Substitution (…and Continued Fractions)

This one’s really only for the Maths teachers of there. But I saw it and just had to reblog it! Awesome. SC.

Continuous Everywhere but Differentiable Nowhere

Today in Precalculus I went on a bit of a 7 minute digression, talking about continued fractions. You see, a recursive problem showed up (we’re doing sequences): Write out the first five terms of the following sequence:

$latex a_{n+1}=sqrt{2+a_n}$ where $latex a_1=sqrt{2}$

So obviously they go like: $latex a_1=sqrt{2}$,$latex a_2=sqrt{2+sqrt{2}}$, $latex a_3=sqrt{2+sqrt{2+sqrt{2}}}$, $latex a_4=sqrt{2+sqrt{2+sqrt{2+sqrt{2}}}}$, and $latex a_5=sqrt{2+sqrt{2+sqrt{2+sqrt{2+sqrt{2}}}}}$

So great. Awesome. NOT. Booooring. So I showed them the decimal expansions:

$latex approx 1.414, approx 1.848, approx 1.961, approx 1.990, approx 1.998, approx 1.999, approx 1.9998, approx 1.99996, approx 1.999991, approx 1.999997647$

WHOA! This is getting closer and closer to 2… Weiiiird…

And then I say I can show them this will continue, and we can find a way to show that $latex sqrt{2+sqrt{2+sqrt{2+sqrt{2+sqrt{2+…}}}}}$ [where the pattern continues forever] will practically become 2.

DIGRESSION WHICH IS ACTUALLY WHY I WANTED TO BLOG ABOUT THIS

To do this, I start with something else. I…

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EDpuzzle. A leap forward for flipped learning?

I’ve been a big fan of the flipped classroom for a number of years now. With many classes I find that lesson time is too often devoted to “delivering content”, as it were, en masse. Typically, unless you are some kind of differentiation god (which I am not) there will be some people who latch onto the new idea pretty quickly and start to get bored as I am required to repeat, re-explain or look at from a completely new angle whatever gem of knowledge I am trying to impart. I can’t go as fast as the quickest, nor as slow as the slowest. It’s a balancing act. This means I only really get it exactly right for a few individuals.

I know there are many ways to try to get round this problem without flipping the classroom. I have great admiration for colleagues who can cleverly structure learning activities so that they are differentiated and meet each learner’s needs much more effectively. I can do this too, sometimes. But I like flipping.

I have gradually introduced my students to the idea of flipped learning and over the years have created a number of videos at www.youtube.com/mrcampbellmaths, most of which have proven successful. There are many advantages of the flipped classroom. For me I would sum it up like this:

  • Students can learn at their own pace using the video, at home or out and about.
  • Students who lose their notes, missed the lesson or want to revise can re-watch the video at any time.
  • In lessons I can spend much more time working with individuals or groups to help move their learning forward.

But there are limitations…

  • How do I know who has watched the video?
  • How do I plan my lesson if I don’t know if they have understood the video?
  • How do I make sure they actively watch the video, rather than passively copying down notes from a computer screen?

This is why I got so excited about EDpuzzle.com

This website provides a simple to use structure in which you can embed a video from almost anywhere on the web (or one you make and upload yourself) and both augment the learner’s experience through questioning, audio comments etc, and track in close detail the activities of each learner. So, basically, addressing the three key limitations listed above.

So the weekend before last my “To do” list (after “Set up blog!”) included “Make video on factorising quadratics and use via EDpuzzle”

But who ever has time for everything on their list? So it got to 10pm on Sunday and I knew I had no time to start recording. I decided to turn to the number one teaching strategy which almost never fails: nick something someone else has already done! A quick search on Youtube found this excellent video by the equally excellent @HegartyMaths. The perfect introduction to this important skill. All I had to do was EDpuzzle it.

The process was so quick and easy I might as well take you through it (I had already set up an account, which is dead easy, at edpuzzle.com):

Click on “Create” then “New Video”

EDpuzzle Create

Choose your source! TED, Khan Academy, Youtube… and the list carries on down the page!

EDpuzzle ChooseI had already found my Youtube video so I simply pasted in the URL and it appeared. You then get the option to crop the video, removing as much as you like from the beginning or the end of video. Very useful if the perfect 5 minutes happens to be in the middle of an hour long yawn-fest!

