[This post is authored by Judy Mousley – Ed.]
I was asked by Terry to write a guest article for the blog in my capacity as President of MERGA, but the words below have been written in a personal capacity. I encourage other MERGA members to add their thoughts, as those below are certainly not representative of anyone’s ideas but mine.
Another “numeracy” review!
The report of the National Numeracy Review (Commonwealth of Australia, 2008), becoming known as the Stanley report after the chair of the review panel, is yet another federal effort that will have little impact, largely because it has little to say that provides specific direction for the teaching and learning of mathematics. It includes many motherhood statements that have been used throughout my 15 years or primary and secondary teaching and 25 years as an academic, without articulating how curriculum and assessment guidelines can give teachers new understandings and actions that would be needed to bring about significant change. Merely stating that teaching people to be “truly numerate, involves considerably more than the acquisition of mathematical routines and algorithms” fails to challenge the status quo in Australian schools where teachers spend most of their time focusing the acquisition of mathematical routines and algorithms.
The report wastes much space discussing the meaning of “numeracy”. This is not unusual, as attempts to define the term have been made in at least five national reports in the last decade. In most other countries, the term is hardly used and people do not know what numeracy is. Those few countries where the term is used, such as England (where it was concocted in 1959 as an analogue of literacy), have varied interpretations (see, for example, Commonwealth of Australia, 2000; Groves, Mousley & Forgasz, 2003; and Willis, 1998). The Stanley report claims that numeracy is the ability to use mathematics but, as it points out, most people think of numeracy as “the basics”. This lack of clarity and agreement illustrates the folly of using “numeracy” to name (and hence shape) a major review of mathematics teaching and learning. It must be recognised by federal and state governments that it is Mathematics that is taught in schools, mathematics curricula that state or federal bodies need to outline for the use of teachers and textbook authors, and mathematics education that needs content and time guidelines. The use of any term other than “mathematics” in reviews, guidelines, professional development and the media muddies the picture for teachers, curriculum developers, parents and others. While Australians are discussing what “numeracy” means, the rest of the world is focusing on better ways to teach mathematics, the content of teacher education courses and the time given to mathematics in these, as well as what research tells us about children understanding and learning mathematics. The 2003 TIMSS results showed that Australian results have slipped down from their comparatively high position, and I await the release of the next round of results at the end of 2008 with interest.
The recommendations of the National Numeracy Review’s report are a mixed bunch, but nevertheless are sensible on the surface.
The first recommendation is that teachers of all subjects should be prepared to teach the mathematical aspects of their different areas. That is, a teacher of Humanities subjects would teach the necessary graphing, map projection, data collection and manipulation, ratio and other relevant skills—while teachers of Dance, Materials Technology, Science, etc, all taught the mathematical knowledge and skills relevant to their subject areas. My university has found that this is not hard to manage, with its Bachelor course for secondary teachers including compulsory studies on “numeracy across the curriculum”. However, there is strong competition for space in initial teacher education programs (and particularly in one-year Graduate Diplomas that follow the discipline degree), so this recommendation would require states and teacher registration bodies to enforce the recommendation when initial education courses are approved; and this is unlikely.
The second recommendation proposes minimum hours per week for primary and lower secondary school mathematics lessons. It is important that these be set because many secondary schools are responding to the shortage of mathematics-qualified teachers by reducing mathematics teaching to less than 4 hours per week. Further, in primary schools, school camps, swimming lessons, concerts, bike education, sports days and other events that eat into the classroom timetable. Some teachers prefer to focus on an “integrated” or a “problem-based” curriculum, but unless these are very well planned by a teacher with a strong knowledge of the traditional curriculum and how to incorporate it well, most of the mathematical demands are low-level procedural arithmetic and simple, repetitive graphing. In fact, the recommendation gives every school an excuse not to timetable even the minimal hours recommended with the statement “This time should include cross curricular learning” (p. xiv, p. 15). Schools without the optimal number of mathematics teachers do not even have to try to attract them now. In any case, in Australian schools there is no inspection and little other control of timetables and their implementation other than state-wide or national testing, and rather than increasing hours spent on mathematics most teachers soon learn to “teach to the test”. It is probably mainly parental demands that keep most schools in check, but these both have more influence in middle-class regions and independent schools. Given this context, this recommendation from the national review’s report is likely to make little difference to what happens in schools and may result in what used to be mathematics time being allocated to cross curricular learning.
