New Cur 23: False Equivalence

INTRODUCTION

The word “equivalent” is one of the most useful in mathematics and one of the most abused in mathematics education. The implication is that this one is not all ACARA’s fault. Nonetheless, the fact that it was predictable that ACARA would make a mess of it doesn’t alter the fact that ACARA made a mess of it. So, here we are. And a warning: this is a long post; there seemed no way around it.

In Middle English (via Latin and French), “equivalent” meant to be of comparable power or worth, and was more often applied to people and their status:

My derivation was from ancestors
Who stood equivalent with mighty kings

(Just for fun, all quotes in this section are taken from the Oxford English Dictionary.)

The usage of the word was generalised, and can now commonly refer to (perhaps loosely) measurable quantities, as well as to more abstract attributes:

 The gayne … wold be at the lease equevolente with the commodytyes the marchantes should reape therby. 

At times, “equivalent” is used effectively as a synonym for “equal”:

I’ll remit the money to you as you direct, or send you equivalents.

More typically, however, the term is used to indicate that quantities are strongly comparable rather than strictly equal:

Thus a white weasel’s skin was an equivalent for eleven sheepskins. 

The best general definition might be “close enough to equal for the purposes at hand”.

 

EQUIVALENCE CLASSES

Whatever the general usage, modern mathematics is much more careful. The term “equivalent” is used in various ways in mathematics, of which the most important, and the most relevant to this post, is the concept of an equivalence class.

An equivalence class makes absolutely precise the notion of “equal for the purposes at hand”. In clock arithmetic (integers mod 12), for example, the numbers 17 and 41 are declared to be equivalent because they give the same remainder of 5 when divided by 12: for the purposes at hand here, all we care about is the remainder. We then write [17] or [41] or, more naturally, [5], for the set of all such equivalent numbers, which we then refer to as the equivalence class of these numbers. That is, 5 ≠ 17 ≠ 41, but [5] = [17] = [41].

The omnipresent and very hidden application of equivalence classes is to the precise definition of rational numbers. (For now, we’re deliberately avoiding using the term “fractions”: we’ll come to that.) We know how the number 2/6 works, but it is not so easy to say what that number is, to give it some concrete identity. For instance, we can’t simply point to the symbols 2 and / and 6, since most definitely 2/6 and 1/3 must be equal, must be the exact same number, but the symbols in “1/3” are different.

We can make this all concrete by a two-step process. First, we consider all ordered pairs of integers: (2,6) and (1,3) and (4,7) and (7,4), and so on. The only restriction is that the second integer cannot be 0. These are intended to be our rational numbers. So, (2,6) is somehow supposed to be our familiar number 2/6. (Which is why we don’t allow the second number in the pair to be 0, so as to avoid zero in the “denominator”.) But as yet we haven’t done much more than different typesetting, and it cannot yet be the solution. For example, the pairs (2,6) and (1,3) are definitely different mathematical objects, but 2/6 and 1/3 must in the end be the same number. This is addressed by the second step.

In the second step, we declare when two of these pairs are equivalent. Precisely, the pairs (a,b) and (c,d) are equivalent if ad = bc. So, for example, (2,6) and (1,3) are equivalent since 2 x 3 = 6 x 1. Similarly, (4,6) and (6,9) are equivalent since 4 x 9 = 6 x 6. (It shouldn’t be too difficult to see that this notion of being equivalent just amounts to “cross multiplication” of our pairs.) Then, finally, we write [(2,6)] for the equivalence class of (2,6), for the set of all pairs equivalent to (2,6).

And that is our rational number. The equivalence class [(2,6)] is the number we more simply write as 2/6. Or, since [(17,51)] = [(2,6)], we could also write the same number as 17/51. Or, more naturally, we could write the same number as 1/3.

There are a lot of details missing in the above, and there are a lot of questions one might ask, top of which is possibly “What the hell is the point?” There is a point for mathematicians, of course. Mathematicians should be precise about the conceptual objects with which they work. But this point also applies, at least to some extent, to teachers. In particular, teachers should always be aware of, and on guard against, the fact that the representation of a number is not the same as the number. Such confusion, for example, is why it is so hard for teachers (and students, and everyone) to understand that 0.999… is equal to 1.

