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A mind once stretched can never return to its original condition. This statement struck me profoundly the first time I heard my mentor, Joseph Antoine, uttered these words. Now thirteen years later, I find myself reflecting on these words, on my journey as an educator and a student. Throughout my career, my mind has been stretched by several philosophies, one such philosophy is a constructionist philosophy. This philosophy states that students should construct their own knowledge. Paul Lockhart in his article entitled A Mathematician's Lament put forth arguments that support this view. Though I agree with many of the arguments put forth by Lockhart, it's naÃ¯ve to think that all students are going to act responsibly, and gain the knowledge they should from their classes without teacher supervision. In this paper I will discuss the issues raised by Lockhart in his article. I will also discuss why there is a lack of creativity in the classroom and examples of how to remedy the problem. I will then explain what I believe Mathematics education should be and what my role is as a Mathematics educator. Finally I will summarize my thoughts on student-centered classrooms and how they can be effective.
Mathematics Education: A Constructionist's View
Paul Lockhart in his article entitled a Mathematician's Lament sheds insight in to the role of Mathematics in public school. He likens Mathematics to music, and argues that musicians are free to use creativity and are not limited to formal music (p. 1). We also don't expect all musicians to play the same music, and they definitely don't have the same ability. Yet we hold our math students to a higher degree. We expect them to all perform at the same level, using the same math language. We take little or no thought for their interests, or how mathematics applies to their future career choice. All too often the students ask, "When am I ever going to use Systems of Equations or Pythagorean Theorem?" I like to say, "Carpenters use Pythagorean Theorem, to build stairs, and you can use systems of equation to determine when you and your friends path will cross on a trip." The reality is not many students are going to be carpenters, and people use estimates to determine when their paths will cross. They usually wait for each other if one gets delayed, so accuracy is not that important. So, is Mathematics no longer relevant to today youths?
I believe Mathematics is very relevant, and while they may never use systems or Pythagorean Theorem, there is something to be said for the mental discipline that math teaches. Good Math students usually develop strong analytical and critical thinking skills. They can often think outside the box, and work backwards to solve a problem. However, being a Math teacher, and having taught special education, regular education and honors students, I am fully aware that all students don't perform on the same level. In fact, in many case the material is way above the cognitive level of my special education students, who are taking 10 grade math when they're on a 4th grade level. In these cases I agree with Lockhart, we should teach the kids basic math, and then allow them to construct their own knowledge (p.1). Students who are going to technical school should learn basic math and money management. While others who are going on to Medical school should take more advanced courses like calculus. Not only should there be differentiation of content, but assessments. Also, the diploma should match their interests. I believe students will perform better, and enjoy math, if it is catered to their learning styles and interests.
Lockhart also alludes to the idea that teachers aren't teaching outside the box (p.2). I believe this is due to 2 factors. The first is that some teachers have a limited understanding of their content. It takes confidence to facilitate a discovery activity. If students are going to be given exploratory assignments, then they'll have a variety of questions. Without a strong Mathematics background it would be difficult for the teacher to answer the questions that be arise during the activity. In an effort to maintain control of their class, and not to appear incompetent, some teachers shy away from discovery activities in these areas. This sometimes results in some classes being mundane, and with the teacher showing the students how to use formula, but not much is said of why we use the formula, and if there're other solutions to the same problem.
The second reason we don't see more discovery learning, is that the curriculum is preset by the states. While some counties and local school district may implement it differently, the teachers have very little input in these decisions. Exploratory activities are often time consuming, particularly when you're dealing with low math skills. If teachers are to allow the students the amount of time they need to explore, then they'll fall behind, and they won't be able to complete the content before the End of Course Exam (EOCT). Like it or not, we live in a test driven society, and teachers are held accountable for their students performance on these tests. If they're going to have a shot at passing then the students must at least be exposed to the material. Ideally, they should have an in depth understanding of the subject before testing, in order to achieve this goal, teachers usually forego discovery learning for more teacher-centered learning. This ensures that they get through the material in a timely manner, thereby ensuring better test scores.
