This is a sample question for middle school math, published by the International Baccalaureate (IB) program. I found it here.
Here is a graph I made of this equation, using Desmos:
Looking at this graph, you see that after about 10 minutes, there are 11 cars per minute going through the intersection – and that’s the most cars. After about 25 minutes, there are zero cars going through the intersection, and after that, there is a negative number of cars (!!!).
I don’t think this equation models anything having to do with any intersection I’ve ever visited. Instead, I think that any intersection controlled by a traffic light is going to be more periodic, that is to say, something like some mix of sine or cosine functions — obviously not middle school material.
Despite the populist hype of billionaire Sci-Fi fanboys and a perpetual stream of Hollywood entertainments to the contrary, humans will never explore the galaxy in person. In fact, we won’t even explore our own solar system up close and personal. This is not merely because robotic missions can do the job 1,000% better for 1/1000th the cost. It’s because of two fundamental biological reasons.
The first is gravity. Everything about our bodies is evolved to function under a gravitational acceleration at sea level of approximately 9.8 meters per second squared (9.8m²). Our hearts pump blood up to our heads, fighting gravity every centimeter of the way. Our muscles and bones are as strong as they are because every part of our bodies is fighting gravity every moment of our lives. Our sense of balance, which orients us spatially, depends on gravity being constant in one direction only: straight down.
Without gravity, very bad things happen: the heart pumps too much blood to the head and too little to the lower extremities, leading to ocular distortions, crushing headaches, and nausea as the inner ear loses all sense of up and down. Our bones and muscles atrophy dramatically, even when hours each day are dedicated to exercises specifically designed with the intention of slowing down this decay. Put simply, our bodies are incapable of handling microgravity and despite the pictures of smiling astronauts merrily enjoying microgravity on the ISS, the harsh reality is that every single one of those astronauts pays a price very few of us would wish to incur.
The Sci-Fi fanboy response to this fundamental problem is either (a) to ignore it entirely, as per Musk and Bezos, or (b) claim that artificial gravity is the answer.
As Musk and Bezos are ignoring the problem we can likewise ignore them. So what about artificial gravity?
There are only two ways to create artificial gravity. The first is called “constant-g” which means that we accelerate our hypothetical space ship at a constant 9.8m² for the first half of the trip and then flip it around and decelerate it at a constant 9.8m² for the second half of the trip. Einstein’s insight that over areas too small to experience tidal effects such acceleration would be indistinguishable from regular gravity means that in theory Earth-style gravity could be induced in such a manner. Better yet, because the acceleration is constant, relativistic speeds will eventually be attained. In just 12 years (in the reference frame of the spacecraft) we could travel across our Milky Way galaxy. In a single human lifetime (in the reference frame of the spacecraft), under constant acceleration, we could reach the edge of the universe that’s observable from Earth. An Earth upon which, in that frame of reference, billions of years would have passed.
So with constant acceleration we get a “twofer.” Earth-identical gravity and the ability to traverse vast distances within a human lifetime. Problem solved!
Except that there is no way, theoretical or otherwise, to achieve constant acceleration of this magnitude. No propulsion mechanism, theoretical or otherwise, can overcome the problem of mass. In order to power the continual acceleration, our imaginary space ship is constrained by Newton’s observation that any action in a vacuum requires an equal and opposite reaction. In other words, to accelerate a mass of X by some amount of velocity we will need to discharge an equivalent amount of energy in the opposite direction. And that energy can only come from fuel. Which adds to the mass of our space ship. So now we need to expend more energy, which means we need more fuel, which means we’re now carrying even more mass, which means we need to expend even more energy, which means…
In other words, even with some imaginary technology that could convert matter into energy with 100% efficiency, there’s simply no way to get to 9.8m² constant acceleration for any meaningful amount of time. Sure, we can talk about things like an Alcubierre drive but then we’re just as entitled to say that Hogwarts will invent the Spaciamus drive to solve our problem instead. In other words, running off to hide inside imaginary “solutions” is no solution at all.
If constant acceleration can’t provide artificial gravity, what about centrifugal force? We all remember the rotating space station in 2001 A Space Odyssey and everyone knows that this was the only Sci-Fi movie ever to have utilized a science-based series of technologies. Plus, it’s easy to find on the Internet lots of schemes to create artificial gravity in this way, from tethering ships together and spinning them around a central axis to building enormous hollow rotating cylinders on the inside of which humans will experience Earth-like gravity. So, problem solved!
