Yes, the “if any” is implicit. Unlike our recent Who Should People Read post, this one is not intended to produce a list of “useful resources”. Rather, akin to this post, it is a challenge to come up with anything. But first, a story.
Some years back, a maths ed person I know was organising a maths day for secondary teachers. The organiser approached me and also a proponent of maths technostuff, asking if we would be interested in taking sides in a debate on the use of technostuff in the maths classroom. After some haggling over the limits of the topic for the purposes of the debate, and some clarification of the ground rules, we both agreed. Then, soon after, the organiser informed me that there had been “a change of plans”, and that there would be no debate. C’est la vie. But then the program for the maths day appeared: the pro-tech person had been paired with another pro-tech person, and these two pro-tech people were allotted a debate’s worth of time to demonstrate the pro-ness of their tech. This decision to change the program was, of course, both stupid and astonishingly rude, and I was, of course, furious.
I later learned that a more senior organiser had overruled the idea for a debate, and I eventually let both organisers know exactly how I felt. The less senior organiser eventually apologised, genuinely, and that is settled, genuinely. I do not know exactly why the senior organiser didn’t want the debate, since they wouldn’t give me a straight answer, but some conjectures come to mind. As to why the senior organiser and my would-be debating opponent regarded my dis-invitation as even remotely acceptable, I don’t know. Perhaps they simply considered insularity and censoriousness and rudeness as business as usual in modern maths ed. They may well be correct.
Anyway, that’s the story, and back to the question: What are the proper uses of “technology” in the school mathematics classroom? And, yes, referring to it as “technology” is perverse, but the theft of a very useful word by maths ed ideologues is probably irreversible. In any case, I’m using “technology” here to refer to anything from basic calculators on through CAS and GeoGebra and Desmos and Mathematica, and spreadsheets and whatever other Software Saviour people wish to propose. (Of course, for example, suggesting CAS is properly used in VCE because CAS is mandated in VCE is funny but is off the point.)
I imagine I’ll be arguing this one without much support, maybe even from myself. Do I really think there is no such proper use? No, not quite, but close. I have a few applications to which I might agree. The more I think about these applications, however, the more I’m sceptical of their value, and am definitely sceptical that more than very little class time should be devoted to them.
So, have it. Prove to me that I’m the Luddite that I am.
UPDATE (15/03/23)
Tony Gardiner has commented at length, below. He notes the key question:
How do young human beings construct our shared mental universe of mathematics in their heads?
It is in the answer to this question that I, and it seems Tony, can see little or no role for digital things.
Tony also linked to a debate he had with Conrad Wolfram on this very topic, here. I haven’t listened to it yet, but look forward to doing so.
UPDATE (16/03/23)
Here is a link to video of the entire event in which the Gardiner-Wolfram debate occurred, but the debate is incredibly well hidden.
Scroll down until you see the video window:
Below that (but above the PDFs) there is an “image” which can actually be scrolled. Scroll down until you see Smiling Conrad.
Click on Smiling Conrad, and the debate will appear in the video window above.
UPDATE (17/03/23)
Simon the Likeable has pointed out to me a relevant post that has just gone up, by Dan “Novices are Kinda Like Experts” Meyer. Dan provides a fine example of, well, whatever the hell the tripe is that he is selling. As I said, …
Data projectors don’t count right?
Let’s say not. One can argue about general techno, such as data projectors and computers and interactive whiteboards, and I’d certainly argue that most of it is pointless, when not outright poisonous. But let’s stick to maths techno.
Capital M Marty? What’s going on?
Fixed! (I wasn’t logged in when I replied to RF, and corrected incorrectly.)
I have spent 20 years providing maths tuition to VCE students, trying to fill in the gaps in their learning, caused by their over-reliance on calculators and subsequent lack of conceptual understanding m. I’m with you Marty!
Thanks, Libby!
But to (pretend to) be fair, one might agree that techno is in general a disaster while still advocating this or that specific application. What I’m suggesting is that even this “It’s generally a disaster, however …” position requires a proper defence.
A few slightly unconnected points.
I teach 7 to 10 maths, but have taught most of 11 and 12 as well.
Bloody Hell, kids just don’t know their tables no more – and they don’t even know that they don’t know. I’ll ask a kid something like so what’s 3 × 7, as part of a problem involving fractions (or something) and half of them will just guess, 19? And the other half will laboriously add 3s on their fingers 7 times (*whispers* “three, six, nine, twelve…”), and half of them will get it wrong anyway, “17?” and half will eventually get it right, and think they are pretty bloody good for being able to “do” this task – rather than just knowing it. And going the other way – asking, “so what’s 21 divided by 3”, well, that’s even worse. I weep internally, and then, full of sorrow and guilt, reach for a scientific calculator (mine, probably), so we can get on with the original problem. So I like plain old scientific calculators (my fave kind – like the kmart one https://www.kmart.com.au/product/scientific-calculator-42727880/?)
I like using desmos and wolframalpha to check/demonstrate ideas (from the front, or on a computer with a kid), but wouldn’t expect kids to use them.
My school currently has all kids working out of the textbook, rather than using the online superintegrated bellsnwhistles environment thing. And I think it’s better.
Occasionally I’ll use excel to demonstrate something – and it’s possibly useful to realise this ubiquitous software can do interesting things – but again really don’t want to teach it or have kids worry about it too much.
Rulers and compasses are still good technology! But the modern compasses tend to be very dull (safety, kiddos!) to the point (har har) that they are pretty much useless.
I’d happily watch all the CAS calcs in the country burn. Just junk. I’ve never reached for one to get anything done or to work anything out – a normal calc, a phone, a computer, a bit of paper – all superior.
