

comparisons and backward computing. (The purpose not being to ‘subtract? or

‘divide?, but to reverse operation.) Above that, it adds an extra diﬃculty, as

a subtraction by a number must be converted into a negative number, before

an outer “subtraction” can be applied on the expression. Likewise, to perform a

division of an expression holding a division by a number, the number must be

converted into its inverse (with respect to multiplication). Those manipulations

have no correspondences whatsoever in real-world, when we solve problems where

inversions occur in the solution. Unlike this, direct inverting of the operations

themselves, reﬂects the structure to be understood.

•

“Operation” has a dual reference to real actions and to symbols in expressions. You may tell

which one by saying ‘reverse an operation’ or ‘invert an operation’, respectively. By substituting

‘invert a number’ for ‘invert an operation’ the established language is maintained.

•

In abstract algebra, the use of inverting operators (≠ and

≠1

)compliesbetterwiththe

structure of comparisons.

178 Leif Önneﬂod Normat 4/2012

Another view on computing ‘change’

Now we are set to attend to the third and last constellation that the known and

sought values in RAY can form. It looks like R

≠≠æ

Y. For instance: +5

≠≠æ

+8.

In the “dog example”, the problem can read:

“A dog weighs 8 kg. Half a year earlier, it weighed 5 kg.

How much has it gained in weight?” (a)

The question has some alternatives as . . .

“How much less did it weigh then?” (b)

“How much more does it weigh now?” (c)

“How much does it weigh now compared to half a year earlier?” (d)

“How much did it weigh half a year ago compared to now?” (e)

Perhaps your attention were drawn to the words ‘less’, ‘more’ and ‘compared to’?

The two ﬁrst words hint at we are computing an alteration (change). This implies

that a comparison is performed. Only the last two question alternatives, make the

comparison stand out clear. Moreover, they shift the responsibility to judge the

“sign” (more or less hidden in ‘more’ and ‘less’), on to the pupil.

•

The RAY mo-

del helps to sort things out. As the answer to a comparison tells how to change a

reference value (root, R) to the value, which is to be described, it follows from the

formula R

≠≠æ

Y that Y

R = A. (In the next section we see how the ‘

◊

’ compa-

rison yields an ‘

’ inversion.) Applying that on the last two question alternatives,

we get . . .

for (d) question: +8

◊

+5 = +8

+5 = +8 ≠5=+3, i.e. “more”.

for (e) question: +5

◊

+8 = +5

+8 = +5 ≠8=≠3, i.e. “less”.

Observe, that the order of elements in the comparing expression, strictly follows

the wording (order) of the comparison.

When words like ‘gained’ in (a), ‘less’ (b) or ‘more’ (c) are given in advance,

the pupil does not need to reﬂect on the structure of the comparison. The order of

calculation will always be “the greater number subtracted by the lesser number”.

In the end, this lack of understanding strikes back. To get the sign right, as when

calculating the slope of a line, an ability to apprehend a direction in comparisons

is helpful. It is my experience, that this notion of “direction” seldom is accessible

to the pupils as a working knowledge.

The fact that the comparison can be calculated by using the inverse of the root

(R), can, but of course, be stated as a rule and lent credit to by cases. Although,

the apprehension that a comparison leads to the calculation of the A component

in RAY, gives us another option.

The two-step method for computing ‘change’

To search for the A component means to search for what turns R into Y. It takes

two trivial steps to arrive at Y. Firstly, we cancel R by applying

R. Secondly, we

append Y itself. Altogether we have that R [

RY]=Y.

•

Mind there is a direction in the comparison!

Normat 4/2012 Leif Önneﬂod 179

Example: Y◊

R =+8◊

+5 = [+8

+5],

since +5 [+8

+5] = [+5

+5] +8 = +8.

Comparisons where “negative numbers” (i.e. reductions) are involved, don’t diﬀer

in complexity level.

Example: ≠7 ◊

≠2=≠7

≠2=≠7+2 =≠5, that is: “ﬁve less”.

Interpretation: A reduction of 7 makes (something) reduce 5 more, than a reduc-

tion of 2. Quite obvious.

Here is a suggestion on how composition of terms may be learned: Perform the

“changes” involved, on a positive quantity of suﬃcient size (which may be zero). A

comparison between the size “before” and “after” the changes have been applied,

reveals the total change. This is a winning concept in didactical situations. It is

easily demonstrated by help of hands-on materials.

Example: Apply [≠7+2] on +10. The result is +5. Finally, as +5 ◊

+10 = ≠5,

this implies that [≠7+2]=≠5.

•

I name this the “two-step method”. In the world of arithmetical symbol expressions,

this is no new phenomenon. The equation ax = b has the solution ba

≠1

•

where

you can identify ba

≠1

with Y

R. The (possible) novelty is twofold:

1. An understanding of how to reason about comparisons in real life (where RAY

is helpful).

2. A simpliﬁed universal method on how to construe the “inverse” by means of

‘

’. (There is more to this.)

