I never taught mathematics beyond Geometry at the high-school level, even though in theory, I am certified to teach all mathematical subjects through Calculus for K-12. I loved teaching middle school concepts and then Algebra and Geometry when I went into the high schools. What tends to happen, as most teachers can attest, is that the subjects you begin teaching when you first start out tend to be the ones you stick with throughout your career. Especially if you are good at it, which, I believe, I was very good at teaching Algebra and Geometry, especially to struggling math students. This is because I focused so much on conceptual understanding and hands-on learning, technology integration (Sketchpad & calculators) and diversifying instruction. Often times, the decision on what subjects you teach is not actually up to the teacher -it is an administrative decision based on number of students, subject needs, and other factors. But – teacher subject assignment becomes ‘tradition’ – in my 15 years in the classroom, which included 4 different school districts, I can honestly say that most teachers (more true in High school) taught the same math subjects year after year.
This consistency in what teachers teach may be good, may be bad, but it is the reality. I can only speak for myself, but I do know as a result of my own personal ‘teacher tracking’, I am less confident in my abilities with higher-level mathematical concepts – i.e. Algebra 2, calculus. Not because I never learned it, but mostly because I never taught it, and as years go by without interacting with these mathematical concepts, the less you remember and the less you can do. Math is a language, and if you don’t use it, you lose it. Could I teach it? Sure. But not until I thoroughly reviewed and relearned it all. All I can say is thank goodness for Khan Academy when my daughters were going through Calculus!
One of the things I am doing as part of my work with Casio is learning more about what the calculators can do to support student learning, and as a result, I am relearning some math concepts I had forgotten and/or not seen in a long time. And when I say long time, I mean not since my college days, so we are talking a good 25+ years ago (yikes!). In prepping for NCSM/NCTM in Oakland and San Francisco in the next couple of weeks, I have learned some cool things that the Casio Prizm can do which are easy for students to do, visually appealing, and dynamic. Something that engages students in complex math quickly and allows them to explore and investigate their conjectures and focus on problem solving, not just calculations. One of these is finding the area between two curves, which, I will admit, I definitely did not remember and still am not sure about, but it is a concept very much explored in calculus. I do remember lots of calculating – by hand – which can become quite cumbersome. While it is important to understand the process and where these calculations come from and why they are necessary, having a visual of what area between curves means, looks like, and being able to show the constraints, etc. provides a powerful tool for helping students grasp and understand all the mathematics going on. With that in mind, I wanted to share how to find the area between two curves using the Casio Prizm (mainly because it looks pretty with color, but you can do this on any of the Casio graphing calculators) and compare it to the same process on the TI-84+ CE, again, because color looks prettier, but you would follow same steps on TI-84+.
(You might ask why I always do a comparison – the reason is TI seems to still be the prevalent calculator in schools, more out of habit and familiarity than anything else. I like to show – when possible – the difference, because students should be using technology to support their learning and using technology that is going to make that process more efficient and understandable. We want them to get to the deeper meanings and connections and not get bogged down in trying to remember steps/keystrokes – having used both, I just find Casio more user friendly, not to mention cheaper. If no one knows there are better options, then tradition and familiarity win out even if it’s a disservice).