The growing chasm

Ok, so maybe the title of this post is a slight exaggeration, but take it as representing my initial feelings as I marked the assessment I’d given at the end of the last unit – equations. Rather remarkably, there was no ‘middle ground’ at all: 15 A or A* (upwards of 23 out of 37) and the rest Cs or less! I started off by telling myself (and the class) that I’d deliberately set the test hard (the hardest GCSE questions get on the topics covered), and some students even recognised that they’d made a poor attempt and hadn’t revised what was necessary, but there’s no getting away from it – while not entirely the disaster I first thought, our approach so far is clearly only successful with about half of the group.

The challenge now is what to do about it. One student even went so far as to write on her test paper words to the effect of “I couldn’t do this and I don’t feel like you’ve taught me any of this stuff”. Hmmm… ouch.

While I could beat myself up about it and feel like I’ve let this particular young person down, I think more interesting is to take that one opinion as an indicator and look at who else didn’t do so well. I might have expected there to be some correlation with learning style (a greater academic than myself would have been over the moon, I suppose, with such clear-cut findings) but there isn’t. My own knowledge of the students gives me a bit more insight though – it seems the learners who might be feeling a bit ‘let down’ by my approach are some who tend to take the more independent learning options in lessons, sometimes against my suggestions. So actually they’re right: no I haven’t taught them this stuff. That I might trust them to make sensible choices about their own learning was part of the deal we hammered out at the start, but perhaps I’ve assumed that these 32 16-year-olds hold my values of what is and is not sensible, rather than their own. Clearly we need a plan to move this forward.

I’m not going to try to fix what isn’t broken – students for whom the system is clearly working as it is will still have the full range of learning options to choose from in any given lesson, subject to input from me and ‘whole-class’ time booked on the learning plan.

Some of you need better guidance – I’m now going to add a permanent ‘teaching table’ to every lesson, by invitation, so that those who haven’t done as well as they/I might have thought so far can be given clearer guidance on how to progress and more rigid modelling of examples from me.

So, there we go. I think my colleagues call it ‘differentiation’…

Answers!

I didn’t even notice it happening, but my students have spontaneously begun to write answers to some of the questions posed on the Post-It wall! Here are some of my favourites:

Q: How do they program pi into the calculator (they haven’t found it all yet) ?
A: Approximation. It only gives it to the amount of decimal places on the screen.

Q: Who discovered / invented pi?
A: Babylonians. See http://www.answerbag.com/q_view/5850

Q: What kind of questions should we be writing on these things?
A: These kind (with arrows pointing to some of the other stickies)

And one from C, as yet unanswered:

In ‘completing the square’, we end up with (x-b)² for example. If x is already an imaginary number and we minus b from it, is it still imaginary? If we then square it, we end up with a negative value (inside the bracket) before adding / subtracting (to find the minimum point). Is this possible or am I just getting something wrong?

Any takers? Come on Mr Stucke, I know you’re reading…

The Factorising Factory

They were on fire yesterday! Not literally, of course – that would be a serious health and safety issue – but in learning terms. Having self-assessed where they’re individually up to on this unit’s roadmap, and after a brief whole-class input from me (I have to ‘book’ their time now if I want to do this – more on that in a future post!) there were at least seven different learning activities taking place around me:

  • A ‘teaching table’ where I led a small group through the basics of factorising quadratics
  • Several students working independently from textbooks or exam questions
  • A peer-teaching group where 3 students read ahead on completing the square – they will be leading some coaching in a future lesson!
  • Three students producing a poster-sized T&I flow-diagram, now on display along the maths corridor (the idea from me was: ‘If humans were computers, what algorithm would they follow?’ – classic CS
  • One going it alone with a card-sort activity to investigate the link between quadratic equations, how they factorise and what their graphs look like
  • Two students recording a commentary (via the voice recorder on R’s mobile) to a GCSE T&I question from last lesson – this will be posted on the wiki as soon as I work out how!
  • A small group working on computers in the library, either contributing to the wiki or researching ahead to the next topic

My role in these lessons is now less and less the classic teacher / ringmaster, and more and more as a resource for the class to use, as they would a textbook or the internet.

Assessment and Review

It was the acid-test today. Make or break for the new method.

I don’t think I came off too badly…

I had warned the class I’d be assessing their learning of the first set of objectives from the Roadmap (Wikilink: Area, volume etc) and it was a fairly standard sheet of GCSE questions – one for each of the assessed objectives. I haven’t marked them yet so I’ll let you know.