A more pictorial view of the two-step method can be obtained by starting from:

Ω≠≠≠ ≠≠

≠≠ ≠ ≠ ≠æ

This shows how the quantities +a and +b are produced from a common origin in

the center – the neutral quantity of +0. To compare +a with +b, is the same as

to ﬁnd what transfers +b into +a. By reversing the right arrow (and inverting the

operation), you arrive at the two steps to take:

Ω≠≠≠ ≠≠

Ω≠ ≠ ≠ ≠ ≠≠

The reason to invert the operation for the comparison reference value +b, becomes

obvious. Other “intermediate values” than +0 are pos sible. As to compare +402

with +397, the intermediate +400 can be an option. Incidentally, in Davydovs con-

cept writing diagrams have proved to help learners get a “picture” of the problem.

More on scalar interpretations

To gain full beneﬁt from the RAY model and the two-step method, one has to

expand on the issue of real-world correspondences to integrated elements. So far

we have come to know that . . .

The terms correspond to quantities with units,

and to some extent that . . .

•

With hands-on materials, the obvious “circle” in this reasoning is broken, as the material

provides the answer to the comparison, not yet another expression of a similar kind.

•

Provided it is commutative, as holds for basic opera t ions.

180 Leif Önneﬂod Normat 4/2012

The factors correspond to relations between quantities.

Thus ‘·3’ demonstrates the relation between a quantity of, say, +3 m and the unit

+1 m. It also illustrates the relation between an inﬁnity of other pairs of quantities

like +15 s and +5 s or +0.189 kg and +0.063 kg. As the concept ‘factor’ is deﬁned

to refer to multiplication, the notion of ‘scalars’ seems to agree better to the use

of both of the two opposite operators, as in ·5 and /5.

In abstract algebra, rational numbers are deﬁned using the ‘equivalence class

concept’. Pairs of integers e.g. (a, b) and (c, d) are said to be equivalent if ac = bd.

(This is the old safe “golden rule” for proportionality in a modern outﬁt. Other

equivalence conditions can be stated, but this is the one of interest for now.)

Pairs that are equivalent all represent “the same” relation. Pairs equivalent to

(a, b) form an ‘equivalence class’, denoted by [(a, b)] or [a, b]. The equivalence

class [(3, 1)] will correspond to the scalar [·3] as . . .

Û The identity scalar ‘·1’ can be regarded as to implicitly provide the number

‘1’ in the pair.

Û The reference to multiplication is explicit and can be deﬁned as to address

the equivalence condition above.

Hence [·3] can be considered an equivalence class. It comprises the inﬁnite many

relations between pairs of quantities. N.B. that the use of square brackets in

equivalence expressions complies with this interpretation of [·3]. In the same

manner [/5] repres ents the equivalence class of [(1, 5)] (numbe r order reversed).

Finally [·

] represents [(a, b)] = [(

)] = [(

, 1)].

In the ancient Greek mathematics (due to Euclid), the relations between quan-

tities (or ‘magnitudes’) always were expressed as pairs of quantities. (In a way,

this is a fundamental kind of view to comprehend.) Their “analogies” have its

modern parallel in equivalence classes.

On combining quantities and scalars

In order to combine quantities and scalars, we have to settle on two matters.

• The scalar primarily shows the relation between quantities. In practise, the sca-

lar also can be interpreted as showing a “change” of one quantity into another.

There is no contradiction between those two interpretations, but the ﬁrst one

has to be stressed upon. (I leave out the arguing in this matter.)

• According to the RAY model, a scalar is both the answer to a comparison

between quantities and an element of quantity transformation (showing the

relation). This duality of scalars is essential to recognize.

One more word about relations. – In symbol expressions, all signed term elements

can be regarded as to show relations between quantities. A term representing a

quantity, shows the relation to a zero quantity, +0. As already mentioned, all

signed scalars show relations between quantities. A scalar being a measure,

•

shows the relation of a quantity to the unit quantity, +1.

The comparison of quantities, yielding scalars, was ascribed a symbol of its own:

‘

•

’, e.g. Y

•

R = +18

•

+6 = ·3. Here, the two-step method is employed to

•

Observe, that ‘measure’ says ‘mätetal’ in Swedish.

Normat 4/2012 Leif Önneﬂod 181

construe the answer as follows: +6 [/6 ·18] = [+6 /6] ·18 = +1 ·18 = +18. From

this it is evident that [/6 ·18] = ·3 holds the answer to the comparison, as it sup-

plies the transformation needed to change +6 into +18.

•

The “pictorial view” of the two-step method may set oﬀ from:

+18

·18

Ω≠≠≠≠≠

·6

≠≠≠≠æ

Here the unit +1 is used in the center, as it makes the factors above the arrows

trivially equal to the measures. The next step is to “go from +6 to +18”, which is

accomplished by means of:

+18

·18

Ω≠≠≠≠≠

·6

Ω≠ ≠ ≠ ≠ ≠≠

Other intermediate quantities may be used as in:

+18

·6

Ω≠≠≠≠

·2

Ω≠ ≠ ≠ ≠ ≠≠

Moreover, this gives an alternative way to infer fractions (see later section) and to

motivate why diﬀerent fractions can work the same way. Still another example:

·5

Ω≠≠≠≠

Ω≠ ≠ ≠ ≠ ≠≠

. . . demonstrates the comparison, that conventionally is written as 5/

. (The reader

may picture the steps before and after.) The division by a fraction adds an extra

diﬃculty to be unraveled. In contrast, inverting a division is no m ore diﬃcult than

inverting a multiplication.

