I told my adviser that I wanted to write a book about learning to teach mathematics meaningfully. He said that sounded great but not to start until after I was done with my dissertation. So my dissertation can’t possibly be about my life-plus somebody already wrote a dissertation about my life as a teacher! So, then, what should it be?

It should be connected to the research I’m doing with my adviser. We are designing a test of Mathematical Meanings for Teaching for Secondary Mathematics. We ask high school math teachers questions about mathematics as well as typical student understandings of mathematics. However our questions don’t look like textbook problems and they are designed to be impossible to answer well without a meaningful understanding of the mathematics in the item. However, everyone, even if they have an unproductive meaning for an idea such as slope, should be able to respond to the question in a way that reveals their thinking to us.

I don’t want to write a dissertation detailing how horrific the results of our assessment are though I’m certain I could. We have tested a few teachers and the people who we share the results with are horrified. My adviser has been carefully building models of student and teacher thinking for 30 years and he knows how to write mathematically straightforward questions about middle school math that show high school math teachers are confused about function notation, rate of change, graphs, and proportions. Teachers have learned tricks to answer the most common questions-and as many math students know, if you memorize where to put numbers you can pass a class without knowing what you are doing.

I don’t want to publish articles warning that the Common Core Standards in mathematics are never going to be taught as intended in many classrooms because the math teachers with math degrees don’t have strong understanding of the content in the standards. I fully expect that they will continue to teach basically the same way as they have in the past and continue to choose similar questions for their students as they did in the past. These are the questions they know how to get the right answers too. These are the questions that students can get the right answers too without doing more than learning where to put the symbols.

Below I give an example of what I mean when I say people are learning what to do with symbols but not the point of it. I talked to a bright student last weekend who had passed College Calculus. She described the integral sign as a squiggle and that the point of class was learning what to do with the numbers in front of the squiggle. I asked her “do you have any idea what the squiggle means?” She said, “No, not at all.” We were out hiking and I gave her a brief lesson about figuring out how far we’d hiked if we only knew how fast we were walking on different intervals of time and it made sense to her. She had no idea that Calculus had anything to do with accumulation and rate of change.

I just refuse to make my dissertation such a negative description of other people’s understanding of mathematics, though I’m sure this will be there if I’m intellectually honest about it. I want to provide some hope of change. Even if it is only one story.

I worked with a pre-service math teacher last semester in a teaching experiment that was designed to promote strong understandings of fractions. Despite having good grades through Calculus, she had some unproductive and sometimes incorrect meanings for fractions and division. Video-taping myself trying to teach her and analyzing the tapes was a career-altering experience that I would recommend to anyone learning to teach. We had the hardest time communicating about the meaning of the word partition in the context of division. She kept interpreting my words into her understandings of fraction and division she developed long ago. I was absolutely aware that she had different meanings for the word partition than I did, and despite multiple direct efforts I struggled to change them. In the end, I think I did make progress, but we fell short of our original goals of developing a strong multiplicative fraction scheme.

Still, there was hope in this story. The student I worked with was happy-she liked thinking about the meaning of fractions and she was able to recognize her own lack of understanding as I posed questions to her about division and fractions. In a one-on-one environment I was able to manage the emotions she experienced as she realized that her elementary and secondary teachers had set her up to be confused in college. As a result of this process she became aware of how many things she didn’t understand and what it meant to understand division and fractions quantitatively. Making people aware of their misunderstandings about basic topics while simultaneously making them believe they can improve is no easy feet.

Believing adults can change understandings they have held for years, is especially challenging because I’m so honest, and as I watched the videos of me teaching it was obvious to me how hard it was to teach adults about something they think they know. Every time I said “fraction” and tried to explain, my student just seemed to imagine whatever she already thought a fraction was and molded my words to fit her image. Another teacher insisted that a fraction must be always smaller than one because how could a whole be less than a part. I asked her about 17/4 miles per hour and she said “well, that is a rate, not a fraction, so my students should know this is different.” I need to believe that adults really can change deeply held mathematical understandings so that I can provide hope to the people I work with. Right now, I’m coding tests that reveal just how sorry the state of teachers thinking can be, and I need a boost of optimism.

Right now, I’m most interested in developing suggestions for what mathematicians and math educators should do when they give their teachers our assessment and realize the depth of their teachers’ confusion. How can we just hand them a test that reveals that their students getting master’s degrees in mathematics teaching don’t know what function notation means and not give any suggestions for improving their classes?

Could I design instruction that would allow teachers to make progress on some of the items on our assessment? I’m taking Geophysics right now, to try to grasp how scientists use concepts on our test-in particular how they think of rates of change. I want to find some sort of situation that will help teachers understand why thinking about rates of change they ways I suggest is actually useful.

Oh my goodness…. this is SO HARD to figure out!