Tag Archives: mathematics

Stop teaching ‘thousands’

If you haven’t already read my rant Stop teaching simile! then I’d suggest starting with that first. However, having started something, now things keep cropping up that I think the same sort of thing about, so here’s an addition to what seems to be turning into a series of “Stop teaching….” posts.

Stop teaching thousands

This will seem silly at first. Of course we need to teach thousands. But I’m coming to the conclusion that we tackle it in the wrong way in some ways. We expect children to work with increasingly large numbers as they go through primary education, and so once they seem to have grasped hundreds, it seems to make sense to move them onto thousands. Except, there’s a difference in the way the numbers work here, and it’s not always obvious when we teach it in that way.

The problem is not so much the thousands, and the next ‘column’ in our place value system. We often call it the ‘ten-thousands’ column, but like with the ‘tens’ column, we don’t often use that language for numbers that include a digit in that place.

Consider the number 54,321.

We don’t treat the 5 as a digit in its own right here; rather it becomes the tens digit of the section of the number that we describe as 54 thousand. It works just like the tens digit.

The same is true as we move over one more column. Consider 654,321

Here the 6 is merely part of the 654 thousands that are needed. It’s why we use commas after every third digit starting from the right. It’s not just a handy number, it actually helps us to read them.

So when we teach thousands, we should teach them as a block. It makes dealing with larger numbers much simpler. Recognising that each section of up to 3 digits is read as a single ‘chunk’ of a number makes it easier to read large numbers, and to avoid the common errors with placeholder zeroes. When a child needs to write four hundred and six thousand, and seventy four, it’s much easier to think of the blocks of 406 in the thousands block, and then 74 in the units section. It even invites the ‘punctuation’ of the number:

406, 074

(I grant you that the last section is lacking a name. I’m tending to prefer to call the very right-hand column “ones” and then refer to the last three digits as units, but there may be a better term. Suggestions welcome!)

The wondrous thing about so much of maths is that patterns are often scalable. The same system now allows us to consider millions, billions (so long as you’ve come to terms with the US billion) and to extend the system in groups of three, rather than one place at a time. Children are then very quickly able to read

123,456,789

as 123 million, 456 thousand, 789.

It also allows them to see the structure of the system so that they can identify any point in the place value structure. So, in the case above the digit 5 is clearly in the tens position within the thousands block: it shows us how many tens of thousands there are.

So, in truth, the argument is not for teachers to stop teaching thousands, but rather to consider thousands as a block of three digits in the numbering system following the HTO pattern.

bmto

Perhaps most importantly, before even thinking about numbers larger than 999, we should ensure that children have a secure understanding the relationships between the first three digits and their positions, fully grasping the nature of powers of 10. Once that is secure, rather than simply extending to thousands, it should be easier to teach the whole thousands block, right up to 999,999 with ease – and then to further extend.


Since posting this, Alex Weatherall has raised some perfectly reasonable points about the use of this structure. As with so many things, there is a risk that it becomes a prop which fails to secure clear understanding. So let me stress, I’m not proposing that we extend place value grids in this way. I present the image merely to make the point. My personal view is that if a child still needs a place value chart to organise numbers, then they shouldn’t be dealing with anything greater than a three-digit number.

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Reading, Writing & Maths Key Objectives (new curriculum)

The Key Objectives for Reading, Writing, Maths and Science for KS1/2 can now be found in the Free Resources section.

Primary Progression Documents for English & Maths

Example progression documentThe nature of the new curriculum documentation is such that the primary section alone lasts for some 200 pages. It makes sense that it is organised in year group order for the core subjects, but it also makes it harder to visualise the progression of concepts and skills. That’s particularly problematic if you’re trying to identify key thresholds for assessment or planning.

Therefore, I have created these simple documents to support schools. They are not revolutionary, but simply present the objectives from the National Curriculum in a sequence of progression strands from Year 1 to Year 6 across Reading, Writing and Mathematics. Hopefully they might help schools in organising their curricula, and also in identifying progression across these very large subjects.

