Transcript Giving maths a bad name - Learning and Work Institute England

Giving maths a bad name
Joan O’Hagan
[email protected]
07515702991
You want to carpet your
room.
How much will it cost?
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The “give maths a bad name” response:
Measure the room to the nearest centimetre.
Add on 10 cm each side for wastage.
Calculate the area, including the wastage bits.
Take that figure with you to the shop.
Look at some carpet and price it up using your area figure.
Ask the shop how much they will charge to lay it.
Add that on.
Add in the price of underlay (return to Step 1...)
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The “real world” response
Measure the room to the nearest foot. You know the shop will
come out and do a more accurate measurement later.
Take those figures with you to the shop.
Look at some carpets and ask the shop to give you a rough cost,
including underlay and fitting.
Haggle. Ask if they’ll throw in the underlay for free. Ask them
why they don’t do free fitting – the shop next door does...
Go away and think about it.
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Oh, and listen to the saleswoman doing the sums
She’s not saying
“6.23 x 4.32 = so many square metres”
She’s drawing a sketch and saying things like
“So we’re using the 4 metre grey fleck? And the same
thing on the stairs? Good choice. Well, that’s a run of 4.3
metres that way, with a join here . . . and then we can use
the other bit for the first run of 6 steps and then that bit
will take us round the corner. . . . .”
Back to problem list
You’re running a playgroup
and you want to take the
children on a day trip.
How many cars will you
need?
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The “give maths a bad name” response:
Count the children.
Divide by however many children you
think can fit into a car.
Round up your answer to the nearest
whole number.
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The “real world” response: Plan A
The last time you organised a day trip, some of the parents used
their own cars. Did this work well? If no, go straight to Plan B.
Check your h&s policies; is it still ok to use volunteer drivers?
If yes, ask some of the adults to help out again.
Ask each adult how many children they’re happy to take, and which
children they’re happy to take. Check your policies; don’t end up
with too many kids per adult.
Find more drivers if necessary.
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The “real world” response: Plan B
Check out the price of a minibus.
Back to problem list
You’re in trouble with your
partner – should you
apologise?
10
The “give maths a bad name” response ?
D [ Rp(Ra + P) + D(Ra – Rp)] = A
Where
A = your “answer”
D = how big a deal it is
Ra = how responsible you actually were
Rp = how your partner perceives your responsibility
P = how pissed off your partner is
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If you want to hedge your bets when buying and selling stocks. . . .
Curtseying – paying
lip service to adults’
needs, aspirations,
ambitions,
mathematical
insights and
creativity
Not sure
I’ve quite
got the
hang of
this yet
Honestly,
it’s easy. . .
P.S. Michael
Flatley eat your
heart out.
Why are they
all bobbing up
and down? Do
they really
expect me to
take them
seriously?
We may find ourselves curtseying
when we
(are required to)
teach maths that our adult students
really really didn’t ask for, and that
people really really don’t use when
solving real-life practical problems.
About this workshop . . .
• A clarion call to “contextualise” or “embed” everything?
• No!!!!!
• A rant against contextualising / embedding?
• No!!!!
• A rant against the “abstractness” of GCSE maths?
• No!
About this workshop .…
• A rant about inappropriate over-mathematicising ?
Well, a little. . .
• A rant about teaching adults about flipping coins when
they’ve (probably) got bigger probability issues to think
about?
Well, a little. . . .
• A rant about bogus contextualising?
Well, a little. . .
A problem about diluting orange juice. . . . or a ratio exercise?
Oughton, Helen, 2009 A willing suspension of disbelief? ‘Contexts’ and
recontextualization in adult numeracy classrooms, Adults Learning
Mathematics Journal, Volume 4(1), February 2009
What happened in the classroom
“Their discussion demonstrated their
understanding that they are expected to extract
numerical information from the arbitrary
referents in the problem … and to perform a
calculation which, if done correctly, will result in
the ‘right’ answer …..”
In discussion after the class with the
researcher. . .
. . . . . the students listed a wide range of methods,
few of which bore any similarity to the one used by
‘Selina’ on the worksheet.
The most commonly mentioned was approximating
a quarter by eye or by markers on the squash bottle,
but other methods included looking at the colour of
the mixed drink, listening to the sound of the liquid
filling an (opaque) container, and tasting the drink.
Back to problem list
All the students denied ever measuring
accurately.
As one of the students, Charlotte, said:
“I’ve more
important
things to do.”
Willing suspension of disbelief.
Yes, it happens,
but. . . if it doesn’t. . .?
Teacher:
Alright class, here is a ratio problem for you. In
order to paint a certain wall pink, a painter uses
a gallon of white paint mixed with three drops of
red paint. How much white and red paint would
he use to paint a wall three times that size?
