Feeling positive about negative numbers

Now sats have finished yet term stretches out before us – what shall we teach year 6 in maths?  The test and the demands of the curriculum behind it dominated the landscape for so long, now that’s all done and dusted, it can all feel a bit anchor-less – what now? What shall I teach them now I am free to teach them anything I want, without anyone checking up on me? It’s all a bit disconcerting.

We usually opt – as I am sure many do – for doing lots of problem solving. I bet the website analytics at nrich show a peak in June and July. And there is nothing wrong with that.  It was difficult to yield precious curriculum time to their more open ended problems when there was so much  stuff  to be got through. But now – now we have 7 whole weeks to fill and spending time on actually applying all that maths we’ve been learning seems like a great idea. Consolidation and application; isn’t that what mastery is all about?

And, to a degree, it is. But 7 weeks of nothing but…I’d go crazy – never mind the kids.  I know maths isn’t all about getting the right answer but its nice sometimes to be able to get the right answer fairly quickly – and be pretty sure that you’ve got it!  And while some activities are great – this is my all time favourite -others are just boring number crunching.  Also – you actually have to be pretty good at maths reasoning yourself to teach these well. To use these effectively, you have to know yourself what would be an effective solution so you can prompt the children appropriately in the right direction.  nrich and other maths problem solving sites are written by mathematicians who find such matters trivial. They don’t supply the correct answers – what would be the fun in that? In the example I dismissed above as boring number crunching – Im sure if I actually sat down and spend a few minutes I would see that  – of course – if I did such and such that would be better than something else. It’s not something I immediately know and frankly would prefer some pointers to save me time.  The same goes for the wonderful Don Steward. His website is aimed at secondary teachers who presumably solve the problems he poses in almost instantly. He doesn’t write for primary practitioners who mostly stopped learning maths aged 16. If only he would put the answers!

Also – all that maths we’ve taught year 6 – it’s amazing how quickly it begins to dribble out of their ears once you stop the highly focused daily practice that marked lessons before sats.  There’s a sort of ‘match-fitness’ to a lot of maths that rapidly declines once you stop.  Consider times-tables facts. Remember how rusty yours were when you first started teaching and how much better they are now? Once term ends, they will probably have 6 maths-free weeks. If we tack on another 7 weeks when they don’t really learn anything new or practice much old, then no wonder secondary school maths teachers sometimes think sats ‘levels’ (50p in the swear box) are inflated. They may have been a 4B/at the expected level/secondary ready in early May; by September they have probably slumped way below that. Spaced repetition doesn’t like spaces quite that big.

So, what to do?  Giving them a flying start to secondary school obviously. So if they don’t yet know their tables or understand place value or can’t do the four operations in their sleep – then you should do those. But I’m taking it as read that nearly all the children can. There is no point in accelerating them through ks3 content because  *mastery* and also because if you have ever looked at a ks3 maths textbook, part from the algebra it’s just just the yr6 curriculum again. Seriously – we bought some k3 textbooks for the level 6 children last year – when level 6 still existed and we had to jump to halfway through year 8 to find anything sufficiently challenging.  I am sure secondary school teachers are well aware of this and use such text books selectively.   They probably even have new ones for the new curriculum.  Although I am slightly haunted by the experience of my first son who spend the first half term (yes the whole half term) revising what happens when you multiply or divide by 10 and multiples thereof. But that was some time ago, I really must move on.

So the obvious contenders are those things we whizzed through prior to sats in the mad dash to cover the new curriculum and suspect the children do not really understand deep down. They just know a few tricks.  As I worried about here.  So possibly we could revisit fractions, especially multiplying and dividing which are so easy to teach as procedures and so difficult to understand conceptually. Hey, we could even do some nrich investigations on them. This is the only one I could find that focus on multiplication. Most focus on equivalence but I am assuming the class is pretty solid on that.  This one from Don Stewart is good – but see what I mean about answers being useful – took me a minute or so before I realised that drawing a bar model made these ridiculously easy.

The other candidate is negative numbers which we whizzed through at the start of the year and then realised some children were confused about when we did algebra. There wasn’t time at that point to go back and address those and actually, the algebra questions in the sats paper were far easier than those we had been practising. However, what  gift to the child and their future teachers to have a rock solid understanding of negative numbers on starting secondary school. Including, of course, a firm grasp of why when you subtract a negative number, you end up adding. Algebra gets really tricky if you are not secure in your understanding of positive and negative numbers – so let’s give our leavers something that will really set them up well for the coming year.

