Geometry for Teachers (GeT) is a course at South Dakota State University (SDSU) that is entirely made up of mathematics majors who are planning to become certified to teach middle and high school mathematics. It is typically the first mathematics course that our preservice teachers take that includes pedagogy. Since I want the students to be well prepared to teach high school geometry, we focus on Euclidean geometry throughout the course. 

This fall when I walked into class on the first day, I heard one student say to another, “we just need to memorize it.” I informed the student that the word “memorize” was not to be used in my classroom—our goal should always be to understand mathematical concepts and to teach our students in a way that develops their understanding. In this article, I will outline some discovery activities that I used with my GeT students to help them understand formulas rather than relying on memorization of formulas. My goal is that they will use these experiences when they are teachers to help their own students learn with understanding rather than just memorizing.

​​Sum of Angles in a Triangle

Throughout the semester my students used the theorem for the sum of angles in a triangle, but they didn’t really understand how we knew it. So in class, I gave each of my students a half-sheet of paper and asked them to use a straightedge to draw a triangle and a quadrilateral and to cut out their figures. Next, I asked them to cut or tear the corners as shown in Figures 1 and 2. Once they had torn off the angles, I asked them to put the vertices of the triangle together so that they are touching. With the triangle, the students immediately saw that together these angles formed a straight angle, which demonstrates that the sum of the measures of the angles in a triangle is 180°. I had them repeat this same process with the quadrilateral in order to recognize that the sum of the measures of the angles in a quadrilateral is 360°. While this was a quick activity to do with the students and required few materials (half sheet of paper, straightedge, and scissors), it gave them a physical representation of facts that they had been using throughout the semester.

Area Formulas

Another discovery activity that I have used involves finding the areas of rectangles, triangles, parallelograms, and trapezoids using only the fact that the area of a rectangle is base*height. I gave students a piece of cardstock with Figures 3, 4, 5, and 6 printed on it, and asked them to cut out each of the figures. Then I had them take the rectangle (shown in Figure 3) and cut along the diagonal so that they could “see” that the area of a triangle is ½ base*height.

Next, they cut the parallelogram (shown in Figure 4) along the dotted line. Then they translated this right triangle to the left so that the hypotenuse lined up with the side of the parallelogram, creating a rectangle. Looking at this rectangle, they saw why the area formula for a parallelogram is base*height.

To understand the formula for the area of a trapezoid, we looked at it in two ways. Using the trapezoid shown in Figure 5, students cut along the dotted line and reflected the triangle over the bottom base of the trapezoid. Next, they translated the triangle left and vertically so the hypotenuse of the triangle lined up with the non-parallel side of the trapezoid, forming another rectangle. The rectangle clearly had height h, but the length of the base of the rectangle wasn’t obvious. They observed that the length of the base was a number between a and b, and then eventually came up with the base length of , the average of the bases. Again using the formula for the area of a rectangle, they

concluded that the formula for the area of a trapezoid is

Now using Figure 6 and cutting along the dotted lines, they created another rectangle by rotating these small right triangles 180° about the midpoints of the non-parallel sides of the trapezoid. Once again, they created a rectangle and “saw” the formula for the area of a trapezoid.

Distance Formula

When we started the discussion of the distance formula, I asked my students how the formula was explained to them. Some of my GeT students said that their teacher wrote the formula on the board and told them to memorize it: . Again, this idea of having students memorize formulas without understanding them is not what we want our future teachers doing. So, I plotted the points and on a coordinate plane, drew the right triangle (shown in Figure 7), and marked the hypotenuse of the right triangle, d.

Then the students found the lengths of the legs of the right triangle and used the Pythagorean Theorem, giving them:

Now rather than memorizing the distance formula, my GeT students understood its origin.

Using these discovery activities in my GeT course to cover basic concepts in geometry gave my students active learning strategies to use in their own classroom. For some of these students, it was the first time that they experienced active learning in a mathematics course. In addition, completing these exercises illustrated the importance of teaching mathematics for understanding rather than telling students to memorize formulas. Mathematics education research indicates that memorizers are the lowest achievers in mathematics (Boaler, 2015).

Throughout our mathematics courses for preservice teachers, we need to model best practices. These discovery activities facilitate meaningful mathematical discourse, connect mathematical representations, and build procedural fluency from conceptual understanding, which are some of the mathematics teaching practices found in NCTM’s Principles to Actions (2014). As we prepare future mathematics teachers, we need them to understand the importance of what they do every day and the impact that they have on their students’ learning. Many of my GeT students dislike geometry at the beginning of my course, frequently because they had had a bad experience in their high school geometry course.

By the end of the course, most feel prepared to teach geometry and some even enjoy geometry.

Sharon Vestal is an Associate Professor at South Dakota State University

Citations

Boaler, J. (2015, May 7). Memorizers are the lowest achievers and other Common Core surprises. The Hechinger Report.National Council of Teachers of Mathematics. (2014). Principles to Actions: Ensuring Mathematical Success for All. Reston, VA: Author.