Points and Angles Online
September 2001

    Table of Contents:

Start Visualizing Mathematics in the Heart of the Chicago Loop
Robert L. Devaney

Although we all know that the Como Inn has made many changes over the past 20 years, nothing could have a much greater impact on MMC than their announcement that they were closing for remodeling. For MMC, it causes a moment of chaos! So, as we embark upon the 2001-2002 year, MMC is going to experience “Adventures in Chaos” at the famous Berghoff Restaurant, which is located in the heart of the Chicago Loop. Robert L. Devaney of Boston University will deliver a talk entitled “The Adventures in the Chaos Club: Online Activities Involving Chaos and Fractals.” Bob’s talk will take us on a journey that will demonstrate a number of different internet-based tools that can be used to introduce topics from contemporary mathematics in both the middle school and high school curricula.

All who have heard Robert Devaney speak can tell you how dynamic he is. Bob has given over 1,000 invited lectures in 49 of the 50 United States. He has also lectured on 6 continents worldwide. Robert Devaney has taught mathematics at several universities and he has held the position of chairman of the Department of Mathematics at Boston University. Dr. Devaney has also authored or edited numerous research papers and books, and he has developed software packages designed to teach students at all educational levels about the mathematics underlying captivating images. In addition to all of these individual accolades, he is also the director of the National Science Foundation Institutes and Projects.

Robert Devaney believes that one of the most interesting applications of technology in the mathematics classroom allows teachers to bring topics, such as chaos and fractals, that generate images of objects which capture students’ interest and enthusiasm. Bob has found that “chaos club” activities allow students to discover mathematics topics! He plans to show us, as classroom teachers, how to find chaotic adventures on the internet. Therefore, MMC will experience chaos in the heart of the Chicago Loop (on September 14th) as we embark on a new year filled with changes while Visualizing Mathematics!

REMEMBER!!  You can earn CPDU credit for attending this!

Developing Proportional Reasoning:  So Important, So Widely Ignored
Steve Leinwand

What do the NAEP, buying straws and porta-pottys in Hartford have in common?  “Not much!” might be your first answer, but Steve Leinwand provided the connection in his May talk at MMC, a fitting final event - for the time being - at the venerable Como Inn.  As always, Steve spoke with flair, humor, and energy - and gave the group significant ideas to take home and ponder.  The main theme of Steve’s talk was proportional thinking - and the lack of it among many US students. Beginning with “simple” problems in rates and proportions from the NAEP, Steve asked the audience to guess the percent of US students who had gotten them correct.  When he noted that only 12% of students could manage these problems, he had the audience’s attention focused on a dilemma faced by all math teachers.  He noted that there is a large chunk of the curriculum that students “simply don’t understand.”  Instead most students fall back on the strategy of cross-multiply and divide, a strategy which leads to mistakes rather than understanding.  Steve noted that proportions and ratio do not lend themselves to drill and practice in order to achieve mastery. Instead, they involve proportional thinking, which is akin to number sense or data sense, and which is developed by a wide variety of experiences over time.  Furthermore, proportional thinking is a unifying mathematical idea, appearing over and over again in the curriculum, as number ratios, fractions, percents, slope, and in many other situations.  Yet, when looking at the time spent in the average text on key ideas, proportional thinking comes up short.  For example, place value is touched on over six years with 20 weeks spent on the ideas embedded therein; proportional reasoning is approached in only two years, for a total of about six weeks of study.  In addition, the problems often presented in standard 7th grade texts are similar to this one: It costs 12 cents to buy 20 straws.  How much will 8 straws cost?  Using the audience as his classroom, Steve asked people to propose a method to solve the problem.  It soon became clear that there would be many different approaches - all of them leading to a correct answer.  He also showed us student approaches - making a table, drawing a picture, and one where the student began a complex method and then simply gave the correct answer.  As the speaker noted “Only the impoverished need to multiply and divide!”  Steve noted that proportions are a representative case for teaching math in a better way.  The speaker posed this question to the audience, “Why does the average student learn to read more effectively that to do math effectively?”  The process of learning to read occurs at many levels of comprehension: literal comprehension (Where did Jim go?), inferential (Why did Jim go?), and the evaluative (Does it make sense that Jim went?).  In math class, we often don’t ask why, engage students in meaningful discussions of the inferences which can be drawn from a problem, or require students to assess their proposed solutions and thinking strategies.  Steve urged the audience to consider how we approach problems in our classrooms and ask ourselves, “Do the students gain the rich experience they need to develop proportional thinking?”  Oh, and what about those porta-pottys?  At the rally to welcome the national champion women’s basketball team back home to Hartford, the crowd expected during the three-hour event over lunch was 200,000.  How many porta-pottys would be needed?  He pointed out that this was a rich and interesting real problem - and that only 100 were actually set up!  Steve ended with this problem which you can give your students to assess their proportional thinking.