The third recommendation is that “greater emphasis be given to providing students with frequent exposure to higher-level mathematical problems rather than routine procedural tasks, in contexts of relevance to them, with increased opportunities for students to discuss alternative solutions and explain their thinking”. I commend this advice but believe that most teacher education courses have been suggesting it for years with little effect. Teachers have experienced 12 years of mathematics pedagogy during their own schooling, and this is hard to overturn during their professional courses. Further, about a third of people who teach secondary maths across Australia have not been trained to do so. There is little support in schools because most textbooks and most other teachers plan a lesson or a learning sequence as presentation of a rule or algorithm, student practice implementing it, and then working through some problems where it can be applied. This recommendation is creditable, but too easily interpreted as doing more of the same. As was shown with the introduction of the Victorian Certificate of Education, it is only major top-end changes such as the inclusion of unfamiliar problems that demand higher-level thinking in Year 12 assessment tasks that will drive such a change, so this should become a linchpin of the new national curriculum framework. Teachers at all levels (from pre-school to Year 12) also need exemplary teaching and assessment materials to support this development. (As an aside, the notion of mathematical modelling is, strangely, omitted from the report but perhaps we can put this down to its focus on numeracy as mathematical application.)
The power of broad-based testing to impact on group of recommendations about assessment is undeniable. However, while I understood the outline of early years assessment programs that preceded it, Recommendation 4 (p. 40) did not make sense to me. “That a balanced view be taken of the relative contributions to effective student learning of systemic assessment programmes and high quality classroom assessment in the allocation of resources to develop and support each”. What is it that the authors are asking for here? Recommendations 5, 6 and 7 certainly make sense, calling for increased use of diagnosis to ascertain learning needs, intervention programs, and assessment-based research to inform curriculum and pedagogical developments. While I agree that teachers need support in the form of diagnostic tasks, I would argue that they need to learn how to probing and challenging questions during every classroom task to diagnose what their students understand and need to learn next.
This is not as easy as it sounds, and a teacher needs a sound knowledge of mathematics as well as subsequent curriculum content in order to know what probing questions to ask. Experience helps. Unfortunately with a shortage of confident and competent primary teachers as well as a dire shortage of mathematics-trained teachers for Australia’s secondary classes, there is generally not sufficient knowledge of mathematics and curriculum content or teaching experience. In fact, I am amazed that the National Numeracy Review made no recommendations about the amount of mathematics to be studied by primary and secondary teachers and no advice for universities of state departments about essential content in preparatory courses! Like every review instigated at the national level during my academic life, it really missed the boat here.
The next two recommendations appear fairly regularly in national numeracy reviews as well as teacher education courses. Recommendation 8 argues for the language of mathematics to be taught explicitly and Recommendation 9 discourages the use of so called “ability grouping” before the senior secondary years. Again, one wonders what federal or state departments will do to ensure implementation of these recommendations. No doubt teachers will continue to focus (or not) on mathematical language and to use (or not) achievement groups as they see fit. With state and federal governments each taking different forms of responsibility for education, and both having a hold on purse strings, it is likely that nobody will take firm action on such matters.
If the federal government can convince teacher registration bodies (currently state-based but soon likely to be subject to the proposed national teaching standards) to take note of its Recommendations 10 to 15, on teacher education, then all of the energy and resources that went into the review will not be wasted.
Recommendation 10 proposes that AMMT’s Standards for Excellence in Teaching Mathematics in Australian Schools be used as a professional framework. Number 11 argues for exemplary research-based professional development programmes be extended to include teachers of students up to Year 10 and lists key elements such programs. I support Recommendation 12, about pedagogical content knowledge (i.e., what is known about the effective learning and teaching of specific content) being a prime focus of teacher education, equally strongly. I am disappointed, though, that no recommendation was made about content knowledge (i.e., knowledge of mathematics). This would not be a problem in secondary schools if all teachers of mathematics had appropriate qualifications, but that is not the case, and it is a continuing battle to win and maintain sufficient studies of mathematics in primary teachers’ courses.
