Teaching Geometry

Piaget and the Three Mountain Task Experiment?

The three mountain test was one of the various tasks designed by Jean Piaget in the 1940s. This one assesses the perceptual perspective of children when seeing the same scene from someone else’s perspective. Originally, the child was sitting in front of a rectangular display that modeled three different mountains and a few objects on them. The child was sitting in front of it, seeing the display from their own point of view. Then, the researcher placed a doll at different sides around the display and asked the child to select the picture that best describes what the doll saw at that moment. The purpose of this task is to see if a child can reconstruct the scene and name the objects from another person perspective.

What is the implication of the Three Mountain Task Experiment in teaching Geometry?

According to the findings in this Piagetian task, children in pre-operational stage, see the world from their own point of view. They are unable to describe or chose the picture a person will see from another angle. An additional example is that children in pre-operational stage draw two eyes inside a head, no matter what direction the head is pointing. However, as the child progresses to a concrete operational stage, the child can construct what is in the other side without necessarily seeing it. There are other features of spatial thinking that progress with time and experiences (aging alone does not guarantee progress).

When teaching Geometry, or geometric concepts, we need to be aware of what stage out students are in relation to spatial thinking through tasks and conversations. Then, we need to select scaffolding lessons, tasks, and assignments that tap into the current stage and give plenty of opportunities to explore and move towards the next stage or the necessary stage. Students must have the hands-on experience to be able to develop their special thinking. For example, when teaching total surface area and volume of solids, starting in grade 7. We must provide manipulatives that allow students to touch, see, and experience the solids from different angles. A drawing by itself is not enough. We must provide and/or create nets to study total surface area and be able to go back and forth (reverse) the net to the solid state and vice versa. Either store-bought or homemade (or both), I cannot image teaching all these concepts, including vertices, edges, bases, lateral surface area, volume, and all relationships and connections without them.

Even in high school Geometry, we need to make sure we know where our students are standing. For example, there is an item is from a released High School Geometry STAAR test from Texas (and other states include a question like this one). In the question, the front, side, and top of a figure made of identical cubes are shown. The question is to choose the diagram that represents the three-dimensional figure. Most of my students were not able to understand the question; once clarified the question, they had a hard time figuring out the answer. Having some blocks to play with could be a great idea to practice spatial perspective as Piaget intended. Yet, using manipulatives in high school is not that common because we assume that students will understand.

For example, when introducing trigonometry and the concept of opposite and adjacent sides, students need to reposition their point of view for each angle to identify the opposite side of that particular angle. For instance, the concept of angle o elevation and depression, is all about point of view!

When moving to coordinate geometry, awareness of spatial perspective and reasoning are imperative. Otherwise, students just memorize formulas without understanding what they are for or what they mean: when introducing the term “foci”, let’s say within an ellipse, students need to be able to operate in the highest level to be able to understand that any point on the line is related to the foci in certain way, no matter which point is taken around the ellipses’ line. For these last two examples, computer-based manipulatives or software such as GeoGebra might be necessary, even a model of a solar system could work!

Assessment should serve the student more than it serves the teacher. Reality check: Keeping track of students’ learning in a middle or high school setting when teacher sees 100+ students a day can be overwhelming or even impossible. For this reason, using a student-centered approach is so crucial to assessment. Assessment should be done often, as an integral part of the learning process at each step of the way, including self-assessment, peer-assessment, and teacher assessment. Once the assessment is done, what we do with the results of the assessment is what takes the student to the next level.

When assessment becomes part of the routine, it helps with student engagement because it fosters self-discipline, self-motivation and encourage students to become independent learners. Most of the knowledge students will need 2, 5, or 10 years down the road, will need to learn new skills that more likely wouldn’t have been taught during their time in school.

There is a saying: we assess what we value. Therefore, I am asking myself the what and the how of assessment. What do I assess? What do I value? Do I value isolated skills or critical thinking and application? How do I assess? How do students will show they are learning.

Traditional tests and quizzes in a short answer or multiple-choice format have a small role in assessment, it is useful for some purposes, maybe to test some skills. Nevertheless, this format seems to be the main character, the star, the only “serious” way to assess. That is not right.