# Category Archives: Mathematics

This category contains my posts related to mathematics, as a teacher and user.

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So, you’re walking around the Sahara Desert, as one does from time to time, and you come across someone who’s very dehydrated. What do you do? Throw them in a swimming pool. Stick a funnel in their mouth and pour … Continue reading

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Detta är texten från min webcast, som lades fram för TeachMeet Framtidens lärande 18 maj 2011. Jag gjorde en origami session med ett universitet matematik klubb. En av utmaningarna jag ger är att vika en liksidig triangel från en kvadrat. En student använde … Continue reading

## What is this #anyqs thing?

Followers of #mathchat and #scichat may be wondering what this #anyqs hashtag is that people seem to be tweeting about.  Dan Meyer’s at it again!  For anyone who hasn’t heard of Dan (), here’s his much-viewed TEDxNYED talk.  He started off WCYDWT and the #WCYDWT hashtag, which he vigorously tries to defend as a brand.  That is an abbreviation of “What can you do with this?” – the prompting of “How much, how big, how long, how many…?” types of questions using a photograph or a video.

The #anyqs hashtag, which Dan launched on 5th May 2011, seems to me, at least, to be a refinement of the #WCYDWT idea.  Dan’s follow-up post Dissents Of The Day: Danielson, Pickford, Scammell says:

The point of the #anyqs challenge is to evoke a perplexing situation so skillfully that the majority of your students will wonder the same question (whatever that is) and the rest of the class won’t find that question unnatural or uninteresting, even if it wasn’t the first question that struck them.

The ‘rules’ are much tighter than #WCYDWT – one picture or one video with a duration under 1 minute, viewers should respond with the first (mathematical) question that springs to mind.  A few others have ‘jumped on the bandwagon’, I have set up my own pages to collect questions for myself, but this raises another question: “What do I do with all of the questions once I have collected them?”

It seems to me there are two main approaches you could take, once you have first tested the visual prompt outside your class:

1. Look at all of the questions and explore which can be answered, maybe choosing one or two to explore first or allow students to work on one or two of their choice from the full list; or
2. If there is too much disparity in the range of questions, then the visual prompt needs to be revised so that only one (or maybe two) question(s) screams out to be answered.

The first approach is probably a return to the confusion or flapping about, the “excessive cognitive load resulting from a unfocused problem space” Dan refers to in his response to a question about the difference between #WCYDWT and #anyqs (comment 14).  I suspect, judging by the comments, advice and friendly prods from Dan, that his intent is to adopt the second approach.

If  (when!) I get approach two working myself, it could/should lead to the situation where my own students are making #anyqs challenges for each other, with me doing the friendly prodding and giving advice.  If students can get caught up in designing these types of prompt, perhaps they’ll increase their awareness of the mathematics around them, and forget that maths lessons aren’t supposed to be fun… And if they don’t get to the stage where they can create their own challenges, maybe they’ll have fun doing mine… (ooh, did I just say fun again?)

## What’s it all about #mathchat?

I am assuming you are here because someone knows you teach mathematics and has got you onto Twitter as part of your professional development.  Or maybe you are a student or parent who is interested in issues related to mathematics and someone told you about #mathchat.

I have still to reach my one-year anniversary on Twitter, but probably a large proportion of those following me know me through #mathchat.  So what is #mathchat?  Basically, #mathchat is one of the many hashtags used by educators to hold discussions on topics of immediate relevance to them.  There is a whole raft of educational chats and hashtags on Twitter, but #mathchat concerns itself mostly with issues surrounding the teaching and learning of mathematics from beginning to end, K-death (for the North Americans!).

I am usually credited with founding #mathchat, but it’s probably fairer to say that I revived it in it’s current form.  The revival started because of a comment made by a middle school teacher who said he wished there was something for mathematics teachers which was like #edchat.  I got the ball rolling, we had a TwtPoll to decide on day and time, and I decided that the same topic should be discussed on a different day at a different time to allow more input or opportunity for those on both sides of the Atlantic to take part.  I’d like to see a third discussion in the Pacific region too – any takers in Aus/NZ?

What is the purpose of #mathchat?  This excerpt from the #mathchat wiki will give you an idea:

The aim is to provide a forum for anyone involved with Mathematics – whether as an educator, a student or an interested party – to discuss and share ideas about issues affecting them at this particular time.

## Some questions you may have:

• When is #mathchat? The new topic is introduced on a Thursday/Friday (we used to start at 11:30pm GMT on Thursday, but this was shifted to 00:00 GMT Friday, by popular demand).  This is now a fixed time, so when daylight-saving hours change, the time of the chat will shift backwards or forwards accordingly.  The same topic is revisited on the following Monday (starting at 20:30 London time, 19:30 GMT in Summer, 20:30 GMT in Winter).  The Thursday chat is one hour and the Monday chat is 90 minutes.  You can find your local time from the link on the #mathchat wiki.
• How are topics chosen? Topics are decided by a weekly vote.  The topics for discussion come from a list of suggestions in a Google Doc, and anyone is free to add further ideas to that list whenever they like.  Each of the scheduled discussions is archived, in case you missed it or want to go back and review it.
• What is a hashtag? Check out this video to discover what a hashtag is and how to use a hashtag.  At the designated time, you only need to add “#mathchat” to the end of your tweet to participate in the discussion.  Some applications will add the hashtag for you, you can find suggestions on the home page of the #mathchat wiki.  The TweetChat one is probably best for first-timers!
• What if I want help? The pages on the #mathchat wiki or the archive which start with “!” will give you extra guidance, or just send out a tweet with #mathchat and you should get a response fairly quickly.
• Who should I follow? You should definitely follow @mathchat to keep updated on new topics, voting and so on.  There is also Pooky Hesmondhalgh’s 20 Top Tweeters for Maths Teachers or you can follow any of the people on my #mathchat list.
• Can I only use #mathchat at the pre-set times?  No, absolutely not!  The pre-set times are just to discuss a single topic.  Lively discussions will often appear at all sorts of times during the day, particularly at weekends.  People use the hashtag to share links, their latest blog post and even the occasional joke!