Student:
Teacher, I know!
My parents run a painting company, so I learned this
from them. If you paint a really big wall, you have to mx
the colour a little bit darker, because the sunlight falling
on a large wall will make the colour appear lighter.
And you would have to mix up the first gallon, and then
mix the other batches to a chip, because there might be
a slight colour difference in different job lots of paint
from the factory.
In any case, you wouldn't mix up three batches of paint
all at once, because the colours would start to separate
before you were ready to use them. You’re usually
better to trust your eye than just to go by the
measurements anyway. . .
Teacher:
Alright! Enough!
What you have to realize is that
we’re not talking about painting
here, we’re talking about ratio!
Paraphrase of an exchange overheard in an
elementary school classroom (taken from
(Keitel, 1989), quoted in (Gerofsky, 1999)
Scoping The Job –
whose agenda?
My agenda:
• Engage students’ interest by setting a reasonably authentic
problem
• Resist the temptation to over-teach (ask a simple question
and give them time to respond)
• Stimulate students to model a reasonably authentic
problem
• Stimulate discussion and learning around measurement,
costs etc
• Feel good about my own authenticity 
So I posed the following question
You’re running a gardening business
A client has asked you to quote for the
cost of cutting a 20 foot hedge down
to about 6 feet.
Think of some questions you should
ask the client before you quote for the
job.
• Write down some questions you think
students might come up with.
• Are there any you’d particularly welcome?
And why?
One student said the first question
he’d ask was ….. . .
“Have you had a quote
from anybody else?”
Squaring the shed
– who’s the real expert?
We needed to shift the shed, so we got
underneath and walked it over to the new
location. Plonked it down on the dirt floor. It
was a bit skew. I started wondering how we’d
go about making the corners right angles.
Meanwhile Phil had measured the diagonals
and was busy pulling one corner of the shed
towards the middle. A few seconds later, a bit
more measuring of the diagonals, and…job
done.
Concrete Experience…
Norman is a builder and a Maths student. This is his story:
‘Coming from the building trade I had to order material. Initially when I first
started doing this I made many mistakes with this, especially if I had to order
concrete in a cubic capacity in order to fill a trench.
‘My first order, when I measured a particular building to put a concrete base on
the floor, I measured this particular area and it was 2 metres by 3 metres, I
could calculate this as 6 metres. Having phoned the company up and ordered
my 6 metres, not thinking for a moment that they delivered in cubic metres, I
got tonnes and tonnes of surplus cement, spewed out onto a person’s driveway,
for which I had to pay, because they wouldn’t take it back and we couldn’t
shove it back into this big, high concrete lorry… so I learnt the formula!’
Adapted from The Numbers, Disc, CTAD, 1997
NIACE Power of Maths Stories
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About genre
• Westerns versus comedies?
• Clear boundaries between styles?
• Geometry / algebra / number theory /
statistics?
Gerofsky on genre
• Dialogic
- between teacher / question-setter and student
• Intentions and Expectations
• Culturally recognizable conventions
Gerofsky, S. G. (2004). A Man Left Alburquerque Heading East. New York: Peter Lang
Publishing Inc.
Gerofsky on genre
“… and this meeting place of text, intentions and
expectations actually constitutes the genre.”
“It allows students and teachers to make sense of
word problems, despite their inherent oddness
both as stories and as mathematical exercises.”
“It allows students to work out. . . what is likely to
be expected of them in their response. . .”
Gerofsky on genre
• “A curricular genre like mathematical word problems
consists as well of the “specific systems of expectation and
hypothesis” which students learn early in their school
careers and subsequently bring with them to mathematics
classes.”
• “For those who like word problems . . . there is a great deal
of pleasure in decoding their hidden mathematical
message, performing the required operations and re-coding
the answer till it matches the one at the back of the book.”
• “For those who hate word problems (…) the familiarity of
the genre may evoke feelings of panic, helplessness and
self-doubt.”
Choose a task / handout
• Who’s being addressed, and what effect do you
think that might have?
• What (funds of) knowledge might learners bring
to the task?
• Is the teacher expecting the learner to use those
funds of knowledge? Is this communicated via
the text?
• Will the learners know which bits of their
knowledge to draw on / ignore?
• What are the teacher’s intentions? How are they
communicated?
Wason Tasks
Choose a Wason task.
Try to complete the task, but at the same time
try to log your reactions, your process, your
emotional response (if any).
Mortgages
• Smith requires a mortgage of £18,000 on
which the interest is to be charged at 12% per
annum. He agrees to pay regular monthly
instalments. Find how much capital he still
owes at the end of five years if his monthly
instalments are (a) £180, (b) £240, (c) £300.