As with all my materials, they are also available at www.primarycurriculum.me.uk/support, and I recommend looking at the other resources available there to support schools’ journeys in implementing the new curriculum.

The documents are all shared here for ease, including easily printable versions:

English progression document (editable Excel file)

Reading progression document (printable A3 PDF)

Writing progression document (printable A3 PDF – 2 pages)

Maths progression document (editable Excel file)

Maths progression document (printable A3 PDF  – 3 pages)

 

Mastery Maths in KS2

Around this time last year I started reading about the work of the Ark group and Mathematics Mastery. So it was that as I moved to KS2 in September, I set about leading my year team – and an adjoining one – on a mastery maths journey. We’ve not reached the end-point yet, but it seems to be a hot topic at the moment, and following on from Bruno Reddy’s great blog about how he’d tackled mastery maths at Secondary, I thought it would be worth sharing what we’d done in KS2.

My initial thinking was led by what I’d read about the Ark scheme, and then built on by what I read in Dan Willingham’s excellent “Why don’t students like school?” about how children learn. It was soon put into context by my early experience in KS2. Having moved from KS3 I had previously taught maths sets; now I would be teaching a mixed ability group at a very different level. In KS3 I had previously moved towards what I’d considered to be longer blocks of two or sometimes three weeks on a unit. That had worked quite well for the high ability groups, but it had become clear for others it had still been too much too fast. I hadn’t previously been dealing with the need to teach and learn tables, or introduce area, but it felt like this was a good way of getting it right!

As the new curriculum was on the horizon, it was a useful starting point, and seemed to fit rather well with the mastery approach anyway. I began by mapping out broad units, using a model based very loosely on the Mathematics Mastery secondary curriculum map. It has’t held fast all year, but it provided a perfectly good starting point.

Year 5 Mastery Overview draft

Year 5 Mastery Overview draft

It meant that the first half term of the academic year was spent almost exclusively on place value and addition/subtraction. Within that we drew in elements which related to those skills. So, it seemed a sensible time to tackle in aspects like calculating perimeter, or finding missing angles on a straight line. Interestingly, there are plenty of similarities between our plan and that of KSA, particularly in that first term. See also what it says about separating minimally-different concepts (such as area & perimeter!)

In the Spring term, we took the step of spending a whole half-term on fractions. I’ll be honest, I was nervous about it. It’s never been my favourite area to teach, and rarely is it students favourite area to study. However, the system seems to have paid off. Knowing that we had weeks to spend on it meant that we weren’t afraid to take the time to secure the basics before launching into the higher level skills suitable to their age. And we weren’t abandoning it for another topic just as they were getting into things.

thinkingblocksWhat’s more, I drew on the things I’d seen of the Singapore bar method to really secure understanding of fractional calculations. We’d been using thinkingblocks.com in school as a general problem-solving tool, but it seems that for fractions this approach really comes into its own. It allowed the children clearly to visualise the problems we were tackling, and to secure a much clearer understanding of why mathematical approaches worked. I cannot speak highly enough of the bar model in the context of mastery!

We haven’t been working on this approach for anything like as long as Bruno Reddy’s school, but initial results look positive. We’ve trialled the approach in Years 4 and 5 and seen a substantial improvement in ability to master the key methods, as well as spending more time to drive a focus on number bonds and tables. It seems that the approach will likely be even more successful in data terms once the new KS2 tests begin with the additional arithmetic paper!

Although it’s early days for us, some of the most significant evidence of success has come from the teams teaching the curriculum. Not all were sold on the idea at the beginning, but it has garnered the support and enthusiasm of those involved because it’s working! You can see it in the progress made by groups who traditionally do well, but perhaps more importantly in the successes of those learners who might traditionally have found making progress more challenging!

