New Cur 29: Negative Definite

We’ve already written about the lack of proof in the Australian curriculum. The problem is much more fundamental than that, however: there is a lack of clear mathematical constructs to even prove things about.

Mathematics, as is any discipline, is an amalgamation of ideas and assumptions and arguments. But mathematics can do it better. The ideas as precise definitions, the assumptions as axioms and the arguments as formal proofs confers to mathematics a soundness and strength shared by no other discipline. It is why the Pythagorean theorem can be as true, and not merely as “accurate“, as it was thousands of years ago.

Not that mathematics has always been so clear about its foundational strength. Mathematical rigour is neither obvious nor easy and it has taken millennia of pondering and playing, and outright cheating, for mathematics to take on its modern form. Arthur Koestler wrote a fascinating (and flawed) book titled The Sleepwalkers, about the history of astronomy. The history of mathematics is not so much sleepwalking than it is a parade of shameless sleight of hand. This dodgy history is captured, in nobler language, by the title and the contents of John Stillwell’s excellent Yearning for the Impossible.

All of this is to make clear that mathematical foundations can be difficult and arcane and unforgiving, even for modern practitioners, and much more so for practitioners of mathematics education, which is the backdrop for this post. To start at not the beginning, consider multiplication. Multiplication begins easily enough, as repeated addition: 3 x 5 means three 5s, means 5 + 5 + 5. Fine. But why, then, does 5 x 3 equal 3 x 5? There’s no obvious reason why five groups of three things should amount to the same as three groups of five.

But then, we have the brilliant idea of arranging these groups in an orderly manner, tilt our heads, and all is clear:

Well, up to a point. One might get away with pictures for a while, maybe into the beginning of fraction arithmetic. But eventually reality or, more accurately, unreality, takes hold. Why, for example does 3 x √5 = √5 x 3? One might accept that 3 x √5 means three √5s, means √5 + √5 + √5. But what does √5 x 3 mean? And, if we don’t know what one side of the equations means, what is it we are trying to prove? The troubling reality/unreality is that, as soon as multiplication ceases to be repeated addition, and in fact long before, we have entered deep and dark waters. Teachers have to navigate these waters, all the while trying to ensure kids don’t fall overboard and hauling them back in the boat when they inevitably do.

How is a teacher to accomplish this? God only knows, and we are not God (even if it sometimes sounds as if we think we were). At least, teachers can read the maths ed gods, such as Hung-Hsi Wu and Tony Gardiner and Ed Barbeau. These guys are incredibly smart and they have worked incredibly hard, trying to figure out how the intrinsically deep discipline of mathematics can be adapted and presented in a clear and engaging and as honest-as-can-be manner to school-level students, and to the teachers teaching them. But teachers, even intelligent and scholarly teachers, are, frequently enough, going to be performing sleight of hand, when not simply, humanly, winging it.

Now, with that introduction, on to the Australian mathematics curriculum. This curriculum was not written by gods; it was written by fallible human beings. Very fallible human beings. About as fallible as it gets.

A curriculum cannot be an exposition of “elementary”, school-level mathematics. The curriculum can and must assume of the teachers, in a general sense, a body of common knowledge and accepted goals. To the extent that this cannot be safely assumed, secondary documents such as glossaries can be there to guide the user; here again, the emphasis of a glossary should be on natural language clarification rather than pure mathematical definition. As such, it is not a hanging offence that the Australian curriculum is almost devoid of definitions, and that the glossary is of so little help in this regard. Still, it is an offence and it is worth documenting this offence before considering the more systemic issues.

The curriculum contains four explicit references to “definition/defining”, the first of which appears in Year 6 Number, as an elaboration on prime numbers:

identify and describe the properties of prime, composite and square numbers and use these properties to solve problems and simplify calculations (AC9M6N02)

using the definition of a prime number to explain why one is not a prime number

This is then unclarified by the glossary:

Prime number: A natural number that is greater than 1 and its only factors are 1 and itself.

We have already posted on this particular nonsense, including that the curriculum and glossary cannot decide what a natural number is, and that the curriculum writers have no sense of when to use a name (“one”) or a numeral (“1”). So, let’s move on.

The next reference to definition occurs in Year 8 Statistics:

analyse and report on the distribution of data from primary and secondary sources using random and non-random sampling techniques to select and study samples (AC9M8ST02)

defining and distinguishing between probabilistic terms such as random, sample space, sample and sample distribution

This is vague and aimless, and it is not properly backed up by the glossary, but at least it’s not meaningless.

