Category Archives: Education

This category contains my posts and comments in relation to education.


So how does learning really happen… Continue reading


Seeing someone leave the nest or walk by themselves for the first time is inexplicable, unless you’ve experienced it for yourself. Being in a learning environment encourages you to continue learning. Having something to care about and share is, almost, … Continue reading


Despite your training and experience, everyone you meet tells you what you are doing wrong and how you can do it better. When people find out you are a teacher, the first thing they mention is the long holidays and … Continue reading


This is just a short post to explain about the 101 manipulative lessons with LEGO® page. By subscribing to comments, you can stay advised of new additions. I will also advise of updates now and again on the #mathchat hashtag in Twitter. Continue reading


One of the analogies I have used in past CPD sessions about teaching and intervention is connected to the idea of maps, car-navigation systems, driving and being a passenger.  I generally use it to illustrate levels of intervention, but it … Continue reading

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 …
[insert your favourite here!]

Many of the participants in #mathchat recommended various online manipulatives, as well as more traditional ones!, during the discussion What are ‘brilliant’ activities with manipulatives?, 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… 😉


imageID 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

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: A Tweet from @MariaDroujkova asking for how it could be usedAnything 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!
(added 6th September 2010)

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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


I want to talk about normalized percentages. “What makes these irrational?”, I hear you say. Their application to examinations and assessments is what.
The big change for students completing A-levels this year was the introduction of the A* – an attempt to increase the value of the ‘gold standard’, as I see it. Unless of course you are taking Maths, Further Maths, Further Maths (Additional), Statistics, and Use of Maths AS. No AS will have the A* grade available.
“We can’t rely on A-levels any more.” [Sir Richard Sykes, Rector of Imperial College quoted in The Sunday Times].
Lower tier students (oops, I mean foundation…) are expected to work proportionately harder to achieve the higher grades than those doing the higher tier.
Does a C at GCSE mean they did enough to get 60% of each question correct but not reach the final answer of any of the problems? Or am I just being silly?
I have one word which I feel would address most, if not all, of these issues: portfolios.
Continue reading


Having never had to run a department, let alone a school, I wonder how apprehensive the management teams are about sudden changes. I also wonder how those of us who are not in management positions can best support our colleagues.

Whilst many situations may be new to the management of schools, people have been managing businesses for years: running meetings, controlling budgets, hiring and firing people, etc. Perhaps the time is ripe for schools to get some wisdom from local businesses, since some of their students may be future employees or business owners. The way things appear to be headed in the UK, to me as a recent returnee, seems to be that school = small business!

Try to empathize with the people around you, whatever their position or status, if you can start to see the world through their eyes, then times will certainly become more interesting and less worrying.

Aristotle summed it up in one word: φρόνησις Continue reading


It seems a natural thing for educators who are like me to spend time reflecting on themselves and their work, and then for them to take action on their reflections.  In my own case, the reflective process is also combined … Continue reading