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Worker Tung
• When worker Tung was 6 years old his family was
poverty stricken and starving. They were compelled
to borrow 5 dou of maize from a landlord. The
wolfish landlord used this chance to demand the
usurious compound interest at 50% for 3 years.
Please calculate how much grain the landlord
demanded from the Tung family at the end of the
third year.
Maxwell, J., in Tomlin, A., (ed), The Numbers Game, 1982(?), Hammersmith and Fulham
Council for Racial Equality
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Freedom fighter
• A freedom fighter fires a bullet to an enemy group
consisting of 12 soldiers and 3 civilians all equally
exposed to the bullet. Assuming one person is hit
by the bullet, find the probability that the person
hit is (a) a soldier, (b) a civilian. If instead the bullet
hits 2 people find the probability that (c) both
people hit are soldiers (d) both people hit are
civilians (e) one of the people hit is a soldier and the
other a civilian.
•
Maxwell, J., in Tomlin, A., (ed), The Numbers Game, 1982(?), Hammersmith and Fulham Council for
Racial Equality
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• If 28 men working 8 hours a day take 15 days
to do a certain piece of work, how many days
would it take 42 men working 7 ½ hours a day
to do the same work?
L.Crosland and M.Helme 1963, Middle School
Mathematics With Answers.
The Holborn escalator
• The escalator at the Holborn
tube station is 156 feet long and
makes the ascent in 65 seconds.
Find the speed in mph.
Maxwell, J., in Tomlin, A., (ed), “The Numbers Game”, 1982(?), Hammersmith
and Fulham Council for Racial Equality
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“Well, my friends, in the research we had done in
the townships and favellas where we were, we
could observe the deficiencies among our
comrades. Then, we realised that what our
settlement companions really need is mathematics.
They also need writing and reading, but, mainly
mathematics. They look for mathematics the same
way they look for a medicine for a hurt because
they know where the hole of the projectile is, by
which they are exploited”.
Knijnik, G. 1997 'Popular knowledge and academic knowledge in the Brasilian peasants'
struggle for land', Educational Action Research,5:3, 501 - 511
To link to this article: DOI: 10.1080/09650799700200038
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Here Gelsa describes and comments on
approaches to the measurement of land. An
“academic” method – measuring the land in
terms of hectares (squares of side 100 metres)
– is contrasted with a measurement based on
the length of time needed to work the land.
The discussion took place in a context where
ideas about the “size” of land are very
significant for people involved in a struggle
over control and ownership of land.
McCafferty, J., Mace, J., & O'Hagan, J. (2009). Developing Curriculum in Adult Literacy and Numeracy Education: a report
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from the NRDC on a research project in Ireland 2006 – 2007. Dublin: National Adult Literacy Agency, p 43.
Two of the peasants used as parameter to
determine the size of a surface the “tractor time
used to hoe”.
One of them explained to the pupils “One places
the tractor on the land. Working with it for 3
hours makes exactly one hectare”
Knijnik, G. 1997 'Popular knowledge and academic knowledge in the Brasilian peasants' struggle for land'
“…. The question of measuring the land with time
was analyzed jointly with the pupils and the farmers.
What, initially, as the pedagogical work began,
appeared to be “inappropriate”, was then more
clearly understood by the group, as examples of
linear distances expressed by measure of time were
examined…..
For farming purposes, the hour of tractor use is more
relevant data than the precision related to square
meters of land.
Knijnik, G. 1997 'Popular knowledge and academic knowledge in the Brasilian peasants' struggle for land'
The alternative to curtseying?
Exploration with adults of their needs, interests, wants,
mathematical methods and insights, aspirations
followed by
negotiation of the curriculum
and discussion of genre
Some areas of maths
that are worth putting
on the (negotiating)
table?
The test is positive.
Have you got the disease?
The doctor says “Yes, very probably.”
Should you believe her / him?
(How) can you check?
Would our adult lives be better if we’d
been taught Bayes’ Theorem?
Interpreting medical test results
Using conditional probabilities:
The general probability that a woman has breast cancer is 1%.
If she has breast cancer, the probability that a mammogram will
show a positive result is 90%.
If a woman does not have breast cancer the probability of a
positive result is 9%.
Now consider a woman who has had a positive result.
What is the probability that she actually has breast cancer?
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Interpreting medical test results (Bayes’ Theorem the easy way)
Using natural frequencies:
10 out of every 1000 women get breast cancer.
Of these 10 women with breast cancer 9 will have a positive
result on mammography.
Of the 990 women who do not have breast cancer, about 89 will
still have a positive mammogram.
Let’s consider some women who have had positive
mammograms.
How many of these women actually have breast cancer?
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The Sally Clark case
• Independent events?
