math …

cannot live with it, cannot live without it

miroslav lovricmcmaster university lovric@mcmaster.ca

What is Mathematics?

* research mathematics (including applications)* mathematics for users (‘everyday math’) * mathematics taught in schools, colleges and universities* mathematics for teachers * recreational, ethno-mathematics, popular math, etc.* mathematics as used in art, architecture, music, etc.* …

Who wants/needs to know math? Who needs math? Why?What math should we teach? What important messages do we get from math?

Two television commercials for Head & Shoulders shampoo have been criticised for implying that the products leave hair 100% dandruff-free.

Procter & Gamble … [100% dandruff-free] claim meant those visible to another person from a distance of two feet.

[source: BBC News Online, Tuesday, 4 April 2006]

What “40 percent” means?

(a) one-quarter

(b) 4 out of 10

(c) every 40th person

What “40 percent” means?

(a) one-quarter

(b) 4 out of 10

(c) every 40th person

*Suddeutsche Zeitung Magazin, December 1998.

In a survey* of 1000 people, about a third got the answer wrong.

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.

The manager asked if simply throwing out 1 in 4 eggs from each packet and use the remaining in four-egg quiches would serve the same purpose. The inspector was not sure but would try and find out.

New Scientist (volume 176, number 2367, 2 November 2002)

Date: 07/02/2003 at 14:49:23From: LisaSubject: Volume to be infused by drip factor

A patient is to receive 1440 milliliters of fluid in 24 hours. The drip factor is 15 drops per milliliter. The formula for drops per minute is:

volume to be infused x drip factor ---------------------------------- time in minutes

The patient should receive about how many drops per minute? I want to pass an entrance exam for a nursing program, but I have never been very good in math.

[Ask Dr. Math http://mathforum.org/dr.math/]

Hi, Lisa.

You've been given the formula, so you only have to apply it to your problem. […]

1440 mL * 15 drops/mL --------------------- 1440 min

[…]

[Ask Dr. Math http://mathforum.org/dr.math/]

I'd like to make a comment, and I hope it will not offend you, but motivate you.

Your question scares me, because I don't want nurses who can just barely do this kind of calculation to work on me! Also, the fact that a high school could graduate a student who can't do this means our schools are practically worthless.

[…] Don't just figure that you got by and can ignore math in your work; make sure you fully understand how to work with units, because it will someday be a matter of life and death.

[Ask Dr. Math http://mathforum.org/dr.math/]

Who makes a decision?

34–59% preferred to leave decisions to their doctor 23–44% wanted to make collaborative decisions 12–22% 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 45:941–950.]

Testing for a medical condition (allergies, SARS, mammography, etc.)

Ideal situation:* test turns positive - the person has it* test turns negative - the person does not have it

Testing for a medical condition (allergies, SARS, mammography, etc.)

Ideal situation:* test turns positive - the person has it* test turns negative - the person does not have it Above are true positive and true negative casesBut there are also …false positive - the person tests positive but does not have the conditionfalse negative - the person tests negative, but has the condition

Facts: * probability that a woman (age 40-50) has breast cancer is about 0.8%* if she has breast cancer, the probability that the mammogram test is positive is 90%* if she does not have breast cancer, the probability that she tests positive nevertheless is 7%

What are the chances that a woman who tests positive actually has breast cancer?

[Gerd Gigerenzer: The Psychology of Good Judgment: Frequency Formats and Simple Algorithms. Medical

Decision Making 16, 273-280, 1996]

correct !

CONVENTIONsize = linear size (units are centimetres or milimetres)

doubling the size means:eight-fold increase in volume

if you want to double the volume, thenincrease the size by about 26%

BMJ 2000; 321: 1071-1073 (28 Oct 2000)

non-palpable, mammographicallydetected cancer

threshold forcbe detection

wrong !!!

… to double the flow of water, double the sides??

issues ...

December 2004 | Volume 2 | Issue 12 | e439 | e440PLoS Biology | www.plosbiology.org

Many good mathematicians will not become math majors!

