My Favorite Teachers and Me

The #YourEduStory blogging challenge question of the week:

How are you, or is your approach, different than your favorite teacher?

I don't have just one favorite teacher. I have lots. Long, long ago, before I started teaching, I made a list of my favorite teachers:

Mr. West, high school biology, and then anatomy and physiology
Ms. Purvins, high school Shakespeare teacher
Mr. A, high school poetry teacher
Mr. X, UM philosophy prof
Ms. Y, UM history of feminism prof
Gisela Ahlbrandt, EMU math prof

 There were probably more on the list at the time. These are the ones I still remember. (And I'm losing the names. Yikes!) When I made the list, I noticed something interesting. There were about equal numbers of men and women on the list, but they were very different sorts of teachers. The men were good performers, and the women were good facilitators. A few did both well (the poetry guy and Gisela). I wanted to do both well. I thought about taking some drama courses to improve my performance skills. I did that while teaching in Muskegon, and realized I needed a different sort of course. Performing in a play is a lot different than performing as a teacher. Improv might be good for me. Hmm... I also learned a lot about facilitation over the years.

I know now that the best performers make students happy to come to class, but that's not enough. We need to get students actively engaging with the material for them to learn much. (Mr. West did that in lab, even though I remember his great lectures.) If you don't know the research done by Eric Mazur on this, check it out.  (This video might include the best parts of the hour-long video I watched a few years ago.)

How is my approach different than theirs? I think it's only in the combination that I'm different. I try to pull in all my students (like my Shakespeare and history of feminism profs did). I ask them multiple times each class to show me with thumbs up, down, or sideways how well they understand what I've just explained. I call on students randomly. (Because teachers tend to call on male students more.) I come in as excited as my bouncy philosophy prof. I suggest my students try strange experiments, like my poetry prof did (he had us write at a cemetery and a mall). I try to be as accepting and as challenging as my best teachers were.

Math Circles at Nueva School

Nueva School, in Hillsborough, south of San Francisco, puts on a math night three times a year, with multiple math circles, along with a puzzle and game room. Nancy Blachman invited me to lead two math circles last night, one for 2nd and 3rd graders and another for 4th and 5th graders.


2nd and 3rd grade Circle
This circle met for just 30 minutes. I know that the Collatz conjecture is dependably fun for kids this age, so that was our main activity. I asked the kids what they thought mathematicians do, and got a reasonable answer, but saw that there wouldn't be time for useful discussion. So I said a bit about math being like a game for mathematicians, and how fun it was to come up with a new puzzle.

In 1937 (I just said it was about a hundred years ago), Lothar Collatz came up with this puzzle/game:

• Pick a number.
• If it's even, cut it in half. Write your new number.
• If it's odd, triple it and add one. Write your new number.
• (We drew an arrow from each number to the next.)
• Repeat until you get back to a number you've already written.

Collatz conjectured (guessed) that the sequence would end up at 1, no mater what number you started with, but he couldn't prove his conjecture. Mathematicians have tried to  prove this for over 75 years, and it is still an open question. (It is very likely to be true. Using computers, people have tested every number up to and past 5 quintillion.)

As I expected, the kids loved it. At the end, I showed them a "mind reading" trick.

• Pick a number from 1 to 31. Don't say it, just keep it in your brain.
• (I pretend I'm sucking their thoughts over to my own head.)
• Now show me which of these five cards it's on.
• (I barely glance at the cards.)
• Your number is ___.

After we did it a few times, I had the parents cover their ears and told the kids how it worked. I had  the five cards on the board, and half-size index cards for them to make their own cards. They loved it.


4th and 5th grade Circle
This circle met for an hour and a half. My plan was to analyze Spot It with them. (I've written at least 4 posts on using Spot It for math circles. Search on Spot It to find them.) We started out playing the game for about 15 minutes, which they all enjoyed.

The problem was, half of them had done this last year in their math class at Nueva! Luckily, one girl had come early and I had shown her the number trick. I asked her if she wanted to teach it to the others. She did.

I split the group in two, and she showed her group the number trick, while my group started thinking about the game. I had one boy who answered every question very quickly, and asking him to slow down didn't help. So, after we had figured out that there would be 57 different pictures, I got out the half-size index cards and suggested they make their own decks, with 4 pictures per card. Or, if they weren't into drawing pictures, 4 numbers per card. They worked hard at trying to make a deck where each card matched every other card on exactly one picture.  Towards the end, they wanted to play with the number trick too.

