This case study was written by Dr. Ferdie Rivera and Dr. Joanne Becker, two professors I learned from in my credential program. I found it on a site that has a virtual treasure trove of math research papers (I would love to find time to read quite a few of them...)

It provides a prime example of what "sociocultural constructivism" is all about. It does have a lot of technical jargon in it which can make it a little tough to understand, but overall I think it is pretty clear what is happening when you read it.

This case involves the authors teaching a math analysis class the topic of solving polynomial inequalities, using the TI-89 graphing calculator as a tool. (If you are of an anti-calculator mindset, read the study - it shows a great way to mediate use of the calculator. The students are guided to understand why knowing how to use the calculator is not sufficient, and when and why they should use algebra for exact solutions - so not only do they know how to do both, they know *when* it's most appropriate to use each).**Our overall concern in this investigation is to provide a sociocultural basis for the meaning objectification of mathematical concepts and processes - that is, by surfacing both the social nature of technological tools and the social transactions that take place in classroom activity which assist students as a collective to establish viable shared practices and collective representations**

Um, right. So that points out one of the fundamental differences between this theory and "discovery learning" - students are not left to explore aimlessly on their own, or even just in small groups. They begin an exploration typically in pairs. This is followed up by a whole class discussion, in which students propose different solutions, ideas, and difficulties that they are having. There then exists a process of negotiation, in which the students discuss and evaluate each others' ideas, which will hopefully lead to further new ideas - the teacher's job at this point is to help move the discussion along by facilitating (if needed), clarifying students' thoughts, and posing key questions. Eventually, ideas, concepts, and notations can become "taken-as-shared", which means everyone in the class ascribes the same meaning to the given concept. The second fundamental difference is that, once this shared meaning has been established, the teacher helps the students generalize and formalize the knowledge (i.e. learn and use traditional algorithms, representations, and techniques).

I find this stuff fascinating. The drawback is that it requires more class time. I do think, however, that the gains can outweigh the losses if this is done right, as the students are more likely to understand and retain concepts in the long run.

I encourage people to read this study and make comments about it. **Below is an excerpt. Notice how the comments of the pairs of students follow the logical process that a teacher would provide directly.***Developing a Model For Solving Polynomial Inequalities. *

The class needed eleven 55-minute sessions to accomplish this task. The model they developed for solving polynomial inequalities graphically could be broken down into three stages below using different types of tools and in which the TI-89 served as the primary tool for the progressive evolution of the two later tools.

I Using the TI-89 as a tool for investigating the following:

(1) graphs of even- and odd-powered polynomial functions in factored form;

(2) graphs of polynomial functions in factored form that contained odd and even multiplicities;

(3) graphs of polynomial functions in factored form that contained imaginary zeros;

(4) solving polynomial inequalities in factored form graphically.

II Using a constructed Cartesian plane on paper as a tool for solving polynomial inequalities in factored form graphically.

III Using a number line as a tool for solving polynomial inequalities in factored form graphically (and the same tool was used later in the case of polynomial inequalities expressed in the general (non-factored) form.

Initially, the students relied on the TI-89 to obtain generalizations about the graphs of polynomial functions subject to certain restrictions (see (I) above). They also used it to solve inequalities and to see the significance of knowing how the x-intercepts played out in the solution process. The TI-89 enabled them to develop their initial ability to describe and to reason perceptually about graphs of polynomial functions and their relationship to solving polynomial inequalities that were all initially expressed in factored form. In establishing a model for solving inequalities graphically, two additional shifts took place, and both shifts were unaided by the graphing tool. When the students were prompted to solve a polynomial inequality

independent of a TI-89, Pair 3 suggested for the class to draw a Cartesian plane, plot the real x-intercepts, and use what they initially learned about the graph of the corresponding polynomial function to draw a sketch of its graph, and then to write down the intervals in which the inequality made sense. It took students some time to accomplish this because they had to calculate specific points on the graph. A number of them obtained values for y by beginning with x = 0, 1, 2, 3, and so on,

which did not make sense in many cases of polynomial functions and, hence, did not gain much support from the class. One collective practice that emerged from a whole-group discussion came from Pair 4 who suggested obtaining points that lie

between x-intercepts that the class immediately accepted. A second collective practice came from Pair 5 who suggested that to solve a polynomial inequality graphically, a rough sketch of its corresponding graph together with all the x-intercepts

was all that were needed and that none of the other points mattered.

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