My goal: to introduce authentic mathematical situations and experiences Agenda: eliminate calculus as gatekeeper/requirement for some programs (business, commerce, economics, etc.) Opportunity to rethink teaching year 1 math courses, especially the so-called service courses Which math is relevant? Why calculus and not statistics?

Year 1 math and stats courses in Canada [First Year Courses in Canada Database.] Why? Province-wide calls for improvement in numeracy

McMaster University initiative to improve students literacy and numeracy In collaboration with a college and with health care institutions (schools, hospitals), Im working on a health numeracy project Math in everyday life ...

Most people are passive users of mathematics Some people are active users (part of their work, or interest) But we need educated people to make it all work Math in everyday life really used? activity

math used/required sleeping none

shower, coffee, breakfast none cleaning shopping

watching tv none elementary (+ - * / %) none financial (taxes,

mortgages, loans, etc.) high school; outsourced; calculators driving, orientation, navigation

none; outsourced Math in everyday life activity math used/required

painting area (?); outsourced building, designing, gardening

elementary (+ - * / %); outsourced using computer, cell phone, appliances

none jogging, sports making food, eating none elementary (+ - * / %)

travel (money, maps, schedules, ...) none; outsourced; online tools

medical, health elementary (+ - * / %); outsourced Everyone is talking about data and algorithms

Math IS everywhere: GPS and driving directions Medical imaging (CT, ultrasound, MRI, fMRI, Xray, etc.) Autopilot, driverless cars Predictions about future are calculated (population dynamics, oil consumption, climate change, epidemics, etc.) Weather forecast, chance of precipitation, humidex, wind chill, UV

index are all calculated Data compression (email, pictures), security and data transmission Search engine results Suggestions for a diagnosis/medical treatment Geological, astronomical history of Earth, and almost

everything about universe are calculated Financial markets, trading and investments Human behaviour is predicted/modified based on data Cannot live without Source:

#ho5EDshvkLq2bEQl.97 As reported by New Scientist an inspector in the Food and Drug Administration (FDA) visited a restaurant in Salt Lake City famous for its

quiches made from four eggs. She told the owner that according to FDA research every fourth egg has salmonella bacteria, so the restaurant should only use three eggs in a quiche. Source: Bureaucrat's Math Makes Dizzy Dozen, written by P. Rolly and J. JacobsenWells and published in The Salt Lake Tribune on 11 October 2002 (Article ID:

100DF2EEC6A2847B) What 40 percent means? (a) one-quarter (b) 4 out of 10 (c) every 40th person

In a survey (conducted by the Emnid Institute, Germany), of 1000 people, about a third got the answer wrong. [Suddeutsche Zeitung Magazin, December 1998 reported in: G. Gigerenzer, Reckoning with Risk. Penguin Books, 2002] Who makes a decision about treatment?

3459% preferred to leave decisions to their doctor 2344% wanted to make collaborative decisions 1222% wanted to make decisions regarding treatment on their own [Davison B, et al (1995). Information and decision making preferences of men with prostate cancer. Oncology Nursing Forum 22:1401 1408. Degner L and Sloan J (1992). Decision making during serious illness: what role do

patients really want to play? Journal of Clinical Epidemiology 45:941950.] Some people use serious math research examples from intersection of math and biology Biostatistics Population modelling, population ecology

Big data in epidemiology Climate change and climate forecasting Nanotechnology Bioengineering, bioinformatics DNA sequencing Computational cell biology

What is numeracy? a.k.a. quantitative literacy quantitative reasoning numeric literacy Quantitative Reasoning is

Quantitative = context is numbers (in broadest sense: 12, 4.5, 100,000,000, a lot, large, very small, roughly three halves, several thousand, etc.) and Reasoning = applying rules of logical reasoning, or understanding why they do not apply or cannot be applied

(and finding alternatives) Whats in blue: rarely, if ever, in academic math courses In quantitative literacy, numbers describe features of concrete situations that enhance our understanding. In mathematics, numbers are themselves the object of study and lead to the discovery and exploration of even

more abstract objects. [Manaster, A. B. (2001). Mathematics and Numeracy: Mutual Reinforcement. In Mathematics and Democracy: The Case for Quantitative Literacy (pp. 6772).] How does numeracy differ from math? Numbers/quantities and tasks have to be constructed A task could have (and often has) different solutions; why

solutions differ is a valuable discussion Accuracy and precision are determined by the context Numeracy tasks contain lots of noise that has to be removed Numbers in authentic tasks are not nice Authentic situations demand the use of mathematics (vs.

reality as pretext to use mathematics) Solutions of tasks have meaningful and practical consequences Numeracy course at McMaster University Numbers for Life

