Points and Angles Online
November 2000

    Table of Contents:

How Can Priciples and Standards for School Mathematics (PSSM) Make a Difference for
Glenda Lappan
University Distinguished Professor
Department of Mathematics
Michigan State University

We will examine the major content and intstructional stances of the document with an emphasis on middle and high school.  Be prepared to do some thinking about both mathematics and teaching as we look at the examples from PSSM and surrounding NCTM activities.

Glenda Lappan has been on leave from the university from 1997 to 2001 to serve as President and Past President of the National Council of Teachers of Mathematics.  Her research and development interests are in the connected areas of students' learning of mathematics and has published over a hundred scholarly papers and numerous books for middle grade students and teachers.  She has served as Chair of the middle school writing group for National School mathematics (1989), and as Chair of the Commission that developed the NCTM Professional Standards for Teaching Mathematics (1991).  Glenda also has served on the NCTM Board of Directors from 1989 to 1992 and from 1997 through 2001.

Please assist me in welcoming Glenda Lappan on November 17!

 John McConnell announced a website to check out of the Chicago Academy of Sciences.

Doctoral Cohort Group in Math Education to begin Fall 2002.

Given sufficient interest, Illinois State University's Mathematics Department will sponsor a second doctoral cohort group for Chicago area teachers: AY classes in Chicago, summers and one semester residency at ISU. For details contact Dr. Carol Thornton:

(309) 452-8636, nights OR e-mail:  thornton@ilstu.edu
Letter of intent required by May 1, 2001.

Answers to October's Mathematical Word Permutations

1. ten, net  2.trio, tori  3. seven, evens  4. share, shear  5. eighth, height  6. recount, counter  7. couplet, octuple  8. contour, crouton  9. aligned, leading  10. enlarge, general  11. inverse, versine  12. triangle, integral  13. highest, eighths  14. centiare, iterance  15. altitude, latitude  16. algorithm, logarithm  17. sixty-nine, ninety-six  18. six-sevenths, seven-sixths  19. nineteen hundred seventy-six, seventeen hundred sixty-nine  20. eleven hundred and four thousandths, four thousand and eleven hundredths

Points from the Interior

        The new NCTM document, Principles and Standards for School Mathematics, had its debut last spring at the annual meeting in Chicago. It was an exciting and well orchestrated event; quite unlike the debut of the Standards in 1989. Over the decade of the 90’s, many of us were busy trying to understand and implement those standards, along with the Teaching Standards and the Evaluation Standards, which came out later. They gave us cause for reflection of our own practices and a means to evaluate their effectiveness. The new PSSM are going to be of even greater value. The six principles (Equity, Curriculum, Teaching, Learning, Assessment, and Technology) provide overarching themes; and the ten standards (Number & Operations, Algebra, Geometry, Measurement, Data Analysis & Probability, Reasoning & Proof, Communication, Connections, Problem Solving, and Representation) provide a “connected body of mathematical understandings and competencies...for all students”. For those who are concerned about the Illinois Learning Standards, you will find much in common between the two documents. The Applications of Learning and the Goals are found throughout the PSSM.

        The new standards are the theme of our workshop in February and of our November meeting. We are pleased that Glenda Lappan is going to join us in November. She will provide us with some valued insight. There is more about her talk on the front page. However, as members of a professional organization that has endorsed the PSSM, we need to be proactive is communicating these new standards to our colleagues, our districts and our communities. We need to present a clear and concise message as to what they say and what they do not say. For example, they are not prescriptive in how we should be teaching; but they are emphatic that all students should be given a quality mathematical education. The opening vision is significant. Imagine a classroom, a school, or a school district where all students have access to high-quality, engaging mathematics instruction. There are ambitious expectations for all...knowledgeable teachers...The curriculum is mathematically rich...Technology is an essential component...Students are flexible and resourceful problem solvers...” The PSSM present a challenge for each of us, the rookies, the not-so-rookies, and the we-can’t-ever-remember-being-rookies. It is an exciting time for mathematics teachers. For those who have not had the opportunity to read the new PSSM, the November meeting is a great place to start to gain an understanding of them.

        For those who could not come to the October meeting, I’m sorry to have missed you, but the wine was good (with much thanks to Rita McGuire of Prentice Hall), the fellowship was great and the talk by John Diehl was excellent. See y’all on the 17th.

Summary by  Don Porzio

         The second dinner program talk of the 2000-2001 MMC season saw local boy John Diehl present an enlightening and sometimes humorous look at “Statistics throughout the Secondary Curriculum.” Currently the chair of the Mathematics Department at Hinsdale Central High School, John has also been, among other things, a mathematics teacher, a College Board consultant, a calculator consultant, an AP Exam Reader, an AP Statistics Teacher, and a member of the AP Statistics Test Development team. It was from the perspective of these latter three positions that John gave us his thoughts on what statistics should look like in the secondary school mathematics curriculum and why it is important. Before John could begin, he wanted some agreement from the audience on what was meant by statistics. Without much in the way of arm twisting on John’s part, everyone agreed that two main themes in the study of statistics are data analysis and probability. Such a viewpoint fits nicely with the viewpoint of the new NCTM Principles and Standards for School Mathematics, which submits Data Analysis and Probability as one of the main areas of study for grades 9-12. But, as John pointed out, statistics should be studied not just because it is included in the NCTM PSSM, or because it is needed to prepare students for AP class, but because it is essential to our students’ development as knowledgeable citizens. People are bombarded daily with information presented in a variety of ways, and if they lack basic understandings of how to analyze this information that the study of statistics can provide, they will have difficulty making informed decisions.

