The Quest for Infinite Depth: Place Value

As ‘Place Value’ is such a fundamental component of the mathematics curriculum I feel it’s worth considering how we can ensure those pupils who appear to have grasped the key concepts are sufficiently challenged and their understanding pushed to ever greater depths.

To be honest I feel that it’s worth considering how we can do this across the curriculum, particularly the oft neglected nether-region known as shape and space, and I do plan on re-visiting this line of thought quite a bit throughout the year.

That being said, my aim is to present my ideas in manageable portions, so for now I’ll set my sights solely on place value.

While I feel we’ve quite recently raised our problem solving game at primary level, perhaps due to the renewed aims of the Primary National Curriculum or the form taken by questions in our most recent KS2 SATs papers, it strikes me that the infinite depth of mathematics hasn’t been given as much consideration as it’s, typically more popular, neighbour.

As I understand it, the minimum expectation of the Primary National Curriculum for maths is that pupils are fluent mathematicians who can reason about and solve problems with their mathematical knowledge. Paraphrased of course but this is what I understand to be the essence of the spirit in which it was written. Thus, taking this as a given, problem solving is the least we should expect from our pupils.

So, what do we do? I believe the answer lies in our subject knowledge.

…keep the concept in tact while changing the context…

In order to provide greater challenge we should keep the concept in tact while changing the context.

Take for instance place value. If a pupil can read, write and order numbers to 10,000, then the addition of TTh, HTh or Millions to the mix won’t make a blind bit of difference to their level of understanding. By that point the pupils understand the concept and they need to be pushed further.

Place Value Base 10

To manipulate our preferred base 10 is to know that we have ten possible digits (0-9) and each column is 10x greater than the last. 1s Tens Hundreds Thousands etc…

Place Value Base 2 Sequence

Say then we move our focus to base 2. In this base we have two possible digits to work with 0 and 1 and each column is 2x greater than the last. I propose that a pupil who, upon receiving simple instruction, can develop a working understanding of this system; reading, writing and ordering as with base 10, has shown a greater depth of understanding than is perhaps provided by the opportunity to solve written word problems.

Place Value Base 2 Sequence

I believe greater opportunity lies in this stable concept/altered context approach

I don’t know. If I’m barking up the wrong tree I’d genuinely like to know. I accept that problem solving is essential to our pupils’ mathematical development, however, I believe that even greater opportunity lies in this stable concept/altered context approach.

I’m reliably informed that there was once a time when children were taught all of the number systems from base 2 – 10, though I can neither confirm or refute this accurately. Yet, for most of us this was not the case, so, in support of any readers who would like to put this idea into practice, here is a run through of base 2 and 3, their relevance to real life and ways they could potentially be used in the classroom.

Every single one of our pupils has the right to access the highest quality education…

I read somewhere (and I apologise to the author as I cannot recall at present) that it was ‘morally objectionable’ for pupils to waste their valuable learning time on content/tasks/exercises/activities they could ‘already do’ and which provided insufficient challenge. So regardless of the accuracy of my suggested approach, I believe this should always be our focus. Every single one of our pupils has the right to access the highest quality education and I believe the consideration of how we can provide greater challenge is no small part of this.

Kieran

@Kieran_M_Ed

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