Learning Stuff About Stuff

For a long time, Constructivism seemed to me to be a theory of learning that didn't have any direct implications for the way that I taught.

Then I read "Evidence Based Learning" by Geoff Petty and the penny dropped.

Constructivism is now the lens through which I view everything that happens in the maths classrooms in my department (on good days!).  Our rooms are full of young people constructing their own versions of maths.  They do this with the support of the environments, inputs and activities which we offer them.  They do it together with those around them, and they do it, in large part, by attempting to connect new learning to their pre-existing model of maths.

I get a kick from seeing this happening. My S4 class today was looking at extending sine and cosine beyond 90º.  It involved playing with a Geogebra applet that showed how the sine and cosine of an angle can be represented by the y and x coordinates of a point moving round a unit circle.

I used show-me board…

My S3 class sat an important test yesterday.  It was set at level 4, and was designed to ascertain whether or not they would be able to pass a National 4 unit assessment.  The class have known about the test for several months, and have known that failure to gain 50% in the test would mean moving to a different class - to focus on passing National 4 by the end of S4 instead of moving beyond level 4 to start working toward National 5.I've written about this class before in Notes for Future Self and Aww. I put in place a major program of support for all those pupils who were not reaching the standard back in September - almost half the class.  Yesterday's test was where we found out whether or not all the hard work had paid off - and there was a lot of hard work!  I contacted the parents of every pupil who had underperformed; the pupils attended support sessions after school from 4-5pm on Wednesdays and Thursdays, completed substantial homeworks every week and spent time correc…

I was very lucky to have the opportunity to attend the "Flying with Geese" masterclass in Glasgow today.  Dylan William and Graham Donaldson spoke. It would be futile to attempt to summarise the day, but here are some interesting points that stuck in my head:

Having identified the strengths and weaknesses of a teacher, the research shows that the best way to elicit improvement in their professional performance is to focus on trying to further improve their strengths, rather than trying to fix their weaknesses (unless the weaknesses are really bad!)Creativity, problem solving and critical thinking are not generalisable. Rather than trying to teach these (or any other) skills separately, we should use these terms as audit tools within disciplines.Content from disciplines must be strong within topic work, and be directed by teachers.  Left to their own devices, learners will not bring worthwhile levels of disciplinary content into their project work.There is no correlation betwe…

Last year, I tried giving feedback to pupils - about their homework - using a detailed marking scheme, highlighted to show successes and failures.  A pupil's feedback would look something like this:



Question Step Comment 1a differentiate sin to cos use the chain rule to produce an extra multiplier of 2 (the derivative of the function inside sin) keep 3 as a multiplier of the derivative know to evaluate f’ at x=0 perform evaluation correctly 1b realise that rearranging to tan(x)=... is the first step take square root to give tan(x)=... remember to use +/- square roots find building block angle from exact values choose correct quadrants for solutions
work with fractions of pi correctly not available, as you did not consider other solutions state solution clearly My hope was that the pupils would use the feedback to see where they had "gone right" as well as where they had gone wrong, and would be able to use it to help with corrections.
As it turned out, these feedback forms (returned to the…

Just thinking aloud...

I'm not the only one with a video camera in my phone.  Most of my pupils have the same technology in their pockets.  I wonder if I could convince some of them to record a video explaining their thinking as they solve one of the questions in their homework?

That would be really interesting. I would receive so much more information about their thinking processes, which would in turn allow me to provide higher quality feedback to them.

Sometimes the language that pupils use highlights misconceptions that are not apparent from their written work. For example, I was working with an advanced higher pupil on Friday, who asked me, when tackling the step below,  "what happens to the dy?"

$\int dy = \int x^2 dx$

I asked her what happens to the dx in a regular integration, and she said "it turns into +c?".  Further discussion showed that she had not grasped the fact that the integration symbol, plus the "dx", represent an instruction to pe…

I have been using my mobile phone to record video feedbacks for homeworks this session. I post them as unlisted videos on Youtube and give the link to the pupils.

I expect all my pupils to attempt corrections once I have given them feedback on their formal homeworks, so I need the feedback to be good quality.  I have used two models:  individual feedback based on the pupil's own homework, and general feedback for the whole class.  The latter is obviously quicker, and is probably sufficient if the same errors are cropping up in most of the homeworks.

Here's why I'm using videos for feedback:

Verbal feedback is much richer than a few scribbled words and symbols in a jotter;I find it easier to give verbal feedback than to try and express myself in writing in a pupil's jotter;I do not have time to give verbal feedback to all pupils during class time. I do have time to record a wee video as I mark each piece of homework*;Video feedbacks are available to pupils as they are at…

Today my S2 class spent a lesson doing some mathematical thinking.  The prompt for this was an Nrich task, More and More Buckets. They are a high level set class, and this particular problem would not have been appropriate for all classes, but there are Nrich tasks to suit all ages and levels of prior learning.

This S2 class is used to the idea of "doing some mathematical thinking".  The pupils appear to understand that I am interested in the process rather than any particular conclusion they may reach.  They work on the task in groups of 3-4. Here are a few examples of the artefacts of their thinking:

This group were trying to find a formula.  They laid out possibilities in an organised list, and spotted some interesting patterns.

This group took a more qualitative approach - they did very well to analyse the problem sufficiently to reach this simple summary.


This group decided that it was helpful to think about the numbers that were missing each time, and this layout (whic…

The S3 class mentioned in the last post is a group which I am aiming to lead towards National 5 maths by the end of S4, but this is a wildly ambitious target.  Based on previous cohorts, I could reasonably expect only about half of the class to get there. This is one of those "could go either way" classes.  There are plenty of pupils in the class who are regularly getting into trouble around the school, but there are also many who are conscientious and hard working across the school.  Given that the class is behaving excellently and making good progress, I'm writing this post as a reminder to myself for the future.

