Learning Mathematics within an Interdisciplinary Context

Miroslav LovricMathematics and Statistics

McMaster University

lovric@mcmaster.ca

 

My experience comes from …

• Teaching math for life sciences (working toward integration of math, biology, chemistry, health sciences)

• Writing a textbook on math for life sciences

• Teaching in Arts and Science Program

• Developing and teaching of Inquiry in Science course

• Creating and teaching in the Integrated Science Program (iSci)

• Studying interdisciplinary programs in Canada and beyond

Integrated science programs in many cases

combine content from several disciplines (math, physics, biology, chemistry, earth science,

psychology, computer science) AND

use numerous teaching and learning strategies (lectures, group work, flipped class, clickers,

problem-based, etc.)AND

provide a variety of experiences (research, project work, thesis, field work, etc.)

Motivation/rationale for this?

“Modern economy demands and rewards complex problem solving and communication skills in technical subjects”

“… lecturing is not an adequate teaching model”

[Carl Wieman, A New Model for Postsecondary Education: the Optimized University www.cwsei.ubc.ca/about/BCCampus2020_Wieman_think_piece.pdf]

Example: • Interdisciplinary setup plus “innovative” teaching methods = success

true value

Until 20-30 years ago, college and university education has been necessary and useful only for a selected elite. Today, college/university diploma is a necessary for almost any job.

How do we effectively teach complex technical knowledge and skills to a large fraction of population?

… are people willing to learn and to know?

Do we truly need science-sophisticated population?

Yes, in small numbers (much less than the total student population) … no reason for panic?

Report by Council of Canadian Academies, 2014

Do we truly need science-sophisticated population?

Technology dumbs things down and demands only basic human skills (Amazon, Uber, jobs on demand)

It is possible to work with sophisticated instruments and know very little about them. Examples: medical imaging, GPS

Interest, motivation?

… mathematics was selected by the smallest group

“I'm happy to see that science stimulates the European youth,” commented European Science and Research Commissioner Janez Potocnik.

[…] survey shows that there is a huge reservoir of interest and support to science in the young generation. However, the low interest in engineering and scientific studies is a major concern, as well as the gender imbalance.

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

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

around 40% said they agreed that scientists did "boring and repetitive work”around 70% said they did not picture scientists as "normal young and attractive men and women”

[BBC News [no author mentioned] (2006). Science ‘not for normal people.’ Available at: http://news.bbc.co.uk/2/hi/uk_news/education/4630808.stm]

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"

Are we interested in “long-life learning”?

But we do need math!

How many people can correctly argue about …

Research Question

What are (extra) qualities of mathematics which is learned within

Integrated Science Program at McMaster?

Background:Tensions between various science disciplines in

interdisciplinary courses/programs

Research Setup:

pre-test / post-test with time distance of 8 months

pre-test and post-tests are identical

Question:what do students know (and how) after 8 months of math instruction in iSci that they did not know when they started?

Two surveys

(1) Material which is explicitly taught as part of curriculum (lecture format, with frequent activities used to engage/motivate students)

(2) Problem-solving strategies, clear thinking, creative thinking, higher-level thinking (transfer of known to new situations, communication of mathematical ideas, etc.)

Results of survey (1)

General improvement, as expected

Sample of questions, survey (2):

(1) Which fraction is larger? 7/8 or 78/87? (looking for creative approaches)

(2) In how many ways can you pay 50 cents in change? (organization, breaking down into cases)

(3) Simple math calculation; provide a context where it could occur (communication, applications, interdisciplinary context)

(4) If the radius of a sphere doubles, how does its volume change? (quantitative thinking, dimensional analysis)

(5) What is the sum of the angles in a hexagon (reducing to a simpler problem)

(6) Given new definition, draw conclusions (applying skills to a new situation)

Pre / post comparison grading scheme, survey (2)

Range from -2 (wrong idea, wrong calculation) to 2 (correct, or almost correct, good idea)0 (no answer)

No change: pre- and post- differ by 2 or less

Change: pre- and post- differ by more than 2 points

Pre- / post- comparison, survey (2)

no change13

post test better3

pre test better1

Problems …

Small data set

Students had no motivation to do well (especially on post-test)

We found …

High robustness of cognitive models developed in high school

Cognitive models are often hostile to new ideas

hexagonnot hexagon

square diamond, not square

Why? Because math was not taught properly!