Mathematics at all levels should be more friendly

College, University Mission Statements

‘developing learners informed about the world’‘developing empowered communicators’‘developing integrative thinkers’‘preparing for the changing world’

McMaster University's vision is to achieve “international distinction for creativity, innovation and excellence.”

… what about knowledge?

How is Education Different

• lot more than academic requirements needed (e.g., volunteer work, community service)• political correctness (demagogy of success … ‘delayed success,’ ‘all students perform above average’)• knowledge vs information management• reference letters - no negative comments (everybody is in the upper 10%)

• ‘babying’ of students (facilitating learning, forming study groups, sample tests, tutorials, etc.)

The Science Learning Centre in London, England surveyed 11,000 students aged 11-15 for their views of science and scientists

around 70% said they did not picture scientists as "normal young and attractive men and women".

around 80% of pupils thought scientists did "very important work" and 70% thought they worked "creatively and imaginatively”

around 40% said they agreed that scientists did "boring and repetitive work".

BUT very few plan to become scientists

… reasons included

”because you would constantly be depressed and tired and not have time for family"

"because they all wear big glasses and white coats and I am female"

between 1991 and 2004: the number of students taking A-level physics dropped by 34%, chemistry by 16%, math by 22%

[BBC NEWS report on global study Relevance of Science Education, based at Oslo University]

based on responses to 250 questions

favourite topics for boys were... * explosive chemicals. * biological and chemical weapons. * black holes and supernovae. * how meteors, comets or asteroids cause disasters on earth. * the possibility of life outside earth. * how computers work. * dangerous animals.

girls, however, preferred to learn about their own bodies;they wanted to know...

* why we dream and what it means. * what we know about cancer and how to treat it. * how to exercise to keep fit. * sexually transmitted diseases and how to protect against them. * life and death and the human soul. * biological and human aspects of abortion. * eating disorders. * how alcohol and tobacco might affect the body.

BUT

the two genders did agree on what they least wanted to learn about:

* modern farming methods * famous scientists and their lives

Who’s interested in math?

• Women make up almost half of today's workforce, yet … one-fifth of all engineersfewer than one-third of chemistsabout a quarter of computer and math professionals

• Complex issue (family, gender, stereotypes, etc.)

• [relatively new] part of the explanation for the gender gap are the preferences of women themselves

• Study relates to men and women that have substantial math background

Ferriman, K., Lubinski, D., & Benbow, C. P. (under review) Work preferences, life values, and personal views of top math/science graduate students and the profoundly gifted: Developmental changes and sex differences during young adulthood and parenthood

• Study of Mathematically Precocious Youth (Vanderbilt U.) ** started more than 30 years ago ** began following nearly 2,000 mathematically gifted adolescents, tracking their education and careers ** it has since been expanded to 5,000 participants ** both men and women in the study achieved advanced credentials in about the same numbers

• Math-precocious men are much more likely to go into engineering or physical sciences than women• Math-precocious women, are more likely to go into careers in medicine, biological sciences, humanities, and social sciences[both sexes scored high on the math SAT]

• Women who are mathematically gifted are more likely than men to have strong verbal abilities as well [highly qualified women may opt out of certain technical or scientific jobs simply because they can]

Susan Pinker, The Sexual Paradox: Men, Women, and the Real Gender Gap, Simon & Schuster, 2008

• In countries where women have the most freedom to choose their careers, the gender divide is the most pronounced (United States, Norway, Switzerland, Canada, UK)

• In countries with less economic opportunity (Philippines, Thailand, Russia) the number of women in physics is as high as 30-35%, versus 5% in Canada, Japan, and Germany

Canadian Teachers’ Federation

% of males among elementary and high school teachers

1900-1950 … 17-27% (17% in 1920)1950-1975 … increase from 27% to 42%1989-1990 … 41%1999-2000 … 35%Latest … 30% (consistent across all Ontario’s publicly funded systems)

15% of students in teachers colleges in Canada are male

Narrowing the Gender Gap: Attracting Men to TeachingJean-Luc Bernard, Pat Falter, David Hill, Doug Wilson (2004)

[http://www.oct.ca/publications/documents.aspx]