About halfway through the girl who led the other group came over and said, "The number trick is done." So I joined their group for a bit, and asked, "Why does it work?" A few parents were there, thinking about it with their kids. I should have asked them to work with all the kids (about 6 of them), but didn't think to say it. A few kids wandered away, to the puzzle room, no doubt.

The kids who stayed worked hard on the problems and had fun. I had a great time.

Days Three and Four

8am
Calculus. Wednesday: Circle area. Archimedes. Zeno. Started Boelkins' Velocity of a Ball activity. On Thursday, we got through most of the Velocity of a Ball activity. The students did not recognize that (s(b) - s(a)) / (b-a) is a slope. So We are working through the parts they need to review. I am goign slower than in other semesters. I hope I'm not going too slowly.


10am
Linear Algebra. Wednesday: Discussed differences between Echelon Form and Reduced Echelon Form. I started with: a matrix in Echelon Form, and got them to tell me the values of the variables. I explained that this way is quicker for computers. We talked about number of possible solutions, and drew examples in 2D and 3D. Quiz tomorrow. (Quiz made and copied.)

Thursday: Most of them aced the quiz. The ones who didn't will be in my office to retake. We finished 1.2. (I hate referring to book sections, instead of math topics. Basically, we are working on row reducing matrices. We've started to think about parametric representation of solutions, where there are free variables.)


11am
Pre-Calc. Wednesday: We practiced an arithmetic sequence (find the nth term) and a geometric sequence. We looked at a problem that used a recursive definition for a _n. I mentioned the Fibonacci sequence, but didn't do much with it. Quiz tomorrow.

Thursday: Only a few aced the quiz. It was harder than what we had done in class. I'll give a retake on Tuesday in class. We reviewed lines. I walked them through my proof that perpendicular lines have slopes that are negative reciprocals. (It's different from the text's proof.) In the process, I also walked them through the proof that the angles in a triangle add up to 180 degrees. I love how the result suddenly pops out of the picture. I asked them to show me with their thumbs (up, down, sideways) how cool it was. They all gave it a thumbs up and I said they were being too nice. The bigger proof (for perpendiculars) gets an 80% coolness rating from me.

1pm
Calc III. (I am sitting in on this class.) Wednesday: Ed showed us how to connect the tops first and use dotted lines for hidden lines. I noticed that it felt like we were seeing the xz-plane from the back. Thursday: Over an hour of lecture. Ed is a good lecturer, but that's too long for me. I fell asleep. I woke up for the quiz. It included drawing 3D surfaces. I understand all of this, but how well did I draw? I'm not satisfied yet.

Day Two

8am
Calculus. I talked about what we had done yesterday with finding a line tangent to y=x² at x=3. In algebra, we find the slope when we are given two points. We know one point, (3,9), and there is no other point that we know. [Last semester, at least one person used the y-intercept of the tangent line they had graphed as their second point. I liked that, but forgot to mention it today.]

I asked them to give their definitions of the word tangent.
First student definition of tangent: A line that touches the curve in one place only.
Sue's counter-example: I drew y=x³ and drew at tangent line at about x=1. They agreed that I had drawn a tangent. Then I extended the curve and the line. They cross at x = -2. I suggested that we could add the word nearby, and maybe this would work.

Second student definition of tangent: A line that touches the curve but doesn't cross it.
Sue's counter-example: I asked them what the tangent to y=x³ at x=0 would look like. They told me it would be horizontal. I drew it in. Hmm. (I told them that later we'll talk about concavity, and showed it with my hand curved. I said that I think the only time the tangent line crosses the curve is when it's tangent at an inflection point. Is that true? I should try to prove it.)

Third student definition of tangent: A line that determines the direction of the curve.
I think this one is about as good as we can get at this point, although it's hard to turn it into something precise. I talked about thinking of the curve as a road, and your point being a car driving along the curve. Its headlights make half the tangent line, and its taillights make the other half.

Talked just a bit about history of calculus, and gravity. Got some volunteers who will drop a heavy and a light object, and see what happens.

Then we started our circle activity. I had a picture of a circle of radius 10cm on the back of the handout. I asked for the radius, a rough estimate of the area, and a more careful estimate of the area. (I asked them to pretend they knew no formulas. Next I had them fold a round coffee filters in half through the middle over and over, then cut it into wedges, and play with them. Tomorrow we'll do the area formula from that. Today I gave the definition of pi (C/D), and talked about how C=2*pi*r comes easily from this definition. I got a few volunteers who will measure around a circle and across it, using string, and will bring in their string tomorrow. Area is different...



