Where to get good math images
You don’t have to get all fancy and techno-savvy to have decent graphics on your blog. It’s all about be resourceful and being open to learning a few simple technological tools.
I have to confess. I use images I find on the web all the time. I link to the sources/pages where these images are housed and give full credit. I figure that this is a personal blog and as long as I’m giving credit where credit is do, I can use and re-use content I find on the web. Also, If I’m using a game or manipulative I take a screenshot (Prnt Scrn on PC or command+function+shift+3 on a Mac) and then cut out or crop the graphic I want in a photo or graphic editing program. Most PC’s have a program installed called “Paint.” It’s usually located in your “Accessories” folder. There are also a number of freeware graphics programs available online. But the program should be able to save any graphic image as a commonly used graphic file (such as a jpeg or gif).
The picture of the Pascal’s Triangle game below was taken by doing a screenshot of my desktop with the game open. I pasted the entire desktop image in MS Paint and then cut out only the portion I wanted to share. I save the image as a jpeg and then uploaded to my blog.
Click on the image to view the actual game on the Shodor.org site.
Shodor Site Rules!
I’ve been reviewing the Shodor site and it’s a virtual treasure trove of manipulatives and pretty cool lesson plans.
- Math lesson plans: http://www.shodor.org/interactivate/lessons/
- Fun Activities: http://www.shodor.org/interactivate/activities/
If you go to http://www.shodor.org/curriculum/subject.php and select Mathematics you’ll find all the links to externally based interactive tools and lesson plans. By the way, there are links to materials on all disciplines in this site.
The Marvelous Maya
I don’t have much time to write today, but I thought I’d write a brief reflection on the Maya. I’m really in awe of the accomplishments of the Maya. They were obsessed with tracking the movements of celestial bodies. Without telescopes or equipment they mapped the movements of the planet Venus and used these observations along with others to establish their incredibly accurate calendar.
It made me think… how spoiled our we we our cable tv, our docu-dramas, and the Internet. They had two sticks they used as sort of cross hairs for pinpointing heavenly objects in the sky, their eyes, and the night sky.
Their understanding of the patterning of celestial events led to calculations that defined their calendar.
Here’s a pretty good example of their calendar: http://www.geocities.com/wwwtimto/gfx/azteccalendar.jpg
When I was a child we had a pottery plate with the calendar on it. I think it was a souvenir from our visit to Mexico. I used to trace and follow the different patterns around the cycle. Of course I had no understanding of what they meant or even the purpose of the disc.
Is it happening? Connections are firing off…
This may seems strange, but I think a bit of a metamorphosis has come over me… no, don’t worry I’m not going to transform into a cockroach or some unpleasant things with more limbs than what’s normal for a mammal. I’ve been noticing that I actually, am finding it easier to explain or think out mathematical terms and concepts better than before. I actually start analyzing experiences or occurrences with Math in mind. As I’m out walking the dog, I notice my shadow and I ask…What angle is my shadow to me in the afternoon vs. the morning? What size rice dispenser should we get if we eat about 2-3 cups of rice a day? We’ve actually been eating more since we’ve made an effort to be gluten free. I want to design a baby blanket or quilt for a friend… but I want the proportions just right. Say I want the width to be 3 feet. To get the proper length I’m going to multiply 3 feet by 1.6 (the Golden Ratio).
Also, strangely, I’ve been finding that I actually am more drawn to reading about or even thinking about math concepts and problems lately. I’ve been reviewing a site called Ravelry which houses a ton of information on knitting patterns, yarns, etc. What I love about this site is that many knitters can contribute what they’ve learned about knitting a pattern or yarn, or using a different technique. They can contribute their own pictures of their finished objects or their process, and others can look at or observe. I often find myself looking at a texture of knitting or a pattern and start to build the chart for that pattern out in my head. I love charts and prefer to refer to them rather than written instructions. It’s funny because we have a good friend that’s a gifted and unorthodox programmer and gamer and he was looking over my shoulder one day when I was looking over a book of charted patterns and he noted, “Wow, that’s pretty cool it’s like a program.”
I’d never thought of it that way, but that’s right. Knitted charts are like numerical patterns or programs. The simple chart below is a great example. It’s actually the chart from the pattern I used to make part of my Mother’s X-mas gift this year. But you can see how the symbols are used to create instructions on stitching. I often find myself reviewing patterns without looking at photos of the knitting to see if I can imagine what the finished swatch will look like. Sometimes, if I’m waiting in line and the person in front of me is wearing a particularly fetching piece of work. I’ll start charting out the pattern in my head. It’s more fun than re-arranging the words on the message board to spell silly things.
I’m starting to think that perhaps my new appreciation of Math and my ability to see connections between Math and other applications or forms of analyzing the phenomena around us is being heavily influenced by my love of knitting (you think?). But it seems that that really what we need to do as teachers or educators, to tap into what our students are interested in and then start helping them build connections that make meaning.
Image of chart from Spindilicity:http://www.spindlicity.com/spring2006/smokering.shtml
More knitting and some algebra
I didn’t have a great deal of time to hash this out, but I thought I would create a few images that describe a knitting project I started. I’m knitting a seamless yoke sweater. It’s seamless because the body and sleeves are attached and then knit together to form the yoke. Other than the sewing in of ends to make things neat there’s absolutely no sewing involved with this project.
