Mathematics (and Physics) and Calculators

This is the third 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, but the comments spanned a range from that topic through mathematics and its applications beyond calculus. My first article merely laid out a common set of definitions, but does include a few assertions about various types of calculators and levels of mathematics that might deserve comment. The second article tried to focus on Developmental math but also included some comments about Algebra. In between these, I posted a shorter article that included a more polemical set of comments about the "modern" Z80-based Graphing calculators. Comments on the second article made me realize I also owe the community a long-deferred article about the math preparation of elementary ed teachers.

My second article limited itself to classes that are remedial in the sense that their goal is to get students to finally learn skills that were supposed to be taught in elementary and middle school as well as the first year or so of high school. College Algebra occupies a fuzzy territory because it is sometimes learned in high school (where it would be Algebra II) but is considered a college-level math class that is sometimes a general education requirement. I included it in my previous article because it is not the only gen-ed math option at our CC and serves many masters. In this article, I will take up the issue of most interest to me: whether students are prepared to use calculators and algebra to do physics, calculus, and (perhaps) engineering problems.

My expectations

As noted earlier, I allow my students to use a Scientific calculator and I expect them to have a decent one and be fairly fluent in its use. I do not allow them to use a Graphing calculator or one that is capable of doing computer algebra. The former is excluded because I do not have time to police all of them for cheat sheets, the latter is excluded because I want a level playing field. They can use MathCAD or Maple or Mathematica when they get into upper division classes where everyone will be using equivalent tools on any given assignment. I expect them to do algebra with pencil and paper in a freshman physics class.

The calculus teachers here have a similar expectation. Many (but not all) give exams where no calculators are allowed on part of the test, but a Graphing calculator (mainly for the numerical integration feature that is on some Scientific calculators as well) is allowed on others. Sometimes they even use a computer algebra program on an exam, but that is rare.

One thing I mentioned in a comment on Dean Dad's blog was the importance of defining outcomes. I forgot to mention that outcomes are best defined so the match the desired inputs for a subsequent class. It is for that reason that our calculus faculty require that students actually know certain derivatives cold, like times tables, and why they were stunned into disbelief when a student transferred here from a school where they used an Algebraic calculator that can do all of the basic derivatives and integrals symbolically. That outcome (being able to take a derivative with a calculator) is mismatched to the requirements of physics and engineering. (True, an engineer taking the "fundamentals" exam has a reference book handy that contains the basic derivatives, but the few minutes you are given to answer each question does not give you enough time to look up every basic result.)

Physics

In general terms, my views on calculators are similar to what Chad Orzel wrote in response to Dean Dad's blog. Real math (meaning math major math classes) have no need at all for calculators unless the topic is numerical analysis, and then you are better off with a programmable computer. Ditto for upper division physics majors classes, although they can have a computational component as well (that is, arithmetic rather than the symbolic mathematics of algebra or calculus). My impression from former students is that engineering expects correct computation as well as algebra, so exams require computation as well as the proper setup of the problem.

I should add that the exam security issues inherent in larger classes, where students are unavoidably sitting within copying range, also requires numerical variations between problems. (Exam fairness has, so far, kept me from putting totally different problems on versions used in the same class.) Most on-line homework systems also do this, although some have symbolic variations as well as numerical ones. This leads to an emphasis on problems with numerical values.

Further, because my students tell me what they do in their first engineering classes, I know computation is only part of it. Setting up the problem algebraically and simplifying before computing is ALSO part of it. For this reason, I require them to state the problem symbolically before plugging in the numbers. However, primarily because of their comfort level, I do not take off if they do the algebra with numbers present rather than keep the symbols until the end. (Having numbers and unknowns makes it easier for most of them to keep track of what is unknown and needs to be isolated or eliminated.) I'll let someone else break them of that habit later on, but I will encourage them to work on it in my class. That said, I do sometimes give exam problems where a symbol like L has to be in the final answer. See below.

Computing

What has surprised me is the degree to which students either cannot compute efficiently or use their calculators inappropriately when solving a problem.

The first problem has only become evident to me recently. I don't think it is a new development; I just happened to see a particularly egregious case last year where the student would evaluate something like A*B*C/D by doing A*B, write down the answer, enter the answer*C, write down that answer, then enter that answer/D. Painful. And slow. And prone to error. I should have suspected this sort of problem because the other version, entering ((A*B)*C)/(D), is a bit of craziness not uncommon in Algebra classes. They don't know order of operations and, even if they do, some have used bad calculators that violate those rules and been burned.

This is, however, a real handicap. They need to use one calculator type and use it enough to understand what it does under different circumstances, but might never have been taught that it is OK (and even necessary) to hit lots of buttons and see what they do under different circumstances. I'm going to mention that this year, going beyond such simple things as whether your calculator does -3^2 correctly or whether it knows automatically that the arcsin of 2 (or the ln of -1) is imaginary.

The second problem is doing algebra with long messy numbers in the equations. This came up in an earlier blog post about algebra, with some nice observations in the comments. This summer I've been thinking about where this comes from, and I am convinced it is because they never use realistic numbers in Algebra classes. Their equations all have numerical coefficients that are small whole numbers, not the 10 digit value for the y component of the velocity, v*sin(theta). There is no penalty for using 3 as a coefficient. There is a penalty for using 34.5619288 as a coefficient. They also seem to have not been exposed much to subscripts, so they are initially quite uncomfortable using Vx as a symbolic replacement for that nasty number.

My preferred solution would be to have pre-calc and trig classes use symbols with subscripts so they get comfortable with that math skill, just as I would like them to work with functions like g(y) or x(t) or even x(y). As we talk more about outcomes at my college, I have to see where those skills fit into the goals of our math curriculum. It might be that these are one-and-done skills (like some skills in physics) because instructors at one level don't know how important it is when you do kinematics in physics or power series in calculus and how much students struggle with those concepts. However, I also know that this is overly optimistic. Instead, I am thinking about ways to work those in from the beginning in my class, perhaps by starting with y(t) motion rather than x(t) motion and using v_y and a_y even when they aren't really required at that point.

Finally, there is the way I model doing problems in class. Comment number 4 on Chad's article mentioned math exams where you could only use a calculator on part two, something some of our math teachers do, but then came up with a nice insight:

it also could be used to introduce the concept of only taking out your calculator when you reach the stage where you've gotten the problem to its simplest state, and need only put in the numbers.

I've seen students do exactly that while taking an exam, just as I do, but I've never thought about really making a SHOW of pulling out the calculator at that point of the problem. I need to model that step as clearly and explicitly as I model algebraic steps when solving a problem. I also need to find or invent more problems where a symbol is in the final answer, like it would be if you were writing a program where a few values are fed in by the user but others are fixed by material properties or whatever.