Wednesday this past week I had the privilege of teaching about rational numbers. I began the lesson with a basic question what are rational numbers?

I had a good thought of trying to make the lesson an inquiry lesson. My desire to have students explore to find the definition was inspired by my background in the GVSU math department. The professors constantly encourage us to let ourselves find multiple ways to a solution, and to provide opportunities for student discovery. As a new teacher I sometimes feel that I must teach this way or else my students will not develop a real deep conceptual understanding of the mathematical topics I am covering! This is my internal struggle right now. A fight between teaching conventionally: learning a procedure that a teacher models and then students begin to use that procedure. Along with some conceptual explanation throughout the lesson that often gets lost in the shuffle of an adolescent mind. Compared to the alternative which is a discovery based lesson: where students explore some numbers or aspect of life and find out on their own the patterns or models present by building from their prior knowledge. This way they first get the concept, and then derive the subsequent procedure. The great war in my teaching mind is between these two tactics. With the first teaching conventionally, I find myself frustrated that students may never understand why they do what they do, and they will struggle to apply what they learn to real life. With the alternative, I find that I feel like a bystander to student learning, and have difficulty determining how to effectively step in and focus student thinking without corrupting it to make it exactly like my own. Currently, I am persevering in problem solving to determine how to best teach students, and the problem is an ever evolving one I feel.

The reality remains that I believe deep down I need to strike a balance between these two approaches to teaching. I am pretty sure the more discovery sided lesson has a better long term impact for student memory and implementation, but the conventional approach has merit too in that humans have always learned already discovered concepts through modeling the behaviors of others. I want to be a teacher that has students learn how to think deeply, but also explains concepts clearly and in a manner which students build a math toolbox. Through the past four weeks, my first four weeks trying to teach 15+ students at a time, I have certainly learned that I have so much to learn and grow in!

Before I digress, back to the lesson. I asked the big question, then displayed some examples and had the students guess whether the numbers were rational or irrational, then I told them whether they were rational or not. I utilized numbers like, 3/2 , √2 , 5, -2/3, and 5/0. Once students began to see which numbers were rational and which were irrational they developed some ideas about what could make a number rational. The idea of having an inquiry lesson could have worked, but students had no definition to work from, so I was using examples of which they had no idea whether they were rational or irrational. The students had nothing to construct their own framework from as they had no experience with irrational or rational numbers. Intriguingly, even with the flaws in my lesson structure the students provided great possible explanations for why certain numbers could be rational and some could not. For 5/0 they said because the denominator is a zero the number is not rational. It is not possible to have 0 groups given 5.

The point at which I had most overestimated my students prior knowledge was when I presented them with the square root of 2. I did not realize that in 7th grade most students have hardly or never seen square roots. The students had little idea of what square root even meant, yet I was expecting them to tell me whether it was rational or irrational. I used this example, remembering my Sophomore year of college where we proved the square root of 2 was irrational. My poor planning had led my students astray. I wonder how often teachers cause their students more challenges than help like I did. Still as I mentioned prior, the students had some solid ideas, one said a rational number must be one where there is a ratio, and another said any fraction is a rational number. In the end, I stated the definition, and as much discussion as the search caused, in the end it seemed to have been wasted.

Later, I taught a bit more on converting fractions to decimals and vice versa, and throughout the class I noticed that when I spent too much time on individual student responses other students checked out and became disengaged. I wonder how in full group discussions I can best keep all students engaged and learning? When students work in smaller groups and engage in mathematical discourse I have noticed that they learn a lot! Hopefully I will be able to develop better methods to assist struggling students while the lesson is occurring.

After being encouraged to talk to a few professors and teachers about my frustration regarding the gap in experience between students. I have learned that I need to model how I think about solving problems, then give students time to try to think about the problem my way or their own way building on what they know. If I verbalize how I approach problems, then perhaps my students will become more comfortable with asking questions throughout the problem solving process. My teacher and professors have also encouraged me to set up more opportunities for students to teach each other. In the past two days, I have tried to connect students with my coordinating teacher by having the less experienced students sit by the more experienced students in table groups. This way the “lower” students at the beginning of the year will grow from the expertise of the “higher” students. A further problem this week I discovered is that when shy and less experienced students sit further back, it is harder for me as a teacher to call on them as often as is necessary to help them to be engaged and learning. Thus, there are many challenges in the classroom, but I think I am beginning to understand how to help struggling students learn.

To wrap things up, I am really enjoying my experience as a teacher assistant though it is very challenging to teach well! I realize that it may take me many, many years to become the teacher I want to be, but I am looking forward to the growth I can and think I will experience. In the coming weeks, I hope to develop a better wider view of the whole classroom as I teach. By this I mean, that I want to have a full awareness of student behavior so that I can prevent problems and consistently address issues that detract from the learning environment of the classroom I teach in. I want to become more proficient at utilizing yellow cards, which are warnings where students talk to me after class about their behavior. Another thing I would like to figure out is how to plan lessons well so that they are engaging and interesting for at least 95% of my students. I look forward to seeing what happens as I strive towards my goal of helping student’s best learn and being satisfied with my performance as I teacher.