Scaffolding student learning is the primary task of teachers of mathematics, but this cannot be achieved without accurate information about what each student knows already and what might be within the student’s grasp with some support from the teacher and/or peers. This requires assessment techniques that expose students thinking, but it also requires an interpretation of what different student responses might mean and some practical ideas to address the particular learning needs identified. This is particularly important in relation to a relatively small number of ‘big’ ideas and strategies in Number, without which students’ progress in mathematics will be seriously impacted. This, in a nutshell, is the purpose of the tools presented here. The tools comprise a number of easy to administer, practical assessment tasks designed to address a key area of Number at each Level of the Victorian Essential Learning Standards (VELS). A hierarchy of student responses is identified for each task and, for each of these, an interpretation of what the response might mean is provided together with some targeted teaching suggestions.
The key ideas addressed at each level are listed below.
- LEVEL 1 – Trusting the Count, developing flexible mental objects for the numbers 0 to 10
- LEVEL 2 – Place-value, the importance of moving beyond counting by ones, the structure of the base 10 numeration system
- LEVEL 3 – Multiplicative thinking, the key to understanding rational number and developing efficient mental and written computation strategies in later years
- LEVEL 4 – Partitioning, the missing link in building common fraction and decimal knowledge and confidence
- LEVEL 5 – Proportional reasoning, extending what is known about multiplication and division beyond rule-based procedures to solve problems involving fractions, decimals, percent, ratio, rate and proportion
- LEVEL 6 – Generalising, skills and strategies to support equivalence, recognition of number properties and patterns, and the use of algebraic text without which it is impossible to engage with broader curricula expectations at this level
The tools are based on a series of highly focussed, research-based Probe Tasks which were developed for teaching purposes at RMIT University and subsequently used in the context of the Supporting Indigenous Student Achievement in Numeracy Project in the Northern Territory to identify learning needs in Number. To support teachers in remote schools interpret student responses, identify learning needs, and choose developmentally appropriate tasks to address those needs , advice in the form of a Probe Task Manual was prepared. The advice was prepared on the basis of the broader research literature, ‘mainstream’ student responses derived from Victorian classrooms, and the responses of a small sample of Indigenous students who participated in the NT study. The Probe Tasks and the Probe Task Manual are currently being used by teachers in Northern Queensland to identify and address student learning needs.
The tools presented here draw on the Probe Tasks and the Probe Task Manual but include a number of additional tasks and resources which have been organised to address some ‘common misunderstandings’.
About the tools
The idea behind the use of the tools is to provide teachers with a set of easy-to-use diagnostic tasks that expose critical aspects of student thinking in relation to key aspects of Number as it is this area that research has shown to be most responsible for the huge differential in student performance by the middle years. The tools also provide advice on targeted teaching responses to the ‘common misunderstandings’ and/or learning needs identified. They are particularly useful in identifying the learning needs of students who teachers believe are ‘at risk’ or likely to be at risk in relation to these important underpinnings. In some cases, this might mean using the tools from the level below the student’s actual year level. They can be used to obtain more accurate or in-depth information about students who teachers feel are under-achieving.
A small number of ‘stand-alone’ tasks are provided at each Level. The tasks have been designed to be administered individually, and generally take between 5 and 10 minutes, which means that they can be used in class without withdrawing students (although sometimes this may be advisable). Before using the tools, it is suggested that teachers read the related advice so that they are aware of likely responses. As the whole point is to expose student thinking, not to assist student’s to get the ‘right answer’, teachers are strongly advised to resist the urge to teach during these episodes and to either terminate the conversation or move on to another task as soon as a student experiences difficulty.
Wherever possible, readily available classroom materials have been used. Cards can be reproduced as required or laminated and some additional resources have been provided to support the targeted teaching suggestions. While it might seem tedious to prepare the cards, it is important to use these with individuals rather than attempt to adapt the activity to a pen and paper worksheet task that might be used with a small group or the whole class. The cards provide ‘thinking space’ as students have the opportunity to move them around and see them in relation to other cards. Some tasks require the cards to be manipulated or sorted, others simply serve the purpose of presenting a problem in isolation from other problems without the expectations flagged by empty spaces to provide a whole lot of working.
For each tool, the advice has been presented in a table that matches an observed response (left hand column), with an interpretation (in italics) and one or more suggested teaching responses (dot points) in the right hand column. Teachers should identify the observed response that best matches the student’s response and consider how they might implement the suggested teaching response.
The tools at each Level have been chosen to address key ideas at that Level which, if not understood, will seriously undermine students’ capacity to engage meaningfully with core aspects of the Number Strand in subsequent years.
It is hoped that the tools will prove a useful resource in addressing the needs of all learners but particularly those that fall behind.