Good question for adding unlike fractions

Can you add 2 dollars and 3 pounds?


Teaching Mathematics Ch 9: Reflections on the influence of technology on the curriculum

Technology inevitably changes the way we do maths: the most obvious is probably the use of electronic calculators replacing slide rules, log tables etc.

Does this mean that the skills that learners require change too? Is there any value in knowing how to compute a log by hand if the calculator can do it so easily? Or, a more up-to-date question, do pupils still need to know how to manipulate algebraic terms in the face of computer algebra systems?

Answering this question relies on making a distinction between functional maths and pure maths, or mathematical thinking. For a learner, they probably don’t need to know how to manipulate algebra in their day-to-day lives (if they ever did!), and technology is further reducing the need for this.

However, in terms of mathematical skill and mathematical ability, the ability to manipulate algebra and, even more importantly, to understand what you’re doing when you’re manipulating algebra, is no less important. It’s part of a package of training and skills that will give you the confidence to go on to study more advanced maths, such as calculus. Making sense of what is actually happening when you’re undertaking these operations is a crucial element of developing mathematical thinking; technology can’t replace this I don’t think.

The question is whether I can successfully engage pupils in the need and enjoyment of developing mathematical thinking, rather than seeing all maths as functional.

A final point on algebra is that computer packages, whilst very reliable, give you no feel for the answer; it is very hard to tell whether the answer it gives is correct, or whether its an error (for example if you mistype input variables). Doing the process yourself allows you to spot these errors more easily.

Looking ten years ahead, the reflection exercise asks me to consider the importance of a range of operations:

  • Calculating a percentage of a quantity: Calculating an exact percentage may be less important than having a feel for the impact of percentage changes in assessing e.g. a sensible loan or a good discount at the supermarket. Performing the exact calculation may be easy on a mobile phone, but having a feel for it intuitively can be very useful too.
  • Measuring an angle: I’d argue that this isn’t enormously useful in most people’s day-to-day life already. However, in the next 10 years it is unlikely that the average piece of technology will be able to do this for you by default, and so I don’t see the importance of this changing greatly as technology develops.
  • Simplify sums of like terms in algebra: For me, this won’t change a great deal in its importance. At present it is valuable for anyone seeking to take maths to a further level, and isn’t much practical use for someone who isn’t interested in doing this. I don’t think that will change, based on the arguments above.
  • Solve linear and quadratic equations: As above. This is valuable for its contribution to mathematical thinking and skill, not its instrumental value in pupils’ day-to-day life.

Using and applying mathematics

A clear priority for me is to develop my understanding of this part of the curriculum, and how to teach it well. Ofsted’s Understanding the Score elaborates on the concept (pp.35-6):

…require pupils to use and apply mathematics in substantial tasks through which they are able to decide what approaches to adopt, use a range of mathematical techniques in exploring the problem, find solutions, generalise and communicate their reasoning.

The report argues that this aspect of the curriculum is currently far too weak (p.36):

Teachers seldom plan explicitly for ‘using and applying mathematics’ and it is very rare for schools to assess this aspect of pupils’ learning separately.

The report also describes the best practice that inspectors found (p.49):

The best practice had ‘using and applying mathematics’ at the heart of teaching and learning in mathematics: pupils were viewed as budding mathematicians and developing their understanding was of paramount importance. This was reflected in a shared ethos, pervading the teaching, learning and curriculum, and focused on approaches that developed pupils’ understanding and their independence in using and applying mathematics… Good curricular planning provided pupils with opportunities to apply mathematics to a variety of interesting tasks, enabling them to choose approaches, reason and refine their thinking in the light of their solutions. Teachers encouraged pupils to discuss mathematical problems in depth and this helped to build their confidence.

The more recent Ofsted report Made to measure adds (p.74, my emphasis):

The schools with outstanding mathematics curricula ensured systematic and progressive development of pupils’ skills in using and applying mathematics. Some secondary schools were taking steps in the right direction by linking rich mathematical activities into each topic to support pupils’ conceptual development and problem-solving skills, but did not always consider explicitly which process skills were being developed.

It’s odd because to me this doesn’t sound like a very difficult thing to do, and it has clearly been a strong priority since Understanding the Score a few years ago. I’m sure once I get practical experience in school I will better understand why it is difficult to bring rich, real-life tasks regularly into the curriculum, but for now I will note it as a priority for my classroom.

Finally, Understanding the Score has a great example of good practice: 

Prime practice: teaching mathematical thinking  

The context of this Year 9 problem-solving lesson was a series of questions about the number of permutations of letters in different names, such as LUCY, ALI or WAYNE.

Rather than show pupils the standard formula, the teacher provided them with an opportunity to find their own solutions. This was not as haphazard as it might seem, because he also had a very clear idea about which kinds of thinking he wanted to encourage and the point he wanted pupils to move towards. This type of problem solving might be characterised as ‘open in the middle’ rather than open-ended. The lesson objectives were: ‘Pupils will learn: the value of working systematically to solve problems; to refine their understanding of the methods they develop; to refine their oral and written explanations of their methods; and the value of reducing a problem to a simpler case.’ For much of the lesson, the teacher’s role was to listen to pupils explaining their ideas, to encourage and nurture any systematic thinking, and to intervene with additional problems when appropriate. Mini-plenaries were used as appropriate to encourage pupils to share their ideas with the class, draw out key ideas that emerged and stimulate further thought about variations on the original problem. By the end of the lesson, most pupils had worked out that the number of permutations of n distinct letters would be n ! = 1 × 2 × 3 × … × n. More importantly, they understood the importance of making systematic lists and therefore understood in a concrete sense the recursive nature of the solution: that a five-letter word could begin with any of the five letters, followed by any of the 24 permutations of the other four letters, giving 5 × 24 = 120, and that 24 arose as 4 (starting letters) × 6 (ways of arranging the other three letters), and 6 as 3 × 2, and so on. Variations of the problem were held in reserve, such as EMMA, ANN, GEMMA and DONALD, leading to the generalised problem of counting permutations when some letters repeat. Many pupils recognised that having two letters the same halved the number of possibilities and that having three letters the same reduced the number further, but realised that this needed more thought.