The next recommendation proposes that teachers learn to cater better for the variety of students and their social/cultural needs, and this is advice that is ubiquitous in national reports and teacher education, but the final recommendation surfaces less often: that systems develop strategies and programs to support out-of-field secondary teachers.
In summary, you have probably gathered that I find the review’s report to be disappointing, mainly because it says no more than previous reviews except perhaps in relation to teacher education. While the report includes much uncontroversial and useful information and advice, it is hard to see how this will bring about the major changes in school mathematics and teacher education and employment that Australia needs.
The report bounces from numeracy (as narrowly defined) to mathematics and back again as it draws heavily on details and even recommendations made in previous significant reports — few of which have been implemented. Major gaps include:
- lack of discussion and recommendations about minimum teacher qualifications for teachers of mathematics at respective levels and advice to governments about how to address this problem;
- little discussion and limited recommendations about the time and content of initial teacher education courses and on-going professional development and renewal of registration for current teachers (other than points summarised above);
- no account of exemplary teacher education or professional development programs that show how sufficient focus time and energy can be given to the development of discipline content knowledge as well as pedagogical content knowledge; and
- lack of a coherent framework for analysis and critique of current teaching and assessment practices, teacher education, and mathematics education research.
My key criticism, though, is that the review focuses on “numeracy”. This was certainly not the fault of the panel. People who are able to apply mathematical algorithms without having learnt to understand mathematics well are not what this country needs. Mathematical application is only one narrow realm of mathematical understanding, and it is certainly not the primary one.
To illustrate other realms, and hence the narrow focus of the panel’s brief, I outline below my view of how students of mathematics come to understand it. It is a framework that arose from my reading of Schoenfeld (1986) as a teacher many years ago. It has not been presented to wider audiences, and is sure to have mistakes and gaps, so I welcome comments in this blog arena.
Developing mathematical understanding
Initially, mathematics students at any level learn about objects. In infancy and the initial years of schooling, these include blocks, beads, counters and other manipulatives as well as various representations of objects. Children have a rich variety of everyday experiences in coming to understand these objects as they learn to manipulate them appropriately. There are traditions to be followed here, such as touching individually, grouping, adding more, rotating, and so on. These two realms of understanding exist not only in early years mathematics, but also in the learning of each new concept throughout schooling. For instance, children might first learning subtraction with base ten materials such as bundles of straws or attribute blocks. To learn factorisation, junior secondary students might also use an object as they draw, partition and rename a rectangle sized by as . Senior students might learn about integral notation using drawings of curves and some rectangles that fit under them. Mathematical objects to be manipulated and mastered include graphs, models, variables, theorems and formulae as well as ideas like vertex, negative number and limit. This learning takes place out of school too — students of any age might learn about bank interest, infinity or measurements through every day, objective and subjective experience. Thus object-based understandings form a network of concepts about the way we operate with things and their representations in mathematics.
From experiences with objects, students develop an understanding of matching symbols. These include typical symbols (including words) that we name and describe objects with — such as “two”, “limit”, or “theta”. They also include symbols used to name and describe actions, whether real or imagined, such as “divided by” and “square root of”. Many traditional symbols represent complex ideas, such as “equality”, “Cartesian areas” of multiplication, and “waves” of trigonometric graphs. Thus children seek a second level of understanding — development of a coherent knowledge network of symbols (including language) and symbols-based operations.
The physical and symbolic realms each need internal coherence, but direct links need to be made between these two realms for the latter to be personally meaningful. At first the symbols are bound to objects and actions on these, but gradually they become an entity in their own right and thus can themselves be used as foundation objects for further operations and subsequently for more complex symbolisation. Students need to understand that the way we use symbols embodies some very powerful ideas.
A vital third realm is abstracted understanding. It involves the abstraction of ideas away from objects: the “threeness” (cardinal nature) of “three” and the ways that this compares with the ordinal nature of “third”, the general triangle, and the idea of a circle as a set of points, for example. Again this type of understanding can starts in children’s earliest years. I have strong memories of a toddler pointing to pairs of things saying “Two, two” and within a few weeks (before he could count three objects) saying “Two, two, four, four, eight. This demonstrated a big and beautiful, but unpredictable leap in his understanding of number as well as delightful appreciation of mathematical patterns and structure — characteristics of the discipline that excite even research mathematicians.
