Ten calculations that A level students should be doing on a calculator: can yours?
Tom Button, Technology Professional Development Lead for the Advanced Mathematics Support Programme (AMSP), addresses the increased emphasis on the use of technology in the new A level specification. Here, he tackles use of calculators.
While proficiency with using a calculator for mathematical calculations clearly saves time and supports accuracy in exams, Tom argues that there are wider benefits for calculator use permeating the whole of teaching and learning of A level maths.
When the NCETM met the exam boards for a podcast in October 2019, the role of calculators in the updated A level Mathematics examinations was discussed. Reference was made to the expectation that A level students use their calculators to perform certain standard calculations, unless the question explicitly indicates otherwise. These processes include solving equations/inequalities, evaluating derivatives/integrals and solving simultaneous equations as well as performing binomial or normal distribution calculations.
When I’ve spoken to teachers there has been a good awareness of the need for students to use their calculators in statistics; however, there has been less appreciation that students should be using the calculators for many routine processes in pure maths, such as definite integration. In the podcast it was identified that students would benefit from being more secure with using their calculators for many such processes.
Why the increased emphasis on use of technology to do the maths?
Ofqual guidance states that the use of technology should ‘permeate’ the study of A level maths. The reason that this statement is in the guidance is based on an awareness that technology is fundamental to how mathematics is used in most areas of academia, industry and commerce. Outside the classroom most mathematical processes are automated by technology and not performed by hand. This can be seen in a variety of situations as diverse as scientists investigating climate change, engineers designing aircraft, or companies such as Amazon analysing their customers’ data.
The new A level is intended to reflect this to ensure that part of each student’s learning experience involves using technology mathematically. It is difficult to fully model this experience of using technology for maths in examinations; however, setting questions which involve students using calculators for numerical processes is an achievable way of partially addressing this. As such the use of calculators for these processes can be viewed as an element of students gaining an appreciation of how mathematical technology is used.
So why do we teach students mathematical techniques, such as solving quadratics, by hand?
This approach to using technology is consistent with how students are expected to work with other mathematical processes. For example, students will learn the concept of multiplication earlier on in their schooling, but by the time they reach A level they do not need to be assessed on whether they can find the product of a pair of three-digit numbers. In many cases learning to perform a mathematical process by hand can be used to support students in developing a deep understanding of the structure of the mathematics. When such processes are being applied in other areas of mathematics it is often more efficient and appropriate to use a calculator. Learning to multiply without a calculator can be very useful in helping students understand the underlying concepts, but in an A level question the expectation is that they would find such a product using a calculator. Many other mathematical processes can be thought of in a similar way. During the A level course, students need to understand the concept of differentiation, and questions can be set that test this. However, if a derivative is needed in a question set in a particular context then using a calculator to find it allows for the focus of the question to be on the modelling skills.
What does this mean for the A level classroom?
Students need to know how to work algebraically, as well as which processes they can outsource to their calculators. They also need to know when it is appropriate to use a calculator and when a more rigorous algebraic approach is needed. Page 6 of AQA's AS and A-level Maths Teaching guidance (free registration required, but available to all teachers even if they are not at an AQA centre) has a very helpful checklist of the processes that students are expected to know how to perform on their calculators:
- solve quadratic equations
- find the coordinates of the vertex of a quadratic function
- solve quadratic inequalities
- solve simultaneous linear equations in two variables
- calculate summary statistics for a frequency distribution
- repeat an iterative process, including the Newton-Raphson method
- find binomial and normal probabilities and find the z-value for a normal distribution
- calculate a definite integral
- calculate the derivative of a function at a given point
- solve equations when the question permits, e.g. when problem solving and modelling.
It is advisable for teachers to ensure that their students can do all of these. And they are all implicitly covered in a comprehensive set of Instructional videos for how to perform various calculations on the Casio Classwiz are provided by Casio. For other brands of calculator, please consult the relevant guide for your calculator.
Teachers should also encourage their students to look out for any phrases in exam questions, that indicate that a calculator method isn’t expected such as “fully justify” or “In this question you must show detailed reasoning”. However, even in those questions where a calculator method will not gain the marks, students can still be encouraged to use their calculators as a checking tool.
As I’ve indicated here, I think that using a calculator for numerical processes is one aspect of using technology effectively in mathematics. This is more powerful if it is complemented by students seeing a variety of uses of technology for mathematics including graph plotters, statistical tools and spreadsheets. I will explore these in more detail in future articles.