And my current school has crap whiteboards and whiteboard markers, which makes it hard to sketch and fiddle and scribble and show – all of which are better than a button sequence.
I do think it’d be interesting to have a class with NO screens, NO calculators, nothing, for a fortnight, and see what comes of it…
Thanks, CASetc. Ignoring the points where I’ll obviously agree, what is an example of an idea you would check/demonstrate with desmos or wolfram alpha or excel? What do you think your students get out of this checking/demonstrating?
Even if there were useful applications of calculators/devices in the mathematics classroom, you would often find that some students in the class do not have the right equipment, and therefore could not benefit from the exercise.
Marti,
I reckon a good grasp of logarithms , trig tables ,slide rules and statistical tables as well as a basic graphing calculator as used in IB , and a sound introduction to programming in any of the modern languages would be sufficient for most secondary education courses.
Old school dinosaur
Steve R
All of “logarithms , trig tables ,slide rules and statistical tables” are obviously outside the discussion.
What is the value of a “basic graphing calculator”? Give an example.
How would “a sound introduction to programming in any of the modern languages” help with school mathematics?
Marti,
As an option in year 11-12 perhaps
Perhaps learn vba scripting used in excel macros for spreadsheet population ,regression analysis,…
Or how to programme in python for example as one of the current open source languages
Eg develop a routine to produce roots of a polynomial using newton raphson’s method
Better than just pressing the buttons on a CAS preprogrammed algorithm IMO
Steve R
Just plain nonsense.
Marti,
The functionalities of the allowable IB graphing calculators and there functionality are listed here
Click to access use-of-calculators-in-examinations-2019_en.pdf
On reflection Whether they assist much with the learning process of curve sketching other than with very complicated functions is debatable
Steve R
Thanks, Steve. I agree, of course, although “debatable” is an amusing choice of words.
@SteveR: Try adding “trig tables , slide rules and statistical tables” to the booklist!
You joke Terry, but I have heard recently of an under-30 Mathematics teacher trying to get their hands on a book of trig tables for class.
Their reason? Students did not understand that sin(A) is a function (Year 9/10 level students, so technically haven’t studied “functions” as a topic).
I had to admit not owning such a book. They didn’t want my slide rule though because the shunt was missing.
Good topic. Thanks for raising it. I believe ‘chalk and blackboard’ are all one needs to teach math. However, I wouldn’t object if a teacher who mastered (the keyword is mastered) a technology using it as an aid device in the classroom. Whatever helps to teach is good if it helps. Howeevr, teachers/lecturers often become ‘the voice behind a projector’ or PowerPoint deck, which is awful. Although, it isn’t the only technology that converts the educational process into ‘scholastic masturbation’. Modern books used in schools have been transformed into texts that teach to answer particular questions with particular answers. No more thinking whatsoever. Pupils learn questions and answers. It is all unfortunate.
Thanks, Dr. M. I agree on the texts. Again, although I loathe PowerPoint, I’ll stick to the issue of specifically maths techno.
You write that you “wouldn’t object [to] a teacher who mastered … a technology using it as an aid device in the classroom”. Can you think of a specific example of such a device being used (masterfully) in a way you think helps?
Marty, for instance, in functional analysis, it might be easier to demonstrate to students computer-generated saddle points than actually drawing them on the board. That would be an example that immediately springs into my mind.
Thanks, Dr. M. The question relates to school mathematics.
GeoGebra, if mastered well, can help in geometry in everything related to geometric constructions. Other than that, I see nothing else.
Can you think of a specific example where you imagine GeoGebra might help?
I can think of an example, Marty, and find more than one. However, your next question would be whether it is possible to do it without GeoGebra? Yes, it is possible.
Almost. My question would have been is it preferable to do it without GeoGebra?
One case springs to mind. GeoGebra allows almost immediate change of scales/stretch of the construction, which often helps to see the relationships between different lines/figures. Doing this in the traditional ways might take an enormous amount of time and sometimes might be even close to impossible.
Yes, that’s the kind of thing people suggest to me when I ask the question of this post, and I’m simply not convinced. I’m not convinced that kids half-watching a computer stretch something, with the teacher going “See?” gets much of anything into their heads. To me it just feels like the “Ooh, it does get in!” from the toothpaste commercial. Except it’s the teacher going “Ooh!” rather than the kids.
It works. I have seen it working. I can get you on Zoom with the child for whom it works. I am sending you an email now.
Um, no. I received your email, it is very interesting and I’ll reply by email. But you are describing a situation that is way, way outside the norm. It is almost entirely irrelevant to what we’re discussing: the proper uses of “technology” in the school mathematics classroom.
Didn’t Churchill say something about a drunk and a lamppost: for support rather than illumination…?
I think the same can be said of technology in any classroom.
If the thinking drives the technology choices (and let’s be honest here, it is most often the school leaders driving the technology choices – “we paid for this so use it”) then maybe it will be of some benefit. An awful lot of thinking is required though.
I would also like to keep the question of “doing exams” separate from learning. Sure, the calculator can help with an exam. The exam was written this way.
It is not easy to “frame” this debate in a suitable way. (Think “mobile phones”, or time spent online – rather than with real people.)
The problem is obvious; but very few people have either the desire or the confidence to go against the flow – even if the dangers of following the crowd are evident. There is an underlying dynamic, which makes the debate unbalanced (even when it takes place:
https://www.cambridgeassessment.org.uk/news/audio/view/maths-viewpoint-debate/ )
and makes it easy to characterise people like Marty (and me) as “stick in the muds”. (I suspect the above debate had a clear “winner”. But it made absolutely no difference! The juggernaut simply marched onwards. But we shouldn’t give up! Such struggles are like decorating; or tidying up after kids/husbands, or changing nappies; or … ; or sticking up for what is right! Unlike mathematics, society does not have a “QED”. The Goodies do not win out in the end. The truth needs permanent advocates, willing to engage in a sequence of endless battles, that nevertheless contribute to a better status quo.)
