The current (May 2008) NOTICES OF THE AMS contains the following opinion piece from Virginia Tech mathematician, Frank Quinn. It bears noting that the mathematics education folks at his university are members of the math department, which must make for some fun faculty meetings.
K–12 Calculator Woes
In the third grade my daughter complained that she wasn’t
learning to read. She switched schools, was classified as
Learning Disabled, and with special instruction quickly
caught up. The problem was that her first teacher used
a visual word recognition approach to reading, but my
daughter has a strong verbal orientation. The method
did not connect with her strongest learning channel and
her visual channel could not compensate. The LD teacher
recognized this and changed to a phonics approach.
My daughter was not alone. So many children had
trouble that verbal methods are now widely used and
companies make money offering phonics instruction to
students in visual programs.
It appears that Professor Quinn has been perusing the Mathematically Correct play book carefully. The attack on so-called "fuzzy" math grew at least in part out of the attack on whole-language. I earned certification as a secondary English teacher in the early 1970s, and taught English at the University of Florida for 3 1/2 years while doing graduate work in the mid-'70s, but it would never occur to me to claim to be an expert on early literacy instruction. Yet folks with mathematics, engineering, and science backgrounds, none of whom are vaguely involved with teaching reading or writing to kids, have emerged as self-proclaimed experts on the best way (and of course, there's only ONE such way) in K-5. And that way just happens to be - (Drum roll, please!) - phonics-only instruction.
Oddly, every K-5 teacher and literacy education professor I've spoken with in the past two decades who is a fan of whole-language states unequivocally that phonics is part of what they teach and/or advocate. It's just not the entirety of that instruction. Does this sound at all familiar? Like the debate about "fuzzy" math supposedly being devoid of facts, not caring about right answers, etc.? If so, don't be surprised. Ken Goodman, one of the pioneers of whole language instruction, identified many of the foundations and think-tanks and the experts they fund that are opposed to whole language and who promote phonics-only literacy teaching. I was not completely shocked to realize that these were often the same groups and individuals who were attacking progressive reform in mathematics education. And the tactics and rhetoric employed in the Math Wars had long ago been developed in the Reading Wars. So it is no coincidence, or at least not much of one, that Professor Quinn opens a piece about calculators with an anecdote about the alleged horrors of and fallout from whole language teaching.
The concern here is with serious learning deficits associated
with calculator use in K–12 math. Calculators may
not be making contact with important learning channels.
Are they the latest analog of visual reading?
See? It's SCIENCE!!! Guilt by association. Tinker (with every progressive effort) to (Bill) Evers to (Leave Nothing To) Chance!
For brevity, connections are presented as “deductions”
(this about calculators causes that in learning). However
the deficits described are direct observations from many
hundreds of hours of one-on-one work with students in
elementary university courses.(1) The connections are after-the-
fact speculations. If the explanations are off-base, the
problems remain and need some other explanation.
Stop right there, please, Professor Quinn. What you're saying seems to be that you've worked a lot with students in lower-division courses (math, I presume) and found many of them to be wanting? Such courses are typically calculus and below, and depending upon the college/university in question may include precalculus, college algebra (high school algebra 2, for the most part, though the level of the class might start slightly lower and still result in college credit), or even lower-level classes that carry no college credit (at the community college level, such courses are, of course, a major part of what mathematics departments teach). So it's not exactly shocking that a lot of the students are not all where we might hope them to be mathematically. Indeed, some are no doubt deeply deficient. And of course that is a matter for concern.
But what, exactly, is new about this situation? On what basis other than ideology do you imply that you are seeing something that is news and that can be attributed directly or significantly to calculator use in K-12 mathematics teaching? Reading the fine print above, it seems even YOU realize you don't have anything that you would accept as proof if someone were to assert just the opposite: that calculators (or other calculation and number-crunching tools, like computers), were improving the quality of mathematics students emerging from our high schools and entering our post-secondary institutions. Somehow, I don't think you'd buy for one second "personal observations" and "deductions" of that sort without a great deal of supporting data with careful statistical and methodological analysis and detail. I give you credit for the honest admission above, even if you rather quickly gloss over it and don't hint at any alternative explanations to what you think you've seen.