Now when I started writing this post, I fully intended to explain at length about using integer counters to teach negative numbers.   If you don’t know what integer counters are, watch this. (There’s the added bonus the teacher sound a bit like Officer Dibble from Top Cat – the original series obviously).  But then I did a bit of pre-post googling, to se what else was out there and stumbled across this from Tess India which is simply brilliant and uses integer counters as well as various other good ideas.  I particularly liked the bench number line  where you use PE benches to make a number line with children describing how they are going to move from say -4 to +3, and the ‘feeling positive’ idea where you ask students to think about things that make them feel positive and things that make them feel negative.  If you add yet another negative thing you become even more negative whereas if someone takes away a negative thought you become more positive. Simple – but brilliant! So there is no point in me explaining much more about negative numbers – read the lesson plans in the link.

Indeed the whole Tess India resource is a treasure trove of wonderful ideas – well the bits I’ve managed to look at so far are anyway. And it is humbling to read the plans and realise they are aimed at a context where a class size of 60 is not uncommon and interactive whiteboards don’t exist.  Alongside the maths there are handy tips about how to make flashcards from old cardboard…rather puts things into perspective. There are English and science resources alongside the maths and it covers primary and secondary. It’s really well worth a look.

But back to integer counters. These don’t seem to be well known about in the UK. Indeed, we use the wonderful Primary Advantage maths scheme in key stage 2 and while they bang on about CPA everywhere else – for negative numbers they state that no concrete materials are possible and go straight to number lines.  Now number lines are all fine and dandy but some students get so confused using them and it all seems a bit arbitrary why you are moving forwards or backwards.  I  love  integer counters because I love being able to see why the maths works.  When I found out you could even model  why -4 x-2=+8, I beamed for days. I kept on showing people my newest party trick. (I didn’t get invited to many more parties after that.)  And here’s a great link showing how to divide negative numbers.  And no I am not suggesting you teach year 6 how to multiply and divide negative numbers. I’m just banking that if you’ve got this far, this sort of thing brings you great joy.

Using integral counters is the final idea in the Tess India resources.  To be able to understand them, children need to understand the concept of a zero pair, made from one positive counter and one negative counter.   These can then be added to any equation without changing its value. I could try and do some badly drawn graphics to explain it properly – but others have done so with so much greater flair I suggest you look at these instead. Unfortunately Officer Dibble’s video on this seems to be missing. However, this from Learn Zillion  is perfectly serviceable although this is a bit more fun, if a bit more complicated. Can’t quite place the accent.

With a bit of practice, children soon learn to just draw themselves + and – signs if they want to check a calculation, rather than need to counters.

One of the problems children face with this topic is that we never make it clear that all numbers have polarity –  that they are either positive or negative – and that strictly speaking we should write 3 as +3 etc. It’s a shame that the polarity signs are the same as the operator signs – I’m sure it would be a lot easier if they weren’t.  When we write 5-3 do we mean

+5 + -3 or +5 – +3?   I’m sure if we did a lot more work with counters showing that they give the same answer but actually represent something different, that might help. No wonder children get muddled and think -3-5 equals 2  or possibly 8 (because you have got 2 minuses and they half know something about two minuses making a plus. How much better to act it out with counters and see the maths before you very eyes. Works for me.

Then we can return to where this post started and have children investigating negative numbers.  Back to nrich.  And here’s a great reasoning activity from maths pad. If they can articulate why certain statements are or are not true then that’s job done. Secondary schools- here we come!

 

 

 

Feeling positive about negative numbers

Milling around in bewilderment: that reading comprehension.

You can see the reading test for yourself from this link

The day started well; dawn casting spun-gold threads across a rosy sky.  The long wait was over; sats week was finally here.  And it looked like summer had arrived. Year 6 tripped in to classrooms  while head teachers fumbled skittishly with secret keys in hidden cupboards.  Eventually teachers across the nation ripped open plastic packets.Perhaps at first their fears were calmed, for the text – or what you can glean about it from reading snippets here and there as you patrol the rows – didn’t seem too bad. In previous weeks children had struggled with excerpts from the Lady of Shalott, Moonfleet, Shakespeare.  The language here looked far more contemporary.

But no. Upon completion children declared the test was hard – really hard. Many hadn’t finished – including  children who usually tore through tests like a…white giraffe? What is more, the texts didn’t seem to be in any kind of order. We had drilled into them, as per the test specification guide, that the texts would increase in difficulty throughout the paper (section 6.2) Yet the middle text was almost universally found to be the hardest.  Some declared the final text the easiest. What was going on?

