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Chris McGrane, Principal Teacher at Holyrood High School in Glasgow, describes some ways to encourage mathematical behaviours and insight in learners of Higher Mathematics.
Chris McGrane, Principal Teacher at Holyrood High School in Glasgow, describes some ways to encourage mathematical behaviours and insight in learners of Higher Mathematics.
Tasks for all purposes in Higher Mathematics
Tasks for all purposes in Higher Mathematics
Chris McGrane
Holyrood Secondary, Glasgow City Council
Twitter: @ChrisMcGrane84
December 2020
A major focus in my Higher lessons this year has been to increase the amount of ‘up time’. That is, the amount of time in a lesson where pupils are actively thinking about and doing mathematics. I’ve increasingly moved away from seeing my role as a ‘provider of notes’. There are many great resources out there where these are available. Of course, pupils do require examples and explanations – but there is much more to teaching Higher maths than providing these. I have tried to reduce the time where pupils are copying, watching and listening to me and, instead, increase the time where they are practicing, sense making, discussing, inquiring and problem solving.
I see my role as doing whatever I can to help pupils develop the procedural fluency required for standard questions, to gain conceptual insight into the ideas underpinning the work and, finally, to help my pupils to develop a sense of themselves as mathematicians. Direct instruction may be the basis of much of what happens in Higher Maths teaching, but I feel the key factor is in knowing when to ‘get out of the way’ and let pupils think and inquire for themselves. All of this is rooted in a belief that it is impossible to tell pupils conceptual understanding. We can tell facts and procedures and even give reasons for things, but it is by interacting with the mathematics for themselves that pupils actually begin to make sense of ideas. The guiding principles of the Association of Teachers of Mathematics (ATM) capture it nicely:
“The power to learn rests with the learner. Teaching has a subordinate role. The teacher has a duty to seek out ways to engage the power of the learner.”
I have been working on task design in relation to these goals. In this blog I share two tasks where I have tried to meet these broader aims.
Task One: Radian Measure Introduction
This task comes after an explanation that there is an alternative way of measuring angles, called radians. I explain how a radian is the angle formed in an sector where the arc length is equal to the radius and show the animation found here: https://www.reddit.com/r/GeometryIsNeat/comments/7rszn1/gif_explanation_of_radians/.
After discussing how multiples of pi seem to be useful, I set the pupils off on the task. I rarely tell pupils rules for converting between radians and degrees. “If it is a radians question do it in radians”. The relationships that exist become clear to the pupils by working on the task. I want pupils to develop a ‘feel’ for radians and to hold onto the idea that they are measures of angle. For this reason, the task includes the diagrams for pupils to draw the angles. I’ve also deliberately included the decimal representation of radians from the outset. Too often pupils think radians are “something to do with pi”. If pupils can make sense of the decimal representation and convert, mentally, to a ball park degrees figure, then they are at a significant advantage when it comes to checking working etc.
On this second page pupils have to work interchangeably between the representations. This is cognitively demanding but, coupled with peer and class discussion can help pupils to develop a better sense of what radians are about and how to work with them. As pupils work on the task my role is to circulate (wearing my mask!) pose questions, identify issues that arise and to reassure. The pupils are very much in control. They are the ones doing the thinking, doing the maths and making the connections for themselves. This is important in developing self-reliance and sense of oneself as a mathematician.
Of course, this is only one of a variety of radians tasks. When I followed this up with a traditional exercise, afterwards, I was pleased to see that the majority of pupils could take their learning from here and apply it elsewhere.
Task Two: Equation of a tangent to a curve
This is another task where I try to get pupils straight into the maths as soon as possible. We discussed the first example together and then I set the pupils the task. This task scaffolds the process of finding the equation of a tangent to a curve. However, it serves several other purposes.
The task makes use of variation to consider the case of positive, zero and negative gradients. Again, I deliberately direct pupils’ attention towards the graphical representation. We aren’t just manipulating meaningless symbols! The task is designed to set prepare the groundwork for later learning on stationary points by incorporating positive, negative and zero gradients with the graphical representation.
This task also gives pupils opportunity to problem solve. They have to tackle, without any teaching, the problem “how do we work backwards given the gradient?”
The task also makes links to finding the point of contact of a tangent and a parabola. It is a very related idea, but requires no calculus, although some pupils chose to use a calculus based approach.
Conclusion
In both tasks there is a limited need for direct instruction and, instead, space for sense making and opportunities to think mathematically. The tasks, once complete, also double up as a selection of examples which pupils can add to their ‘notes’. These tasks and many others can be found on my website: Starting Points Maths – Tasks and Blogs from Chris McGrane.
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