Disconnect from mathematical structure. Calculators
lead students to think in terms of algorithms rather
than expressions. Adding a bunch of numbers is “enter
12, press +, enter 24, press +,…”, and they do not see
this either figuratively or literally as a single expression
“12+24+…”. Algorithms are less flexible than expressions:
harder to manipulate, generalize, or abstract; and
structural commonalities are hidden by implementation
differences.(2) The algorithmic mindset has to be overcome
before students can progress much beyond primitive numerical
Intriguing, in that one of the oft-repeated complaints from the folks on the anti-progressive side of the Math Wars is that "fuzzy" math doesn't teach the current "standard" algorithms of arithmetic, or presents or encourages kids to develop their own alsternative algorithms. Now we hear from Professor Quinn that the very mindset of mathematics as calculation comes NOT from teaching kids to think that way (which has long been the contention of progressive reformers) but rather from letting them use calculators. Who knew? This will no doubt shock the heck out of the leading spokespeople for Mathematically Correct and NYC-HOLD, should they actually notice what you've said. (Of course, they'll also be able to spin it to mean something else. If by some miracle they cannot, you've seriously risked being drummed out of the club!)
Disconnect from visual and symbolic thinking.
Calculator keystroke sequences are strongly kinetic. But
this sort of kinetic learning is disconnected from other
channels: touch typists, for instance, often have trouble
visually locating keys. Many students can do impressive
multi-step numerical calculations but are unable to either
write or verbally describe the expressions they are evaluating.
Their expertise is not transferred to domains where
it can be generalized.
That's a really interesting assertion, and if it is supported by research data, I'd be truly fascinated to look at it. Absent such studies, however, you appear to be indulging in some convenient speculation that ALMOST sounds like it's grounded in the work of the multiple-intelligences and differentiated instruction folks that is so thoroughly dismissed by those who hate progressive reform in K12.
I also like the nifty use of the word "transferred to domains where it can be generalized." That sounds really scientific, too. Except that it begs a lot of questions. Does WRITING mathematical expressions and equations with pencil-and-paper that one doesn't understand for purposes of calculations using algorithms that one doesn't understand either transfer in the way you mention above? Indeed, the whole issue of transference of learning is a thorny one with a history that suggests that its VERY difficult to pin down. Not all that long ago, I blogged about a study that purported to show that kids couldn't transfer from using concrete objects to model ideas, with the claim being that this study called the use of "manipulatives" in math instruction into question. The problem was that the study seemed rather rigged to produce the desired conclusions, and the tasks appeared to have little, if anything to do with mathematics or justify the desired beliefs.
Even among high achievers calculators leave an imprint
in things like parenthesis errors. The expression for
an average such as (a + b + c )/3 requires parentheses.
The keystroke sequence does not: the sum is encapsulated
by being evaluated before the division is done.
Traditional programs also encourage parenthesis problems(3),
but they seem more common among calculator-oriented
I must be more English-language impaired than I thought. In my experience with calculators, (a + b + c)/3 gives the average of three numbers when entered into a calculator and evaluated; a + b + c/3 gives the sum of a, b and one third of c. The calculator forces the student to think about order of operations very consciously if the correct answer is going to result.
Is Prof. Quinn saying that calculators "know" what the student intends? If so, he's wrong. Is he talking about a statistics function on a calculator where the three numbers can be entered into a list and then one-variable statistics can be run on that list, giving, among other things, the mean? In that case, obviously no parentheses are required, but then, neither is the formula for the mean (or the median, or the variance, the standard deviation, or a lot of other statistics such devices or computer programs can spew out simply by entering all the data points and a few relevant commands).
I fail to see how ignorance of or incompetence with order of operations and proper use of grouping symbols can be ascribed to use of calculator or other computing tools. Nor is failure to use mental arithmetic and estimation excusable in students whether they use computational aids or simply figure a simple average on paper or even in their heads. If a student doesn't ballpark results, silly errors are more likely to be taken as correct. But it's still perfectly possible to MAKE the silly or careless errors. The difference is that regardless of HOW those errors are introduced (and I don't think Quinn really knows), students who think are more likely to CATCH and CORRECT such errors than students who do not. Checking one's work is another way to employ intelligence and care that is something that should be mandatory for all students, but which was never popular for many students before the advent of wide-spread use of calculators. Is that now also to be blamed on them? It wouldn't surprise me in the least to hear that argument made by those who oppose these devices and alternative tools.