The third reference to definition appears in Year 9 Algebra (sic), and is not so benign:

apply the exponent laws to numerical expressions with integer exponents and extend to variables (AC9M9A01)

simplifying and evaluating numerical expressions, involving both positive and negative integer exponents, explaining why; for example, \boldsymbol{\color{OliveGreen}5^{-3}=\frac1{5^3}=\left(\frac15\right)^3=\frac1{125}} and connecting terms of the sequence \boldsymbol{\color{OliveGreen}125, 25, 5, 1, \frac15, \frac1{25}, \frac1{125}\dots} to terms of the sequence \boldsymbol{\color{OliveGreen}5^3, 5^2, 5^1, 5^0, 5^{-1}, 5^{-2}, 5^{-3} \dots} 

relating the computation of numerical expressions involving exponents to the exponent laws and the definition of an exponent; for example \boldsymbol{\color{OliveGreen}2^3 \div 2^5 = 2^{-2} = \frac1{2^2} = \frac14} and \boldsymbol{\color{OliveGreen}(3\times 5)^2 = 3^2 \times 5^2 = 9 \times 25 = 225}

In sum, this is kind of sort of in the ballpark. But not really, and not really close.

The key point here is that negative exponentials/powers are being defined (as reciprocals), and then the “exponent laws” for general integers follow; they can be proved. But any sense of this definition and reasoning appears only weakly, only in the optional elaborations, and it is ass-backwards. One cannot explain, for example, “why” \boldsymbol{5^{-3} = \dfrac1{5^3}} except to refer to the definition, which only occurs, vaguely, in a separate and subsequent elaboration. Plus, while we’re here, the equating to \boldsymbol{\left(\frac15\right)^3} in the elaboration is entirely irrelevant. Finally, and it is a relatively minor point but it really grates: the proper “definition of an exponent” is “those little numbers that appear up the top, which mean powers and then other things”. The curriculum writers really mean exponential or exponential expression (neither of which the glossary defines).

The final explicit reference to definition in Year 10 Space (sic). It is so vague as to be meaningless, and requires no comment, but we include it for completeness:

design, test and refine solutions to spatial problems using algorithms and digital tools; communicate and justify solutions (AC9M10SP03)

defining and decomposing spatial problems, creating and applying algorithms to generate solutions, evaluating and communicating solutions in terms of the problem; for example, designing a floor plan for a department store that limits congestion at key areas such as checkouts, changing rooms and popular sale items

That’s it. The above four whatevers is all there is in the way of explicit reference to definition. As we noted, the absence of such explicitness is not itself an issue, but of course that the very few such references are so meh or so garbled is not a hopeful sign. Indeed, the non-references are worse.

With or without formal dressing, any mathematical concept will make its first appearance in some manner. In the Australian curriculum, such an introduction is often framed as simply a matter of “recognition”, or the concept simply appears, as if by magic. Here is a very small sample of such first occurrences, with the introduced terms in bold:

recognise ways of measuring and approximating the perimeter and area of shapes and enclosed spaces, using appropriate formal and informal units (AC9M4M02)

recognising that perimeter is the sum of the lengths that form the boundary of a shape or enclosed space; choosing suitable units from a range of objects to measure around the boundary of a shape such as a garden bed; comparing the results to say which unit was an appropriate choice for the context; using a piece of string or rope to measure the perimeter of irregular shapes and enclosed spaces, including those that have curved sections

recognising that area is the space enclosed by the boundary of a shape or the surface of an object; measuring and comparing the area of shapes, using an array of paper tiles or mosaic squares, including part units to fill gaps at the edge of the shapes; comparing the total areas by combining the fractional parts to make whole units

approximate numerical solutions to problems involving rational numbers and percentages, including financial contexts, using appropriate estimation strategies (AC9M6N08)

describe the relationship between perfect square numbers and square roots, and use squares of numbers and square roots of perfect square numbers to solve problems (AC9M7N01)

recognise, represent and solve problems involving ratios (AC9M7N08)

using diagrams, physical or virtual materials to represent ratios, recognising that ratios express the quantitative relationship between 2 or more groups; for example, using counters or coloured beads to show the ratios 1:4 and 1:1:2

solve problems involving the volume of right prisms including rectangular and triangular prisms, using established formulas and appropriate units (AC9M7M02)

building a rectangular prism out of unit cubes and showing that the measure of volume is the same as would be found by multiplying the 3 edge lengths or by multiplying the area of the base by the height/length

developing the connection between the area of the parallel cross-section (base), the height and volume of a rectangular or triangular prism to other prisms

recognise irrational numbers in applied contexts, including square roots and π (AC9M8N01)

recognising that the real number system includes irrational numbers which can be approximately located on the real number line; for example, the value of π lies somewhere between 3.141 and 3.142 that is, 3.141 < π < 3.142

recognise terminating and recurring decimals, using digital tools as appropriate (AC9M8N03)

identify the conditions for congruence and similarity of triangles and explain the conditions for other sets of common shapes to be congruent or similar, including those formed by transformations (AC9M8SP01)

developing an understanding of what it means for shapes to be congruent or similar

establishing that 2 shapes are congruent if one lies exactly on top of the other after one or more transformations including translations, reflections and rotations, and recognising that the matching sides and the matching angles are equal

solve spatial problems, applying angle properties, scale, similarity, Pythagoras’ theorem and trigonometry in right-angled triangles (AC9M9M03)

investigating the applications of Pythagoras’ theorem in authentic problems, including applying Pythagoras’ theorem and trigonometry to problems in surveying and design

understanding the relationship between the corresponding sides of similar right-angled triangles and establishing the relationship between areas of similar figures and the ratio of corresponding sides, the scale factor

Again, to emphasise, these are the introductions, the (in)effective definitions of the various concepts. There are dozens more.

