This weekend has seen a focus on mathematics in the Weekend Australian.
On the front page we have the heading “In summary, maths is too simple” then there is a followup comment piece by Kevin Donnelly as well as a further article by Donnelly in the Inquirer section of the paper titled “Don’t leave the kids alone”.
Apparently “the National Numeracy Review report criticises the national benchmarks in mathematics, which assess students against minimum standards rather than requiring a desirable proficiency”. Without having seen the report myself, I can only speculate what this comment actually means. What, exactly, has been determined as ‘the minimum’? By whom? It is always important, when making comparisons, to establish the measuring device and what it is actually measuring.
The report is quoted to recommend that “curriculum emphases and assessment regimes should be explicitly designed to discourage a reliance upon superficial and lowlevel proficiency” and goes on to recommend the phasing out of streaming of students according to their ability. I could not agree more. Readers of this blog will know that I have ‘soapboxed’ on these issues frequently. The latter issue of streaming is something I have commented on in a previous post on a recent article put out by Barbara and Doug Clark.
As for the former point, students can learn how to do well on assessments that require lowlevel thinking and simple regurgitation of procedures. The ‘good’ results they attain on such tasks then actually mask a lack of real understanding of the mathematical concepts behind the procedures. Assessments that don’t challenge common misconceptions, require students to transfer knowledge from one context to another or apply their knowledge in unfamiliar situations are not performances of understanding. Consider the following examples of questions that could be used in tests. How are they different from ‘typical’ test content? Note that each question is linked to a specific understanding that has been identified as being ‘essential’ in that particular topic. I strongly believe that we cannot write meaningful assessments until teachers have thought about what these understandings are for each topic we teach. Welldocumented curriculum is important because it sets the agenda, articulates the vision and provides an enforceable structure, on which improvements to teachers’ practice can be based and referred back to. But we have to simultaneously and continuously work with and on teachers’ beliefs about teaching, pedagogy and methodology otherwise the curriculum design will be largely ignored. As we seem to be heading towards a national curriculum, I hope that its designers will focus on the understandings to be engendered rather than a listing of content to be covered. The government body responsible for this has a responsibility to provide the ‘pedagogic capital’ necessary to drive and support such an approach to teaching and learning.
Concept being tested 
Test for understanding question 
Topic Area : Quadratics 

Understanding that TP of the parabola is halfway between the xintercepts and that its shape is symmetrical 
A parabola has a yintercept of 12 and one xintercept of –6. Its TP is at the point (a, –4) The value of “a” could be A. 0 B. –2 C. 6 D. –4 E. 3 
Understanding of Null Factor Law and what is means to be a solution to a quadratic equation 
A quadratic equation is x(3x + 1) = 4 Which one of the following statements is true? A. x = 4 ; 3x + 1 = 4 B. x = 0 ; 3x + 1 = 0 C. x = 1 is a solution to this equation D. x = 1 is the only solution to this equation E. x = 0, x = and x = 4 are all solutions to this equation 
Topic Area : Number 

Understanding that fractions can be ordered by first writing them with a common denominator 
Give a fraction that would lie between Explain your reasoning. 
Understanding of rational and irrational numbers 
Which one of the following statements is false? A. All irrational numbers are surds B. All finite decimals are rational C. All surds are irrational 


Understanding of index notation, the index laws and how they can be used 
One third of 3^{18.9} would be equal to: A. 3^{6.3} B. 1^{18.9} C. 1^{6.3} D. 3^{17.9} E. 129140163.3 
Understanding of directed number operations
Understanding of factors
Understanding of equality and how numbers behave 
By what number must (4 +3 –2) be multiplied by to give an answer of 6 A. 3 B. 2 C. –2 D. –3
2. 3´5´7´17´29´31´j = 3´5´17´29´31´77 j must equal:
A. 7 B. 11 C. 15 D. 21 E. 3. a ´ b = a + b whenever
A. a = b = 1 B. a = b = 2 C. b = 0 D. a = b

