[QOTD] Hans Freudenthal’s “Major Problems Of Math Education”

The tenth of Hans Freudenthal’s “Major Problems Of Mathematics Education“:

I am obliged to say something about calculators and computers. You would protest if I did not. I could refuse because I can prove I am incompetent. I know almost nothing about calculators and computers. It is a lack of knowledge that prevents me from tackling any minor problem of calculators and computers in mathematics education. It does not prevent me from indicating what in my view is a major problem.

Technology influences education. The ballpoint, Xerox, and the overhead projector have fundamentally changed instruction. But this is as it were unintentionally educational technology. Programmed instruction, teaching machines, language laboratories, which were intentional educational technology, founded on big theory, did not fare as well, to say the least of it.

Calculators are being used at school, and they will be used even more in the future. Computer science is taught and will be taught even more. How to do it – these are minor questions. Computer assisted instruction has still a long way to go even in the few cases where it looks feasible.

What I seek is neither calculators and computers as educational technology nor as technological education but as a powerful tool to arouse and increase mathematical understanding.

Thirty years ago.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.


  1. How prudent! Very true for science ed, too. While I love the smell of Vernier probes in the morning, kids still have to know why the data they’re collecting is significant.

    In my teacher prep program, we had to take a number (three, I think) of “technology” courses. One of the available courses was an entire quarter on using Palm Pilots in the classroom. I worry that iPads (perhaps pre-loaded with Khan Academy?) are the new thing that are supposed to fix all of the classroom woes.

  2. Interesting that I only bumped into QAMA last week for the first time and was intrigued by its implications in “Math Wars”-style debates, not that I would expect a sane, reasonable response from extreme foes of modern technology in K-12 math classrooms. But more neutral folks might be mollified by this sort of device, though it seems somewhat limited as currently construed when it comes to high school mathematics.

  3. On a quiz recently, one of my students had to graph y = -2x-1. She got it wrong, and then on the half-credit re-take she called me over to show me that what she got on her TI-83 calculator, after punching in the equation, exactly matched what she had on the quiz. The problem was that she had the calculator zoomed out too far, so when mimicking the graph on the quiz sheet, the slope and intercept were off. She sure knew how to use her calculator, but she was not even close to having the concept of slope and intercept down. She was trained to rely on the calculator and was shocked that she was wrong even when all of her inputs were correct. Even if she had been zoomed in far enough to accurately copy the graph, she still would not have been showing me that she understands the concepts.

    An anecdote, to be sure. But a damning one nonetheless. Had she not been using a calculator, she may have had no choice but to work harder to truly understand slope and intercept. As it is, she is reliant on that TI, come what may.

  4. @mathteacher: damning, yes, but whom, exactly, does it damn? I would say that either the student was not effectively taught or she willfully resisted learning. Calculators or any other graphing tool are not the cause of her lack of understanding.

  5. True enough, in that the calculator is an inanimate object and, yes, she could have used it more effectively. In the absence of a calculator, though, she would have to at least learn the concept first…and then, perhaps, use the calculator to speed up the process.

    (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.)

  6. As for the student who mistakenly trusted the window on–that’s just a teachable moment. Forgive her trespass, teach her the reason it’s wrong, and tell her go and sin no more.

    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.

  7. Most teachers I come across who allow students to use the calculator have little idea of how to use the calc as a learning tool, hence they do not teach how to learn for understanding using a calc. Even without a calc, students have and will continue to create bloopers in all sorts of Math areas. Stop blaming the calc and start thinking about ways to assist you do your job.

    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.

  8. @Ken Ellis:
    “Sort of like letting me use my glasses to see once I can prove I can read the stuff posted on the far wall.”

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

  9. Wow…a fair bit of blowback to my calculator posts. I happen to believe that, absent a true learning disability, calculators hinder one’s ability to learn and internalize mathematical concepts. It becomes a crutch, as it did for this student. (And, yes, I have forgiven her sin.)

    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.

  10. I agree the student should be able to graph without a calculator, but also should be taught the most effective way to graph with the calculator.

    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.

  11. math teacher@: I have to question the notion that calculator use or that of any specific technology (and again, you need to recognize that pencil and paper and similar tools are technology) inevitably become a “crutch.” If “crutches” are so horrible, of course, we should all go back to mental math exclusively. That even early humans realized that this was an ineffective way to calculate suggests, however, that we are naturally inclined to try to find more efficient ways to do donkey arithmetic and the like.

    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?

  12. Interesting points Mr. Goldenberg…especially viewing paper and pencil as technology (which renders just about everything outside our person as technology I suppose). Calculators certainly have a place, and you and Dan (above your latest reply) have good ideas.

    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.


  13. While I think these ideals are great, and are completely truthful, I wonder if maybe the calculator has taken classrooms too far? Often times in the classroom, especially younger grades, the main focus of the teacher is to teach the fundamentals (how to arrive at the answer), and some of them are just taking the easy way out and going directly to the calculator, because it is readily available to the students and the teacher. Math is often scary to elementary teachers, and they do not fully understand some of the concepts, so they will “cheat” their way through by allowing the use of a calculator, rather than teaching the method that it takes to arrive at the given answer.

  14. Calculators-in-education posts tend to get really flamey real fast and I appreciate you guys keeping it cool. Everything seems to have been said so I’m going to close this up.