Vibrant Watercolors in Twelve Easy Lessons

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


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


Subtraction Property (of Equality)


Distributive Property

Let h=(k-j)

Then, nh=(p-q)


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!

Why Make Art?


I was visiting our local craft studio last week. I have a gift certificate from my former employer (you can buy their awesome socks here.)  I have been wanting to turn my prints into jewelry and I wasn’t sure which jewelry making class to take, so I talked to the teacher in charge of the metals studio. Well, I must have said something annoying because she came out with some harsh opinions, one of which was:

You had better know why you want to make jewelry otherwise you’re just making stuff, and the world doesn’t need more stuff.

Strangely, this made me really want to take a class from her. It also got me thinking. Why make art?

I doubt that I will make lots of money selling my art and I no longer harbor illusions that I will become famous for my art. I’m not even sure I have the time and money necessary to learn another skill well. So why make art?

1.  I feel like it’s important to see and express beauty in the world. It’s almost like making a daily gratitude list: the action of doing it changes my thoughts and the way I perceive the world. I love the Bertholt Brecht quote:

Art is not a mirror held up to reality, but a hammer with which to shape it.

I would argue that it’s both. By reflecting selectively (as we must, or we’re merely recording devices) we can choose what we remember or where our minds rest. We can stop time and record an idea into human consciousness. We have a responsibility to be intentional about that because we are creating our reality and affecting others.  Also, what we choose to create informs us about our values so we can shape our futures with greater integrity. Which brings me to my second reason . . .

2. I create art to connect with other people. The reason I walked into that metals studio is that even though I am primarily a printmaker, I feel a strong desire to wear my artwork. I want to wear my artwork because I want people to know something about me when they see me. I want them to know who I am, what I stand for, and what’s important to me. I want to remember it too (it’s so easy to forget!) So making art is a means of self expression, but it’s also a way to connect with others–to find my tribe.

3. The third reason I make art is one I am newly cultivating, but I think it’s an important one. I make art to hold a space in our present culture for art, because it enhances my appreciation for the work of others, and to pass on my skills to others. When my children first began taking violin lessons, this was an idea brought up by the owner of the local strings rental studio. I sort of knew my kids probably wouldn’t become professionals, so I wasn’t sure how much musical training made sense for them. What this man told me was that we give our children music lessons not only so they can perform, but also so they can be part of a generation of informed/appreciative audience members. That made a lot of sense to me. I also came across this idea last weekend at a singing workshop I took. The leader spoke quite eloquently about the music we were learning being an oral tradition, and our responsibility in carrying it forward. I know for a fact that even though I may not be a perfect practitioner of art, music, or math, I can be a conduit of knowledge for the next generation.  And that is reason enough for me!

What are your reasons for making art?

Kahn Academy is Distance Learning

There is a trend emerging in our local elementary and high schools to use Kahn Academy and other online learning resources to supplement or replace classroom instruction in math. I understand the attraction. At the small schools in the region where I live you tend to see a wide range of abilities in students. There are simply not enough resources to create different classes for kids who learn at different speeds. By using an online curriculum, kids are free to work at their own pace. The faster students can barrel ahead, while the slower ones can re-play a lecture to reinforce a concept that didn’t come through the first time. There is the potential to minimize classroom management issues and maximize everyone’s productivity–sounds like a win-win.

However, what teachers and administrators who jump to embrace this new way of teaching need to realize is the importance of the real time teacher-student relationship in learning. Relationship is another important way to make information sticky. A teacher, teaching in real time, can intuit moments of confusion, enlightenment, or boredom in a way that a computer never can. A good teacher can leverage those moments to propel or pause a lesson, to engage or disengage students from on another, in order to broaden the education of the whole class. I’m not saying it’s easy, but I believe it’s an art worth striving to master–and that is lost when we rely upon online learning.

Most of the students I’ve talked to emphatically do not like learning from Kahn Academy or other video lectures, whether they find math inherently easy or difficult. The online lectures are good as a back up or reinforcement, but as the primary source for content they are disorienting. Most teenagers (especially the ones who struggle with math) are not good at knowing whether or not they have mastered a concept, and what they need to learn next. The computer algorithms that attempt to figure that out from them are flawed and impersonal. There’s also the HUGE problem that when a student doesn’t understand something in a lecture, he or she can’t ask the video a question. Most teachers know that when your student doesn’t understand what you said, repeating it word for word is generally not going to help. There are also a million other factors that affect how receptive someone is to learning–human relationships are all about meeting and welcoming the ineffable. And that’s all part of education.

I think that Kahn Academy and other online learning tools are a great resource as long as you see them for what they are: distance learning. They can be really helpful if you’re in a situation where you don’t have access to a classroom and a live teacher: studying while working full time, recovering from an illness, or traveling. But if you have a choice, it makes much more sense to choose live human teachers as the means for teaching math and all other subjects. In addition to classrooms, textbooks, and instruction, I believe human contact is critical to quality learning.

Art, Education and Math–What’s the Connection?



In addition to being an artist I am a math teacher. I often struggle with deciding whether I am more of an artist or more of a math teacher, but at the moment I am surrendering the struggle to explore areas where the two overlap.

I was working with a tutoring student last year who was having a hard time memorizing vocabulary words on flashcards. She was getting frustrated and she said to me, “The words go in but then they fall right out–they’re just not sticking.” Immediately I thought of Malcolm Gladwell’s book on social trends, The Tipping Point, where he explores the importance of stickyness in making a message memorable. I also thought of a class I took called Teaching to the Adolescent Brain where I learned that for information to be stored in deep memory it has to pass through the region of the brain called the amygdala–also the processing point of emotion and basic fight or flight instincts. These two different ideas combined and I realized that my job as a teacher is to help make information “sticky.”

There are many ways to do this–some better than others. And different sticking techniques will work for different people. I happen to be a very visual person, so color is one of my favorites. After learning a little about Goethean color theory at a summer class on Anthroposophy and Art, I also believe that color carries archetypal emotional content and meaning. Blue is peaceful, red is dynamic, yellow is uplifting. My student tried color coding the vocabulary words to correspond with the part of speech (blue for nouns, red for verbs and green for adjectives.) It takes more work to make information sticky (especially if it’s a topic you don’t particularly care for), but a little effort in the right direction will go a long way.

I’ll explore more in subsequent posts–meanwhile I invite you to ponder: What makes information sticky for you?

Registration for March 15th Spring Into Artistic Action Block Printing Class is Now Open!

Break through the ice, snow, and winter doldrums with a burst of creativity. Spring is almost here and your muse is calling: learn something new, take a block printing class! Learn basic design, carving and printing techniques in a relaxed, supportive, and playful atmosphere. You will come away with a completed print suitable for framing, embellishing, and/or mailing as a card. This class is for people at all levels of artistic ability–complete beginner through seasoned professional.

The cost is $40 per person and classes are taught at my studio in South Strafford, VT. All tools and materials are provided. To register, email me at the address in my contact information page.

Spring Creativity Breakout Class


Block Printing Class with Tracy Gillespie

Sunday, March 9th, 3-5 pm

Break through the ice, snow and winter doldrums with a burst of creativity. Spring is almost here and your muse is calling: learn something new, take a block printing class.

No experience is necessary! Learn basic design, carving and printing techniques in a relaxed, supportive, and playful atmosphere.

All tools and materials are provided, and you will come away with a completed print, suitable for framing, embellishing and/or mailing as a card.

This class is for people at all levels of artistic ability—complete beginner through seasoned professional.

This class costs $40 per person, and is taught at my studio in South Strafford, VT. To register contact me at the address on my contact page!