Julie Vaccaro
EDSS 531
Learning like a Mathematician
When I think of what type of math teacher I want to become, I often ponder about how math as taught before my time. How did the great minds of mathematics come up with their theories taught in schools today? These men and women were all brilliant in their own right, and almost all of them have a story which explains why they were successful, crazy or in most cases both. After my own education, I consider myself to be a mathematician, not as refined as those before me, but I have picked up skills and ways of learning that classify me as such. I want my students to have the same opportunities, and be able to think, act and learn like a mathematician while in my class. I want them to feel worthy and confident enough to be called a mathematician.
Almost all great mathematicians had a few things in common, which are things I want my own students to foster and have access to. These things include a love and understanding of mathematics, a good learning environment, persistence and high expectations (usually for themselves) which includes respect, working against all odds, and were able to teach themselves what they needed.
“It is not knowledgeable, but the act of learning which grants the greatest enjoyment.” This is a quote said by Karl Fredrich Gauss, who is still one of the most respected mathematicians and lived in the 19th century. He was known to be trying new concepts and proving brilliant ideas which have created a good foundation of our modern mathematics. It is all about his curiosity that made him successful. It was known that when he was in grade school that a teacher gave the class a “busy work” problem to sum the numbers from 1 to 100 and only he was able to find the solution so quickly. Instead of summing the numbers one by one, he found a pattern of symmetry in the sequence of numbers, (Weisstein, Karl Fredrich Gauss, 2007). This is something that I want my own students to be encouraged to do in their learning. As a math student, you shouldn’t be bounded by what is taught but encouraged to think beyond the “rules” to gain a deeper understanding of what is happening.
So many people don’t get this or are not exposed to an environment of learning mathematics where this is acceptable. There are so many students that are being faulted for getting problems wrong on a test for not doing it the way that a teacher wanted them to even though a method was not specified. Algebra is one of the subtopics of mathematics that gets the worst rap about rules. “Many view algebra as having rather rigid boundaries; they see algebra as rule-based manipulation of symbols to model problems and solve equations. However, algebraic boundaries are permeable, and algebraic habits of mind can appear in a variety of settings,” (Driscoll, 1999). As a math teacher, I want to be able to allow my students to explore mathematics and understand the beauty of the subject, just as Gauss did. This could take the form of hands on activities, ideas that are applicable to everyday life or something that is needed for understanding of other topics.
Another thing that is important amongst these famous mathematicians is their ability in logic and reasoning. So many students these days don’t have a good basis of number sense. What is the difference between one million of something and one billion of something? We use these numbers in mathematics all the time, but what do they mean? It is rare to see a million of something, if you own a dog, there are about a million hairs on their body. Finding a billion of something is even more difficult to conceptualize (Sawyer, 2012). The problem is that sometimes math has become so abstract that students have nothing to physically connect to it, which can make it difficult to reason with and apply. “A focus on reasoning and sense making , when developed in the context of important content, will ensure that students can accurately carry out mathematical procedures, understand why those procedures work and know how they might be used and their results interpreted,” (National Council of Teachers of Mathematics, 2009). Simply having a good understanding about the world around you and the concept of numbers as well as logic, it makes mathematics a lot simpler. Mathematics after all is built upon itself and more difficult topics can be derived and understood based on things that have been learned earlier on. This is something that I know I will have to work on with my students and show them that mathematics isn’t as scary as they may perceive it to be.
Sometimes the development of mathematics is all about the learning environment. In an extreme case we have the example of Pythagoras and his development of the cult-like group of the Pythagoreans in about 500BC in Greece. It was group based on the environment of learning about philosophy and mathematics which also included religion. This group was so productive in their learning that one of their known theorems developed was the Pythagorean Theorem, amongst other lesser known ideas in Geometry. They all had common ideas, goals and a comfortable environment, it was even said that they banned beans due to the evilness of flatulence, (Weisstein, 2007). Even though the Pythagoreans were a cult, they have a good basis of learning and some of their basics can be applied to a classroom.
I envision my classroom to be comfortable, colorful and someplace where the students feel as if it is their own. One thing that is important is hanging student work on the walls. This is a difficult thing to do with a math class since there aren’t a lot of “art work” related projects to hang on the walls. However, that is not a difficult thing to connect to. Students can draw to help them learn, create diagrams to help support a project or even have a project that uses math to create the art, like with the use of fractals. “There is a long-term benefit to working toward making your students feel safe and comfortable in the classroom and helping them learn to take ownership of their space is an important part of building the classroom community,” (Rethinking Schools, LTD, 2004). If the students are part of the classroom, then they will more likely be comfortable in the learning environment since they will take ownership of the room. On top of using students work to make them feel comfortable, I envision playing music to put students at ease and bringing in soothing colors to make it feel “homey”. This will all help in creating a student centered atmosphere. This also means group work and creating a strong student community.
