I’ve had a problem lately. In my mind I have this clear image of a rant I want to post, but each and every time I try to write a post about said rant it turns into a rambling mess. Maybe that’s okay, maybe that is just what a rant is, but there will be no effectiveness. And if the post really wasn’t effective at anything, what was the purpose in the first place.
What, exactly am I supposed to be teaching? No, seriously….
Well, I teach math, but what, exactly is math? No, seriously….
Math is a beautiful exploration into the discovery of the structures that explain the universe. Or something like that. The point is, for me, math has intrinsic value. A few days ago, after school, I spent 45 minutes just playing with the idea of the sum of distances compared to the distance between foci and the triangle it forms in conic sections. There was no set answer, no goal. There was no “real life” purpose. The only purpose was because I wanted to. The explanation in the textbook felt too formulaic, too cookbook. I think what I did was what Paul Lockhart describes in A Mathematician’s Lament. I explored and inquired into the recesses of my mind. I created knowledge that is not beholden to any other authority. It can’t be taken away by another teacher or a different textbook. I can’t be told I’m wrong. I can be proven wrong, but I can’t be told I am wrong. And that is math in its most basic form. When math is authentic, it contains a simplistic elegance at it’s core. It is amazing and awe inspiring.
Well, to me math is amazing, but to many other people this is art. Beauty is in the eye of beholder and I am certainly glad I wasn’t coerced into taking up drip painting. That’s the catch, coercion. Intrinsic appreciation is achieved through an intrinsic fulfillment of an intrinsic purpose. Coercion frequently masquerades around as different forms of extrinsic motivation, movie days, pizza parties, grades, scholarships, detentions, suspensions, etc. The problem is extrinsic motivation doesn’t really work, especially in a setting like education.
Ever since I started teaching, which is 10 years ago, I feel that there has been an increase to push more students into more advanced math. I have an intuitive feeling that the push for creating new standards (Common Core), for standards in the first place, for the push against rote learning, is all stemming from a desire to create the type of mathematician that Lockhart describes.
How do we know do we know, as educators, if we have been successful? How do universities and employers know what kind of mathematicians schools are creating? Are these good future academics or employees?
Assessments, projects, PARCCs, MAPs, ASVABs, PSATs, ACTs, any number of devices we create as educators to quantify a student’s qualitative abilities as a number or letter, that’s how we know. I try to design assessments that can’t be coached or trained, but ultimately I realize that I don’t know if it is truly possible to make an unbiased, reliable, valid assessment that is perfectly uncoachable. The problem is that the students who have learned math through rote, by turning out exercise problems full of procedures but devoid of concepts, take the same assessments as those students who have internalized math. When I look at an ACT score of a student how do I know if that student just memorized a plethora of examples or is really a great mathematical mind that is just lacking experience?
I propose that the intrinsic motivation is necessary to comprehend math at the level that I feel pressured to create is an impossibility to extrinsically create. This would be acceptable as long as math is entirely voluntary. Conflict arises when the math students are coerced into learning has no relation to their desired outcome. It is difficult, if not impossible, to internalize something when the process itself serves no purpose. Most high school students I encounter are acute pragmatists. They might realize that to be a pharmacist they will have to take a Calculus course or two, but from their perspective those courses are only necessary because a university has set Calculus as a prerequisite. Students realize that the label of Calculus is much more valuable in society than actually comprehending Calculus.
The list of prerequisites keeps growing for my students and I feel pressure to ensure students keep getting past the barriers that are placed in front of them. I feel that my job is becoming less about teaching math and more about making sure my students obtain certain ACT scores, have certain GPAs, and have impeccable transcripts. I don’t feel pressure to teach Calculus, I feel pressure to teach how to get good grades in a Calculus class. We have created a labor supply that is hardwired to hoop jump. We have created a labor supply that is all accepting of school functioning as nothing more than a gatekeeper, but the gate is really the colander used to drain the bag of tortilla chips before I serve them for a snack.
Do tortilla chips really need to be drained in a colander? Of course not, but is a universal math requirement really the best way to determine the competence of a student? There is a movement out there to completely change the paradigm of education, but I think it is too fringe to go mainstream. For most of us we live in the cruel reality of that chip colander that consists of prerequisites, test scores, and GPAs.
And our reality has some harsh consequences. Books like Academically Adrift are written. Employers complain about college grads lacking workplace skills. The reality of our school system is that it is full of students who don’t really learn, yet have Pavlovian responses to terms like “honor roll.” That is the reality of school and I am the problem.
I wish I could inspire my students to love math and see the subject the way I do. I empathize with Lockhart. Every so often that passionate mathematician surfaces in class, which really gives my class a bipolar atmosphere some days. But as much as I have developed a passion for math, I have humanistic tendencies that dictate much of my behavior in class.
When I started writing this rant I had intended for it to be a indictment of the state of mathematics education in our schools. I wanted this post to be about how I want to teach pure mathematics, and how I can’t. I could teach math from a more stringent standpoint, but I couldn’t live with the self-image I was creating. I can’t sacrifice the economic futures of my students so that I can go grind my own personal vendettas about how I feel the educational system is failing. As much as I enjoy a good mathematical discussion and how some days I veer off topic into more theoretical mathematical explanations, the reality of my classroom is that the core is procedural drill. As much as I would be thrilled if I could inspire a student to take up research mathematics, I am proud that many of my students score high enough on the ACTs to avoid college remediation. I am happy when I hear from former students that they felt that their general education math requirement was relatively easy and that they felt ready for the class. I like to hear students who normally struggle with math say that they like my class because they feel successful.
Part of the reason that I can create that successful environment is because I have created grading scale that casts a wide net. I am able to catch several students who would have normally slipped through the cracks. A couple of years ago I had a miserable time and a large portion of my students ended up taking summer school. Often summer school simply acts as a credit recovery process more than a learning opportunity. Because I was trying to make my class rigorous by common conceptions, many students were struggling. Before I began the next year I decided to change how I evaluated students because I really didn’t like the fact that the quadratic equation was standing in the way of students taking up electrician classes at the vocational school. I really didn’t like that it was my personal adherence to the quadratic equation that was keeping students from taking vocational classes. I really didn’t like families had to spend money on summer school because I placed the quadratic equation on a pedestal.
Though I am happy that my changes in pedagogy and grading has broadened the number of students that find success in class, there is a sacrifice that is made to create this environment. My safety net has the unattended consequence of lowering the standard for all students. Really my success story means I am just a grade inflator. I should feel awful about myself, but I don’t. I felt worse when students would fail my class, but I could claim an ethical and academic high ground. I still expect a basic level of competence, but my students won’t be graded on compliance and obedience. Sure I don’t have the ability to push the intrinsically motivated math students, but they will be fine. Because they are intrinsically motivated they will find success someplace, usually in college when they can finally group with other intrinsically motivated math students.
I started out by asking what exactly am I to be teaching. I wear the label of math teacher, but I don’t teach much of anything that Lockhart would probably recognize as mathematics. In cruel acceptance of reality I teach survival skills. I teach students skills that will hopefully help them navigate a world of prerequisites, Accuplacers, and ACTs. We practice those skills in class. I wish that I could create an intrinsic love of mathematics in my students, but I can’t. By teaching students how to survive the system of hoops and gates I am part of the cycle that creates hoops and gates.
I am the very thing that I wanted to rail against in education.