EDpuzzle CropNext you are taken through three options designed to enhance the video and make the experience interactive. The first two (which I skipped on this occasion) let you record either a voiceover for the full video, or audio notes to appear at times of your choosing while the video pauses. The third option is to add Questions. These can be added at specific points in the video and come in 3 formats:

  • Open. You type a question to which the student must respond with a simple text answer. These are not marked/verified automatically – you log in and do these yourself later.
  • Test. I used this. You type a multiple choice question and any number of correct or incorrect answers from which students will choose. See image below. These are immediately marked for students and the correct answer revealed.
  • Comment. This is simply a way to add a typed comment which students will see but do not have to respond to. Very useful if you want to clarify something from the video or for many other reasons.

EDpuzzle Add Question (blank)

Now use it!

The whole process was very quick – taking only as much time as it took me to think of the questions. My next job was to ‘deliver’ the video to the students. First I created a ‘Class’ by going to ‘My classes’ and clicking ‘Add Class’.

EDpuzzle Add Class After some thought I called it “10 set 2”. I then found my freshly EDpuzzled video in ‘My Content’, selected it and clicked ‘Assign’, after which I had the following choice:

EDpuzzle AssignI chose my class from the list, then (of course!) chose “Prevent skipping” – so simple but so brilliant! Students can navigate backwards and forwards through the clips as much as they like except that they can never skip ahead beyond what they have already seen.

Once assigned to “10 set 2” I had to give my students instructions for how to join the class. Sign up is very simple – they all went to edpuzzle.com (at home) and followed instructions to “Sign up” as a student, and once in they simply chose “Join Class” and input a short code I had given them (specific to my class) and that was it!

What happened?

So I set this “homework” on a Monday, to be completed before the next lesson on the Thursday. By the Tuesday night, when I logged in to check how they were getting on, I could see that around half the class had completed the task. I had a quick scan through the available data and I knew straight away that this really was going to prove an invaluable tool for flipping my classroom.

Planning perfection

On Wednesday night, late(ish), I logged in again to take stock and plan my lesson, the aim of which was to consolidate the knowledge gained at home through the video.

I could see that…

Four students had still not logged in. Not a problem! I went to the school portal and booked a few laptops for tomorrow’s lesson (Ok, it was lucky there were some available, but still). Then in the lesson I sent them off to fetch them straight away and after 20 mins they were done.

Three students had achieved full marks. Great! These would be my Peer Tutors and for the first fifteen minutes of the lesson they would circulate and help the others. (They did. It was great!)

One student had clearly struggled much more than the others. Ok, so I planned to spend the first 5 minutes with her while the Peer Tutors looked after everyone else. What a luxury to have the time!

The majority of the class scored between 50% and 83%, meaning that while many would need a decent amount of practice, a large number of them would probably be ready to move on before the hour was out. Knowing this, I planned a suitable follow on task, to which students could “self-refer” once they felt “green” on the main task.

I was buzzing by the time I finished planning my lesson. And in particular I was impressed by all the data at my fingertips. On the top level, the first thing I saw was this snapshot of results and (at the top) the questions ordered by difficulty:

EDpuzzle Results overviewUseful. But now drill down – look at a particular question. I can see how many students gave each answer and who gave each answer, so I get a good idea of what the misunderstandings were (especialy if my multiple choice wrong answers were carefully chosen). Useful for deciding how to group students for follow on activities.

EDpuzzleGreat! But now drill down more! The data you get on an individual student was what blew me away. Click on a student’s name from any page and you go straight to their analysis. You can see all their answers and whether they were correct or not. Take this example:

EDpuzzle Question comparisonFrom question five I could see that she had not quite grasped the need to ‘cancel down’ each bracket, getting the second one wrong. But then, having seen that this answer was incorrect, she was able to correctly complete the next one! Brilliant detail.

And this is just awesome:

EDpuzzle video viewsThis girl watched each part of the video at least twice. But she watched the section with which she struggled most (evidenced by the red question marks – incorrect answers) eight times! I would never have had this information from students watching videos on my youtube channel. I was really happy to be able to praise and reward this student for her determination.

Phew!

I realise this blog is already excessively long! Time to stop. If you’ve got this far, I hope you’ve found it interesting. I would encourage you to try out EDpuzzle, and I’d love to hear what you think about it. I know there are other similar options out there (e.g. Educanon) – have you used any of them?

If you want to experience what my year 10 experienced, I have created a new class which, if you log in as a student, you can join! The code is e92Qk4. Tip: Set up an account as a teacher, but then log out and back in as a student (using the same credentials) – this lets you join classes. The quadratics video is there, along with a shorter one entitled “How far away is the moon?”. Knock yourself out!

Thanks for reading and please feel free to leave comments.

Getting Students’ Hands DIRTy

Making feedback work.

