Just got a card from a student who has been having a rough year. She was having a great deal of trouble until I started showing her how to really USE a calculator. (It's algebra and geometry not arithmetic, and she knows the algorithms just makes SO MANY little mistakes)
Since then she's really taken off. I find very few people who could benefit from calculators know how to use them effectively, and there is a lot of snootiness and stigma in the way of this happening.
I generally agree that up to around grade 7 there isn't much use for calculators. But from there out if you understand place value and can estimate being consistent and accurate doing long hand calculations is kind of overrated-- I'm always a little shocked that once I tell the grade 9 students they can use calculators they somehow keep making arithmetic errors... (HOW) not this student, though!
You have planted a seed. You have increased the chances of her liking math. And if she likes math, she increases the number of career paths she can take.
It is one of things I sort of get bummed out about is that schools (in general) do not teach how to use tools. Tools that are widely used in industry.
I've heard teacher complain "my students put things like 6*8 in the calculator"
I always think "so?" I mean if they know it's around 50 and are worried they've mixed it up with some other fact why does it matter?
You are teaching calculus who cares?
(It's 4 ... 12s which is helpful if you know the 12s but if you didn't learn them young... anyway it's kind of a distraction from the problem at hand. Get the correct result. Use the tool. )
@futurebird@snacktraces
Yes! Normalize redundancy, verification, and checking external sources, instead of encouraging the arrogant practice of refusing help to demonstrate knowledge! It’s a fragile ego that refuses “measure twice, cut once.”
Recently my son had CS coursework that involved converting a bowl-of-spaghetti logic diagram into a C++ program. Me, with 40 years of C experience, independently wrote my version of the program, he wrote his in C++. I asked him to send me his output and I sent him mine.
They didn't match, and sure enough I had messed up an AND gate.
@stacey_campbell@futurebird@snacktraces Joe Groff (of the Swift compiler team at Apple, wow do I miss him on here) once remarked that unit tests and static types have in common that they are a form of •systematized redundancy•: both ask engineers to express their understanding / assumption / intent in two forms, and then a second actor (the machine in both those cases) checks that the two forms agree. I love that way of seeing it.
@inthehands That nicely generalises an observation I once made when switching backend dev from Python to Swift: a lot of my usual Python unit tests fell away because they were essentially type checks in data processing pipelines.
@finestructure Yes. I’ve noticed again and again, in multiple directions, that developers dislike unfamiliar p-langs / tools because they bring all their old assumptions and habits with them. “How can I do [practice I’m accustomed to] with [tool designed around different practice]?!? This sucks!!”
Tools have ecosystems, which encompasses both tech things and human things. They only make sense in their natural habitat.
@futurebird
And bonus points (no one keeping score, but you deserve them anyway): teaching her to look at the answers and see if they make sense. This is priceless. Too many times I have let the calculator or program answer for me. Knowing how to "sense" if that answer is right, that is extra special.
@snacktraces@futurebird Yes, the ability to see whether a result is reasonable is important (and it is whether or not you use a calculator or computer to do the actual math)! Take that 68. Well, 6 is close to 5, and 8 is close to 10, and 510 easily comes out as 50; therefore, 68 should be close to 50. The calculator says 528. Something must be wrong; try again! (That's 688, btw. Easy mistake. 🙂)
On the flip side, I've encountered grocery store cashiers who couldn't do simple subtraction.
@futurebird I often think there are assumptions that people know how to use calculators too. So easy to make mistakes if you don't use brackets nor the calculator's memory, or have it on the wrong setting.
I give my students difficult quadratics with rational roots to solve with the calculator. I give them two dozen and expect them to all be correct (no decimal answers)
I also teach them to write programs that will list the factors of any integer (not prime factors, just a list of every valid divisor) and have them program the quadratic formula.
Using those together they can find the roots and factors.
It's not possible to avoid the calculator, nor will it do all the work.
I should really make them do a prime factors program too, but the school limits how much programming I can sneak into math class and they need to be more comfortable with while loops and such for that...
The other programs at least get them to notice that the calculator is pretty powerful and can make life easier. Which I feel like some math teachers hide.
@futurebird I'm always surprised when I learn about the things my calculator can do. But most of the time I'm barely scratching the surface of what it's capable of.
It will probably add fractions for you and reduce them.
It will try to make decimals into fractions for you too, but most calculators are limited ... that's part of the point of my "terrible 24" exercise. With some of the roots you can put them in the quadratic formula and guess or convert them to improper fractions, but with others, you need to use factoring by grouping to get the (ax+b)(cx+d) form and name the roots as -b/a -d/c
@futurebird
I used to love math including algebra until we got to calculus, and specifically integrating functions. That we had reached a part of math that was essentially trial and error made no sense to me, and I hated it.
