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Professor's Page Is understanding “The proficiency strands describe the actions in which students can engage when learning a proFIciency? and using the content. While not all proficiency strands apply to every content description, they indicate the breadth of mathematical actions that teachers can emphasise.” (Australian Curriculum Assessment and Reporting Authority (ACARA) 2011) Mike Askew Professor of Primary Education, Monash University I like this quote from the Australian Curriculum: it. We need continuously to challenge this Mathematics; firstly, because it describes the ‘fluency first’ view as otherwise building the mathematical proficiencies as actions and, other proficiencies into mathematics lessons secondly, for drawing our attention to the idea may be put off to some later (and again often that learners need to enact the proficiencies mythical) time when learners are ‘ready’ to when learning mathematics and not just when engage with them. applying it. Just in case you haven’t yet learnt the In other words, taking the stance of the Australian Curriculum: Mathematics by heart, a proficiencies as actions means moving from reminder that the four proficiencies are fluency, seeing school mathematics as a body of reasoning, problem solving and understanding. knowledge for learners to acquire, to seeing it The everyday use of ‘proficient’ carries as an activity to engage in; or, in the words of connotations of having reached a level of Brent Davis, moving from seeing mathematics expertise; my edition of the Oxford dictionary as preformed to mathematics as performed. gives the definition as “Proficient (adjective): adept, expert” (with Proficiency labelled as Performing proficiencies an adverb). We would not describe someone stumbling through a rendition of “Chopsticks” So what do these proficiencies look like in as a proficient piano player; but novice pianists action? ACARA documents several examples: work on musical proficiencies—practicing scales or playing a polka—in parallel. They Fluency: do not put off playing the polka until they can • choosing and carrying out procedures, play scales fluently. • recalling factual knowledge and concepts Like learning to play the piano, becoming So far, so good. I can engage learners in mathematically proficient means engaging in the action of choosing between procedures certain actions even before one displays full and observe them doing so. I can see competence with these actions. Becoming a whether or not they can recall things (when proficient mathematician means working with appropriate), like multiplication facts or all of the proficiencies from the beginning— names of quadrilaterals. and by ‘mathematician,’ here I mean anyone using mathematics in his or her life. Everyone Problem solving: is a mathematician. • making choices This challenges the popularly held view • interpreting, formulating, modeling and (myth even) that children first become fluent investigating problems situations, in adding, creating equivalent fractions, • communicating solutions naming shapes, or whatever and only then can Again, active verbs that I can imagine apply this to solve problems, or reason about learners doing and could observe in action. APMC 17 (1) 2012 19 Askew I can see what choices learners make in how Is understanding a proficiency? to turn a problem into a mathematical model, and communicating solutions has to be done I have difficulty describing ‘understanding’ as for someone, which I can also observe. an ‘action’—I can develop understanding, I can demonstrate understanding, but I am not Reasoning: clear how I ‘do’ understanding. If I make a • analysing • inferring connection between, say, decimal fractions and • proving • justifying place value, isn’t that likely to be an outcome • evaluating • generalising. of engaging in some problem solving and • explaining reasoning about the relationship? I prefer More good, strong, observable verbs. A child to think of understanding as the result of who conjectures that adding two odd numbers doing the other proficiencies—engaging in must always be even might justify this by some problem solving, reasoning about the ‘why’ sketches showing how each odd number can of mathematics and being fluent in the ‘how’ be drawn as having a ‘one’ sticking out and so of mathematics provide the building blocks of putting them together makes them fit to make understanding. A bit like nourishment: I can an even number. Good observable stuff. buy food, prepare it, eat it and digest it. Yes, digestion is an action, but it is of a different understanding order to the first three, particularly in the • Building robust knowledge of degree of control that I, or others, have over it. mathematical concepts. I am not the only one to take this position. • Making connections between related ideas. Ron Ritchart and colleagues from the Project At first glance these look like actions: Zero team, in their analysis of thinking, arrive building, making, describing. But what does at the position of “understanding not to be building ‘robust knowledge’ or making a type of thinking at all but an outcome of ‘connections between ideas’ look like? I can thinking” (2011). Arguing along similar lines, see what people are doing when building, say, Wiske (1997) concludes that “understanding a robust tower or constructing connections is not a precursor to application, analysis, between riverbanks, but can I actually observe evaluating and creating but a result of it.” the actions of building knowledge or making I am inclined to this latter view—paying connections? I can only see the results of this. attention in mathematics lessons to a good Children can demonstrate that they have made balance of the actions involved in fluency, connections, that they have built knowledge, problem solving and reasoning will lead to but these are the result of internal process that connected, robust, related understanding. I cannot directly observe. OK, perhaps I chose It’s great that the proficiencies highlight poor examples. What others can I find? understanding but let’s hope that it does not • Describing their thinking get sidelined if teachers find it difficult to turn • Develop an understanding of the it into actions. relationship between the ‘why’ and the ‘how’ of mathematics. References Describing their thinking—that is an Australian Curriculum Assessment and Reporting action I can both encourage and observe. But Authority (ACARA) (2011). The Australian how does that differ from the ‘explaining’ Curriculum: Mathematics F–10. Canberra: ACARA. Ritchart, R., M. Church, et al. (2011). Making thinking or ‘justifying’ in the reasoning proficiency? visible. San Francisco: Jossey-Bass. Or the ‘communicating solutions’ in Wiske, M. S., Ed. (1997). Teaching for understanding. San Francisco: Jossey-Bass. problem solving?And to say that an action of understanding is to develop an understanding is, well… taking us round in circles. 20 APMC 17 (1) 2012