A place for everything and everything in its place

Thanks to Deputy Head, Nick Hart, for the latest maths blog post where he looks at the importance of getting the basics right in place value.

Place value is very often one of the first units of work for maths in most year groups and is absolutely fundamental to a good understanding of number.  By getting this right and giving children the opportunity for deep conceptual understanding, we can lay solid foundations for the year.

For the purpose of this blog I’m going to assume that children can count reliably and read and write numbers without error. If these things are not yet developed to the appropriate standard then targeted intervention needs to happen without the child missing out on good modelling and explanations of place value.

Children need plenty of practice constructing and deconstructing numbers, first using concrete manipulatives like base ten blocks or Numicon.  This is to show that 10 ones is equivalent to 1 ten etc.  While they’re making these numbers they should be supported to talk articulately about what they are doing, perhaps with speaking frames: ‘This number is 45.  It has 4 tens and 5 ones.  45 is equal to 40 add 5.’

Clearly, when numbers get bigger as children get older, such concrete manipulatives are impractical. Place value counters can still be used to represent numbers for children that still require it though.

Using this chart, larger numbers can be represented and provides a scaffold for the same style of talk as above regarding the value of the parts that make up the whole.  Also, there being nine spaces in each column makes it easy to model adding and subtracting multiples of a hundred, thousand, ten thousand etc. and the associated exchange.

It is important that they get sufficient practice in constructing and deconstructing numbers so that they keep up with age related expectations.  Varied questioning can prompt desired mathematical thinking, first through talk:

“What is the value of the 6 in 567?”

“In the number 567, which digit is in the hundreds column?”

Notice that the questions deliberately focus in on a particular aspect of place value – the first probing children’s proficiency in recognising a digit’s value and the second recognising the column in which lies.  Working pictorially may still be needed, using a place value grid.

Then, children can work with abstract representations:

Notice the variation in the position of the whole and whatever is unknown, in the former the whole and in the latter a part – we do not want children thinking that the equals sign means ‘here comes the answer’.  Again, children may need to represent this pictorially to ensure that they are thinking mathematically and avoid, in the latter question, saying that the unknown is 5.

For many children, this will be enough to meet the objectives relating to place value in the National Curriculum but is it enough? I’d say no – children could be thinking a lot more deeply and so our subject and assessment knowledge needs to be good enough to be able to get children to reason and problem solve. Underlying all of the following suggestions for deepening understanding is the fact that children will be doing additive reasoning – they’ll need a good understanding of the fact that a whole is made up of its parts. They’ll need to know that to calculate an unknown whole, you add the parts and to calculate an unknown part, you subtract known parts from the whole.

Change the order of parts

Simply by changing the order of the parts, we can get children to think more carefully:

The temptation in the first question will be to say that the whole is 675 because the hundreds always come first, then the tens, then the ones, don’t they?  There’s an important lesson here about the associative law.

Notice and continue patterns

This is one of Mike Askew’s big ideas in maths – concepts that are applicable across the age range and that help to make connections.  If we combine this idea with getting children to partition numbers in different ways, we could task them with continuing a pattern:

Physically manipulating base 10 blocks or presenting this pictorially as well as abstractly can help children to see the equivalence and the repeated use of the equals sign is an important teaching point for what it actually means.

Explain the odd one out

Comparing multiple examples for what is the same and what is different requires deeper thinking than basic constructing and deconstruction of numbers.

Which is the odd one out? Explain.

40 + 16          30 + 26            30 + 16          20 + 36

The more subtle the differences, the better.

Change what’s unknown

What is 100 more than 3952?  What is 100 less?

100 more than a number is 3952.  What’s the number?

100 less than a number is 3952.  What’s the number?

Notice here the deliberate choice of number where children will have to think very hard about what happens when boundaries are crossed. The place value chart and place value counters could be used as scaffold if needed. Also, in the first question, the whole is known and a part is unknown but in the other two questions, it is the whole that is unknown. Children need to get in to the habit of representing this pictorially, perhaps with a number line, so that they glean the correct meaning from the question.

Use a context

Up until now, problems have been based on the underlying structure of additive reasoning.  This was to ensure that children’s working memory wasn’t overloaded with distracting contextual information and to guide children into thinking mathematically when layers of context are added. Creating problems for place value can also be quite contrived but these seem to work:

• Pens come in packs of 10 or you can buy them individually.  Sam bought 7 packs of 10 and 3 individual pens.  How many pens did he buy altogether?

• Pens come in packs of 10 or you can buy them individually.  Sam bought 64 pens.  He bought 4 individual pens.  How many packs of 10 did he buy?

• Pens come in packs of 10 or you can buy them individually.  Sam bought 72 pens.  There were only 5 packs of 10 available.  How many individual pens did he buy?

Notice that each question asks for a different unknown – this is a development of the part, part whole thinking that children will have done earlier. The way that we expect children to go about these problems is important too.  Simply by asking children to classify problems based on whether the whole or a part is unknown can prompt them to think mathematically instead of just looking for an answer – after all experts see the deeper structure of problems rather than the superficial features that novices see and by expecting this thinking of children, we can move them towards more expert thinking.

By lingering a little longer beyond children being able to meet the basic objectives, we can foster a secure understanding of place value and ensure that subsequent additive and multiplicative reasoning is just as successful.

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