Mastery in Mathematics (5)

An Action Research Project by Elizabeth Drewitt (Mathematics)

Focus

In this report I aim to share how our departmental research into Mastery in Mathematics has impacted on the students I teach.

Context

There is no argument to the value of mastery as a life skill:

Director Dr Helen Drury says, “In mathematics, you know you’ve mastered something when you can apply it to a totally new problem in an unfamiliar situation”¹.

What better way to prepare our students for life after school than to give them the confidence to approach new situations and problems with confidence.

Mastery enables students to:

  • Develop mathematical language
  • Articulate their reasoning
  • Share ideas on approaches to problem solving
  • Grow in confidence when discussing ideas

I decided to focus on the techniques we can use as teachers to get our students ready for the road to mastery.

All teachers have experienced that:

‘Students learn better when they are curious, thoughtful, determined and collaborative.’ (Nrich)

We spend vast amounts of energy nurturing these traits within our classes. But for some students, the experience of failure or fear of failure shuts down any chance of curiosity. Expecting failure often means students cannot even consider an alternative outcome and therefore determination, thought and collaboration are pointless and avoided. It is a self-fulfilling prophecy. I feel this is particularly the case in mathematics, often voiced by parents at Parents’ Evening, ‘I can’t do maths’. Here I propose that maths is not something that ‘can be done’ or ‘cannot be done’. I would like to challenge the parents as to whether they know their times tables or not. It is highly likely the case that it is not maths these parents struggled with but their times tables. They did not have the basic tools to face the rest of the subject with and so encountered difficulty at every turn. I believe that for many, it was not the PROCESS of expanding brackets that caused a problem but actually the MULTIPLYING.

DEVELOPING BASIC MATHEMATICAL SKILLS

The importance of times tables within the mathematics curriculum cannot be underestimated, yet the importance of learning times tables is still under debate amongst professionals:

Jo Boaler argued that the UK Government position, that every child must memorise their times tables up to 12 x 12 by age nine, is ‘absolutely disastrous’. In contrast, Charlie Stripp stated knowing the times tables supports mathematical learning and understanding.

“Here at Mathematics Mastery, we believe children who have a strong grasp of their times tables are more confident when learning new mathematical concepts and, importantly, enjoy the subject more.” But note here I’ve said ‘strong grasp’ and not simply ‘memorised’.

Here I put forward the view that the road to mastery must start with each student being equipped with a tool box and in that tool box must be curiosity, resilience and……times tables!

Some students cannot/have not/will not memorise the times tables. Some think that if they do not know the answer then that’s the end of that. Full stop! If we do not give these students tools and tricks to work out the answer then we are closing the door on Mastery, opportunity and the growth of a learning identity.

SOLUTION

Never accept I don’t know my times tables. WORK it out. ‘Not knowing’ does not equate to ‘can’t find out’.

TRICKS

Explicitly TEACH how to work them out.

  • count up in two’s on your fingers,
  • count sticks/dots,
  • write out the times tables each time,
  • use your fingers for the 9 times table.
  • Do 10 x {?) then add 2x [?] for the 12 times tables.

ACTIONS

  • Time must be dedicated to times tables each week if we are to provide each student with a fully operational toolbox.
  • Bell work: fill in 5 x 5 times tables grids with random numbers. Students self-differentiate by choosing different coloured grids that represent basic times tables, reverse times tables, lots of mystery headings, larger numbers, decimal numbers.
  • For KS3 or lower ability classes, 10 minute multiplication and division challenges, results recorded and tracked.

RESULTS

  • Practice makes perfect, whether they are memorised or worked out.
  • Students get familiar with the method they choose to work it out.
  • Students see an increase in speed, ease at completing grids and see their own scores improve over time.
  • Take control of their own Bell Work, empowering, safe and challenging.
  • Pupil ‘A’ counting up in twos on his fingers. yr11!!!
  • Pupil ‘B’ in Year 8 showing a peer the 9x table trick using your fingers

Ultimately, we have removed a massive stumbling block that lurks on the road to mastery!