Creativity: Why it's missing and how to find it
Undoubtedly the pressures of testing stifles creativity in the classroom, however I believe that the lack of creativity in the classroom is also due to the fact that most Mathematics teachers don't know how to be creative because they were not taught in a creative manner. According to Lockhart this "cultural problem is a self-perpetuating monster: students learn about math from their teachers, and teacher teachers learn about it from their teachers, so this lack of understanding and appreciation for mathematics in our culture replicates itself indefinitely" (p.6). Most Mathematics teachers learned mathematics is a teacher-centered way. Their teachers lectured and demonstrated how to apply the formula. They in turn go home and memorize the formula. After a series of practice problem, eureka, they finally get it. This worked for them, so automatically they think it will work for their students. Unfortunately, this philosophy is detrimental in Georgia's public schools. The standards based curriculum in Georgia requires students to apply their knowledge.
In order for students to solve application problems, they must possess a high level thinking skills that will not come from rote memorization, but rather from a deep understanding of the material. It therefore behooves the teacher to find alternate ways to approach the concepts, so that their students can be successful. I have come up with innovative ways to help students with the memorization of formulae. I do this through songs, chants and games. I offer extra credit to students who can put any Mathematics laws to song. We beat on the desk and sing together. My hope is that they'll feel free to be creative. I also make the class more student-centered by having the kids do think-pair-share activities. In these activities students are presented with a problem. They are given a few minutes to think about it individually. Then they come together and talk about how they would solve it. They can then call on me if they have thought and talked about it but they are still stuck. I usually ask them leading questions that point them in the direction of the solution. Often times I also have the student work in groups solve problems. This approach works really well, because the groups vary in ability and the weaker students can gain knowledge from the strong students.
Ingredients for a Successful Mathematics Classroom
As a Social Reconstructionist I believe that classes should be student-centered, with the teacher acting as a facilitator. The main goal of the educator is to help students become critical thinkers and problem solvers. I believe it is essential for students to becoming critical thinkers because it is only though this process that students will be able to analyze social issues and come up with viable solutions, thereby improving the world in which they live. As educators is our responsibility to not teach about Mathematics in and abstract way, but rather use Mathematics can shape the minds of our students. To teach them not accept any idea as absolute truth, but encourage them to question ideas and to reason. This can be achieved by presenting students with open ended Mathematics questions. By giving them open ended questions, students will see that there is more than one answers to a question, and also that there are multiple ways to approach a problem. This approach to Mathematics education will be beneficial in the real world because we will be producing critical thinkers, who will ultimately become productive citizens.
I am a firm believer that effective learning occurs when the students enjoy the lesson; therefore I believe it is my job to be creative in the presentation of the material. This is why I think technology must play a significant role in my lesson planning. I also believe that students absorb the material better if the subject matter is practical and true to life, it therefore becomes necessary for me to give concrete examples that the students relate to when I presenting the lesson. All this planning would be in vain if I lack classroom management, so maintaining an orderly class is also vital. I think it is very important for me to keep in mind the 8 different kinds of intelligences, and to build the lesson so that it caters to the different learning needs of my students.
Ultimately, my goal is to create a positive learning environment. Students should feel secure enough to ask and answer questions without fear of ridicule. There must be provisions made for both groups and individual work. Many students are extrinsically motivated, but my goal is to get them to be intrinsically motivated for only then will they become lifelong learner. There's an old saying ignorance is bliss. Whoever said that has obviously never taught. There's no feeling like seeing little "light bulbs" go off. There's nothing like the look in a kid's eyes that says oh now I see, I've got this, this is easy. I have seen this and that is why I believe knowledge is bliss, and that it's an honor to be one who imparts this gift and that's why I am a high school Mathematics teacher.
I have faith in our students and in our education system. I believe that the move toward student-centered classrooms is good decisions. I believe that most educators want to be creative but that they lack the skills. These skills can be acquired through staff development and a willingness to change. A student-centered classroom requires a lot of trust in the students. Students are expected to explore and construct their own knowledge. However, teacher should constantly monitor students to ensure that learning is taking place. Also a student-centered classroom can easily become chaotic, so good classroom management will be essential. I believe that we'll begin to see a change in the students' attitude towards Mathematics as we change our instruction to meet the needs and interests of our students. I also believe the stress of teaching to the test will be alleviated because there will also be an increase in student performance. This is because students will have a more lasting and deeper understanding of Mathematics because they were active participants in acquiring and formulating their knowledge.
Lockhart, Paul. A Mathematician's Lament, 2002 .