Calculators and Basic Math

This is the second of three articles concerning calculators and mathematics triggered by a 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. I have already commented on the blogspot version of this blog (more than once), which has collected a huge number of comments, but there are also a large number of comments on the IHE version of the same posting. I'll write as if you have at least read Dean Dad's article. The discussion has been quite wide ranging, often not bothering to make a distinction between the various levels of "calculator" available to students or the many levels of math classes they might be used in. (Definitions are given here to provide a common reference.)

In this article, I will take up the specific issue raised in Dean Dad's blog and the one I know the least about as an instructor - calculators and developmental mathematics - but also look at college algebra. I'm mainly interested in putting some of my thoughts on paper and seeing feedback I get from others about their opinions of the problem.

I'll start with a particularly telling comment from Dean Dad:

...part of me wonders if we’re sacrificing too much on the altar of pencil and paper. It’s great to be able to do addition in your head and long division on paper -- yes, I know, I’m old -- but is it worth flunking out huge cohorts of students because their high schools let them use calculators and we don’t?

This isn't about LETTING them use calculators. Part of it is about spending time teaching them how to use a specific brand of calculator rather than how to do algebra. However, as I wrote in my first comment on DD's blog (9:11 AM time stamp about halfway into the 50 or so comments that are there now), I think this is mostly correlation without causation. The real problem lies elsewhere.

I think the problem starts in K-5 and gets compounded by pushing kids along into the next class and lying about the content level of that class. That is, I don't believe for a minute that a student I advised had passed a REAL pre-calc class in HS just a week before I talked to her. DD writes about similar cases:

...students who have passed algebra and even pre-calc in high school frequently crash and burn when they hit our developmental math, because the high schools let them use calculators and we don’t.

I don't buy it, and here is why: Our placement test will put them into Intermediate even if they can't do arithmetic, provided their algebra score is high enough. And you can't work with logarithms and exponentials or trig identities (a given if it is really pre-calc) if you can't solve a simple linear equation written symbolically like I asked the student to do.

You might forget 6 months of math in a week, the newest stuff, but not 3 years of it. And if you do forget that much that fast, you should have failed that pre-calc class. You can't get to much new material if you spend most of the year re-teaching three years of previously-taught material de novo.

K-5 Curriculum:

I don't want to belabor this, but one reason they can't do arithmetic might be that they never learned it. I'm convinced this is the biggest problem we face because it also lies behind the existence of pre-calc classes that are really teaching basic algebra. I'm sure part of it is that teachers who never understood math and hate it with a passion are teaching it by-the-book following a curriculum none of you could possibly imagine anyone would use.

My analogy is to the "look say" approach to reading, where guessing replaced phonetic decoding of words and Johnny (not to mention a cousin with a high IQ who is now an senior engineer at the VP level) couldn't read. Using guessing to construct your own mathematics might work with someone like me (I feel eternal guilt for, AFAICT, being an unwitting subject in a math ed research project that was run before the days of IRB and informed consent where they deduced that this curriculum worked really well), but it is unlikely to work with someone who was not going to get a PhD in physics. Really good algorithms were developed 12 centuries ago and survived for a reason. As is illustrated here, for the Everyday Math curriculum, the most efficient methods are not taught first or (in some cases) are not taught at all in some schools. The starting point for Dean Dad might be to get out into the feeder systems for his CC and find what they are doing in fourth and fifth grade.

If I know anything about learning, it is that students always favor the first method they get taught. (That is one reason you have to really emphasize when conservation of energy or momentum should be used instead of Newton's Laws: they learned F=ma first so it is the first thing they want to try. I'm the same way.) That means it is a really bad idea to start with an inefficient method, but which some people find useful when doing 'mental math', and teach the more efficient one last.

Now think about how to teach synthetic division or multiplication of polynomials to someone who only knows partial quotients division or the lattice method for multiplication. Not pretty. Then consider that they might never have even heard about "invert and multiply". Not pretty at all.

Outcomes:

The answer to Dean Dad's fundamental question, whether students should be allowed to use calculators in an Arithmetic class, starts at the beginning - the first step of course design. What are the desired outcomes for this math course? If the outcome is to be able to do a certain amount of arithmetic with pencil and paper (not in their heads), then the only use of a calculator is to check your own work as you make up your own problem and solve it. Ditto if the purpose is to simplify fractions involving simple whole numbers as preparation for a similar skill with symbols. You need to change the outcomes before changing what you do in the class.

Personally, I'd be happy if they taught them how to do arithmetic on their Basic calculator. Seriously! My biggest complaint when teaching physics isn't that they can't do arithmetic (they can't), it is that they can't calculate worth a damn. Digital natives my ass. I was 22 when I got my first calculator and I am faster than most of them are, and I'm slower than I used to be. (Sure, I've been using one longer than they have been alive, but that only proves they haven't used the thing enough to be competent with it.) I mean, I've watched a student work out a product by multiplying two numbers, writing it down times all of the others, entering it again !!! and multiplying it by the next, etc etc.

I know they use their calculators a lot in our Algebra (meaning College Algebra) classes, but it must all be with simple whole numbers like we used back when there were no calculators. That is the only possible explanation for their struggles with 3 and 4 digit decimal or scientific notation numbers or their mysterious belief in rounding intermediate answers.

Algebra:

There really shouldn't be many numbers in an algebra class, IMHO. Somehow the appearance of Graphing calculators changed the curriculum to emphasize numbers and, curiously, de-emphasize graphing. Since you can't actually read a graph on a TI display screen, let alone interpolate on it using a ruler, they don't appear to know how to make or read an actual graph rather than a cartoon of a graph. This is a nightmare in the physics lab, but also in the classroom when data are supplied in a graphical representation.

