Interactive: Polyrhthmic fractions

I've been coding again! See my latest maths game below...

I was inspired by how some of favourite musicians (Joanna Newsom)
overlap simple rhythms containing quarter, sixth and eighth notes and make something that is subtly complex and interesting. The above version is a much more advanced version of these polyrhythms.

I thought that this was an interesting way to look at how fractions add up. I also wanted the game to be a little more informal and sand-boxy than the last, so it allows you to try things out and see what works. Have a play:

 

Direct students to a fullscreen version with this memorable address: thedrummer.rocks.it

Its not perfect. Some of the problems are solvable, others aren't (because my original file has corrupted!!!)

• there's no way to tell which achievements you've done (i was implementing this when the file corrupted)
• you can't save and share your beats, which would have given an incentive to make a good one.
• the tradeoff with it being informal and intuitive is that its gamable and you can get away with using little fraction knowledge - I would solve this by having a worksheet of questions to answer (but only once people have already got in to playing the game). It feels like a cheap cop-out though. What could I have done to make it organically more challenging/rigorous? I would love your thoughts on this.

Great Posts of March

• Inquirist's Cookbook by Think Thank Thunk: Shawn's rough and ready guide to creating inquiry based learning.
• What value Khan? by Angles of Reflection: A look at why khanacademy is still useful and powerful (even though its a bit rubbish).
• Coding IS a super(edu)power by Gealgerobophysiculus: Another edu-blogger takes to coding to create the exact tools they need.
• Being an SBG teacher in a non-SBG world by Continuous Everywhere But Differentiable Nowhere.
• If you like learning, could I recommend analytics? by e-Literate: Great read which explains clearly why you cannot apply similar analytics systems in education as are applied in online shopping.
• Foil Prank by Yummy Math: Lovely little lesson about surface area.

Teaching Maths Without Words

These are bold statements from http://www.mindresearch.net for what is essentially another raft of online games, and we all know online games are abominable.  They have not made many of their applets available to try for free, but what's there is definitely interesting (to check them out click here and here).

**UPDATE - Before posting this I decided to contact MINDResearch to give them a chance to comment. Not only did they reply, answering many of my fears about the shortfalls of these applets, but also included a much better demonstration (here) of the way that progression between levels is handled.



Players must guess how far along the number line the fractions (here two thirds) will take them.

Positives

• The emphasis here is definitely on making the maths intuitive using the information on the screen. This is a great first-step in comparison to the majority of online learning, where repetitive maths is tacked on to an existing (not very) fun game. The maths feels natural and even vital in this game world, whereas usually it is a chore to get through before you get to 'the fun bit'.
• The feeling of progression is the main/only motivating factor. In fact, the way that more difficult questions are linked to earlier questions is outstanding (It gave me huge pleasure to see the aliens start zapping some of your fractions to force you to find an equivalent)! Progression and 'acquiring new skills are intrinsic motivation and means your reward is the feeling that your maths is improving. Many studies have shown that intrinsic motivators such as this are more effective AND have fewer side-affects than the usual extrinsic (badges, etc.) motivators. (See 'Drive' by Dan Pink).
• Teaching is happening. There is a feedback loop on your initial guesses, showing what your guess means in this world. This is a great way to give pupils an idea of what the maths problem is really asking.
• It is a low-risk environment (no stigma for failure), so you feel free to try things out and lose a couple times. Allowing students to guess and try again can cause the problem of pupils gaming the system by spamming guesses until its right. These games get past that in two ways: you only have a few lives on each level, and the teacher is alerted to any student making too many mistakes so they can help with the misconception.
• This alert system, along with an 'ask for help' button, implicitly admits that this software will not be able to teach everyone and other interventions will be necessary. This is something that few other sites (in my experience) admit to.
• The fact that there are very few words will be a great boost to a huge range of pupils with EAL, dyslexia, etc.
• The game environment and interface is clean and doesn't include unnecessary distractions.



1. Choose how to split the square (for the denominator)
2. Choose the numerator visually
3. Compare it to how much the aliens remove from the bridge



Negatives

• The trade off with having a clean, uncluttered game environment (and sticking to progression as the only motivator) is that the puzzles can feel a little dry and samey. My guess is that this is not much of an issue for their core, primary school audience, but would have more of an effect with older pupils.
• Only one motivator; I like the fact that progression is the core motivator, but there are other intrinsic motivators that could have been employed as background motivation. I feel that this would help give a more rounded gaming experience. These could include feelings of mastery (on optional, extension levels or mixed question levels) and feelings of exploration (of the mathematical concepts, e.g. through sandbox style environments).

