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“Mastery” of Mathematics

December 11th, 2014 (by Nicky Bishop)

The new National Curriculum for Maths is built upon the concept of Mastery. The D of E notes that countries with the highest attainment in maths use a mastery approach to teaching mathematics. So what is it?

“Mastery” is a compound of “deep understanding of concepts” and “procedural fluency”. It’s clear to me, as a tutor of many children struggling with maths, that conceptual understanding is absolutely vital to making progress.

For example, children can be taught how to calculate fractions of a given quantity, but without understanding they quickly forget how to do this, and are completely stuck if they meet a type of problem they haven’t dealt with before. Without understanding they won’t be able to apply the same technique to help them calculate percentages of a given quantity. And without understanding they won’t be able to work from a specified percentage or fraction back to the original quantity. As D of E puts it, “conceptual understanding is essential to enable connections to be made between mathematical ideas.”

Developing understanding in school children requires teachers who themselves are totally secure with the concepts, and also have the skills to communicate those sometimes abstract ideas to young people. Many non-specialist primary teachers sadly do not have this confidence, and who are therefore unable to pass on understanding to their pupils. The result is that many of my pupils can carry out laborious processes (such as the grid method of multiplication or the chunking method of division) without the faintest clue how or why those methods work. No wonder they come unstuck later on and need specialist tutoring.

Of course, deep conceptual understanding is only half the story. Mastery also requires “procedural fluency”, in other words the ability to apply a calculation process quickly and accurately, again and again, to a range of different questions and problems.

The D of E knows there’s only one way to achieve this and that is through lots and lots of practice, with carefully designed variation built in. This means giving up a lot of lesson time for simply using new skills, trying them out on a variety of familiar and unfamiliar questions. And it means practice at home as well. Five minutes individual practice every day for a week will really help to consolidate a new procedure, and is more effective than 1 hour all at once.

The new National Curriculum for Maths recognises that conceptual understanding provides the foundations for progress. In practice, this means that steps must be carefully sequenced, and that children should not move on to the next step until they have mastered the earlier steps. It also means that teacher must provide quick feedback to pupils, along with high quality interventions to help all pupils keep pace with the rest of the class. And it means creating an environment in which pupils are unafraid to grapple with new mathematical challenges. Without these, specialist tutors will continue to be in demand.

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Good tutors, bad tutors

December 7th, 2014 (by Nicky Bishop)

A recent article in The Telegraph criticises private tutors, claiming they don’t help children to achieve higher grades and can end up doing more harm than good to the child’s confidence.   The writer tells us of her 7-year old daughter’s miserable experience being tutored by a bright graduate from a good university who came to help the child with maths.

Rather than concluding that all private tutors are harmful, or at best a waste of time and money, the sensible conclusion is that – as with most professions – there are good tutors and bad tutors.  A young graduate with a 1st class degree, but little or no teaching experience, will often turn out not to be a good tutor for your child.

Tutors (just like teachers) need a good understanding of child psychology (it helps if they are parents themselves).  A good tutor needs experience of coaxing the reluctant or the timid to “have a go” without fear, and endless patience when the child is slow to grasp the concept or master the process.  A good tutor understands why your child doesn’t “get” fractions or algebra (for example), and needs a repertoire of different ways of explaining and demonstrating each tricky topic.  The best tutors have empathy.

I always say to myself (and often say to parents): “If the child doesn’t understand something, that’s my fault for not explaining it properly.”  Hearing that is tremendously encouraging for the struggling pupil – it’s no longer a criticism of them if they haven’t got to grips with ratios or cumulative frequencies.  The best tutors are good listeners and great communicators.

Here are some tips for selecting a good tutor for your child:

  • Have a long chat with the tutor on the phone and decide for yourself whether this person is likely to provide the gentle and patient support your child needs.
  • Ask about their experience with teaching – and even more importantly – tutoring.
  • Does the tutor have experience with your child’s age group? A tutor great at helping 7 year olds may not succeed so well with a 15 year old. And vice versa.
  • Ask for testimonials from other parents. A good tutor will be able to provide dozens.
  • Always ask for a trial session, to ensure your child feels confident and happy with the tutor. If the relationship doesn’t work, then find a new tutor.
  • After the lessons, ask your child whether the tutor explained things clearly.
  • Is your child finding the lessons enjoyable? If not, then find another tutor.

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