I want to use this post to detail thoughts that has been bugging me during lesson planning as well as try to convince myself by putting down some of my thoughts on paper. The main focus will be on the Math 9 classes, but some of these ideas can be applied to the other grades as well.
For me, teaching mathematics should be closely linked to building intuition and induction/deduction using logic. The process of trial and error when playing with math is the backbone of learning. However, it is also important to recognize that students come with different skill sets, especially in subjects such as mathematics. The most important aspect of this teaching style is to find the balance between guided inquiry and concrete instructions to ensure all the students (especially the weaker ones) will not fall behind.
The problem of “over-instructing” in math is frequently discussed in literature as a teaching style that has the potential to deprive students the opportunity to think. It is important to note that this is a generalization that may not apply to every math class. However, the underlying idea behind most of these arguments is that by giving students more instructions, students are losing out on opportunities to think, which leads to (with some exaggeration) students slowly losing the willingness (and potentially their ability) to think. The result would be a self-reinforcing cycle where students have less opportunity to think would think less, teachers who observe students thinking less may be tempted to fill in the gap by providing direct instructions. Students who receive the instruction would learn that the teacher would be willing to feed them the information regardless of whether or not they think, which would lead them to do less thinking and simply wait for the teacher to give them instructions.
This leads to the problem I have, despite wanting to design lessons that are more inquiry based, my lessons tend to be very much teacher-centered. I believe the problem is the preconceptions I have for my Math 9 classes. The class average of all three of my Math 9 periods is the mid-60s. The strength of my students are widely distributed, some are very strong, while others are very weak. In my mind, this translates to “if I were to build an inquiry based lesson, how would my weaker students perform? Would they find it useful? Would it be helpful for them? Would they be motivated to do it?” While the answer to all of the questions I given above is “It depends on how you frame the inquiry and how you present it,” I still feel some sort of reservation to let the weaker students go and explore in fear that they might give up and waste the class. While it is possible for the teacher to help weaker students and provide more scaffolding for their inquiry, what do you do if only two-thirds of the class is fairly weak and require extra support for the inquiry? Can I justify spending a class where (I believe) only a small number of students will benefit? Can I break the class into two groups, where the stronger group performs inquiry while I guide the weaker group through the inquiry together?
I do not have any answers for the questions I propose and I do not know if it is reasonable to pose some of the questions. However, I do know that when confronted with those questions, I tend to back out and take the safe way out; since stronger students can thrive in a more teacher-centered classes, but the same may not be true for weaker students in student-centered classes. While I find solace in the logic behind my arguments, I do not believe this teacher-centered class is beneficial to building the sort of intuition and logic driven math classes I aim to create.
As an attempt to amend the lack of student-centered classes (for Math 9), I planned a “Math Lab” that explore the functions of the variables in linear equation “y=mx+b.” The lab is divided into three different phases: phase one explores the effect of variable b on the graph of the linear equation; phase two explores the effect of variable m on the graph of the linear equation; phase three puts together phase one and phase two and examine how variables m and b affect the linear equation together. The lesson is set up so students would be working in pre-arranged groups of threes, where they would be graphing different linear equations and comparing their graphs with those of their group members and discuss the differences and similarities. The teacher would circulate around the classroom and guide students as they work through the lab and provide assistance when needed.
The result of this experiment was quite interesting. First, some students are very reluctant to share/ask/talk to their group members. Groups are organized so that there is at least one comparatively stronger student who would “guide” the other group members through the lab. The unwillingness to communicate was an unexpected factor that slowed down students’ progress through the lab. Second, students were not use to the more exploratory format of the lesson so many students were lost and did not know what to do. In order to ensure all the students were on the same page, the next lesson was spent debriefing the lesson and going through the questions in the lab. Overall, I believe the stronger students benefited the most from this lesson and were able to make connections between the lab activities and the questions posed during the debrief; the weaker students were able to work through some of the lab and was able to get practice with graphing linear equations. I enjoyed this experience and I do plan on planning lessons with similar format should the appropriate topic arises before the end of the practicum.
—OLD POST (I will not remove for the sake of documenting progress)—-
Despite having this belief, I find my lesson plans are heavily focused on simply “telling” students, or giving students too much scaffold. I sense that I might be afraid of letting go of the agency over how content is conveyed as the teacher. I do not know why I feel this way, I can only speculate this feeling is the result of how I was educated, over-protective of students (underestimation of students’ ability), trying to keep with the teaching schedule, the need for active class management, and perhaps the arrogance of my content knowledge (seeing how I present the content as the “best” for the students).
John McCarthy writes an informative blog post on student centered learning and addresses some of the things I am experiencing. I want to make it a goal to include some student-centered component at least every other class. This could be in the form of giving students instructions and time so students can work out something according to the given instructions or have students in groups to discuss and find patterns. With this in mind, I will look over the lesson plans I have made so far and make changes to them so that it includes student centered components so students can get hands on and engage with the lesson as opposed having me “telling” them information.
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