Giving maths a bad name

Joan O’Haganjoanohagan@btinternet.com

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?

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The “give maths a bad name” response ?

D [ Rp(Ra + P) + D(Ra – Rp)] = AWhere A = your “answer”D = how big a deal it isRa = how responsible you actually wereRp = how your partner perceives your responsibilityP = 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

P.S. Michael Flatley eat your heart out.

Honestly, it’s easy. . .

Why are they all bobbing up and down? Do they really expect me to take them seriously?

We may find ourselves curtseying when we

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.

(are required to)

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. . .

Oughton, Helen, 2009 A willing suspension of disbelief? ‘Contexts’ and recontextualization in adult numeracy classrooms, Adults Learning Mathematics Journal, Volume 4(1), February 2009

A problem about diluting orange juice. . . . or a ratio exercise?

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 - 511To 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 from the NRDC on a research project in Ireland 2006 – 2007. Dublin: National Adult Literacy Agency, p 43. 50

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 resultsUsing 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 Statisticshttp://straightstatistics.org/home

Understanding Uncertaintyhttp://understandinguncertainty.org including an article on the Sally Clark case at http://understandinguncertainty.org/node/545

The Cochrane Foundationhttp://www.cochrane.org/about-us/funding-support

More sources and resources

+plus magazine on breast cancer screeninghttp://plus.maths.org/content/understanding-uncertainty-breast-screening-statistical-controversy

BBC News article about breast cancer screeninghttp://www.bbc.com/news/magazine-28166019

Nuffield Reasoning about Uncertainty (teaching and learning resources)http://www.nuffieldfoundation.org/key-ideas-teaching-mathematics/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-and-cognitionhttps://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/evidencedocument/business-innovation-and-skills-committee/adult-literacy-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-the-macguffin.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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