A Weighty Matter:  Three people are dieting.  Here is a chart of their recorded weights over the first month of the diet.   Who is doing best?
Week    Tom’s Weight     Jane’s Weight      Rhonda’s Weight
0               210 lb.                    158 lb.                    113 lb.
2               202 lb.                    154 lb.                    108 lb.
4               196 lb.                    150 lb.                    105 lb.                        -Virginia Highstone

2001 NCTM Central Regional Conference

On behalf of the National Council of Teachers of Mathematics and the Wisconsin Mathematics Council, you are invited to attend and participate in a great professional development opportunity, the Central Regional Conference 11-13 October 2001 in Madison, Wisconsin!  The 2001 Conference, Forward: Shaping the Future, will offer opportunities to envision mathematics for the future and practical ideas for the present. You may get a preregistration form or lodging form by visiting the NCTM website:
<http://www.nctm.org /meetings/registration>
or by Fax on Demand at 800-220-8483, document #413 or by calling NCTM at 800-235-7566.
Also check out <http://www.wismath.org>
Lodging can also be secured by calling 800-765-1726.  Note: Preregistration ends September 14, 2001.  Don’t delay - Preregistration saves you time and money!  After September 14, registration is on-site only.


...the Principia not only gave us the three laws of motion that still bear his name; it defined gravity, and provided precise mathematical equations by which it could be measured.

Source:  Jennifer Lee Carrell, "NEWTON'S VICE", Smithsonian, Smithsonian Institution, Washington DC, December 2000, Volume 31, Number 9, p. 132.

A. Comitial  B. Attitude  C. Rhomboid  D. Rhapsody  E. Equality
F.  Leonardo  G. Latitude  H. Nuthatch  I.  End point  J. Whenever
K. Theorems  L. Offsides  M.Negative  N. Shanghai  O. Verbatim
P.  Implants  Q. Crabwise  R. Epicycle

Points from the Interior

Welcome to a new and exciting school year!  I know for myself that summer was a time for reflection about who I am as a teacher and what I can do to become the best I can.  When I interview math candidates, I always ask them to think about a teacher they have had who stands out.  What is it about this person that you remember and then to have the candidate make this connection to describing what characteristics a great teacher displays.  So, I want to ask you the same reflective question, what continues to bring you back every year as a mathematics teacher and what qualities do you consider a great mathematics teacher possesses?  We heard John Benson last February talk about ideas of making a good teacher better.  This summer, there was an article in the newspaper about a Chicago high school math teacher who just completed his first year and is coming back for his second year. He no longer needed his goatee, which he shaved off, because he figured out who he was as a teacher and did not have to hide behind the goatee.  There was another article about a month later about a woman who taught a primary grade for two years at a Catholic school in the inner city.  She realized that getting to know the students and showing them that they can trust her and learn was what she was all about.

I often wonder, do my students really retain and make connections with the material I encouraged them to learn until I have a former student come back and let me know what they are currently doing.  I then realize that, yes, they have learned something from me and from everyone else who had the opportunity to teach this student.

We are in the best profession ever!  It is not because we get the “summers off” but it is because we get to work with the youth of today.  We get to facilitate their learning.  We get to challenge them.  We get to watch them grow and succeed.  HAVE A GREAT YEAR!!
                                                                                                             -Fern Tribbey

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