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RE: Possible death of calculators in education? - Vtile - 05-21-2017 11:45 PM
While I'm mr.average engineer and former average student. I can still understand the no CAS for that level of education. In the end, it is level where all the base of mathematical thinking is laid out. When you have CAS and use it for everything you outsource the very basics of mathematical thinking, since most if not all mathematics are eventually just applying the basic arithmetic’s and geometry. If you do not have those in your intuitive understanding (muscle memory/low level of consciousness) then you lack the ability to follow what more abstract mathematical construction (I purposely use this term) will work.
I sometimes find some basic things that I suddenly see in a whole new light, the last such thing is/were the fraction (seen on deltadays (JDN)) while I can mechanically use them they still did hide some finesses that I never thought of or/and teachers failed to point out (or I just slept on those classes). Another interesting personal finding were the realization of sin and cos as a complementary (which translated to my own language do not say anything if you are not familiar with Latin or some other language that contains lots of butchered Latin ie. English.) again while mechanically proficient, only after I stumbled to old Sin-Cos tables in about 80 years book it did strike to me what it means (well of course there had been much use of them between first learning and getting the stone out of my shoe so to speak. A small RPL script to convert between Cosϕ and Sinϕ did play a small role of reviving the interest of the subject). That is the calculators and computers gives nice opportunities of getting deeper understanding, but I can see how it have also hurt my understanding of the basics, at high school with algebraic calculator with its plug the problem in and solve didn’t necessary give insights of how the mathematical structures work (as an example such as operator sine).
The study books are often also pretty poor, unfortunately, they lack of the bigger scope, lack of reasoning (not proofs) why and where this particular tool is useful and the retrospection of the core basics needed in that mathematical tool. I can remember from elementary school or early high school the question some folks did ask, "I don’t need this or that while baying a milk from market", so why we must spend time learning this or that.
I also think that the importance of the basics will be on rise with ever growing digitalization and computerization. If you do not understand the basics and see how the mathematics work, then you never can program anything or need just blindly trust what some program tells you (if it can). in UoAS where I had opportunity to be a math student of one almost retired a bit old fashioned (you learn these or I make sure you do not graduate) PhD. I vividly can remember early trigonometry course where calculators were allowed (all makes and models, but the review were based on the capabilities of used calculator model), but exam were filled with simple plotting questions which were formed in a way that calculators overflowed and weren’t useful at all. Lesson learned, calculators aren’t always useful.
Everything in programming eventually returns to four basic operations addition, multiplication, division and subtraction. The computer and calculator is good slave, but a terrible master. That is something that is too often not put out in those public conversations I have seen in this subject (in regular media mostly). In that sense, I underline In my mind insightful and philosophical thought of that "Only those who can calculate shall use calculator", which is something similar I can recall from the early days of my school path. If the basics are outsourced in early on and as a wide the future generation lose big part of their freedom as form of being reliant in even basics for what a few CAS etc. provides. It is transition from creators and designers to blind users.
(Ps. Sorry of nonclarity. My do it all phone only shows two lines and one line must be written blindly :-\ PPS. Now edited with proper machine)
(05-21-2017 06:24 PM)Maximilian Hohmann Wrote: Hello
(05-21-2017 01:24 PM)SlideRule Wrote: This seems very normal to me.
To me as well. I am not a mathematician. For me mathematics is a kit of basic - quite sophisticated though - tools required to get other jobs done. A carpenter does not need to know how to solve Maxwell's equations in order to use an electrically powered circular saw (whose motor can be described with these equations). As long as I know which tools are best for which task I don't need to know how these tools are built.
Recently I talked to a former colleague of my father and friend of our family. Initially a mathematics teacher and later mathematician in the nuclear research environment. He has come to the conclusion that we should not torture our children/grandchildren at school with mathematics (the most hated subject for most of them) the way we do now. Teach them what is needed for life - fractions, percentages, some statistics - and leave the complicated stuff for the 2% who want to be physicists, mathematicians or other scientists which rely on an in-depth understanding of mathematics. Teach all the others languages or sociology instead if we want to have a future.
My son has just now been sitting his last exams (called Abitur in Germany) to finish secondary school. He was part of an experiment - which will end with his class - where students were using a CAS calculator (Ti nSpire in his case) throughout secondary education. The idea was to raise the overall level of mathematics education by leaving the routine tasks to the machine and letting the students do more abstract work. From what I see this experiment has been declared a failure and they will return to "traditional" mathematics teaching with pen an paper. So to say "the death of calculators in education" as the thread title goes, but not in terms of hardware.
Regards
Max
RE: Possible death of calculators in education? - Dan - 05-22-2017 06:21 AM
Interesting discussion. As a teacher of mathematics for the last 10 years, my 2 cents worth:
Memorization of times tables an absolute must. As someone mentioned, expecting students to fill in the gaps in their knowledge with a calculator does not work. e.g. 99,000/4 student says 396,000 and doesn't bat an eyelid (pressed * instead of /). Or (6+4)/2 = 8 (enters it left to right with no consideration of order of operations). You are laughing but I see this ALL THE TIME.
A 12-year old I am tutoring took over 10 minutes to do his timetables when I started with him. (used a HP50g program to time him). A girl around his age was doing it in under 3 minutes, I've got him now doing it in under 4 minutes. We are doing fractions now and if his timetables skills were still bad if would be impossible to teach. 3/4 + 1/3, "well you find the lcd and find equivalent fractions. How many times does 4 go into 12?" I've had students stare at me blankly. "O.K. use you calculator". 3! "O.K. multiply the numerator by 3" (goes for calculator again). "Now do the same for 1/3". The student keeps going for their calculator, gets lost in the steps, asks "hang-on what are we trying to do?" It's ridiculous. When we get to algebra they are completely lost.