Except the movies and the Sci-Fi books mislead us, as is the way of popular entertainments.
First, the good news: if a person stood perfectly still and did not move in any way whatsoever, then centrifugal force could seem to mimic Earth-style gravity. Unfortunately, here’s the bad news: if they made any movement whatsoever, they would instantly be overcome by nausea and be disoriented.
Why is this? Imagine throwing a ball up into the air here on Earth. If you throw it straight up, it will come straight down, pulled by gravity toward the center of the Earth we’re standing on. But under conditions of “gravity” induced by centrifugal force, a ball thrown straight up will arc and fall away from the person who threw it because unlike here on Earth there’s a second force acting on the ball: centripetal force. As our inner ear orients us by means of reference to the constant downward force of gravity, this means that any movement at all — even something as minor as turning one’s head — would result in signals from the inner ear (responding to the centripetal force) jarring dramatically with the signals from our eyes. At best this would lead to our hypothetical human vomiting in a majestic arc; at worst it could render them incapable of any controlled movement whatsoever.
The diagrams below show the difference between gravity (or constant acceleration at 9.8m²) and a rotating object. On Earth there’s only one force acting on us: gravity. On our imaginary rotating artificial gravity environment there are two forces: centrifugal, and centripetal. And that makes all the difference in the world.
Perhaps this is why Bezos prefers to ignore the problem; it can’t be solved just by throwing money at it. As for Musk, he makes people with ADHD look like paragons of sustained concentration so he probably doesn’t even know the problem exists. But even if you don’t know a brick wall exists, it still kills you if you slam into it at 1,000 kilometers per hour.
Gravity, therefore, is one reason why human beings will never be a space-faring species. It’s also the reason why it’s highly unlikely any other species capable of developing suitable technologies would ever become space-faring either. All organisms are highly adapted to the environments in which they evolve and it is extremely difficult to sustain organisms outside of their natural environments for any significant period of time. Add it the problems of solar radiation, the deleterious effects of microgravity, and everything else associated with space travel and it’s apparent that Sci-Fi fanboy dreams are a very poor guide to the future.
There is a second major reason why we humans will never be a space-faring species: psychology.
Our brains are as much the result of selection pressures as our bodies. Like our bodies, our brains are highly adapted to life on Earth. As a primate group species adapted to foraging, we’re not well-suited to being cooped up in tiny cages. We become obese and we develop all manner of mental problems. Without access to natural cues like water and grass and trees, we become stressed. When forced to interact with the same small group of people for years without respite, we become irrational and angry, or conversely withdrawn and depressed. Worse still, our emotional hardwiring makes us competitive even when cooperation is the optimal strategy, and our intellectual limitations lead us to acquiring and then strongly defending irrational and harmful beliefs.
Imagine, therefore, a space ship upon which 200 hapless humans attempt to exist for years or even decades. Instead of looking to Star Trek as our inspiration, a more probable vision is depicted in One Flew Over The Cuckoo’s Nest or perhaps the concluding episodes of some trash reality TV show.
It is difficult to imagine any species capable of making spacecraft not having equivalent psychological limitations, albeit likely somewhat different from those that control our own behaviors.
There are many other reasons why humans will never spread across the galaxy, but these two should suffice to prove the contention. This does not mean, however, that there won’t be money to be made in enabling space tourism. A few days in microgravity, ensconced in a modestly comfortable environment with a small number of others, could be a very congenial way for the wealthy to break up the monotony of holidaying in the Hamptons or on a private island in the Bahamas. Sheltered in low orbit by the Earth’s magnetic field, the dangers of solar radiation are reduced to a perfectly acceptable level and likely no worse than a dozen trips in a private jet. Microgravity sex will no doubt become this century’s equivalent of the Mile High Club that was so popular among the early jet-setters of the 1960s and 1970s.
But beyond a few amusing days spent orbiting the Earth while watching one’s champagne bubble around one’s head, and after the inevitable disaster of Mars Colony One, we will accept the fact that robotic missions are the real future. And then we will expand our knowledge of the universe exponentially instead of wasting hundreds of billions of dollars on futile dead-end fanboy dreams.
A few days ago, we silvered an 8” diameter 43” FL mirror that had previously been aluminized, and applied the Angel Guard coating.
We did a Ronchi test and some Foucault-Couder knife edge tests before stripping the aluminum and after the silver was applied.
