Marking and feedback

Great couple of posts from the ever-inspirational headguruteacher: Marking in Perspective and Making Feedback Count. Lots of good stuff in them but the key point that stands out for me is making time for and creating a culture in which students act on feedback. The mantra of ‘closing the gap’ is a great one – closing the gap between the work that has been done and the work that could be done at a higher level with the benefit of feedback.

I will add this into my list of habits to develop in the classroom – I can feel a Friday skills lesson or similar coming on!

It also further strengthens my desire to have a space for ongoing dialogue in pupils’ exercise books – perhaps a space in the first few pages at the start of the book?

Finally, the resource below provides a range of practical tips for closing the gap – brilliant.


Teaching Mathematics Ch 10: Continuing Professional Development

  • When assessing your lessons, focus on the quality of teaching and the quality of learning.
  • As you become more experienced, reflection will become less formal but no less important for improvement.
  • Generally, trainees feel that they most in the early months of training their: ability to structure a lesson; and class management. To a lesser extent they develop time management, relating to pupils, and learning to pitch the lesson at the right level.
  • The Career Entry and Development Profile (which you complete at the end of your first year) is designed to help bridge the gap between your training and your first year in teaching.
    • You can use it to reflect on your strengths and weaknesses at the end of your training.
    • It helps the school to coordinate your induction year.
    • It also offers you a chance to specify where you would like further experience to develop your expertise.

Teaching Mathematics Ch 9: ICT in Mathematics Teaching

  • In planning, you should always consider:
    • is ICT enhancing learning or could learning objectives be more efficiently achieved without using it?
    • have you chosen the most appropriate form of ICT?
  • In plenary, it is useful to ask pupils to evaluate the use of ICT.
  • Most effective use of ICT is when it is used selectively to support some pupils’ learning, according to Ofsted.
    • Important to integrate the results into the maths being learnt, so that pupils see ICT as a facilitating tool not an end in itself.
  • ICT can also be used functionally, to reduce the time spent calculating etc.
  • IWBs can be excellent resources for the maths classroom, particularly e.g. with dynamic geometry software to bring geometry alive.
  • The key to successful IWB use is the link between the screen representations, pupils’ activity at their desks, and their internal representations of the mathematics. This depends on teacher skill.
  • ICT is reported as improving motivation and self-esteem because of the quality of the output.
  • However, the key message from research is that all ICT is a tool that the teacher must combine with effective pedagogy to get its best effect.

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.

Teaching Mathematics Ch 9: Reflections on the use of ICT

Using ICT has many and obvious benefits in the maths classroom: it can aid pace, help learners to test out and refine conjectures, assist with graphical presentations of data, make analysis of data much quicker and more rich, can make the lessons more fun, can encourage personalised learning, amongst many other uses.

However, in this reflection I’m asked to consider some of the ways in which ICT might contribute to a lesson being ineffective. I can see several ways this might be the case:

  • Technical issues mean that the focus is on getting the computers to work/log in/run a program rather than focusing on the maths.
  • Less chance for group work and discussion and so pupils not benefitting from peer learning and peer teaching.
  • ICT as a distraction – lots of other programmes/objectives on the computer that don’t relate to maths.
  • Work not being saved/printed to refer back to, so being lost as a reference.
  • Less challenge for more advanced learners, if the ICT package doesn’t allow for differentiation.
  • Poor resources that allow a pupil to follow a process without really learning the maths behind that process

Getting around these issues seems to depend on: a very intuitive system for setting up and using ICT (as I saw at KSA); really good resources available that enable a pupil to challenge themselves, and keep them interested and on task; and an integrated system of recording results and assessment (or allowing the teacher to assess work); a lesson plan that mixes ICT with group discussion.

Teaching Mathematics Ch 8: Teaching different topics

  • One of the most important thing to learn as a maths teacher is that your way of learning maths is only one of several methods you need to know.
  • You need to make a distinction between developing pupils’ mental methods, written methods and calculator methods and help them to make sensible decisions about which to use.
  • Confidence in mental methods is vital for general confidence in numbers. This can be achieved through structured practice in progressively more difficult mental skills.
    • Confidence will only come through structured discussion and support, such as ‘think of a number’ type activities and feedback.
  • Whilst pupils should have developed efficient written methods by end of KS2, some won’t have so you need to be able to return to informal methods with them.
    • Partitioning a number on a number line is a valuable way to approach addition, for example.
  • Subtraction is generally approached on a number line, either counting on or taking away.
  • Ensure you reflect language used in primary school, and refer to place value not just digits.
  • Multiplication is introduced as repeated addition, and then often progresses to the grid method. The grid method can then progress to standard long multiplication.
  • In division, short division and long division are both in common use. Chunking is also used regularly in primary schools.
  • It is crucial to be able to explain any method mathematically, rather than just explaining the process followed.
  • Promote the notion of inverse operations, doing and undoing, as this is very useful for algebra.
  • Work on decimals benefits from an approach that stresses the size of numbers, rather than routines e.g. give me a number between 6.1 and 6.2
  • Fractions often cause problems, because they refer to different things i.e. portions of a whole, parts of a set or places on a number line.
  • Equivalence of fractions, decimals and percentages should be emphasised wherever possible e.g. by asking pupils to match up different ways of expressing fractions, decimals and percentages.
  • Algebra should be matched to equivalent number work whenever possible.
    • It should also be presented as the answer to a problem, rather than simply an exercise without context.
    • Arithmetic sequences can be a great way to introduce the need for algebra.
    • Getting students to create their own algebra problems is much more valuable than presenting them with problems to complete.
    • Also, get them to express algebraic expressions in as many different ways as they can and identifying where they are equivalent.
  • Simultaneous equations should be introduced first using common-sense methods e.g. identifying x and y in x+y=30, x-y=2.
    • This can then progress to elimination, and possibly substitution.
  • It is better to represent a number as x rather than the first letter of the word e.g. p for pence.
  • Three themes run through geometry: invariance, symmetry and transformation.
  • Similarly with algebra, pupils can benefit from building up their own diagrams before analysing them geometrically.
  • Orientation of shapes can confuse pupils.
  • Benchmarking measures against real things e.g. weight of a paperclip, height of a door, can be valuable.
  • Statistics is a relatively more cross-curricular element of maths.
  • Mathematical thinking consists of ‘specialising, generalising, conjecturing and convincing’. These skills can be developed through:
    • Listing possible ways to tackle a problem
    • Setting tasks where a particular key process is clearly needed e.g. a task that is very difficult until simpler cases are investigated.
    • Introducing a problem and getting small groups to share strategies, before tackling it.
    • Getting pupils to reflect on how they have successfully solved a problem.