10am
Linear Algebra. I used a desk corner as the origin, drew the x and y axes with my finger along its edges, and the z axis coming up from the corner. I asked them to figure out (in groups of four) what the equation x+y+z=1 would look like. I heard someone say circle. It is not at all obvious to most of them yet that it will be a plane. But we got there.

Was that before or after we worked on the definition of a linear equation? Yesterday I had asked for their definitions from their heads. I got four volunteers today (yay!) to give me their definitions to put on the board. They were all different, and none matched the official definition. So, after I went over the official definition from our textbook, I asked them to use that to prove or disprove each of the statements given by students. I think this will help them with proofs and with what a linear equation is.

Next I continued with the problem we had done, algebra style (no matrix), yesterday. I talked about computers, and representing it with just the coefficients, and wrote the matrix. I showed them the matrix that would represent the solution, and said our steps will be similar to those we used yesterday, but our order will be different. We did our same problem matrix-style, and I identified the three elementary row operations as we used them. (We never used the swap rows operation, but I talked about when it would be needed, and how you'd never do that with the algebra-style method.)

I finished up with one book problem.


11am
Pre-Calc. Stamped their homework. Had them share with their group the list of 5 problems they couldn't do. Had them each pick a problem from their partner's list, that they would later explain to their partner. Some people working hard; others feeling unsure what to do. (Everyone willing to participate.)

Showed them y=mx+b on desmos, but got caught up in another problem. We'll come back to this tomorrow.

They worked on finding a_n, with the hint that it might be good to find a₁₀₀ first, for the sequence 12, 17, 22, 27. (I got starting value and jump size from students. Good it was five - some people struggle with arithmetic.) We worked on that a while, and then did a problem from 12.1 (Stewart) that turned out to be geometric. It was good to see the similarities.

I loved my day. Now I'm off to the chiropractor.

First Day of Class

So I have this health problem. It's seems to be a GI problem, but no one has managed to diagnose it. It used to happen about once a year, for a week. Now it happens more often. Always before my tummy hurts, I notice that my back is tight. I've had lots of ultrasound scans - no gall stones. And a GI scope - no ulcer. It is not clear what's happening. The chiropractor seems to help, but it still lasts a week or so, just milder. The first few times, I was in the emergency room, in agony. This time, I made it through my first day of classes, able to ignore it, and am distracting myself by writing a blog post. Digestive enzymes may be helping. Stress is a factor: the last time it happened was the beginning of fall semester. The time before was when we were interviewing candidates for a position in our department. I think it's time to figure out how to notice stress, and not internalize it this way. Meditation? Yoga? Definitely, I need to get more active.

I was very pleased that I was well enough to ignore my body while I was teaching. I hate not being chipper on the first day.  I was probably less prepared, and less organized in some ways. But that meant that I did new things that I liked.


8am
Calc I. I had 42 students. I talked about math not being as much about memorizing as many students think. We listed some of the things that do need to be memorized, and then I talked about which things might not belong on the list. Then we did the tangent line activity I always start with. Time was a little tight, and I didn't finish attendance. (I'll have to attend to that tomorrow.)  ;^) Tomorrow we'll look at area of a circle, think a little more about today's activity, and start on the first activity from the Boelkins text. I've put together a coursepack with all the handouts for this first unit, so students won't feel so scattered. (We do very little from our Anton textbook in this first unit, and then use the textbook much more for the rest of the course.)

Even though the way I started wasn't anything exciting, I feel excited just because I did something new with the students. It feels like today was a good start.


10am
Linear Algebra. I used the same warmup activity from the last time I taught the course. I don't need to do anything new to be excited about this course. What's new it that I'll be grading their homework (with help from an assistant), instead of just stamping it.


In one of my classes, I talked about neuron development during learning. But I don't remember which class. I am nervous about having two classes in a row with no break between. I hope I can keep track of what I've done and want to do next.


11am
Pre-Calc. I only had 15 students. There is some chance the college will cancel the course. I think I have a great bunch of students, so I am already feeling very invested in doing this course. I have no control over whether it gets canceled, though...

This is the class I wanted to change up some. Our first day hadn't been very memorable in the past few semesters. I couldn't think of anything better than asking them to add up the numbers 1 to 100 (without adding one by one). They worked in groups of four. They looked very stuck for a long time, but seemed willing to keep at it. One student knew a formula, and had the answer written. I asked her to hide that, and try to figure it out without the formula. In another group, I heard something about adding up pairs. When I finally called them all together, I asked for volunteers and got none, and then asked M to explain. They  had started out by adding up the numbers one to ten. (I had suggested that might give them some ideas.) They did the sum one by one at first, so they knew what number it would be. When they noticed 1+10 = 11, and the total was 55, they looked for a reason to multiply by 5. Their reason didn't make much sense to me. (There are five ways to add two numbers to get 10, one of which is 5+5.) So I agreed that multiplying by 5 made sense, and asked if we could think again about that 1+10 pairing. I had to nudge more than I would have with a math circle, but they still did most of the work. They found the sum of one to a hundred in their groups. And then we came up with a 'formula' for summing the numbers one to n. I liked how it went.