@DenOfEarth@futurebird I love the infinite loop between "phone calls to mathematicians" and "oh no". It's like you're in a pit of despair but at least you got company. 😂
@DenOfEarth@futurebird I find algebra and some of the early calculus useful. Modeling and higher-level integration were… interesting, to borrow my professor’s ambiguous use where interesting covered everything from “that’s useful/cool” to “oh, shit, no” and “no one can solve this”.
@futurebird In my Ed Psych class that all future educators had to take, I was literally told that knowledge of subject matter was completely irrelevant to effective teaching; I refused to believe it, because elementary and middle school teachers can encourage their students to explore/experiment/discover mathematical meaning if the teachers themselves understand connections between the physical world and the mathematical objects that we use to make sense of reality around us
Subject mastery gives pedagogical freedom, doesn't it. The ineffective teacher obsesses with meeting the syllabus, precisely because they fail to fully understand the subject matter in its foundations. They use didactics to hide the superficiality of their content. The effective teacher is free to focus on student learning, precisely because they are able to provide natural links to the subject matter, in spontaneous response to the student's trials and mistakes.
@the_roamer@maryErudis@futurebird At school I had a maths teacher who was clearly reading two chapters ahead of the book, and one who knew this stuff inside-out and had trouble thinking down to our level. The latter was much more fun, and I remember what he taught better even 40-odd years later.
I sometimes help with the grade 6 students and (maybe a little arrogantly) thought that my math background was overkill for grade 6. I've taught graduate level courses.
But the students discovered they could ask me anything about math no matter how crazy the question. We talked a lot about infinity, and I was able to talk to them about series convergence (can infinity many things at up to a finite thing?) in a way I WISH someone had done for me at that age.
It's good for teachers to have deep knowledge of the subject young people have such flexible minds and they want to know more.
I don't think their teachers are always ready to provide that since we don't invest in continued education for teachers and think that kids don't need to know very much...
Teachers should always be learning more about the subjects they teach. After all that's what I'm really trying to show my students: how to learn. I need to model how to be a person who never stops learning new math.
Math only gets better and better the more that you know about it.
And when I say teachers should be learning I mean new material not just education oriented stuff. All grade 6 math teachers should study a little higher math as a treat.
Another reason for continuing ed is that the subjects are moving targets, particularly in science.
One might not think it matters in grade school, but I learned stuff then that has been obsoleted. And more often, simply not taught things that are fundamental now.
Whole disciplines have been invented since 1980: molecular biology, extrasolar planets, AI, computer vision, and even object oriented programming. All in their infancy at best, then.
Yes, there are different stances for being a teacher. When I started teaching I was learning with them. (You never know how little you know something until you try to teach it.) This is how I operate in engineering and electronics.
In math I get to be more of a playful sage kind of teacher. Every topic reminds me of so many other topics, deeper experience makes the material like old friends who can still surprise me.
@notsoloud@the_roamer@maryErudis@futurebird
The issue mostly seems to be elementary teachers who don't feel comfortable with math and are reluctant to explore outside the prescribed curriculum, afraid to make mistakes.
I'm horrified when k-7 teachers say "I hate math it's impossible" or "calculus never made any sense to me"
It's bad that you had a bad experience with math... but we gotta DO SOMETHING about those feelings. Even if you never say it those ideas are transmitted to students.
I'd love to teach a calculus or linear algebra course for k-7 teachers ... provided they were being compensated for the time sink. I think it would make a big difference.
@moirearty@futurebird@dendari@notsoloud@the_roamer@maryErudis Michelle Manes (who taught middle school before going into academia and who by all accounts is an excellent teacher) used How Not to Be Wrong by Jordan Ellenberg when teaching a math course geared toward future math teachers; her students, upon seeing it was priced like a normal book and not like a corporate-published college textbook, thanked her profusely.
@llewelly@moirearty@futurebird@dendari@notsoloud@the_roamer@bstacey “Calculus Made Easy” is a classic, not sure who the intended audience was. The mistake people make is in believing that they require a lot of remedial math work before trying to dive into something like Calculus. The book I’m writing is designed for small group study, w/ a facilitator/SME guiding conversations
Recommending a math book for self study is a tough one. It depends on where the person is starting and how they like to learn. There are what my mom calls "beach math books" ... because you can read them on the beach and don't need to have a notebook and memorize definitions and such to read them. This could be great for many teachers (The book "e the history of a number" comes to mind also "embodied mathematics")
But for others a more math-y interactive book might be better.
