Music, Motion and Learning

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We have family visiting from Germany so it’s the perfect time to brush up on my German. For me, German is the topic that I can study and study and try to memorize things but the words just fall right out of my head. How to make them sticky?

I remember never being able to remember the Dative prepositions until a friend (from a different school) sang them to me. In that instant I could finally remember them perfectly–and I remember them still (alas, not all my endings.)

Something about music lends an emotional quality to information and enables us to store it better in our brains. This works especially well with verbal information, and it’s the reason we sing nursery rhymes with small children.

For example, everything you need to know about baking in German, you will find in this video:

There’s also an element of play in nursery rhymes and songs that puts our brains in the right state to retain information. I can find many examples of how this works with music, but far fewer examples with math. The closest I can find are these (some of my favorite math music videos):

 

Although I can’t say I’ve learned much math from having watched them. It’s more like they make me appreciate the math I do know, and they certainly make me want to learn more.

When have you used music, movement or play to learn math? Please share your comments below–I’d love to hear from you.

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Vibrant Watercolors in Twelve Easy Lessons

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Day 8: Asparagus

It’s my firm belief that painting and drawing are not only talents, but also practical skills which can be taught to everybody. I also believe it’s never too late to learn (yay, neuroplasticity!) So I bought the book You Can Paint Vibrant Watercolors in Twelve Easy Lessons by Yuko Nagayama and I’m very proud of myself for having made it all the way to Day 8. I want to add more vibrant colors to my block prints. For me color = emotion and although I love the crisp, clean, orderly look of black and white block prints, I’m ready to add more. Here’s my painting from Day 1 so you can see how far I’ve come!

Day 1: Bananas
Day 1: Bananas

Girls on the Threshold of Math and Science

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I remember so clearly standing in the doorway of the computer room as my teacher told me about the new computer club that was forming at our school. I was 13 years old. I had just finished 4 semesters of computer language classes: BASIC, Advanced BASIC, and LOGO (remember those days? Remember the little turtle? And Atari computers?) I really enjoyed those classes, but at that time there weren’t many more I could take—the next one up was something called “Assembly Language” which involved nothing but numbers. So I came at lunch time to check out the computer club. I stood in the doorway and what did I see? Rows of computers and lots of middle school kids, nearly all boys, huddled around screens making explosion noises (how do boys do that?) My eyes got wide and I slowly backed away. This was not a world I wanted to be a part of. I didn’t take another computer language course until college.

As the mother of teenage girls and a high school math teacher, I think about this moment all the time. What was going through my mind in that split second? What can I learn from that moment about how to encourage girls (and boys) in math and science?

For me the issue was (and to some extent still is) that I love math, I love computer programming, but I want my world to be bigger than that single activity. I want more connection with others, more emotional vibrancy and more color. And for some reason I was not sensing what I needed in that room at that time.

I also remember that that year was difficult for me socially. It wasn’t until high school that I made more friends and ended up finding my lunch hour home in the art room.

I think when we think about kids and learning, we need to remember the importance of context. There are so many environmental pieces that play into our decision making and our ability to absorb information. For me, the creative and artistic side must be fed. It’s only when I am emotionally grounded in the arts that I have the bandwidth and the bravery to take up the difficult and rewarding intellectual tasks of math and science.

In the years since, I have come to that doorway many times. Sometimes I back away and sometimes I step through and commit myself to delving deeply into the pursuit of intellectual knowledge. The gift of experience is that now I do so with awareness, and I can bring this awareness to my teaching as I help my students on their journeys.

Circles, Divisibility, and Wonder in Mathematics

Watercolor Meditations on the Divisibility of 12
Meditations on the Divisibility of 12

I’m working with a student who is having a hard time memorizing his times tables. Indeed, I suspect he does not have a concept of numbers as representing quantities–I think sometimes he sees them as separate numerals, some kind of squiggly code that doesn’t mean much to him. In performing operations, he does his best to memorize the algorithms, but of course they do not stick. I’m working on ways to get him to internalize some of the qualities of numbers, hoping that this will give him a scaffolding on which he can attach deeper learning.

The Page of 12s
The Page of 12s

We started working with a circle. I marked off 12 equally spaced marks on the circumference (similar to a clock face.) I then had him draw lines connecting marks when counting by different numbers. For example, when counting by 3s, the lines form a square. Counting by 4s you get a triangle. Then something crazy happens–, when you count by 5s, you get a star that crosses itself over and over again until you reach home. This is where the real math starts happening: Can you look at your data and predict what will happen for other numbers?
When I started working with this problem, I knew it had something to do with divisibility, and I thought it would be a cool way to use colored pencils and get away from numerals and algorithms. It turns out this problem is a lot richer than I anticipated.
I was surprised to notice that numbers that add up to the base create the same pattern. For example, 3 on a circle of 12 will make a square as will 9 on a circle of 12. 8 on a circle of 20 will make a 5-pointed star, as will 12 on a circle of 20.
I got excited to prove my observation that if n divides p and n divides q, then n divides p-q. Here’s my proof:

Definition of Divisibility: if n|p that means there is some number k for which kn=p (I remember this definition from an old college number theory class somehow . . . )

Given: n|p and n|q

Prove: n|(p-q)

n|p kn=p

n|q jn=q

Definition of Divisibility

kn-jn=p-q

Subtraction Property (of Equality)

n(k-j)=p-q

Distributive Property

Let h=(k-j)

Then, nh=(p-q)

Substitution

Therefore, n|(p-q)

Definition of Divisibility

OK, I know this is a pretty simple proof, but it made me inordinately happy to remember how to do it, and I truly was surprised to see the shapes on the circles match each other the way they do.

You can also use these circles to illustrate which fractions can be reduced and which cannot (the same idea as showing whether two numbers are relatively prime, or share a common factor.) If the number of marks around the edge of the circle corresponds to the denominator, and the number you count by corresponds to the numerator, then irreducible fractions will be the ones where the lines meet every mark. For fractions which can be reduced, the greatest common factor is the number of untouched marks between the lines.

I’m not saying this should be an algorithm for reducing fractions, but I’m excited because it is a visual way to understand a concept that baffles many of my students.

My hope is that no matter how much we do with this problem, I can demonstrate that numbers have qualities that can be observed and generalized. Observe, form a hypothesis, make a prediction, test your hypothesis . . . repeat as necessary. In my opinion, this is where the real learning is!