Back to NCTM San Diego: Presentation Summary
Tom Edwards - Wayne State University; Detroit, MI
(tedward@cms.cc.wayne.edu)
(mcruz@servidor.unam.mx)
(erickson@selway.umt.edu)
Slide | Slide | SlidePart II: Some Comments From a Mexican Perspective (Armando M. Martinez-Cruz)* the 7th item in the menu calculates a definite integral and shades the corresponding area under the curve.
* first one must set the limits of integration. On the TI-82, this is similar to the TRACE function.
* we began with the interval [-1,1].
* it was interesting to me to note the power the visual image had for these (mostly) mathematically unsophisticated students.
* it is important to pose questions, because the power of using the technology resides not in the technology itself, but in the students interactions with it.
* some of the QUESTIONS that I posed:
* THEIR response: The proportion of the total area which is shaded.
- What is the calculator doing?
- How do you interpret the numerical read-out (.68268949)?
* this area is of interest for the purpose of comparison.
* the calculator treats the definite integral which has been estimated as an "Ans" which can be stored to some variable (e.g., A, B).
* I think this particular pair of questions is important, because it connects a numerical relationship with a geometric relationship.
* in considering this graph and the one on the next slide, we began to discuss the relationship between this curve and the statistical data it represents.
* QUESTIONS:
Slide
- if the graph represents the proportions of a population whose value for some variable (height, weight, income, IQ, ...) falls in a particular interval, then where should we place the mean? what do the values x=1, x=-1, x=2, x=-2 represent?
* having established x=0 as the mean and a scale in standard deviations, we were then able to interpret this graph in terms of a particular variable.
* suppose the graph represents the results of a large scale administration of a test on which the mean is 500 and the wstandard deviation 100.
QUESTIONS:
Slide
- where should we place a score of 500?
- what scores do x=-3 and x=3 represent?
- what per cent of the population can be expected to score between 200 and 800?
- do you know of any tests which report scores like these?
* throughout this investigation, I did not explicitly state the function, because I wanted the students to focus on properties of the graph of the function, rather than be intimidated by its algebraic definition.
* this is a so-called "friendly" window, because in TRACE mode, moving the trace-point one pixel to the left or right causes a change of 0.1 in the value of x.
The second example is from a content course for elementary education students who are completing a teaching major or minor in mathematics.
* the system y = f(t), x = t is equivalent to y = f(x).
* in this window, -2[[pi]] <= x <= 2[[pi]] and -2 <= y <= 2.
* I like to graph inverse relations parametrically, because I believe that doing so emphasizes the interchange in the roles of the x, y variables.
* to produce this graph, some care needs to be taken in setting up the window.
* this set-up is not difficult if one thinks of the inverse in terms of interchanging x and y.
* so in this window, -2 <= x <= 2 and -2[[pi]] <= y <= 2[[pi]].
* QUESTIONS:
* when students respond (as they invariably do), "Because it fails the vertical line test," it is important to ask some additional questions:
- does this graph depict a function?
- why/why not?
Slide
- why does it mean that it's not a function?
- what part of the definition of "function" is violated?
* at this point, the power of the parametric representation can be utilized, because the calculator will produce the graph only for those values of t (and consequently y) which have been defined in the WINDOW set-up.
* QUESTION: how could we resrtict the y-values so that the graph would represent a function?
* I see this as a "guess-and-check" activity at first.
* I use whatever the students suggest and let them make refinements as necessary.
* typically, they will suggest something like the interval [0,2[[pi]]].
* what if a student immediately responds, "Set Tmin=-[[pi]]/2 and Tmax=[[pi]]/2"?
* before looking at the resulting graph, I would ask someone else, "Why do you suppose (s)he said that?"
* BUT:
Slide
- why is it the graph of a function?
- why not use [[pi]]/2 <= y <= 3[[pi]]/2?
- how many potential intervals are there?
- why do you suppose the interval [-[[pi]]/2,[[pi]]/2] was chosen instead of any other possibility?
* a natural extension ...
* can you explain why this graph depicts a function without referencing the "vertical line test"?