The central point (which often gets overlooked, but which will not go away) is not
*What glitzy gizmos are there which might appear to enliven what the teacher does?*,
but
*How do young human beings construct our shared mental universe of mathematics in their heads?*.
Once this has been achieved, technology can dramatically amplify *some* aspects of what is then possible. But, as far as one can tell, there are no short cuts.
To summarise: The first steps are to master the poetry and rhythm of number names and the number sequence (counting!); then to embed *place value* as a routine tool for taming the endless infinity of different numbers, and to use this to juggle numbers and calculations in one’s head with minimal storage and error. Learning ones multiplication tables is then part of a sequence of steps that allow one to store the basic facts and construct/strengthen the neural connections and mental short cuts that leave mental bandwidth available for serious *numerical thinking*; etc. Much the same applies to measures, geometry, algebra, etc.
Thanks very much, Tony. I’ve updated the post with a couple elements of your comment.
Thanks; there are numerous pocasts and videos on this website that are interesting to me:
https://www.cambridgeassessment.org.uk/
Red Five wrote:
“Didn’t Churchill say something about a drunk and a lamppost: for support rather than illumination…?
I think the same can be said of technology in any classroom.”
This reminded me of another mathematical joke about a drunk which may be relevant.
(Drunk on hands and knees crawling round and round a tree. Officer asks: “Come now Sir. Why don’t you just go home and get on with your life.” Drunk replies: “I’d love to Officer. But everywhere I look, I seem to be surrounded by this bloody tree.”)
Love it!
I might have to use that line in the near future, perhaps when going through my annual rectal exam (by which I mean performance appraisal, with sincere apologies to anyone who has to go through both in the same year).
On the topic of drunks, trees and lamp posts …
A policeman sees a drunk searching for something under a streetlight and asks what the drunk has lost. They say that they lost their keys. Policeman and drunk look under the streetlight together. After a few minutes the policeman asks if the drunk is sure they lost them here, and the drunk replies, no, and that they lost them in the park. The policeman asks the drunk why they are searching under the streetlight, and the drunk replies, “this is where the light is”.
Often quoted to illustrate the type of observational bias that occurs when people only search for something where it is easiest to look …
If I was being pedantic (which is pretty much always, I know…) I could suggest that “technology” is not the same as “digital” or even “electronic”.
That said, I think the central point remains completely valid: technology (be it calculators, slide rules or Eddie Woo videos) only really gives the illusion of understanding and can never replace the actual doing of Mathematics (thinking, trying, perhaps some swearing and then some more doing) and anyone who claims otherwise is deluded at best and outright harmful at worst.
If I were being pedantic …
Apologies. I should have learned last time…
There is a lovely movie called “Denise calls up”. The entire movie is conducted on phones.
I had a group of students playing with sine functions for the first time. The exercise was to come up with a formula to match a given sine graph. I had them use Desmos to check their work as they went. What is the mean of the given graph? Modify the basic sine graph on Desmos and see if the Desmos graph looks closer to the given graph, what is the altitude on the given graph? Etc. They were using Desmos for feedback as they went to build some understanding of which number in the formula does what to the graph. I think using Desmos for immediate feedback was helpful in this situation.
Thanks, MW. Similar to my reply to Dr. M, I’m not convinced, but I want to think why precisely this example bothers me. I’ll reply in substance once my brain returns.
Thanks again, MW.
I am still not convinced, and I think this is a very good example to illustrate my general concerns. Clearly, what is valuable here is that “technology” can provide answers very quickly. But the truism is, technology can tell you what is true, but not why. I think your example illustrates how the “why” so easily gets lost in the techno.
You wrote
“[The students] were using Desmos for feedback as they went to build some understanding of which number in the formula does what to the graph.
But that seems to me a highly inaccurate use of the word “understanding”. What it seems to me is that the students were gaining some knowledge of how the various parameters work, which is very short of understanding why the parameters work the way they do. I also question the value of Desmos even for students’ gaining the knowledge of such function workings, but first things first.
Let’s say we want the kids to understand sine functions enough to understand their averages. (Perhaps one wants a different word from “average” or “mean” for pre-calculus students, but that’s a detail.) What do the students need?
First, they need some intuition for the zero average of your basic
guy: there’s as much and in the same way above the x-axis as below. (Hopefully, this intuition comes from considering the unit circle origins of 
, rather than from brutishly plonking the sine function on the board or screen, but I’m also hoping for a pony this Christmas, and I’m aware there’s little chance of either occurring.)
Then, sixthly, you want kids to be able to decipher functions such as
. In between, you want to consider the various standard transformations, and what they do to the average, if anything. But how do you get that understanding across? How do you compare, for example, the average of a function f(x) with the stretched function 3f(x)?
Well, you need to get across that 3f(x) is simply tripling the output, the y-value, which the majority of Year 12 Methods students do not properly understand, and I wonder why that might be. And, you need some intuition of the “average” of a function as summing up the y-values, so that the average of 3f(x) is triple that of f(x). And so on.
I can see variations on how to try to convey this combination of understanding and intuition. What I cannot see is any role of technology whatsoever in doing so.
