Curtseying to adults’ lives Joan O’Hagan email@example.com 07515702991
When is it helpful to mathematize our real life decisions? (sub-text: what maths do adults really really want?) Is it rational to be emotional / intuitive about these decisions?
Curtseying – paying lip service to adults’ needs, aspirations, ambitions, mathematical insights
Not quite sure I’ve got the hang of this yet Well, if you want my advice. .
Let me show you. P.S. Michael Flatley eat your heart out.
Call that a curtsey? In one’s day one did things properly! or perhaps… Why are they all bobbing up and down? Don’t they realise how silly they look?
Curtseying = teaching maths that adults really really don’t ask for
Some maths that probably is useful when we’re making adult decisions? • Curtseying = teaching maths that adults really really don’t ask for
About this workshop . . . • A rant against the “abstractness” of GCSE maths? • No! • A clarion call to “contextualise” or “embed” everything? • No!!!!!
About this workshop .… • A rant about inappropriate over-mathematicising ? Well, a little. . . • A rant about teaching adults to flip coins when they’ve (probably) got bigger probability issues to think about? Well, a little. . . . • A discussion about rationality and intuition and mathematizing? Yes ….. There’s a time and place for everything.
Got a problem? Maths to the rescue? • Carpeting your room • Taking the kids on a day trip • Diluting orange juice • Flipping coins and tossing dice Forward to MacGuffins? Forward to some really useful maths ?
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. (“let’s think about carpets / orange juice / taking the kids out for the day”) • We use them to curtsey to adults’ lives whilst pursuing our mathematical agenda. Forward to some really useful maths?
You want to carpet your room. How much will it cost?
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...)
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.
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?
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.
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&spolicies; 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.
The “real world” response: Plan B Check out the price of a minibus. Back to problem list
You’re buying detergent. There are three different-sized bottles. Which will you buy? “Maths lesson” response? Convert the volumes of all three bottles to centilitres. Divide the price of each by the number of centilitres. Pick the lowest answer. Real world answer? Look at the label on the shelf – the unit price is usually there. Buy the one you can afford this week. Or buy the one that fits in your cupboard. Back to problem list
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.’ Back to problem list
From a recent cpd programme • Two coins are flipped. . . . • Three dice are tossed. . . • Back to problem list
As one of the students, Charlotte, said: “I’ve more important things to do.”
“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
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.
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”
“…. 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.
The alternative to curtseying? • Respectful exploration with adults of their needs, interests, wants, mathematical methods and insights, aspirations followed by • negotiation of the curriculum
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?
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?
Interpreting medical test results Using natural frequencies: 10 out of every 1000 women get breast cancer. Of these 10 women with breast cancer 9will 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?
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?”
Emotion, intuition, logic The Wason Tasks
Wason tasks • Syllogistical reasoning • The effect of context • Does the task have a “social contract” element?
The Wason tasks • 1: The D, 3, 8, E cards • 2: Donal, John, Joan and Anne in the pub • 3: John, Paul, George, Ringo at work
D 3 8 E You’re looking at four cards, each of which has a number on one side and a letter on the other. I’m asserting that the set of cards obeys a rule, which is “Any card with a consonant on one side has an even number on the other”. To check if the rule is being adhered to, which 2 cards should you turn over?
John is drinking alcohol Joan is drinking apple juice Donal is aged 18 Anne is aged 29 This time you’re in the pub, looking at four people. You want to know if anybody’s breaking the “nobody under 21 is allowed to drink alcohol” rule. Who do you want to question? (You’re only allowed to question 2 people) fwd to choices
George is not wearing safety gear John is doing something dangerous Paul is wearing safety gear Ringo is not doing anything dangerous This time you’re a Health and Safety manager responsible for four staff. You know that if they’re doing a dangerous task they should be wearing safety gear. You know John is doing something dangerous but Ringo isn’t; and Paul is wearing safety gear but George isn’t. To make sure they’re all sticking to the safety rule, who do you want to talk to? Talk to as few of them as possible.
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/funding-support
More sources and resources +plus magazine on breast cancer screening http://plus.maths.org/content/understanding-uncertainty-breast-screening-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-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-cognition https://www.mpib-berlin.mpg.de/en/research/harding-center