Diagrams as those above combine well with the use of e.g. cuisenaire rods.

•

Firstly, the pupils learn how to reason about comparisons and other matters

by help of the rods. Next they learn to write diagrams to show the structure.

Lastly they learn how this is written in mathematical expressions. Each step to

be settled on, before advancing to the next.

Observe that the scalar transformation is written after the quantity to be trans-

formed. E.g. +24 /8=+3;+2·5 = +10 (and accordingly +10

•

+2 = ·5). Hence

all scalars need to have preﬁx sign notation.

For a ‘unit fraction scalar’ like ‘/5’, only a preﬁx division operator is possible.

For historical and linguistic reasons an irregular use of the multiplication sign as

“postﬁx” frequently occurs. (That is: the ‘multiplicator’ is put before the ‘multi-

plicand’, also refe rred to as ‘premultiplying’.) This can become quite convenient

at times, but here only the preﬁx notation will be used. (Also referred to as

‘postmultiplying’.)

•

Naturally, a leading quantity can be followed by many scalars, but (without

using parentheses) only one leading quantity submits to its scalar successors.

E.g. +3 +5 ·2=+3[+5·2] ”=[+3+5]·2=+8·2.

This kind of relation is met with in diﬀerent shapes in abstract algebra. E.g. vec-

tors can be transformed by scalars. (Vectors also can be composed, i.e. added.)

•

Pupils experience this model as quite conceiva b le and straightforward.

•

Those are coloured rods of diﬀerent lengths. They must only depict quantities,neverscalars.

It is important to discriminate between the two.

•

Ileaveoutthelengthydiscussion,neededtofullyjustifythepreﬁxpreference,andtoexplain

why and when postﬁx notation is an option. Some advantages with the ‘preﬁx’ alternative may

still become apparent. After all, it’s extensively used.

182 Leif Önneﬂod Normat 4/2012

A general concept is ‘group action’, where a group of elements (here: scalars) by

means of its operation, transform elements in another set (with another opera-

tion, if any at all). Quantities transformed by scalar elements, will be an example

on ‘group action’.

When I ask pupils to combine two scalars into a single one as in x · 12/3=x· a (a is

asked for), they get puzzled.

•

They even he sitate about that x · 12/3 w ill equal

x/3 · 12 (only reversed order of scalars). Apparently they have never been as ked

to compose “scalars”. Such shortcomings lay the ground for trouble with algebra.

Still, composition is easily understood, if the transform starting from ‘+1’ come

into use:

In order to compose the scalars · 3 and ·2, start from +1[· 3 ·2] = [+1 ·3] ·2=

=+3·2=+6. Then pose the question “What s ingle scalar turns +1 into +6 ?”.

Apparently ‘·6’ does the work, as +1 ·6=+6. (Even this simple approach will

turn out a “new way of thinking” to quite a lot.)

As even other combinations of elements (other than quantities and scalars) obey

patterns very similar to those studied above, I introduce two concepts in order

to ease generalization. The elements that perform the transformations (show the

relations), I call agents.Theyact upon an opening element. The opening “target”

for the agents I denote anchor, as it constitutes a ﬁx starting point for the agents

to act on.

•

They are, so to say, “tied up” to an anchor. This anchor is supposed

to diﬀer from the agents in some feature(s), e.g. kind of operation.

The anchor—agent relationship comes about in many shapes. In abstract algebra

we have ‘group action’. In education we meet concept pairs like ‘multiplicand—

multiplier’ and ‘dividend—divisor’. The multiplicand and dividend is always de-

monstrated as something with a size (and measure) that is subject to a (scalar)

change by a multiplier and divisor respectively. It stands out clear that this has

its exact correspondence in an anchor (target quantity) and an agent (scalar at

work). The advantage using the anchor—agent concept is that it emphasizes a

general kind of relation, equivalent to group action. The concept is applicable to

powers (base—exponent relation) and yet another relation, both to be covered

later on. This follows the principle of “Ockham’s razor”. It is rational to avoid

an excess of concepts. (As in the ‘multiplicand’ family of concepts.)

The two kinds of division

The anchor and agent concept help to supply a theoretical setting to the didactical

concepts of partitive and measurement division. In theory, we have but one kind of

division, here exempliﬁed by 12/3=4. In real world we can diﬀerentiate between

two kinds of division. The partitive division follows the idea of the dividend—divisor

relation. With bound operations we can interpret this as +12 /3=+4,where+12

and +4 are magnitudes having a unit. Measurement division, on the other hand,

is about computing a relation – “How many pieces of size 3 is contained by (goes

into) the size 12 object?”. The “division” is between measures of ob jec ts. The RAY

model reveals that this follows the pattern of a comparison. The calculation will

be +12

•

+3 = ·12

·3=·12 /3=·4. Here we make two observations.

•

Despite I don’t use variables! It can look like 5 · 12 /3 =5 ,wherethecontentsofthe

left hand box is to be substituted for one number and one operation in the right hand box.

