Four Strategic Reversals
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| Our content and curriculum analysis has led to four fundamental strategies that reverse the traditional approach, which requires algebra as a prerequisite, that is formula and technique-centered, and that begins with continuous (indeed smooth) functions. |
- Phenomena before formalisms
- Discrete variation before continuous variation
- Accumulation and integrals before rates and derivatives
- Graphs before algebraic symbolism
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| Perhaps the best way to explain our approach is to illustrate with a pair of early activities that embody our starting point, which make sense to students at almost any grade-level (although, for younger children, we begin with their own physical motion as described in our sample activities). |
| The first asks a question based on the Elevator simulation: Given the velocity graph pictured (2 floors per second for 3 seconds) where does the elevator end its trip? |
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| The second provides the downward staircase velocity graph and asks first: What will the red elevator do? The third question asks the student if it is possible to create a graph for the blue elevator so that it gets to the same floor as the red one in exactly the same time, but by traveling at a constant velocity. A fourth option asks the student to create a velocity graph for the blue elevator that has a smoothly decreasing velocity (no steps) and that travels as closely as possible to the red on its entire trip. |
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| The first task illustrates the integration and area-based beginning, as well as the graphical start. The second points to a common position-velocity fusion difficulty—the elevator is not GOING down, it is SLOWING down. The third involves creating a mean-value for the discretely variable velocity. Note that the computations involved here are essentially executable within whole number arithmetic, which is one reason for confining ourselves to discrete variation (these are called “Jerky Elevators”). Here the elevator ends up on the 7th floor after 7 seconds, so the mean-value is easy to determine. Often, this task is posed with “snap-to-grid” turned on, which constrains all velocity and time values to be whole numbers. If one more downward velocity-step were added, then the integer-constraint would make the task impossible. |
| Our experience confirms what is widely known: while the ideas of rate and velocity are intuitively and perceptually available, they are quite difficult to understand quantitatively, which makes the idea of derivative a tough place to start. On the other hand, students can appreciate the difference between slow and fast velocities, even when quite young, and can even determine that the net (signed) area under the velocity graphs determines where the elevator will end its trip provided you know the starting point. The last question involves approximation— of a discretely varying function by a continuous one! If snap-to-grid is on, then a linear approximation through midpoints of the tops of the “velocity rectangles” is not possible. But, critically, the students can try to drag the downward sloping line up and down and test their attempts by running the simulation (something you, the reader, cannot do unless you download the software). Another variation on this activity has the students subdividing steps to approach a straight line approximation. |
| It should be apparent that even in this tiny sample of activities, much other mathematical ideas are being engaged in quite a natural way in the service of learning the powerful ideas of rate and variation, mean velocity, and approximation: Area, signed number arithmetic, slope, among others. |
| Next steps beyond the velocity-based starting-point illustrated here involve position vs. time graphs that describe and control motion, and student creating position graphs to match given velocity graphs, and vice-versa, with the motion acting as the first arbiter of the success of one's construction—does it produce the same motion? These activities are available both on the materials download as well as elsewhere on this web site. |
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