DEVELOPING STUDENT CONFIDENCE

So many students do not know their times tables and believe that is the end of it…but now we have challenged this idea. Just as some people say they can’t do maths…now we can move on and challenge the idea that ‘I can’t do maths’. Mastery teaches students to move away from these barriers and

  • Develop mathematical language
  • Articulate their reasoning
  • Share ideas on approaches to problem solving
  • Grow in confidence when discussing ideas……..

BUT to articulate their reasoning they must first have an opinion. To discuss their ideas they must first have an idea. To solve a problem they must first want to find a solution. They must first form an identity as a learner. Self-worth and confidence play an enormous part. Teaching lower ability classes can often (but not always) mean the students are largely disaffected. Through perceived/experienced failure their confidence has been eroded. We must challenge the perception of mathematics being all about right and wrong answers to build up a self-esteem that is positive enough to support the mastery platform.

When I asked a new group of Year 10s to GUESS the size of ten angles they were shown, half the group did not commit to paper, stating that they did KNOW the answers. Therefore these learners denied themselves the chance to feel good – others who guessed were thrilled when their guess was close but interestingly were not crushed when their guess was way off. Their learning identity was positive and it grew in a very simple exercise. I too joined in to prove that I do not KNOW all the answers, but have the tools to either guess or work it out.

Year 8 Extension task: (LOWER ability) Having studied the rule for adding and subtracting directed numbers, I asked students to write down what THEY THINK the rule could be for multiplying and dividing directed numbers. Some students wrote, ‘I don’t know, we haven’t done this yet’. Again, they didn’t have an opinion and again these students reinforced the negative image they have of themselves as learners. They needed choices pointing out to them and then they were able to take ownership of their choices and make it their idea by giving an example. Imagine their delight when some had predicted the correct rule. Again, those who had predicted in error were not crushed – it was just an idea. The students who had developed their own idea were keen to tell everyone what their prediction was, irrespective of being right or wrong, purely because it was their own idea.

TRICKS

  • Give students opportunities to GET IT WRONG and show it doesn’t matter.
  • Insist (‘encourage’) students commit an idea to paper- to have an OPINIION. Having an opinion gave them a vested interest in outcome which in turn made them more likely to come up with an outcome AND remember it.
  • Admit that as a teacher/ human/ adult we don’t know everything. I am not expecting my students to KNOW everything, the joy is in the working it out.

RESULTS

  • Students are prepared to guess, think, form an opinion, take risks.
  • Students are more likely to see a method through to the end to see if they were right (a win-win situation)
  • Students are more likely to have confidence in the next unfamiliar learning episode.
  • One Year 10 pupil could not even say true or false to a probing question. She has no confidence in maths and so does not think about maths, has no ideas about maths, cannot possibly articulate maths………I sat down with her and asked her to guess (we’ve been working on this idea). She chose False. I encouraged her to use an amount of money to see if she was right or not. We worked through the calculation and proved it to be False. She was thrilled, smiled (!!!!!) and wrote in her book ‘so I was right!!’. Anna believes she has been very successful and her confidence and enjoyment of maths has changed enormously in just a few weeks.

REFERENCES

¹ Drury, H. (2014) Mastering Mathematics. Oxford University Press, pp8.

Department for Education (DfE). (2013a). National Curriculum in England: Framework Document. London: Department for Education.

Kilpatrick, J. Swafford, J. & Findell, B.(eds.)(2001). Adding it up: Helping children learn mathematics. Mathematics Learning Study Committee: National Research Council.

NCETM (2014a). Developing Mastery in Mathematics. [Online] Available from: https://www.ncetm.org.uk/resources/45776 [Accessed: 28th September 2015]

NCETM (2014b). Video material to support the implementation of the National Curriculum. Available from: https://www.ncetm.org.uk/resources/40529 [Accessed 28th September 2015]

NCETM (2015). National Curriculum Assessment Materials. [Online] Available from: https://www.ncetm.org.uk/resources/46689 [Accessed 28th September 2015]

Ofsted  (2015) Better Mathematics Conference Keynote Spring 2015. Paper presented at the Better Mathematics Conference, Norwich, Norfolk

Featured image: ‘Central City Times Tables’ by Derek Bridges (www.flickr.com) CC. BY 2.0

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