The Joy of Tutoring

Artemis* is 8. He arrived for his first tutoring session last week ready to learn more about trigonometry. He doesn't yet do algebra, and a few days ago said "I can't subtract", but trigonometry is what he's enamored of right now. (He can subtract, he just doesn't know how to use the standard algorithm, yet.) He's full of extremes like this. He was reading before he was 2, but had very little control of his body until recently.

For the past year he's been coming to the math salon I host, along with his parents and twin sister. The first time he came, he was so excited he just had to twirl around and get his whole body moving. (He reminds me of myself. When I'm really excited, I just have to wiggle and wag my tail.) He's still excited, but now he can be part of a group of people working together on a math problem. Part of his excitement during our tutoring session was that he got to have me all to himself. He snuggled up next to me on my sofa, and we dove in.

I started with the Pythagorean Theorem. He knew there were hundreds of proofs, but I don't think he'd really walked through one before. The proof I'm most familiar with involves a bit of algebra, and for him that was the complicated part. (Maybe he's ready for lots of heady stuff but not yet algebra? We'll see.) Just now I looked up proofs to try to find the one I used. Didn't get a good link for that one, but here are two I'll show him next Monday, both completely visual: One with the triangles hinged, the other with them sliding.

[In a previous post I mentioned mathematical holes that can cause students grief for years and years, like not learning your times tables in 3rd grade because you were out sick. I was unsure whether I wanted to say that because Artemis and others like him were in the back of my mind somewhere. When a student isn't expected to know things in a particular order, it's not too hard to work around them, and get to them later.]

I showed Artemis a few more basic geometry proofs, like the angles in a triangle adding to 180 degrees. In the middle of our one-hour lesson, he got so excited by it all he just had to move, so he took a 10-minute break on the trampoline. At the end of our lesson, I lent him Geometer's Sketchpad, Who Is Fourier?, and Mathematics: A Human Endeavor. He's been reading the Fourier book since then, and came in this week excited about one of the formulas he saw in it.

[Ooh, this is my first time doing that. I like it! I used codecogs.]

It seemed to me that he was intrigued by the fact that sine isn't additive. So I played the mystery box (or Guess My Rule) game, where I have a function in mind, and he figures it out by giving me inputs to see what outputs I give him. It gave me a fun way to talk about functions, input, output, domain, range, etc. With each of the functions I used, I then drew a graph, and we looked at whether it would be additive. I talked about it as linearity.

He wanted to think about , so I pulled out my TI calculator. I tried to keep chatting with him, but I found myself saying "Look!" a few times, and belatedly realized he was too entranced by the calculator to do anything else. So we looked at things like , which I knew went with what he'd been reading in the Fourier book. I let him borrow the calculator, and later that day his mom went out and bought him one.

Next week he wants to take a walk and find math all around us. Sounds fun to me. (It took me a moment to let go of the notion that we had to do something more industrious.) ;^)

I am like a kid in a candy shop myself, getting to work with someone who loves math so much. It feels like jazz improv, taking his lead and doing a riff on it. Wow! I'll be taking on a few more students in the coming months. I wonder if any of the others will lead me as well as he does, so I can learn more about how to teach by following.


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*He decided to use the pseudonym Artemis for my blog posts because he likes the Artemis Fowl books.

First Day of Class

I had the kids use 12 square tiles to make as many different rectangles as they could. I showed them that the shape is written like this: 3x4 ("we say 3 by 4"). Then we talked about the area being 12 square inches, and that the x usually means times. By golly, 3x4=12, too. Our newest student, who is 7, thought that was totally funny! She laughed for each one. The 6 by 2 rectangle has an area of 12, and, yep, 6x2=12. Hee hee hee.

The size of the rectangle turns out to be the same as a multiplication problem! Imagine that... She made my day.

First Day of Class Coming Up? Some thoughts...

I saw a few comments in my blog reader on first day activities over at dy/dan, First Day Wiki ('07) and This New School Year ('08). I loved his ideas. I have a few of my own you all might like if you're headed back to teaching. (The sabbatical work I'm doing is marvelous, but I am going to miss teaching!) Dan taught high school and I teach at a community college; perhaps that has some bearing on what we do differently first day, but there's lots of overlap.

• On my last first day, in January, I put this on the board: “My ideal math class is a learning community. My goal is to help you become a community of learners." I also asked them to each add something to their choice of one of 3 lists on the board: 'Something I know in math', 'something I don't get', 'something I'm curious about'.

• I have them fill out 3x5 cards with their name, phone, email, and maybe a question or two about their math background. I ask them to leave a space for their photo. I'll use the cards to do attendance and call on people. (In the future, I'd like to jot notes on the cards: things they say and do, so I can remember.)