•

To be precise: ‘op ening’ = ‘leading’ = positioned ﬁrst.

Normat 4/2012 Leif Önneﬂod 183

1. The quantities of +12 and +3 have the measures of ·12 and ·3, which is an

important thing to notice. It is not the quantities (with units) that we “divide”,

but their (scalar) measures. This is why we receive a scalar as an answer.

This idea was stressed upon by the Oxford professor John Wallis, in the 17

century. Due to Wallis, if the “genus” of the (quantity) number was left out, this

allowed for scalar operations to take place between (remaining) “pure numbers”.

That is to say that “pure” scalar measures were separated from the quantity unit

(“genus”). At that time the Euclid notion of numbers as quantities was preva-

lent, and the only ways permitted to compos e quantities was to add or subtract

them, or setting up analogies, i.e. equalling “ratios” between them. The multi-

plications and divisions that accompanied analogy computations, was regarded

operations and did not infer that there were any independent numbers of the

kind we nowadays identify with scalars. Wallis made way for this new idea.

2. The answer is a scalar. The size +12 holds the size +3 “four times”, that is

“·4”. This is quite diﬀerent from ‘+4’ that tells the size of a third of size +12.

By means of bound operations we can make the diﬀerence, between the two ways

of perceiving division, visible in all aspects.

Frequently, the measurement division is written as a division betwee n quantities.

The quantity feature is apparent from the units supplied.

For example:

10 apple

2 apple

=5 or, as I would put it:

+10 apple

+2 apple

= · 5

The idea of ‘comparison’ is so much more general than the idea of ‘containment’.

As a matter of fact, most divisions deal with comparisons in some way. I propose

we s ubstitute ‘measurement division’ for ‘comparison division’. One advantage will

be that the expression follows the wording order: “How much is 12 cm compared

to 3 cm?” is written “+12

•

+3 ” and so on. After some practise, one can go

directly for the ‘·12 /3 ’. The idea that ‘measurement division’ computes a kind

of measure, can be substituted for that a measure is computed by comparison

between quant ities (possibly to the unit). Another advantage shows when dealing

with fractions, as will be seen.

‘Level’ – another real-world correspondence

Not only quantities can have a unit. We use numbers to denote diﬀerent kinds

of states or levels. I will use the term ‘level’ throughout. Examples may be water

level; temperature; position (in a coordinate system); direction (e.g. in degrees );

altitude (above sea level); date; point of time et cetera. Their measures all share

the property of having double references. An ordinary quantity needs to reference

nothing else but a unit. By comparing to the unit, the measure will be set. That

will not do for ‘levels’. A second reference to a “zero state” will be nescessary.

Example: To determine a number (measure) for the altitude of a mountain, you

have to settle on a zero altitude, usually chosen to be the sea level. In southern

Sweden the “highest” mountain is ‘Tomtabacken’ (“Brownie Hill”), 378 mabove

sea level. I you go there, you won’t be impressed – it is but a small hill rising

about 20 m above the surroundings. On the other hand is Mauna Kea on Hawaii

said to be the worlds highest mountain – measured from the nearby sea ﬂoor it

rises from. It is apparent that the measure of “altitude” is a matter of how to

choose, not only the unit, but also the reference level.

184 Leif Önneﬂod Normat 4/2012

With a reference level established, the combination of the measure and the unit

tells the “distance” from this level.

A watch tells the “time elapsed” from midnight on. A coordinate tells the distance

(and direction) from the origin (of coordinates).

The “double reference” is not the only feature of levels. Another is that they cannot

compose. (e.g. get added). “What would ‘4

of July’ + ‘14

of July’ add up to?

What is the meaning of the sum of two positions? (Ulf Persson wrote an article on

this matter in ‘Nämnaren’ no 2, 2010.)

As those level elements cannot operate on each other (= compose), they will

not be equipped with a preﬁx operator. This will help to distinguish levels from

quantities in expres sions. The quantity [+12 h +15 min] diﬀers from the point of

time 12:15 (no leading ‘+’). The length ‘+3 m’ diﬀers from the altitude ‘3 m’.

The mechanism of creating level values can be demonstrated by 0 m +3 m =3m,

or simpliﬁed as: 0+3=3.

•

Here 0 m is the reference level. This and the quantity

+3 m form an anchor—agent relation.

This kind of mathematics is established. It can be found in Lie algebra, topolo-

gies, and so on. It is an example of ‘group action’.

Some signiﬁcant features of levels are . . .

• Their “double reference” feature;

• Their lack of operation. And accordingly . . .

• Their conﬁnement to work as ‘anchors’, not as agents.

• They can indeed be compared! (This being the only “operation” between levels.)

A remark on the use of ‘◊

’ and ‘

•

’: The ◊

symbol is used for comparisons bet-

ween agents, producing an agent of the same kind as an answer. (It may also

be used as a gene ral comparison symbol.) The

•

symbol denote comparisons

between “anchors”, producing a corresponding agent. Moreover, this agent is

intended to show the “numeracy” of the reference anchor. (This rules out using

•

for level comparison. There ◊

is used.) This restriction makes

•

more useful,

e.g. +5x

≠15x

•

≠3=· 5x. Such an expression can be quite accessible to

learners, compared to a quotient between polynomials.

