A couple of days ago I posted a very brief article, Dismantling the calculus pyramid. The post has been seen 2,746 times, thanks mostly to being noticed by 'timwiseman' who posted a link to the article at Y Combinator Hacker News. The posting at Y Combinator has gotten 45 comments while, here, it's gotten four comments. The conversation at Y Combinator is quite lively. Check it out.
A while ago I received an email out of the blue from Texas Instruments (TI). One of their marketing people discovered this blog and offered to send me a TI-Nspire calculator to review. I quickly accepted, after all, who would turn down a free fancy calculator, right? Once I received the calculator I realized that this was no ordinary calculator; it was a visual Math learning system. I did nothing with it for a couple of months until I finally realized that I was not the best person to review it as it would take me quite a bit of time and effort to learn and appreciate its power. Sure, I could read the manual and run some demos but I didn't think that would give me enough experience to write a very in-depth review.
In discussing my challenge with TI, I learned of some teachers who were successfully using the TI-Nspire in the classroom. One person in particular, Eric Butterbaugh, was teaching Math in Harlem, New York. It occurred to us in that conversation that readers of this blog would appreciate hearing about Mr. Butterbaugh's success with the Ti-Nspire system. I created some interview questions and received back the interview you're about to read.
Brian Foley runs a web-site, Math Mojo and a blog, The Math Mojo Chronicles. The web-site aims to make, in Brian's words, "Math meaningful." While I enjoy the site, I struggled to explain what Math Mojo was about until I found this description in the What is Math Mojo page:
Math Mojo is a way of looking at math that fosters a sense for numbers. The more new ways you learn and practice, the more of a feel you will get for manipulating and understanding how numbers work. They will become less of a mystery, and you will feel better about your ability to do math. These methods are based on several different speed-math techniques. They all work at least as well as the methods that you were taught in school. In fact, schools that teach these methods do much better than the national average.
With summer fast arriving (yeah, I know we have to get through winter and spring first) kid's thoughts turn to summer Math camp. "Yeah, right", you say. Well, for those of us who really enjoyed Math in high school, Math camp was a lot of fun.
This post is not intended to be a list of Math camps. I may create one later but you can Google and find a bunch. The list from the American Mathematical Society (AMS) is a good starting place for your research if you're looking for camps for high school kids.
In the late '70s I attended the Ross Program at the Ohio State University. It was the toughest summer I ever had and the most rewarding one at the same time. I was pretty good at Math in school but this camp really worked me. While I can't say that I was mature enough at the time to fully appreciate what I was learning I will say that the camp opened my eyes to how cool Math is at a much deeper level than ever before. The camp left me with a deeper appreciation for the beauty and rigor of mathematical thinking.
It's reminiscent of those Planet of the Apes movies. Apes are smart. Humans are stupid.
ScienceDaily just ran this article: Young Chimps Top Adult Humans in Numerical Memory. Young chimps and adult humans played this game where some numbers were displayed on a screen in some, I'm-sure, hard to remember order for, I'm-sure, not very long. Then the player had to identify which numbers were where, and in what order. Gulp! I'm glad I wasn't recruited for this game. I'd have put the whole human race to shame. I wonder what consolation prizes the humans got.
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.
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.
The tips are based on the Dunn & Dunn Learning Styles Model developed by Professors Rita and Kenneth Dunn to assist the New York State Department of Education in improving the effectiveness of instruction for students not demonstrating sufficient progress. The model is based on 20 elements, divided among five categories (called stimuli in the model), that affect learning. Use of the model has been statistically proven to improve student performance.
The model's five stimuli are environmental, emotional, sociological, physiological, and psychological. A nice illustration of the stimuli and elements is here. A very similar model, the Learning Styles Pyramid Model is illustrated here.
The first half of the tips are not specific to Math and apply to a broad range of learning situations. The second half are Math-specific.
Science News Online just published a fascinating article: Good Stories, Good Math. The article is subtitled: "Preschoolers who can tell good stories develop good mathematical skills by the first grade." Writer Julie Rehmeyer reports on a new study which reveals that there's apparently a very strong connection between mathematical ability and the ability to tell stories from different perspectives.
The researchers found that 3- and 4-year old preschoolers who were able to tell stories and switch perspectives while doing so performed better in mathematics 2 years later. An example of switching perspectives is this comment from a child:
"The waiter was mad when the frog jumped in the soup."
The child was managing multiple relationships, simultaneously keeping track of how the waiter was feeling and of what the frog was doing.
While I had never considered such a relationship, it makes sense to me. Mathematics is all about managing relationships between "things." It's about being able to think abstractly. Storytelling shares these characteristics, even though the storytelling abstractions are about people and what they might be thinking and feeling, and not mathematical abstractions.
An interesting refinement that the researchers made was one that allowed them to make a distinction between mathematical skill and skill in arithmetic. They are not the same skill and many people who are mathematically gifted perform arithmetic poorly.
"...that language arose when humans acquired the ability to visualize complex relationships among different objects when the objects themselves are not in view. The ability to do mathematics arises from that same ability to manipulate abstractions."
The implications of this study are huge. Can students be taught abstract thinking skills early in life through storytelling and improve their future mathematical ability?
I was recently asked how I deal with silly mistakes many of us make in algebra or arithmetic, especially in the context of a tutoring session. Common errors include forgetting to carry numbers when adding, getting confused about operating with plus and minus signs in one problem, and multiplying two digits incorrectly.
I consulted Michael Sheppard, director and lead educator for Big Sky Learning Center in Santa Fe. Michael shared a number of suggestions based on his 19 years of experience teaching kids in one-on-one and group settings. I've combined Michael's excellent suggestions with some of my own: