The quantitative reasoning conference redefined the notion of what constitutes research by broadening the definition of mathematical experience to include any experience that happens while trying to teach and understand math.
I spend most of my time building mental models of student thinking, and constructing logical arguments, yet I know that the forces that kept me teaching and pushed me from the profession were largely beyond logical description. Due to the unexpected death of a young friend, and a subsequent reconsideration of myself, I’ve realized that logic only explains half of my life and by focusing on it almost exclusively I’ve missed some major insights into who I am and why I struggle in particular aspects of my life.
Given the difficulty of implementing reform instruction in mathematics classrooms and the dismal teacher retention rates, studying emotion as an explanatory construct for the state of affairs seems reasonable. (There, I go again, juxtaposing emotion and reason.)
Here is the story of constructivism and my emotions. I entered math education because I realized through tutoring that students didn’t understand why math was true and as a result spent agonizing amounts of time memorizing things they didn’t understand. I read a few articles on conceptual learning of mathematics, entered Teach for America, and began teaching Algebra and Pre-Calculus to students who were years behind in mathematics. I brought with me strong mathematical connections and deep understandings of secondary curriculum but very little understanding of student thinking. The best I knew how to do was to ask and explain why a mathematical statement was true and to try to get my students to “discover” what I wanted them to learn. Discovery often took the form of guessing at a pattern, and in hindsight probably didn’t promote the meanings I intended. It was a start to showing my kids that math was not just something to memorize.
The emotions, which were largely expressed in tears and anger and exhaustion, were a result of the conflict between what TFA told me I could achieve with my students and the reality of what happened when I tried to teach math meaningfully without the support of anyone with knowledge of student thinking. I was instructed to teach algebraic concepts for which my students had no foundation and told that I was lowering my expectation and promoting the achievement gap when I suggested that students who didn’t understand fractions and multiplication would not be able to make sense of algebraic reasoning. I had no research to back me up, just a sense that was I was attempting to teach couldn’t possibly make sense and that the best I could hope for was that kids would be inspired to memorize. I remember telling my program director that I didn’t think that the district Algebra test was a reasonable goal for my students. She made me feel guilty for giving up on them before the semester began and I made it my goal. All of my students failed. Later I found out that 93% of the district failed. I still cried while hearing the click of wrong answers as I ran the responses through the scantron machine.
Students who have failed at math for their entire life are resistant to trying. It opens them up to more failure. It’s easier to dismiss and insult a teacher than to open yourself up in a class that can’t make sense, because you didn’t learn the math that would allow it to make sense. As I tried to teach “why” my students interpreted my instruction as confusing and they revolted. I felt strongly that my role was to inspire and help them, not to constrain and restrict them. My all-boys freshman Algebra class fell apart at the seams. The first lesson that wasn’t an abject failure involved teaching them how to multiply fractions by rote at the suggestion of a mentor teacher. She asked “what can you teach that they will understand and do well on a test.” After a day of multiplying fractions (which probably had little meaning for them) on a worksheet the class aced a quiz that only required multiplying fractions out of context. This was the most emotionally satisfying day of my first semester with them because they listened, they tried, and progress was made. As a constructivist educator, I see that I contributed to the construction of the sense that math was moving senseless symbols around and contributed nothing to developing productive meanings for fractions. But it saved the week and kept me teaching. In my PreCalculus class I instigated “practice tests” which mimicked the real test almost exactly and probably allowed students to earn high grades without a strong understanding of what they were doing. The 80% class averages, earned by mimicry, made life seem more bearable. I had some success to share with my mentors who I assumed thought I should never have been admitted to this elite program.
Most of the time I worked with my mentors I cried about the boys who were sexually harassing me in my classroom. As a young, insecure woman, who hadn’t come to grips with earlier abuse, most of my instructional decisions revolved around a sense of self-preservation. I gave worksheets that everyone could do so that they wouldn’t insult me. I knew they were not learning what they were supposed to but I couldn’t do endless battle. A primary consideration in lesson planning was making sure I was always facing the class. If I let my guard down for a second and wrote something on the board something would get thrown at me. I didn’t put questions on the test that required more than memorization because of the cries of “you never taught us this” were too much to bear. Critical thinking was all in the extra credit for the kids who cared and sustained me. I was faced with a group of boys who I hated because they harassed me yet I felt guilty because TFA told me I was supposed to care about their future. I didn’t, and I did all at once and it ate me up all year. Lest I appear to exaggerate, let me share the day I went to the doctor because I’d stopped menstruating, and he told me that I might be infertile. When life calmed down I realized stress was probably the cause of my health problems that year. I’d lost my voice for days, became overweight for the first time in my life, only felt okay when I wrote my bike for 70 miles and all I could feel was hunger, and prayed to get mono so I could leave school for a month. I realized one day that year that I used to be happy but couldn’t remember what that felt like anymore.