Sophie Germain was an early successful woman in mathematics in the early 19th century in France. Her parents had low expectations of her as a person who is educated, as women at the time weren’t allowed to learn but were supposed to keep up a household. Sophie, however, wanted to learn and despite her parents many attempts at stopping her; she eventually created a pen name representing that of a man in order to converse with other mathematicians including Gauss. Because of her respectful nature conversing with Gauss and others, they accepted her after finding out she was a woman, and taught her things that she didn’t know. Sophie is an exception of what happens when low expectations happen when placed upon students, since she persevered and continued learning math in secret, but for a normal student, they might not have been as successful. However, because of her respect towards other experts they were willing to work with her and teach and learn from her. Sophie chose this state of mind, and due to this, she was successful. This is the same thing that I want to simulate with my students. I want to have high expectations of them so they will be able to think highly of themselves as well. “The only way to hold students to high and rigorous expectations is to gain their respect and their acknowledgement that your class will lead to real learning that will benefit them,” (Rethinking Schools, LTD, 2004). A part of having high expectations of my students is giving them respect to learn. I feel like the more that they get respect the more they are willing to give it back, unless they do something that makes you take it back.
Unfortunately I couldn’t find a mathematician that could illustrate the importance of classroom, but I have learned about classroom management from an unconventional source. Steve Farber, who is known for giving management advice in the business world is now applying the same advice to education. The thing that resonates most with me after seeing him at a conference, was that he discussed love. As a teacher you need to love your students and show them what they can do. It is all about the attitude of believing them and treating them like adults, (Farber, 2012). This works well with understanding expectations and with high expectations, the behavior is likely to be better. When behavior is not up to expectations, which is bound to happen, then I believe that you have to be clear and consistent. “Always have clear consequences and never threaten to take particular action if you are not willing to carry it out. Talk to students as mature young adults,” (Rethinking Schools, LTD, 2004). If you are not consistent then students will either see your loopholes and walk all over you or feel as if they are treated unfair and behave worse than before. To be clear and consistent with correcting behavior goes along with treating students with respect, and teaching them how to be respectful depending on their age and maturity.
The famous Albert Einstein was an under achiever in school, had developmental and social issues and rumored to be dyslexic. He had many issues and school was a place that didn’t work out for him since they didn’t think that was a normal student. He even failed out of the Zurich Polytechnic entrance exams the first time around. He eventually got to college after being recognized as have a great understanding of math and physics, and became the successful mathematician and scientist that we have remembered him for, (Wolff & Goodman). The moral of Einstein’s story is that you can’t judge a student based on their “issues”. Einstein had many of them yet he was a complete genius. I need to support my students in every way possible and get them the help that they need in order to succeed. This could take the form of teaching students how to take notes, work in groups, and vocabulary support. This will allow the students who are a little disadvantaged to boost ahead to get on par with the rest of the students. The important thing here is that students need to be able to learn on their own. That is what ultimately made Einstein successful; he was uniquely knowledgeable about math and physics.
The best gift to students is to teach them how to teach themselves, and while they are with me in the classroom I can model it by asking questions, and teaching them how to ask these same types of questions on their own. “If a primary goal of instruction is to promote higher level thinking, then teachers need to be prepared in their classrooms to ask questions that lead to this important result,” (Baldwin , Keating, & Bachman, 2006). By asking proper higher level questions, these students can go beyond learning and begin to connect it to the outside world, making it easier to understand and remember.
I want the best for my students and with a combination of each of these aspects in the classroom, I believe that they will be successful. They need to be influenced to do well and have high expectations from the very beginning, and know that I sincerely care that they do well and understand the material. It isn’t about regurgitating the material but understanding and deriving it yourself. The more the students do this the more that they will learn like a mathematician and be successful in all subjects.
Baldwin , M. D., Keating, J. F., & Bachman, K. J. (2006). Teaching in Secondary Schools. Colombus: Pearson.
Driscoll, M. (1999). Fostering Algebraic Thinking: A guide for Teachers Grades 6-10. Portsmouth, NH: Heinemann.
Farber, S. (Performer). (2012, February 2). Re-Energize Education. USS Midway, San Diego, California.
National Council of Teachers of Mathematics. (2009). Focus in High School Mathematics Reasoning and Sense Making. Danvers, MA: NCTM.
Rethinking Schools, LTD. (2004). The New Teacher Book. (K. D. Salas, R. Tenorio, S. Walters, & D. Weiss, Eds.) Milwaukee, Wisconsin: Rethinking Schools, LTD.
Sawyer, D. R. (Performer). (2012, Febrary 3). The difference between a million and a billion. Marina Village Conference Center, San Diego.
Weisstein, E. (2007). Karl Fredrich Gauss. Retrieved February 18, 2012, from Eric Weisstein's World of Biography: http://scienceworld.wolfram.com/biography/Gauss.html
Weisstein, E. (2007). Pythagoras of Samos. Retrieved February 18, 2012, from Eric Weisstein's World of Biography: http://scienceworld.wolfram.com/biography/Pythagoras.html
Wolff, B., & Goodman, H. (n.d.). The Legend of the Dull-Witted Child Who Grew Up to Be a Genius. Retrieved February 18, 2012, from The Albert Einstein Archives: http://www.albert-einstein.org/article_handicap.html
Comments: “First, your use of quotes showed reflection and passion. In some places, I cared less about what others said, but what you think, or at least why you agree with the source. You were broad and encompassing in some areas, but in your section on the learning environment, the specificity added to the strength of your philosophy. What you have here is great, but if I could make the slightest suggestion, it would be to specifically mention an educational philosophy or mission statement you have, or relating the nature of the learner/teacher/etc. This, in addition to what you have, would serve as a fine piece to represent you professionally.” -William Rhein