        According to John, the College Board recommends that the following general topics be addressed as part of the secondary school statistics curriculum: Exploratory Data Analysis, both univariate and bivariate, Planning a Study, Patterns – Probability, and Statistical Inference. The remainder of the talk mostly focused on three specific areas of study taken from these topics: graphing, data analysis and probability. The topic of graphing univariate data was divided into two subcategories, content and skills. The content areas for graphing included the different types of graphs that might be used to display the data (e.g., dot plots, stem plots, box plots, histograms, and bar graphs. One set of graphing skills included those skills needed to present graphical information clearly and intelligibly, such as accurate sketching, appropriate scale, and correct labels. Key here was having students recognize and understand what is needed for a graph to communicate its information well.
 A second set of graphing skills included those skills needed to interpret graphical information, such as looking for shape or symmetry, centering, and spread. One of the highlights of the evening came during this discussion. John presented a (PG) Top Ten list of best answers to a data analysis question from the 2000 AP Statistics Exam that concerned interpreting two graphs of data on the flexibility of young and middle-aged men. I’m sorry to say that due to confidentiality restrictions, the Top Ten list cannot be shared with our loyal readers, which is really too bad since it invoked more laughter and applause than Letterman ever gets with his lists. However, if you get the chance, be sure to have John tell you about AP students’ thoughts on the shape and flexibility of older men, on how older men compare to younger men in terms of scoring and climaxing, and on what stubbornness has to do with flexibility.

        Once everyone caught their breath, John moved on to briefly discuss content for univariate data analysis (mean, standard deviation, median, quartiles, and so on) before moving on to graphing and analyzing bivariate data. John noted that because of the improvement in “number crunching” capabilities over the past 30 years, what is more important now are not the skills needed to generate statistical values like correlation coefficients, but instead are the skills needed to interpret relationships between data sets displayed graphically. Key here also, as it was with graphing univariate data, is developing students’ ability to graph data that clearly communicate the desired information. From graphing and data analysis, John moved on to probability. You could tell from the passion in John’s voice the importance that John places in having students experience probability properly. The study of probability was divided into five categories: Probability as Relative Frequency, Probability as Ratio, Sample Space, Geometric Probability, and Simulations. Numerous examples of probability problems were presented or suggested. Among them was a problem where you need to roll a one on a six-sided die to place a head on the Cootie Bug (remember that game, don’t you) and want to know what the chance is you will be able to place that head during the next ten rolls. Another involved choosing two random numbers between 0 and 1 and determining the probability that their sum is greater than 1.2. John pointing out the importance of giving students the opportunity to “play with” probability and suggested that it did not matter if the context of the situation used was contrived (has anyone ever actually taken colored marbles out of an urn) or realistic as long as students just did it. He also mentioned that he viewed simulations as a great way to expose students to probability and provide examples of some of his favorite simulations, like the Birthday Problem and Let’s Make a Deal.

        The talk finished with one of the other major highlights, a demonstration of the Key Curriculum Press software program FATHOM?. Using a drag & drop interface, this data analysis software has the capabilities to quickly and easily create a variety of different plots or graphs from sets of data and import data sets directly from Internet web sites. While the audience stared on in rapt amazement, John provided a number of different fascinating examples of ways in which the software could be used to analyze different sets of data interactively. One example demonstrated how the software allowed the user to generate a best-fit line that included a visual representation of the squares of the residuals and then interactively manipulating the line while viewing how the changes impacted up of the sum of the squares of the residuals. It was a very powerful demonstration for anyone who had never seen it done before. And throughout the demonstrations, John pointed out that the power of this software lies not just in its abilities to make real data readily accessible and to quickly manipulate data, but also in its ability to free students to focus not on mechanics but on interpreting data and understanding concepts. When he finished with the demonstration, John could have easily sold 30 or 40 copies of the software, so powerful was the allure of its potential use in the classroom.

        In conclusion, John provided us with a list of key elements to consider when including statistics in our secondary school mathematics curriculum. Key among these were using real data, incorporating current topics, analyzing graphical and numerical information, interpreting graphs, using mathematical models, giving students lots of opportunities to explore probability, and making sure to have fun. I know that I had a great deal of fun listening to John’s illuminating and edifying presentation, and I greatly appreciate him taking time out of his busy schedule to share his thought-provoking insights about the teaching of statistics with us. In the end, the only things missing were reprintable copies of the Top Ten List and free copies of FATHOM?. Thanks John, and you too Fern for going two-for-two in choosing great speakers this year.     — Don Porzio

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