To infer how comparison between levels works, we set oﬀ by taking a look at how

quantity comparison could be performed:

The comparison +18

•

+3 = ·6 can be outlined as an “extraction” of measu-

res (i.e. scalars) from the quantities, followed by a comparison between scalars. As

+18 = +1 ·18 and +3 = +1 ·3,where·18 and ·3 are measures, we substitute the

comparison between quantities for its scalar equivalent: ·18

◊

·3=·18

·3=·18 /3=·6.

Now we apply these patterns on ‘levels’. To perform the comparison 7

◊

5,we

begin by extracting the enclosed “inner” quantities. As 7=0+7 and 5=0+5,the

quantities will be +7 and +5. Then we have that +7

◊

+5 = +7

+5 = +7 ≠5=

=+2, telling that 7 lies 2 (unit) steps from 5, in the positive direction.

From comparisons of this kind, level values can be ordered. Nevertheless, levels

have no inherent “size” attribute. The position 7 is not “greater” than the position

denoted by 5 on the number line, it only comes “later” than 5 in the order.

•

...by regarding the unit as abstract and as such, implicit.

Normat 4/2012 Leif Önneﬂod 185

The comparison of levels follows the usual pattern, that an agent tells how to

transform (translate) the R anchor into the Y anchor. From 5+2=7 we see

that 7

◊

5=+2. This can be demonstrated on a number line, where levels are

coordinates and quantities are depicted as arrows between coordinates.

We can see that levels mathematically work very diﬀerent from quantities. This

is established knowledge used in e.g. top ology. Nothing is told about those ‘level’

features in compulsory school education.

•

Textbook authors even compromise

how levels work, e.g. by multiplying temperatures with scalars. Typically, negati-

ve numbers are exempliﬁed by “levels” like temperature, altitude and coordinate

points. As a next step, it is demonstrated how to add, subtract, multiply and

divide negative numbe rs, in spite of the fact that a pair of real-world “levels”

cannot be subject to a binary operation. So what do the pupils learn from those

examples? Are they of any help? The only operation a level submits to, is a

“translation” to another level by means of a quantity.

The concept of level elements complies with the structure given by the RAY model.

The anchor and agent concepts help sorting out the component roles.

Powers

I will not go into details about powers. Only a short survey of how the RAY model

and anchor/agent concepts are applied will follow.

• The power base works as an anchor, and is a scalar.

• The exponent is an agent of its own kind. The ‘ˆ’ (hat) is used as an operator.

• Following the pattern of 5 · 3=+5·3 = +5 +5 +5 = +15, we have that

= ·5 ˆ3=·5 ·5 ·5=·125.

You m ay say that scalars “repeat” quantities and exponents “repeat” scalars.

This must not be confused with “repeated addition” or “repeated multiplica-

tion”. For those concepts, what is being repeated is frequently mistaken to be an

operation, not an element.

• The comparison ·125

•

·5 yie lds ˆ3, as is obvious from ·5

≠≠ ≠æ

ˆ3

·125.

This corresponds to computing the “ﬁve logarithm of 125”. (The comparison

·125

◊

·5 yie lds ·25, N.B.)

• The opposite operation to ‘ˆ’ is denoted ‘

ÕÕ

’ and returns a root. Accordingly, if

·x ˆ3=·125,then·x = ·125

ˆ3=·125

ÕÕ

3=·5, is a solution via “back-ward

reckoning”. Mind that R in RAY stands for ‘root’ or ‘reference’. (Regarding R

a “root” in the RAY interpretation of a power makes sense.)

• The associative law is applied as [·2 ˆ3]ˆ4=·2[ˆ3 ˆ4] = ·2 ˆ12,wherethe

composing of exponents by multiplication, easily can be derived.

• One form of the distributive law applies according to [·2 ·3] ˆ4=·2 ˆ4 ·3 ˆ4

in the very exact parallel to [+2 +3] ·4=+2·4+3·4.

•

The “transfer” of university level mathematical insights to compulsory school, seems vanish-

ingly small.

186 Leif Önneﬂod Normat 4/2012

The other form of the distributive law looks like · 2 ˆ(+3+4) = · 2 ˆ(+3) ·2 ˆ(+4)

or simpliﬁed like · 2

+3+4

= · 2

·2

. Regrettably, treating the theory behind

this is an essay on its own.

• A change of the “inner sign” of the exponent, as between ˆ(≠2) and ˆ(+2) is

coupled with a sign change for the base (anchor), that is

·2 ˆ(≠3) =

·2 ˆ(

≠3) = /2 ˆ(+3) = /2 /2 /2=/8.

The transformation ·2 ˆ(≠3) = /2 ˆ(+3) can read “3 less ‘·2’ equals 3 more

‘/2’ ”. Seen as agents working on ·2 ·2 ·2 ·2 ·2 this corresponds to

·2 ·2 ·2 ·2 ·2=·2 ·2 ·2 ·2 ·2 /2 /2 /2.