@futurebird@moirearty It sounds like there is room to develop an accessible book/app for math self-study.
Some years ago, I decided to apply for a grad program in secondary education with a deadline about 8 weeks away. This being many years after getting my undergrad degree, I had to study and take the GRE and the Praxis II in math. So I reviewed calculus, linear algebra, geometry, statistics. I found the Head First books engaging and helpful.
@hedwyg@futurebird@moirearty brilliant dot org has a lot of good math classes for app based self study :)
(disclaimer: my spouse worked there and helped build a lot of their tools)
I'm generally impressed with brilliant's content. I wonder what their content creation tools are like... are they easy to use? Could they ever share them?
I've always dreamed about a hypercard-like math educational software creation tool-- I make many little interactive games for my students but it is labor intensive and I wish I could do more.
@futurebird@chrisamaphone@hedwyg@moirearty I feel that maths Wikipedia has lost this a bit; I'm not a mathematical incompetent but usually when I look something up it's [white noise wall of maths] even for something I understand, perhaps because they're so keep to get in the really fun high-level stuff they don't really cater for anyone else.
@futurebird@dendari@maryErudis@the_roamer@notsoloud there are NSF programs called “research experiences for teachers” (RETs) which are this but for research. i feel like some remedial “stop being afraid of math” education could be integrated into one.. though it would be better if there were a dedicated program probably.
Imagine hearing a teacher say "I hate reading books are impossible." or "Who can understand poetry it makes no sense." etc.
We expect English and history teachers to keep reading new English and History books, to write and explore their subjects, but somehow it's OK not to do that with grade school math?
@futurebird oh so much this. My relationship with mathematics went from “I’m bored with all this manual arithmetic grinding” in primary school (k-6) to “OMG I love it” in secondary school (>=7) because suddenly we were taught to use calculators and were doing algebra, geometry, mechanics, stats woohoo!
@futurebird I've actually been reconsidering trying to learn math, despite having dyscalculia, outside of formal ed because there's a lot of stuff I'd like to learn, but I never, ever want to learn with stakes or to make sure I pass a test again
and being able to use memory aids, to go back and read how to do stuff over again, to not have to remember everything in my head and calculators is one reason why I'm even considering
@mybarkingdogs@futurebird There are several things you need to make math fun. A good calculator helps. You can get an excellent Casio for a bit over $20. Don’t get a graphing one. Next learn to use a computer graphing app. My current favorite is desmos.com. Next is learning to use a symbolic calculation app. A really good one is symbolab.com. These are both online. The next is good math books. You can get used textbooks that are inexpensive since you don’t need the most recent editions. A really nice series is by Elayn Martin-Gay. Have fun.
Desmos is great and I agree if you are talking about independent study for fun... but for students who are allowed to use graphing calculators on exams? Get the best one you can and learn to use all the graphing functions. Graph everything.
And if you can get a graphing calculator for cheap? go for it.
They aren't worth the prices, but they are really fun and powerful tools.
@futurebird@merrillholt Thanks for the advice, both of you :) It's not a priority (current hobby financial priority for me is getting my Ableton account active again - it costs $ - or learning Reaper -it's free - and buying a new keyboard with the $ I'd save - so I can create music again, then getting a macro lens for my DSLR so I can shoot closeups since I love my flower and bug photos etc) but this will be helpful for when/if I get the chance
@futurebird Because I'm interested in old calculating machines I've been reading up a lot on slide rules, and while they fascinate me, there's this aggressively nostalgic "calculators rot your brain" angle to the whole fandom that I'm really skeptical of.
The main reason being that I was a kid who HATED math in elementary school because I hated manual arithmetic. But it was the exact moment when fairly powerful affordable pocket calculators were first hitting the market, and my dad got me one.
I was fascinated by it and started pondering questions like "why do some of these mysterious function buttons settle down on a fixed value if I hit them over and over, but others blow up until the calculator shows an error? Why do some fractions have interesting digit patterns in decimal?" Which in hindsight were real mathematics questions.
Yes, kids will sometimes punch buttons on their calculator and trust whatever nonsense they get, especially if they don't care about the assignment. But anything that works at making you care is good.
@projektionsyta@futurebird And inevitably, physical calculators are now the focus of the same kind of nostalgia since for so many people they've been superseded by smartphone apps or web apps (standalone scientific calculators mostly still exist because kids are made to use them in school).