Conclusion

After trialling the fractions demo, I am very excited about this project, as I am in total agreement with a lot of their core principles. Unfortunately it is not yet available outside the U.S. and is more appropriate for a primary school.

Quotes from a MINDResearch representative

Indeed, our aim is to present math concepts in a simple, uncluttered way, so that students aren’t faced with distractions and they are able to purely focus on solving the puzzle. The feedback they receive is informative; the approach stems from neuroscience research involving the perception-action cycle, or rather learning by trial-and-error.

...our focus is to create intrinsic motivation instead of extrinsic motivation. The former compels students to challenge themselves because they enjoy the challenge, not because they will be rewarded with a badge, or prize (extrinsic motivation). Intrinsic motivation is much more powerful in the learning process...JiJi is the centerpiece of the puzzles, and you probably wouldn’t be surprised that students want to help JiJi get across the screen, so much so that they continue to persevere until they do

...Class-level and student-level [live] reports allow teachers to see which students are struggling (the report highlights a student who is “stuck,” and has tried a game/level a number of times) and thus intervene to help the student using questioning strategies or other intervention. The software also allows teachers to freeze the student’s screen, go back to the puzzle the student answered incorrectly, so that the teacher can pinpoint the exact issue that was causing the student to struggle.

Motivation in computer games

Here is an article at Gamasutra outlining (some of the) motivational techniques computer games use to keep players hooked. Try to ignore the horrendous use of algebra. Some of these include:

• Exploration and mastery of the game world (and set of rules)
• Rewards (Better equipment, experience points)
• Setting up a need (New equipment, resources, skill) then giving it
• Challenge (Tournaments, etc)
• Player state (levelling up etc)

Achievement badges do not get a mention.

Also of note is the mention of negative motivation:

However, there are opposite motivations based on negative characteristics as well, such as addiction, alienation, anger, frustration, etc. It can be interesting to exploit these feelings sometimes, for deeper needs, but to build a complete system based on this would be destructive. At the end, the player would be left feeling bitter and would be repelled by the game.

Many online math learning tools rely heavily on repetition and the human desire for collection and completion, which falls easily in to this category.

Here is another article arguing that acheivement badges are very much negative motivators, stating: "I think that any system that is based on endless collection, or any system where there is collection without a larger purpose is exploitative and uninteresting (and therefore unfulfilling)."

Related posts

Lesson Sketch: Introducing Infinity at year 8

I recently gave a cross-school lecture to Gifted and Talented year 8 and 9 students on the subject of infinity. I got great feedback from it, with most of the kids hungry for more (after a one-hour, after-school lecture no less!), so I thought I'd share it in case anyone else wanted to use/adapt it.


Powerpoint file                     Worksheet

It contains:

• Prompt to discussions about the difference between large numbers and infinity
• A proof that 1=0 using infinity
• Modified Zeno's paradox
• Fractals
• Fractal-like behaviour in nature and its uses
• Puzzles using infinitely nested shapes
• Introduction to the concept of calculus and some of its uses
• Links to (excellent) videos about infinity

P.S. There are notes inserted in the slide-comment section of the powerpoint that explain how I presented it.

P.P.S. The speed slide refers to this distance-time graph creator, which I used to briefly introduce the concept of Calculus.

P.P.P.S. There are so many links at the end because there were so many topics I didn't have time for (Different types of infinity, Infinite/fractal dimensional space, etc.)

How computer games can teach complex skills AND make you feel like a bad-ass

Below is a short video playlist showing how a new game mechanic, Rope Arrows, is introduced and then used in Tomb Raider:

Video 1 - You first get rope arrows and the game shows you an example of how to use them.

• Notice how the gameplay does not stop.
• There is good reason for introducing the mechanic now; you just found the rope and there are several immediate (and fun) uses for it.
• At no point is this dull; you are learning a new skill, but using it makes your character feel instantly more powerful. See how excited the guy is, first when he realises that he has a new tool to play with, then when he discovers another place he could use it: "Oh, this is awesome!"

Video 2 - Over the next hour or so of gameplay there are multiple basic puzzles using solely rope arrows (but in different ways)

• No step-by-step tutorials here, you just try things out in a low-risk setting and see what happens.
• The puzzles very slowly rack up in difficulty and introduce new ways to use the rope arrows.
• Many of the puzzles are optional and for extra rewards (makes you feel like you are a good player).

