Pacific Institute for the Mathematical Sciences - PIMS

8:00 - 8:45 Registration

8:45 - 9:00 Welcome

9:00 - 10:00 Plenary Talk: The Study of Living Things: So, What's Math Got To Do With It??
Leah Keshet (Mathematics, University of BC)

10:00 - 10:30 Coffee break

10:30 - 11:30 Small Group Discussions: Can biology be a major context for math classes? How do visualization and logic interact in mathematics? Is applied math easier than pure math? How can people be taught to like mathematics?

11:30 - 12:00 Report from groups

12:00 - 1:30 Lunch

1:30 - 3:15 Panel Discussion: To what extent is an appreciation of mathematics possible without mathematical training?
Mike Fellows (Computer Science, University of Victoria)
Kanwal Neel (Mathematics, Steveston Secondary)
Jeremy Quastel (Mathematics, University of Toronto)

3:15 - 3:40 Coffee break

3:40 - 4:40 Small Group Discussions: Continuing from the morning.

4:40 - 5:00 Coffee break

5:00 - 6:00 Public Talk: Ingenious mathematical amateurs: M.C. Escher (artist) and Marjorie Rice (homemaker) Doris Schattschneider (Mathematics, Moravian College)

Saturday, February 20, 1999

9:00 - 10:00 Plenary Talk: Would Pythagoras have liked Mozart?
Adrian Lewis (Mathematics, University of Waterloo)

10:00 - 10:30 Coffee break

10:30 - 12:00 Panel Discussion: Mathematics and the Arts: where do they meet?
Ron Coleborn (President, BC Assoc. of Math Teachers)
Doris Schattschneider (Moravian College, Pennsylvania)
Owen Underhill (Contemporary Arts, SFU).

12:00 - 12:30 Summing up, group reports and closing comments

The study of living things: So, What's Math Got To Do With It??
Leah Keshet, UBC
Abstract:
Mathematical methods have been applied to many exciting areas in the life sciences. I survey some of these applications, with emphasis on developmental biology and pattern formation, the motion and aggregation of animal groups, and the dynamics of biological polymers. In many of these areas, mathematical ideas allow us to use knowledge about the behaviour of the individual (or unit) to understand the behaviour of a population. My lecture concentrates on some of the classic examples of topics in biology to which mathematics has made a contribution.

Would Pythagoras have liked Mozart?
Adrian Lewis, University of Waterloo
Summary (by Klaus Hoechsmann)
An abstract by the lecturer is not available for this lively and polished talk with guitar accompaniment. The well-tempered scale was "derived" from the search for a rational number r = m/n such that the r-th power of 2 approximates 3/2 (the relative frequency of the "dominant" or "perfect fifth"). Via the continuous fraction expansion of the appropriate logarithm, this leads to the good old 7/12 of classical music, but also to such oddities as 24/41 and 31/53 -- the latter allegedly known to a certain emperor of China. Of course, all these rational powers of 2 are irrational and would have displeased Pythagoras -- which did not stop Mozart from starting one of his quartets with a "geometric" bisection of the octave. Though smooth and elegant, the talk ranged widely and came to rest on the Method of Archimedes.

It is said that statistics consistently show mathematics at or near the top of student preferences up to Grade 4 -- and at the bottom of this list by the end of Grade 9. What do you think happens in the interim? Why and how?

Could mathematics become an elective beyond Grade 7 or 8? If so, what common core should be emphasized before this divide, and what attractive areas could be opened up after it?

Much mathematics existed before modern notation (c2=a2+b2, etc.) became commonly used just over three centuries ago. For which aspects of mathematics is it really necessary -- and where is it only complicating things by introducing a language barrier?

Can biology be a major context for math classes?

Over many centuries mathematics has grown up in symbiosis with astronomy and physics. Today other sciences and activities are being "mathematized". One example is biology. Can you identify others?

In physics, certain equations (Newton's, Maxwell's, etc.) actually constitute the theory (i.e., the medium is the message). In biology, the mathematics usually provides a "model" which can be adjusted and changed. How does this complicate and/or enrich its use in school?

Try to identify the parts of the math curriculum which can be taught in the context of biology or other newly mathematized fields. Please be specific and consider questions of methodology.