To my amazement, we found that the mirror’s figure was about the same in both cases. How that works, especially how the Angel Guard coating is laid down so even and smooth over the entire mirror, is beyond me. But it DOES work.
This is a video of us washing off the Angel Guard coating.
Here is a video of the finished mirror after drying. Notice that the very edge of this mirror did not take the silver coating, but the area uncoated is probably on the order of one or two percent of the total area.
Ive been doing the aluminization process for telescope mirrors at the NCA ATM workshop with a 55-year-old military surplus aluminizer at a DC rec center for about 20 years. (I’ve had a lot of help!!) This involves high vacuum, a noisy pump, voltage both very high and very low, and quite a lot of time.
Today, I had the opportunity to silver a random piece of glass, in my driveway, with the aid of another longtime ATMer and some chemicals from Angel Gilding. I had seen this demonstrated at Stellafane by Howard Banich and Peter Pekurar in 2019.
Doing it myself was quite eye-opening.
Almost finished, except for the spray-on Angel-Guard, which we didn’t have
Here’s what I wrote on FB:
Success with our first attempt at silvering a piece of glass under a tent canopy, and then stripping off the silver quickly and easily with PCB etchant (FeCl?).
I’ve aluminized many mirrors with the NCA’s vacuum chamber, using a modified version of John Strong’s method from the 1930s.
I must admit that this method was faster, easier, quieter, and much more low-tech, compared to depositing aluminum. In the latter case, sometimes you have to wait an hour or more for the dual-stage vacuum chamber (the primary pump is VERY noisy!!) to finish all the preparatory steps and pump down low enough that a hot atom of gaseous aluminum can travel two or three feet before striking any other remaining air molecule! (That is one hell of a vacuum!)
With the silvering process, you can do any size mirror you can fit on your cleaning jig — and you can make it out of pieces of scrap wood, a few nylon chair legs, two old hinges, and some screws! !
With our NCA-ATM-CCCC aluminizer, we are limited to 12.5” max. I’m currently working on a 16.5” thin Pyrex mirror; the price I’m quoted for aluminizing it is about $600 at Majestic Coatings, which is about three times what I paid for the blank!! And that doesn’t even include shipping!
Today Alan T and I tested the silvering process in my driveway using the screen tent canopy that we use at the ATM workshop to stop dust particles from landing on mirrors that are being polished. (After getting permission to go retrieve the canopy from the Covid-closed rec center, we immediately went into the parking lot to hose off a decade of dust!!)
We unfortunately do not have Angel-Guard overcoating on hand. It should arrive Wednesday. As most folks know, bare silver, unlike bare aluminum, tarnishes very quickly (in weeks or months) when exposed to ordinary air, whereas a I have seen many bare Aluminum layers last a decade. This overcoating is said to extend the life of the coating to about a year, but obviously conditions will vary.
We used a 6″ float glass mirror blank to try out the process today – not an actual, parabolized mirror.
How does it work?
This is basically a five step process:
1. Get prepared and mix the tinning solution afresh;
2. Clean off the mirror properly with precipitated CaCO3 and/or Alconox; rinse;
3. Sensitize the mirror with an invisible layer of tin (Sn); rinse;
4. Spray on the silver solution and its reducer at the exact same time with two separate brand-new one-pint hand squirt bottles, until fully silvered & shiny; rinse;
5. Spray on the Angel-Guard overcoat; rinse; dry.
The amount of chemicals used is minimal. The nastiest stuff was the reducer. I’m glad we did this outside.
Btw: a number of people have bench-tested mirrors before and after this process. Some report no change in figure; somebody I trust, who has a Zygo interferometer, says there is a little degradation, but not much: a mirror that was 1/10 lambda (excellent) might go to 1/4 lambda, which is certainly usable for a big Dob, iirc.
And it’s cheap! And fast! And easy! And quiet!
We were able to fully and completely strip the brand new silver off with the PCB etchant in under 3 minutes. Aluminum takes much longer.
One spray for the silver solution, one spray for its reducer (chemistry)
It’s the tent canopy that’s bent, I think. But this is not a telescope mirror; just a 6 inch disk of float glass.
That’s Alan
I made the jig to hold the mirrors out of some scrap two-by-fours, some screws and reinforcement plates; two old hinges; some nylon chair feet; some 1/2″ PVC pipe; and some 1/2″ PVC end caps.
