Posts Tagged ‘maths’

Australian Poverty Line

October 17, 2016

Recent reports of 3 million Australians below poverty line (where defined as below 50% of median income) – currently $426.30 per week for a single person – have started some public response. One person commented online that increasing welfare wouldn’t help, as it would drive up the average income and thus leave them still below par – another voter who does not know the difference between mean and median. Depressing that they can vote…

My immediate thought was different: have a major depression, and weaken Unions so more workers join the 32% of below-current- poverty-line whose main income is paid employment. Then the dole will be above that definition of poverty, while the executives stay on salaries giving over the poverty level weekly income per executive hour!

To compare with cost of basic needs: The March 2016 Henderson poverty line for a single person, including housing, is $425.61 for a single not in work, $524.89 for a single in the workforce. (The Henderson poverty lines are based on a benchmark income of $62.70 for the December quarter 1973 established by the Henderson poverty inquiry. The benchmark income was the disposable income required to support the basic needs of a family of two adults and two dependent children. Poverty lines for other types of family are derived from the benchmark using a set of equivalence scales. )

Australia’s Newstart Allowance (single person over 22  unemployment benefit) currently is at best about $335 per week, including rent assistance, and the Government is proposing to cut the Energy Supplement from it – about $8 per week. That is why I keep calling for those on welfare to have the right to surrender 90% of their income for guaranteed, supervised basic living provided by the Government.

Literacy and maths geek : QED

September 30, 2016


Cruz Iowa “big victory”?

February 7, 2016  said “Ted Cruz won a big victory Monday night at the Iowa caucuses.”   Most Australian media had American talking heads referring to a clear victory and Donald Trump coming second, with little talk of Rubio.

From, Marco Rubio took 23.1 per cent, Mr Trump 24.3 per cent and Mr Cruz 27.7 per cent of the vote.

Less than a 5% difference?  In polling terms, that’s experimental error.  In USA political terms, at the start of the long chain of preliminaries in  other – less farm-based – States, this is neck-and-neck.

I think the media have not done a good job of reporting here.  We have the right to feel insulted, and the responsibility to wonder about their hidden agendas.



Do the politicians think we have no memory? Part 3

March 23, 2014

After an Australian election, if  one party gets a majority of the whole population vote but another party wins the majority of seats the losing  politicians regularly grumble, throwing around words like “gerrymander.”

Politicians say they want schools to teach students to understand and value our way of government.  They say they want schools to emphasise teaching of history, and it is an important part of our history that a great deal of care was put into setting up our system, which started peacefully and by negotiation well after the hasty and violent starts of the main European countries and the USA.   They say they want these things in the curriculum,  but I wonder whether they want voters to remember their schooling when they come to vote.

Background  to  the Australian Electoral System

(Skip this if you know it already)

It was a deliberate choice to have States’ federal Senate numbers equal regardless of population and representing proportional votes within each State, to prevent the tyranny of the majority.   They were certainly influenced by John Calhoun’s ideas on concurrent majority as an approach to the problem, ideas still discussed this century .   It was also a deliberate choice to have each voter  have as many preferential votes as there are candidates up for election in the State,¹ a change made in 1949, even though the mathematics and vote tracing were horribly curly in the days before computerised  counting.    A voter may vote for all one party first, or one Green, one Independent, one Labor, one Liberal, and one Euthanasia party candidate, then mix up the remaining candidates in any order as long as each candidate has ves preferred number on the paper.  If a candidate has more first preferences than ve needs (one-sixth-plus-one of the votes is the quota if there are 6 seats), ves surplus votes are distributed as first preferences in proportion to the preferences of the voters who gave ver the votes.  Candidates who get less than the fraction needed to get a seat are knocked out from least votes up, and at each step the loser’s votes next preferences are distributed and the scrutineers check whether someone has got the quota.    (Messy!  I’m not making this up – check with the Australian Electoral Commission)  No wonder they introduced “Or you can tick one party’s box and we will distribute all their preferences the way they have told the us to.”

