December 2015

Features

  • Influential focus
    Six months after the General Election, we have a much clearer understanding of how the political land lies. Whatever the challenges may be, there are numerous opportunities for us to influence policy makers both in government and in opposition, says Brian Lightman. More
  • Making maths add up
    When specialists are scarce, leaders need to understand the fundamentals of mathematics in order to ensure it is well taught throughout their schools, says Julia Upton. More
  • Central lines
    John Banbrook looks at the advantages of centralising services for schools in a multi-academy trust (MAT) and the issues for leaders to consider in terms of governance, staff and costs. More
  • Off the chart
    The obsession with tracking and recording data threatens to annihilate joyful learning and teaching, says Dame Alison Peacock. To make assessment truly meaningful, we need greater expertise throughout the system. More
  • No barriers
    Baroness Tanni Grey-Thompson brushed aside societyís low expectations of disabled people to achieve multiple golds on the athletics track and a seat in the House of Lords, yet still faces prejudice in everyday life. She talks to Julie Nightingale. More
  • Recruitment drive
    The number of graduates entering teaching is falling and the confusing plethora of routes into the profession isnít helping. Dorothy Lepkowska looks at how schools are tackling the problem themselves. More
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When specialists are scarce, leaders need to understand the fundamentals of mathematics in order to ensure it is well taught throughout their schools, says Julia Upton.

Making the maths add up

What is the problem with mathematics? As you may expect from a mathematician, I shall start with some numbers.

The number of entries in GCSE mathematics in 2015 was 761,230, an increase of 3.4 per cent. The number of entries at A level also increased in 2015, to 102,844, which means mathematics overtook English as the most popular choice of A level.

At primary level as well there has been an increased emphasis on numeracy skills, driven by the new curriculum.

If we add an increasing focus on mathematics to a growth in the number of students in the cohort, what does that amount to for schools and colleges? A desperate need for mathematics teachers with more and more schools turning to non-specialist teaching to cover mathematics lessons.

Therefore, I believe that itís increasingly important that we as leaders understand the fundamental skills that students need in order to succeed at mathematics.

I shall start with three questions taken from a Foundation GCSE paper. They do not require particularly challenging mathematical techniques but they do make you think about the skills that students need to be successful.


A bag contains only red, blue and yellow counters. There are three times as many blue counters as yellow counters. There are 43 counters in the bag. Some red counters are added to the bag. There are now 50 counters in the bag. The number of red counters has doubled.

How many yellow counters are in the bag?


The arithmetic mean of a list of six numbers is 20. If we remove one of the numbers, the mean of the remaining numbers is 15.

What is the number that was removed?


The Government says that we should eat 5 portions of fruit and vegetables a day. A portion is an item of fruit or a portion of vegetables of 100g or more. Dita has £10 to spend on fruit and vegetables for one week. She wants to buy at least two different fruit and two different vegetables.

Show one way of buying her fruit and vegetable for a week.

Price list:

 Fruit  Vegetables
 Apples 30p each  Broccoli 75p per 100g
 Bananas 25p each  Carrots 20p per 100g
 Oranges 20p each  Cauliflower £1 per 100g


A student who will gain a confident C grade or better will tackle these with ease. Yet a student who ends up with a D grade may well have struggled on these questions, because of lack of the building blocks that underpin mathematical understanding.

What makes the difference between a D and C grade mathematician? I suggest that it is four things:

Mastering the basics

Unlike other subjects, mathematics is progressive. You cannot grasp more advanced concepts without having mastered the basics. With that in mind, we first need to identify those essential building blocks.

William Emeny, a British teacher, identified 164 topics that are included in GCSE mathematics. From this he created a diagram in which each topic is represented by a dot with links between them. The bigger the dot, the more topics draw on it for prior learning.

For example Ď2D view of 3D shapesí, at the end of Emenyís diagram, is singularly linked. For an architect this would be quite an important skill. In the world of GCSE mathematics if you were away the week they taught this, it wouldnít make much of a difference.