Suggested exercises

1. A species of algae living on the surface of tropical ponds grows so quickly that the area it covers is doubled in every eight hours of daylight. An experimental pond seeded with these algae was found to be totally covered at 4 p.m. on February 18.
* At what time was it half covered?
* How much of it was covered at noon on February 18?

2. Most drugs are removed from the blood-stream at a constant rate, i.e., by a fixed percentage every so many hours. The blood level of alcohol, for instance, decreases by 50% every 4 hours. Therapeutic drugs, however, are designed to "decay'' more slowly.
A patient was given a high dosis of metronidazole (an antibiotic). Its concentration in his blood, hour by hour, is shown on the clock dial in the diagram on the left. At 3 a.m. it was 10 mg/l, at 11 a.m. it is down to 5 mg/l.
* What was it at 7 a.m., 3 p.m., 7 p.m.?
* How can you compute these concentrations using only the simple calculator provided?
* What about all the other hours?

3.The nautilus is a tropical marine mollusc, shown here on the right. How would you account for the similarity between its shape and that of the blood-level graph above?
If its volume grows exponentially should the radius of its shell not grow more slowly?

4. Male bees have only one parent (a queen) while female bees have both a mother and a father. How many great-great-great-great-great-grandmothers does a female have?

5. Phenylketonuria (PKU) is a genetic disease afflicting people who receive a certain recessive gene from each parent. Although none of their four parents suffered from it, Bill and Jane know that it runs in their families: Bill's sister and Jane's brother both have it. What is the probability that their first baby will be born with it?

Report by Robert Camfield and Malgorzata Dubiel

The Green Group was asked to consider two questions: "Can biology be a major context for math classes?", and "Can people be taught to like mathematics?". However, we never really got to deal with either of these questions: the first one did not generate much interest, and the discussion of the second quickly became an attempt towards defining and clarifying more fundamental issues of curriculum content and delivery in the context of the purposes and problems associated with math education in schools.

What is the teaching of math meant to accomplish at high school and how are we going about achieving these aims were questions underpinning our discussion. There was general agreement that the goals of a math education should include the development of abstract, creative, structured, disciplined thinking and the ability to do multi-step analysis and cope with applications to "real world" problems. There was no consensus, however, on how to best achieve these aims. A series of subsidiary questions were explored:
(a) What IS mathematics vs. the image of mathematics we create in schools;
(b) Is there a need for a "basic toolkit" of mathematics knowledge and ideas, and if so, what is it? What is it the students need and why it is important?
(c) Should math education remain concrete for as long as possible or should abstract thinking be emphasized at early age (e.g., at grade four level)?
(d) What are the relative merits of curricula based on math "applications" as compared to "pure" math?
(e) Why students get turned off math?

The recent changes in the Ontario system created a lot of interest, in light of the BC experiences with the new curriculum. A wide ranging discussion about the current BC high school math curricula followed (the content and the skill sets emphasized in Grade 12 "Principles" - as compared to Applications - courses, for instance), and what needs to be done to reduce the high attrition rates and growing math incompetence. There was some discussion regarding the resistance universities have to the growing trend towards applications type courses because of the deficiencies in math and cognitive skills that seem to accompany this emphasis.

The question what students should learn and how (proportion of drill compared to "play", and rules versus intuitive approaches, for example) was the one unifying the discussion throughout. And, while we did not come at all close to answering "what" (defining the "toolkit"), we did have some suggestions on "how". To achieve the goals stated above, the value of teachers who love math and can pass their enthusiasm to students was underlined; working with rich, absorbing problems and developing skills and tools as needed was suggested.

The associated problem of how to best evaluate math competency at the various levels was likewise discussed, but not answered.

How do visualization and logic interact in mathematics?

Most people can manage (for a while) to think entirely in pictures, but few seem to be able to think blindly by logic alone. Try to come up with mathematical problems which are as image-free as possible. Watch your mental processes carefully as you ponder them (e.g., notice when visualization passes from objects to symbols).

Traditionally, geometry has been the training ground for mental image processing under strict logical control. What other mathematical activities are now available to play that role? Do they play it as well? How and at what stage should they be integrated into the curriculum?

How much detail do you actually "see" in the visualizations surrounding mathematical activity? Is there a difference between imaging and imagining? Discuss the pros and cons -- for the mathematics learner -- of the very explicit detail in modern computer graphics.