A couple of years ago I was lucky enough to take part in a professional development programme linking several schools in Yorkshire from both the state and independent sectors. It was a good example of how getting teachers together and letting them share ideas can make some of the best CPD.

I took a lot of ideas away from the programme (for example “snot” from my earlier blog) and one thing I remember really loving during the session at Horizon Community College was DIRT. I remember it as “Dedicated Improvement and Reflection Time”, thought I have since seen it referred to as “Directed Improvement and Reflection Time”. There is a subtle difference between the two, and actually I think that including both the D-words is essential.

When coffee is not enough

When coffee is not enough

So we are talking about the hours we all spend marking work, and what good it actually does. As an NQT 8 years ago my head was full of Assessment for Learning and I carefully avoided awarding marks for homework, instead spending a good deal of time writing individual comments which would allow my students to see how to improve. It took me a while to see the bigger picture, however: what was happening next? I knew enough to know that some lesson time should be given over to reading my comments but it had not occurred to me to assign time to doing something about it.

DIRT by @TeacherToolkit

DIRT by @TeacherToolkit

This is where DIRT comes in. To me, what you do in this time might vary enormously according to your subject, the type of task, etc. But the essentials, to my mind, are:

Improvement and Reflection. This is time in which all students are expected, and helped, to reflect on their work and consider both what makes it good and what can be improved. But they do not only consider it, they actively make those improvements.

Dedicated. Set aside, and ringfence, some of your lesson time for this process. In some instances, where work can be endlessly redrafted and improved upon, this might even be a whole lesson. But not a token 5 minutes.

Directed. Give the process some structure. This will vary with subject and task as well, just don’t leave students with 15 minutes to fill and only your written comments to go on.

Time. Um…

Quite apart from the benefits to students’ work, the fact that everyone is expected to use this time to improve strongly reinforces the idea of a growth mindset. If we return work and immediately follow with “right, now we’re all going to see how we can improve”, this sends all the right messages about effort and achievement, drawing attention away from the ego-involved “what mark did I get?

There are tons of great blogs and articles about DIRT out there. Most recently I saw this one from Carol Stobbs (@littlestobbsy), in which she shares the poster below as well as her interpretation of DIRT as a History teacher (see the ppt within the blog).

Thanks to @littlestobbsy for this great poster

Thanks to @littlestobbsy for this great poster

I also saw this excellent post this week from Shaun Allison on the use of very carefully structured DIRT in the Maths department in his school. This looks like it would take a good deal of planning but the benefit to students is obvious.

Twitter is also full of #DIRT at the moment. @TeacherToolkit tweeted the image I used above as well as this interesting (and acronym-filled) blend of DIRT and the 5 minute lesson plan:

5 minute DIRT?

5 minute DIRT?

So, when you are next marking work and carefully considering your feedback for each student, ask yourself “what is the point?”. Then either plan some dirty lesson time so that the effort you are going to will mean something, or just get an early night instead.

You’re either SNOT, or you’re not

Fostering habits of independent learners.

Do students need a lot of help in lessons in your subject? I should re-phrase: Are students constantly asking for help in your lessons?

I get a LOT of questions from students in my Maths lessons. A good deal are related to self-esteem/confidence: “Mr Campbell, is this right?” Not indicative of a growth mindset, but sadly pretty common. My natural response would be “You tell me!”. The same goes for “What do I do next?” and “Can you tell me what I’ve done wrong?”. Usually, when prompted, students can reason their way through the problem they are encountering. Sometimes a little scaffolding is needed, but it is amazing how often the students do all the work themselves.

Of course, there are always some who have not yet grasped the idea or technique with which they are grappling (in fact, this should apply to every student at some point in a good lesson) – these students might be in urgent need of my time, but at the back of a queue of “is this right”-ers.

So in the status quo each of the protagonists described above (including me) is being prevented from working to their maximum potential by the whole setup. There are two questions: 

  1. Why do students so often need me to prompt them to think for themselves?
  2. How can I make them more independent learners (thus making more time for those who really need teacher input)?

There are some great practical ideas for prompting the sort of habits which we would wish to see, my favourite two being:

Thanks to E Salton (via TES) for the poster

Thanks to E Salton (via TES) for the poster

“3B4ME” – written in classic (and brilliantly outdated) text speak, this is a quick slogan to remind students that they should be consulting (at least) three sources before the teacher. I’ve seen a few interpretations of this, my favourite being “Brain, Book, Buddy, Boss”

 

Snot logo

Ok, so I got a bit carried away making a logo…

“SNOT” – along the same lines as 3B4ME, but probably more memorable to a 14-year old and a lot more fun to say! Stands for “Self, Notes, Other, Teacher”. Thanks to Kerry Wade of Clifton School, Rotherham for that one. She said she would ask her students “Have you snotted?”. Brilliant!

