LA ÚLTIMA CENA (13,1-17,26)

Such Generative Activities as described above are at the heart of the GenSing projects. This section will weave together the design principles upon which Generative Activities are built and examples of what those theoretical principles provoke in actual practice.

In 1995, Jim Kaput outlined what he felt would need to happen to make algebra more accessible to more students and how that change would most likely occur. First he laid out three dimensions of reform for algebra, breadth, integration and pedagogy (Kaput, 1995). To achieve breadth one must interweave the many different facets of what it is to do algebra; modeling, working with functions, generalization and

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abstraction. In addition to breadth within mathematics, he felt it would be important for algebra to be integrated across other subjects. Finally, he stated that the pedagogy for teaching algebra, especially as supported by new technologies would have to change. Kaput then went on to outline three phases of reform. Near term, where existing curriculums were enhanced by the use of new technologies, mid-term where algebra was more significantly implemented in the middle grades and long term where the mathematics curriculum would be totally restructured across all grade levels and algebra as a specific course would disappear.

This section will discuss the ways in which the GenSing project has used Kaput’s vision of reform to shape classroom implementation in Singapore. Specifically the theories and practices surrounding breadth, pedagogy and near and mid-term reform.

5.1 Breadth

The GenSing intervention uses a curriculum of function-based algebra supported by a classroom network (TI Navigator), and is grounded by the belief that most algebraic topics can fit within three key areas; “equivalence (of functions), equals (one kind of comparison of functions), and a systematic engagement with the linear function” (Stroup, Carmona, & Davis, 2005, p 3). The classroom integration of the TI Navigator network were described in the narrative. This section will focus on large structural ideas on which the curriculum was built.

Within typical introductory algebra topics there are three big areas of instruction; ideas of equivalence, ideas of equals and ideas of the linear functions. Equivalence is the idea that you can have two equations that look completely different in their algebraic form but graphically they are the same (Figure 2). Equivalence encompasses (but is not limited to) simplifying, factoring, combining like terms and expanding polynomials.

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Figure 2. Equivalence: Two expressions that look nothing alike create the same graph

These ideas are usually completely disjoint for students for most of their mathematic career. Other than exactly following the rules they don’t understand why what they are doing works, why it is correct. The big idea of equivalence gives students the ability to see that for expressions, there are different ways of writing the same thing.

Figure 3. Equals: A special relationship between two expressions where they intersect. “Doing the same thing to both sides”, preserves the solution set. Even in the extreme example of multiplying both sides by sin(x), the solution set is preserved (new solutions

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Equals and or Inequalities are the second big part of introductory algebra. For example, solving systems of linear equations, solving inequalities and finding where one expression is greater than another. Here the focus is not on expressions that are everywhere the same. Here the focus is on the place (or places) where two different expressions are the same; the intersection(s) (Figure 3). Taking a function-based look at the concept of equals helps students answer; Why am I supposed to do the same thing to both sides?, Why do I flip the inequality if I multiply or divide by a negative? and What does it matter if the X and the Y are different sides of the equals sign? Take for example the two expressions in Figure 4, 2x and -.5x+2. If we focus on where 2x is greater than - .5x+2, we are looking at points that are in the first quadrant. If both sides are multiplied by -1, the X-values which had been greater than, are now less than. The rule of flipping the inequality when you multiply by a negative ceases to be an obscure rule that is just memorized. The graphs literally move, the region of greater-than less-than has to be changed because the graphs aren’t in the same relationship to each other any more. Without a visual representation of why the rules they are using to solve algebraic problems work, students easily confound the rules for Equals and Equivalence.

Figure 4. Graphs of Y=2X and Y=-.5X+2 and the same graphs multiplied by -1

The final idea is of linear function, specifically doing activities to separate for the student the ideas of intercept and slope. The project starts by tying ideas of physical motion to slope. Much work is done with motion detectors. This allows students to see that moving towards

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the motion detector is a positive slope, moving away is a negative slope, faster motion is steeper, and slower is flatter. We believe that starting with a linear function to explore ideas of slope and rate is like starting with black to explore colors. There is not enough variety or richness of information available for the students to build understanding. It is in the slowing down and the speeding up, its in the changing points in the graph that you start to make sense of where the fast parts and slow parts are. If it is always moving the same speed, if it is always a straight line, there is no complexity in it to see the changes in speed to make sense of it. For this reason, we start with messy graphs and qualitative verbal descriptions of motion.

Figure 5. Motion detector graph and expression

After the students have become proficient with describing changing rates, we look at constant rates. As an example of an activity, one student will act out a rule in front of a motion detector, start at a point away from the motion detector and move at a constant moderate pace towards it. The network is used to collect the one graph and send it out to all the calculators in the classroom. The students then fit a function on top of it (Figure 5). A series of these rules are done to tie the ideas of what the person acted out to features of the expression. We start out with wiggly graphs to give students a rich environment to explore the faster and slower parts, then we move to constant motion connecting up with functions to model that motion. In this way the student is mathematizing the motion and quantifying the slope and intercept.

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5.2 Pedagogy

The new pedagogy that the GenSing project employs is that of Generative Design. There are four key principles for designing for the generative space. Activities should have a dynamic structure, open up a space for mathematical play, allow greater agency for the students and increase participation (Stroup, Ares, & Hurford, 2005; Stroup et al., 2002).