Photo of Renee LaRue available here.
Co-author Nicole Engelke Infante
Classic problem: Fence along a barn. Minimize amount of fencing given a fixed amount of fencing.
Literature: 4 versions of this problem. More difficult when students have to set up problem from words.
Carlson and Bloom (2005) Problem-Solving Framework
Tall and Vinner (1981) Concept image
7 students (pilot with 3 students), just before final exam involving optimization.
Recordings of students thinking aloud.
Questions about rectangles – what happens to perimeter if area changes?
5 students solved without intervention, 3 perfectly, 2 forgot about barn. Other 2 needed much help.
Responses showed gaps in reasoning.
Six key maths concepts that played a role:
- Use 2x + y or 2y + x? 2l + 2w = 2y + 2x (matching variables to what they think they must mean).
- Function notation. Haphazard use of equal signs. Student said you can’t write 2x + y as f(x) because y = f(x). Writing P(x) = 2l + w shows lack of understanding of function notation.
- Function composition. Didn’t connect the objective function (perimeter) with an expression that was only in terms of y. Not sure what the in-terms-of-y-only expression represented now or how it related to fencing.
- Properties of rectangles and the relation between them. Some understand that if you change area, perimeter has to change.
- Role of optimizing function.
- Graphical representation of the optimizing function. Many couldn’t plot on a graph of perimeter where the minimum value is (even though they should know it will be the lowest point).
Summary
All students had difficulty with the properties of rectangles and the relation between them.
Suggestions:
- Provide example of the problem situation (e.g. get students to draw a few examples of solutions that at least match the constraints)
- Discuss the optimization function in detail
- Rectangle activity. How many rectangles can you draw with perimeter 100m / area 100m^2?
- Discuss the need for constraints
- Emphasize the meaning of the answer
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