Tony Guttmann, AKA Mr. Very Big, is a member of or is associated with just about every mathematician organisation in Australia, excepting the Geelong Primary School Mathletics Squad.* Tony has signed the open letter and, much more importantly, has been working hard within the various organisations, arguing against the draft mathematics curriculum. He has been quoted in the recent, excellent reports of Rebecca Urban (Murdoch, paywalled).
As part of his efforts, Tony prepared a statement on the draft mathematics curriculum. With his kind permission, we have reprinted Tony’s statement here.
*) Membership pending.
COMMENTS ON ACARA’S DRAFT MATHEMATICS CURRICULUM
Tony Guttmann, AM, FAA, FTSE, FSIAM, FAustMS
As Australia is falling behind its international peers in basic mathematical skills, it is important to remedy this. A major problem is the lack of an adequate supply of trained mathematics teachers at the secondary level, and the too-frequent occurrence of math-phobia among primary school teachers, which risks alienating their students from an appreciation of, and facility with, basic numeracy. This problem has been recognized for at least 40 years, and piece-meal attempts to address it have been made, but little has changed in four decades.
Australia has frequently sought inspiration from successful countries, such as Finland and Singapore. Unfortunately, their success is founded on a highly skilled and well-trained cohort of teachers, along with an appropriate curriculum. Australia lacks both.
One aspect of the Singapore curriculum is the application of mathematics to so-called “real world problems”. But this application only takes place after the mastery of basic skills. These are not learnt by trying to solve problems, and picking up the basic skills by some sort of osmosis.
So while the present Australian Mathematics Curriculum definitely requires review, ACARA’s current draft is systemically flawed, being based on this flawed premise of teaching through investigation, problem solving etc. While these have their place, the basic skills need to be there first, so that the focus can properly be on applying those skills to the problem at hand.
The power of mathematics, and our ability to learn and to apply mathematics, comes from its simplicity and its precision. Mathematics simplifies and abstracts the real world: from “three apples” to 3, from “three apples and five oranges” to 3a + 5b, and so on. With particular abstractions sufficiently understood, then, and only then, mathematics can feed back, to be applied to and to help us better understand the real world. This two-step process is long and sometimes difficult, but it is natural. A beautifully clear description of such an approach to mathematics education is given in the just-released UK Government’s Ofsted mathematics review.
The current ACARA approach is much more difficult and convoluted. The draft proposes that students learn fundamental mathematics, and come to understand the way it works, largely through “investigation” and “modelling” and “problem-solving”, much of it open-ended and poorly defined. Even if such investigations were totally within mathematics, this would be a flawed approach. What is critical to learning the arithmetic of fractions, for example, is practice on arithmetic with fractions. What the draft curriculum offers Year 6 students is a range of activities, heavy on vague modelling (pp 87-88). The draft curriculum is further flawed by situating “problem-solving” and the like in real-world scenarios (p 14). This may be where one eventually wants to apply mathematics, but it is not how one should attempt to learn the mathematics to be applied.
The strong emphasis on exploration in the draft curriculum leads to many other troubling issues. In brief:
1. The draft curriculum is difficult to read:
A consequence of the draft’s mixing mathematical content with applications is the loss of a simple and coherent structure. The framing is around thirteen “core concepts”, many of which are poorly defined and overlapping (p 15). In an apparent, and failed, attempt to make sense of this mix of concepts, the long introduction to the draft is prolix to the point of incomprehensibility (pp 1-16). Equally vague are the content descriptors in the draft, which are not infrequently a mix of the key facts/ideas/skills to be learned and the typically exploratory and ineffective methods proposed to learn them (p 27, for example).
2. The delaying and dilution of the “basics”:
Critical examples of this deficiency are the decisions to delay the learning of multiplication tables until Year 4 (p 64), and the solving of linear equations until Year 8 (p 115). ACARA’s arguments for these changes are quite unacceptable (pp 8-9). The dilution is also systemic. An emphasis on real-world modelling is a very poor mode for the practice of fundamental skills and, inevitably, it takes emphasis and precious time away from the proper practice of these skills. It denies students the opportunity to develop the fluency to apply the mathematics in a proper and rewarding manner.
3. The devaluing of mathematics:
For History or for English, the subject is largely taken to be its own worthy goal, but this is too seldom done for mathematics, and the idea is almost entirely ignored in the draft curriculum. The strong emphasis on premature exposure to real-world contexts squanders the opportunity for students to gain an appreciation of mathematics as its own beautiful discipline. In doing so, the draft also squanders the opportunity for students to gain a rich understanding of mathematics, which, in the long run, is what will best serve the students. Well-constructed problems within mathematics, and posed after the mastery of the appropriate basic skills, can be highly engaging, and can elevate students’ approach to the level of sophisticated mathematical thought. Such problems are barely hinted at in the draft curriculum.
The stated intention of the Curriculum review was for it to be modest, with an emphasis on “refining” and “decluttering” (p 1). The review, however, has been radical, the absolute antithesis of modest. Moreover, it is radicalism doomed to failure; every top-performing country on international mathematics tests has a fundamental emphasis on the mastery of basic skills, which the draft simply lacks (p 8, p 7). The draft curriculum is an abject failure, on its own terms and on any terms. The draft should be withdrawn, to make way for a fresh review, which includes the proper participation of discipline experts.