• I think it's important to try to call on people as equally as I can. (See Failing At Fairness for research that shows how much more boys are called on than girls, and how differently they're responded to.) I tell them the first day: I don't want to intimidate you, so you can always say 'pass' if you don't want to answer. But if you want to be brave, you shouldn't pass just because of not knowing the answer. You can say "I don't know", and I'll ask you easier questions and we'll work back up to the original question. I still end up asking questions to the class as a whole sometimes, and when the same 3 people raise their hands all the time, I start telling them I'll wait for other hands, in order to ‘spread the wealth’.

• It's vital to learn your students' names, but I have a bad memory. In recent years, I've brought my digital camera, and taken pictures. I found it used up too much time if I took the pictures, so now I get a volunteer to be the photographer. They can do a photo of 3 people at once, and afterward I organize in iPhoto, copying and cropping, so I have head shots of each student. I print and have the students put their names on their photos during the next class. Then I get to cut them out and use a glue stick to put them on those 3x5 cards. That work with the photos helps me get started with learning the names.

• I send around a sheet of paper with my name, phone number, and email. I suggest they add theirs to the list if they want, and tell them no one has ever had problems from it that I know of. Then I copy the list, and give it to them. I talk about how studies have shown that students who work in groups do better.

• I offer ‘donut points’ for catching me in mistakes, so that they’ll question what I’m presenting, instead of assuming it’s right just because the teacher said it. After the class has caught me 30 times, I bring in donuts.

• I use ‘thumbs up-down-sideways’ to find out the level of understanding. (And have found out that it’s important not to ask people with their thumbs down to elaborate, or else fewer people will use their thumbs at all.)

• I ask them if they think you need a good memory to learn math. Most think so. I tell them about my terrible memory and say it's all about connections and understanding why things work. I made a poster that says: ‘Real mathematicians ask why’. It's in my classrooms and my office.

• Last year I was working way too hard to have time for correcting homework. They have answers in the book of the book, so they don't need me to correct it. They just need me to give them credit for doing it, so that it gets high enough in their priority lists to get done. So (I learned this from a high school teacher at Middle College High School which is housed inside our college. Thanks, Eric!), I stamp their homework. (Buy a self-inking stamper for this.) If the homework is complete, or close to it, they get two stars. At least halfway there gets one star. They turn it all in at test time, and I record the number of stars. I can do that while they're taking the test.

• Some classes get the math autobiography assignment. Lately, though, I've gotten tired of it, and just offer it as extra credit.

• College textbooks are outrageously expensive - generally over $100 for math texts. My goal is to use the textbook as little as possible. So I told my beginning algebra students that they didn't have to get the official textbook, but could buy any Beginning Algebra text. I changed my homework sheet to show the topic names, and told them to pick 10 problems from their book on the right topic. Lots of students have tried to come to class with no book, because they couldn't afford the $100-plus. Now they didn't have an excuse to have no book. We discuss where used bookstores are, and online sources for used books.

Here were some ideas I heard at a Great Teachers Seminar that I liked and hope to try in future:
• Send email before semester begins. 2 days before gives them a day to get their book.
• Put students in the position of teaching what they’re learning.
• Ask at end of class: ‘What’s the most interesting thing you learned this week?
• Ask: What are you curious about? (What are they already interested in?)
• Have a question for them each time you walk into the classroom.
• One teacher pairs students up and has them fill out a form which starts "I am my brother's and sister's keeper. I will help my partner succeed. I commit to..." Then there are 3 blanks, and th pair discuss how they can help each other. If one is absent, she'll ask the other about it.

• The Math Students' Bill of Rights is included in my syllabus for all lower level classes. [Added on 8/25]

[Note to self: Dan's First Day Wiki has a great stacking cups activity. He posted a different cup stacking activity here.]

Math Student's Bill of Rights, by Sandra L. Davis

I have the right to:


Learn at my own pace and not feel stupid if I'm slower than others,

Ask whatever questions I have,

Need extra help,

Ask my teacher for help,

Say I don't understand,

Not understand,

Feel good about myself regardless of my math abilities,

Not base my self-worth on my math skills,

View myself as capable of learning math,

Evaluate my math teachers and how they teach,

Relax,

Be treated as a competent adult,

Dislike math,

Define success in my own terms.

Adapted from the Math Anxiety Bill of Rights by Sandra Davis, in Resource Manual for Counselors/Math Instructors: Math Anxiety, Math Avoidance, Reentry Mathematics, ed. by Donaday & Auslander (1980).

I put this into my syllabus for all my lower level classes. I also have it on my office wall.

I tell students that schools aren't always the best way to learn math, because they can make it pretty hard to learn at your own pace, if the course is paced faster than you need to learn it.