Then, back to the knowledge of such function workings, I am deeply suspicious that Desmos or whatever can even help with this. I do not believe that trying to impart that knowledge without a cement-solid foundation in the understanding can possibly work, can possibly keep the knowledge stably in students’ heads. And, I do not believe that the techno-experimentation with a few examples even properly demonstrates the knowledge to be learned.
The only thing I can see working here is a teacher patiently explaining, to the class as a whole and individually as required, why the functions at hand work the way they do, and then students working throw many carefully chosen and progressively more difficult and challenging exercises. You know: traditional teaching.
I’ve updated the post again (and stealth-updated the second update), with a link to a page with the video of the Gardiner-Wolfram debate, and with instructions on finding the (well-hidden) video. The debate is well worth the watch, and Tony is correct: there was a clear “winner”.
Thanks for linking it. I watched it and it was interesting, though I feel deflated now. I don’t have much to add, but I thought Conrad Wolfram’s point about photography actually worked against his case in the debate. The stuff he was describing as what we don’t do anymore (working with film and developing from negatives) is technology. Isn’t it analogous to the stuff that he was arguing we need to include more of? I think calculating is not as trivial as he makes it out to be and it’s worth learning what
is when you’re young because it won’t ever change.
Thanks, wst. Luckily for us Victorians, almost no one here takes Pastor Conrad seriously.
I listened to the debate. Let me support Tony’s closing comment about evaluation. It is often the case that governments set up a scheme, let it run for a while, then close it down – without any evaluation. Indeed, it seems that evaluation is not even considered at the outset. To take a topical issue, how will the purchase of AUKUS submarines be evaluated? (This is just an example; it is not meant to change the direction of the original post.)
Well, perhaps by accident, you’ve come up with a great analogy and made a very important second point. Paul Keating gave a perfect evaluation of the submarine madness, and nothing further needs to be pondered or said.
Of course you are correct, that before implementing any halfway large or radical change, there should be very solid arguments for that change, and there should be planned very solid procedures for subsequently evaluating that change. And evangelist “I Am The Word” clowns such as Wolfram have no such concerns.
BUT, some proposals are such blatant horseshit that it is absurd to be asking for evidence or to try to muster counter-evidence. That was exactly the situation with Keating and all of the press gallery: these stooges wanted to debate the nature of the Chinese “threat” to Australia, Keating wasn’t buying into that lunacy for a minute, and he was absolutely correct not to.
There are plenty of such proposals, some now orthodox practice, in education in exactly the same category. Tony Gardiner had the opportunity to debate Wolfram in a prominent forum, he was right to take the opportunity, and in my opinion he ripped Wolfram to shreds. They debated (in a fashion). But, primarily, Gardiner was able to rip Wolfram to shreds because Wolfram was talking horseshit. The only reason the debate was valuable, and necessary, is because so many people have bought into so much horseshit.
To be clear, I don’t think to consider using “technology” in the mathematics classroom is at that general level of horseshittiness. For my own, vetoed, debate, I would have had to work hard, to consider carefully such uses and their purported purpose. Similarly, I’m trying to and willing to think hard about the suggestions in the comments on this post. Nonetheless, the underlying premise, that “technology” is generally, essentially automatically, beneficial to the classroom is undeniable horseshit. It is now orthodoxy, but it is horseshit. Perhaps authorities need to gather evidence on this. But, to steal from Keating, anyone with a brain requires nothing of the sort.
Just about all the top (and not even near the top) chess players use computers (chess engines) to improve their game. Yet, you are not allowed to use computers in a tournament.
Maybe it is similar in mathematics. Computers can assist students in improving their understanding, but they should not be used in examinations.
What is an example of computer use in the school classroom where they would assist students’ understanding?
I can’t give a universal answer; the impact of an explanation (with or without a computer) can vary with the student.
For a couple of my students, who are quite good at mathematics and computing, when faced with a mathematical problem, their first instinct is to calculate.
Related to this, when I ask students to prove something like
, it’s not unusual for students to show that this is true for particular values of 
and 
. Eventually most (but not all) students understand that this is not sufficient.
I’m not asking for a universal reply: I’m asking for specific examples. If you have none, that’s fine.
When I say that I did not know of a universal answer, I meant that I could not think an example of the use of technology that would enhance the learning of *all* students. On the other hand, some demonstrations or explanations help some students, but not others.
If I am struggling to understand something, I do a simple worked example, or look up books and papers, or look at online resources, or ask a human being. I find many different explanations end eventually the penny drops, not necessarily because of a single resource. Rather this may be the cumulative effect of many resources.
Jesus. Move the goalposts wherever you wish. I can wait.
Isaac Asimov’s 1958 short story, “The Feeling of Power.” might be an interesting take on this discussion. Here is the free link.
Thanks, Dr. M. Is this effectively enough public domain? If not, then I’d prefer a link to an upload.
That story was reprinted in the 1959 collection Nine Tomorrows. It is the second story in the collection. Interestingly, the lead story in the collection is “Profession” (first appearance July 1957 Astounding Science Fiction) and is also a very interesting take on this discussion.
Thanks for mentioning will give it a read
Sure, almost all modern master-level chess players use engines to improve their game, and so do many expert-level players as well. But the analogy for mathematics there is that most modern mathematicians with PhDs use technology, and so do many grad students.
School mathematics, if you were to convert it to chess would probably be about in the range of 300-1700 elos. At those elos, using an engine is just like using a black-box which tells you that ‘a, b’ moves are good, ‘c’ move is okay, and the rest of the moves are blunders. It’s useful for checking to see where you went wrong, which is probably the best use of calculators in school mathematics.
But of course, a GM’s analysis would be infinitely more valuable than an engine for a 300-1700 elo player, just as a teacher’s guidance should, as they both can point out specifically what their students are having trouble with.