Suggested exercises

1. Two glasses each contain 100 ml of wine, one of them red, the other one white. A 5 ml spoonful is taken out of the red wine glass and added to the white wine -- giving it a faint blush. Now 5 ml of the blush wine are put back into the red wine glass. Which of the two glasses contains more "foreign" wine at that point?

2. You are led into a room with four doors, one of them hiding a car (the prize you hope to win), the other three nothing of value. The Quizmaster asks you to point to a door, and when you do so, marks it with an X. Then he opens one of the other doors, and shows you that there is only junk behind it. "Take another guess", he says, and when you do so, marks the new door with an O.
* What is the probability that the car is behind the door marked X?
* What is the probability that it is behind the door marked O?

3. The square shown here on the right has an area of sixteen square units.
* Draw in a square having only eight square units of area.
* Repeat the exercise with ten square units instead of eight.
* How many squares are there of each type?

4. Let us say that a pyramid is "perfect" if it has a square base and all eight edges are equal in length. Now imagine taking six perfect pyramids of equal size, and stacking them in such a way as to outline a pyramid whose edges are twice as long.
Around the base of the big pyramid, this will leave four sizable gaps. Describe the shape of such a gap. Compare its volume to that of the original pyramids.

5. Vaclav's apartment in Prague (shown as P in the diagram below) has a big southern window, where he likes to sit and admire the sky.
His latitude is 50 degrees north relative to the equator E, but his elevation from the earth's orbital plane (shown here as a horizontal line) is constantly changing, because of the 23 degree tilt of the earth's axis with respect to that plane.
At what angle of elevation does he see the sun at the beginning of each of the four seasons (spring, summer, autumn, winter) ?

Report by Pamela Hagen

Although not directly followed, the two questions provided for Group Two were an excellent basis for discussion. How do visualisation and logic interact in mathematics? And can people be taught to like mathematics?

The ensuing speculations about possible links between visualisation and the development of reasoning, beginning with the very young, led to examining the contexts in which mathematics is learnt. The importance of connecting formal study with the real world was a point of agreement, but the means of getting context and relevancy into the classroom raised various open-ended questions.

Clear for all levels of mathematics education was the importance of qualified teachers. This was identified by a number of participants as an area in which PIMS could have a role -- specifically by sending letters emphasising the importance of a mathematics qualification for all teachers, since mathematics is considered a core subject.

It was also felt that teachers needed ongoing support from a variety of sources in order to continue being active learners. Exactly how and by whom this should be provided was not spelled out, but again it was thought that PIMS might help connect teachers across and between different academic spheres (i.e., schools to colleges to universities).

Again and again the curriculum was mentioned as an area of great concern, the current one being seen as too packed with extraneous things to allow for enrichment or the development of a holistic view. Unfortunately there was not enough time to discuss in detail the appropriate topics to be included or excluded.

Time was considered to be a big constraint on opportunities to become more "creative" with the material -- with quality regarded as much more important than quantity. A variety of suggestions were discussed including having a core content which would cover a large portion, but not the whole, of the curriculum. It was also felt that teachers should have some freedom to create enthusiasm and not be driven so much by the way in which topics would appear or not appear on provincial exams.

Comments by Djun Kim

Points made in the discussion
* Stressing "numeracy" does not exclude "visualization" as an important mathematical skill.
* Visualization helps build intuition -- but unaided can also lead one astray. It is important to couple visualization with logic. Example -- the common fallacy that area varies linearly with scale.
* Visualization seems to be an important component of teaching in primary grades but diminishes in its application in later elementary/secondary grades. This seems to coincide roughly with the enjoyment rate for the subject of mathematics among school children.
* "Real-world" applications such as drafting and design (e.g. using AutoCAD) require good visualization ability.
* It's not enough to present visualization as a "module" -- it must be integrated at all levels. This will take time...

Critique of organization and structure.
* not enough structure for discussion groups
* not enough focus provided by moderators
* certain participants had their own agenda to present

Suggestion for a framework.
* 5 min. introductions/icebreaker
* 10 min. solve problems in smaller groups
* 5 min. discussion in smaller groups
* 5 min. report to main group
* 10 min. assign one "large" topic to each smaller group to discuss
* 10 min. report back to main group
* 10 min. summary and conclusions.

These recommendations might help with the problems identified above. It seemed that nobody really knew what to say, so all were expressing issues of "general" concern, or going off on their own topics.

Is applied math easier than pure math?

