Math 512, Blog Post #3

I spoke a little in class about what I'd do if I had a million dollars to spend on improving a school district, but I never did post those thoughts here. Allow me to correct that.

In class, I'd said that if mathematics was never specified, I don't think I'd be able to justify spending the million dollars only on improving mathematics education. Especially in the more urban areas of New Jersey, there's just far too much disrepair to ignore the option of spending money on improving the physical structure of school buildings. When doors and windows are broken, hallways have leaks and bathroom stalls are missing doors, there are problems much larger than the state of math textbooks or a lack of SmartBoards in each room.

Of course, in newer and more affluent districts, problems like these wouldn't arise nearly as often. That's why I posited the idea of, at the upper end of the spectrum, purchasing an electronic book reader such as the Amazon Kindle for each student. In class, I gave the excuse that it eases the burden on the students' backpacks; in reality, it also eases the burden on the schools' budgets. Purchasing an electronic textbook license will cost much less on the whole than to purchase a completely new set of textbooks each time a newer edition of a math or science book is released. The electronic readers themselves can then be returned to the school after a student has completed high school and loaned anew to the incoming freshmen.

That said, I have to agree with my classmates who came up with the idea of using the money to hire additional teachers. Classroom sizes continue to spiral out of control, not only increasing the student-to-teacher ratio but also forcing teachers to work more than ever before, both in terms of class coverage and grading. With more teachers, it opens up the possibility for more prep periods to be spent on teacher collaboration rather than teacher relaxation. (Not that relaxation is bad, mind you, but it's not the entire point of having the prep.)

Math 512, Blog Post #2

So, calculators.

Let me begin by saying that I love my TI-83+ with all my heart. We've been through a lot together -- the SAT four times over, the Calc AP test, the GRE, and oh yes, my entire undergraduate career in the sciences. I credit it for keeping me sane in high school chemistry, both with the molar mass program I wrote to easily calculate the mass of any carbohydrate and with the 5-card-stud poker program I used to zone out of particular easy lectures. After 9 wonderful years, the poor thing is showing signs of being on its last legs; I am devastated in the same manner as a small child whose favorite toy has broken. So strong is my devotion to the 83+ that, when it comes time to purchase a replacement, I would eschew the [admittedly much sexier] TI-89 without a second thought if I was certain that the students I will eventually teach would still be using the now-comparably-ancient 83+.

And I don't think they should be used in a classroom.

My view is that calculators should really only be allowed in class for tasks that would waste inordinate amounts of time and paper otherwise, such as basic operations on large numbers (e.g. 674.5 times 325.2). Allowing them for use in all scenarios causes students to rely on them too heavily, to the point where "obvious" math problems such as 12/1 or 15x0 cannot be seen without the aid of a liquid crystal display. Concepts like this are central to an understanding of math, and while the calculator is just one of many tools for solving a problem, it does little to aid the understanding.

The issue that I come across is that my personal rapport with the 83+ comes from years of experience and an innate knowledge of how I need to input any particular expression. Say, for example, that you wished to find the square of -2. On the scientific model TI-30, you input the 2 first, then press the negative sign, then hit the x² key (or x^y, then 2, then enter). However, on an 83+, it's a different beast entirely: open parenthesis, negative sign, 2, close parenthesis, x² key (or ^2). Without this exact combination of keys, odds are good that you will be getting -4 as your answer.

And many students will submit -4 as their answer without a second thought, since they are certain that the calculator will always find the correct answer. It has no reason to lie. This blind trust is the great tragedy of calculator usage. The calculator will always do the math perfectly, but it requires that the user also be perfect with the input of the question. To this end, teachers must devote entire blocks of time to the instruction on how to input the expression into a calculator, rather than focusing the discussion on why the answer comes out the way it does.

I understand that calculators, like the algorithms we use to solve expressions, are merely tools that humans use in order to definitively calculate the relationships between numbers. My issue comes with the fact that the calculator is a tool which requires that the user master it first before the user can move on to mastering the skill it is supposed to help with. (The issue is compounded further if a student replaces their calculator with another model, forcing them to develop the skill set of proper calculator usage all over again.)

For understanding things like zeroes of a function, maxima and minima, and other tools which are useful when seen graphed in the context of an algebra-2-and-higher setting, I agree that calculators can be a great help in furthering knowledge. Essentially, I suppose this means that I am a fan of calculators when they're not being used as calculators, and instead as things such as graphing or trigonometric tools. Until then, I don't feel that the use of calculators in a classroom adds anything to the student's basic understanding of math. It may save the student time, but it also saves them from having to think about what is truly being done.

Math 512, Blog Post #1

My name is Mike Hergenhan, and of the three Michaels in the class, I suppose I'll specify myself as the blonde one. I finished undergrad in 2007 with a degree in chemistry. Halfway through my senior year, I realized I'd made an egregious error in switching away from math in the first place. Two years and five undergrad math courses later, here I am pursuing my grad degree in math education.

I believe that I'm the only person in the course without any classroom teaching experience, either past or present. While I do currently work in the basic skills lab at Montclair, that's really more of a tutoring gig than teaching. I do, however, have some knowledge of the technology implemented in math classrooms -- I worked for five years as an IT technician in the Linden school district. Many of the programs and websites brought up in the first class, such as Geometer Sketchpad and Study Island, are staples of Linden's curriculum.

I will admit that my opinion towards technology in the math classroom is that of reservation. My high school calculus teacher, who is my inspiration for wanting to teach, taught his lesson plan straight out of handwritten notebooks. The most technology we ever used in the class was a graphing calculator. He was universally praised by my classmates as being a wonderful teacher and presenting the material more clearly than anyone we'd had previously or we've had since. If a standard lecture methodology (when planned properly) can accomplish its task so easily, I see no need to add extraneous technologies to the mix.

And, admittedly, my opinion of technologies as "extraneous" comes from my experiences with them. From what I saw in my home district, a technology-based curriculum was rarely implemented because of the opinion that it would a better job than the lecture-based alternative. Mostly, adding technology was done for the sake of having it.

I'm hoping that this course breaks me of my aspersions towards technology in the math classroom. Seeing technology used effectively in a math curriculum could go a long way towards helping me determine how much I would use advanced technologies in an actual lesson plan.

Until then, though, I intend to continue to swear by the notebook and the occasional TI-83+.