I don't believe anyone has tried teaching algebra with a Scientific calculator and graph paper in decades. I doubt if anyone other than the textbook and calculator companies have studied it, and studies like that are notorious for the difficulty in controlling the student mix or the instructor effect. However, stories about students who finally got algebra in a class where only symbols were used - no calculators needed - are common enough to make one wonder how it would work. The studies (see some comments toward the end) seem to lack a smoking gun in favor of the primitive Graphing calculators used today.

There is another side effect. Since they don't know how to use their calculators, particularly concerning order of operations, they use parentheses like they were the only operator known to man. ((3)(2))/((5)). One result is they don't see the key role of the parenthesis to denote "function of". I've seen calculus students who think x(t) means x*t, although this could be partly due to the fact that x is never a function in calculator-based Algebra. I have to wonder out loud if this would improve if they all used HP calculators instead of TI calculators. Also see my next comment below.

Commenting on the comments:

Some Anonymous, writing at 5:56 PM, about 3/4 of the way into the comments, writes:

A kid that is getting good marks in algebra screws up their physics equations. i check with their math teacher, and they don't make those mistakes in math class. So I test them myself, and they can manage algebra just fine when x, y, and z are variables and a, b, c, and d are constants. Anything else and they're lost.

This could be a result of using graphing calculators. The TI-83 will only plot Y(X) unless it is in one of the other modes (where it is similarly limited). Parametric mode, the only place where you can do X(T), does not appear to be used at all until they get to Calc III. This really bugs a chemistry colleague, because they are always plotting the log of this versus the sqrt of that, neither of which is X or Y. Similarly, we start out in physics by plotting x on the Y axis and t on the X axis and it blows their minds. I'm tempted to start by doing only y(t) problems at the start of fall, then moving to x(t).

Mthgeek, aka timfc, writing at 7:45AM of the second day of comments, listed several references. The first of these was

The Arithmetic Gap
Educational Leadership, v61 n5 p55 Feb 2004
Summary: The students using calculators in school classrooms result in lower math scores than students who never use them.

I'd like to know what grade level this was, but it sounds like K-8 from the title. As for the second one that was listed, I don't ever pay attention to something like a meta analysis of 42 other papers that span middle school through calculus. Apples, oranges, confounding variables, design differences, and systematic errors make a tasty goulash but don't help with teaching Basic algebra. One other reference, discussing "computer assisted instruction" would appear to be irrelevant to this discussion. You can use computers as an instructional aid (instant HW feedback, for example) without using a graphing calculator - or any calculator at all.

However, the last reference from Mthgeek is rather interesting. It is a link to an article (Refocusing Introductory College Mathematics Courses) that has a link to a new textbook (Contemporary College Algebra: Data, Functions, Modeling) that implements some of the ideas from the study. That is, the study and the textbook are closely coupled, but I recognize some things in that report that are reflected in what we do in our Intermediate class with a different book. (I'll have to ask around, but we might have made this choice because Intermediate is a pseudo-terminal course for many majors in our curriculum. The situation discussed in the article does not apply as much to our college Algebra course, because it normally leads to business calculus or trig. The statement in that report that biological sciences don't go beyond Algebra is patently false in our curriculum. They have to take Calculus even if they don't ever use it.) That said, I strongly criticize the textbook author for conflating a graph on a Graphing calculator with a graph produced on a computer. There is no comparison in detail or quality.

Some Anonymous, writing at 11:21AM on the first day, said (in part):

1) middle school math is more focused on algebra as early as 7th grade. So students don't have enough mastery of fractions, percents etc
...
3) More students attending college- so the lack of good high school prep is more evident.
4) Content of dev math courses in college are aimed at preparing students for a precalc/calculus track. But those going into sociology or psychology ...

Developmental math at my CC is about preparing students for 9th grade math, not pre-calc. The Intermediate course barely prepares them for real Algebra, and certainly not for the calculus track. (Our failure rate is spectacular at every one of those steps.) Besides. students going into Psychology need a real, college-level statistics class that has college Algebra as a pre-req. Criminal Justice, on the other hand, has no real math requirements and our statistics show that the combination of our Developmental and Intermediate classes does a GREAT job of preparing them to pass the basic financial math class that constitutes their "college level" math requirement while teaching them about compound interest.

The fact that they have not learned arithmetic or fractions by the time they get to 7th grade (which is when we started Basic algebra when I was growing up) is the real problem. Three years should be enough if the curriculum and teachers were any good, but if they aren't or the kids don't learn it in 3 years, our schools track those kids away from Basic algebra for another year, or more. But this does help strengthen my point that the problem is really in the K-5 classroom.

I don't buy the "more students" argument because the fraction going to college has not changed that much in the last few decades.

Finally ...

If you have read this far, thank you. I want to close by saying that the problem really is deeply rooted in our educational system and very frustrating for all involved. The high failure rate in Developmental classes is a major problem that no one is ignoring at our CC.

However, many students fail because they never attend class, or don't attend frequently enough to engage with the instructor. With any instructor, no matter how talented ze might be. I've written about that in an old bit of wishful thinking about new student orientation. Coming straight out of HS, they believe they were taught pre-calculus or Algebra II, so they just don't believe they need to go to class and actually learn math. Older students, out of school for years or decades, know they don't remember anything from school so they take it seriously and often do quite well. An age-based breakdown of performance in Developmental classes might be worth looking at, Dean Dad.

Or Dean Dad might only need to walk by a classroom or three on a regular basis and take a sort of visual attendance. Is the room still full after 4 weeks? Maybe that, rather than calculators, is the real problem.