A typical application of math is like a sandwich, with at least one extra cognitive layer on each side of the mathematical contents. How can these layers be used to enhance the latter instead of distracting from it? Give examples.

Do you think it likely that all common applications are (or will be) incorporated in readily available software? If so, how would this affect the teaching of applied mathematics?

Some topics are not much fun but useful (mortgage tables), some are fun but probably not very useful (fractals), while others seem neither fun nor useful (factoring trinomials). Make a list of math topics/examples which are both fun and useful.

Report by Klaus Hoechsmann

Instead of considering the detailed questions suggested for the first round of this discussion, the group immediately launched into a debate on the "Applications of Mathematics" programme proposed by the BC Ministry of Education. Was it yet another euphemism for the A-stream or -- as the Ministry representative claimed -- a brave new attempt to meet the challenges of the computer age?

One result was a clarification of the words "application" and "applied". It seems that they are not intended to have the same meaning as they do in professional mathematics, but rather to signal an emphasis on contextual problems instead of abstract operations. A sample problem from the ministerial web-page illustrated the distinction: as Wilbur strays from his house in an expanding rectangular spiral, one is asked to find his distance from home when he turns from east to south for the n-th time; this is clearly not a pressing real-life problem, but it does provide a context for Pythagoras and for induction.

In the afternoon, the questions suggested for the second round were at least touched upon. The idea of making mathematics an elective after Grade 7 or 8 was rejected as politically unacceptable: it would be difficult to agree on the basic numeracy core for all students, and to resist demands that all should be pushed to the highest level of math within their reach.

The remedies proposed for the distressing downward trend in students' attitude toward math (from excellent in Grade 4 to very poor by the end of Grade 9) were fairly standard: creating a more interesting curriculum, using more varied teaching techniques, and providing materials for hands-on exploration. Admittedly, the pursuit of interesting sidelines would require a thinning out of the traditional curriculum.

There was some disagreement between the two university professors in the group about the importance of rote algebraic skills -- though neither considered them to be the foremost aim of secondary mathematics. It was generally conceded that their central position in the curriculum is at least partly due to their suitability for drills and tests (e.g., 40 questions in 50 minutes).

With a second strong advocate of the new "Applications" programme having joined the group, it was led to a resumption of the morning's debate -- which eventually turned around the perceived injustice of the universities' admission policy.

The latter was defended with the argument that, in the absence of entrance examinations, colleges and universities must rely on mathematical competence acquired and certified elsewhere. As they also wish to avoid high failure rates, they are understandably wary of experimental programmes -- even promising ones -- especially when there is some doubt as to whether these can be carried out without the retraining of teachers. (It was admitted that successful runs of "Applications " had been prepared by summer institutes for interested teachers.)

On the other hand, it was pointed out, this conservative attitude of post-secondary institutions hinders the necessary periodic renewal of the school curriculum -- thus encouraging the persistence of outdated practices. With only 37% of Math 11 graduates enrolling in Math 12, whence only half go on to university, the influence of university policy can truly be called disproportionate. The reason for this double bind is the stubborn refusal of parents and students to risk exclusion from higher education.

Comments by Nataša Sirotich.

Interestingly, nobody mentioned that a plausible possibility for the "applied" math would be to incorporate it more systematically into scientific and technical subjects. Instead of dragging, say, woodworking into math, one can more naturally drag math into woodworking -- and have it really applied there. The transferability of mathematical knowledge would, I think, suffer a lot if it were "attached" all the way through, that is, if there were no pure math at all. People with a mathematics background are the most adaptable I know: they work as engineers, programmers, DB admins, teachers, economists ... you name it. (Maybe this flexibility is the most important thing people need to be taught.)

Another topic strangely absent from the discussion was "evaluation". I really believe that the core reason for children's growing dislike of school (together with math) is the way we have decided to gear them into the world of work. I mean that marks -- which later get substituted by paychecks -- do change their understanding of why they learn, they sort of spoil their innocence. Up to grade 3 it's to satisfy their own curiosity and maybe also to please the teacher -- but later they start working for marks. That's the thing that motivates them, for a great majority at least: you can't get them to do the work unless some mark is attached to it. It looks like an addiction, whenever you ask them to do something you hear, "are we getting marks for that?" If not, a good portion of them won't even bother doing it.

1998

For more information, contact conference organizer, Malgorzata Dubiel, dubiel at math.sfu.ca