Lack of kinetic reinforcement. It is ironic that calculators
might be too kinetic in one way and not enough in another,
but this seems to be the case with graphing. In some
K–12 curricula, graphing is now almost entirely visual:
students push keys to see a picture on their graphing calculators
and are tested by hand math actually connected
with ways our brains learn, and the way calculators are
used to bypass drudgery has weakened these connections
and undercut learning.
This is more absurdity. Any competent teacher has students learn how to sketch graphs by hand. the calculator is used first as a check of one's work and as a way to explore a lot of graphs in a short time to see the relationship between changing parameters and the resultant graphs (when that is what is being focused upon). As new sorts of functions are graphed, hand techniques are still introduced. This is true all the way into calculus, and most students can readily appreciate the improved power of the methods taught in first semester calculus for sketching graphs.
However, over the long haul, and especially as graphs become increasingly complex, having calculator and computer tools available is enormously useful for most students (in know they have been and continue to be for me). But as good teachers are quick to note, it's vital to THINK about the graphs produced with these tools. They can be misleading. So just as with number crunching, students have to use their brains, and good teachers make this fact clear and push students to do heed it) (often by giving problems that highlight the dangers of being overly-credulous).
If the explanations offered are correct, then there are
several further conclusions. First, the learning connections
in traditional courses are largely accidental, and a
more conscious approach should significantly improve
learning. Second, calculators are not actually evil, but we
must be much more sophisticated in how such things are
designed and used.(4) But most of all, learning must now be
the focus in education. Not technology, not teaching, not
learning in traditional classrooms, but unfamiliar interactions
between odd and variable features of human brains
and a complex new environment.
1 At the Math Emporium at Virginia Tech, http://www.emporium.
2 For further analysis see “Beneficial high-stakes math tests: An example” at
3 See the Teaching Note on Parentheses at http://amstechnicalcareers.
4 See “Student computing in math: Interface design” at the site in
footnote 2 for an attempt.
The last paragraph above conveys a considerably different tone from most of the rest of Quinn's piece, particularly the title. They aren't exactly all in keeping with my own views (Calculators aren't "actually evil"? Gee, thanks for damning with faint praise!) but by the end, Professor Quinn actually seems to be advocating that we study the impact of new technology in classrooms on people and attend to how learning is enhanced, hindered, or simply done differently.
It would have done much more good for everyone if the piece had been written with a more open spirit of inquiry in mind from the beginning, with a good deal less of the usual anti-calculator, anti-technology, and anti-progressive tone. The problem is, in no small part, I think, that Professor Quinn starts with a very doubtful assumption about whole language, draws an analogy to math, and then jumps to a host of conclusions, none of which necessarily comprise anything that couldn't be said of mathematics teaching and learning prior to the invention of affordable hand-held devices and personal computers. Further, the idea that technology needs to be used intelligently is true but hardly new.
Perhaps I've erred in my suspicion that Professor Quinn is just shilling for the MC/HOLD camp. He may genuinely believe on his own that he's found some serious problems with calculator use and is merely calling for careful studies of their use and of other technology in mathematics teaching and learning. While the first part seems doubtful (at least from what he gets specific about) the second notion is always reasonable. It simply would have been better if he'd made clear from the beginning (and through a less inflammatory title) that he was calling for a bit less panic than would seem to be the case. I hope that in the future, when mathematicians decide to offer up calls for caution, they themselves exercise their own care in how they do so. Because the idea that technology and texts and much else that has been promoted by progressive reformers in mathematics education are just a "bunch of fuzzy crap" is all-too-easy to find on the Internet and elsewhere, and such hysterical claims that the sky is falling merely produce a great deal of heat without shedding much light, if any, on the real issues of improving teaching, learning, and achievement in US mathematics classrooms.