“The award of satisfactory completion for a unit is based on a decision that the student has demonstrated achievement of the set of outcomes specified for the unit. This decision will be based on the teacher’s assessment of the student’s overall performance on assessment tasks designated for the unit”

Any Victorian VCE teacher will recognise this paragraph. It is from the Mathematics Study Design for the final two years of schooling in this state.

A conversation today based around whether we should include the final examination in our determination of whether a student has or has not achieved the outcomes (to ‘get an S’ and be labelled as ‘Satisfactorily Completed’) has initiated this post.

Mathematics, more than any other subject, I think, presents students with the stark alternatives of ‘success’ and ‘failure’. Not just in terms of ‘passing’ a test or subject but with each individual problem attempted by a student. Students ‘succeed’ if they solve the problem and ‘fail’ if they don’t. Many students, however, have ‘learned’ – as opposed to ‘learning’ – difficulties with the subject.

These can be affective difficulties, difficulties with understanding, difficulties with the actual skill level involved etc. Sometimes (those readers who struggled with mathematics when they were at school would probably say ‘most times’!) the way that the subject is taught is mainly to blame for these difficulties becoming entrenched. For example, by being taught that there is one, acceptable way of doing problems to obtain the ‘right’ answer without developing the method of solution themselves, students can remain disengaged, passive recipients of handed-down knowledge rather than active, creative thinkers and knowledge creators.

Furthermore, in previous posts on Creating Resilience in Students and Templating in Mathematics Teaching and Learning, I have expounded on one of my pet subjects in relation to the teaching and learning of mathematics: that of how the actual methodology of teaching mathematics can assist or hinder students’ affective difficulties with the subject. The ability to respond to failure by exerting more effort (or trying different approaches) is the ability to persist. This is linked to academic resilience. We must exert more effort, ourselves, as teachers – and therefore as people who have the potential to be great sources of behaviour modelling and inspiration – to help and support our students develop the belief that they have the cognitive and behavioural controlling capability to develop their intelligence.

So – how does all this relate to getting an S for a VCE unit or not?

If a student is successful in the performance of an assessment task, this can sometimes say more about the task itself than about whether the student understands sufficiently well for a teacher to determine whether the student has performed ‘satisfactorily’. I would like to be able to say that every student to whom I have awarded an ‘S’ for a unit of mathematics has **understood** the mathematics to a specified degree…not just handed the work in or sat the task. In order to authentically determine this level of understanding, a number and variety of types of assessment tools should be used in order to give students the maximum number of opportunities in order to demonstrate said understandings.

Yet, as I have indicated above, being able to demonstrate understanding on authentic (not just skill and drill memorisation exercises) performance assessments in senior mathematics can sometimes take a while to develop as students have to grapple with affective concerns that occur in this discipline more than others. Hence, the determination of whether a student can or cannot be afforded the dastardly label of ‘Satisfactorily Completed’ must be done over as many tasks as possible.

Exams have had bad press in ‘teaching for understanding’ circles but a properly constructed exam that doesn’t just test accurate memorisation of facts, recall of standard procedures and the student’s ability to ‘plug and chug’ can be a real tool to test not only for understanding over a range of topic areas but also the ability to transfer knowledge between contexts. The exam (in fact any task that has questions that compel learners to assimilate knowledge from various sources and apply this knowledge in unfamiliar contexts) can be a wonderful tool. It can focus students’ thinking and compel them to think more widely than individual topic tests as they must interpret what each question is asking them to do and connect this to an area of content that may not be stated in the question.

What would you consider to be evidence of determining whether a student had a satisfactory understanding of mathematics?