Tests safely dispatched, I decided to take a proper look. It didn’t take long for it to be apparent that the  texts contained demanding vocabulary, and some tortuous sentence structure. The difference with the sample test material was stark. Twitter was alive with tales on sobbing kids, and angry teachers. Someone said they had analysed the first paragraph of the first text and it came out with a reading age of 15. Debate followed; was this really true or just a rumour? Were readability tests reliable? I tweeted that it was a test of how middle class and literary one’s parents were, having identified 45 words I reckoned might challenge our inner city children.  After all, as a colleague remarked, ‘my three-year-old knows more words than some children here’. Other people drew groans by mentioning how irrelevant the texts were to the kind of lives their children lived. I seemed to be implicated in this criticism…although it’s difficult to tell who’s criticising who sometimes on Twitter. Still, I was put out. I don’t care if texts are ‘relevant’, I retorted. I cared that the vocabulary needed to answer questions  favoured a ‘posh demographic.  Apparently, this was patronising. I saw red at this point! It’s not that poorer children can’t acquire a rich vocabulary but that since it is well known that a rich vocabulary is linked to parental income and the domain ‘rich vocabulary ‘ is huge (and undefined), it is not fair or useful to use tests that rely on good vocabulary for accountability.  And then I put a link to this previous post of mine, where I’ve explained this in more depth. If accountability tests over-rely on assessing vocabulary as a proxy for assessing reading, this hands a free pass to school choc full of children like my colleague’s three-year-old, since such children arrive already stuffed to the gills with eloquence and articulacy. Whereas the poorer the intake the greater the uphill struggle to enable the acquisition of the kind of  cultural capital richer children imbibe with their mother’s milk.

Flawed as the previous reading tests were, they did not stack the cards against  schools serving language-poor populations. The trouble with using vocabulary as a measure is that it that each individual word is so specific. Usually what we teach is generalisable from one context to another. Learning words however has to be done on a case by case basis. I recently taught year 6 somnolent, distraught and clandestine, among many others. I love teaching children new words, and they love acquiring them.  But unless there is some sort of finite list against which we are to be judged, I’d rather not have our school judged by a test that is hard to pass without an expansive and sophisticated vocabulary. With the maths and SPAG tests, we know exactly what is going to be tested. The domain is finite. We worry about how to teach it so it is understood and remembered, but we do not worry that some arcane bit of maths will worm its way into the test.  Nautical miles, for example.  Not so with reading. Any word within the English language is fair game – including several that don’t regularly appear in the vocabulary of the average adult. There may be very good reasons for the government to want to ascertain the breadth of vocabulary acquisition across the nation. In which case, they could instigate a vocabulary test – maybe something along the lines of this.  But that shouldn’t be confused with the ability to read. To return to our earlier example, my colleagues three-year-old may have an impressive vocabulary but she can’t actually read much  at all yet. Whereas our 11-year-olds may not know as many words but are happily enjoying reading  the Morris Glietzman ‘Once’ series.

It is becoming accepted that  reading is not just the orchestration of a set of skills, but requires knowledge of the context for the reader to make sense of the bigger picture.  But that’s not what happened here.  It’s not the case that children found the texts difficult because they lacked knowledge of the context. The context of the first text was two children exploring outdoors. True only 50% of our present year 6 knew what a monument was at the outset – a bit tricky since this was pretty central to the test – but by the end of the story they sort of worked it out for themselves. The second text featured a  young girl disobeying her grandmother and taking risks. And a giraffe. Well I reckon this is pretty familiar territory (grandmothers and risks, I mean) and while we do not meet giraffes everyday in Bethnal Green, we know what they are.  The third and final text told us all about dodos and how they may have been unfairly maligned by Victorian scientists. So that was a bit more remote from every day experience but no so terribly outlandish as to render the text impenetrable. The third text is meant to be harder. The children are meant to have studied evolution and extinction by then in science anyway.   So it wasn’t that the Sitz im Leben was so abstruse as to render comprehension impossible. The problem was the words used within the texts and the high number of questions which were dependent upon knowing what those words meant. The  rather convoluted sentence structure in the  second text didn’t help either – but if the words had been more familiar, children might have stood more of a fighting chance.

According to the test specification, questions can be difficult in one of five different ways. These five ways are based on research commissioned by the PISA guys. It’s an interesting  and informative read – so I’m not arguing with the methodology per se.  I don’t know nearly enough to even attempt that. Amateur though I am, I do argue with the relative proportions allocated to each of the five strategies in this test.

With three of these, I have no quarrel. Firstly,  ( and my ordering is different from that in the document)  questions can be made easier or harder in terms of accessibility; how easy is it to find the information? Is the student signposted to it (e.g. see the first paragraph on page 2).   Or is the question difficulty raised by not signposting and possibly by having distractor items to lure students down dead ends?  I think we have little to complain about here. e.g. question 30 has clear signposting…’Look at the paragraph beginning:  Then, in 2005…’ whereas  in question 33  the relevant information is much harder to find – it’s a ‘match the summary of the paragraph to the order in which they occur’ question.

Secondly, questions may vary in terms of task-specific complexity. How much work does the student have to do to answer the question?  Is it a simple information retrieval task or does the pupil have to use inference?