## What makes #mathchat special?

Like most of the other educational hashtags on Twitter, the people who contribute to #mathchat are passionate about teaching mathematics and helping others come to terms with what is involved in learning mathematics.  What is really special for me, though, is that not long after its revival, #mathchat expanded beyond the ‘topic of the week’ to become a place where teachers and students can ask questions at almost any time of the day or night and get helpful responses, usually within 5 or 10 minutes…  If you want to know what #mathchat means, just send out a tweet “What does #mathchat mean to you?” and wait for the answers!  And if you have any further questions or need help, feel free to tweet me.  I’ll look forward to tweeting you some time!

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OK, before everyone starts screaming about sexism, I used to go to my father for help with my maths homework, and this post is a reflection on the problem (grand)parents of my generation and/or younger are facing, or going to … Continue reading

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“It’s the things we take for granted that we need to identify and question.” (Sir Ken Robinson in a speech to the RSA) Continue reading

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A tale told through twitter about connections – broken and repaired. For Maria Droujkova, Happy Math Storytelling Day! Continue reading

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## My first online manipulative – virtual origami

Compared with way-back-when, when I was learning computer programming by using Hollerith cards and making bootstraps (complete with chads) to accept the input from a keyboard and display it on a monitor as a 3rd-year machine-coding project, the idea of online manipulatives for algebra, geometry and so on were just a gleam in Wolfram Research’s eye!  I wonder how many students since the 1990s could imagine life without something like Mathematica or GeoGebra or the amazing virtual manipulatives from McGraw-Hill or …

Many of the participants in #mathchat recommended various online manipulatives, as well as more traditional ones!, during the discussion and I thought it was about time I stopped playing with them and tried to make one myself…

Now, what to choose?  I decided I wanted to do something virtually that could also be done in ‘real life’, with ‘normal’ classroom technology, being something of a 2.0 Luddite!  Since I am a great lover of both 折り紙 (or origami to you!) and LEGO, I decided I’d try to represent folding a square of paper by dragging a corner down.  That, of course, would have been sufficiently interesting in and of itself, but Mr Maths Teacher said: “You should be doing something educational with this, preferably related to Mathematics”…

So, enter Haga’s Theorems, more specifically the First Theorem. [OK, here’s the link in English if you must 😉 ]  芳賀 和夫 (Kazuo Haga) a retired university professor, presented his theorems at the International meeting of Origami Science and Technology. Subsequently, he published two volumes about what he named: オリガミクス or Origamics.

• オリガミクスI [幾何図形折り紙]
• オリガミクスII [紙を折ったら,数学が見えた]

Both are published by 日本評論社, but may be out-of-print or difficult to obtain.  Fortunately, his work is also available in English!  The mathematics becomes quite complex, and there is an interesting puzzle to solve too.  Definitely worth a read, if you’re a fan of origami and mathematics!

Basically, each of his three theorems says that a particular set of constructions can be used for dividing the side of a square of paper into any arbitrary rational fraction.  You can use this to divide a square into fifths, say, in three simple steps.  I’ll leave that for you to work out.  The table below should help…

Haga’s First Theorem is a neat little piece of geometry, which only needs Pythagoras’ theorem, and some algebraic manipulation to prove.  So, ideal for KS4, and for keeping Mr Maths Teacher happy… 😉

##### ID is always a rational number if CE is.

Let x be EC, then a number of other lengths are also rational functions of x.

For example:﻿

﻿﻿﻿﻿﻿

Table showing some examples of the generalized 1st Theorem

I was going to explain how I put the manipulative together, but the process I used was very similar to that described by Guillermo Bautista, so there’s no point in reinventing the wheel!  The differences between our two simulations are basically:

• I used a square, not a rectangle
• I used GeoGebra 3.2, not version 4.0
• I intersected AD, rather than AB, when creating the perpendicular bisector

I also skipped over a few other steps, when creating EFGH, but the overall effect is similar!

Click on the diagram showing the theorem to visit my GeoGebra page and download the manipulative!  Alternatively, you can access them from my Mathematics Resources page on this site.

Addendum: Anything to oblige, Maria!  I added a screenshot which you could use in conjunction with the table above to check things out.  This is also available on the Mathematics Resources page on this site.  The sides of the square were set to 360 to allow for many different fractions to be investigated (elevenths may be a problem though!).  I expect I will expand the manipulative itself to encourage exploration sometime in the future… Anyway, enjoy!

OK: Here is the dynamic worksheet if you want to try it out online!

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OBSERVATIONS

As a data-collection activity, I feel this was reasonably successful. For future experiments to be successful using Twitter, I think:

a more specialized hashtag would be better.
instructions should be in a tweet about 120-characters long to reduce chances of re-editing.
a longer time period could/should be applied with RTing at timed intervals to allow for global participation.
anyone who ‘hijacks’ the instructions should be contacted and asked to RT original as far as possible.
Google Reader did not retrieve all of the Tweets sent during the experiment, so it is best to have two or three archives available for analysis.
graphs and data charts provided by archive software will also contain extraneous ‘noise’, if people ask questions or discuss the activity using the hashtag.
any archive will probably need to be edited before analysis.
So, over to you now… let me know if you want me to take part in your data-collection activity! Continue reading