It was also a deliberate choice to have each House of Representative seat linked to its own area (and electorates other than islands are single patches of land), and that the voters from that area  vote  for  individual candidates as individuals, though the candidates  could ally to parties.  That way, local interests could be well represented by someone known to the locals.   Also, in each area, the voter has preferential votes as in the Senate – so that if they like Alan but would rather have Jan than Ursula if they can’t have Alan, they can try for Alan but know that Jan will get their vote if he fails.  They just number the order of preference in the candidates’ boxes.  This means that you don’t get someone hated by 60% of the electorate into the seat just because the 60% have slightly different ideas about the best way to do things and vote for 3 other candidates first.  If they all prefer a 4th to the 40%er, they get their way.


For philosophical reasons, State governments have been selling off State-owned housing in expensive areas, buying housing in less expensive locations,  and subsidising private rentals for those in need – who can seldom get private rentals in the prime locations.   In addition, those short of money sell out of high-value areas to free up the money, and the wealthy seek houses close to well-known exclusive schools and other valued social resources.  This has led to the service-providers (shop assistants, teachers, police, cleaners, etc) having to travel long distances to work, and tertiary students having to travel hours to their studies, with the associated travel costs – while the wealthy are within easy foot or  public transport access of resources.  This is fair in the  eyes of those benefiting from the user-pays  approach, and they see its good points:  after all, if the State provided enough low cost housing in the  upmarket areas, the dregs of society would lower property values.  An additional benefit is that the local State schools have a better class of student and parents and thus better outcomes than in the more difficult suburbs..

You got over half the total but not enough seats.  Problem?

True, there are many reasons people vote their different ways, but let’s pretend that wealth-aligned interests are usually enough to swing the vote.  Let us assume that the electoral boundaries are fair, with pretty similar numbers in each electorate, and thus there is no real gerrymander.  Our Electoral Commission does work at being fair that way.

Pretend there are 10 electorates.

Rich party has 90% of the votes in each of 4 electorates.

Poor party has 60% of the votes in each of 6 electorates.

% of total voters                   %  of total vote             seats / 10

R 36%        P 4%                                 40%                      4

R 24%        P 36 %                              60%                      6

total votes by  party                 R 60 %        P 40 %

Total seats by party                 R   4             P  6

Don’t complain.  This was part of the design of the Australian system, deliberately included to control concentrated power groups with regional agendas inimical to the wider society.   This is in the curriculum – the intersection of History with Society and Environment.   Why don’t the journalists call the politicians on this, rather than just quoting them?

I am so annoyed that I am going to shout.  

If  you want a greater proportion of the seats, have a better distribution of your supporters across electorates. 

A good start would be:  Get out of your enclaves of power, and make housing available for the “lower orders” closer to the places that they work.  If you can’t stop the worsening inequality, at least reduce home address’s value as a predictor of socioeconomic status.  


¹ I know, it is really “a preferential vote” but they used be allowed to number only a limited number of preferences and I wanted to make the distinction .

Graphing climate change: an activity examining persuasive graphing / writing

October 31, 2011

The New Scientist had a graph, reproduced below.  It struck me as a good teaching example, both as a source for examining the effects of presentation choices on interpretation and as a trigger for discussion on the distinction (if any) between persuasive writing and biased writing.

The  graph’s title as published was vague, as it related the graph also to an added range of possible effects  cut from this image.   I think of this graph as “Temperature increase in °C for given CO2 concentration, by climate sensitivity to CO2 in °C per doubling of CO2 level”


  • What does this graph show ?
  • How does it make you feel about increasing CO2 levels?
  • What title do you prefer at present?
  • Can you imagine the following re-graphing:

– Put temperature increase (the dependant variable) on the Y-axis and CO2 (the independent variable) on the X-axis  (This is, after all, the customary arrangement.)

– double the size of the scale for Atmospheric CO2, so that “100” is as far from the zero point as “200” is at present;

– place the in-graph  labels for “likely Scenarios” “Outside possibilities” and “most likely scenario” so that they  are in the same visual spaces as in the original.

  • Make the new graph you tried to imagine.   It does represent the same data.  Look at it:
  • What title would you give it?
  • Does it make you feel the same way about increasing CO2?
  • Compare the two graphs.  Which do you prefer?  Why?
  • New Scientist has been accused of being biased  in its presentation of the science concerning climate change.   Does the published graph  cross the line between science reporting and biased writing?
  • Is there a distinction between persuasive and biased writing?
  • Is there /should there be  a line between science reporting and persuasive writing?  Why / why not?