People thought CAS was the answer, focus on problem solving instead of tedious algebraic manipulations but it doesn't work like that.
RE: Possible death of calculators in education? - pier4r - 05-22-2017 05:42 PM
interesting input Dan, and it makes sense to have an idea of the results if one mistypes.
RE: Possible death of calculators in education? - c785 - 05-22-2017 07:06 PM
Thanks for your valuable input, Dan, it has reminded me of a lot of situations I encountered in the past.
This is also why I mention slide rules so often: even if it gives you the mantissa of a product of two numbers reliably, you still have to think to work out the exponent (order of magnitude), and even that simple task will make you THINK and be much less likely to make errors.
Regarding memorization: I used to say that the difference between a physicist and an engineer is that the former knows one Ohm's law, the latter knows three.
RE: Possible death of calculators in education? - SlideRule - 05-23-2017 12:14 AM
(05-22-2017 07:06 PM)c785 Wrote: ... a physicist and an engineer ...
Shame, shame, shame; everyone knows the standard expression is a mathematician, a physicist and an engineer ... (if you experience uncertainty, GOOGLE™ words in italics).
BEST!
SlideRule
RE: Possible death of calculators in education? - JimP - 05-23-2017 12:57 AM
(05-22-2017 06:21 AM)Dan Wrote: Interesting discussion. As a teacher of mathematics for the last 10 years, my 2 cents worth:
Memorization of times tables an absolute must. As someone mentioned, expecting students to fill in the gaps in their knowledge with a calculator does not work. e.g. 99,000/4 student says 396,000 and doesn't bat an eyelid (pressed * instead of /). Or (6+4)/2 = 8 (enters it left to right with no consideration of order of operations). You are laughing but I see this ALL THE TIME.
A 12-year old I am tutoring took over 10 minutes to do his timetables when I started with him. (used a HP50g program to time him). A girl around his age was doing it in under 3 minutes, I've got him now doing it in under 4 minutes. We are doing fractions now and if his timetables skills were still bad if would be impossible to teach. 3/4 + 1/3, "well you find the lcd and find equivalent fractions. How many times does 4 go into 12?" I've had students stare at me blankly. "O.K. use you calculator". 3! "O.K. multiply the numerator by 3" (goes for calculator again). "Now do the same for 1/3". The student keeps going for their calculator, gets lost in the steps, asks "hang-on what are we trying to do?" It's ridiculous. When we get to algebra they are completely lost.
People thought CAS was the answer, focus on problem solving instead of tedious algebraic manipulations but it doesn't work like that.
Absolutely spot on. This illustrates not only the need to understand what one is doing before being given tools with which to make the task easier (ie prove you can do it without a calculator, then you can have the calculator), but also a parallel in the world of English composition. Spell checkers are for people who know how to spell, not for those who don't. One doesn't want to be lead [sic] down the wrong path!
RE: Possible death of calculators in education? - AlexFekken - 05-23-2017 02:07 AM
And of course that means they need to understand rounding too.
For example, rather than expecting rounded percentages to add up to 100, they should suspect fraud when they see that happen too often.
What an educated world that would be...
RE: Possible death of calculators in education? - Dan - 05-23-2017 04:39 AM
(05-22-2017 07:06 PM)c785 Wrote: This is also why I mention slide rules so often: even if it gives you the mantissa of a product of two numbers reliably, you still have to think to work out the exponent (order of magnitude), and even that simple task will make you THINK and be much less likely to make errors.
Love it. Teachers who used them in school have said the same thing, gives you a feel for numbers. I bought a "Sun Hemmi" slide rule a few years ago. Ingenious devices and exquisite craftsmanship.
(05-23-2017 12:57 AM)JimP Wrote: Spell checkers are for people who know how to spell, not for those who don't. One doesn't want to be lead [sic] down the wrong path!
A colleague gave a similar reply to an English teacher who thought students don't need to know arithmetic, calculators do it all for you. She didn't know what to say.
RE: Possible death of calculators in education? - EugeneNine - 05-23-2017 11:23 AM
(05-23-2017 04:39 AM)Dan Wrote:(05-22-2017 07:06 PM)c785 Wrote: This is also why I mention slide rules so often: even if it gives you the mantissa of a product of two numbers reliably, you still have to think to work out the exponent (order of magnitude), and even that simple task will make you THINK and be much less likely to make errors.
Love it. Teachers who used them in school have said the same thing, gives you a feel for numbers. I bought a "Sun Hemmi" slide rule a few years ago. Ingenious devices and exquisite craftsmanship.
(05-23-2017 12:57 AM)JimP Wrote: Spell checkers are for people who know how to spell, not for those who don't. One doesn't want to be lead [sic] down the wrong path!
A colleague gave a similar reply to an English teacher who thought students don't need to know arithmetic, calculators do it all for you. She didn't know what to say.
Years ago the spell checker helped me learn to correctly spell a lot of words, however the modern version of spell check as you type tends to derail the train of thought and when I stop to make corrections I forget what the rest of the sentence was about. So there is good and bad to it.
Similar, the HP solver on my 48SX taught me a lot of math that the school refused to.