Fiddling+with+Math+-+Virtual+Manipulatives+and+a+Homeschool+Journey

This page started off as a wiki for the Create and Share Math Interactives course, but somewhere along the line it changed direction a little and seems to be turning into a story of the math side of the homeschooling journey I've shared with my son. He's fifteen now,so a lot of this will also be a bit of a trip down memory lane for him, and I hope some of you will find our journey of some interest.

We start with the first exercise in the course - a GeoGebra interactive exercise to discover the triangle with the greatest area possible that would fit inside a circle.

I explained to my son what we were supposed to be doing and he immediately said "I guess that's going to be an equilateral triangle then." I told him to wait and see then set up the exercise and 'Jinged' it as he experimented. He immediately set about making a roughly equilateral triangle which touched the edge of the circle. Then he fiddled about with the controls a bit to figure out which of the readings on the left gave the area. Then he spent a while tweaking the triangle to make sure he had the greatest possible area. He concluded that 5.2 was the maximum reading he could get for the area, which coincided with the most equilateral triangle he could draw, which wasn't perfect but was pretty close.

He appeared to take very little notice of the information on the left until he actually needed any of it. After he'd roughly found the maximum area and was tweaking it, he kept an eye on the info on the lengths of the lines, I think to try to get them all exactly the same and see if it co-related to the maximum area, but I don't think he bothered to get his head around any more of the info than he needed to to solve the problem. He's very much the sort of person who will just jump straight in and fiddle without worrying about all the details, assuming he'll be able to fill in any gaps in his knowledge as he progresses.

media type="file" key="geogebra_triangle_in_circle.swf" width="680" height="480" align="center"

For the paper folding exercise, which involved folding a circle of paper into an equilateral triangle, we did a 'thought experiment' and he concluded that what he would do is to imagine the triangle drawn on the circle and then fold along the imaginary lines.

Interestingly, my husband, who is in his seventies, has had very little formal education and claims to have no imagination whatsoever, was listening to the 'lesson' and announced that to fold the paper into an equilateral triangle you'd have to start by folding the bottom up to the centre. I queried how he knew that and said that he could 'see' it when he was listening to the two of us talking about it. For someone 'with no imagination' I was pretty impressed - he hadn't even *seen* what was happening on screen and was relying on what he could overhear. I asked him more about this later and he said that he could see the circle in his mind and 'just split it into thirds', which makes sense but I can't see how that relates to folding the bottom of the circle to the centre, which seems more like quarters/fourths than thirds to me. I guess he's imagining it as a clock face, with 12 at the top, and 4 and 8 at the 'third' markers. Presumably, when you fold 6 to the centre, then the fold lines touch 8 and 4. I guess I'm going to have to go and fiddle now to double check...

Linda, the course organiser, commented about the way Alan will jump right in and fiddle with stuff and wondered how that could be encouraged in other children, so I began to remember the earlier parts of Alan's education and thought that maybe some of the early software packages he used had played a part. One in particular I remembered was this one - Dr G from Edmark's Mighty Math Calculating Crew.

media type="file" key="doctor_G.swf" width="599" height="599" align="center"

We dug out the record book and found out that Alan was eight years old when he used this software. I'm amazed the disc runs on Windows Vista, but it did, so were able to Jing a video of it and I think we're going to try the same with a few more of his old favourites, and maybe a few not-so-favourites to see if we can figure out what worked best and why.

I noticed the way that the introduction was *not* presented as a lesson. It's like you've been invited into a secret lab to assist with important discoveries.

"Today, I'm examining nets..."

and a net appears as soon as the new word was introduced,

"...that fold up into solids."

And the net obligingly folds itself up. It's up to the player to use their own powers of observation and figure out what's going on, just like kids do in real life.

"You are here to assist in this crucial endeavor."

They player is an honoured and vital part of the game. Not a student in a classroom.

"Does this net match this solid?"

Dr G obligingly points to the 'net' and the 'solid' at the appropriate moments, without lecturing, and tells the player how to input their yes or no answer. There are loads of lovely enticing buttons just asking to be fiddled with, and Dr G sits back, folds her arms and waits eagerly for your response without hurrying you. It's like Alice in Wonderland, finding a bottle labelled 'drink me' - kids just *have* to jump in and fiddle.

There is a 'print' button in the explore section too, so you can create nets, print them out and use the print outs to make your own solids to your own designs. We didn't do that this time, but seven years ago we had a whole load of colourful paper 'solids' cluttering the place up.

The next morning, we watched the video for the 'construct a square based on a diagonal' exercise, which gave us enough confidence to start fiddling with GeoGebra, so after repeating the exercise as given, then fiddling around for a while experimenting with other ways of making squares, I decided to see if I could fiddle about with a clock face and find out if 4 and 8 really were on a line half way between 6 and the centre of the circle. It turns out they were!

The same diagram also neatly demonstrates the way that the Sun's rays are spread out over twice the area between the lines of latitude represented by 4 and 6 as they are between 3 and 4. No wonder it's colder down there...



