Thursday, July 29, 2010

Calculators - Background Info

This is the first of three articles concerning calculators and mathematics triggered by a blogspot and IHE blog article by Dean Dad, a community college dean who appears to be writing from another part of the country yet has the same problems we have at our CC. The original article concerned calculator use in "developmental" math classes that typically cover fractions and 7th grade algebra. I have already commented on the blogspot version of this blog (more than once) and the two together have generated more than 80 comments. I added some more in my second article of this series.

I won't actually comment on this topic here. My purpose is solely to set the terms of the debate, as it were, because the wide-ranging discussions of this topic by Dean Dad and others are seldom clear about which of the four or more levels of "calculator" available to students are being discussed and/or which of the three or more levels of math classes (plus physics and chemistry) provides the context for the discussion.

The divisions I make are somewhat arbitrary and perhaps idiosyncratic, so I want to spell them out somewhere without cluttering up a discussion of the teaching and learning issues as I see them. That way I can link here for future discussions of this topic and not have to repeat myself.

Although I think three levels of "calculator" suffice for most classroom use, and hence for later discussion, I think I need to list at least five to make the definitions as sharp as possible.

  • Basic - Here I have in mind a wide range of very cheap calculators that can do arithmetic, including parentheses and scientific notation, but cannot deal with trig functions.

  • SCIENTIFIC - These calculators can evaluate all of the basic functions (trig, hyperbolic, log, exponential, power) but cannot store text or programs. Some can work with complex numbers and/or hexadecimal numbers. At the high end, some can numerically evaluate definite integrals or derivatives or solve simple equations, but they cannot show any intermediate algebraic steps or work purely with symbols.

  • GRAPHING - Here I have in mind several calculators that are similar in capability to the TI-83Plus. They can do all of the calculations of a top end "Scientific" calculator, but can also make graphs and store programs (including large amounts of text that can serve as a sophisticated crib sheet). They can store text, but cannot work with symbols. Functions are limited to y(x) except in the rarely-used parametric or polar modes.

  • ALGEBRAIC - These calculators can solve equations written symbolically and can, in some cases, even show step-by-step the algebra or calculus used in the solution. They are typically somewhat limited in how much calculus they can do symbolically, but they make it unnecessary to learn any of the derivatives typically encountered in calculus.

  • Computer Algebra - Here I have in mind small computers that can run computer algebra programs like Maple, Mathematica, MathCAD, etc. Now you might say "a laptop is not a calculator", but there is actually a rather modest size difference between a notebook-sized laptop and the top end TI "calculator" that comes with a full keyboard and a wide screen. Besides, these are widely used in classes at the Junior level and above so they help frame the discussion.

The three in the middle, in all caps, are the ones I will refer to most often within the context of lower division classes taught at a community college.

For the record, I allow Scientific calculators in my introductory physics classes but do not allow formula sheets or cell phones or Graphing calculators to be used on exams. I encourage students to get one of the high-end Scientific calculators that can be used throughout their engineering career, including on licensing exams, so they become fluent in its use.

The four levels of mathematics classes are defined as follows:
  • Developmental - The content here ranges from arithmetic and fractions (what I characterize as 4th and 5th grade math) to basic algebra (the first class where "x" is used, taught in 7th grade when I was in school). These do not carry college credit. A well-calibrated placement test determines where a student starts, and some have an exit exam to verify competency at a certain level.

  • Intermediate - The content here is algebra through what I knew as the 9th grade level (the quadratic formula, for example, but no logarithms). This might earn college credit at a community college, but not at a university. It is not considered to be at the college level. A well-calibrated placement test is used to place students in or through this level of math.

  • College Algebra and Trig - I group all of the pre-calculus "college level" courses here but exclude other "college level" classes that exist mainly to ensure that liberal arts majors can graduate even if they can't do college algebra. (Those other classes usually cover enough about exponential behavior to understand compound interest on credit cards and enough probability so you should know better than to play the lottery, both very valuable life skills!) At our college, College Algebra serves many masters so skills not needed for the pre-business curriculum are put in an "advanced" college algebra class (pre-calc) and a trig class. (I know that some colleges, like my high school and undergrad university, combine these into a single course but I will use our curriculum as my reference point.) A different, also well calibrated, test is used to place students above this level although most students take the class.

  • Calculus - Although my students will usually take everything through differential equations and linear algebra, I'm mainly thinking about first semester calculus because that is where the bulk of students fail.

The distinction between Developmental and Intermediate might seem unnecessary to some readers, because both levels are usually non-credit classes at a university. Indeed, some universities define college algebra as a remedial course. I make the distinction because our math department teaches classes at the Intermediate level and above, while the Developmental classes are taught by a separate department that specializes in teaching those skills. I know that smaller colleges do not make this distinction, but we are not a small college. (We have more t-t faculty in our Developmental math department than a private school like Union College has in its regular Math department.)

If I just say "Algebra", I mean College Algebra. I will say "Basic Algebra" or "Arithmetic" when I am talking about Developmental skills classes.

For the record, our Developmental classes use a Basic calculator for some things but some exams must be taken without any calculator. (The placement test and exit exam do not allow use of a calculator.) I believe they allow the use of any calculator up through a Graphing calculator when they allow a Basic one, but that might depend on the instructor. Our Intermediate classes all use calculators. Our Algebra classes require a specific Graphing calculator that is also required for statistics. Our calculus classes are a bit less picky about which Graphing calculator students can use, but ban Algebraic calculators and computers except in some special situations.

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