Ever since starting this blog a couple of months ago I have felt that text and graphics could only go so far in helping people to get wild about Math. I knew that video was the next step in helping to explain mathematical ideas and in getting more people engaged in Math. But, until a few days ago I didn't know what it would take to produce a Math video.
Well, as luck would have it, some searching on the net led me to a fellow Math lover, Tim Fahlberg (Tim.Fahlberg@mathcasts.org), who just happens to have tons of experience producing mathcasts, which are broadcasts (videos) aimed at teaching Math. Tim, his Math professor sister Dr. Linda Fahlberg-Stojanovska, colleagues and students have produced tons of mathcasts. Tim, in fact, is a pioneer in producing mathcasts and I'm sure there will be many more of them produced in the future, especially as the needed technology gets better, cheaper, and easier to use.
Tim's business-related Wiki is here and his Math247 Wiki, with lots of mathcasts plus instructions on producing them, is here. I'll be writing more about Tim and what he's doing in future posts as there's tons to write about Tim and mathcasts. For now go ahead and check out his web-site and Wiki.
The very first mathcast is about how to quickly multiply together two-digit numbers if they meet two conditions. Check out the video. (Flash is required to watch the video.)
This video is pretty primitive but I think it works. There's an error in one of the Math problems. See if you can find it. And, error aside, tell me if you find this video helpful, if you'd like to see more of them, and what subjects you'd like to see videos about.
I've got plans for more impressive productions but what's interesting about this video is how little technology it took to produce. While I was learning all about screen capture using Camtasia, and researching Whiteboard software and graphics tablets for producing mathcasts, Tim urged me to check out something I had never heard of, a free service called voicethread. Voicethread allows you to take one or more pictures, a slide show, and apparently some kinds of videos, and annotate them with voice, text, and by drawing on the screen. And, you can allow people to comment on your voicethreads with voice, text, and handwritten annotations ; Don't worry your original isn't damaged.
The neat thing about Voicethread is that its pretty easy to learn and use. I made this mathcast with my Windows XP laptop, an image file of the Math problems that I created with a whiteboard program called NotateIt, and an inexpensive microphone. Tim got me excited about the idea of encouraging kids to comment on mathcasts and to make their own. Voicethread does have some weirdnesses but once you get over them it's quite easy to produce fun videos quickly.
So, kids (and adults), do comment on the video. And tell your friends to check it out. Also, do tell me what kinds of Math problems and explorations you'd like covered in a video as I plan to make lots more, and more impressive looking ones as soon as I gather up everything I need and get some experience. But, don't ask me to show you how to do that Math problem that's due tomorrow. I can't be that responsive!
p.s. to fellow WordPress bloggers. Embedding videos in your blog can be a real pain as I'm sure many of you have discovered. The way to go is to find plug-ins that know about your particular video service and generate the proper HTML for you. Well, there's a plug-in for Voicethread. It's here. I discovered it at this blog article: Getting Voicethread and WordPress to play nicely.
If you enjoy this video check out all of the Wild About Math! mathcasts.
This is hilarious. There's a new web-site, Rejecta Mathematica, that claims to give new hope to many. They've got a call for papers in the mathematical sciences. They're looking for papers specifically rejected from peer-reviewed journals for reasons like "mapping the blind alleys of science", "applications of cold fusion", and my personal favorite, "squaring the circle." And, they've got an RSS feed so you can keep abreast of those rejected papers that meet Rejecta Mathematica's standards.
Vlorbik, are you reading this? This site is the kind of stuff you live to write about
Here's something fun, not too heavy on the Math unless you want it to be, and quite remarkable to those who haven't seen this before. You're going to create a Moebius strip, and variations on it, and surprise yourself and others when you see what you get when you cut the Moebius strip in various ways. A Moebius strip is a strip of paper that you tape together at both ends, but before you do, you make a half twist in the strip. Your Moebius strip should look like the one in the picture from Wikipedia.
Here's the first thing you can do that'll be interesting. Take a pen and draw a line along the strip lengthwise, starting anywhere you want to start on the strip. What happens? You end up back where you started, right? That's the first interesting discovery - the Moebius strip has only one side.