However, if you canít multiply and divide negative numbers you will be in more of a pickle. That is not just because we need to use it with numbers, but we need to use it in algebra, in shapes and in context.

So what comes out on top?

  1. Multiply and divide whole numbers (prior knowledge for 90 topics).
  2. Add and subtract whole numbers (prior knowledge for 73 topics).
  3. Brackets, Indices, Division, Multiplication, Addition and Subtraction (BIDMAS) (50 topics) Ė the order in which functions should be carried out (would you know that 3 + 10 x 4 = 43 NOT 52?).
  4. Multiply and divide decimal numbers (43 topics). 
  5. Understand place value and identify the value of digits in a number (38 topics) (could you do 0.3 x 0.2 as easily as 3 x 2?).

Building blocks: what can we do?

 

  • Get students to tackle open problems with more data than is needed so that they have to select information and decide on operations.
  • Ensure that students routinely and frequently practice the basics.

Estimation

The second requirement for success in mathematics is estimation. John Allen Paulos, in his bestseller, Innumeracy: Mathematical illiteracy and its consequences, argues that an understanding of figures, large and small, and an ability to estimate quantities, length, area, volume and time, is at the heart of mathematical understanding.

Estimation: what can we do?

  • Give students ways of working with numbers in real-life contexts.
  • Manipulate units of measure to give a feeling of size in length, area, volume, time and compound measures. 
  • Compare object size and hence build concept of scale.
  • Question students on things that are hard to estimate. For instance:
  • How long is the room you are sitting in, in metres? If you placed iPhones across the room, how many would there be?

Proportion

The third requirement, proportional reasoning, involves thinking about relationships and making comparisons of quantities or values.

Students use proportional reasoning in early mathematics learning, for example, when they think of eight as two fours or four twos, rather than thinking of it as one more than seven.

They use proportional reasoning later in learning when they think of how a speed of 50 km/h is the same as a speed of 25 km/30 min. Students continue to use proportional reasoning when they think about slopes of lines and rates of change.


If one dog grows from 5kg to 8kg and another dog grows from 3kg to 6kg, which dog grew more?



Proportion: what can we do?

  • Offer problems that are both qualitative and quantitative in nature. For example, which shape is bluer?
  • Help students distinguish between proportional and non-proportional situations.
  • Encourage discussion and experimentation in predicting and comparing ratios and recognise that mechanical procedures for solving problems do not develop proportional reasoning.

Resilience

Finally, mathematical resilience describes that quality by which some learners approach mathematics with agency, persistence and a willingness to discuss, reflect and research. Learning mathematics can be difficult, but struggling and overcoming those obstacles brings satisfaction with success.

Having a prevailing fixed theory of learning (Dweck 2000), where students are led to believe that they have a ceiling to what they can do, runs counter to the idea that with effort and the right sort of help, learning can grow. In our classrooms we must give students problems that challenge them and help them to consider how they can use a range of resources to solve those problems.

So what does this all mean as a school leader? As a leader of teaching and learning?

  • Is there a mixture of routine and discovery in mathematics lessons?
  • Is knowledge seen as facts or skills?
  • Do you expect progression to be linear?
  • Does your mathematics team understand the underlying concepts? Are you dealing with non-specialists teaching mathematics who donít have a full understanding of the building blocks to mathematical development?

It is striking that adults are happy to say, ďIím rubbish at mathematics,Ē but very rarely will someone comfortably admit, ďIím no good at English.Ē By ensuring that students are mastering the basic concepts, I believe we can overcome the notion that it is OK to be not very good at mathematics and to help all students to achieve.



Julia Upton is Headteacher at Debenham High School, Suffolk and author of Teach Now! Mathematics: Becoming a great mathematics teacher.

Your professional development

Book your place on our Conference for Heads of English, Maths and Science on 2 February in London. See here for more: www. ascl.org.uk/ emsconference 

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