Excellent reply.
Hi anonymous,
I like the chess analogy
Speaking as a “retired” chess player of average minus Elo grade from my uni days where the bi annual informator from Eastern Europe was the main source of opening innovation For the ambitious player.
I agree that these publications too were more than useless to 90% of the club chess playing population .
These publications have now been made redundant by chess databases and AI trained engines playing to a level of over 3000 Elo where opening innovations are lucky to last one day.
Steve R
Ah, I see. Some PDFs display and some do not, and I’m never sure why. I’ll tidy your comments later with whatever linking works best.
Who pays for calculators? Parents. Is the increasing emphasis on technology in the classroom part of the move to privatise public education?
As an adult going back to learning Maths B, I pretty much steer clear of technology, find the textbook is the best source. Sometimes general websites are useful, for example when the textbook had a perfunctionary explanation of parametric equations, I was able to look up some useful websites on that. Other than that, its mainly just get a good textbook and a good teacher. I think all the other stuff they’ve added just distracts a lot.
I agree with you Frank – although getting a “good” text book is not so easy these days. Modern school textbooks are full of colourful photos to excite – or to use your word distract – the reader, and short on solid mathematical content. Even then, I doubt that many students read or study the text book.
I came across one text book which had what I thought was a serious error in the solutions to some exercises. I happened to meet the lead author one day – text books are now written by teams of people – and I mentioned this over a coffee. The reply was “The solutions are outsourced to other people – I don’t check them.”
Text books are big business.
Although there aren’t many algorithms taught in school mathematics, there are a few.
A teacher implementing the Sieve of Eratosthenes, or Newton’s method, or even something like polynomial long division or Gaussian elimination could be a worthwhile use of time: you could show the Sieve of Eratosthenes and Gaussian elimination for much larger problems than that you can do by hand, you can show that Newton’s method really does converge, and you could show that all these methods just require following a sequence of rather simple steps.
Of course, that would only be if the teacher wants to and is able to program; forcing a teacher to program those algorithms (especially the latter two) could only end in disaster. And this would only be useful the scarce times where (non-trivial) algorithms are used in school.
And if you’re willing to extend the definition of school mathematics to include 1st year linear algebra, there’s a lot of interesting stuff you can do regarding computer graphics.
Thanks, A2. Brief replies:
*) Eratosthenes is a very standard thing to have kids do in upper primary and lower secondary. But how could a teacher “show” the Sieve of Eratosthenes for larger ranges? Sure, they could naturally show the output of having done the sieve. But you really want to have the kids watching some machine crossing off numbers? What’s the point?
*) Ditto for Newton’s and Gaussian and polynomial divison.
*) Getting a computer do Newton does not “show that Newton’s method really does converge”: it only suggests that Newton converges. You might think I am nitpicking, but I very much think that I am not.
*) I’m not willing to extend school mathematics to include, e.g., Gaussian elimination, and consciously included “school” in the title. There are obvious and sensible uses of Matlab or Mathematica or whatever in some university subjects, although even here it is typically overdone, with lecturers seemingly feeling obligated to techno their lectures whenever remotely feasible, rather than when it is properly wise.
“you can show that Newton’s method really does converge”
More importantly, you can show when it fails:
1) Newton’s method converges to another solution x=b such that f(b)=0 instead of converging to the desired solution x=a.
2) Newton’s method eventually gets into a never ending cycle, oscillating between the same two approximations xi and xi+1.
3) Eventually, each next approximation xi+1 falls further from the desired solution xa than the previous approximation xi.
4) Newton’s method is not able to find the next approximation xi+1 because f'(xi)=0 or f'(xi) does not exist.
“What’s the point?”
It’s a fun little demonstration that is harmless, and could further understanding in some way. I do think that my understanding of Gaussian elimination was improved after I implemented it myself on a computer. I’d say it’s a lot more useful than whatever “estimation” exercises, “games” and attempts to connect mathematics to the “real world” litter school mathematics at the moment. Of course, it shouldn’t replace actual teaching of the content, but from what I remember there was a lot of free time during school mathematics classes, where we were left to our own devices: supposedly to do exercises, but most people played games or used social media or whatever. This is certainly a better use of time than that.
My “what’s the point” was directed at watching a machine doing Eratosthenes. My question stands.
Well, what’s the point of doing the sieve of Eratosthenes by hand? I’d say that the reason is to show a method for generating prime numbers.
Showing the sieve of Eratosthenes in action by using a script shows how:
(1) to decompose the algorithm into a number of precise operations
(2) the process of translating precise operations into a language that a computer can read
(3) the sheer power of a computer for doing precisely set out algorithms (and how much faster they are than humans) and how people can easily utilise this power.
Also, it’s just interesting and engaging to program a computer do something in seconds that would take a human days. As a child I had a lot of fun implementing an algorithm that generates primes (not using the sieve of Eratosthenes, but just by simple iteration.)
I suppose this isn’t all mathematics, and it is related to ever-dreaded “algorithmic thinking”, but I think that concept is completely valid in this particular scenario.
Yes, I think it is very important, particularly in maths classes, for teachers to demonstrate the power of computers. Otherwise students would have no inkling of this.
You may jest, but consider that the average 10-year old’s interactions with a computer probably only consist of gaming, (unfortunately) social media, music, and reading and creating documents. The most they have probably used a computer for related to math is the calculator app, or the graphing app. Perhaps they might know that a computer can quickly execute algorithms, but they might not know how accessible it is for the average Joe, the amount of freedom you have in specifying algorithms, and how powerful of a tool it is to solve structured problems. Maybe it’s because I’m primarily a computer science guy, but I think it’s a worthwhile lesson to teach, even if it perhaps shouldn’t be the primary focus of a maths lesson.