For example, question 7 is easy in this regards.’ Write down three things you are told about the oak tree.’   The text clearly says the oak tree was ‘ancient’.  I haven’t checked the mark scheme as it’s not yet published as I write, but I am assuming that’s enough to earn you 1 mark. Whereas question 3 is a bit harder. ‘How can you tell that Maria was very keen to get to the island?  Students need to infer  this from the fact that she she said  something ‘impatiently’.   There are far fewer of this kind of question under this new regime – but we were expecting that and the sample paper demonstrated that. Again – no complaints.  Indeed the test specification does  share the relative weightings of different skills (in section 6.2.2, table 9), but the bands are so wide its all a bit meaningless. Inference questions can make up between 16% and 50% of all questions, for example.

Thirdly, the response strategy can be more or less demanding, a one word answer versus a three-marker explain your opinion question.

The two final ways to make questions more or less difficult are by either varying the extent of knowledge of vocabulary required by the question (strategy 5 in the specification document) or by varying  the complexity of the target information that is needed to answer the question.  (Strategy 2) The document goes on to explain that this means by varying

• the lexico-grammatical density of the stimulus

• the level of concreteness / abstractness of the target information

• the level of familiarity of the information needed to answer the question

and that …’There is a low level of semantic match between task wording and relevant information in the text.’

I’m not quite sure what the difference is between ‘lexico-grammatical density’  (strategy 2) and knowledge of vocabulary required by the question (strategy 5), but  the whole thrust of this piece is that texts were pretty dense lexico-grammatically and in terms of vocabulary needed to answer the questions. When compared with the sample test for example, the contrast is stark. Now I’m no expert in linguistics or test question methodology. I’m just a headteacher with an axe to grind, a weekend to waste and access to google.  But this has infuriated me enough to do a fair bit of reading around the subject.

On the Monday evening post test, twitter was alive with people quoting someone who apparently had said that the first paragraph of the first text had  a Flesch Kincaid reading ease equivalent to that of a 15 year old. I’d never heard of Flesch Kincaid – or any other of the readability tests – so I did some research and found out that indeed, the first paragraph of The Lost Queen was described as suitable for 8th-9th graders – or 13-15 year olds in the British system. But there was also criticism online that the readability tests  rated the same texts quite differently so weren’t a reliable indicator of much. (Someone put a link up to an article about this, which I foolishly forgot to bookmark and now can’t find – do share the link again if it was you or you know a good source.)*

Anyway, be that as it may, I decided to do some readability tests of various bits and pieces of the sats paper.  And this is what I discovered. (texts listed in order of alleged difficulty)

The Lost Queen first paragraph:  13-15 year olds

Wild Ride first paragraph:              13-15 year olds

Wild Ride ‘bewildered’ paragraph     18-22 year olds

Way of the Dodo first paragraph          13-15 year olds  (and lower score than The Lost Queen)

Way of the Dodo 2nd paragraph         13-15 year olds.

So there you have it, insofar as Flesch Kincaid has any reliability, the supposedly hardest text was in fact the easiest, the middle text was the hardest.

I did the same with the the sample paper. The first had a readability level of a 11-12 year-old and the second 13 – 15. I had lost the will to live by then so didn’t do the third text – but it is clearly much more demanding than the previous two  – as it should be.

I also used the automated reading index and while this gave slightly different age ranges, the relative difficulty was the same and all the texts were for children older than 11, the easiest being… the first part of the way of the dodo.

However,  it was also clear from my reading that readability tests are designed to help people writing, say pamphlets for the NHS, make the writing as transparent and easy as possible. In other words, they are intended to make reading simple so people who aren’t very good at it can understand stuff that may be very important. It struck me that maybe this wasn’t exactly what we should be aiming for in a reading assessment. After all, we do want some really challenging questions at some point. We just want them  at the end, where they are meant to be. We need readability tests because previous generations have not been taught well enough to be presented with demanding information. We want better for the children we now teach.

Which brought me to discover this site, which ranks words by their relative frequency in the English language.  If we are going to be held accountable for the sophistication of the vocabulary are children can comprehend, then surely there should be some bounds on that.  While the authority of this is contested, it seems to be generally held that the average adult knows about 20,000 words. You can test yours here.   How many words the average 11 year old does or should know I did not discover – so here are my ball park suggestions.

For the first text – the one that is meant to be easier – there should be a cap on words ranked occurring below 10,000. (I’m assuming here we understand that as words are used less frequently their ranking falls but the actual number rises: a ranking of 20,000 is lower than a ranking of 10,000. If this is not the correct convention for such matters, I apologise). Definitions should be given for low frequency words, especially if understanding them is critical to answering specific questions. In the same way in which Savannah was explained at the introduction to Wild Ride

Then in the second text words could be limited to 15,000, and in the third 20,000 – representing the average adult’s vocabulary. I have plucked these figures from the air. I would not go to the stake for them. But you get my meaning. We need to pin down the domain of ‘vocabulary’ if we are to be held accountable when it is tested.