DEFINITION!!!

What is …

Hexagon, square, continuous function, derivative

Planet, habitable zone

Climate change

Sodium free?

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 … not visible to another person from a distance of two feet.

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

We found …

High robustness of attitudes attained in high school (surface vs. deep learning; memorization vs. understanding)

Memorization: search for a right formula, or matching a problem with a memorized “template”

We found …

No improvement (or slight improvement at best) in using logical thinking in new situations (in most cases incorrect in both pre- and post- situations)

This is difficult, needs time to develop

So it seems that not much is learned

But there are opportunities to change this!

Also … it takes time to learn

Creative tensions …

Sequential development in math vs other sciences

“On a closed interval, a continuous function assumes its absolute maximum and absolute minimum values …”

“Animals need to maintain fluid balance …”

Whereas non-linear (non-sequential) thinking is suitable for some disciplines, math requires linear (sequential) approach

but within it, plenty of non-linear thinking!

Interdisciplinary context: usually non-sequential wins (non-linear thinking!), and so math loses

Creative tensions …

Nature of evidence in math vs other sciences

In math, we require proofsIn other sciences?

Interdisciplinary context: math-type evidence usually the first casualty (students taught algorithms, formulas, etc. without understanding)

4=2+26=3+38=5+310=3+712=5+714=7+716=11+518=11+720=13+722=13+5

24=17+726=13+1328=17+1130=19+1132=29+334=17+1736=29+738=19+1940=29+1142=31+11

44=31+1346=41+548=41+750=37+13...

what?

Goldbach Conjecture (1690-1764)Every even positive integer greater than 2 can be written as the sum of two prime numbers.

The conjecture has been verified for all even numbers at least up to 100,000,000,000,000BUT no definitive proof has been given so far

Prime number distribution

p(n)= number of primes less than or equal n

p(2)=1p(3)=2p(4)=2

p(10)=4p(100)=25p(1,000)=168p(10,000)=1229p(100,000)=9592

etc.

Prime number distribution

p(n)= number of primes less than or equal n

p(2)=1p(3)=2p(4)=2

p(10)=4p(100)=25p(1,000)=168p(10,000)=1229p(100,000)=9592

etc.

Gauss:p(n) approximated by l(n)=n/ln(n)

l(10)=6.1l(100)=30.1l(1,000)=177.6l(10,000)=1246.1l(100,000)=9629.8

etc.

Prime number distribution

… so, Gauss’ estimate is an overestimate

In 1955, it was discovered that it becomes an underestimate sometime before n reaches ...

10 to the power of

1010,000,000,000,000,000,000,000,000,000,000,000

Hype about integration of mathematics

“Mathematics is everywhere” and “[should be] easy to integrate”

“[we should] teach applications which are meaningful to our students” (from elementary school to university)

Not like this …

There are plenty of meaningful applications

• Big data• Population models (on all scales)• Feeding patterns of animals• Environmental issues (water pollution)• Stochastic modelling in health sciences• Nanotechnology (hype is over!)• Bioengineering, bioinformatics• DNA sequencing• Computational cell biology• Astronomy (asteroids are very popular)

etc.

Many math education papers talk about the value and need for integration of math (“meaningful applications”) at all levels, however, very few contain explicit suggestions on how to do it

Need to integrate math education research with practice!

As well, funding agencies are biased (no funding for “lesson plans”)

Major issues in integration of mathematics…

• Breadth vs depth, i.e., “black boxes” (algorithms geared toward specific problem) vs understanding (ability to expand the model used)

• In many cases, (advanced, challenging) math is actually avoided, and only used as interpretation (pre-made applets, diagrams, web sites, etc.)

