A square pyramid is a pyramid with a square base and four triangular lateral faces. The slant height is the distance from the vertex of the pyramid along a lateral face to the midpoint of a base edge. If the slant height is 10 and an edge of the square is 12, what is the altitude of this pyramid?

This is an example of a problem that probably is closest to what a traditional textbook reading would do for a student on a typical night’s homework. We do assume that students have had past experience with three-dimensional solids in middle school curriculum and have heard the term ’square pyramid’ before, although this is their first encounter with it in our book (it’s also defined in the reference section of our book too). The problem comes complete with a few other definitions just in case they aren’t familiar with slant height as well. However, although the definitions are in the problem, it can often seem that students don’t use their logic or problem solving skills to really understand what these new terms mean in the problem.

When I gave this problem last week, the student that put it on the board did it incorrectly because she assume that the slant height was actually the lateral edge (the length of the congruent sides of the isosceles triangle that is the lateral face of the pyramid). So even though the defintion of the slant height was right there in the problem, why was this student unable to transfer this definition to the problem? Well, there could be many reasons. When students read a problem for the first time, I theorize that new material takes time to become a part of their toolkit, or set of skills that they would use. So in this case, even though the definition is inthe problem, and possibly she even understood the definition, transfering that to the problem situation was difficult since the new idea of slant height was “too new”. She actually new to use the pythagorean theorem, just used it with the wrong triangle.

I also theorize that it is easier for students to visualize the right triangle that is formed by the lateral edge, the altitude and the square base because the lateral edge is physically drawn in (in a diagram that students generally draw) already. We are going to change that for next year and actually draw in the slant height in a diagram in the text.

There are a number of problems in our curriculum where a great deal of assumptions are made about students reading the definitions and applying them to the problem at hand. This result is more positive on specific problems when students have the motivation to complete the problems on their own. The kids who are finding the curriculum instrisically challenging and interesting don’t have a problem realizing that applying the definition is an expectation. I’ve also found that the more you expect from them on a regular basis, the more they begin to fulfill that expectation too.