Students learn through exploration, challenge, and dialogue.
So often we forget that students have dreams. Each child has at some point looked up to somebody as who they want to be. Each child has at some point thought about being a certain profession when they grow up. Some children have lost this. Students are trying to have fun when they're messing around and as adults we look at them as needing to grow up. How do we expect children to grow up if they're not given the chance to explore and discover? Challenging students to participate in activities of discovery promotes growth. The dialogue that students have amongst each other and share with family/friends furthers their understanding of the world around them. Students should be given the opportunity to explore, challenge, and converse as they need to create a path for themselves "grow up" and become adults.
My students are able to solve problems, given the opportunity.
Doesn't this go back to exploring and challenging? If I don't give my students the chance to think about a solution, of course they will never reach that conclusion. When I teach material and teach a process, I'm robbing my students of the intellectual growth of an opportunity. As adults, we look for opportunities to solve problems. Lack of household income - find a job. Miss your best friend in England - Skype them. No matter the situation, we are always looking for solutions. Kids must be given the chance to also learn how to problem solve. It is within a free context that the most creative answers are produced. With different ways to solve a problem, the more unique perspectives and better answers.
Math is based on concrete principles made abstract.
One of the most intriguing things that I've heard today is that "Math is abstract. You don't walk around the world seeing 2s like you do chairs." This is wholly true but we are founding our desire for abstract math upon the concrete principles that we are trying to solve. For example, we needed math to build the Hoover Dam. We could have created a unique mathematical principle for the Hoover Dam to prevent cracking at the Dam's particular size and the amount of water it will hold. From that principle, we can then take it out of context and apply those concepts to more abstract situations around the world. This is how discover in the real world. We look for examples and ask questions about that specific circumstance. Why are beehives shaped a certain way? Why do arches hold so much weight? From that point of discovery we then apply the same concepts abstractly. It's beautiful. Stone Arches are some of the longest surviving structures of nature... and now some of the longest existing bridges were built with arches.
Math is a body of knowledge, not a checklist of skills.
This is hotly debated. You needs skills to do math. True, necessary, but definitely not sufficient. If math was strict skill, we would not be able to apply addition and subtraction to every situation we come across. I wouldn't be able to balance a check-book because I was never taught. I can add my money in my pocket, but I can't track how much money is in my bank account. Knowledge is the ability to use skills in different contexts. We look at knowledge as abstract, but it is only abstract because it is applicable widely. Math is useful in everything we do, therefore is must be knowledge. That knowledge is supported by the skill that is required to actually perform the action we need to accomplish (If I know we need to subtract 3 from negative 5, I have to know how to actually perform the action versus just say it is necessary in my situation)
Math is beautiful, useful, and exciting.
I think most people have the biggest issue with this complex. We're raised in a society where we were taught PEMDAS, FOIL, SOHCAHTOA... acronyms by which to remember how to perform operations. We're forced to memorize these, otherwise we will not succeed. The truth is that math is so much more than these mundane, sometimes useful, sayings. Let's anaylze the beauty of math. Take a look at this picture:
Beyond your inclination to say "I don't like cities" or "I love the city. OMG So Pretty"... What is beautiful about it? The sky-high buildings? The lights that showcase the calm streets? Is it the calm, tree-lined streets?
Would it have been possible to build this city without math? The beauty of it all was founded on ideas that required math to develop. Every additional level on a building required engineering to prevent collapse, financing to ensure completion and even architecture to develop beautiful structures.
Math's uses turn into beautiful things. We can envision every swipe of the pen(cil) on the paper as one more movement toward beautiful creations.
The utility is clear. With math we create living spaces, we create movement in the world and even more now-a-days, we see the need to calculate our remaining resources and developing solutions to prolong depleting our nature.
Every advancement is exciting. "New Drug Prevents Death From Over-Caffeination" (I'm hoping they develop this for my sake). Our lives are getting increasingly better and we are living better and better. We see happiness as we raise the levels of those around us. Happiness breeds happiness, which breeds excitement for more happiness. It's a loop, but nonetheless a useful one.
My commitment rests on how I will use all of these beliefs to better those lives of my students in context of math. My commitment is to use beauty in every lesson, to allow students to develop knowledge and to never forget my students are young adults waiting to grow by bounds and bounds. I will give them the chance to explore, create solutions and create more beauty in the world. You can see this in my vision for my students and the discussion-based method of teaching I will use in class. I'll post that vision in a few hours after I complete it more.