I was just looking into the whole alternate world of ten-key desk calculators with adding-machine logic, which are still a product with a market in accounting. It's like a whole other universe of computing devices.
And, yes, it wasn't hard to find older accountants grumbling that the kids these days just rely on Excel for everything or whip out their phones instead of using a proper ten-key printing calculator, and it's so unprofessional--how does that look when you're closing Facebook and hunting for the calculator app with a client in the room? You want to be able to punch up the numbers on your ten-key and list them on that paper roll, like a real accountant...
@mattmcirvin@projektionsyta@futurebird Well, you use Excel for anything involving floating-point, I know you don't care about correct results. So many ways that can go wrong.
@RogerBW@mattmcirvin
I've taken that complaint for conventional use with a pinch of salt ever since I saw someone complain that Excel is wrong, show how to do it "properly" in Python, then after quietly showing that their imported package did the exact same thing, told everyone to just use the round() function to turn 10^-15 into 0, and ta da that now proved it was correct.
@Rhodium103@mattmcirvin Scratch a little deeper and I'll just excoriate floating point anywhere that precision is important, but Excel has had some big public failures in the last few years.
@futurebird Do you have any tips/suggestions about how to teach good calculator skills? It's something I see my students struggle with, but I often find it difficult to explain what they should be doing differently.
You need to make exercises, and problems with rational or integer answers that have numbers so large that you need to use the calculator.
I think we send mixed messages too often "use it if you need it" can make some kids avoid it as if it's a crutch. Put the whole class in a position where they have to use it to have any hope of getting the problem done.
At the same time you have to help them understand that decimals can be imprecise. Otherwise they stack rounding errors.
@futurebird@ptrsinclair I don't know if this is in any way helpful, or if I'm missing the point, but I remember teachers taking the time to teach us how to estimate the answer before picking up a calculator.
Stuff like rounding off all numbers to a single significant digit, and solving mentally or on paper, so that when you used the calculator to solve for the actual values, you would instantly catch any errors you'd made by punching the wrong button or skipping a digit (easy, on the terrible keyboards some calculators had). So, for example, if the estimated answer was 45,000,000 but your calculator said 12, it would be obvious that you'd made a mistake somewhere, and could try again.
@futurebird@ptrsinclair
it took me all night to remmeber how to do these, and then when I finally remembered, I did them at the painfully slow rate of about 1 every 5 minutes. 15-yr-old me would be ashamed. No pictures, bc I did them on post-its, so the work is all crammed together in the most haphazard way. Also I don't own a camera.
@futurebird my high school algebra teacher used to give us credit for most of an answer if we did it right but made arithmetic errors. "You'll have a machine at your desk to do the arithmetic," she said. "The important thing is knowing how to set up the problem." This was before calculators were a thing much less all the computers we have now. I often marvel at her foresight, not to mention her teaching.
@futurebird Making so many little mistakes exactly describes my 13yr old daughter. Can you give a few more hints about how you showed her how to really use a calculator? (I love my HP 50g in RPN mode, but am not a good teacher at all).
If she is always allowed to use the calculator she should always have it out and ready when doing homework and studying. Like any tool it takes time to adapt. It's important it's the exact same model calculator she can use on tests and in school. (ideally the same calculator)
From here out I'd need to know more about the math she's doing and the model of the calculator.
@futurebird Thank you very much for your reply! She has a Casio fx-991CEX, although so far she hasn't done more than using it as a 4 function calculator and starting to use the memory. They've just done pythagoras and are currently learning how to multiply & add powers and roots. They also have to be able to use tables of powers and roots (rounding to 3 digits).
Lastly some of my students didn't know "shift" would get at all the little yellow functions printed above the keys, alpha too. However, you DO NOT press shift or alpha at the same time, press shift first then "ln" and you'll get e^x for example.
For kids raised on computers that's counterintuitive.
Beyond that, it's about practice and not being shy about using the calculator to do simple things. It's there keep your mind placid and untroubled. If you want to put 3*8 in do it.
That all might seem very basic, but most of my students had no idea their fancy graphing calculators had all of these functions, especially the arrows. Also important:
Parenthesis: These allow you to control order of operations.
X^n button: (above sin) this will raise whatever you just entered to an arbitrary power. Can be used to do roots too, eg 32^(1/5) is 2 because that's the 5th root.
@futurebird “you won’t always have a calculator with you” lol guess what year 8 maths teacher?
Related, my year 12 physics teacher gave us all the formulas that we needed for the test. Just the formulas, not which one was for what, so that if you mostly knew which formula did what you could look it up and get it exactly right.
Add comment