Teachers aren’t in it for the money: doing the maths

June 27, 2011

From time to time various Australian States claim to have “The best paid teachers in Australia”.  I think they would be more honest to claim “The least worst paid teachers”:

Take a four-year trained Primary teacher in Western Australia on a starting salary of $56 122, rising to $61 567 after a year of service (the latter is the starting salary for a five-year trained teacher).

I will compare ves effective hourly payment with that of a starting-level Education Assistant/Teachers Aid on a starting salary of $33 484 / 19.75 per hour  (2010 WAIRC 00742)

They have the same school hours, but the teacher is expected to be present before and after school to open and close the class, meet parents and students, do required bureaucratic work, deal with emails, write up the daily workpad,  and so forth.  In addition, a teacher has to work out of school hours to complete important duties such as: prepare and change programs and lessons to meet the students’ needs; attend professional development; represent the school in out-of-hours activities; do professional reading;  explore, collect, and document useful materials for  activities;, contact parents; mark assessments; and write student reports.

I have heard teachers say that they regularly do over twenty hours per week out of school – that is, beyond the minimum of five hours at school before and after class, and the DOT time in school.  This does not include preparatory work done over school “vacations”.  First-year teachers are told to expect much longer hours, often eighty hours a week.  To be conservative, let us assume the new teacher is a genius and can get away with a term-time weekly average of the five hours at school and twenty-five out of school, and donates vacation time.

If teachers were paid the extra school hours at single time and the away-from-school  hours at time-and-a half,  and paid this  at the EA hourly rate of $19.75 per hour; if this were paid for 39 weeks a year; and if a first-year teacher received the EA base annual of $33 484 for the school year hours the EA attends,

 Then  the new teacher would get

$33 484 + (39 * (5 + 25 * 1.5) * $19.75)

= $33 484 + $32 735

=$ 66 219

So, for the first two years (ending on $61 567), the new genius teacher can expect an effective hourly rate below that of a new Education Assistant.  Less organised and gifted teachers would receive much less.

Note that I have made conservative assumptions on hours worked;  have had the teacher donate some holiday time;  have not made deductions for the purchse of professional reading, professional training,  and class materials;  and have not included double time for work on Sundays, although many conferences are scheduled for weekends and most teachers do some work on Sundays throughout term.

Teachers aren’t in it for the money.

It reminds me of the story of a Martial Arts master who received many gifts from those he taught, gifts according to their income.  A very rich man said: “Come and teach me, I’ll pay you well.”  Sensai was offended and said “You can’t afford to pay for my teaching!”

As I noted previously, Australian teachers’ pay relative to Average Weekly Earnings (and relative to backbenchers’ pay) has dropped massively over the past thirty years.  If the schools can’t afford to pay teachers more per work hour than unskilled workers, then Teachers’ Unions must demand that society as a whole regulates for extreme courtesy¹ towards teachers:  students, parents, and bureaucrats must recognise teachers’ years of training, specialist knowledge behind decisions, advanced diagnostic skills, and out-of-school workload,  and the complex challenges teachers face, and reflect this in their approach to individual teachers on individual issues.  

You can’t afford to pay the teachers what they are worth.  Recognise the imbalance in  social obligation:  you owe them, big time.


¹: Courtesy is the behaviour, respect is an internal state.  It is impossible to coerce people to have respect – one can only coerce them to produce the socially defined behaviours indicative of respect.  I believe that it is time for some coercion.

Teaching Mathematics: not wrong, just differently right

March 4, 2011

In the usual order of presentation of fractions, the concept of ratios is left until the upper primary years.  I am not sure that approach is wise.

I have noticed that a fair proportion of those learning fractions  (even at 11 years old)   if asked to write the fraction shown by  [X X X X X O O O] (really by a corresponding set of black / white circles, an image which I am too lazy to insert here)  will write either 3/8 or 3/5.

A computer would mark both of these as incorrect.

Many teachers explain to the class that they expect students to count the coloured circles as the fractional part, and accept the 3/8 at least once, but mark the 3/5 as wrong.   Few explain that the student has seen the ratio relationship – in the time constraints of class they say  just that it is not the fraction.  The unintended lesson hits – the ratio-perceivers’ perception is flawed, they do not “see” maths.