Video 3 - A later puzzle involving rope arrows and earlier learned skills

• This is a fairly complex puzzle with multiple steps and multiple pre-introduced skills used in combination (although still a pretty early puzzle).
• Throughout the rest of the game you use rope arrows along with other learned skills. Many other skills are then added and introduced in a similar way.
• Each puzzle requires you to think about how you can use your learned skills in the environment, what the effects will be, and how you can combine these effects.
• The pay-off for solving the puzzle is fully satisfying, excellent cinamatography - you feel like a bad-ass!

Conclusions

This is not how maths games (currently) work: either they focus solely on practicing a pre-learned skill or include a dry written explanation of the concept (probably copied from some revision website). This is because maths games are not made by people who understand much about computer games (and not always much about teaching), and their budgets are miniscule in comparison to the blockbuster franchise of Tomb Raider.

Focusing of practice does serve a purpose to a certain extent, but there are some problems:

• As you focus on one skill, all the problems become exactly the same (and easily gameable)
• It is difficult to incorporate any feeling of progression or mastery (one of the main rewards all computer games use)

So can maths games teach maths in this way? As a gamer who now possesses a wealth of completely useless knowledge of moves and button combinations and tactics, I am inclined to say yes. There are obviously big differences between what a game will teach and what is included in the standard maths curriculum, and taking the steps to understand how to intuitively introduce these topics will not be easy. However, there is an online company who seems to have solved at least part of this puzzle. More on that company and my thoughts next week. Stay tuned!


Bonus - See how the gravity gun is introduced in Half-Life:

Interactive: Live Distance-Time Graph Creator

For a recent gifted and talented session I ran, I created This Excel File, which creates a distance time graph live. I gave pupils a pre-made distance time graph and pupils had to recreate it by walking across the room. I used it last week and it worked really great; pupils loved getting up and trying to time their movement. I'm sure it could be used in plenty of other ways too.

 

Setting up the room

• Make a clear path along the length of the room.
• Lay strips of tape in parallel lines spaced one meter apart across the length of the path.

Using the file

Open the file and click 'Enable Content' at the top.

Sheet1 - Creating the distance time graph

• Press reset to delete old values and set the view.
• The 'S' Column contains the timing of the 'Original' line. Change these for a different target graph. (Edit: Times have to be written in this format to work - hh:mm:ss)
• The 'T' Column contains the distance values that are shared between the 'Original' graph and the pupil graph. Change these if your room is a different size.
• Press prime to activate the distance time graph creator.
• Once primed, press enter (on the numeric keypad) every time the student steps across each line of tape, and a new point on the graph will created. (Note: first point created is at (0,0) and also starts the timer)

Sheet 2 - Working out speed
 

• 'View' sets the view
• Each button below this creates the triangles needed to calculate average speeds, but currently must be clicked in order (from top to bottom) for them to work. I will fix this eventually; at the moment the file is limited to my one use session, but I'd love to expand this to be a more multi-purpose graphing tool.

Great Posts of February

To celebrate finally figuring out how to share my starred posts on Google Reader (see the new widget in the sidebar), and to make this feature more prominent, I will also describe some of my favourite writings each month. This one is for February up until now:

• Differentiating Maths Homework by I Speak Math:  A list of great links to posts from teachers describing their methods for differentiated homework.
• Visualizing Concepts by I Hope This Old Train Breaks Down: Some examples of how Mimi makes concepts more concrete by introducing them visually/kinaesthetically.
• Oscars 2013 by Yummy Math: Using the flawed 'top grossing movie of all time' stat to motivate learning compound interest.
• Two Tens for a Five by f(t): A well considered way to introduce irrational numbers.
• The First Relativistic Video Game by Starts With a Bang: Actually from ages ago, but this is amazing! An open-source video game which allows you to experience relativistic physics. Try it out!

Back to Basics: Intervention

My head of department recently showed me a chart she's created to help with intervention for pupils who are working below their target grade. This is what they look like:

The idea is that you put each student who is below target in to a ring of the concentric circles. You also give them codes (from the second sheet) as to why you think they haven't made their targets yet. Now you have a list of suggested intervention strategies you can use!

Some people in the department felt that this was too basic and didn't really give them anything they didn't know.