One of my favorite subjects to write about is the use of multiple intelligences (MI) in teaching Math. A while back I reviewed Math for Humans and I wrote a couple of articles that touch on MI: 10 ways to get wild about Math, and 11 tips for building a strong Math foundation for kids. I also reviewed a really nice Math history book that helps engage interpersonal intelligence. And, I wrote 26 tips for using learning styles to help your kids with Math, that relates to MI.
Multiple Intelligences in the Mathematics Classroom, by Hope Martin, is a great book with very practical ideas for incorporating MI into activities that homeschool parents could guide and motivated kids could work out on their own.
Yesterday Mathmom (is that her real name?) posted a great video on her blog, Ramblings of a Math Mom. The video is of a working Lego mini-car factory made out of Lego Mindstorm pieces. Some people have way too much time on their hands. I love it!
I have to say I really enjoy Mathmom's blog. It's one I check on every couple of days or so as Mathmom has a way of finding humorous Math-related stuff on the Internet and when we're not looking she sneaks "serious" Math into some of her posts too
Anyway, after watching the amusing video and reading on Mathmom's post that their next project is to build an airplane factory I got to thinking, what would the ultimate Math geek Lego construction be? Well, it so happens that I collect lots of Math-related URLs (yeah, weird hobby, I know) and I remember having run across this Lego construction:
So, what is it? It's a Babbage Difference Engine, and apparently a working one. It's an early calculating device. You can read all about it at Andy Carol's web-site.
Can anyone top this for Math geekiness?
Here's a game that's easy and leads to a nice exploration of number theory for those so inclined. Two people play. All you need is a sheet of paper and a pencil or pen. Here's how to play:
- Each person thinks of a number between 1 and 50 without telling the other person what the number is. Then, each person writes their number on the sheet of paper.
- Decide who is going to go first, by tossing a coin or in some other mutually agreeable way.
- Players take turns writing down the positive difference between any two numbers on the sheet of paper.
- Numbers cannot appear more than once on the paper.
- The player who cannot write down a unique positive difference loses.
Here's an example of how a game might go between Sol and his friend Michele.
- Sol thinks of the number 5. Michele thinks 3.
- They write 5 and 3 on the paper.
- Sol goes first.
- 5 minus 3 is 2 so Sol adds 2 to the paper.
- The paper now has these numbers: 5 3 2
- Michele notices that 5 minus 2 is 3 but 3 is already on the paper.
- Michele also notices that 3 minus 2 is 1 so she writes 1 on the paper.
- The paper now has these numbers: 5 3 2 1
- Sol notices that 5 minus 1 is 4. He writes 4 on the paper.
- The paper now has these numbers: 5 3 2 1 4
- Sol wins as no more unique differences can be calculated.
Here's another sample game:
- Sol thinks 8. Michele thinks 6.
- The paper has: 8 6
- Michele goes first.
- Michele notices that 8-6=2. The paper now has: 8 6 2
- Sol notices that 6-2=4. The paper now has 8 6 2 4.
- The game is over and Sol wins as no more unique differences can be calculated.
Here are some interesting exploration questions:
- Once both numbers are written down is there a way to determine who will win?
- Once both numbers are written down does strategy matter, other than who goes first?
- For starting numbers of 5 and 3 all numbers between 1 and 5 got written down but when 6 and 8 were the starting numbers only 2, 4, 6 and 8 were possible differences. What determines whether all numbers get used and if not which ones are used and which aren't?
This game is related to Euclid's algorithm and to the greatest common divisor of two integers. At Cut the Knot there's a Java version of this game, Euclid's Game, that you can play alone against the computer. In the computer game the computer picks the two starting number but you can practice determining who should go first.
Judging by the comments I've received on the blog there's a good amount of interest in techniques for simplifying and speeding up basic arithmetic. This is great because I enjoy learning and writing about these techniques.
Jakow Trachtenberg was a Russian Jewish mathematician who, while imprisoned in a Nazi concentration camp during World War II, developed a system of speed mathematics, no doubt to help preserve his sanity. The Trachtenberg system is particularly good at allowing one to multiply big numbers by small numbers although it teaches a number of other techniques as well.
Amazon and other online booksellers have multiple editions available, at quite different prices, so shop around if you want to own a copy of the book. Here's one edition.
