Designed for prospective secondary mathematics teachers, this activity has students predict, test and justify the effects of changing parameters d and b for the logistic function family given by f(x) = a/(1+b(e)^(cx)) + d. Reflection questions draw attention to the role of claims and evidence, interpreting symbolic expressions and results, the role of the dynamic environment, and the many ways in which parameter explorations fit with secondary school mathematics topics.
Before the Activity
This activity should accomplish two general goals. First, it allows prospective teachers to generalize and reason with graphs and symbols to reach a conclusion that likely is new to them yet within the domain of secondary school mathematics. Second, it provides experience with a type of activity that can be used at many places in the secondary school mathematics curriculum.
FILE AND HANDOUT:
Prior to the lesson, students will need to have a copy of LogitFuncParam081229.tns on their handheld devices. This activity may also be done with the computer software, perhaps with minor changes to relocate the sliders to accommodate differences in screen size. The key questions in that file are captured in the optional handout.
The introduction to the lesson suggests five places in which parameter ideas might be found in secondary school mathematics. Students return to these ideas at the end of the activity and discussion. This sheet is presented at the beginning of the lesson to provide motivation for looking at what some students might see as a kind of function and activity that does not connect with what they see themselves teaching in their current or future field experiences. Prospective teachers with field experiences prior to this activity often have seen lessons about slope-intercept forms, which make a natural connection to this activity.
During the introduction, students encounter the idea of "post-punch" and "pre-punch" questions. The alliterative terms, despite the extent to which "punching" keys is no longer the totality of how students interact with their handheld units, tend to be memorable. The terms draw attention to the need to have teachers and students think about mathematics and use their technological tools thoughtfully and purposefully.
During the Activity
Prospective teachers work with prepared sliders and graphs in the LogitFuncParam081229.tns file. LogitFuncParamCAS081229.doc illustrates possible CAS work for page 1.14 of the tns file.
The activity provides exposure to the TI-Nspire's Q&A note format. Prospective teachers' responses might be gathered electronically and used during the post-activity discussion as examples of evidence used to develop and justify claims. Particular things for which to look while they work on the activity are: the values they choose for the parameter under consideration, whether they change values for the other parameters, whether they argue for the general case. LogitFuncParamActivity081229.doc is an optional handout with the questions in the file.
Prospective teachers have new tools by which to create or critique parameter explorations across the school curriculum (see LogitFuncParamAssign081229.doc). This is a good time for prospective teachers to learn how to create sliders and to induce the equivalents of sliders for non-numeric needs, such dragging a point to control an angle of rotation. Understandings developed through this activity form a useful base for many TI-Nspire logistic regression activities.
After the activity and discussion prospective teachers time, perhaps in small groups, outline a parameter exploration or to analyze an existing activity for another setting (i.e., linear functions, quadratic functions or quadratic formula, trigonometric functions, conic sections, or isometries). It is particularly productive to encourage them to develop questions that focus on how they might have their potential students reason with symbols, consider a relevant range of cases, and justify their claims. Ideally, the prospective teachers have an opportunity to try these explorations with others, perhaps college students or students they tutor if not in a secondary school mathematics classroom context.
After the Activity
Discussion includes mathematics, technology and learning goals. See the Additional Attachment section for implications for practice.
Through the activity, prospective teachers (PTs) will be making claims about the effects of parameters d and b and relative effects of parameters a and c. In talking about claims and evidence, draw attention to mathematically appropriate statement of claims and the nature of evidence. The evidence to generate and support conjectures should include a variety of cases, such as positive, zero and negative values of the parameters. The symbolic evidence to justify the claims should be as general as possible. As one of my students said, "we are making generalizations and more general generalizations" as we conclude first what the effects of changing one parameter are as other parameters are held constant and then we look at whether those conclusions hold up for all combinations of the other parameters.
The CAS evidence can be deceiving. Most PTs draw on their past experiences with asymptotes and look for zero denominators. If they have a non-negative value for b, they might be surprised when they receive "false" upon setting the denominator equal to 0 and solving the resulting equation for x. The important point here is not that the value is negative but that the negative value depends on the values of the parameters and the existence of a "break" can be predicted by reasoning about the symbols without using specific values.
The parameter exploration can be done without a slider, as PTs often note. However, it is the intensity of seeing the "break" suddenly appear and the ease of changing and managing four parameters that are clear advantages of using sliders. One way to bring this point home is to have a few students (PTs or others) try the parameter exploration in a static setting and share their work with the PTs after they encounter the dynamic setting.