Not many primary, secondary and perhaps even tertiary teachers have thought about their roles in relation to the development of abstracted (as opposed to merely abstract) understanding. Many make good links between concrete experiences and abstract ideas, but don’t realise the importance of taking the next step. Abstracting mathematics is not just moving children’s understandings out of particular objective experiences, but away from them. If teachers focus on “numeracy” that is defined as mathematical applications to everyday experience, as suggested by the Stanley report, they will not even see the need for this step. The challenge is to have students be able to recall enough relevant experience for the abstract to remain experientially meaningful, but to make the children’s understanding independent of particular experiences. It is in the fact that mathematics is removed from particulars that makes it powerful.
Throughout their learning, students should be encouraged to see mathematics as an interlocking network of ideas, and this provides a fourth realm of understanding. Teachers assist in this development, such as when they explain that multiplication is repeated addition, but this is usually within components of the curriculum, such as within the Number area. Focusing a review on applications of specific mathematical skills (or on basic numeracy skills if “numeracy” is interpreted that was) does not help curriculum developers, teachers to reflect on new ways of teaching that will develop a strong sense of inter-relationships between concepts.
Where is “numeracy” (mathematical application) in this framework? Primarily, mathematics is much more than the ability to apply maths to everyday situations. Mathematics presents ways of thinking and working as well as a language for interpreting, exploring and recording a comprehensive network of ideas. That is, it is a discipline in its own right — one that offers students of all ages intellectual excitement and opportunities for creativity. The usefulness of numerical, spatial, graphical, statistical and algebraic concepts and skills that are taught in schools is not only about immediate application in everyday situations (as the review panel’s interpretation of its brief suggests), but in developing a facility with higher level skills such as abstraction, generalisation, and prediction. Students at all levels need to see themselves as people who are capable of abstraction and generalization, creative problem solving and modelling. People capable of such activity are the young mathematicians that Australia needs; but the Stanley report, with its focus on numeracy, gives little sense of this possibility.
The above realms of understanding provide one framework for thinking about how to bring about improvements in teachers’ work through making coordinated changes in teacher education, curriculum and pedagogy. Of course it is not the only possible framework, and there would be others that could be used to create excitement and appropriate activity nationwide. The point is that we need to think about how to bring a coherent new direction to mathematics pedagogy. However, one does not achieve this aim with yet another national report with a narrow brief, despite its sensible recommendations that nobody at the highest levels — and few teachers in schools — will attend to.
It is time that mathematicians, mathematics educators, teachers and curriculum developers across Australia demanded that governments stop reviewing different interpretations of “numeracy” and put the spotlight on how students of all ages best learn mathematics and hence what teachers of mathematics need to know. Improvement in “numeracy” is the agenda of previous governments, and it has not served Australia’s best interests.
- Central Advisory Council for Education (1959). A report of the Central Advisory Council for Education (England): Crowther report. London: HMSO.
- Commonwealth of Australia (2000). Numeracy, a priority for all: Challenges for Australian Schools. Canberra: DEST. Available August 25 from http://www.dest.gov.au/sectors/school_education/publications_resources/profiles/numeracy_priority_for_all_challenges_Australian_schools.htm.
- Commonwealth of Australia (2008). National numeracy review report. Canberra: DEEWR. Available August 25 from http://www.coag.gov.au/reports/ – numeracy.
- Groves, S., Mousley, J., & Forgasz, H. (2003). Primary numeracy: A mapping, review and analysis of Australian research in numeracy learning at the primary school level. Canberra: DEST. Summary available August 25 from http://www.dest.gov.au/sectors/school_education/publications_resources/profiles/primary_numeracy.htm.
- Schoenfeld, A. (1986). On having and using geometric knowledge. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 225–263). Hillsdale, NJ: Erlbaum.
- Willis, S. (1998). Which Numeracy? Unicorn, 24(2), 32–42.
Use can be made of this phenomenon for teachers’ professional development. A recent question on a nation-wide test asked students to identify a pentagon from a set of figures that included a regular hexagon and other shapes. The pentagon was irregular and concave. Hopefully, teachers and children across Australia learned about the properties of a pentagon.
Associate Professor Judy Mousley
School of Education
Geelong, Vic. 3217
Phone (on leave): 0417157815