Through her short time on earth, Alice has of the importance of sharing because of the constant reminders she receives on a daily basis. Now that she’s a toddler, she is constantly learning how to speak like a big girl, being rewarded with verbal praise when she does and asked how to speak like one when she gets very demanding. The goal of these lessons is to help her progress developmentally and to instill a sense of courtesy and respect for others. In this situation, there is not only a parent-child relationship but also a teacher-student one, where the student is Alice and the teachers are her parents. Occasionally, however, someone else may step in and serve as the teacher. The lessons are ad-hoc and occur anywhere and at any time. Because of the routine her parents provide Alice, the lessons often occur when getting ready for the day, at meal times, or in the car. There are no specific, tangible artifacts that reappear in each lesson.

The artifacts of each lesson, whether it’s food, a toy, or a person, are specific to the situation. The intangible artifacts are the values and morals learned from previous lessons. The learning activity is usually a conversation, based on the rules set up by her parents she still struggles to learn as she balances her self-centered instinct to survive she’s had since she was board and her need to relate to and share with others. In each conversation, Alice must apply prior knowledge that she has learned through these conversations that have occurred since she started to speak. Everyone who has an active role in Alice’s life values these lessons because of how it helps her to grow into a person and because it reduces conflict and stress at dinner time!

]]>Fifth grade math was all about fractions and suddenly I got it. I’m not really sure what triggered this understanding but as I reflect on this experience, I feel it was the process of learning was much more investigative than anything I had ever experienced. Granted the questions asked me to find the greatest common multiple of two numbers, but I had to break each number down to its prime factors. Through this I was able to explore the makeup of numbers just as chemists study the makeup of compounds. I did not think I was the smartest kid in the class, but was always among the top. I really liked it there and enjoyed learning from my colleagues.

This experience in fifth grade put me on the path as a mathematician. In sixth through eighth grades, I was in the advanced math class, taught my a wrinkly, old nun. She beat ratios and proportions into my head and at the time I thought they were useless. As I advanced in my studies of math, I soon realized that when in doubt on how to solve a given problem, find ratios, build a proportion and solve said proportion. From this, I would be able to solve the problem. This helped me on the SAT and the GRE, and it was something that I learned in grade school.

In high school, I was clearly the best math student in my grade. When I graduated, I received the math award for the highest GPA in mathematics. It meant everything to me, representing how hard I worked in school and celebrating my natural talent. When I arrived in college, it was a different story. I wasn’t the best, but I wasn’t the worst. I was average, but I still enjoyed learning, especially when I was able to find the mathematics in computer science. I was on my way to “do math” every day as a grown-up with a career and an office.

Now that I am back in school eight years later and plan to teach math every day, I constantly reflect on why I loved math (and continue to love it) when I was younger. Quite frankly, I’m still not the best in my math classes, but I still enjoy it. So do I enjoy it because I am naturally good at math? Is it because I have to work hard to understand it? Is it because my teachers are enthusiastic? I’m pretty sure it is a combination of all three and that’s probably how it has been through my career as a mathematician.

This entire experience beginning in the fourth grade has shaped how I see myself as a mathematician. Sometimes, I am the best and most times I’m just an average student, but all the time, I really enjoyed math because of the challenge it presented and the enthusiasm many of teachers had. For me, it’s really about the journey of learning mathematics instead of one particular course or day in school that has made me the student I am today.

]]>I really like to idea of bookending a unit with a discussion on how or why one would solve a specific type(s) of problems before giving the algorithms to solve such problems. Such an approach would allow me to see what creative ways they students would use to solve the given problems based on their existing skill set. It reminded me of a lesson about solving systems of equations. I found on NCTM’s website (http://illuminations.nctm.org/Lessons/CandyProblem/CandyProblemAS.pdf). The actual solution to this system of equations cannot be found using any of the algorithms taught in algebra (substitution, graphing, or elimination). I wonder if presenting the students with a novel problem such as the Candy Problem at the beginning of a unit would stimulate and open their minds to new concepts in mathematics.

]]>This was especially poignant when understanding the need for creating and presenting a question at the beginning of the lesson. The idea that said question does not need to be clearly formulated. I felt at times when writing lesson plans in my first student placement, I was just forming a question out of the keywords of a lesson instead of how the skills learned in the lesson may be applied to solve real-world problems. Even while writing this, I still struggle with the idea of how much information must be conveyed to students before presenting them with a real-world problem. For example, when I presented my eighth-graders a lesson on how to solve inequalities using absolute values, I posed the question, “how are inequalities using absolute value solved?” I wonder what the impact would be if I just presented them with an example of such an inequality. Would it allow the students to investigate and learn more through discovery or trial-and-error? While I realize that there isn’t a single method in presenting an inquiry-based lesson, I still wonder if one way is better than the other.

As an aside, I also really appreciated the author’s diagram on how to structure an inquiry-based lesson. It illustrated simply what the building blocks of strong lesson rooted in inquiry are. I feel like I should have it blown up and pasted inside my lesson planner or made as the wallpaper of my laptop!

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