You could say that it ideally should be taught instead in a dedicated computer science subject, but those don’t teach pretty much anything about algorithms up to and including VCE, with the exception of Algorithmics (which is a niche subject that most students won’t have access to.)
It obviously shouldn’t be the primary focus of a maths lesson. The question is, should it be anywhere near a maths lesson. I believe the answer is no, and you’re not doing much to make me doubt it.
Maths lessons are about learning maths. Why is this hard?
Well, I guess that’s where we fundamentally disagree. I think there’s enough overlap between math and computer science that it’s okay to have a bit of computer science in school level math, and a lot of math in school level computer science.
Ugh! Maybe if maths teaching were in good repair, I’d consider what you’re saying. But maths teaching is currently a disaster, for all manner of reason, and one reason being that everybody keeps shovelling in irrelevant, distracting nonsense. “Oh we could have this cute little demo …maybe the kiddies could model World Hunger …”. All the goddam time.
What you’re suggesting is not quite as irrelevant and not quite as nonsensical. But it is not remotely the main point, and until the main point is being properly, or even vaguely, addressed, I have zero time for “we could also teach …”
There is a wonderful movie called “Razzle Dazzle” about dance schools. As I recall, there is a scene where the teacher is trying to explain to the children how dance could be used a protest against animal cruelty. This was too much for the children to even contemplate. In the end, the little children did a dance which was a protest against the invasion of Afghanistan.
Did their dance have any effect on the invasion?
This is irrelevant to my suggested uses of technology above, but I see a few people on this thread suggesting that graphing calculators can help with learning sketching.
LOL
Graphing calculators are the biggest crutch and the greatest thing stopping people from learning how to sketch properly.
There’s not much point (for the purpose of maximising VCE results) in learning how to sketch properly, because:
(a) the sketches you do in tech free are all very simple, and just a bunch of cases you can memorise without knowing how to sketch. As an aside, I think this is where the stupid idea of never crossing the asymptotes came from: extrapolating from the y=a/(x-h) + k graph, one of the graphs my teachers made us memorise.
(b) the sketches you do in tech active are all just copying the CAS.
Teachers know this, and at least from my experiences they didn’t teach sketching properly. Graphing calculators are the cause of this.
@Anonymous2: “the sketches you do in tech free are all very simple”.
My experience working in a tech free environment.
I recall that when I had finished my honours degree I applied for a job and got an interview. One of the questions I was asked was to draw the graph of a function. Maybe it was something like
. It had been several years since anyone had asked me to draw a graph, and I suppose that I was nervous, but I struggled to answer the question. (Real pure mathematicians don’t draw graphs! We prove theorems.) Now with much more experience – from teaching – under my belt I would not find it difficult. But, at the time, the question was no so simple for me.
They also asked me to multiply two
matrices with entries that were small integers. Now that was more like it! I could do this! However I made some arithmetical errors as the panel looked on. Anyway I did not get the job.
Decades later, when I was more confident, at an interview I was asked a question about how would I go about solving a particular computational problem about matrices. I struggled even to grasp the question which was quite wordy. The other members of the panel also asked questions of the questioner so that they could understand what the problem was asking – this was heartening for me. We all beat around the bush for about 5-10 minutes while I struggled to find something sensible to say. Eventually, the questioner admitted that he had been working on the problem for 20 years! I didn’t get that job either.
As a wise colleague once told me, “We should be grateful for the jobs we didn’t get”.
Hi,
James Stewart (calculus) not as rigorous as Spivak , Spiegel , etc .. but still used in under grad courses has a section in the appendix D about “ lies my calculator taught me” .
An interesting read IMO as it was written long before CAS been came common place .
“ thinking before doing” is his conclusion
Steve R
Click to access LiesCalcAndCompTold.pdf
We were given the same advice by a computing lecturer! Think before you compute.
TM,
That’s true too
Have a look at the slow convergent
p series summations of 1/1.001^n for n = 1 to 10^6 say mentioned in Stewart’s link above and test your favourite calculator’s processing speed ,memory and inaccuracy
Steve R
one proof of convergence of a similar p series below
https://math.stackexchange.com/questions/2020988/does-the-sum-from-1-to-infinity-of-1-n1-01-converge
Although I have not followed this thread diligently, I suspect Marty may be being a bit rough on Terry Mills. (Maybe they are old sparring partners, and so are used to this.)
I read Terry’s inputs as: “We cannot simply ignore this stuff. So how can we make a reasonable fist of it?”
The fact that there may be no easy answer does not make the question go away. But it may imply that it is a mistake to distort the whole mathematics curriculum while ignoring the psychological realities of the beginner’s internalising processs.
Terry and I know each other well, but I think I would have replied the same no matter the commenter.
Yes, we have this invasion of technology, and of course we have no choice but to deal with it. But “this impact will surely increase” is way too passive a description of what is occurring. The problem is not primarily the technology but the worshippers of that technology, and presenting the ‘impact” as if it’s simply the result of a natural, non-human progression is ignoring that.
Technology affects every aspect of our lives, including the experiences of teachers and students. Calculating machines enable new mathematical experiences just as “a child who learns to speak has a new facility and a new desire” (Dewey, J. (1938/1956). Experience and education. Macmillan, p. 37). We should strive to maximise the value of the impact of technological developments in learning about mathematics because this impact will surely increase. There is more to come.
For Christ’s sake. That’s not an argument for anything.
Not sure about resources, but this blog post produced, so far, 33 (not counting this one) messages more than “What People Should Read” 🙂
33+ messages, and no defensible answers to the actual question.