This is perfectly paralleled by

+2 ·(≠3) =

+2 ·(

≠3) = ≠2 ·(+3) = ≠2 ≠2 ≠2=≠6,

saying “3 less ‘+2’ equals 3 more ‘≠2’”.

Perhaps this gives you a hint on the issue of what I call “inner quantities” in

scalars and exponents, along with their contribution to simpliﬁed “sign rules”.

Those are not to be treated upon, more than this in this paper. Here the focus

is on real-world structure. Anyhow, my little “hint” on sign rules might have put

you on the track of seeing that the interpretation of a negative number depends

on what “role” the numb er plays – does it represe nt a level, quantity, scalar or

exponents?

On fractions

There’s much to be said about fractions. Only a fraction will be treated here. One

thing is that they form rational numbers, deﬁned as equivalence classes (of the

kind, earlier mentioned). Another is that positive rationals form a group under

multiplication. That hints at that rationals can be associated with scalars.

The main point has already been touched on. A general comparison of type

•

+b, produces the composed scalar [·a /b] as an answer.

From +1 [·a /b]=+a /b =+

=+1·

, it follows that ·

•

This

interpretation of a fraction, as a combined multiplication and division, is sometimes

referred to as “the op erator mo de l”. In a misguided ambition to have but one model

for fractions, the operator model is questioned. On the contrary, it is the basic

notion of fractions. This is why . . .

A s calar can always work as a measure. Just as ·5 in +5 s =[+1s] ·5 is the

measure of ﬁve seconds, so is ·

the measure of +

. The operator model and

scalar fraction is indeed the basis, as the scalar works as a measure that creates

the quantitative fractions.

Many models for fractions, use quantities in the shape of rectangular or circular

(pie) areas, or stretches with length. To demonstrate

4 of a rectangle, it is divi-

ded in four parts, whereof three may be shadowed (or otherwise marked). What

is neglected, is that a mathematical creation of

4 of an area is not performed by

means of a ruler and a pencil. It is done by the two operations /4 and ·3. They

are applied on a area being a quantity. Here the language used about fractions

•

The equality ·

=[/b · a] can as well be stated as a deﬁnition of a scalar fraction. Observe,

that the unit +1 works as a quantitative anchor to the scalar agent, being the measure.

Normat 4/2012 Leif Önneﬂod 187

set a trap. After having divided ‘the whole’ (e.g. a rectangle area) by four (i.e.

applied /4 on +1 leaving +

4 ) we name the part a ‘quarter’. Then we take 3 of

them: +

4 +

4 =+

4 · 3=+

4 .

•

The trap is, that if understanding fractions makes use of quarters, ﬁfths and

other parts as a basis, attention is drawn away from the role of the denominator

to work as a divisor. It reduces into a “nomen” (name) for the part. Only the

multiplicator role of the numerator is properly recognized. Parts will be created in

mind or by practical arrangements, not by division. The parts will be perceivied

as ‘unit fractions’, that is: quantities. (The us e of leading +’s, like in +

2 , +

3 ,

4 and so on, would expose this property.)

With fractions understood this way, it will be hard to construe the meaning

3 ·

4 as you cannot multiply two pieces of a pie, to use a metaphor. When

the pupil is asked to write an expression showing “three quarters of a”, the trap

deﬁnitely shuts.

•

Teachers are told in their education, that the ‘operator model’ is not central

and has the draw-back of not ﬁtting in with other models. (Those models seem

to be classiﬁed after what kind of real-world object the fraction is applied on.

•

To me, that is not mathematics.)

The RAY model provides the structure needed. The scalar ·

is to be under-

stood as [/b ·a] or [·a /b]. It shows the relation between quantities. It is construed

with the two-step method as in +5

≠≠ ≠æ

·2

+2 implying that +2

•

+5 =

=[/5 ·2] = ·

. The method and reasoning is, due to my experience, easy to convey

to pupils, with a helping hand from hands-on materials. You can apply a (scalar)

fraction on “a whole”, that is +1, regardless of the unit will be a ‘pie’, ‘square’,

‘glass of water’, etc. It makes no diﬀerence to apply a fraction on a number of

objects, i.e. +n. In either case the application of a scalar on a quantity yields a

quantity: +1 ·

or, if +n = +35: +35 ·

= +14.

•

When you tell apart representations for quantities and scalars, you make way for

attending the parallel between (concrete) division by a knife, and (mathematical)

division by a number. (The quantitative “pie anchor” is, so to s ay, submitted to

the operation of a surgeons knife or a mathematicians divisor.)

The interpretation of

as [+

] ·a manifest the widespread view of the denomi-

nator as a kind of “unit” (telling the “size”) and the numerator as the “number”

(i.e. works as a multiplier). As mentioned before, this leaves out the mere creation

of +

as depicted of +1 [/b ·a]=[+1/b] ·a =+

·a. Indeed, this initial step

is crucial for getting a ﬁrm grip on how to manage fractions.

•

Conventionally this is written

4 +

4 =3·

4 =

4.Herenothingpreventsthequantity

property of the (unit) fraction from being overlooked. Mind: things being added are quantities.

•

The division by the denominator will not likely be an active part of the pupils model of a

fraction. How to make four parts of “a”bymeansofarulerorby“piedividing”techniques?