There’s still plenty to iron out and tweaks to be made over the coming years as different cohorts come up with different experiences. I still don’t think I’ve spent enough time and effort on securing number bond and tables knowledge – despite finding myself in every week’s work saying at some point “Now, can you see why it helps to know your tables?”. I still think we can do more to incorporate the important stages of concrete and representational development before the abstract. It’s not perfect yet.

But I can no longer imagine teaching any other way. Five years ago I was arguing that we needed to move away from week-long planning for maths; now I’d argue that anything less than six weeks is probably doing our students a disservice!

Ask me next summer how it’s paying off in terms of KS2 results!

Why is Mastery just for Maths?

The trouble with failing to lay proper foundations.

The trouble with failing to lay proper foundations.

With the new National Curriculum, and a whole host of new players in the field of education, it is certainly a time of innovation of sorts in our schools. I have been interested in the work being done by Ark and others looking at mastery in mathematics. It seems that their approach – based in some part on that used in Singapore and like places – is built on the premise of covering fewer topics in greater depth each year, with the intention that over the course of a child’s education they receive a thorough education in each stage of the process.[1]

This strikes me as sensible. Too often I have taught children at KS3 who have raced through the curriculum, picking up bits of skills, but for whom the basics of number knowledge and calculation are still insecure. The comparison to the end-moments of the game, Jenga, is too often fitting: students who lack the secure base on which to build their higher knowledge soon come crashing down.

It has meant that this year I am approaching my teaching of maths with something of a mastery model.

But I’ve got to thinking. Why does it need only to apply to maths?

I’ve also, this week, seen students in my class complete an unaided writing task in which it seems they ignored everything they have been taught this half term and just jotted down notes at random. After some initial frustration (as is common), I soon realised that the fault here was mine (as is also common).

I have taught them a good deal over the past few weeks in terms of writing skills. But I’m not convinced I’ve given them enough time to really securely practise and secure their use of those skills. And so, just like the kids who can’t do their tables in Y10, I’ve got students who haven’t applied even half of what they’ve learned.

I suspect that my model of teaching is not unlike that of many other primary teachers. We’ve looked at a particular genre, linked to a theme we’re studying, over a couple of weeks, and I’ve used that vehicle to teach some appropriate structures and techniques. However, I fear that the downfall of the process has been the movement on to another genre and another set of techniques for the next fortnight. Indeed, I know many schools where each block lasts a week before moving on.

What I’ve begun to consider is not yet a fully-formed idea, so excuse my thinking ‘out loud’, but I’m wondering now if maybe I need to re-think how I tackle these things. What if next half term I identified just a handful of core skills that I wanted to really allow the children to explore and embed. My initial thoughts are to select just three issues from text, sentence and word level (à la Literacy Hour 1998)

So, for example, I might decide that next half term I’m going to focus on:

  • Developing fuller/more detailed paragraphs
  • Variety in sentence length
  • Use of verbs

Those key ideas can be woven through the themes and genres we’re looking at in a variety of ways, but importantly, in ways which complement one another, and which allow the children to become more proficient at each of them, rather than flitting from one idea to the next. They’re sufficiently broad to allow for a sensible amount of development and differentiation, while still providing a sense of connected learning and practice for all.

My units planned so far for next half are likely to be ghost story-writing, creating a narrative from a comic strip, and then some form of descriptive writing about the locality. Each of those would easily lend itself to all three of those skills – with some particularly strong in different areas – and so perhaps by the time we reached Christmas I might have some students who were really secure in some of the elements of that, rather than having had a taster of lots of techniques, few of which have stuck.

Like I said, it’s not a fully-formed idea yet, so I’d be exceptionally glad of any thoughts and experiences from others who have tried similar things – or think it best avoided. All comments welcome!

[1] If you aren’t already familiar with the Ark Mastery Project, it’s worth taking a look at their website for a brief insight: http://www.mathematicsmastery.org/