The tent-screen canopy needed staking and tying down to prevent it flying off. Putting up the frame took three people (me, Alan, and my wife, Gail).
‘Shruti” will explain how to do it, but she doesn’t explain why it works:
Personally, I had never come across this one in my over 40 years of teaching math. She is correct, it does work, but WHY?
I’ll try to explain.
994 is 6 away from 1000, or 10^3 minus 6.
And 997 is 3 away from 1000, or 10^3 minus 3.
If we multiply (10^3 – 6) by (10^3 – 3), we get 10^6 – (6+3)*10^3 + 6*3
So as she explains, we count down from a million by nine thousand, and then we add on 18.
or, if we take away any small number from a thousand, say, a, and subtract any other small number b from a thousand, you have a situation like this if you use the area model for multiplication:
Dear Flat Earthers, Many people have been derogatory of your belief that the Earth is flat. Please note that they are belittling your belief, not you per se. You, personally, are an idiot, but that is probably not your fault.
Here are any number of accessible approaches for discovering the shape of our beloved planet. Enjoy!
* * *
Use Your Phone! On Christmas Day, here in Chicago, I expect there to be snow on the ground because, well, it is winter. On Christmas Day I can pick up my phone and dial up anyone in Australia and ask them “What season is it?” They will tell you that it is summer in Australia. You might want to ask your flat Earth mentors how it could be winter and summer simultaneously on a flat Earth.
Use Your Phone! Go to a globe and pick a spot half way around the Earth (I know it is a false representation in your belief, but humor me.) In the middle of the day, phone somebody at or near that spot. Call a hotel, they are always open. Ask whoever responds “Is it light or dark outside?” They will tell you that it is dark where they are. You might want to ask your flat Earth mentors how it could be light and dark simultaneously on a flat Earth.
Look Up What Local Time Was In the US there was this concept of “local time” which was that “noon” was when the sun was at its highest point in its arc. You could call up people on the telephone who were not that far away and ask them what time it was and they would tell you something different from what your clock was telling you. The farther away they were, the greater the difference would be. On a flat Earth the time would be the same everywhere.
Look Up What Time Zones Are I am writing this in the central time zone in the U.S. These zones were created at the behest of the railroad industry whose dispatchers were going crazy making up schedules for trains when every place had their own times. By creating these “zones” everything would be exactly one hour off from those in neighboring zones, two hours off for the next over zones, and so on. If you don’t believe me . . pick up your phone and dial up a friend who lives a considerable distance (east-west) away from you and ask them what time it is. The time they state will be a whole number of hours away from your time. Heck, even the NFL knows this. When I lived on the left coast, the games started at 10 AM and 1 PM. Now that I live in the central time zone, the games start at 12 Noon and 3 PM. Over New York way the games start at 1PM and 4 PM. Do you think those games are replayed in one hour increments? Nope, time zones!. You might want to ask your flat earth mentors how it could be that simultaneous games start at different times on a flat Earth.
Watch the Video Astronauts in the International Space Station (ISS) have made continuous videos of an entire orbit of the Earth. It takes only about an hour and a half about the length of a typical Hollywood movie. During the whole movie the earth appears round, and yet it is clear that different continents are passing in our view.
Now you may argue that NASA made this movie as propaganda for the Round Earth Conspiracy. It is certainly within our CGI abilities at this point, but you may want to ask why NASA would want to do such a thing? Plus, many astronauts have taken their own cameras aboard and taken pictures for themselves and they show the same thing. How could the Round Earth Conspiracy have allowed that to happen? It must be incompetence! Conspiracies aren’t what they used to be!
Da Balloon, Boss, Da Balloon Many amateurs, unaffiliated with the government, have launched rockets and balloons high up into the atmosphere to take pictures. Every damned one of those pictures shows that the Earth is round. How come all of those cameras ended up pointed at the curved edge of your round and flat disk Earth? Such a coincidence!
An Oldie But Goodie #1 Occasionally, during a lunar eclipse, you can see the shadow of the earth falling upon the Moon. The shadow is always circular. This would be true if the flat earth were always dead on to the Moon, but the Moon orbits the Earth and wouldn’t a flat Earth be edgewise, often as not, and wouldn’t that create a non-round shadow on the Moon? Inquiring minds want to know.
An Oldie But Goodie #2 It was claimed that one of the first demonstrations of the earth being round was the observation of ships sailing west from Europe/England could be observed for a while but the ship itself was lost to sight while the mast was still visible. This would not happen on a flat Earth. The whole ship would just get smaller and smaller as it sailed west.