•

The use of a square to model fraction multiplication is considered diﬀerent from using a

rectangle! Somehow someone somewhere seems to have missed the point.

•

This extensive use of ‘+’ signs (and others) might seem to clutter up the expressions. The

intention of using them is only to make the role of numbers in an expression explicit, and to

facilitate reasonings abo ut numbers and their correspondences in reality. Nothing prevents the

experienced user of mathematics to leave out signs where they seem to be of minor use. The more

complex expressions, the more the need for simpliﬁcation.

188 Leif Önneﬂod Normat 4/2012

Composition of scalars

Understanding calculations with fractions is but a part of understanding scalar

expressions in common. How to make a pupil understand a rearrangment like the

following?

· 5/4 ·

3 · 4

In 2002 Wiggo Kilborn presented (due to a study) that as many as 42 % of the

pupils in form 9, couldn’t calculate

/3 ! If the pupils had been able to regroup

the scalars as into

/5, there would not have been a problem.

To facilitate rearrangements of scalar expressions, one needs to identify what aﬀect

each number has on the expression as a whole, with respect to its operation. As

of the expression above, the roles the numbers posess can be demonstrated by

·1 /3 ·5 /4 ·6 /5. (Leading signs, N.B.) After having left out the redundant ‘·1’

and done some regrouping, this equals [·5 /5][·6 /3] /4=·1 ·2 /4=/2.

In parallel with the concepts ‘plus term’ and ‘minus term’, that are useful tools

to classify term agents, we need similar concepts for scalars. I propose we name the

factor ‘·a’ and its converse ‘/a’ a ‘profactor’ and an ‘antifactor’, respectively.

•

Other terms would be possible, for instance ‘sizefactor’ and ‘partfactor’. The

scalar ‘· m’ can be regarded a “measure” of something, as scalars often are. The

signs themselves can read ‘times/part’, ‘pro/anti’ or some other short forms.

Cumbersome wordings like ‘multiplied with’ and ‘divided by’ are not likely to

be used in practise when long expression are read aloud. To avoid “baby talk”

in mathematics, the short forms should be settled on.

Here is a suggestion on how to demonstrate the scalar “roles” in expressions. For

“one-liners” like ·3 ·4 /5 /6 ·7 a consistent use of preﬁx interpretation of signs

tells the role of each number. For “two-liners” in quotients the interpretation is

straightforward: Above the fraction bar (in the numerator), the aﬀect on “the

expression as a whole” is unchanged. An ·a will even in a wider scope (outside the

numerator) work as a profactor and /a as an antifactor. Below the fraction bar

(i.e. in the denominator) the role of a factor will be reversed. Here the fraction bar

(a vinculum) takes on the role of grouping factors and submit those below to an

inversion as by the sign ‘

’.

·6 ·7 /10

·7 /5 ·3

=[·6 ·7 /10]

[·7 /5 ·3] = [·6 ·7 /10][/7 ·5 /3]

Naturally, a vinculum can be regarded as a “token” for regular division and,

concurrently, may imply some sort of inverting, following less transparent rules.

A subsequent, more transparent and immediate reasoning follows. This can, in its

turn, be used to prove how the division operation can be used to invert factors.

From a theoretical standpoint, the inverting property of division is quite ob-

vious, as division is derived from the basic concept of inversion. (The same goes

for subtraction.) In school mathematics, whe re division as a concept is brought

up ﬁrst, the other-way-round inference of the inverting property becomes a dif-

ﬁculty later on. (This inverting property is introduced for managing comparisons

and backward reckoning.)

•

Naming e.g. /2 a‘factor’ismotivatedbythatyoucannottellitapartfrom· 0.5 when you

Normat 4/2012 Leif Önneﬂod 189

Instead, the idea behind why reversing (of operations) is needed in the ﬁrst

place, should be brought into daylight using the

operator. The use of ordinary

operators for this purpose should be introduced only as a second step.

In eﬀect, the role of a factor above or below the vinculum can be derived by means

of comparison. Any pupil can learn that a comparison with the unit quantity +1

works according to +a

•

+1 = ·a since +1

≠≠ ≠æ

·a

+a. Now use (what I call) the

“trivial fraction” +

=+1. Make it “change” by inserting factors in the nume-

rator and/or denominator. Then determine the overall “change” by comparing the

result with the origin +1.

For example does an /5 inserted in the denominator of +

yield . . .

1 /5

0.2

=+5

Then +5

•

+1 = · 5 tells that the insertion of /5 made the expression as a whole

5 times larger. The general conlusion will be that an antifactor inserted in the

denominator works as a profactor. The pattern of that factors in the numerator

keep their roles but have them reversed in the denominator is simple to learn

and derive.

With those insights the pupil can perceive

/3 as

·6 /5 /3=[·6 /3]/5=·

/5=·2 /5=·

The problem of “rearrangement” melts down to identifying the factor roles in

the one expression, and write the other with their roles maintained. The ‘neutral

factors’ ·1 and /1 can be left out or inserted as one pleases. Exercises in rewriting

make room for the pupils creativity, and ease the way to algebra.