• Physics suffers even more than math

• Computer programming not even taught

Major issues in integration of mathematics…

• Learning content of a particular discipline vs working on skills development (scientific reasoning, clear thinking, communication, etc.)

• Assumptions vs reality about our students

(knowledge, time spent studying, motivation, attitudes, maturity, etc.)

do we know our students?

Average student in Canada spends about 14 hours a week studying, down from 24 hours several decades ago.

(Similar stats for other countries)

[New Report Shows Students Are Studying Less But Getting Better Grades http://www.edudemic.com/new-report-shows-students-studying-less-getting-better-grades/

The decline of studying http://www.macleans.ca/news/canada/the-decline-of-studying/

Is college too easy? As study time falls, debate riseshttp://www.washingtonpost.com/local/education/is-college-too-easy-as-study-time-falls-debate-rises/2012/05/21/gIQAp7uUgU_story.htm]

‘Only about 11 percent of full-time students say they spend more than 25 hours per week preparing for their classes - the amount of time that faculty members say is necessary to succeed.’

[National Survey of Student Engagement: http:/nsse.iub.edu

Hoover, E. (2004) Undergraduates Study Much Less Than Professors Expect The Chronicle of Higher Education, November 15, 2004]

Students spend less time studying, and we expect less from them (in spite of claiming the opposite)

So, students do know less …

Knowledge is the precondition for creativity, critical thinking and problem-solving

When we try to solve any problem, we draw on all the knowledge that we have committed to long-term memory

The more knowledge we have, the more types of problems we can solve

(Residents coming out of McMaster medical school)

What else is going on?

The percentage of young people (under 30) who own their own business is the lowest it has been in 24 years

[Wall Street Journal, Endangered Species: Young Entrepreneurs, 2 Jan 2015]

Hard to predict how our students will change

[…] millennials prefer print because it’s “easier to follow stories.”

Pew studies show the highest print readership rates are among those ages 18 to 29, and the same age group is still using public libraries in large numbers.

[Naomi S Baron, Words Onscreen: The Fate of Reading in a Digital World, Oxford University Press, 2015]

Don Kilburn, North American president for Pearson, the largest publisher in the world and the dominant player in education, said the move to digital “doesn’t look like a revolution right now. It looks like an evolution, and it’s lumpy at best.”

[Naomi S Baron, Words Onscreen: The Fate of Reading in a Digital World, Oxford University Press, 2015]

ebooks … 13%paper … 87%

Students’ attitudes, beliefs and expectations

• Some lectures are boring (what are the objectives of a lecture? Is it a good idea to make everything non-boring?)

Why are students bored? Because many do not want to be there in the first palace, but have to?

Why is there a need to replace lecturing (“sage on the stage”) instead of embracing the best strategy for a particular learning objective? Whenever someone criticizes lecturing they do so by lecturing

Students expect that we (instructors) do more for them, thus further shortening time they study:

posting lecture notes and/or summaries of lecturesprepare study sheetshold review sessionspodcasts of lectures

… and we do it!

why don’t students do it themselves? they are the most connected, tech savvy of all generations

Landscape:

Students expect justification (or we tell them to demand it) for many things they do in their math courses (and elsewhere)

“Why do I need this?”

“When will I ever use this?”

J.M.W. Turner (1775-1851) “my work was not to be understood but to show what such a scene was like”

Hamilton, ON sunset

CTH

Hamilton, ON sunset

CTH

Steve Jobs, 2005 Stanford Graduation Speech

I decided to take a calligraphy class ... I learned about serif and san serif typefaces, about varying the amount of space between different letter combinations, about what makes great typography great.

None of this had even a hope of any practical application in my life. But 10 years later, when we were designing the first Macintosh computer, it all came back to me.

And we designed it all into the Mac. It was the first computer with beautiful typography. If I had never dropped in on that single course in college, the Mac would have never had multiple typefaces or proportionally spaced fonts.

And since Windows just copied the Mac, it’s likely that no personal computer would have them.

Thank you!lovric@mcmaster.ca