For what it is worth, I asked our year 6 after the test to tell me which words they did not know. There are 30 children in the class. Words where half the class or more did not know the meaning included  from the first text: monument, haze, weathered (as a verb); from the second text: jockey, dam, promptly, sedately (zero children), counselled, arthritic, nasal, pranced, skittishly (zero children), milled, bewildered, spindly, momentum; from the third list haven, oasis ( they knew this was a brand of drink though), parched, receding, rehabilitate and anatomy. My Geordie partner tells me they would have known parched if they were northern because that’s Geordie for ‘I’m really thirsty.’  Here we can see again that the middle passage had the highest number of unknown words in my obviously unrepresentative sample. In fact, it was the first paragraph on page 8, which henceforth shall be known as the bewildering paragraph that seemed to have the highest lexico-grammatical density.  As the mud flats entrapped the Mauritian dodos, so did this paragraph ensnare our readers, slowing them down to the extent that they failed to finish the questions pertaining to the relatively easy  final text.

Maybe I’m wrong. Maybe when the statistics are finally in, there won’t be a starker-than-usual demarcation along class lines. I’d love to be wrong. Let’s hope I am.

And finally, what you’ve all been waiting for – what was the lowest ranking word?  Well yes of course, it was ‘skittishly‘;  so rare it doesn’t even appear in the data base of 60,000 words I was using. But suitable for 11 year olds, apparently.

In case you are interested, here’s the full rankings. Where the word might be more familiar as a different part of speech I have included a ranking for that word too, in italics. The words I chose to rank were just those my deputy and I thought children might find tricky.

Word

(organized by rank lowest to highest)

Part  of speech Ranking Text Necessary (N)/useful  (U) for question number
skittishly adverb <60,000 WR
parch adjective 46,169 WD E29
sedately adverb 38,421 WR
clack noun 32,467 TLQ 4 distractor
sedate verb 23,110
prance verb 22,360 WR
skittish adjective 21,298
sedate adjective 20,481
arthritic adjective 20,107 WR
mossy adjective 19,480 TLQ U8 distractor

E9

misjudge verb 19140 WD
spindly adjective 19025 WR
bewilderment adverb 17,410 WR E16
squeal noun 17,103 WR
sternly adverb 16,117 WR
plod verb 16,053 WR
prey verb 15,771 WD
dismount verb 15,601 WR
hush noun 15,394 TLQ N4
burrow noun 14,900 WR
hush verb 14,295
nocturnal adjective 13,755 WR E14
enraged adjective 13,378 WR
mill verb 13,378 WR E16
sprint noun 12,187 WR
squeal verb 12036
folklore noun 11,722 WD
rehabilitate verb 11,496 WD E31
oasis noun 10,567 WD U29
stern adjective 10,377
moss noun 10142
evade verb 9759 WR
sight verb 9730 WD
jockey noun 9723 WR
weather verb 9568 TLQ U8
blur noun 9319 WR
murky adjective 9265 TLQ U6
inscription noun 9164 TLQ U8
rein noun 8793 WR
sprint verb 8742
slaughter noun 8494 WD
anatomy noun 8310 WD E32
haze noun 8,307 TLQ 4 distractor
counsel verb 7905 WR
nasal adjective 7857 WR
recede verb 7809 WD
intent adjective 7747 WR
stubborn adjective 7680 WR question E15
slab noun 7585 TLQ U8
arthritis noun 7,498 WR
promptly adverb 6762 WR
blur verb 6451
haven noun 5770 WD
vine noun 5746 TLQ U6
defy verb 5648 WR E15
startle verb 5517 WR
drought noun 5413 WD
remains noun 5375 WD
devastating adjective 4885 WD
rehabilitation noun 4842
prey noun 4533
dam noun 4438 WR
momentum noun 4400 WR E18
ancestor noun 4178 TLQ N1.
monument noun 4106 TLQ U8
dawn noun 4044 WR N12a
intent noun 3992
click noun 3822
counsel noun 3441
indication noun 3401 WD
prompt adjective 3142
mount verb 3012
urge verb 2281 WR
cast verb 2052 WR
judge verb 1764
unique adjective 1735 WD E25
weather noun 1623
sight noun 1623
Word (organized by where they appear in the texts) Part  of speech Ranking Text Necessary (N)/useful (U) for question number
monument noun 4106 TLQ U8
ancestor noun 4178 TLQ N1
clack noun 32,467 TLQ 4 distractor
hush noun 15,394 TLQ N4
hush verb 14,295
haze noun 8,307 TLQ 4 distractor
vine noun 5746 TLQ U6
murky adjective 9265 TLQ U6
weather verb 9568 TLQ U8 distractor