I am still standing by GeoGebra. Yes, one might go without it, no doubt. However, it might tremendously help the class by demonstrating complex alignments. It might take plenty of time that lesson doesn’t allow drawing it by hand, while GeoGebra can speed up things dramatically. Also, GeoGebra helps to learn different tactics in proofs. I would not recommend using GeoGebra in problems where the answer is numerical, but when it comes to proof, one can demonstrate very interesting things almost effortlessly. Again one can go without it, but it speeds things up dramatically.
Provide me with a concrete example, and I’ll consider it. As I’ve tried to make clear, I’m open to the possibility of such uses, albeit sceptical. The suggestions made so far in the comments have only made me more sceptical.
Here we go. I have asked a certain someone in my house who says that GeoGebra helps to provide a concrete example.
That reminds me of an puzzle game app I saw once where the puzzles are solving some of the constructions in Euclid’s Elements, and once you complete a puzzle, you can use that construction for a future puzzle. I don’t think it was too educational in of itself, but it was a cool concept.
One of my first instinctual reactions to the use of technology in mathematics was proof verification, but I quickly disregarded the question was for school mathematics. But proof verification for geometric proofs (not sure if GeoGebra can do that, I’d be pleasantly surprised if it could) would be an interesting use of technology.
Regarding this: I think that doing a computer simulation of compass-and-edge construction is probably a better option than doing it with a physical compass and ruler. Physical compasses are very finicky and annoying to use, not to mention slow, and there are definitely some imprecisions with them that you can’t really avoid. And using a computer simulation allows you to much more easily “undo” mistakes than with an eraser that would also require you to redraw collateral damage.
However, I’m skeptical that the example in your video actually shows much, and I doubt that using GeoGebra would actually help in solving the problem as stated any more than a quick sketch would.
I agree that the use of technology can vary and also might help one and not another. I am against making it compulsory. However, this example was demonstrated by my son. He solved the problem without GeoGebra but then came home and showed me quickly how GeoGebra could help to run through the alignment. Using a compass and ruler would have taken much more time. The video I posted is 2.26 minutes.
Thanks, Dr. M and A2.
You’ll correctly regard it as a dodge, and a weird dodge, if I say I’m not thrilled or much concerned with Euclidean Proof. (I know most mathematician types in this ed space are pretty strong advocates.) So, I won’t try that dodge. Still, as a separate question for another time, the proper role of Euclidean Proof in modern mathematics education is worth pondering. Maybe I’ll post on it, just so the few people who haven’t yet thought of lynching me can reconsider.
I was aware from the outset that if I were to concede some proper use of such technology in the school classroom then it would probably be in the realm of geometric construction. I agree there is some arguable merit to Dr. M’s assistant’s specific suggestion, and to A2’s general argument, but I’d still include some very large qualifications.
(1) Let’s be clear that what the computer is doing here is suggesting/indicating to us what it is true, not why it is true. Of course, as I already conceded to MWare, the speed at which a computer can provide or point to an answer is valuable. There is more here, as I indicate below, but you have to be careful.
(2) Following on, this is perhaps a reasonable example of a teacher demonstrating, after some student effort, how to go about seeing the way to a proof. If there is any suggestion that classroom students should be playing around this way, that is much more questionable.
(3) What is GeoGebra really demonstrating for us here, and is GeoGebra the best way to demonstrate it? There are two key points to Dr. M’s assistant’s proof outline: (i) the semicircle touches the midpoint of CD; (ii) the second semicircle passes through A. I’m not convinced that GeoGebra is the optimal way to get either of these truths into the minds of the students. (And, again, GeoGebra cannot tell you why these truths are true.)
(4) Following on from the final point of (1), in the hands of Dr. M’s assistant, GeoGebra can be seen as not just a source of quick answer but as a tool for experimentation. Such computing is a genuinely new and valuable tool for mathematicians, and, following on from (2), it is reasonable to ask whether there is a consequential place for such experimentation in the school mathematics curriculum. But, critically, the question must be asked, not the answer assumed. In particular, if such experimentation is going to infiltrate some other curriculum element, the costs as well as the benefits must be considered. And there clearly are costs. In particular, although A2 notes the slowness and errorness of doing ruler and compass constructions with, um, a ruler and compass, this slowness and errorness, and tactileness, also has obvious and genuine value. There is no royal road.
In sum, I concede the point. Just. I reserve the right to change my mind.
Thanks for your reply Marty. My ‘assistant’ is on the moon because his demonstration made you to concede the point 🙂 In any case, I agree that use of GeoGebra, if brought in, is not instead but in addition to regular construction.
You have a very able assistant.
It is always hard to enter into the mind of someone with a different background in order to see genuine *advantages* – as opposed to necessity enforced by one’s particular circumstances. (Is the support I value really useful? Or is it the only support with which I happen to be familiar?)
For others to judge, here is the response of someone who has observed Geogebra being used, but has *never* themselves used Geogebra. (The point here is to consider a *whole class* of students, rather than to applaud the particular approach achieved by A2: How can one help large numbers of students to work their way into such problems – and easier problems of a similar type?)
1. The problem involves
(i) a square ABCD (not abcd!);
(ii) a SEMIcircle, with diameter XB with X on AD
(iii) which is tangent to CD.
2. Others may be better placed to compare the two relevant ways of engaging my “mental mathematical universe”.
(a) Setting up Geogebra to draw the figure (where I have to specify the constraints, but the drawing is *done for me* – though I am then free to *vary* the position of X and observe experimentally(!) what happens).