Guided by the questions we routinely ask in mathematics (What? Why? How do we know?), we conceptualize numeracy to include critical and evidence-supported thinking, as well as logical reasoning. Numeracy courses -- having no outside constraints on

their content, unlike calculus or linear algebra which are prerequisites for upper-level courses -- seem to be the right place to discuss authentic (true, really real) mathematical applications. Numbers for Life course (Math 2UU3) McMaster Based on mathematical habits of mind, which

let students in on the process of creating, inventing, conjecturing, and experimenting It is a curriculum that encourages false starts, calculations, experiments, and special cases. Cuoco, A., Goldenberg, E. P., & Mark, J. (1996). Habits of mind: An organizing principle for a mathematics curriculum. Journal of Mathematical Behavior, 14(4),

375402. Class dynamics: Friday, or weekend before Monday lecture: visit 2uu3 web page WEEKLY SCHEDULE AND HOMEWORK. Read the assigned problem set questions and do all advance readings.

In lectures: listen, participate, take notes you will be able to identify answers to some questions from the problem set Immediately after the lecture: visit the web page to see what else you are asked to do (could be extra practice, additional readings, etc.)

Later, as things happen: complete answers to all questions from the problem set Friday, or weekend before Monday lecture: cycle repeats Tests and the exam will be based on class discussions and problem sets.

Course content most of Language of maths and everyday language: contrasting definitions, theorems, proofs and algorithms in mathematics and in real life 'Quantitative' in quantitative reasoning: numbers and related concepts; elementary relationships between numbers; proportional reasoning; currency conversion

and units conversions as examples of proportional reasoning Scientific notation; visualizing numbers in narratives (developing a feel for numbers and their size); Reasoning with numbers; analyzing narratives which involve quantitative information and/or logical

reasoning Patterns of change, in particular, linear, quadratic, cubic, and exponential Models involving exponential functions, such as human population dynamics (demographic and economic ramifications) and exponential decay; logarithmic

scale, Richter scale Scatter plots and building models using linear regression; non-linear regression Climate change parameters: modeling the amount of carbon dioxide in atmosphere, surface area and thickness of Arctic ice, average global temperature

Discussing appropriate and inappropriate use of visual representations of information/data; interpreting information presented visually; dynamic visualization Finance: managing credit card debt, loans, mortgages; financial calculators; interest, inflation, CPI (consumer price index); financial market and indicators (Dow, S&P, TSX, NASDAQ)

Economy and social indicators: GDP and gross national product/income (GNP/GNI); human development index and Gini index Case studies of randomness: lottery and games of chance; probability in contexts Basic probability, stats and data: law of large numbers;

mean, spread; normal distribution, and reasoning based on standard deviations Intuition about total probability and Bayes' theorem, applied to testing for a medical condition (false positives and false negatives); Basics of statistical hypothesis testing, null hypothesis, p-value

How is it done? Activity: Problem: Create a mathematically sound argument

showing that Starbucks Venti (20 oz) blonde roast cup of coffee (which has 475 mg of caffeine) can be claimed to be 99.9% caffeine free Activity Do you turn the tap off?

Estimate the amount of water wasted in Canada every day by people who do not turn the tap off while brushing their teeth Using online clickers, we gather information on how many times each day they brush their teeth,

and whether or not they turn the water tap off while doing so. (In future iterations, students will be given the problem only, and will have to decide what data they need; that data will be collected on the spot by using online clickers.)

As no other information about the problem or data were given, students had to decide what else they needed, and how to obtain, or calculate, or estimate these quantities. As a conclusion of the activity, members of two groups were called to present their solutions

We see evidence of students engagement in a variety of mathematical habits of mind: they looked up population of Canada (34 million in Estimate 1 and 35.16 million in Estimate 2) to scale up the sample (classroom) proportions to the entire Canadian population

they estimated how long, on average, one person brushes their teeth (90 seconds in Estimate 1 and 2 minutes in Estimate 2; these were rough averages based on groups personal experiences). Another piece of information was missing water flow rates.

Estimate 1 is based on scaling an estimate (which was a consensus of the group members but not checked independently) that a 500 millilitres glass of water can be filled in 15 seconds, whereas Estimate 2 was informed by the water faucet flow rates provided by Lowes Canada (hardware store) web page.

Additional issues were identified Students questioned whether the sample size (classroom size) is large enough to merit reliable statistics about the entire population. As well, there is an issue of bias: can the data taken from a sample of university students be viewed as

representative of the entire Canadian population? Related to the 10-fold difference in the two estimates, there was a suggestion to test their sensitivity. This amounts to asking if a certain numeric piece of information is increased or decreased by 5 percent, how would that affect the final estimate for the

amount of water wasted. (This kind of thinking is rarely, if ever, done in calculus.) Thinking about creating a narrative based on the obtained estimates -- how does one visualize large

quantities of water (47 million litres in Estimate 1 and 401 million litres in Estimate 2)? Here, students recalled their previous experience, where we discussed narratives including numbers. Here is a sample of a such a narrative.