E9

weather noun 1623
mossy adjective 19,480 TLQ U8 distractor

E9

moss noun 10142
inscription noun 9164 TLQ U8
slab noun 7585 TLQ U8
dawn noun 4044 WR N12a
cast verb 2052 WR
jockey noun 9723 WR
dam noun 4438 WR
startle verb 5517 WR
nocturnal adjective 13,755 WR E14
promptly adverb 6762 WR
prompt adjective 3142
stubborn adjective 7680 WR question E15
defy verb 5648 WR
sedately adverb 38,421 WR
sedate verb 23,110
sedate adjective 20,481
plod verb 16,053 WR
arthritic adjective 20,107 WR
arthritis noun 7,498 WR
nasal adjective 7857 WR
squeal noun 17,103 WR
squeal verb 12036
burrow noun 14,900 WR
prance verb 22,360 WR
skittishly adverb <60,000 WR
intent adjective 7747 WR
intent noun 3992
enraged adjective 13,378 WR
squeal noun 17,103 WR
squeal verb 12036
mill verb 13,378 WR E16
bewilderment adverb 17,410 WR E16
spindly adjective 19025 WR
evade verb 9759 WR
momentum noun 4400 WR E18
urge verb 2281 WR
sprint verb 8742
sprint noun 12,187 WR
blur noun 9319 WR
blur verb 6451
dismount verb 15,601 WR
mount verb 3012
sight verb 9730 WD
sight noun 1623
haven noun 5770 WD
slaughter noun 8494 WD
unique adjective 1735 WD E25
prey verb 15,771 WD
prey noun 4533
folklore noun 11,722 WD
remains noun 5375 WD
drought noun 5413 WD
oasis noun 10,567 WD U29
parch adjective 46,169 WD E29
recede verb 7809 WD
rehabilitate verb 11,496 WD E31
indication noun 3401 WD
anatomy noun 8310 WD E32
misjudge verb 19140 WD
judge verb 1764
devastating adjective 4885 WD

By way of contrast I did the same with the sample text. In the first text there were no words I thought were hard enough to check. In the second there were 4: cover (15,363), pitiful (13,211), brittle (10,462) and emerald (12,749).  In the third and final passage there were 8:triumphantly (16,3,43), glade (20,257), unwieldy (16,922), sapling (16,313, foliage 7,465, lurch (9339), ecstasy (9629) and finally, ranking off the scale below 60,000 gambols.

Milling around in bewilderment: that reading comprehension.

KS2 reading comprehension analysis teaser

I have just completed an in depth analysis of the difficulty of various words that appeared in the ks2 sats reading comprehension.  Because of the embargo for children who have still to take the test, I can’t publish the post yet – as it identifies the actual words – until next Friday 20th.

But, as kind of teaser advert I can reveal that

  • text 2 was by far the hardest
  • followed by text one
  • text three had the easiest vocabulary
  • the hardest word (i.e. least frequently used) was in text two…can you guess what it was?
  • this word had a frequency so low it didn’t appear in the ranking of 60,000 I used
  • 6 words ranked lower than 20,000 – 4 of which were in text 2 and 1 was in text 1…again – have a guess.

That’s not what the test specification says should happen.

Presuming I can get the wifi working in the hotel  (I’m away at a residential on Friday), I will post by blog on Friday morning.

KS2 reading comprehension analysis teaser

Fast, good or cheap? The ‘Triple Constraint’ in Education.

‘Fast, good or cheap? Pick two.’ Originating in software design, this aphorism nicely challenges our ‘have it all’ delusions. Make something quickly and cheaply and quality suffers. High quality products made quickly are expensive. You can have both quality and affordability, but you will have to wait for it. Known in business circles as the ‘triple constraint’, it also lends itself to a great pub game – finding examples in everyday life. Home cooked food can be tasty and easy to make (bacon sandwich) but not terribly nutritious, or easy to make and nutritious (a salad) but not terribly tasty, or tasty and nutritious, but not easy to make (anything in a ‘Anna Jones’ recipe book). I first came across ‘triple constraint’ in this article by Oliver Burkeman and he cites a blog by the entrepreneur Ben Casnocha who has some amusing, if provocative examples. For example, holidays can be exotic and/or cheap and/or relaxing, and our partners can be hot/smart or emotionally stable.

