To be sure: I’m not looking to return to a mythical golden age of rough and tumble paper and pencil work. I just believe that there is value in learning how to, say, multiply by using traditional algorithms (or others like the lattice approach). I guess I agree with your third paragraph. While I still like introducing graphing the old-fashioned (if you will) way, analyzing and graphing data (mean, median, line of best fit) is best done on a calculator or spreadsheet.

In any event, I suppose that on the continuum from purist to new age/hippie (and I’m kidding with the latter label), I probably fall closer to purist than you.

G’day!

There’s also the chicken & egg conundrum here: without research evidence, the assumption that the proper order is to start with no calculating aids (and again, there are far more powerful things available to many students these days, free of charge, than hand-held calculators) and then allow the tools, I’m afraid you don’t have evidence other than appeal to tradition to back your position.

I suggest that it’s reasonable to allow calculators, etc., first for some things, paper-and-pencil methods first for some, hands-on representations for still others, and so forth. I can see situations (and graphing is one of them) in which it makes sense to have the paper/pencil tools and more powerful ones introduced together. Purists, of course, can never allow such radical departures from tradition, I know (having been actively engaged in the Math Wars since ’92 and in fights about calculators and computers since the late 1980s).

Your assumption that seeing an electronically produced graph first is debilitating to learning begs a rather obvious question: why is it that many of those “thinking” students using paper/pencil long before there were hand-held or desk-top graphing calculators and computers, not only didn’t get it, but continued long after their early exposure to the subject to not get it? Were they just “dumb,” “lazy,” etc.? Or could it be that there are many students who could benefit from more approaches than were dreamed of in the philosophy of earlier generations of math teachers?

Why was Leibniz, one of the greatest geniuses of all time, intent on creating a mechanical device for proving formulas, crunching numbers, and in other ways investigating scientific and mathematical truths? Why did Babbage think it a worthwhile project to do something similar?

I’m no opponent of having students learn mathematics as a thinking activity. I’m simply not convinced that making rapid, accurate hand-calculation a shibboleth for pursuing mathematical ideas is a wise practice. Intelligent use of tools seems to be part of the nature of being human. “No pain, no gain” may be all well and good for building muscles, but does the analogy hold up quite as well as some think for learning mathematics? And who exactly wants to be Arnold Schwarzenegger, anyhow?

Instead of using the picture of the graph, it would be much more informational to teach the student to use the table of values that the calculator provides for such a function. This would eliminate the students error and show them the points that needed graphed. And allow the student to see the pattern of the y values as the x values increase.

Using the technology more effectively is the key. Also, to take a page from Dan Meyer, have them graph one by finding many ordered pairs with the equation first (annoying and tedious). Thus showing them the need for the tool (the calculator) and how it make it easier. Creating a need for the technology will make them want to learn how to use it and remember it.

The glasses analogy is not quite apt, as that is a true disability that requires correction. Why introduce a calculator before a student has had a chance to try, say, graphing an equation on a piece of graph paper?

Of course students will make mistakes with or without a calculator. But at least without the calculator they are using their brains and not a pre-programmed machine that will do much of the work for them.

That analogy is fantastically better than anything I’ve used previously. Thank you for sharing it!

Maybe it also has a place in assisting to make some areas of Math a more level playing field by allowing them to think mathematically or learn different algorithms to solve problems than the ways taught a hundred years ago.

The poorest approach I’ve seen is only allowing students to use calcs after they can prove they don’t need it. Sort of like letting me use my glasses to see once I can prove I can read the stuff posted on the far wall.

I have a strong suspicion that even in the decades before graphing calculators were widely available (the 80s, for instance), there were students who made mistakes graphing y=-2x-1.

(Kind of like learning your multiplication tables first and then using the calculator to take care of the rote material when you advance to higher level math classes.)