Consider a different approach:  ask the class to “write the fractions shown by the image”.  Touch on the darkened ones traditionally being the fraction numerator, and thus the one they should use for teachers.  Welcome the 3/5,  or 5/3,and explain that the student has noticed the ratio relationship – but that it should be written as 3:5, not 3/5 (we traditionally write the smaller first.)   By using the vinculum we are saying that the bottom represents one set  divided into that many parts and the top shows how many of these parts are in the subset we are examining, whereas the ratio (:)  form says that the colon-separated sides add to make the whole.  (This format allows for cooking  ratios such as 1:1:2, basic biscuit and cake weights of butter / sugar/ flour.)

Aside: a topic for another time – like the technical terms “phone” ,”phoneme”, and “morpheme”, does the term “subset” belong in class before upper – primary?

Ask the class whether they want to investigate  ratios as well as fractions, even if they are not on the curriculum for the year (a conspiracy of learning [1]).

What can this mean to the students? The non-standard forms are seen as mathematically sound, but not the traditional (and thus  preferred) form  – just a matter of presentation, and thus not a big error.    The ratio perceivers have opened up an option for the class to explore – their perception is affirmed as being of an important mathematical relationship, though it is not the fractional one.  They just have to learn which label we use for which relationship – again, a matter of presentation – and they can go home saying “I saw ratios, and most of the others hadn’t noticed them!” .  Isn’t that better than “I suck at Maths!”?

In addition, there is the opportunity to re-emphasise that the key to fractions is that they are the relation of a current subset to the theoretical unit set, related by division of the unit set (into the number of parts shown in the denominator) and multiplication of the unit fraction by the numerator.     This is a difficult concept, but can be examined using the excellent physical approach to fractions reported by Doug Clarke [2], explicitly forming fractions by sharing (i.e. division and multiplication).  This is the key to grasping equivalent fractions  rather than “doing the sums” without understanding.

Aside:  this links to the concept of fractions as pure numbers  (How big is a quarter?  How big is one?  One what? 1/4 cup is larger than 1/2 teaspoon.)   Eventually, it also links to the concept of  percentages as special fractions where the unit set is divided into a hundred parts, so we can have fractions or decimals as numerators.  The excellent and widely used First Steps in Mathematics/Number [3],  says that percentages are special ratios – and indeed, being a subset of the special ratios called “fractions”, they are – but I think it makes more sense to link them in the first place to the immediate superordinate set.

1. Louden, W, Rohl, M, & Hopkins, S. (2008).  Teaching For Growth:  Effective teaching of literacy and numeracy.  Department of Education and Training, Western Australia

2.  Clarke, D.  Fractions as division: The forgotten notion?  Australian Primary Mathematics Classroom 11 (3) 2006

3.  Willis,S., Jacob,L., Powell,B., Tomazos,D. & Treacy,K. (2004).
“First Steps in Mathematics: Number – understand whole and decimal numbers, understand fractional numbers.” Port Melbourne, Victoria: Rigby / Harcourt Education, 2004


From Polygon Song to Michael Jackson

January 25, 2011

Thoughts on “Polygon Song” by Peter Weatherall

I was looking for ways to help children remember the polygons’ names, and was struck by a performance of “Polygon Song”,  (with students in costume singing and acting the parts) at an assembly.   At first I thought this was an unexceptionable, really  good teaching song.

Then I considered the subtext:

“Nah. nah nah, nah, nah, just a boring Square.

I wish I was a pentagon, but I am just a square.
I wish I was a pentagon, but I am just a square.
My sides equal four, but if I had one more
I’d be a pentagon and not a square

Nah. nah nah, nah, nah, just a boring Square.”

So far, not bad.

The song continues, with desire to have more sides (triangles are not mentioned) and dissatisfaction with the main player’s squareness repeated.

So, where does it go – to the amazing properties of the tesseract?  To the practical designer choosing the square above the rest?

“(Play act  the square going behind a screen with a surgeon, and bits of paper tossed out)

Well now I am a decagon, and not a square.
Now I am a decagon, and very rare.
I won’t complain again ‘cos my sides equal ten,
I am a decagon, and not a square.

When I was just a square, I thought it wasn’t fair,
So I had surgery to my geometry ….
Now look at me!

Nah. nah nah, nah, nah, not a boring Square.”

The subtext I see is :  strong social acceptance of plastic surgery to change a standard but socially less valued appearance – with bust enhancement and nose reductions being normalised, the full Jackson option anyone?

I think I may still  use the song, but as a piece to introduce the idea of subtext to older students.