There were moments in my first two years that were more than this suffering. Some of my brighter students, who typically rebelled at boring worksheets, loved it when I spoke of the beauty of math. I remember tossing in the Goldbach conjecture into a lesson on prime numbers. One student spent all period trying to write numbers as differences of primes while the rest of the class worked on basic arithmetic skills. I saw students blossom and grow and go on to major in math, study physics at MIT, and tell me I’d changed their mind about education. In the midst of the coping by subverting my inclination to teach math meaningfully, I still explained why enough to reach a few students. For better or worse, I don’t think I knew how ineffective many of my “why does this work” explanations were and kept giving them over and over and over. I get the sense that some of my kids learned because they kept studying math successfully in college. Maybe it wasn’t me-who knows. But, I still keep the nice notes they wrote to me in a scrapbook.
I did care about the kids I worked with in TFA. I cared about the school and the teacher and starting making impacts and seeing successes. But the feeling I got when I imagined the emotional energy it took me to deal with the meanness directed towards me was enough to make me move on to a fancy private school with minor behavioral issues.
Again, emotions were a primary reason, that I changed my math instruction. I entered the school with the idea that I must teach these above average students the meaning and beauty of mathematics. They were going to fancy colleges and needed to solve hard problems and understand what they were doing. The first weeks of school I realized how much they knew about math! I dreamed of all I could do with them.
I was using Hughes-Hallet’s Calculus and assigned them problems that didn’t look like examples. They revolted. They insisted that I assigned work I hadn’t taught them. They told the principal that everything they had learned about Calculus they learned from friends and Kahn Academy and that I wasn’t teaching them anything. I thought that these bright students could clearly comprehend the delta epsilon definition of a limit because I had in Calculus 2. I saw them as younger versions of me. Yet again, no one with a background in student thinking was there to point out to me the obvious. Most of my students had completely different conceptions of rate of change and function as I did and didn’t assimilate my instruction in the way I intended. At this fancy wonderful school that promotes critical thinking many of them had memorized their way up to Calculus. They had impressed me the first week with loads of memorized procedures and formulas and I’d attributed my own understandings of those concepts to them because we used the same words at the same time.
There were a few brilliant, notable exceptions who told me that my instruction was wonderful. I remember one student saying “I love how you teach because you explain the idea and then I can solve all the problems myself.” As I didn’t see my students thinking as different than my own, I attributed their complaints to laziness. A logical conclusion given how disinterested in work my former students often were. My students were offended I thought they were not working hard enough and attributed their confusion in switching from a traditional to a conceptual textbook to me and my teaching. After all, I was young and new to their school. The knew the school, the administration, and the climate much better than me. They told the principal these things, and although he was a former math teacher he didn’t believe that math should be meaningful either. I was once shocked when he said “I never figured out why you need the quadratic formula or what the point of quadratics are. To me it is just a song.” I was annoyed when I was trying to teach the definition of the derivative and he came in to observe and impressed the students by using the power rule to compute in his head what they had spent lines doing. Many probably thought I was an idiot for teaching them the hard method when the power rule was so simple. To them, I should be teaching them to find answers quickly. That was what math was. And some saw exactly why the definition was more important and continued to back me up and provide me the joy I needed to believe in myself.
The observations and meetings turned into a formal letter-they were observing me twice a week to see if they wanted to keep me at the school. The principal did not attribute my difficulties to my lack of awareness of student thinking but to my personality. He told me “teaching is an art, and some people just don’t have it.” He told me “the kids don’t respect you.” He asked me to watch sporting events and make friends with my students to solve the problems he saw. Yes, relationships are critically important. To learn math meaningfully, I needed my kids to trust me enough to risk failure and express confusion. Yet, as a woman, who was still unaware of how fundamentally insecure I was, I just believed him that I was bad with people while maintaining the sense that my mathematical knowledge was much stronger than his.
