The Complexity of Confidence

After a given lesson, each student's understanding of the material is unique; thus, each student represents a point on what could be called the "Spectrum of Confidence" related to the taught material.

The "Spectrum of Connnnfidennncee" according to Spongebob

Let Spongebob's right hand represent no confidence in understanding a given concept, his left hand represent complete confidence in understanding a given concept, and each "star" represent an arbitrary student in the classroom (sound like the start of a proof anyone?). From my experience of serving a SLA Facilitator (for those unfamiliar with the role, a description can be found here), I can tell you that it is not hard to find two points along the spectrum: complete confidence and complete lack of confidence. To visualize what is going on internally with these types of students, I call upon my friends Robin Hood (complete confidence)

Oh please, I got this concept.

 and Harry the hedgehog (complete lack of confidence). 

If I can't see you, you can't see my lack of understanding.

As teachers, we need to identify if our students fall into either of these levels of confidence especially. First, we will discuss what it means to look like Harry the hedgehog. While going over a topic with my SLA class, I can often identify these students by looking for the face depicted below. 

I look like I got it, right?

This blank stare signals to me that I am speaking what sounds like a foreign language to these students. For most students, it is very difficult to be honest with themselves and admit that they need help with understanding the material. Often, they choose to give up because they believe will never understand it since, as some of them say in frustration, "math is stupid and too hard for me to understand."

Did you really just say that?

I believe that every student has the capability to not only understand the material, but also should feel confident that he/she can learn and understand anything presented to them Thus, in my future classroom and currently as an SLA Facilitator, I want to instill self-efficacy into each each student, especially those who utter that dreaded phrase that sounds like nails on a chalkboard. However, I must be cautious in doing so as it is important they do not lose the ability to think critically. In other words, a student should not be so confident as to think their initial understanding will always be the correct one. An example of a student who is confident in his understanding, but not have an accurate understanding of the material is Colin from "Quadratic Functions: Students' Graphic and Analytic Representations." In the article, Colin, a tenth grade honors student, is interviewed on his knowledge of quadratics. He confidently completes each task presented to him without questioning his answers, since he believes the connections he has made are correct. However, his understanding of quadratics is very flawed. Whether going from an equation to the graph or vice versa, he made connections that he believes are mathematically sound, but are actually completely false. As teachers, we must also pay attention to the students who seem to understand the lesson because they could be just like Colin. These students will probably look like Lea Michele below, but without further investigation, there would be no way of discovering this false sense of confidence and severe misunderstanding of the concept.

See Mr. Schweitzer, I understand the material!

In other words, I want to prevent myself from finding out that a student has a severe misunderstanding of a concept before a summative assessment, and looking like this guy when seeing their test or quiz.

But it seemed like you were getting it!?!

Therefore, on my quest to instill self-efficacy into my students, I must use an abundance of formative assessment and check-in with each student. In doing so, I hope that when my students leave my classroom, they will be ready to take on not only the material in their other courses, but also everything they encounter during their lifelong learning. 

Learning from Mistakes

Throughout my elementary and high school years, I cannot recall more than a handful of times being asked how I thought through a given math problem. On the other hand, in my math education class at GVSU, one of the most frequently asked questions is, "Well (insert name), how did you get that?" In this class, the question comes after being prompted to solve a problem we would have encountered in one of our high school mathematics courses or to the desired value in a counting circle. (For more information on what a counting circle is, see Jennifer's post.) My good Game of Thrones friend, Margaery, does an excellent job of depicting how I feel when I am asked to answer the question.

Oh, let me tell you how I solved this!

But much to my surprise, I am often met with the question, "Ok, but how exactly did you do that?" Wait, you want me to explain every little detail about my thought process? My other good GoT friend, Tyrion, perfectly captures my feelings.

But, but don't you already know what I'm thinking?

As tedious as it may seem to articulate every single step in solving a problem, some of the most profound understandings or insights can come from this. One of the few times I was asked to show my thought process in high school was in my AP Calculus class where my teacher, Mrs. Cortes, asked the class how to solve a revolution of solids problem from the previous night's homework. Since the problem was an odd and the answer in the back of the book matched the answer I had found, I gladly raised my hand to volunteer to work out the problem on the board. Yes, Mrs. Cortes chose me!

Yes!!! I get to show Mrs. Cortes that I understand this tricky topic!!!

Jumping out of my chair, I walked up to the board and began to write the integral and draw the graph that I used to solve the problem. Once completed, Mrs. Cortes asked the question I hear so often nowadays, "Well Nick, how did you get the integral from your picture?" I began to explain how certain features of the graph led me to the integral I had created. Seeing there were serious gaps in my understanding, Mrs. Cortes stopped me from going any further. In my mind, I was thinking, "What? Why would she need to stop me? I'm pretty sure I got this. She must want to point something out to the rest of class so they can understand it too." She went up to the board and started to point out where I went wrong. Any excitement left from being called up was definitely gone now as I could not believe how I could be so confident in being so wrong.

Whoa, look at all of those mistakes!

After she explained how I should have analyzed the features of the graph, the process of creating the integral need to calculate the volume became much clearer, but now in the right way. In retrospect, I should not have been so process oriented and algorithm dependent. 

You mean the algorithm in the notes
does not apply to every problem?

Since the lesson over revolution of solids didn't go over so well, I can tell I was trying to memorizing a process instead of trying to conceptually understand what was going on with each problem. Clearly, I can now tell that this form of learning is not the type of learning a student should be employing.

Cheer up Nick, there's hope!

What I need to keep in mind, and wish I knew back in high school, is a mantra of Dr. Karen Novotny's: "Often, you learn more from wrong answers than you do from right answers." Now that I am in the process of becoming a secondary mathematics teacher, I plan to ask the question, "How did you get that?" frequently in the hope that it can be a type of formative assessment in the classroom. In asking this question, I hope that my future students will be able to learn from their mistakes, since I will be able to check for conceptual understanding, and they will not have to revert to a memorized process to answer a problem. In other words, I believe that if we as teachers make the effort to see what is going on in our students' minds, we can make a profound impact on their learning. Particularly, I think that if math teachers employ this practice, we can positively affect the student's experience with mathematics by increasing their self-efficacy. In the process, we might even begin to break down the stigma that math is always difficult and hard to understand. Also, we may help some students begin to think that math can be just as much fun as this shark had dancing at the Super Bowl this past weekend!

I think it's time for a math dance party!