Saturday, November 26, 2011

Why We Study Math

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" ( 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.

Friday, November 25, 2011


One of my goals in starting this blog is to address (and ultimately overcome) many of the fears I have as a first-year teacher. Before I go into detail, let me set up a little background for you.

I graduated with my B.S. in mathematics with minors in biology and chemistry about a year and a half ago. Having a vocal music/performing arts background as well as a passion for the sciences, I decided to pursue education rather than research for my career. I'm now 3/4 of the way through a master's degree in secondary education with a focus in math education. I also teach at a small (~140 students) charter school that emphasizes career-readiness and project-based learning.

Sounds positive, right? Well, I'm teaching six different classes (regular and honors Algebra I, regular and honors Algebra II, Geometry, and Statistics). My classes, while small, contain mostly at-risk students and several fully-immersed special education students (without support from a special education teacher, which violates several IEPs). I also have fewer resources and more administrative duties than I expected because we're such a small school. Add the stress of a complete administration overhaul and two weeks of medical leave taken after surgery (stories for another time, perhaps...), and it all adds up to what sometimes feels like the First Year from Hell.

And there you have the source of my many fears. I'm afraid that I can't relate to many of these kids. I'm afraid I can't reach some them no matter what I try. I'm especially afraid of failure despite what I tell my students about failure as a necessary element of creating knowledge.

I'm afraid that I can't relate to many of these kids.I don't have a rebellious bone in my body. Conflict makes me nervous (probably not a good trait to have as a high school teacher). I've always respected authority, even when I've questioned it. I've never been in a fight. I've never done drugs. I didn't drink until I was 21. I've never even gotten a traffic ticket. So, how do I compassion toward teenagers who aren't afraid to throw or take a punch, who don't fear consequences like Fs or suspensions, who approach problem-solving with emotion first and logical reasoning second (if at all), and who probably think I'm a giant pansy. How do I show them that I am, in fact, a real person and not a nerd-bot?

I'm afraid I can't reach some of them no matter what I try.
The students and I are fortunate to have such small classes because it makes differentiated instruction much easier to implement. However, that does not guarantee that I'm any good at implementing it. I'm really a terrible multitasker, so up to this point in the school year I haven't done well running different activities simultaneous. I've focused instead on differentiating the level of complexity and the presentation of material (auditory, visual, verbal, hands-on, etc.). I try to give students some choice in the style and context of questions they answer (though I believe nudging students out of their comfort zones is necessary for personal growth). Despite this, I have some students (and a good 2/3 of one class) who I struggle to engage in learning. I've tried creative projects. I've tried authentic contexts. I've tried structured activities. I've tried open-ended questions. I'm having trouble leading them to water, much less making them drink. On Monday/Tuesday, I plan to use the following as a warm-up activity in 3 of my classes: "Write me a letter. In it, tell me what your goals are for this class and for this school year. What do you enjoy about school? What do you wish were different? Suggest an activity for us to do as a class some time this year." I hope this will allow some of the "unreachables" and I to communicate with each other a little better. What else can I try to engage these kids?

I'm afraid of failure despite what I tell my students about failure as a necessary element of creating knowledge. The scientific method requires failure. We form hypotheses. We test hypotheses. We measure the results and compare them to the hypotheses. Very often, we analyze the failure in our results and use the failure to refine our hypothesis for a new test. Despite my understanding of this, there's still a stubborn part of me that feels like failure is something final that I can't recover from. FALSE. Recovery can be difficult, but it is possible. I assigned a project to my geometry class to analyze the logic of advertisements in which they write and study conditional statements and their converses, inverses, and contrapositives based on ads. This is the only project I've ever found on the topic of logical statements, and in hindsight it's a pretty awful one. Is analyzing the logic of advertising a useful activity for real-world decision-making? Absolutely! Are the converses, inverses, and contrapositives of advertising slogans useful tools for such analysis? Hardly. My students recognized this far faster than I did, and their response was to do anything other than complete the project. My confidence was temporarily crushed. My content knowledge of geometry is much weaker than I'd like (meaning I haven't had a geometry class since I was a freshman in high school; I focused on linear algebra, statistics, and numerical methods as an undergrad), so I was completely lost as to how I should approach the study of logic with this group. This situation leads me to several questions. How do I find a new way to do things when I don't really feel like I know what I'm doing? How do I make logic feel more like "math" in my students' eyes? How do I help my students grow from both my failures and their own failures?

Hopefully I'll answer all of these questions over the course of the year. I appreciate any constructive input I can get. Both the challenge and beauty of teaching is that there is no single right answer to any question. All I can do is approach it like a scientist: do my research to collect options, experiment, refine my hypotheses, and try again until I get it right.