For pity’s sake, I live 22 stories up and the shores of Lake Michigan and I cannot see anything directly opposite me in Michigan. All I can see is water, with any kind of magnification I can muster. And I am not looking across the widest part of this lake! If the earth were flat, the lake would be flat and I could see the Michigan shore.
And Finally . . .
All of the fricking satellites! Do the math. What kind of orbit is stable around a flat disk earth? Answer none! And there are hundreds of the danged things in orbit.
Also, just for giggles. Look up what a Foucault pendulum is, And explain its behavior based upon a flat Earth.
PS You may be getting good vibes in your special knowledge that you know something other people do not. However, would not that special feeling be more worthwhile were you to volunteer at a food bank or a day care center or senior center? Wouldn’t doing something worthwhile be more rewarding that making a statement about how those pointy-headed intellectuals aren’t so smart?
PPS I have seen the cute models with the Sun and Moon on sticks rotating around (see photo above). If that were the case, everyone could see the Sun and Moon all day, every day. (There is straight line access to both objects in that model from everywhere on the flat disk.) Do you see the Sun and Moon all day, every day? No? Maybe someone who had more creativity than knowledge came up with those models. They do sell well, I must admit, so maybe their interest is commercial.
PPPS Regarding the 200 foot wall of ice that supposedly exists at the “edge of the disk,” supposedly so all the water doesn’t flow off and be lost into space. By now don’t you think someone would have sailed next to that wall all of the way? That distance would be somewhere in the neighborhood of a 28,000 mile trip. Has anyone ever report such a thing? Hmm, I wonder why not
Hopewell Observatory has three WW2 or Cold-War aerial spy camera optical tube assemblies, including a relatively famous Fairchild K-38. No film holders, though. And no spy planes. The lenses are in good condition, and the shutters seem to work fine.
We would like to give them away to someone who wants and appreciates them, and can put them to good use. Does anybody know someone who would be interested?
They’ve been sitting unused in our clubhouse for over 20 years. Take one, take two, take all of them, we want them gone.
We are located in the DC / Northern Virginia area. Nearby pickup is best. Anybody who wants them shipped elsewhere would obviously need to pay for packaging and shipping.
Here are some photos.
This one is labeled K-38, has a special, delicate, fluorite lens in front, and is stamped with the label 10-10-57 – perhaps a date. The shoe is for scale.
The next two have tape measures and shoes for scale.
Let me know (a comment will work) if you are interested.
I just did the math in two ways: if each person infects 5 people who have never been infected, it only takes a bit more than 14 cycles from “patient zero” (whoever that was) to infect the entire living human population.
Obviously the real progress of an epidemic isn’t that simple.
Being a retired math teacher I figured this was a perfect case for using logarithms, so I did. (For me, that’s fun!) I went like this:
I’m trying to find n such that five to the nth power equals 7.5 billion, or in math-lingo,
5^n = 7.5*10^9
One takes the logarithms of both sides, and because of the wonderful properties of logs, I get n*log(5)=9+log(7.5) which we can solve for n by dividing both sides by log(5), obtaining
n = (9+log(7.5))/log(5), after which my calculator said n was about 14.1.
But if you have a cell phone you can confirm my result much more easily by asking it work out 5^14. I think you’ll find it’s about six billion; if you try 5^15 you’ll get an enormous umber, over 30 billion, which is much too high. We have only roughly seven and a half billion humans…
I’m copying part of this from Ali Kayaspour at Medium.com. I’ve read some of these but not all. A while ago I made a list of math-related books for my students to read; maybe I should resurrect it. Here is the link.
Here is some information that teachers at quite a few different levels could use* for a really interesting geometry lesson involving reflections involving two or more mirrors, placed at various angles!
Certain specific angles have very special effects, including 90, 72, 60, 45 degrees … But WHY?
This could be done with actual mirrors and a protractor, or with geometry software like Geometer’s Sketchpad or Desmos. Students could also end up making their own kaleidoscopes – either with little bits of colored plastic at the end or else with some sort of a wide-angle lens. (You can find many easy directions online for doing just that; some kits are a lot more optically perfect than others, but I don’t think I’ve even seen a kaleidoscope that had its mirrors set at any angle other than 60 degrees!)