• Prosecutor’s fallacy
http://understandinguncertainty.org/node/545
The Sally Clark case
• The two deaths were inappropriately treated as
independent events; hence the “1 in 73 million” figure
• Prosecutor’s fallacy
http://understandinguncertainty.org/node/545
The fact that it is unlikely that a particular event will occur is not
relevant when, after that event, one is trying to work out the
cause.
Once it is known that the two children are dead, the relevant
question is not: “what is the probability that these deaths were
natural?” but “is it more likely that these deaths were natural
rather than deliberate?”
Sources and resources
Straight Statistics
http://straightstatistics.org/home
Understanding Uncertainty
http://understandinguncertainty.org including an
article on the Sally Clark case at
http://understandinguncertainty.org/node/545
The Cochrane Foundation
http://www.cochrane.org/about-us/fundingsupport
More sources and resources
+plus magazine on breast cancer screening
http://plus.maths.org/content/understanding-uncertainty-breastscreening-statistical-controversy
BBC News article about breast cancer screening
http://www.bbc.com/news/magazine-28166019
Nuffield Reasoning about Uncertainty (teaching and learning
resources)
http://www.nuffieldfoundation.org/key-ideas-teachingmathematics/reasoning-about-uncertainty
More sources and resources
• Gigerenzer, G. (2008). Rationality for Mortals - how people cope
with uncertainty. Oxford: Oxford University Press.
• Gigerenzer, G. (2014). Risk Savvy - How to make good decisions.
London: Allen Lane.
• Gigerenzer, G., & Muir Gray, J. A. (Eds.). (2011). Better Doctors,
Better Patients, Better Decisions - Envisioning Health Care 2020.
Cambridge, Massachusetts. London, England: The MIT Press.
Gerd Gigerenzer websites:
https://www.mpib-berlin.mpg.de/en/research/adaptive-behavior-andcognition
https://www.mpib-berlin.mpg.de/en/research/harding-center
More sources and resources
McCafferty, J., Mace, J., & O'Hagan, J. (2009). Developing Curriculum in
Adult Literacy and Numeracy Education: a report from the NRDC on a
research project in Ireland 2006 – 2007. Dublin: National Adult Literacy
Agency, p 43.
O'Hagan, J. (2012). (When) can we trust ourselves to think straight? –
and (when) does it really matter? ALM18 Proceedings.
http://www.alm-online.net/alm-publications/alm18/
O'Hagan, J. (2014). Written Evidence to Business, Innovation and Skills
Inquiry.
http://data.parliament.uk/writtenevidence/committeeevidence.svc/ev
idencedocument/business-innovation-and-skills-committee/adultliteracy-and-numeracy/written/5770.html
Calls to action?
• Say no to mathematical MacGuffins
• Campaign for Real Maths
• Respect and Negotiate with learners
Is this just Joan ranting?
• Here’s a quote from a very respectable source. . .
• If xxx xxxxx xxxxxxxx xxxx xxxxxxxxxx is to be
successful, it is important that:
• the learner is clear about what they are learning
and what the activities they are undertaking are
designed to teach – a clear and consistently
delivered curriculum helps with this;
Joan ranting?
• the learner brings the context that will
be the ultimate ‘proving’ ground for
their improved skills;
• the learner is sure that the skills and
knowledge that they are learning are
helping them to use their numeracy in
the range of ways they want.
(my emphasis)
Joan ranting?
• So what was this very respectable
document?
The Adult Numeracy Core Curriculum
BSA, 2001, The Adult Numeracy Core
Curriculum, page 8
The MacGuffin and the Curtsey
• MacGuffin = a dramatic device that helps propel the plot in a story
but is of little importance in itself.
http://www.openculture.com/2013/07/alfred-hitchcock-explains-the-plot-device-he-called-themacguffin.html
• Many maths “problems” are MacGuffins:
We say “Let’s think about carpets / orange juice / taking the kids out
for the day” when we really don’t care about those issues but just
want to use those contexts as vehicles for teaching various bits of
maths.
• We use these MacGuffins to curtsey to adults’ lives whilst pursuing
our mathematical agenda.
Forward to some really useful maths?
Of course, anti-MacGuffins also exist.
. . . from a recent cpd programme…..
“Two coins are flipped. . . .”
“Three dice are tossed. . .”
We introduce problems like this, telling the
students that they will use these probability
ideas “later” to solve “real” problems.
• Back to problem list
Dan Meyer:
Math class needs a makeover?
•http://www.ted.com/talks/dan_meyer_math_
curriculum_makeover.html
•Sorry about the advert
•0.00 – 4.15 and 6.15 – 11.00
Do you see this in your classes?
What Einstein said
Dan Meyer’s pedagogical hints
•
•
•
•
•
Use multimedia
Encourage student intuition
Ask the shortest question you can
Let the students build the problem
Be less helpful
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