Measurement division and proportionality

The correspondence between the functionality of a quotient and a comparison is

obvious:

Y◊

R = Y

R =

As earlier mentioned, the use of a quotient for comparison is often referred to as

‘measurement division’. That is an old notion. The ancient Greeks said that one

quantity measures another if it (due to our speaking:) divides into it. Two thousand

years later John Wallis (Oxford prof.) stated that diﬀerent kinds of quantities

may be compared by means of a quotient, provided the ratio was between “pure

numbers” without any units or other quantity features.

The underlying reason is that a combination of scalar me asures (“pure numbers”)

in a formula shall reﬂect how one quantity depend on others. E.g. if ‘velocity’

of s ize +1 is deﬁned as that of a +1 mmovein+1 s, then the measure ·v of

velocity will be directly proportional to the strech measure · s and reciprocally

proportional to the time measure ·t. Accordingly the formula v =

is all about

measures, not quantities. The operation “strech divided by time” remains to be

performed.

190 Leif Önneﬂod Normat 4/2012

Textbook authors often emphasize on the idea, that comparison division is between

quantities, by spelling out the units in the quotient as in

10 m

10 cm

= 100 or

35 km

5 h

=7km/h

Nevertheless, the pupil gains from being aware of that in those expressions, the

vinculum only is a symbol for comparison. That is . . .

+35 km

+5 h

= +35 km

•

+5 h = ·35 ◊

·5=·35

·5=·35 /5=·

= ·7

Now ·7 is the measure of the velocity !

The expression ‘·35 ◊

·5’ above does w hat Wallis told us to do: it compares

“pure numbers”.

•

Behind all this lies that the relation between the quantities is

the one of proportionality. The relation itself constitutes a quantity – a velocity.

Its measure equals that of the strech traversed during the time unit (+1) measure

(·1). When velocity is constant, strech and time are proportional, implying they

change in step. In our example the factor /5 yields unit time: +5 h

≠≠ ≠æ

+1 h.

Applied likewise on the strech, it yields +35 km

≠≠ ≠æ

+7 km. Now the strech

measure ·7 is the velocity measure, per deﬁnition.

The Wallis method is a short cut. Cumbersome proportionality reasonings

are left out and he goes directly for the measure that may be determined from

the strech and time measures in a quotient. It can be questioned wether pupils

shall learn this short cut without understanding the underlying proportionality

reasoning. I have tried and found the “long way round” to be a short cut to

comprehension.

The use of a quotient for direct proportionality actually reﬂects that two measured

phenomena change the same way.

Example: If the strech 12 m is prop ortional to the time 5 s (constant velocity),

then a 3 times longer strech is accompanied by a 3 times longer time. If the

quantites are arranged in a quotient, those changes will leave the quotient un-

changed:

12·3

5·3

or, fully featured:

+12 m

+5 s

+12 m

+5 s

·3

+36 m

+15 s

Written with quantities, the quotient gains from being interpreted to show a

relation, rather than a division. A quotient between plain measures is another

cup of tea. It’s extended like:

= ·12 /5=·12[·3 /3] /5=[·12 ·3][/5 /3] =

12 · 3

5 · 3

In eﬀect: we multiply and divide by the same number, which explains why

the quotient remains unchanged.

•

Measures for reciprocal proporitionality are usually set up as a product. Here, too,

we multiply and divide by the same number. It looks like:

a · b = ·a ·b = ·a[·c /c] ·b =[·a ·c][·b /c]=ac ·

It becomes evident, why a product is suitable to show reciprocal proportionality.

•

It is, but of course, impossible to ﬁgure out “how many times 5 hdividesinto35 km”!

•

Understanding connections, thrills pupils.

Normat 4/2012 Leif Önneﬂod 191

To conclude

The RAY and anchor—agent models combined with bound operators c omprise

what I call “Change and Comparison Mathematics” – CCM. The concepts lead

considerably beyond what is outlined here.

Among other things: Proportionality and negative numbers become straight-

forward matters. The angle concept can be settled on. Ways to interpret and

generalize complex numbers are pointed at.

Basically, the conce pts in CCM are aimed at making it more simple to understand

basic arithmetical structures. This is the CCM raison d’être. There’s no need for

alternative theories like this one, just to make out calculations. What justiﬁes its

existence is the exte nt to which it clariﬁes on diverse issues, and does so in a

simple, consistent way, that readily lends itself to be demonstrated by hands-on

materials. This is of major importance, as pupils frequently are able to perce ive

how to handle mathematic s in practical situations. Almost equally frequent, they

are less able to transfer their insights from practice over to written expressions and

general structures.

Many features of CCM can be found elsewhere, in a slightly diﬀerent outﬁt.

There, those features are aimed at solving complex problems and, accordingly,

not well adapted to be conveyed to children. In Russia, the Davydovian way of

mathematical instruction in form 1–3, have several outﬁts in common with CCM.

This, too, talks in favour of CCM.

CCM lays a foundation, that complies with advanced modern theories, but avo-

ids all their complexities. It is meant to provide a tool to bridge the gap between

abstract symbol expressions and real-world mathematical structures. To my expe-

rience, it does so surprisingly well. Will this be the expe rience of others, too?