How many Canadians actually live up North? Approximately 118,000. Thats one-third of one percent of the national population. To put it another way, about as many Canadians live in Australia as live in Nunavut. If the entire population of Northwest Territories decided to attend Edmonton Eskimos game, Commonwealth stadium

would still have 10,000 empty seats. Thus Olympic pool = 25 * 50 *2 m^3 -> 2.5M litres 47M -> about 19 pools 401M -> about 160 pools

What did everyone else get? This simple activity contains lots of good, interesting, relevant math! Numbers for life course is, in that way non-traditional does not follow linear progression of development of

math (which is of course, needed in some math courses) Can we design a math course differently say, based on a collection of good problems? Framing

Side effects of Prozac: you have a 30 to 50 percent* chance of developing a sexual problem patient interprets the above as: something will go wrong in 30 to 50 percent of

my sexual encounters *G. Gigerenzer, Reckoning with Risk. Penguin Books, 2002. PERCENT You have a 30 to 50 percent chance of developing a sexual problem if you take Prozac.

RELATIVE FREQUENCIES Out of every 10 people who take Prozac, 3 to 5 experience/develop a sexual problem. Framing

1/10 = 10/100 = 100/1000 1 out of 10 10 out of 100 100 out of 1000 Reasoning learning from math

Math teaches us to be critical about things, and to always be on alert What is it? (definitions) How did we arrive at it? (assumptions) Why is this true? (evidence)

How does it relate to ? (broader picture) Two television commercials for Head & Shoulders shampoo have been criticised for implying that the products leave hair 100% dandrufffree.

Procter & Gamble [100% dandruff-free] claim meant not visible to another person from a distance of two feet. [source: BBC News Online, Tuesday, 4 April 2006] What does this mean?

Source: However, in reality we need to declare certain correlations to be causations so that we can act appropriately Am I

convinced? Should I worry? What do I do about it? Optional reading:

Just because it says farms does not mean its from a farm Optional reading: environment/2017/dec/13/tescofaces-legal-threat-over-marketingits-food-with-fake-farm-names

CBC News optional reading: Important!

Questions we must ask: Is there evidence? What is acceptable as evidence? What exactly does the evidence show? optional reading:

[ c_id=6&objectid=11749101] Financial matters Currency exchange:

one currency = conversion factor * another currency conversion factor is called buy rate or sell rate depending on whether were buying or selling currency or, conversion factor could be some kind of average rate

Rates change frequently, many times a day as well, different institutions offer different rates Example: Euro Scotiabank:,,7662,00.html

Which is buy rate, which is sell rate? Buy and sell rates BMO site:

What are buy and sell rates? Example: BMO, exchange rates between Canadian Dollar and Euro (data from 3 Oct 2018) 1.4211 1.5447 Which is buy rate, which is sell rate?

Hint: compute the conversion that you need using both rates, and the one thats worse for you (better for the bank) is the right one Optional: Among other pieces of advice:

Optional: Remember: Called dynamic currency conversion which is often a rip-off

Make a decision Sometimes not at all clear Optional reading:

Gini Index A = area of inequality Gini index = 2 * area of A World Gini index in 2014

Gini index = wealth distribution Historic Gini Index in Canada

Table data source: World Bank Can Canada do this?

traffic tickets, fines proportional to income Faulty reasoning Growth of cancer cells if you want to double the volume, then

increase the size by about 26% doubling time: time T needed for cells to double in number; i.e., to double their volume Doubling times of breast cancer

age median doubling time in days (doubling time interval) < 50 50-70

> 70 80 (44 - 147) 157 (121 - 204) 188 (120 - 295) T = very large ductal cancer in situ

source:PG Peer et al, Age-dependent growth rate of primary breast cancer. Cancer. 1993 Jun 1; 71(11): 3547-51 Cancer growth, doublings 27, 28, 29, and 30 BMJ 2000; 321: 1071-1073 (28 Oct 2000)

non-palpable, mammographically detected cancer threshold for cbe detection

wrong !!! Thank you! [email protected] Diagnoses of kidney cancer in 3141 counties in the

U.S. Lowest incidence: mostly rural counties, sparsely populated, in the Midwest, the South and the West Highest incidence: mostly rural counties, sparsely populated, in the Midwest, the South and the West Source:

Diagnoses of kidney cancer in 3141 counties in the U.S. Lowest incidence: mostly rural counties, sparsely populated, in the Midwest, the South and the West Highest incidence: mostly rural counties, sparsely populated, in the Midwest, the South and the West

Statistics: extremes are lot more likely to occur in small samples

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