For me though, I think its really good to have all these strategies written down somewhere as a reference or a checklist of things to try out. Maybe its because I've moved around so much from school to school, but I think that this is an area where I could definitely improve. In future I plan to use these sheets to review the interventions I have done at least once a term.

Are there any other strategies you would add? Is it a little too formulaic? I already changed some of the wording of the sheet to make it a little less algorithmic.

Lesson Sketch: Who Wrote This?

I'm excited about this lesson idea! So excited that it had me tweeting J.K. Rowling and Stephen Fry (though I didn't get a reply yet from either). I hope you like it as much as my pupils did last week.

Act 1

Get pupils to read this letter:

Dear ___________,

I'm so sorry to hear that life is getting you down at the moment. Goodness knows, it can be so tough when nothing seems to fit and little seems to be fulfilling. I'm not sure there's any specific advice I can give that will help bring life back its savour. Although they mean well, it's sometimes quite galling to be reminded how much people love you when you don't love yourself that much. I've found that it's of some help to think of one's moods and feelings about the world as being similar to weather:
Here are some obvious things about the weather:

It's real.
You can't change it by wishing it away.
If it's dark and rainy it really is dark and rainy and you can't alter it.
It might be dark and rainy for two weeks in a row.

BUT

It will be sunny one day.
It isn't under one's control as to when the sun comes out, but come out it will.
One day.

It really is the same with one's moods, I think. The wrong approach is to believe that they are illusions. They are real. Depression, anxiety, listlessness - these are as real as the weather - AND EQUALLY NOT UNDER ONE'S CONTROL. Not one's fault.

BUT

They will pass: they really will.
In the same way that one has to accept the weather, so one has to accept how one feels about life sometimes. "Today's a crap day," is a perfectly realistic approach. It's all about finding a kind of mental umbrella. "Hey-ho, it's raining inside: it isn't my fault and there's nothing I can do about it, but sit it out. But the sun may well come out tomorrow and when it does, I shall take full advantage."
I don't know if any of that is of any use: it may not seem it, and if so, I'm sorry. I just thought I'd drop you a line to wish you well in your search to find a little more pleasure and purpose in life.

Very best wishes, ____________

Tell them that it was written by one of the following people:

J.K. Rowling	Lewis Carroll	Stephen Fry
Russell Brand	Franz Kafka	Fyodor Dostoevsky

Let them discuss who they think it might be and why. Ask if there are any we could discount straight away. (note: I feel that I should have more women up there. Can you think of any women authors that all the kids would know?)
They will talk about writing styles and uses of certain words and things they know about the lives of these authors (I've purposefully used people for whom it would be understandable that they would have written such a letter).

Act 2

We are going to try to use their writing styles to determine who wrote it. What sort of things could we measure about their writing styles? (Some) Possible answers:

• How often they use certain words/punctuation marks (make sure the words chosen are common enough to be in other writings and not specific to the context).
• Average sentence length.
• Average paragraph length.

We will also need some samples to measure these things on. I'd use the previews you can get on Google Books:

• The Beadle and the Bard by J.K. Rowling
• The Life and Letters of Lewis Carroll
• The Fry Chronicles by Stephen Fry
• My Booky Wooky by Russell Brand
• Letter to my Father by Franz Kafka
• Notes from the Underground by Fyodor Dostoevsky

This task works best if each group measures and compares a different word or aspect of the writing as it creates for some great discussions about which measurements hold more weight and why.

Act 3

Write a table of the techniques each group used and the conclusion they made from it. Decide as a group who we think it is. By this point my students were desperate to know who wrote it. Talking points:

• Accuracy and how we could have obtained better estimates.
• Which techniques should hold most weight? Why?
• Why is "The Beadle and the Bard" probably not as useful to use as "The Fry Chronicles"?
• Where might this technique be used outside the classroom?

Reveal the answer: Tadaa!
After this you might also want to show how historians have used this exact same technique (and its name: Stylometry)!

Final Notes: When I used this in class, with student written essays rather than a famous letter, it worked exceptionally well. As I said, pupils were desperate to know who wrote the essay by the end. It fell down a little at the end as I didn't have a good way to show the answer. Making it about celebrities (I think) makes the answer more interesting.
The reason I tweeted Fry and Rowling was to try to get a school essay from them (the story would be that I have an old school essay from a celebrity and I could sell it if only I could remember which of these people had written it). This letter works well and is a great and interesting bit of writing in itself. The only problem I have with it is that students could quite easily Google part of the text and get the answer.

What about you? How could you improve it?