It is no easy job to write a clear mathematics curriculum, and the proper introduction and relating of the concepts is perhaps the most difficult aspect. But you gotta try. You at least gotta try. And they haven’t. The Australian curriculum writers have simply thrown stuff together, so that one can never be sure what is being defined or assumed or argued: it is all just “recognised”.

The curriculum writers have reduced the beautiful structure of mathematics to a heap of stuff, a meaningless rubble of half-ideas and half-facts, with which nothing of worth can possibly be taught or learned. In this sense, and in every sense, the Australian mathematics curriculum is a disaster.

10 Replies to “New Cur 29: Negative Definite”

  1. I wonder if the issue is ‘axiomatic’ :-).

    As opposed to the Singapore Curriculum, they have chosen a very complex structure. And then not been really rigorous (which a complex structure requires). You can get away with far more if you keep it simple (not to mention that ordinary mortals can comprehend it).

  2. This is terrible and has a harmful impact on children. Sometimes they [kids] think the authors of such questions obfuscate definitions because they want kids to guess right what they mean. Then many useless teachers who couldn’t provide the correct definitions and fix this gibberish will be guessing along with the kids. These things destroy trust in teachers, beat the love of the subject and discredit the whole concept of education.

  3. One cannot explain, for example, “why” 5^(-3) = 1/5^3, …

    Why not? The equation is a definition. So, the question is, why did mathematicians choose this definition? Discussing and answering this question can lead to a deep(er) understanding of mathematics, in partular about “definitions”. It becomes really interesting a few lessons later, when the question arises “why” (-8)^(1/3) is somewhat messy.

  4. Christ, no.

    hjm, you’re confusing the question for no purpose except, as does the curriculum, to invite teachers to screw up. Which the vast majority already do.

    Of course you can ask why mathematicians chose that definition, and almost any teacher will address that “why” in some manner, even if they do it in a screwy manner. BUT, that “why” is a Subsidiary Question. The Main Question is, “Why is \boldsymbol{5^{-3} = 1/5^3}?” And the answer, as I noted, and as you noted without noting that I noted it, is: “Because I said so”. It’s a definition. That’s all.

    You must make clear when a definition is a definition. That doesn’t mean, in school, that the word “definition” need be used, but you must fundamentally distinguish the definition from motivations for or arguments with that definition.

    This stuff drives me spare, and in particular this stuff on index laws (new and improved as “exponent laws”). My very first WitCH was on an idiotic textbook “proving” that \boldsymbol{9^{\frac12}=\sqrt9}. See also:

    Of course you, hjm, are not claiming to prove the definition: I know you know what you’re talking about. But your muddying of the two questions creates exactly the quagmire in which teachers will get stuck. Which then also means, when one asks about, for instance, \boldsymbol{(-8)^{\frac13}}, no one has a hope, because students are taught to treat the index “laws” as if they were commandments from God. See here and here.

    1. Question (I may well have asked it back on WiTCH 1, I do not recall): you say to distinguish the definition from the motivations for that definition.

      Is it acceptable, in your opinion, to show the motivation first and then conclude the index law?

      For example, 3^{2}=9 and so if we want 9^{x}=3 it makes sense that x=\frac{1}{2} because of existing index laws. “Internal consistency” is a phrase often used.

      1. Yes, I think it is acceptable, and natural.

        When I first wrote the comment I labelled them Question 1 and Question 2, but I realised that this was misleading, for exactly the point you make: of course it is natural to begin with the motivational question before getting to the critical answer question. That’s why I changed the names to Subsidiary Question and Main Question.

    2. (Just call me Jesus.)

      No, seriously. I know what you mean. And I know teachers (typically fresh from university) who will definitely [sic] screw this up when they try to teach it. So, I completely agree with you that it is extremely dangerous to write something like this in the curriculum without clarification of the meaning of “why”, i.e what meta-level it refers to. I just wanted to state that there is a reasonable way to ask and discuss this question, and this is how I usually do it when teaching powers. I also share the suspicion that the authors of the curriculum did not really understand the issue (or mathematics in general). This is because I know some authors of curricula in Germany and we have the same issues here.

      1. Dear His Jesus Majesty,

        OK, we agree, which is not surprising. But I reiterate the point, that you have to be very careful with the wording of such “why” questions. You wrote “this question”, but teachers and students will not know which question “this question” is, unless you’re super-explicit. And even then, the majority of teachers and students will not get it.

  5. I wasn’t familiar with the books of Hung-Hsi Wu. However, after you mentioned him here, I checked on his work. I have to say it is excellent. I know there are PDFs on the web, but I will purchase several of them—really great stuff. They are on par with AoPS but exceed those, in some cases, regarding the depth of explanations. When it comes to problem-solving, AoPS is doing a slightly better job [my subjective opinion]. However, using these as complementary would be priceless.
    Why schools aren’t using these outstanding books is beyond my comprehension.
    Thanks for posting the link Marty.

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