I am reproducing a couple of the images and text that Angel Gilding provides on their website(which they set up to sell silvering kits (about which I’ve posted before, and which I am going to attempt using pretty soon)).
At 72º you see 4 complete reflections.
When two mirrors are parallel to each other, the number of reflections is infinite. Placing one mirror at a slight angle causes the reflections to curve.
* assuming that the teacher are still allowed to initiate and carry out interesting projects for their students to use, and aren’t forced to follow a scripted curriculum. It would be a lot better use of computers than forcing kids to painfully walk through (and cheat, and goof off a lot) when an entire class is forced to use one of those very expensive but basically worthless highly-centralized, district-purchased computer-managed-instruction apps. God, what a waste of time – from personal experience attempting to be a volunteer community math tutor at such a school, and also from my experience as a paid or volunteer tutor in helping many many students who have had to use such programs as homework. Also when I was required to use them in my own classes, over a decade ago, I and most of my colleagues found them a waste of time. (Not all – I got officially reprimanded for telling my department chair that ‘Renaissance Math’ was either a ‘pile of crap’ or a ‘pile of shit’ to my then-department head, in the hearing of one of the APs, on a teacher-only day.
Keep in mind: I’m no Luddite! I realized early on that in math, science, and art, computers would be very, very useful. I learned how to write programs in BASIC on one of the very first time-share networks, 45 years ago. For the first ten years that my school system there was almost no decent useful software for math teachers to use with their classes unless you had AppleII computers. We had Commodore-64’s which were totally incompatible and there were very few companies (Sunburst was one) putting out any decent software for the latter. So when I saw some great ideas that would be ideal for kids to use on computers to make thinking about numbers, graphs, and equations actually fun and mentally engaging, often I would have to write them my self during whatever free time I could catch, at nights and weekends. Of course, doing this while being a daddy to 2 kids, and still trying to teach JHS math to a full load of students (100 to 150 different kids a day at Francis Junior High School) and running a school math club and later coaching soccer. (I won’t say I was a perfect person or a perfect teacher. I believe I learned to give better math explanations than most, didn’t believe that you either have a ‘m,ath gene’ or you don’t, at times had some interesting projects, and at times was very patient and clear, but had a terrible temper and often not good at defusing things. Ask my kids or my former students!) Later on, I collaborated with some French math teachers and a computer programmer to try to make an app/program called Geometrix for American geometry classes that was supposed to help kids figure out how to make all sorts of geometric constructions and then develop a proof of some property of that situation. It was a failure. I was the one writing the American version, including constructions and tasks from the text I was currently using. There was no way I could anticipate what sorts of obstacles students would find when using this program, until I had actual guinea pig students to use them with. Turns out the final crunch of writing however many hundreds of exercises took place over the summer, and no students to try them on. Figuring out hints and clues would require watching a whole bunch of kids and seeing what they were getting right or wrong. In other words, a lot of people’s full time job for a long time, maybe paying the kids as well to try it out so as to get good feedback, and so on. Maybe it could work, but it would require a lot more investment of resources that the tiny French and American companies involved could afford. We would have really needed a team of people, not just me and a single checker.
I find that none of these computer-dominated online learning programs (much less the one I worked on) can take the place of a good teacher. Being in class, listening to and communicating logically or emotionally with a number of other students and a knowledgeable adult or two, is in itself an extremely important skill to learn. It’s also the best way to absorb new material in a way that will make sense and be added to one’s store of knowledge. That sort of group interaction is simply IMPOSSIBLE in a class where everybody is completely atomized and is on their own electronic device, engaged or not.
Without a human being trying to make sense out of the material, what I found quite consistently, in all the computerized settings, that most students absorbed nothing at all or else the wrong lessons altogether (such as, ‘if you randomly try all the multiple choice answers, you’ll eventually pick the right one and you can move on to some other stupid screen’; it doesn’t matter that all your prior choices were wrong; sometimes you get lucky and pick the right one first or second! Whee! It’s like a slot machine at a casino!).
By contrast, I found that with programs/apps/languages like Logo, Darts, Green Globs, or Geometer’s Sketchpad, with teacher guidance, students actually got engaged in the process, had fun, and learned something.
I find the canned computer “explanations” are almost always ignored by the students, and are sometimes flat-out wrong. Other times, although they may be mathematically correct, they assume either way too much or way too little, or else are just plain confusing. I have yet to detect much of any learning going on because of those programs.