In our scenario here, it is critical for teachers, and then students, to understand that 2/6 and 1/3 are equal, that these are the exact same number. That understanding need not come through equivalence classes, and no one is suggesting that primary kids be introduced to this stuff via ordered pairs: pieces of pie is still where you want to begin. But understanding the equality underlying these numbers is critical, and it is not so easy or as much of a given as some might imagine. This very common confusion is highlighted by the question,

   

The majority of, say, secondary maths teachers and secondary students get the answer to this question wrong. It is telling. (Tony Gardner raised the question here, and we raised it here and here.)

 

EQUIVALENT FRACTIONS

Now, a word about “fractions” and, in particular, “equivalent fractions”. Both terms cause trouble. To begin, it is important to realise that “fraction” is ambiguous. The term can refer to either a number, or the representation of a number, or both:

The fraction 3/2 is greater than 1.

The denominator of the fraction 3/2 is 2.

We shall add the fractions 3/2 and 1/3 by first finding a common denominator.

Just to hammer the point, note that we can only add numbers, not representations; conversely, we can only talk about the denominator of a representation, not of a number. So, when we say “the fraction 3/2”, what we really mean is something like “the rational number represented in fractional form by 3/2”. Which is such elegant phrasing, it is sure to catch on. But, until it does, we’re stuck with the double-usage. (This double-usage occurs with other mathematical terms as well, but for various reasons “fraction” seems to cause more trouble.)

Now, what about “equivalent fractions”? What does it mean? Are 2/6 and 1/3 equivalent? Are you sure you want to reply “yes”?

If we are talking about numbers then the precise and natural description is that 2/6 and 1/3 are equal. On the other hand, if we are talking about representations, then what exactly are we saying is “equivalent”, and why? The language is clearly a reflection of the “equivalence class” notions discussed above, but it is not an accurate reflection.

To be sure, it essential that we have some terminology to refer to the various fractional forms of the one and same number. But “equivalent fraction” is clumsy and is often actively misleading. It is hard to enough to give kids a sense of these numbers without employing an expression that actively undermines confidence in the their equality.

But again, we’re stuck with it. All that can be advised is that the expression “equivalent fractions”, and “equivalent” more generally, be used sparingly and with care, and always with an eye open for the possibility that “equal” or some other term might be clearer and more accurate. Unfortunately, carefulness and being open eyed are not ACARA’s strong suits.

 

“EQUIVALENT FRACTIONS” IN THE AUSTRALIAN CURRICULUM

Finally, to ACARA’s curriculum. Unsurprisingly, the curriculum often employs the term “equivalent” and its variants to describe equivalent fractions and their manipulation:

find equivalent representations of fractions using related denominators and make connections between fractions and decimal notation (AC9M4N03)

representing addition and subtraction of fractions, using an understanding of equivalent fractions and methods such as jumps on a number line, or diagrams of fractions as parts of shapes (AC9M6N05)

Such usage is mostly standard. It is clumsy but only because, as discussed above, the standard terminology is intrinsically clumsy. (Yes, the content is also of a painfully low standard, but the focus of this post is on the language used; see our previous hundred posts for discussion of the fundamental poverty of the curriculum.)

Any step away from bread and butter fractions, however, and the language in the curriculum tends to go haywire:

using a number line to represent and count in tenths, recognising that 10 tenths is equivalent to one (AC9M4N04)

This is awful. The truth to be grasped is that 10 x 1/10 = 1, here represented by arriving at the exact same spot on a number line, and it screams for a declaration of equality. The writers, however, hold steadfast to a perverting “equivalent”. (The curriculum writers also have no clue when or why to use a numeral (1) rather than a name (one), but this has already been documented.) There are a number of such, very bad curriculum lines.

Similar, though perhaps not quite as egregious, is when the curriculum is concerned with comparing fraction, decimal and percentage representations.

recognise that 100% represents the complete whole and use percentages to describe, represent and compare relative size; connect familiar percentages to their decimal and fraction equivalents (AC9M5N04)

This may not be exactly wrong, but it must be close: the word “equivalent(s)” is performing no special function, is inviting confusion with equivalent fractions, and the word/notion of “corresponding” would appear to work much better. Again, there are a number of such lines in the curriculum, and all read awkwardly.

 

“EQUIVALENCE” IN THE AUSTRALIAN CURRICULUM

Away from fractions, the curriculum’s use of “equivalent” and its variants is an absolute mess, encapsulated by the self-contradictory and illogical definition provided in ACARA’s glossary (Word, idiots):

equivalent: Equal in value or meaning. Something such as an expression or statement that is essentially the same. 