The second half of the lesson was more freeform. I sat back as the circle discussion got under way and managed itself pretty well. They were asked to create two lists on flipchart paper (excellent scribing from S and K) – the results are below:

The Happy List (aka things that have worked well so far)

  • Different ways to practice
  • The ‘teaching table’
  • Small groups
  • Unconventional learning
  • Not wasting lessons (on things we know already)
  • Dessert (there had been a good analogy about a lesson as a meal, but I forget the details…)
  • Fun makes the learning stick
  • Learning style-based lessons
  • Appropriate way to learn for each person
  • It’s nearly Christmas!

 The Wish List (aka things we’d like to change or new things to try)

  • The Pi Song
  • Be taught first, then practice
  • Too much freedom, could we be given more structure?
  • More discipline on working
  • CHOCOLATE MATHS! (their caps, not mine…)
  • More maths-oriented games e.g. Countdown

Mostly sensible contributions. I’m not going to answer any of these points straight away, but if any of my lot are reading (and I know some of you have been) why not post a comment and we’ll start a discussion!

Dropping a Mouse Down a Well

Drawing from Galileo's If I drop a mouse down a (dry) well, it will probably walk away. If I do the same with a polar bear (ok, so think of it as a scaled-up mouse), it will go splat and make a large crater. Galileo knew this (the picture is from his “Two New Sciences“) – it’s very similar to his ‘why giants can’t exist’ proposition.

Now I bet any maths teachers out there are thinking “Yes! That’s exactly how I would teach volume and area scale factors” – but just in case, here’s a selection of useful quotes from pupils in today’s lesson:

“A mouse has more outside than middle; a polar bear has more middle than outside”

“If this big pink cube was the polar bear, it would smash apart when you dropped it”

“Arrrrrr. So 1m³ isn’t the same as 100cm³? That knowledge is worth more than any gold!”

The last is from the one-and-a-half minute play that two of my pupils (abstract randoms, of course…) produced to demonstrate their learning. It was about pirates.

Today’s Learning Options

The objective we covered in the lesson today was being able to distinguish between formulae (for length, area and volume) by considering dimensions, which we immediately re-wrote into everyday language:

“If I gave you a strange formula, how would you know if it was for length, area, volume or none of these?”

A typical GCSE dimensional analysis question was on the IWB throughout the lesson, and the learning options were based on the Gregorc styles:

CS – Complete a card sort activity, create a flow diagram

CR – Brainstorm all the formulae you know and investigate links / similarities between them

AS – Independent library work, research, note-taking, self-guided learning

AR – Produce a short play(!), record a commentary for the GCSE question, peer-group learning

By the end, once we’d come back together as a whole class and been able to solve the exam question, I could tell they’d enjoyed the lesson – is that enough? The proof of this particular pudding will be in next Monday’s lesson – I’ve given them advance warning that I will assess them on the first ‘unit’ of work then!

Gregorc’s Learning Styles

There’s a chap called Anthony Gregorc who has conducted research into brain hemispheres and mind styles since the 70s – anyone who’s trained as a teacher in the UK over the last decade or so has almost certainly heard of him and his Style Delineator, but he was a new name to my Y11 class. The learning style model used most often in my school is VAK – fine as a broad-brush approach but to my mind it lacks the precision needed for truly personalised learning, so the next layer of complexity I’m going to introduce will be tailoring any ‘menu’ of learning options to the preferred Gregorc learning style of each individual pupil in the class. In broad terms, the four styles are:

  • Concrete Sequential learners want step by step instructions using real examples you can touch.
  • Concrete Random learners want real examples, but want to browse through the knowledge in a trial and error manner
  • Abstract Sequential learners want clear visual material that is well organised
  • Abstract Random learners however are happy to find a trial and error approach to visual material.
  • The half-lesson that the style delineator activity took up was fun enough (although I lost bets on one or two individuals whose preferred style I tried to guess), but the worth of the end result depends on how I use it. At one level, simply reflecting on learning preferences will have been helpful for some of the group, while others I know will appreciate being guided towards activities suited to ‘their’ style. There were 6 predominantly CS learners (7 if you include the teacher!), 5 predominantly CR, 6 predominatly AS and 4 predominantly AR, with the rest tending towards a combination of two different styles – as can be expected – and one pragmatist, scoring equally in all four areas!

    Finally, here’s a list of the sorts of activities I might guide each learning style towards:

    • CS – checklists, outlines, flow diagrams, lecture-style lessons
    • CR – simulation, problem-solving, experiments, open-ended questions
    • AR – media/music, group discussion, peer-group work, creative expression
    • AS – independent study, research, working alone, note-taking