* part of the experience involved experiences with inservice high school teachers in Mexico.Part III: Electronic Journals in Teacher Preparation Courses (David Erickson)
This presentation is divided into three parts. The first outlines the current reform in mathematics education in the U. S. and in Mexico. The second part presents two courses for inservice mathematics teachers which illustrate how teachers cn be introduced to technology. The closing part offers some conclusions.
As in the U. S., Mexico is going through a reform in mathematics education. In both countries, teachers are seen as key figures in the change, andthis change needs to be supported. The NCTM Professional Teaching Standards in the U. S. outline what teachers are expected to do. Related Mexican documents contain ideas in the same vein.
In this section, we outline two courses in which we have teachers learning about technology and learning and teaching with technology. We share Bright and his colleagues' (1994) thoughts ....
One of the courses we have been teaching is titled "Educational Technology." The objective of this course is to discuss the impact of graphing calculators in mathematics education. In order to discuss such an impact, teachers need to be familiar with graphing calculators. To accomplish this, teachers were asked to choose and present one of the menus of the TI-85. Their presentation had to involve examples. Being familiar with the calculators provided an opportunity to ask teachers to write activities for their classrooms centered around a topic of their own interest and need. Later, these activities were presented to their classmates as a means to evaluate the activities formatively.
Still later, teachers were introduced to the CBL and asked to get organized in teams and present an experiment in class. As a final paper, teachers were asked to write a document to be submitted for publication in a mathematics education journal in Spanish (e.g., Educacion Matematica) in mexico. This requirement is really two-fold. On one hand, teachers are asked to write a final paper that incorporates all the ideas, if possible, and the technology presented in class (including the CBL, TI-85, and Graph-Link software). On the other hand, we want to increase the number of publications in Spanish dealing with graphing calculators. Currently, there are relatively few such articles, with some specific topics and levels hardly represented at all.
The second course is a caluclus course and teachers are introduced only to the graphing menu of the TI-85. From a graphing perspective, teachers revisit (and experience) calculus topics. This course intends to discuss the following situations:
Slide
- graphing as a means, not as an end,
- rethinking the content,
- new opportunities to teach and to learn, and
- reflection about students' learning.
* Teachers are involved in the change.
* Developing activities and involvement in the reform of mathematics education gives teachers a sense of ownership.
* Of particular importance is that training is conducted by people who know the classroom and what teachers need.
Three teacher prep courses:elementary mathematics content
secondary mathematics methods
instructional media
Move into groups of 2 or 3 and discuss what comes to mind from these three words: Montana, mathematics, technology
Answer: excellent mathematics program: SIMMS
or
unabomer!
* electronic journal groups of 3 or 4 or 5 students
* methods students grouped by school
* students write to each other, not just to professor
* sharing of experiences from field experiences and from class activities adds another dimension to the "coursework"
Student voices through their journals communicate the value as seen by students.
Note that communication of mathematics is important!
Although ideally we would be able as teachers to talk with each student each day, we are not able to hear them all; a journal may provide an insight into more of the students for the teacher.
Megan uses the same elementary math content book Tom mentioned by Billstein, Lott, and Libeskind.
Variety of methods for solving problems is an attraction of mathematics.
Journals with others allow us to learn from others, as well as oneself.
Sunny does not like journals, but still searches for reasons that support using journals.
Value in asking questions as the framing of a good question leads to finding solutions.
Julie recognizes the value of asking questions, and modeled that in her own journals.
* Electronic journals have the advantage of:
- immediate feedback
- response at a convenient time.
Eli focuses on the value of journals promoting thought, and how promoting thinking is very difficult.
* Rachael responds to a journal prompt I gave regarding involving parents in the child's learning: she reflects on having written journals with her students that parents read and respond to also.
* Before proceeding, separate into groups and have them respond to each other on the following setting:
In your current position, you could encourage reflection from your students or colleagues by using journal prompts such as ___.
Collect comments from audience after several minutes
Definitions of reflection.
Dewey, 1933, suggests teachers need to be reflective. The process of a learning loop is described.
Schon, 1983, fifty years later, encourages reflective practice on the part of teachers.
This slide summarizes the entire presentation.
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