As new teachers, we need to be ready for the implementation of the Common Core Standards, which will replace the California Content Standards.  Being ready for this means knowing how these standards are similar and how they differ.  One way in which they differ is that the Common Core Standards are more succinct, but aim for more depth. 

For example, in the CA Standards for 8th grade algebra, six separate standards focus on various ways to solve and use quadratic equations.  This was reflected in an Algebra class that I tutored in, as months were spent focusing strictly on factoring, solving and graphing these types of equations.  In contrast, the new Common Core Standards do not even mention the word "quadratic".  Here, the emphasis is heavy on functions: what they are, how they are used, how to build them, how to visualize them, etc. 

In my opinion, this is much more important for building the students' mathematical understanding.  Using functions to understand relationships, model reality and make predictions is one of the most applicable components of mathematics to all other areas of study and life.  While many students do not re-visit factoring quadratic equations by hand in higher mathematics, knowing how to recognize, build or analyze a function is a valuable skill in calculus, engineering and other sciences. 

In this way, the Common Core Standards focus on the "forest" of mathematics, teaching students how to think and reason mathematically, where the CA Standards focus on the "trees", algorithms, techniques and formulas.  Because I think a higher emphasis on mathematical reasoning and problem-solving skills will prepare students better for college, careers and daily life, I am excited for the transition into the Common Core Standards. 

Find my breakdown of the how the standards intersect here.

We all use math on a daily basis, whether we are calculating tip at a restaurant, doubling a recipe or figuring out how much that t-shirt would be with a 40% discount.  Just this past weekend, I had to use math multiple times in order to complete tasks.  In my apartment, I am in charge of bills and household concerns.  Our carpet has been looking really bad lately, but we got a flier on our door advertising a special for carpet cleaning. I needed to use math to sell my roommates on this idea!

The deal was $40 for the dining/living room combination, with an extra $10 for each heavily soiled spot.  Because my three roommates are on a budget, I had to project the price range we would each pay before they approved booking an appointment.  I went around the room, and counted the various areas they might consider "heavily soiled".  Because I did not know how many square feet would count as one heavily soiled area, I had to come up with the various possibilities.  The area by the door was pretty bad, and I figured it would count as one area at the least, and two at the most.  This would be an extra $10-$20.  The area around the living room couch was also pretty bad, and it was slightly larger than the one by the door.  This could be considered 1-3 heavily soiled areas, equating 10-30 dollars extra.  Together, this meant that the minimum "extra charge" would be $20, and the maximum would be $50. 

Because the base charge was $40, I added this to the min and max extra charges to get a price range of $60-$90.  This sounds like a big price tag for young college students. However, because there are four of us, I divided the range by 4, to get a cost of $15-$22.50 each.  I rounded this up to the next multiple of 5 (a little extra wiggle room never hurts) and told my roommates that we could get our carpet cleaned for as low as $15 each, with the absolute max being $25 each.  They agreed!

The actual estimate came out to be the minimum projection, $60.  I was stoked, because the carpet looked great and the price was a steal. Many people are shocked when getting home repair estimates, but with a few math skills and basic information about the cost of services, this can be avoided. I also used math as a way to successfully present an argument.  My roommates may not have agreed if I simply told them I wanted to book a carpet cleaning that would cost between 60 and 100 dollars, but presenting what the individual costs were made it sound more reasonable. (Especially when comparing it with the $125 security deposit we want to get back!).

How do YOU use math in everyday life?  

The little pig with my name on it was always at the front of all the other pigs on the wall of my third grade classroom.  This represented that I was the leading my class in times tables.  This is my first memory of learning math in school, and it revolved around memorization.  As I got older, math basically remained the same in my mind: you learn the algorithm, use it on your test and homework, and earn an A! I had my classes down a formula! The one time I remember actually being frustrated in math was when I did a project in 8th grade Algebra where we could use any method we wanted to estimate the number of marbles in an oddly shaped vase.  It was madness! How was I supposed to get an A when nobody was telling me what to do?

Now, after completing my degree in mathematics, I thrive on having to use my own reasoning to arrive at a solution.  Getting into the upper division classes, I was aware that they would be focused on proofs, but I was not prepared for the learning curve I faced as I transitioned from memorize-er to mathematician.  However, this is when I fell in love with the subject.  I began to see the beauty in the logical foundation of mathematics and the creation of a proof from prior knowledge, intuition and creativity.  My love for mathematics started to come from my ability to act as a mathematician, struggles and all, and not from my ability to receive an A in the class.  I am confident in my claim that I am driven intrinsically in my study of mathematics. Math is definitely my favorite subject now.

One teacher that helped me to come to my current view is Professor Laura Stevens, who taught "Foundations of Teaching and Learning Math".  After 3 years of calculus courses and real analysis, she helped me to actually understand what the subject was used for and where it comes from (intuitively and logically).  She would always start classes by throwing a question at us and giving us time to think of the various ways of solving it.  We would then present some of the solutions to the class and then rigorously build new theorems or definitions together as needed for deeper exploration. 

As a teacher, I hope to intrinsically motivate my students and provide them with a deeper understanding of mathematics.  I want to go beyond the formulas and algorithms and teach students in the way that I was taught by some of my best teachers.  I hope for my students to understand that math is approachable, something they already do, and a skill they can apply to all facets of life.  

I recently posed a problem on my social networking site: "How do you personally calculate the tip at restaurants?".  This question reaped a plethora of responses, each friend sharing their own strategy to find the tip without using a calculator.  Among the responses, I would like to highlight three.  The first one, which was the most common response, was "doubling the tax".  This is a strategy that I use as well, as many of us know that the tax is somewhere between 7.5% and 10%.  Simply doubling that number will yield a tip between 15% and 20%, the recommended range for tipping.  Another strategy that I found quite interesting was multiplying the total bill by two, and then taking 10% of that by moving the decimal one place to the left.  This would yield a 20% tip, but the author also recommended rounding up or down to the nearest dollar depending on the service.  Another variation of this was to take 10% of total, and then double that number.  Lastly, one friend shared that he tips the total amount divided by 5 in order to yield 20%. 

Calculating tip is one of the many ways that everybody uses math in their daily lives, and the variety of ways that people go about this task provides insight into formal, classroom mathematics.  Students come into math classrooms with prior knowledge and "street smarts" for arriving at solutions.  Sadly, formal mathematics often fails to make connections between students' prior knowledge and the learning goals of the class.  Rather, teachers should build on this knowledge, encouraging students to share how they might use their reasoning and logic to find solutions.  Lastly, this example also highlights another important goal of teaching mathematics: multiple routes to the same solution.  Each "tip strategy" I shared would roughly yield a 20% tip, yet each person had a different way that made sense to them.  These multiple routes should be taught and encouraged in secondary math classrooms, providing more opportunities for students to be intellectually creative and to find a way that connects with their own reasoning.