As we began our unit on logic, my geometry class assaulted me with the question "How is this stuff math?!" I explained that logic is the foundation upon which effective arguments are created, and in mathematics it would be how we show that ideas are true or false.
They refused to buy into it. They couldn't understand how something that didn't involve numbers and equations could be mathematics. They'd never seen anything else.
I think we as educators must completely rethink our attitude toward the purpose of mathematics. We spend so much time emphasizing the importance of particular skills--counting, solving equations, or measurement, for example--that the overall goal of studying math is obscured. Math, like any science, gives us a specialized way to investigate our world and create knowledge. When students leave any of my classses at the end of the year, I have the following expectations:
1. Students should have the confidence to take on complicated problems rather than avoid them.
2. Students should be able to use a variety of resources (calculators, computers, the internet, reference materials, collaboration with others) to investigate problems and form strategies for their solution.
3. Students should use logical reasoning to analyze information and make decisions.
Do I care if my Algebra II students memorize the process for finding the roots of polynomial equations? Not really. I didn't even touch the topic after high school until I reached a 400-level numerical methods class. I certainly don't expect the majority of students to need that blurb of information. However, I do care that they know how to develop a budget and that systems of linear equations can be a useful tool for optimizing their budget. I don't care if my Statistics class can recite the formula for standard deviation. I do care that they know to use standard deviation as method for determining if a data set is precise (and likely reliable). I don't care if my geometry students can prove that a particular triangle is congruent to another triangle. I do care that my students know how to prove ideas are true in a variety of contexts.
I care that my students become more logical thinkers with the ability to analyze ideas and effectively communicate their thinking. That is why we study math.
I'm excited about New Mexico's adoption of the Common Core mathematics standards next year. They do a much better job of addressing the "big picture" goals of mathematics as well as describing the combination of skills students need to accomplish those goals. The current New Mexico standards separate desired skills into "process" and "content" knowledge, with a majority of the document addressing the mathematical content students should know. What little the standards address about process is vague, including things like "Students should solve problems that arise in mathematics and other contexts" (Ok...how?). The New Mexico content standards include "Students should use mathematical models to represent and understand quantitative relationships," which is followed by several sub-standards. The Common Core standards explain, "Modeling is best interpreted not as a collection of isolated topics but rather in relation to other standards." So many features of mathematics are intertwined with the idea of modeling real-world phenomena that it's difficult to accurately discuss modeling as a stand-alone topic.
Even subtle wording differences between the sets of standards imply a greater depth of understanding required by the Common Core standards. The New Mexico standards want students to write equations. The Common Core standards want students to create equations. I often see students write and manipulate equations without really knowing what's going on. However, the creation of equations requires that students recognize the meaning of variables, coefficients, constants, and symbols in order to put together an accurate mathematical sentence.
Of course, standards mean nothing if we can't implement them effectively. That's where teachers must take the lead.