I decided I didn’t like talking to students very much. That I hated noise. Sometimes my notions were contradicted by some of the amazing, meaningful and fun lessons I created with those wonderful, curious students(not the girl who didn’t understand rates who tried to get me fired-the others). I started to understand my students as I watched them try to estimate the area of baked goods using Calculus. I started to realize that my meaning for function and derivatives were not theirs as their knowledge broke down in application of math to the real world projects I’d devised with their help. I realized that some of my students could solve and pose problems more complicated than I’d ever dared to suggest. For some, what I was saying made sense. My guess is that these students were the reason I wasn’t fired. One told me after scoring a 5 on his AP test and an 800 on his math SAT 2 that I was the reason he’d been able to do that and that I’d made it all clear. (I’m sure it was not just me, but the emotional boost I took from that allowed me to read and process the weekly emails about how bad at teaching I was.)
Getting those emails about how bad my class was from the principal was brutal. They tended to point out that I’d added two numbers wrong or that a kid who’d stayed up late the night before working on an essay was sleeping and I didn’t do anything about it. When the principal asked me what I needed I told him “positive reinforcement.” I didn’t get it except for once after the year was done when he congratulated me on how well my students had done on the AP calculus test. I cried under my desk. I wrote angry responses to my criticisms and sent them to my mom. I relied on the love of my friends and tried to fix my boyfriend’s life to distract me from my own failure at work. I started changing my instruction. I started giving examples that looked precisely like the questions on the homework and telling the students which example corresponded to which problem. I made the tests easier and eliminated problems that I knew required thinking that only some of the students had. I got rid of epsilon–delta descriptions of limits(based on further reading, that was a cognitively sound decision!). I tried really hard to get my students to like me instead of getting them to think critically. I ignored topics that I sensed would be hard. I dumped extra time into writing challenge problems for the students who wanted to do hard math-they kept eating them up. They did amazing projects. The second year at the school I realized I had to build student confidence before asking them to grapple with hard ideas and didn’t sacrifice the quality of my instruction as I had the year before.
Although most of this was subconscious I imagine that I was forming enough of a sense of student thinking that I could predict what would be easy and hard without fully being able to explain this why in terms of constructs such as understanding of function and graphs. I don’t know if all of these modifications to reduce the demand of my class really worked to improve the problems my principal saw in the class culture. At that point, I really still thought that the problem was that I just wasn’t likable as a person and I didn’t see the huge rift in thinking between me and my students. None of that made sense until I worked with Pat Thompson in graduate school and learned about student thinking in his Calculus class and could reconsider the problems my students found challenging and what that might have said about their thinking. From this standpoint, I developed the theory that different conceptions of math, rates, functions and fractions led to many of the difficulties that were written up as a problem with my “likability as a person.” My kids did try to do the novel problems I assigned them but they didn’t have the meanings I did and therefore didn’t know what to do with them. This frustration got directed at me.
Emotions were part of the reason I was working with Pat. The most memorable moment of my graduate school search process was a conversation with Pat Thompson. He commented that I’d had realized more about student thinking while teaching than most of the teacher’s he worked with. I said “I had some particularly difficult circumstances and had to reflect daily on my instruction to decide if it was really as bad as my principal told me it was.” He said “I’m impressed you didn’t just believe the principal.” And then he turned serious, sat me down, and said “This is important. No matter which school you choose, I think that you need to be in a PhD program where people can help you grow and support you.” He somehow realized how horrible things were, without me saying anything. I wasn’t about to tell my future advisors that I was under consideration for being fired for a year. What if they thought I was actually a bad math teacher? Although choosing a PhD program involved a lot of rational thought on research goals, money, geography, ranking and more, the moment where Pat cared about me, seemed to read my mind and understand my pain for a second, stands out to me as a decisive experience. It’s the only I’m most likely to explain when asked why I chose ASU.
So take this as an existence proof that emotions do matter- especially to a young, insecure female math teacher. I had the knowledge and passion to teach math meaningfully. Once I was orientated to build models of student thinking and given some assistance I was able to do it productively. I had creativity and passion and love for math and lesson planning. I cared deeply about my students. Some of them even thought I was funny and wrote wonderful notes about how I was their favorite teacher. My teaching career could have been so much more, yet the tears, depression, self doubt turned my classroom into a place where thinking wasn’t valued as much as I thought mattered. Boosting content knowledge is important. Providing curricular support is important. But for some teachers, a giant piece of the puzzle in teaching math meaningfully is how they feel about themselves. I know I’m not alone-I’ve mentored many teachers from my role in TFA and realized that teaching math meaningfully is an exhausting, stressful experience for many and that despite their best intentions they are up against an institutional and emotional mountain that few have the energy to climb.