Perhaps teachers can be popular and/or effective and/or emotionally stable whereas with leaders the trade off is between being visionary and/or consultative and/or effective. Pupils’ writing can have good ‘SPaG’ and/or be neat and/or be interesting…although this year’s end of key stage interim assessments are only bothered with the first two. While I hate and detest that dull and boring writing which is neat and has good spelling trumps imaginative, thoughtful work – particularly the perverse incentive to use easy-to-spell vocabulary rather than take a creative risk – I would be willing to trade a little bit of creativity for better spelling and handwriting. Or we could go all out for all three by increasing the curriculum time for English by cutting time for something elsewhere. You might be able to cling onto a broad and balanced curriculum by your fingertips – but depth of coverage in English will have to be paid for by superficiality of study somewhere else. You just might not admit it. At least, not publically.

Project managers describe the triple constraint as defined by choices between time, cost and scope. By scope, performance specification, and/or quality is implied. Project managers use this as a tool to stop kidding themselves that there are no limits upon what can be achieved within a given set of finite resources. It’s a refreshing blast of realism in the face of aspirational, ‘whatever it takes’ woo.   No one gets to have to all. We’ve all made choices along the way – using the model just makes us honestly own the downside of our decisions. If you want something quickly (or frequently) and high quality, it is going to cost. That cost may not be in cash terms, it may be in terms of opportunity cost – you can only spend the time of your teachers once, so make sure you spend that time wisely. So with marking, a set of books can be marked and returned to the class very soon after the initial lesson and the marking can be very effective in that it enable great progress – but this will be at a huge opportunity cost to the teacher. All other calls of their time will have to be rejected – including the calls of their family commitments and personal wellbeing. Here the high quality of the marking and quick turn around is achieved at the expense, or cost, of the teacher’s time. Whereas in days of yore when ‘tick and flick’ was the norm, marking cost relatively little in terms of teachers’ time and could be turned around quickly – but didn’t have much impact of pupils’ learning. Its scope was limited. The Holy Grail of course being finding a system that effectively accelerates pupil progress (scope) whilst still occurring frequently (time) without incurring too large an opportunity cost on the teacher’s time. (Although I suppose theoretically one could reduce opportunity cost by increasing financial costs by employing more teachers to do the marking – not a route likely to catch on in the present funding climate).

The Marking Policy Review Group certainly makes some interesting suggestions and claims that its triplet of ’meaningful, motivating and manageable’ marking is relatively easily achievable – no triple constraint here. Maybe, because meaningful and motivating cover the same ground? Marking’s hardly meaningful if it is not motivating, is it? Clearly, we all want to reduce the opportunity cost to teachers that marking in its present form is extorting. So either we accept that we will have to reduce how frequently work gets marked, or reduce the scope of marking.   So, for example this primary school uses codes and symbols which direct action the next day. A different solution is to only explicitly marking a couple of pieces per class – and sharing these, via a visualiser, with the rest of the class – leaving them to then ‘mark’ their own work by extrapolation. The trail blazer schools are reporting that this approach is working really well; better in fact than the old distance ‘deep’ marking of all pupils’ work ever did.  This of course has its own opportunity cost in terms of curriculum time – curriculum content not covered because lesson time was spent improving and deepening what has already been taught – teaching less but in more depth – in other words, a mastery curriculum. Maybe this is a price well worth paying – for what it is worth I think it probably is – but we should not flinch from owning our choices. There is a shadow side to every decision.

One of the things I really admire about Michaela School is the way it is so up front about its choices. Accepting that it is impossible to do everything – it doesn’t try to. But rather than sweep under the carpet the corners it has cut, it advertises its omissions as a badge of pride. No distance marking here, no siree and no display neither. No computing, or DT or PHSE. Joe Kirby explains here how ideas can be either hornets or butterflies. Hornet ideas are high-effort, low-impact, whereas butterflies are vice versa. Reports and homework are hornets. I don’t think computing and DT are seen as such – just collateral damage in the struggle to teach an exacting and demanding curriculum in the other subjects. You pays your money and you makes your choices. Costs and time being relative fixed within schools – the only give in the system is to reduce scope somewhere along the line. Even when you’ve honed your systems to be as effective as possible – no school can do everything – so choose what you don’t do or what you do less well consciously and not by default.