As a gentle introduction to the Trachtenberg system I'll demonstrate how to multiply any number by 12. Trachtenberg has the notion of neighbor, which is the digit to the right of the digit you're applying a technique to. Also, when multiplying with Trachtenberg we move from right to left and keep track of carries just as we do with the approach to multiplication most of us are familiar with.
Here's a really easy Math trick you can impress your friends and family with. All you need is 4 index cards and a pen. Here's what you do:
- Copy what's on the four cards on the right to your four index cards.
- Ask your friend to think of a number between 1 and 15.
- Show your friend all 4 cards.
- Ask your friend to tell you which cards the number appears in.
- Count 1 if he tells you card 1,
2 if he tells you card 2,
4 if he tells you card 3, and
8 if he tells you card 4
- Add up the counts.
For example, if your friend thinks of the number 6, then he'll tell you it appears on cards 2 and 3. Card 2 counts as a 2 and card 3 counts as a 4. Add up the counts, 2+4, and you get 6, the number your friend thought of.
Another example, if your friend thinks of the number 9, then he'll tell you it appears on cards 1 and 4. Card 1 counts as a 1 and card 4 counts as an 8. Add 1+8 and you get 9.
Can you figure out why this trick works? Hint: It has to do with the binary number system.
I'm always impressed to see a new way to do something familiar. Recently, I happened upon a fascinating video, titled Weaving Numbers at the IsAllAboutMath web-site, which has some instructional Math-related videos.
Weaving Numbers demonstrates several non-traditional ways to do multiplication. I found the Napier's bones approach depicted fascinating as well but the one I want to focus on today is the visual approach to multiplication.
The video goes a bit fast for my tastes but since I already had a sense of what visual multiplication would be like I was able to follow it. Here's a nice explanation of the approach from Mudd Math Fun Facts if you can't figure out what's going on in the video or if you want to understand why this technique works.
What I particularly like about this number weaving approach is the visual nature of it. Kids who have a hard time memorizing the multiplication table can simply count the number of points of intersection between the lines that cross. After a while the idea that 2 rows of 3 dots = 3 rows of 2 dots = 6 dots will come naturally to them. What's also wonderful about this approach is that kids can do multiplication by doing addition! So, as soon as kids are comfortable with addition, including carrying, they can learn to multiply. Also, kids can use different colors, as in the illustration above, to engage more fully with the numbers they're multiplying.
Once kids get grounded in this approach to multiplication, and as their confidence builds, they'll learn more quickly, and with better understanding, the approach most of us are taught in school.
A final point, as a Math fanatic, I am delighted whenever I see something like multiplication, which is pretty much taught as an algebraic function, seen from a geometric perspective.
It's been a month and a half since I started this blog and I'm delighted with the nice reception it's received. Thank you all for the kind comments you've left. I wish each of you a warm, safe and wonderful Thanksgiving. Enjoy good times with your families. Oh, and it's ok if you don't crack the Math books for a few days!
I've been enjoying the Carnival circuit, publishing articles recently in the carnivals of healing, education, and homeschooling. I particularly like the homeschool community as I appreciate the commitment it must take to not only raise children but to educate them yourselves. And, I imagine, that if Math isn't your forte it can be a challenge to provide that part of the education. I'll be doing what I can to continue writing articles and pointing everyone to helpful resources. That's my commitment as expressed in this blog.
In this week's Carnival of Homeschooling there are three Math articles. Yeah, Math!
While you're all making preparations for the big feast Thursday here are some ideas for sneaking Math into your kids.
- Bottled juice algebra. If adults each drink 16 ounces of juice, and kids each drink 8 ounces, how many 64 ounce bottles will you need to buy given the number of adults and children you'll have Thanksgiving dinner with? How much juice will be left over from the last bottle?
- Time arithmetic. If the turkey takes 57 minutes per pound to defrost how long will yours take to defrost?
- Money calculation. If cranberry sauce cost ____ per pound and you buy five pounds, how much did you spend?
- Ratios and fractions. If you need to mix ___ cups of water with ___ ounces of mashed potato mix to feed 4 people, how much water and mashed potato mix do you need to feed everyone at your dinner table.
- Percent calculation. If everyone uses two forks, one knife, and one spoon, what percent of the utensils are forks? What percent are knives? What percent are spoons?
Happy Thanksgiving, Everyone!