Here’s an example of what a yoke sweater looks like click on the link to enlarge:
Here are the slides where I thought out the math for starting my yoke pattern:
BTW: I may seem like I spent a lot of time creating the images, but not really. It was easy to reproduce those little dots. I just selected a whole row of them and then copied them over to the next section of the grid. I’m finding that by using shortcuts and pasting repeats of patterns makes creating illustrations much more easy.
I forgot to post my finished sweater:
My finished yoke sweater
Math writing
Arghhhh… finding a math language editor that’s easy to work is hard.
I can’t wait until they start applying handwriting recognition technology for math/science applications (and make it available to the general public at free or little cost).
I did find this, a nifty tool call DragMath:
http://www.dragmath.bham.ac.uk/
If you click on the “Demonstration Link” and download the Java applet you can play with a demo of the tool. I’m including an image/screen shot of the tool here.
You can select copy expressions to a clipboard however it converts them to the Math Markup language. In order to paste them on to a webpage like this blog post, you need to paste the code into the HTML code of your page. Actually, I’m still fiddling with this to get it to work the right way.
2+ \sqrt{3+13} =x
I’m actually pretty stoked that this application can be used with the Moodle Learning Management system. This means that courses housed in Moodle can use this tool to allow students to build their own mathematical expressions.
Graphing linear equations
I found a pretty good applet that allows you to graph and view changes in equations/function:
I found some drawbacks with this program.
You cannot copy/cut and paste former expressions from the application itself. I tried typing the the expressions in a word document and then pasting them, and this didn’t work either.
Also, you can’t place lines from different functions on the same graph to compare.
I’m sure there are other graphing tools out there that do this. I just have to find them.
Click on the thumbnail to see a larger version of the image.
The importance of building our own visual examples
Years ago, I worked with a science teacher to teach a which encouraged students to examine and then draw magnified images of natural objects such as sea urchin shells, pinecones, leaves, etc. I introduced the students to drawing by referencing lessons I adapted from Mona Brookes’ Teaching Children How to Draw.
I discovered this book when I was in teacher training in New York, and it was funny after reading the book I found myself drawing all sorts of mundane things… the seat back in front of me on the bus, the window view outside my apartment window. I suddenly felt empowered to draw just about anything because the Brookes method taught me how to visualize the basic shapes in everything I saw. I suspect this is the general method taught for drawing anyway.
I believe that we really should teach and encourage children to draw anything and everything because it allows them to both internalize and then begin to analyze what they see. The same holds true for constructing visuals for understanding Mathematics and Math concepts. As a student, I think I was on the tail end of educational pedagogy that made us sit in our seats and listen rather than experiment and explore. I was forced to memorize formulas without really understanding how they worked. Perhaps this is why I loved Geometry once I discovered it because here was a Math where we had to ‘prove things’
As a student in a middle school Algebra or Pre Algebra course I think I would have been too impatient and impulsive to see the value in proving simple things such as the Multiplicative Inverse property, but perhaps proving why rule like this worked would have helped me internalize these crucial concepts more so that I could apply them more readily in the more complex Math courses.
Pretty nifty applet: Pascal’s Triangle
I found this great applet today on Pascal’s Triangle
You can actually take a look at the patterning on the triangle when prime numbers are selected.
Click on the image to view a larger version.
Representing algebraic equations visually
I’ve been experimenting with the use of Algebra Tiles. I like the idea of visually representing values and variables. When I was learning Algebra in middle school and high school many of the things we learned seemed so abstract. I always felt like we were learning rules about abstract things, but in Catholic school we had no exposure to mathematical concepts such as the Properties of Numbers, the Properties of Rational numbers. We were drilled on how to do the ‘mechanics’ of math, addition, subtraction, multiplication and division plus a thing or two on how to use the same operations on fractions. By the time I got to high school I didn’t know what hit me. The math teacher was speaking a muddle of a language that I could barely understand, and considering my upbringing and former educational experience, I focused only on memorizing equation types. It was a very short sighted and limited way of learning. I know it was awful, but I really do think that the words of that fifth grade teacher (I wrote about her briefly in a previous post) really stuck with me. She sent this message that basically told me that it was okay to give up. I should understand now that she was only human and she had her own fears when it came to math (she was a Baby Boomer teacher and most likely grew up during the Donna Reed age where understanding and mastering higher math wasn’t a goal expected of most women).
I muddled my way through Algebra and then found my way into a Calculus class. Our teacher in this class was a very caring and methodical woman who really had a passion for her discipline. She even took us to see “Stand and Deliver.” Incidentally, Jaime Escalante became a sort of role model for me and his work actually inspired me to become a teacher myself. However, by the time I got to Calculus, I still had a really bad or superficial understanding of some of the number sense basics (Integers, Rational Numbers, Properties, Operations, etc.). I wonder how I was able to even manage a C in Calculus.
I actually created a set of my own “eTiles” for Powerpoint. I’m including images here.
Ooops…. the operation should read 2x2+4x+12. Ahhh the joys of using graphic tools that don’t have superscripting. Thanks to my commenter for the correction.
I wasn’t sure how to represent negative numbers so I did so by circling the depicted value with a dotted red line. It might be embarrassing to admit this, but I feel a sort of healing and relief as I worked to develop these visual images. Perhaps actually creating these pictures help re-reinforce the concepts I probably did get, but only half way.
