As seen in the quote on the previous page – numeracy is not about abstraction, but rather the use of mathematics in life’s contexts and situations. That is, numeracy is about getting the job done with the mathematics that is already in place. As such, the tasks on the previous page have been designed to require a lot of mathematical thinking without relying on a lot of mathematical knowledge. This may seem like a bit of an oxymoron, but when you start to play with these tasks you will see that it isn’t. These tasks really do allow for a great deal of mathematical activity and discussion without relying to heavily on specific pre-requisite knowledge.
As a result, these tasks are a great resource for many aspects of teaching and learning that you may wish to focus on in your classroom. From formative assessment to group work, from persistence to writing, from estimation to reasoning these tasks provide a context within which you can invite learners into the discourse of mathematics in general, and foster numeracy proficiencies in particular.
In what follows I outline the characteristics and qualities of these numeracy tasks that allow for these affordances.
Four Types of Numeracy Tasks
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The Numeracy tasks I have been involved in developing tend to fall into one of four categories:
- Fair Share Tasks – These are tasks that involve a scenario in which something needs to be shared. How this is to be best accomplished is always problematic as the notion of fairness is not well defined. Goody Bags, Sharing Cookies, and Giving out Bonuses are some examples of Fair Share Tasks.
- Planning Tasks – These are tasks that involve a situation in which some sort of planning must take place. They could be budget type problems or they could be spatial problems. Going Bowling, Trip to the Waterslides, and Playland Adventure are some examples of Planning Tasks.
- Estimation Across a Large Number of Variables Tasks – These are tasks that require the solver(s) to make several estimations in order to arrive at a possible solution. Although often being restrained to one unit of measure (like time) the estimates range across a wide spectrum of contexts. The Concert in Toronto is an excellent example of this kind of task.
- Modelling Tasks – These tasks require the solver to develop a strategy for action based on one set of data or information. They are then required to apply this strategy to another set of data and then make adjustments as necessary. The 2004 Summer Olympics Results task is an example of this type of task.
Of course, many tasks are actually a blending of these types requiring students to plan and estimate, or estimate and share, etc.
There is a 5th type of task that I am just now beginning to experiment with. This type of task will be structured around the concept of subjective probability – a skill that is vitally important in a numerate society. Stay tuned!
Qualities of a Good Numeracy Task
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Each of the numeracy tasks on the previous page have all been developed to have the same five qualities:
- Low Floor – There is something in each of these tasks that every student in the intended grade range can engage in. By necessity, this means that careful consideration is given to the prerequisite knowledge that is needed to begin a task.
- High Ceiling – There is room at the top to go beyond the trivial. That is, the task can be interpreted or extended in such a way so as to allow some students to incorporate high degrees of mathematical thinking.
- Huge Degrees of Freedom – There are lots of opportunity to make decisions as to how to proceed. Although this always pertains to chosen strategies, these tasks are designed more specifically to allow students many options to choose from.
- Fixed Constraints – When faced with huge degrees of freedom students can become paralyzed by the overwhelming number of possibilities available to them. Constraints are therefore used to give them something to organize their thinking around – thus liberating them to begin.
- Inherent Ambiguity – Deliberate ambiguity is used not only to create situations where the students must MAKE decisions but also to create situations where they become AWARE that they are making decisions. These places of decision then become excellent points of discussion and prompts for writing about their thinking.
Attention to each of these five qualities allows us to create tasks that become effective vehicles for a number of different teaching and learning outcomes. What it does NOT do, nor should it, is allow for the creation of tasks that encourage students to work towards ONE best answer. These are open-ended tasks that allow for many interpretations and many different solutions.
Designing Numeracy Tasks
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The numeracy tasks that are available on this website have all been designed following the same set protocol.
- Engaging Context – The design of each task starts with the identification of a context that may be of interest to the target age group. I say may because the interest inherent in the task is sometimes difficult to gauge.
- Locate the Problematic Situation – This begins with deciding which of the four types of numeracy tasks this is going to be most closely alligned with. Once this has been determined the problematic nature of the task can begin to take form – how can we make planning or sharing or estimation more complicated here? Often the answer is through the use of ambiguity.
- Finalize the Pilot Task – The first draft of the task is finalized through a process of attending to each of the aforementioned five qualities of a good task.
- Pilot Test – The task is tested on a class of students and evaluated according to its potential to engage students in thinking, discussion, and written output. It is important NOT to judge the task on its ability to drive the students towards a normative single answer.
- Refine – Based on the pilot test the task is refined for more general use.
Once the task is complete it is important to resist the urge to think of it as perfect. The task needs always to be thought of as open to further refinement and development. This is why the tasks are offered as word documents rather than PDF’s.