Um, two things being of “essentially the same” value is not the same as them being of “equal” value: that’s the whole point. And, what does it mean for two statements to be “equal in meaning” or to have “essentially the same” meaning? Yes, it might be intended to mean that the statements are logically equivalent in the formal mathematical sense; as we shall see, it does not mean that, or anything.

The use of “equivalent” in the curriculum proper is, in turn, bad and weird and illogical. First, the bad. The curriculum has two notable lines on comparing units of measure:

recognising the equivalence of measurements, such as 1.25 metres is the same as 125 centimetres (AC9M6M01)

using models to demonstrate the number of cubic centimetres in a cubic metre and relating this to capacities of millilitres and litres, recognising that one millilitre is equivalent to one cm³ (AC9M8M02)

These are absurd. Similar to the number line example above, if you have moved the exact same distance, or if you have the exact same quantity, then the proper notion is of equality, and the terminology should reflect this.

Now for the weird. Apart from (pseudo)formal usage, one can accept that the curriculum might also use “equivalence” in a more vague, general language sense. Nonetheless, the following sentences, from the Year 1 and Year 2 level descriptions, struck us as, well, weird:

develop a sense of equivalence, fairness, repetition and variability when they engage in play-based and practical activities.

develop a sense of equivalence, chance and variability when they engage in play-based and practical activities.

Maybe it makes sense. Maybe it really means for kids to develop a sense of “equal for the purposes at hand”. But how, in Year 1 and Year 2, that can be any more precise than a sense of fairness, God only knows.

And finally, and to end this Proustian-length post, we come to the illogical. There is, it seems, a phrase that pops up on occasion in the maths ed literature:

equivalent number sentence

Once can imagine various meanings of the phrase, but they all appear to be wrong: as near as we can tell, the phrase is meaningless.

We won’t document our search of “equivalent number sentence”, since the references appear to be few and unscholarly. On occasion the phrase is simply used as a synonym for “equation”, which is plain nuts. More often, and reflecting ACARA’s glossary definition, the phrase is intended to indicate that two equations have the “same meaning” or “essentially the same meaning”, but with no clarity or consistency, and seldom with even a clear purpose. The usage of the phrase in the curriculum is not one ounce better.

There are many uses of “equivalent number sentence” in the curriculum, and on every occasion the usage is blatantly absurd or utterly incomprehensible. We had thought to make a selection and then comment upon them, but we are tired, and the absurdity is so glaring there is clearly no point. So, here they are, all of them. Think of it as like exit music, and roll the credits. We’re done.

using balance scales and informal uniform units to create addition or subtraction number sentences showing equivalence, such as 7 + 8 = 6 + 9, and to find unknowns in equivalent number sentences, such as 6 + 8 = □ + 10 (AC9M4A01)

using relational thinking and knowledge of equivalent number sentences to explain whether equations involving addition or subtraction are true; for example, explaining that 27 – 14 = 17 – 4 is true and using a number line to show the common difference is 13 (AC9M4A01)

using materials, diagrams, number lines or arrays to show and explain that fraction number sentences can be rewritten in equivalent forms without changing the quantity; for example, is the same as (AC9M5N05)

using knowledge of equivalent number sentences to form and find unknown values in numerical equations; for example, given that 3 x 5 = 15 and 30 ÷ 2 = 15 then 3 x 5 = 30 ÷ 2 therefore the solution to 3 x 5 = 30 ÷ □ is 2 (AC9M5A02)

using relational thinking, an understanding of equivalence and number properties to determine and reason about numerical equations; for example, explaining whether an equation involving equivalent multiplication number sentences is true, such as 15 ÷ 3 = 30 ÷ 6 (AC9M5A02)

constructing equivalent number sentences involving multiplication to form a numerical equation, and applying knowledge of factors, multiples and the associative property to find unknown values in numerical equations; for example, considering 3 x 4 = 12 and knowing 2 x 2 = 4 then 3 x 4 can be written as 3 x (2 x 2) and using the associative property (3 x 2) x 2 so 3 x 4 = 6 x 2 and so 6 is the solution to 3 x 4 = □ x 2 (AC9M5A02)

constructing equivalent number sentences involving brackets and combinations of the 4 operations; explaining the need to have shared agreement on the order of operations when solving problems involving more than one operation to have unique solutions (AC9M6A02)

applying knowledge of inverse operations and number properties to create equivalent number sentences; removing one of the numbers and replacing it with a symbol, then swapping with a classmate to find the unknown values (AC9M6A02)