It would be really useful if schools had to be really honest about the downside of their choices. Particularly in these days of school to school improvement, where we look to schools with amazing results and then try and copy what they do, its really important we are aware of the hidden cost in the choices they’ve made, so we can decide whether the strategies being employed are really replicable, sustainable and ethical. For example, some schools burn through young staff by working them to exhaustion at great cost to the individual teachers concerned. It gets results…but is this sustainable long term? Obviously it’s unethical. (Maybe that should be marked up as an increase cost…to one’s mortal soul!)   But less dramatically, how useful it would be to hear about the things people have decided not to do. School A decides it won’t have a library, thus reducing both financial and curriculum time costs – no more time consuming book-choosing time. The downside is some children who don’t have parents who either buy them books or take them to the library don’t get to read much for pleasure. Maybe their intake means they don’t have many parents like that, or few enough for some different, cheaper strategy to expose those children to a rich selection of books. Or maybe that’s just how it is. Instead, all the children get quality musical instrument teaching. At School B, the priorities are reversed. School C teaches maths in a way that means almost all children make rapid progress. The cost? Children on p levels become more and more isolated, hardly ever working in class with their peers, never taught maths by the class teacher. School D withdraws poorer readers from humanities lessons for extra phonics. The downside is while their phonics improve, their general knowledge suffers, so later on they find it harder to understand what they read.

One problem is often the effect of the downside is displaced a few years, so the school in question does not pay the accountability-price of their choice. The school without a reading-for pleasure strategy doesn’t pick up the tab when that child effectively stops reading fiction. As I’ve written about before, teaching maths with an over-emphasis on the procedural at the expense of the conceptual might engender short term results but at a cost to longer term comprehension of the basics which comes back to bite (some other teacher’s) bum. And the outcome of some choices will make itself felt many years down the line, well into adulthood. An adult drowns; her primary school cut swimming provision to the bone. Another goes to prison; he never received help with his anger management when he was little. Yet another has poor health due to obesity; PE was a Cinderella subject. Obviously the lines of cause and effect aren’t anywhere near as clear-cut as this. But let’s be honest with ourselves. We say we come into teaching to transform children’s lives. Yet the reality of it is, we have to choose which bit of their life it is we are trying to transform. In other words, we have to be clear about the scope of education; what it is we can do well, given the other constraints of limited time and money. What really matters, what will we go to the stake for?

With the coming National Funding Formula, us London schools are bracing ourselves for cuts on an unprecedented scale. The triple constraint reminds us that if there is less money, then either scope will have to be reduced or timescales will increase. In an education context, timescales are fixed. Whether SATs or GCSE’s, those annual results wait for no man – there’s no potential to ask if year 11 can take their Maths GCSE at the end of year 12 as we’ve had to reduce the frequency of Maths lessons due to staff cuts. (Although I love the idea of ‘when –ready’ exams along the piano grades model, I can’t see the government adopting this any time soon). Age-related expectations set tight delivery timescales. Failure to meet them is, well, failure.

So the scope of what we offer will have to take the hit, or several hits, meaning we’re reflecting as rigorously as possible on what is absolutely essential and what is potential cut-able. Not being an academy, I can’t ‘do a Michaela’ and decide we are just not going to teach certain subjects. Obviously English and Maths take centre stage. The time devoted to the rest is already less than ideal, except for music and French – which being taught by subject specialists and taught whilst class teachers have PPA – get an hour per week come what may. Who knows if we will still be able to have subject specialists. Are they the cheapest way of covering PPA? Are they the best use of ‘spending’ precious curriculum time.

We are very proud of our pastoral provision. As well as a (part time) learning mentor and a (part time) home school liaison officer, we also have our own social worker half a day a week. She’s invaluable. They all are. I could reduce this team, or salami-slice their hours. But the inevitable effect would be to reduce the scope of support for the most vulnerable. Maybe, longer term, the scope of the kind of pupil that comes to our school will have to be reduced as we have to cut the resources that enable them to stay in mainstream. They won’t cope. We won’t cope. But hey, there’s always permanent exclusion.

Cutting back on the arts is pretty much inevitable. Swimming provision: how little is too little? We have already cut back on our intervention programmes. Children who are way behind in Maths would have, in previous years, had half an hour a day catch up with a specialist teacher, following a programme proven to be highly effective over the long term. Well, those children won’t get that opportunity anymore. But at least our deputy head teacher is able to run intervention groups – for now.

No wonder oldprimaryhead wrote this heart felt blog recently. That’ll be me next year. I don’t deny that funding needs to be fairer and that in Tower Hamlets we’ve been generously funded compared with everyone else. But it’s not like we’ve been burning fivers; the money’s been used very effectively.

Of course, we are not allowed to admit that scope – quality and breadth of provision – have taken a knock. Since we are held accountable for standards in English and maths, we will move heaven and earth to maintain quality there, while wondering what we can pay lip service to, while maintaining a veneer of quality? What can we get away with? What can we live with and still sleep at night?

I didn’t start this blog to moan about education.   There’s plenty of people doing that already. I’m not decrying moaning. It’s necessary. Done well, it galvanizes us to change things. But I wanted to mull things over, suggest solutions, share what I’ve read. This post seems to be a bit scarce on the sharing suggestions front. Sorry about that.

 

 

 

 

Fast, good or cheap? The ‘Triple Constraint’ in Education.