(b) Drawing/sketching a figure by hand – perhaps discarding my initial attempt – (with no other distractions or aids) would seem to be an art we have simply thrown away without thinking.
Software that delivers an “accurate” figure (i.e. a very good approximation!!) runs the risk of being like a calculator that can work out 7×9, or “evaluate” (13^2 – 1)/(7×24). In theory it allows me to engage with “calculations” that might otherwise be beyond me – which is wonderful. But in practice, it is tempting
– to become dependent,
– to use it for easier problems, and so
– to never to develop the “internalised universe” which we call “mathematics”
– to land up creating something more like “engineers” (users of mathematics who want *answers*).
3. My guess is that the key step in either case is not assisted by 2.(a) – namely the thought that:
“Nothing in all my experience of geometry suggests how to work effectively with a SEMIcircle
(though I do know something about *angle in a semicircle*).
Geometry studies circles – a locus with a centre and a radius.”
Until I had this thought, I wasn’t sure how to proceed.
4. As soon as I had this thought, I tried to complete the circle – to move away from the world of “half circles”, or rather to embed the only relevant thing I knew (about angles in a semicircle – or rather its *converse*!!) into the world of circles.
5. I can’t be sure, but then the power of Geogebra might then well have been a distraction (backing up Marty’s point 2.).
Instead of focusing on the apparently *asymmetric* data (fixing B and “moving X on AD”),
there is now good reason to switch attention to the *symmetric* constraint (circles through A and B
expanding until they become tangent to CD – at the midpoint M of CD).
6. But I am asked to locate X. There are lots of ways to proceed.
(E.g. X is where the perpendicular bisector of AM meets AD. Treat A as the origin, and let the square have side 4. If L is the midpoint of AM, then L = (1,2). Since A to M is 2 units right and 4 units up, L to X is 4 units [which must be quarters] left and 2 units up, so X = (0, 5/2) – and the centre of the circle O = (2, 3/2).
(And if K is the midpoint of AB, then AKO is a 3,4,5 triangle.)
Tony, the ‘voice’ behind the Geogebra video, is in primary school right now. I will ask him to respond to your post when he returns. Despite his ‘tender’ age, he firmly believes that GeoGebra should be used as a discovery and demonstration tool. The problem in the video was question 3 in the last week’s Maclaurian Olympiad. He solved it without GeoGebra but then came home and visualised a solution very quickly, demonstrating possible ways of thinking about it. The speed with which one can create geometric constructions in GeoGebra is incredible. The amount of time a teacher can save demonstrating different geometric alignments in class could be very significant.
Dear Mr Gardiner,
My solution differed somewhat from yours when first solving this question (without GeoGebra).
I have split it into parts:
1. The first important insight is that the tangency point is the midpoint M of side
CD. I first noticed this when hunting for similar triangles. I found that the ratio of
XO to OC is equal to the ratio of SM to MC.
Finding this with GeoGebra is not very effective, and it is likely easier to see it with more ‘traditional’ tactics.
However,the following insight can be found much faster with the help of GeoGebra.
2. The idea that A is always on the semi-circle is essential to solving the question. I found this by thinking about important facts about diameters and semi-circles.
Since there were many right angles, this led me to think about Thales’ Thereom. You can now quickly notice that angle XAB is 90, which suggests extending the circle.
GeoGebra is beneficial here as, like you said, once you complete the circle, the correlation becomes clear. If students do not know Thales’ Thereom, this can be a good way for a teacher to let them feel like they are discovering it themselves, and, in my opinion, this promotes education.
3. The problem can now be finished quickly with power of a point and some
algebra.
In conclusion, GeoGebra can hugely speed up and help to learn; however, it is true a way must be found that can stop GeoGebra from becoming a ‘calculator’ to make questions easier.
Tactics should be taught as they are, but I think GeoGebra can help the students learn.
Returning to the original question … as soon as you try to apply a method for data analysis from the school curriculum, you would wonder if this could be done more easily with a calculator … and it can. Of course, you can show that you understand the method with a small sample, but a small sample is usually not helpful in suggesting a hypothesis in a practical problem. Interesting problems in data analysis generally require samples of reasonable size, and the associated calculations are facilitated by a calculator.
My impression is that there’s essentially no need for “technology” in the math classroom.
First, it really isn’t NEEDED. When the kids are learning arithmetic, they need to do that. And then later when they are doing algebra-calculus, the problems are not arithmetically complicated. This is in contrast to chemistry classes, where scientific calculators moved in, without debate, around 1980.
Second, it might screw the kids up. The crutch argument. I don’t really think it’s that much the danger people think it is. It’s hard to learn math with or without the silliness.
Third, technology is EASILY LEARNED WHEN NEEDED. Like, do we really think the kids need help mashing buttons? If you are strong at math on pencil and paper, you will pick up technology extremely quickly (Excel or Fortran or whatever) very quickly as needed. The converse is NOT TRUE.
Fourth, it is prone to a lot of ed appearance silliness. Gotta do that them thar technology so our kids don’t fall behind the Japanese. But you’re not really cracking the hard nuts (e.g. getting kids to do multi step algebra). You’re just blowing money on “technology” and not even knowing what you’re doing. It’s all appearance.
Fifth, if you don’t have facility with algebra and arithmetic, you will struggle in chemistry or physics. It’s too much extra cognitive load to follow a derivation or work a homework problem if you need a crutch. Like trying to read a foreign language text with a dictionary. [Maybe this is related to “second” or I’m sequeing to push back against the general argument that kids don’t even NEED algebraic ability. I think it’s a slightly different aspect than the crutch point…because here there’s actual acceptance and saying, kids don’t need something.]
P.s. I’m OK with protractors and straight edges though.