Ambitious mathematics: Ratios, decimals, fractions and time for Ta’ovala:‘Low floor, high ceiling’ tasks: Ambitious mathematics for mixed ability groups
The Ta’ovala mathematics problems
These students demonstrate collaborative problem-solving skills and show the depth of learning already established. They enter into their group work with no intervention from the teacher and high levels of engagement are sustained as everyone participates.
The lesson has been launched and students know how to proceed. They bring their developing understandings about Ta’ovala weaving to bear as they read the problem together and are engaged in collaborative problem solving from the outset: questioning, reflecting, listening, checking and requestioning.
While the standard expectation at Year 8 is for students to achieve curriculum level 4 by the end of the year, this ratio problem sits at level 5 of the curriculum, with Problem (b) being high order due to the use of time as one variable and all the numbers not sharing a common factor. DMIC tasks at times may be designed to scaffold the children from an initial straightforward/simpler problem into one or more further problems that will challenge everyone in the class: a ‘low floor, high ceiling’ approach.
Problem (a) How long will it take Sia’s family to weave 12 Ta’ovala?
Figure 1: Problem A
Figure 1 Photo by Indira Stewart
The groups build on their prior knowledge, especially around identifying the common factors. Their prior learning with fractions – that you need to multiply or divide both parts of a fraction or ratio by the same number to maintain equivalence – is an important resource for the problem-solving endeavour.
Problem (b) How long will it take a bigger group of ladies to weave 12 Ta’ovala?
Figure 2: Problem B
Photo by Amy Heeemsoth ©Khaled bin Sultan Living Oceans Foundation
The teacher’s role is to observe closely to understand the children’s thinking and where they are experiencing challenges. In Problem (b) the groups are grappling with the challenge of dividing that won’t tidily resolve into a whole number. David is listening carefully to the children, asking questions and noting that all groups have moved to using decimal fractions. He notices that most groups are not yet attending to the dimension of time.
At this point in the lesson David intervenes, but not to tell the students a correct way to solve Problem (b). Rather, he reflects and values their thinking and directs the students to reflect again on the wording of the problem. In this way the students learn to grapple with productive failure, and work collectively towards the correct solution.
One group, who attended to the time element and broke the problem into hours and minutes, has made the most progress towards solving the problem.
The success of DMIC group work depends on the expertise of the teacher at every step of the process. Because the teacher has followed the thinking and progress of the different groups and has himself reasoned through the possible solutions they may select, he is well-placed to make a strategic choice for the order in which the groups will share their progress with the rest of the class. This will matter so that the sequencing builds on and extends the learning of all the students.