Sunday, January 29, 2017

fork in the road options and Direct Instruction

An attempt at a pithy critique of Direct Instruction:

In Direct Instruction there is no script for those who depart from the script or who desire to write their own script.

I'd rather be a Robert or a Lauren, than an Alice.
Alice in Wonderland

“One day Alice came to a fork in the road and saw a Cheshire cat in a tree. ‘Which road do I take?’ she asked. ‘Where do you want to go?’ was his response. ‘I don’t know,’ Alice answered. ‘Then,’ said the cat, ‘it doesn’t matter.”
- Lewis Carroll
The Road Not Taken

Two roads diverged in a yellow wood,
And sorry I could not travel both
And be one traveler, long I stood
And looked down one as far as I could
To where it bent in the undergrowth;

Then took the other, as just as fair,
And having perhaps the better claim,
Because it was grassy and wanted wear;
Though as for that the passing there
Had worn them really about the same,

And both that morning equally lay
In leaves no step had trodden black.
Oh, I kept the first for another day!
Yet knowing how way leads on to way,
I doubted if I should ever come back.

I shall be telling this with a sigh
Somewhere ages and ages hence:
Two roads diverged in a wood, and I—
I took the one less traveled by,
And that has made all the difference.
- Robert Frost

Saturday, January 21, 2017

the pitchfork solution to world inequality

Each year Oxfam delivers a report about how the inequality in the world is getting worse and how this needs to stop. This year, I was encouraged by a marginal note from Nick Hanauer, one of the Super Rich, who warns his fellow billionarie's that:
‘No society can sustain this kind of rising inequality. In fact, there is no example in human history where wealth accumulated like this and the pitchforks didn’t eventually come out.’
- the pitchforks are coming ... for us Plutocrats
Now, what did that Oxfam report say?
  • Since 2015, the richest 1% has owned more wealth than the rest of the planet.
  • Eight men now own the same amount of wealth as the poorest half of the world.
  • Over the next 20 years, 500 people will hand over $2.1 trillion to their heirs – a sum larger than the GDP of India, a country of 1.3 billion people.
  • The incomes of the poorest 10% of people increased by less than $3 a year between 1988 and 2011, while the incomes of the richest 1% increased 182 times as much.
  • A FTSE-100 (Financial Times Stock Exchange 100) CEO earns as much in a year as 10,000 people in working in garment factories in Bangladesh.
  • In the US, new research by economist Thomas Piketty shows that over the last 30 years the growth in the incomes of the bottom 50% has been zero, whereas incomes of the top 1% have grown 300%.
  • In Vietnam, the country’s richest man earns more in a day than the poorest person earns in 10 years.
more details here

Oxfam report: AN ECONOMY FOR THE 1%
the strengths and weaknesses of capitalism
Land of the Free, Home of the Poor

Sunday, January 15, 2017

RAMR Deadly Maths: Adding Fractions

The RAMR cycle originates from Chris Matthews, an indigenous man who has a PhD in maths.

RAMR is a new model for teaching maths, or at least, new to me. I've appended a link to a video at the bottom where Tom Cooper from the YuMi Deadly Centre at the Queensland University of Technology (QUT) explains the RAMR cycle. I adapted his talk to a lesson I developed about adding fractions. YuMi is a Torres Strait Islander word meaning "you and me" and Deadly is an aboriginal word meaning smart.

The RAMR acronym stands for Reality, Abstraction, Mathematics, Reflection as illustrated by this graphic:


Teach maths the way it is created or invented. Start with a problem. The problem I posed to the class was how do you add one half and two thirds.

The elements recommended in this Reality phase of the cycle are start with a real life problem, draw on local knowledge and construct kinesthenic activities.
PIZZA DIAGRAMS half and two-thirds

Present this problem as a real life exercise. John eats half a pizza. Jermain eats two thirds of a pizza. How much pizza have they eaten altogether?

Bring real pizza into the room and cut it up. It is a recommended part of the RAMR cycle that the teacher constructs such an activity, preferably kinaesthetic.

Prerequisites: The teacher needs to be aware of prerequisite knowledge required for the problem. In this case the denominator (bottom number) tells us how many equal parts the pizza is divided into. The numerator (top number) tells us how many part of each pizza are being eaten. This had been covered in a previous lesson.


Abstraction means moving from the real world (a pizza cut into various pieces) to a representation of that reality in words, pictures and / or symbols.

We have already begun this above by using the words, symbols and pictures for half and two-thirds. In practice the various phases of the RAMR cycle overlap as well as having some distinctiveness.

One aspect I need to improve on is that of adding in creativity by inviting students to create their own representations of fractions. I didn't do this in the class but later when running the session for trainee teachers it did energise the session with some imagination.

Can you develop your own representation of fractions?


In this phase of the cycle we stress the formal language and symbols of mathematics, practice the concepts a lot (most students need lots of practice) and connect to other maths ideas that have been taught earlier.

When asked how to add ½ plus 2/3 many students will add the numerators and denominators to get the answer 3/5ths. Explain why this is wrong. You don't add denominators because they don't represent something that ought to be added to solve this problem. Rather they represent how many pieces each pizza has been cut into.