Topic Area : Geometry 

Understanding of the properties of quadrilaterals 
Decide whether the following statements are True or False. Give reasons for your choice in each case. 1. The only quadrilateral that has diagonals that cross at right angles is the parallelogram. 2. The diagonals of an isosceles trapezium bisect each other. 3. The diagonals of a rectangle bisect the corner angles. 4. There are two pairs of equal angles in a kite. 5. There are two pairs of equal angles in a regular trapezium. 6. There are two pairs of parallel sides in a parallelogram. 7. The diagonals of a parallelogram are equal in length. 8. The angles of any quadrilateral add up to 360 ^{0}. 
It is by assessment that we show students what we value. If we want to authentically assess for meaningful understanding of the mathematics studied then we need to deliberately teach in ways that engender such understandings. Many textbooks, no matter how welldesigned or how well their examples are explained, are still set up on the precept that mathematics is taught according to the paradigm of ‘explanation of procedure, examples of procedure, setting of questions to practise the procedure’. Unfortunately, too many mathematics curricula are determined by the chosen text. Textbooks are essential resources but they shouldn’t determine the course or the methodology employed. I would suggest the following layout of curriculum for mathematics (example given is taken from a Y10 topic on Quadratics). It starts with an overview (taken from the State’s set of standards) then divides up into three columns – Concept and skill objectives, Methodology and Work Set & References. When assessments are being set, questions are directly and deliberately related back to the objectives.
Topic 8 : Quadratic Relationships Dimension : Structure Students construct tables of values and draw graphs for functions specified by rules constructed from arithmetic operations, for example, y = x^{2} – 3. They identify rules for such functions from tables of values and use these functions to models for practical situations. They solve simple equations, such as 4x^{2} – 3 = 13. They apply the algebraic properties (closure, associative, commutative, identity, inverse and distributive) to rearrange and simplify algebraic expressions involving real variables, and verify the equivalence or otherwise of quadratic algebraic expressions. They recognise and apply simple compositions of transformations to graphs in the plane. Students identify and represent linear, quadratic and exponential functions by table, rule and graph (all four quadrants of the Cartesian coordinate system) with consideration of independent and dependent variables for a given relationship, domain and range. They distinguish between these types of functions (constant first difference, second difference and ratio between consecutive terms of the dependent variable in a table of values; shape of their graphs) and their use and interpretation as models for data where a functional relationship is anticipated. They recognise and explain the roles of the relevant constants in the relationship y = a(x + b)^{2} + c
Extension : Expansion of three brackets by hand, completing the square with coefficient of x^{2} other than 1, factorisation by grouping, solving quadratic inequations. 

Concept / Skill / Technology Objectives 
Methodology

Work Set 
Students should be able to understand…. 


· how to recognise a quadratic rule from a table of values (by establishing it has a constant second difference) and that it has degree 2 · how to substitute values into a quad. expression · that a quadratic rule produces a parabola when graphed. · that a quadratic rule can be given in various forms – expanded, factorised, power form · that “correct power order” means to write the expanded form in decreasing powers of x · how to simplify algebraic expressions to correct power order · the meaning of terminology such as turning point, axis of symmetry, Maximum TP, Minimum TP, upright or inverted, axial intercepts · how to find the above for parabolas given in various forms 
Lessons 1 & 2 Start with a ‘whole game’ scenario involving a quadratic rule so students can see how a quadratic rule produces a parabola and that aspects of the equation are needed to find out things about the real situation…eg. the yintercept gives…the xintercepts give….the TP gives. Calculators can be used to find the axial intercepts and coords of TP for these real contexts. Students should understand that parabolas occur in real life in many different contexts (eg parabolic nature of any curve of flight of an object under gravity, parabolic shaped dishes, lights) Link back to other function work done previously on linear and exponentials by comparing patterns in tables of values (constant first diff for linear, constant second diff for quadratic, constant multiplier for exponential)

SSM Introductory Activities and Ex 1, 2

I am with Kevin Donnelly in his article “Don’t leave the kids alone” when he calls for more direct instruction in classrooms…up to a point. (There are many other points I disagree with in this article but this post is too long already – more on these later!!) I have never liked the idea of teachers as ‘facilitators’. Teachers are important. The work we do is important. The work we do in classrooms has to involve thinking – teachers’ thinking as well as students’ thinking. The framing of purpose for each lesson is essential. We need to have a set of concepts in mind that we want to target for each lesson and deliberately set about ensuring these are added to students’ knowledge bases. What is not essential, however, is the compelling of uniformity in approach to teaching to that purpose. The diversity of perspective, experience, context, language and ways of working that indiviudal teachers bring to their teaching can only enrich the classroom experience. Methodology should not be dictated by a central, controlling government body.
I apologise for the length of this post. More to follow later…and best wishes for the start of a new term!!