Here are the strategies I have consciously employed with the class:

Consistent use of AiFL techniques in class - lollipops, think-pair-share, celebrating wrong answers  etc etcExplicit discussion about growth vs fixed mindsets (Carol Dweck's work).Zero tolerance of disruptive behaviour, backed up with generous praise, particularly for those who mend their w…

No, I haven't got a video of pets wearing hats.

This "awww" moment came this afternoon as my S3 class were queuing up outside my classroom.  I had asked a general "how has your day been today?" and had a few responses from others, then one of the girls said "good now we're at maths!" with a big smile on her face.  "Yes" said her friend.  "We were saying at lunchtime how much we are enjoying maths now you are teaching us." "Yes" said the first girl. "You make it so easy.." She saw the expression on my face freeze, and hastily added "no, not easy - but we have learned so much with you." "Yes, I've learned so much maths this year." I mumbled something about it having been their hard work that had made the difference.

What a delightful moment!

A two part question to determine if you "think like a mathematician," from Prof. Eugene Luks, Bucknell University, circa 1979.

Part I: You're in a room that is empty except for a functioning stove and a tea kettle with tepid water in it sitting on the floor. How do you make hot water for tea?

Answer to Part I: Put tea kettle on stove, turn on burner, heat until water boils.

Part II: Next, you're in another room that is empty except for a functioning stove and a tea kettle with tepid water in it sitting on a table. How do you make hot water for tea?

Non-mathematician's answer to Part II: Put tea kettle on stove, turn on burner, heat until water boils.
Mathematician's answer to Part II: Put the tea kettle on the floor.

I recently bought a class set of overlapping decimal dart cards for the department - this pdf link is essentially the same resource.

I used them today with a class of 14 yr olds, all of whom have been working with decimals for at least two years, but none of whom were able to correctly order this list of decimals at the beginning of the lesson:

0.26
0.102
0.2
0.8
0.13
0.7
0.34

I did not tell them the correct order - just that none had got it right.

I handed out the decimal cards and started by asking them to hold up the one that would go exactly half way between 0 and 1 on the numberline I had on the board. 100% success.

I then asked them to show me whereabouts on the line 0.1, 0.01 and 0.001 would go.

We discussed this for a while, corrected some misconceptions, and moved on with the idea under our belts that 0.1 is small, 0.01 is tiny and 0.001 is incredibly tiny!

I then asked them to hold up the decimals that represented 4 tenths, 7 hundredths, 3 thousandths etc. All got there after a bit of di…

I shared these two versions with a class recently. I'm sure you can guess which model better matches their perception of what is supposed to be going on in the classroom.

Model 1

"Mr Jones know loads of maths. His job is to transmit some of the stuff he knows to me, so I can use it in my life, my further studies and the exam at the end of S4."

Model 2

"Mr Jones has a version of maths in his head, that he has created over the years, in response to teaching, reading, activities he has undertaken etc. I also have a version of maths in my head, which I have been building since I was born. Mr Jones's job is to provide me with experiences and feedback that will help me to construct a bigger, better version of maths in my head."

Working after school with a pupil...

Me: How many of these [10cm lengths] would you need to make a 30cm length?
Pupil: 3
Me: Ah okay, so back to the problem, how many would you need for 300cm?
Pupil: [with a "got it now!" tone of voice] would it be 30?
Me: [with a slightly questioning tone of voice] What makes you say that?
Pupil [utterly deflated] I don't know, I don't think I understand.
Me: Really? You sounded quite confident before. 
... we spiralled around and back to the same problem, and the pupil eventually went with 30 and told me how they had got it.

Me: before, when I asked why you thought it was 30, with a tone of voice that maybe seemed like I disagreed, you went to pieces a bit didn't you, even though you really thought you knew why it was 30. I wonder why that happened.
Pupil: [pause as tears well up in eyes] I think it's because when my Mum helps with me with my maths she gets frustrated with me if I don't get it straight away - like she thinks I…

If you gave an elite athlete the choice between the use of a heart rate monitor or a stopwatch to train with, which do you think they would choose?

An HRM I'm sure.  Doesn't that tell us everything we need to know about the kind of feedback we should be providing to learners?

Are you a Teacher, Principal Teacher or Senior Manager in a school?  Are you putting time into developing assessments in line with Curriculum for Excellence - "say write make do"?  If so, allow me to ask a provocative question: are these basic techniques of AfL being used every day in your classrooms?

wait time
no hands up
random selection of pupils for responses (lollipop sticks or the like)
think pair share
comment only marking
self and peer assessment

If not, then I think you are focussing your energies in the wrong direction.

Curriculum for Excellence clearly  demands a richer, deeper assessment of learning than can be delivered by giving a percentage mark on a written test.

I have been exploring the implications of this fact over the last few years, as we all have in Scottish education.  This post isn't really about the issue of assessment under Curriculum for Excellence, though.  It is about the way that being a Connected Teacher has facilitated my exploration.

Last week, I came across a review "Evidence Based Teaching" by Geoff Petty in my Google Plus stream.  I follow hundreds of maths teachers on Google Plus (very few of them from the UK yet) and they provide me with a rich stream of new ideas about maths education.

I bought Geoff Petty's book, and was very interested in the potential of Biggs's SOLO Taxonomy as a framework to assess learning in mathematics more meaningfully.

I searched on Google, but couldn't find anything specifically relating to the use of the SOLO taxonomy fo…

I recently posted on Google Plus about a lesson using a new resource to help pupils to improve their understanding of the decimal system.  Read it here.