Planning and Teaching a Rich Mathematics Curriculum Based on Proficiencies
Peter Sullivan
Emeritus Professor, Monash University
July 2025
An important question teachers ask themselves when planning is “What do I want my students to learn?” This can be readily answered by reading and choosing from descriptions of the content in the curriculum. A further question “Why do I want them to learn that?” can be considered by reading ways the relevant curriculum content develops in subsequent years
A more important question “How will they come to know?” is answered by choosing how we want students to approach unfamiliar mathematics tasks and problems not by asking themselves “What have I been told about how to do this?” but rather, by asking “What do I know that I can use to do this?”
These two questions represent quite different mindsets. Many of the resources currently being advocated by jurisdictions and systems explicitly assume the optimal approach is for students to rely on what they have been told to do by the teacher. Drawing on what students already know is much more desirable, and all approaches to teaching mathematics should seek to develop their confidence and orientation to approach new tasks in this way. It is where the mathematical proficiencies can guide planning and teaching of lessons and sequences.
The Australian Mathematics Curriculum (ACARA, version 9) includes:
An expectation of mathematical proficiency … that students develop mastery in mathematics through the development and application of increasingly sophisticated and refined mathematical understanding and fluency, reasoning, and problem-solving skills.
Similarly, the Victorian Mathematics Position Statement (Department of Education, 2025) includes:
The VTLM 2.0 provides a framework to develop and connect the mathematical proficiencies – understanding, fluency, problem solving and reasoning.
Formal definitions of these four proficiencies can be found at:
The following are brief explanations.
The goal of Understanding is that students learn why and how, not just how. For example, multiplying by 10 means that the digits stay the same but move one place value (10 times) higher. This is much more meaningful that being told to move the digits to the left.
The goal of Fluency is not speed, but flexibility. For example, we want students to come to know that one way to calculate 6 × 9 is by thinking of it as 6 groups of 9 which is the same as 6 groups of 10 less one group of 6. This readily leads to students mentally working out for themselves calculations like 6 × 99, 8 × 19, etc. Fluency is much more than remembering.
An exercise is a problem for students if they have not been shown how to solve it. Learning Problem Solving is about having experiences that allow them to apply what they know to unfamiliar tasks. Carefully selected problems can also build understanding and foster fluency.
Reasoning is the process of extending the other proficiencies in more general, adaptable and flexible ways. Reasoning is done by students when prompted by suitable tasks and in discussions with teachers and peers around meaning. It is not about being told.
To give a specific example, suppose we are hoping students learn angle measurement using protractors in an upper primary class. Consider these three ”lessons”.
A: A page of various angles that students estimate, then measure with a protractor.
B: A page of various diagrams such as a 500 angle with arms of 10 cm and 8 cm. Students are invited to reproduce each diagram and describe them as fully as possible.
C: A set of tasks such as “A triangle has one angle of 500 and sides of 8 cm and 10 cm. Draw what the triangle might look like”.
All three options have value although they offer quite different opportunities. Some key questions teachers could consider when planning:
What experiences do each of A, B and C offer students?
Which would you do first? Second?
Are they each accessible? Do they each extend student thinking?
Which proficiencies do they each address?
In my opinion, A and B are about fluency, although B has the advantage that students are using measuring tools.
C is about solving an unfamiliar problem that, in addition to fostering fluency, builds understanding of the ways a given angle and two sides define the triangle. C also illustrates reasoning in that if the angle is between the given sides only one triangle is possible, but if it is not, there may be other possibilities.
Different teachers may have opinions of the order of presenting such experiences to students. Whatever order is chosen, C has much more potential to engage students in thinking about angles than either A or B. If C is done first, B then A can consolidate the learning.
Such discussions and decision making reflect the process of planning sequences.
Reference:
Department of Education (2025). Mathematics position statement. Downloaded from
