CPD – An alternative to lectures – Part 2

In a previous post, I outlined plans for an action research model of CPD rather than a traditional lecture style followed by an expectation that colleagues implement ideas.

Since then, we have used a staff meeting to discuss what has worked in our classrooms as well as what failed miserably in order to refine the teaching of place value across the school. The findings are as follows:

Approaches to refine the teaching of place value – findings from the action research

Which models and images worked best for which children? Additional notes from the maths team
Key Stage 1
  • Dienes and coins worked best for all children.
  • Arrow cards worked for children working at L2 but not for children working at L1.
  • Number lines linked work on place value and ordering.
  • Numicon did not work for teaching place value.
When introducing place value and through counting, present the concept of exchange (i.e., 10 units can be exchanged for 1 ten) – Understanding Mathematics for Young Children (Derek Haylock) page 12 onwards

 

Use the hundred square and number lines to show ‘where numbers are’ when partitioning, and when talking about 1 or 10 more or less.

 

Bundle straws in groups of 10 and practise counting in10s and 1s.

Lower Key Stage 2
  • Modelling the links between models and images (mainly dienes, coins and arrow cards) and then getting children to link them has worked best.
  • Dienes were accessible by all.
  • Money worked best after children had understood dienes. This was particularly important for children working at or below age related expectations.
  • Children working above age related expectations found arrow cards and the abacus more useful, but these were not so appropriate for other groups of children.
  • Place value charts and digit cards worked well for children working above age related expectations.
  • Place value charts were useful for children working at age related expectations but only after they had understood dienes.
In explaining place value to children, use the language of ‘exchanging one of them for ten of those’ as you move right to left along the powers of 10, and ‘exchanging ten of these for one of those’ as you move left to right.

 

When children are introduced to decimal numbers, the shift from 50 meaning 5 Tens to meaning 5 tenths needs to be made clear. The same clarification is needed when using dienes: 5 small cubes no longer means 5 Units, but 5 hundredths. Use a place value chart alongside these models.

Upper Key Stage 2
  • Dienes were most effective for most children.
  • When teaching decimals, money was the most useful model.
  • The abacus was useful for some but not for children working below age related expectations as it was deemed too abstract.
  • Arrow cards helped children working below age related expectations to see the value of digits in numbers.
 

Approaches to refine the teaching of place value – findings from the action research

How have you linked place value to other areas of maths? Additional notes from the maths team
Key Stage 1
  • Comparing and then ordering numbers by placing numbers on a number line

 

More on the link between place value and number lines – Mathematics Explained for Primary Teachers (Derek Haylock) – page 74-75

 

Lower Key Stage 2
  • Links to money and shopping have worked well.
  • As soon as children can make a number with dienes, they can begin to add and subtract tens and units.

 

Upper Key Stage 2
  • Ordering numbers – looking at the value of the largest place value column; placing numbers on a number line with landmark numbers labelled.
  • Rounding – finding the multiples of 10, 100, 100, 0.1 that a number lies between.
  • Estimating – rounding a calculation to find an approximate answer.
  • Calculating with all four operations – partitioning method of addition; decomposition method for standard written method of subtraction; grid method for multiplication.
  • Clear links can be made when looking at multiplying and dividing by 10 etc for example: ‘What will be the value of the 4 after you multiply 14 by 100?

Approaches to refine the teaching of place value – findings from the action research

What tasks did you set children working above age related expectations and what progress was evident? Additional notes from the maths team
Key Stage 1
  • Children working above age related expectations in Y1 could cope with partitioning TU numbers and talking about the value of digits. This was often in a small guided group and saw quick progress.
  • In Year 2,some children were ready for the vocabulary and understanding of 3 and 4 digit numbers.

 

Children need to be explicitly taught how to reason. Good questioning will help. See:

  • Active Learning in Formative Assessment – Shirley Clarke
  • Mathematics Explained for Primary Teachers – Derek Haylock (Chapter 4)
Lower Key Stage 2
  • Useful, applicable games such as ‘making the highest number’ with available arrow or digit cards, were open ended and easily extended by expecting a greater emphasis on talk and reasoning, as well as the use of decimals.
  • Some children worked in a guided group to learn the vocabulary of larger numbers and then applying the skill of partitioning.
  • Reasoning questions were set such as ‘The 4 in 4589 is worth more than the 4 in 76,409. Agree or disagree?’
Upper Key Stage 2
  • Greater emphasis on reasoning and explaining for more able children, particularly the depth and clarity of responses.
  • Learning the vocabulary of very large and very small numbers
  • Children working above age related expectations moved quickly on to applying place value understanding to calculations.

Approaches to refine the teaching of place value – findings from the action research

What tasks did you set ‘invisible’ children and what progress was evident? Additional notes from the maths team
Key Stage 1
  • Lots of support staff in KS1 meant that all children were assessed regularly. Having ‘invisible’ children working alone or with or with less dominant partners worked best, as in larger groups they stepped back and let others dominate.
  • These children, and others working at or slightly below age related expectations were not ready for partitioning and place value. They needed more work on number recognition and the value and relative size of digits as well as number formation and ordering.
  • Clear success criteria helped focus children’s thinking.
Assessment of Prior Knowledge is key. Ask good questions to reveal misunderstandings.
Lower Key Stage 2
  • It was important to keep ‘invisible’ children close to monitor their understanding.
  • Useful, applicable games such as ‘making the highest number’ with arrow or digit cards revealed whether or not they understood concepts more so than a list of answers to questions in maths books.

 

Upper Key Stage 2
  • Tasks that were not simple and repetitive, rather deep and probing revealed well hid misconceptions (for example: odd one out and why?)
  • Slight changes of context helped teacher assessment of whether the children understood the concept.
  • Expecting explanations for answers worked well to develop thinking and reveal gaps in undeerstanding.

 

Approaches to refine the teaching of place value – findings from the action research

What is your understanding of progression in place value? Additional notes from the maths team
Key Stage 1
  • Required knowledge for work on place value: number recognition; matching numbers and quantities (Numicon); and knowing the concept of more and less.
  • Partitioning numbers to 20 into T and U was the next step.
  • After that, any 2 digit numbers (including place holders)
  • Then 3 digit numbers (including place holders).
  • Children need to know the vocabulary of numbers, before ordering numbers, then partitioning.

 

Although awareness of larger and smaller numbers is perhaps the starting point for progression, there are many other ways.

 

Bare in mind that children moving from concrete models and images, through symbols and on to language is also key for progression. See:

  • Understanding Mathematics for Young Children – Derek Haylock (Chapter 1)
  • Mathematics Explained for Primary Teachers – Derek Haylock (Chapter 2)
Lower Key Stage 2
  • Progression comes from the range of vocabulary used and children’s verbalisation of their understanding.

 

Upper Key Stage 2
  • The vocabulary of larger and smaller numbers is the starting point.
  • After that, the appearance of multiple place holders challenges understanding.
  • Then, expecting deeper, clearer reasoning and making wider links make the topic more complex.

Naturally, it doesn’t end there. This is intended to be used as a reference to support quality first teaching so that children have a secure understanding of place value. These findings may well change as colleagues refine further. It also sets a standard for thinking about the effect of the teaching strategies and tasks that we set children.

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