Class discussion of Ball’s ‘The Subject Matter Preparation of Teachers’
One of the issues discussed in class was the ability to field students’ questions. Concern was raised about maintaining students trust if you can’t answer their questions. I don’t believe you need to or should be able to answer every question. Many of the best discussions and investigations in my classes have started with a question I could not readily answer. Students do not need a teacher to be omniscient. In fact, I’ve found students trust me more when they know I won’t tell them something I’m not entirely sure is accurate. They appreciate my saying “I really don’t know; let’s see if we can figure it out.”
One of the major issues I face in my math classes is getting students to let go of algorithms and tricks they have learned in previous classes. Ball states that “critical knowledge about mathematics also includes relationships within and outside the field - understanding the relationship among mathematical ideas and topics and knowing about the relationship mathematics and other fields.” Many teachers have a good understanding about the standards and concepts in their particular math subjects and teach the tricks necessary to solve problems in those areas. However, due to a weakness in overall or higher level math, they do not see where their subject fits within mathematics as a field. Many do not see that by teaching students one algorithm, they limit conceptual understanding and hinder learning extensions of their subject taught in later years. One example is teaching students to multiply binomials using FOIL without ensuring understanding of the concept. Students will then have difficulties factoring and multiplying polynomials with more than two terms. Another example is teaching and allowing students to change every fraction to its decimal equivalent. Students do not understand fractions conceptually. They are then unable to manipulate fractions which will affect their Geometry in proportions, Algebra 2 in rational functions, PreCalc and Calculus in limits and differentiation.
Outside experience and subject matter knowledge
Unlike many in the class, I have little recollection of my high school math classes. I remember my Geometry teacher was the basketball coach, and that I sat behind the varsity first baseman in Elementary Functions. I was a fan. (Of the first baseman. Not the Elementary Functions.) What I do remember is the feeling that girls weren’t of any value in the upper level math classes. I was one of very few girls in Elementary Functions and PreCalculus. All the math teachers at my high school were male. I remember a relaxed atmosphere in classes, and the easy rapport the teacher had with the male students. I also remember the teacher not knowing my name at open house in May, after the entire year in his class. This feeling was exacerbated in college. I was in the third coed class after my private catholic school began accepting female students. Some of the professors were still obviously resisting the change. My Differential Equations professor was a Benedictine monk who refused to acknowledge any of the female students. When test score distributions were shared, our scores were not even considered. I responded by making sure I destroyed his curve on every exam, but most female students dropped the major. So while Ball states “teachers influence students through their own engagement in ideas and processes”, their influence on me was less subject specific. I left high school and college with a drive to prove I was as good as anyone else in my field.
So almost all of my conscious acquisition of subject matter knowledge has come from actual teaching, professional development and non-school experiences. I am extremely lucky to work in a department of well educated, experienced professionals. We have daily lunch room discussions about the best ways to teach particular concepts as well as ‘spirited’ conceptual debates about mathematical concepts. Our last argument/discussion the last week of classes was about the derivation of the equation of an ellipse.
My two other outside school experiences that have contributed to my subject matter knowledge are my years as a software engineer and my lifetime of figure skating. I have drawn several lessons directly from programming - sorting exercises to order/understand rational and irrational values, cryptography to learn functions and inverses, binary/hexadecimal systems practice functions and number sense. Students are more engaged when I can get out of the textbook and answer their incessant “When are we ever going to use this?” with a concrete, real example. I can also honestly tell my students that I use math while teaching skating. (I don’t usually tell them about my skaters’ reactions - rolled eyes, groans, “school’s out” exclamations - to my using vectors to describe how to correctly perform a turn or jump.)
Line of Inquiry
I have narrowed my line of inquiry down to two general possibilities.
I am interested in how to get students to problem solve and actually think for themselves. Many students do not begin unless they know how to solve a problem completely. If they are not sure of the answer by looking at the problem, they do not even try. If there are more than four words in the directions, they skip the problem, and they do not even read the word problems.
My second possibility is how to prepare a math club. My math club has attended several competitions over the last 4 or 5 years. My bright, high achieving seniors have been completely demoralized by 8th grade students from Silicon Valley schools. I want to know how to bring math club, and by extension all students, up to a level that will be competitive.
I am behind in research, but the skating competition ends tomorrow, so I plan to be back on track after the weekend.