Another old favourite was Edmark's Carnival Countdown, especially Annie the Aardvaark's geometry saloon. Alan played this when he was five and loved every moment of it, but he couldn't bring himself to do a demo as he finds Annie a bit too annoying these days, so Mom had to step into the breeching. Again, it's very 'hands on' and interactive, and note the 'growth slides' which can be set to be adult-only controlled so you can monitor your child's progress and adjust as necessary.

media type="file" key="carnival_countdown.swf" width="674" height="528" align="center"

In the evening we tackled the stopwatch challenge. We watched the how-to screencast, copied it for practice, then Alan fiddled about trying to make a two-handed version that timed minutes as well as seconds. We managed to post it up but it wasn't exactly perfect - the minute hand went a little fast, and we had forgotten to re-set the hands to 12 before saving the file, but we were both pretty pleased with it anyway. Just for practice, we had another go, adjusted the size of the circle so the 6 wasn't so squashed up and re-set the hands to 12. The minute hand still runs a bit fast though..

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We installed another piece of Edmark software, Cosmic Geometry, aimed at High School kids. This could have been so good, but it didn't quite work. Alan 'tested' the GeoMovies lab, with no more success than he had around eight years ago. It went roughly like this...

media type="file" key="geomoviefail.swf" width="700" height="506"

The first problem seemed to be based on the incredibly annoying arty tetrahedron - Alan couldn't even bear listening to her and clicked her to make her shut up. But even if he had listened to her unsufferable wafflling, she didn't really say anything inspiring. After she'd shut up, Alan clicked around a bit, declared that he never had figured out what on Earth he was supposed to be doing, unenthusiastically clicked a few buttons and then departed to find something more interesting to do. There is a tutorial button, but somehow that was just as uninspiring as everything else. It's a pity, because for it's time, this was quite an advanced 'hands on' type of software, but it failed in it's presentation. I went back and Jinged a quick recording of the annoying babble and a few representative exploratory button clicks so you could see what it was all about.

I decided it was time I really got my head around what the GeoMovies Lab was all about, so I fiddled with it until I realised that the tutorials were movies, and so were the ideas, and that you had to tell them to start. It was a bit unintuitive, and the tutorials really needed either prior knowledge or someone on hand to explain them to you. I think this software would actually work well given the appropriate back up, but as something to give to kids to let them just play with, it just didn't provide enough of a helping hand to get them started.

media type="file" key="geomovies_ideas.swf" width="712" height="503" align="center"

Cosmic Geometry also had a robot lab! It's a bit arty, but the word 'robot' means that it is more easily forgiven. And this time, it's a lot more obvious what you're supposed to be doing, and there are hints if you get stuck. After constructing your robot, which involves learning about coordinates and transformations, you program him to dance, then send him out decorate the plaza. It

media type="file" key="dancing_robot_beta.swf" width="682" height="522" align="center"

The software was aimed at High School kids, but at the age of seven, Alan had a whole plaza full of dancing robots, and a pretty thorough understanding of coordinates and transformations that have stood him in good stead as he has progressed from dancing robots to 'real' 3D graphic packages.

And then there was Mind Power Math. I believe it was originally designed as an on-line course, but I bought it as a set of CD-ROMS for Alan to work through. It's not terribly interactive, but it's quite fun in it's way. Definitely 'education' rather than 'entertainment', but it had it's good points. Here's the first part, where Digit introduces fractions.

media type="file" key="digit_first_lesson.swf" width="664" height="509" align="center"

All well and good, and there is plenty of guidance on what to do so you don't get stuck. Unfortunately, there is rather too much help - it's impossible to get stuck, and perfectly possible to click your way through a lesson without absorbing any math whatsoever, not to mention that tempting 'skip' button.

media type="file" key="digit_cheat.swf" width="618" height="458" align="center"

Alan loved the first few discs, but all it took was a couple of 'off days' where he had to get a few too many hints and he soon learned to skip his way through his lessons and we gave up the idea of a digital math course and went back to book work for 'real' math learning.



This morning, Alan had a bit of a problem with his maths exercise. He was learning about sin rules (something I've never heard of) and the exercise was to calculate the possible values for the angle PQR in a triangle where PQ = 10 cm, QR = 8 cm and the angle QPR = 48 degrees (not sure how to do that superscript thing - guess I'll figure it out sooner or later...). He ploughed his way through using the example given in the book, but really couldn't understand why sin(180 degrees - x) = sin x. And without understanding it, he was reluctant to use it. Very reluctant. Unfortunately, I wasn't much help as I'd never heard of sin rules, or ambiguous triangles. So I suggested we attempt to draw it on GeoGebra. I constructed the bits that were known, and then drew a circle to represent all the points 8 cm from point Q, and, lo and behold, it did cut the line extending at 48 degrees from PQ. So there *were* two solutions. But what did that have to do with sin(180 degrees - x) = sin x. Then Alan noticed the two possible angles for QRP - they were supplementary, adding up to 180 degrees.

media type="custom" key="8225380" width="250" height="250" align="left"

And what's more, if you grab either P or Q and swing them around a bit to form other pairs of ambiguous triangles, the two choices for QRP always seem to add up to 180 degrees. We're still not sure why, but we both now have a pretty good visual on the way they do. I guess why comes a bit later, with yet more fiddling...

I did a little research about ambiguous triangles and discovered that using one angle and two sides (ASS) to prove a congruent triangle is known as the donkey theorem, and for the most part it doesn't work. Strangely enough, I have a donkey who doesn't work very much these days either...

This page is getting rather unwieldy, so I've started on Fiddling wih Math - Page 2

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