Gisselle Guzman, Lecturer
Motivation and curiosity are two strong factors that contribute to student learning. Students can approach a maximum level of engagement when the instructor designs lessons that incorporate alternative ways to present the course material or when class time is devoted to discovery-based learning. Students are able to understand and remember better when allowed to discover solutions to problems rather than having the instructor tell them the answer.
My lesson plans are mainly structured in 3 stages. Stage 1 is primes student’s curiosity by using hands-on activities, tricks, or error analysis. In Stage 2, I guide them through the reasoning process through carefully chosen prompts. At this stage, students are encouraged to think critically which helps them to develop logical thoughts along the subject material. This step also helps them to bridge the gap between existing concepts and new material. Stage 3 is when students achieve awareness of their own learning process. At this level, the student’s conjectures will lead them to their own conclusion. At this final stage, they are aware of their own learning methods and mechanics.
As an illustration of these stages in practice, I used this technique in teaching Elementary Algebra with solving equations. Throughout the course, students needed a strong notion of how to solve a variety of equations and they needed to be able to check their answers. One challenging concept for students was to understand when solving linear equations, we are not always going to get a distinct solution for the equation. Students will encounter examples that have “no solution” or “all real numbers” as solutions. At this level, students are often confused when they get an answer like “0=0” instead of a simple, more familiar answer like “x = 2”. They often wonder, “what does this mean?” or “what did I do wrong? “They don’t recognize that their answers are right because they are used to getting an exact number as a solution. They aren’t able to process the answer to see that it means that we have no solution or infinite solutions. To help them understand, I started the lecture with Stage 1 and introduced the topic with a trick: “Think about a number. Multiply it by 2. Add 7 to it. Subtract 1. Divide the result by 2. Subtract the number you first thought of.”
Then I ask them to compare their answers with each other. If they did not make a mistake, they all get the same answer, which is 3. They’re usually amazed that they all got the same answer and they don’t understand why at first. This is where Stage 2 comes in. I give the class several different prompts to facilitate the discussion: “In order to get the same answer, did you all start with the same number? If each one of you selected a different number, why did you all of you get the same solution? Is possible to have more than one value of x satisfying the same equation?” This puzzling trick often ignites a student’s innate curiosity to unravel the mechanics behind the underlying mathematics.
This activity opens a great class discussion and it also helps to simplify the abstract mathematical concept of having “infinite solutions” for them. It is a tangible example illustrating that by selecting any number, each one will satisfy the equation. In the final step and my Stage 3, the “trick” instruction is written in a mathematical form and solved collectively to further prove their result and to explain their conjectures. After this activity, it makes more sense to the students what the abstract solutions mean.
Curiosity is a powerful tool that helps students to improve learning and retention of new material. Overall, “the trick” is a fun activity and it is highly rated by my students. It allows me to increase student participation my classroom. Students are more productive, engaged and the learning process becomes easier when I guide them to explore the concrete rules to further lead them to a formal abstract concept. For students, complex mathematical definitions become simpler with activities that promote exploring their own innate curiosity. Not all material can be approached this way, often due to time constraints. But there are ways to promote curiosity and engagement. When there are opportunities to use their curiosity, it really enhances their learning process and retention of the material. It makes Mathematics seem more like their own discovery instead of something created by the teacher and that makes the class much more fun and much more productive for both students and the teacher.