The trick to solving this problem is to cut both pizzas in such a way that the parts are equal. This can be solved either
(a) visually or
(b) arithmetically by multiplying the denominators or
(c) by finding the lowest common number in the two times (2, 4, 6 ...) and three times tables (3, 6 …)

So, we divide both pizzas into 6 equal parts

Now we can add the fractions 3/6 + 4/6 = 7/6 = 1 and 1/6

The denominators are not added since they represent how many pieces the pizza was cut into. The numerators are added since they represent the parts of the pizza which are eaten.

Transforming ½ into 3/6

How do we get 6 from 2? Multiply by 3.

Now if we multiply the denominator by 3 we then have to multiply the numerator by 3. ½ x 3/3 = 3/6

This doesn't change the value of the fraction since 3/3 = 1 and multiplying by 1 doesn't change the value of the number. The technical name for this is compensation, the numerator 3 compensates for the denominator 3, etc. This technique is really valuable and can be used over and over again in the future, so it needs to be reinforced. Multiplying by 1 doesn't change the value of a number.

Repeat this process to transform 2/3 into 4/6 by multiplying by 2/2

More Prerequisites: (which were covered in earlier lessons) Equivalent fractions: ½ = 2/4 = 3/6 etc.

Improper fractions (7/6) and mixed numbers 1 and 1/6


The goals here are to:
  • set problems that apply the new idea back to reality
  • enable students to validate and justify their own knowledge
The concepts being covered can be extended at any point throughout the cycle. It doesn't have to be confined to the end.
Extension: Represent 7/6 or 1 and 1/6 on a clock? Answer: 70 minutes or 1 hour and 10 minutes.

Inverse: How else could the pizza have been divided b/w John and Jermain to get the same answer? ie. What other two numbers (fractions) would add up to 7/6 or 1 and 1/6?

Generalise: How could you add any two fractions? Looking for answer here about achieving a common denominator.

For the class I was teaching I found that they struggled to understand what was required for the inverse section but with prompting they got it. Overall, I was happy about the response to the challenge posed by the Reflection section.


Maths textbooks are notoriously dull. Direct Instruction as developed by Rhonda Farkota (link to her PhD thesis) is very useful. But how do we further develop maths curriculum in a rich way once the basics are established?

I found that the RAMR cycle challenged me as a teacher to develop my delivery further. There were some elements in the cycle which made me think hard before I could deliver them. I did find that students responded well to those elements of the cycle in which I harboured a hidden belief that they may not cope with. The cycle integrates real world, creative elements and traditional elements of maths in a manner which I found very satisfactory. I think it is a very good model.

Is this maths which incorporates indigenous culture or simply good maths teaching? Good question! I think both but mainly I lean to the latter view. But really good teaching adjusts itself to take the individual needs of all the current students in the class into account. This requires more analysis and thinking. LINKS
Tom Cooper explains the RAMR cycle here: Professor Tom Cooper - YuMi Deadly Maths

Wednesday, January 11, 2017

Some books I am reading in 2017

Appiah, Kwame Anthony. Cosmopolitanism: Ethics in a World of Strangers (2007)
Dennett, Daniel. Intuition Pumps and other Tools for Thinking (2013)
Engelmann, Siegfried and Carnine, Douglas. Could John Stuart Mill have saved our schools? (2011)
Farrington, Benjamin. The Philosophy of Francis Bacon (1964)
Hofstadter, Douglas R. Metamagical Themas: Questing for the Essence of Mind and Pattern (1985)
Kenny, Robert. The Lamb Enters the Dreaming: Nathanael Pepper and the Ruptured World (2007)
McIntosh, Dennis. Beaten by a Blow (2008)
Minsky, Marvin. The Society of Mind (1985)
Osborne, Barry (Editor). Teaching Diversity and Democracy (2001)
Monk, Ray and Raphael, Frederic (editors). The Great Philosophers (2000)
Putnam, Hilary. Ethics without Ontology (2004)
Putnam, Hilary. Words and Life (1994)
Roughsey, Dick (Goobalathaldin). Moon and Rainbow: the autobiography of an aboriginal (1971)
Sarra Chris. Good Morning, Mr. Sarra: My life working for a stronger, smarter future for our children (2012)
Shapin, Steven and Schaffer, Simon. Leviathan and the Air Pump: Hobbes, Boyle and the Experimental Life (1985)
Van Fraassen, Bas. The Empirical Stance (2002)
Willingham, Daniel T. Why Students don't like School: A cognitive scientist answers questions about how the mind works and what it means for the classroom (2009)

Sunday, January 01, 2017

Donald Trump's twitter feed

I guess one of my new goals in life is to try to understand Donald Trump's twitter feed.

I think this guy gets it: Staking Out The Far Edge and he predicts that President Trump will continue to twitter.
As to the tweeting of the Donald, there’s little chance that his style will become more “Presidential”. He used that style to defeat 16 Republican candidates for the nomination and to defeat Hillary Clinton. Yes, his tone may change, the Presidency has changed the tone of every man I’ve seen take on the job.

But the tweet is far too useful to Trump to ever give up. It is his direct connection to the American people, one which cannot be changed, misquoted, or slanted by the media whether favorably or unfavorably. In addition, he employs it as a most potent negotiating tool, using it to good advantage in a number of ways including staking out the far side of a discussion. So I expect little change in his use of Twitter.