 Why is this on my mind right now? On Saturday night when I was meant to be marking but was instead lurking on Twitter I saw this tweet:

This was something I used to do all the time, but which had somehow been squeezed out of my mental teaching toolbox. So I had to reply…

And a brief, but inspiring, conversation ensued and I resolved to bring this technique back to my classroom.

Doing it

So what happened in school this week? Well, I tried the 5 min thing with my Year 8, 9, 10 and 11 classes, and I threw in SNOT for good measure! Some of the things I learned:

SNOT

  • If you get everyone’s attention and say “I need to tell you a little bit about snot”, you really have their attention. Helps if you write “snot” in green pen on the board.
  • All the classes embraced SNOT pretty quickly. Some Y9s rolled their eyes at first, but soon one girl was asking “Please can I move next to Emma as I’ve got no one to snot with?” – brilliant!
  • In response to a request for help, use Kerry Wade’s line “Have you snotted?” – it works a treat!

5 Mins no teacher support

  • I gave them 5 mins warning each time and this worked well – they know they have to make sure they understand the task properly in that time
  • Combines really well with SNOT – I had written a horrible green “snot” on the board and so I rubbed off the “t”, telling them “you only have “sno” for the next five minutes
  • It was good to see girls who are usually VERY dependent on encouragement finding other resources or simply getting on with it.
  • Students got up and moved round the classroom to discuss the work. It worked.
  • You have to choose the task, and the time within the task, carefully, otherwise you can leave them out to dry!

The main thing I noticed after each 5 minute period was what I  mentioned in my tweet at the weekend: When you tell them you’re putting the “t” back into “snot”, you never get the equivalent of 5 minutes’ worth of questions fired at you.

Conclusion? At least some of them have been more resourceful and determined than they might otherwise have been, and figured it out for themselves. It is not a revolution in the classroom, but it is surely a good start.

What do you do to encourage independence in your students? And what is the answer to question 1 above?

 

 

Remembering the basics

This week I was reminded quite how much of behaviour management is about making the lesson interesting.

At the start of September I picked up a new group in Year 10. It can be difficult taking over a class which has previously been taught by a colleague for some time (in this case over two years). In some cases it might come as a welcome change for the students. In our school, however, more often than not classes become quite attached to their teachers and might resent the change. This was certainly the case with my Year 10s.

Starting out with this new class, I very quickly began to hear “But that’s not how Mr XYZ used to do it”, for example when I imposed a seating plan in the second lesson. There was unrest. It made me nervous because I didn’t know any of their names, a serious handicap when it comes to dealing with unrest. Furthermore, a class like that carries its own confidence, built up over time, which makes them feel like I am intruding upon their territory. And I confess, with hindsight, that I underplanned my first few lessons: I had taught the topic many times before, I knew what examples to give, which bits they would find tricky. It was all fine, except that the lessons were, well, boring.

The result was the classic “low-level disruption”. Not how I wanted to start out with a new class.

I realised I had to engage and interest my students much, much more. So I called upon an old favourite starter of mine, picked up on my PGCE, which I refer to as “silent algebra”. As the girls came in and got settled, I put my finger to my lips and made a theatrical shushing sound. Any utterance from them was met with the same gesture, each time more over the top. Meanwhile I began drawing on the board. A number “1”. An arrow to the right. Shhhh!  A “7”. Meaningful look around the room implying something impressive is happening (it isn’t, yet). Then underneath, a “2”. Shhhh! Arrow. “10”. Another meaningful look. Underneath: “3”. Arrow. Then stop. At this point I held out the drywipe pen, lid off, eyebrows raised, looking for a volunteer. They quickly got the idea. The first volunteer correctly wrote “13” – she had spotted the pattern – and I immediately gave feedback by drawing a large smiley face next to the number. This went on for a few iterations, getting harder as I started missing numbers out, then going for massive ones (1,000,000 is always good), then finally “n”. Keeping the shushing going all the while, as they were enjoying it by then.

Mathematically, they were spotting patterns, solving particular examples, then generalising (a higher level skill). But more importantly they were waking up and buying into my lesson.

It may be coincidence, but the next day in my pigeon hole I found a small but delicious cupcake, accompanied by an anonymous note:

Okay, maybe your lessons aren’t THAT boring…