Goal-free Problems in Mixed Attainment Classes
Posted 22nd May 2019
The vast majority of Secondary schools teach mathematics in setted groups, usually based on prior attainment. Often these sets are decided at the very start of Year 7. Decisions about which set a student goes into could have monumental impact on how they access the mathematics curriculum. This year I have been teaching both a mixed attainment Year 7 and a middle set Year 9 class. For this blog post I’ve reflected on what the students and I have learnt together in these two very different environments.
Year 7 Mixed Attainment
To allow all students in my Year 7 group to be able to contribute to lessons, “goal-free” problems have become an integral part of my planning. These give the students the opportunity to reason mathematically without the added stress of “finding the answer”.
This visual prompt is part of a lesson on coordinates. The students have met coordinates in all four quadrants in Year 6 and this task allowed me to assess their current knowledge, and to get a feel for how they think.
This simple collection of points was a textbook question that has been stripped of axes, numbers, context and wording to lessen the restriction of a pupil’s thought processes. The labels allowed for students to refer to specific points immediately, though even these could have been removed.
Some students might simply notice things (A is above B), some might want to impress with vocabulary (the line formed by joining C and B is parallel to the x-axis), others might allocate numerical values for each point. I then write all their contributions on the board for the whole class to discuss and learn from - there are no right or wrong answers here. Having established shared vocabulary and concepts and built on pupil’s understanding, the next task could delve a little further into their understanding using sentence stems to guide students:
With a mixed attainment group, this exercise allows for reasoning and accessibility at all levels. Some students might think about how their answers would change for specific types of triangle, say right-angled or isosceles. For some, writing in sentences is a good start, which then might turn to a more mathematical answer with input from others in the class or the teacher. Some schools might even have iPads or laptops to allow for digital exploration. Most importantly, all are able to access the question and contribute to each other’s learning in some way.
Material from the White Rose Maths Secondary Schemes has similar tasks that promote discussion, sharing and group work, all of which are have proven invaluable in getting the most out of every student in a mixed attainment group, and building student confidence. These Schemes of Learning and resources are currently available for Year 7, all of which are FREE to download and use in class. Release of Year 8 Schemes will commence shortly for Autumn 2019 onwards.
Year 9 Middle Set
In Year 9, as my students start to prepare for GCSE work, their thoughts turn towards scores, ratings and rankings. This fixation is natural, given that they will be given a number from 1 to 9 to work towards for the end of KS4. I find myself making a concerted effort to disregard any talk of sets, “What are top set doing?”, “Will I move down?”, “What do I need to do to move up?” as this is seldom productive or helpful discussion.
Some students already had a fixed idea of how good (or not) they were at maths, and I soon realised that goal-free problems would work well here too, removing pressure of right or wrong and allowing everyone into the conversation. Using a similar structure to the tasks I had used with my mixed attainment Year 7s, wide gaps between student knowledge were revealed, even within a supposedly similarly attaining cohort of pupils.
Given the same collection of points on axes, and the question, “Which two points are furthest apart?”, conversations started about which pairs easily expose the distance between them, how many times longer some distances were than others, some measured and some used approximations. A handful postulated that the addition of a right angled triangle might reveal that Pythagoras had hidden yet more of his work amongst the coordinates. All from the very same set of dots that Year 7s had looked at previously.
Reflecting on these two cohorts, I find that goal-free problems not only open up maths for all the students, regardless of the make up of the class, but also reveal hidden links to other areas of maths for us teachers. To strip a problem of its question and its identity seems such a simple technique, but allows for so much scope and allow children to see the same image grow up alongside them, maturing in its mathematical richness and possibilities, just as they do with theirs.
Tim Chadwick
Secondary Maths Specialist for White Rose Maths