I think this class may take off this semester...


1pm
I'm sitting in on Calc III. I loved Ed Cruz's talk, about how bad a student he was at first, and how he evolved to become a better student, and then a teacher. No math yet.

Tomorrow my 8am and 11am classes meet for an hour and a half. From 8am to 12:30pm, I'll only have half an hour outside the classroom. Yikes! But being done so close to noon sounded too great to pass up, so I didn't try to change this strange schedule. I see my chiropractor tomorrow afternoon.

Share #YourEduStory Blogging Challenge 2015

I decided to join the Share #YourEduStory Blogging Challenge 2015 this morning. Last month was the first month I ever skipped completely on this blog, since I started in 2009, and last year I posted less than any other. So this is a good time to push myself a little.


One Word
The group that set this up (folks from edcamps) has created a list of topics for those who need them. The first week's topic (last week) was:

What is your "one word" that will inspire you in your classroom or school in 2015?

I'll choose 'Connect'. I want to connect with my students in the classroom and out. I want to connect with my colleagues as I work on creating a new course. I want to connect with friends more - I need more of that in my life.

I'd like to get photos of my classroom up, and videos of lessons. But I won't make a commitment to that. Just one new thing at a time...


A Better World
Week Two's topic question:

Inspired by MLK: How will you make the world a better place?

I hope my book, Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers, will help people to enjoy math, fear it less, and overcome the hurdles math requirements place in their way. It's coming out within the next month or two.

Join Us

I've been following hundreds of math teacher blogs, and, though I haven't pared down my list, I'm seeing very little writing from math teachers these days. Will you join the Share #YourEduStory Blogging Challenge 2015? Each week you write a post and tweet it, and read a post and comment. That's it.

Think: The Five-Letter Word Game

Linda taught me this game when I was visiting her in Muskegon. She just called it "the five-letter word game." I thought it needed a name, and Think is as good as any.

Preparation
Each person thinks of a five-letter word with all five letters different, and writes it down in a hidden spot.

Goal
Guess your opponent's word.

Play
The players take turns offering a five-letter word (all letters different), as guesses, and as clue-gathering. If the guess is not the right word, then the opponent tells how many letters are right.


The game is similar to mastermind, but there's no information about where the letters go. (So you could get all five letters right, and still have the wrong word!) It's not a math game, but it is a logic game, and I think logic and math are twins.


Example
I'll share my guesses:
STOMP 0 (No letters right. Great. I can cross all of those off my alphabet where I'm keeping track.)
BLAND 2
QUOTE 1
FLASH 2
DIRTY 1
TRUCK 0 (Now I know 9 letters that aren't in the word.)
VOWEL 2
EXITS 2
QUICK 1
LARDY 2
BINDS 1
NIGHT 2
LINER 3

Would you have picked other clue words? Do you have enough information from these clues to guarantee any of the letters in the word?

Calculus: Planning My "Standards", Calculus Skills to Master & Mini-tests

I don't like the term 'standards-based grading' (SBG), because it reminds me of standardized tests. But I like the ideas behind it, and it's the term used by most of the teachers who are grading based on the goals they have for student learning. (Hmm, would 'learning-based grading' be any better? I like it better, but maybe it loses something for others?)

I know that most people who use these methods have more "standards" than I do.  I thought about all of the content of each course, looked at what's been on my tests, and portioned it out into what I have been calling mastery tests, though I think I'll start calling them mini-tests. I came up with 15 (later 17) for calculus. (Precalc has 16.) Each time I give a test, it includes about 4 of these mini-tests. Each one gets a grade, and can be retaken. The ones I'm willing to make lots of version of, I allow students to retake in my office when they are ready to show mastery. (Below the skills list is a description of how to show they're ready to retake.) The ones I can only make a few versions of, we schedule a time for them all to do the retake together. They get at least two chances on each mini-test and the final and as many chances as they want on quizzes.

Here's what I've been handing out on day one for the past few semesters, along with the syllabus. I fine-tune it each semester:



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Calculus Skills to Master

The test portion of your grade will be based on the 17 mini-tests described below. You will have more than one chance on each of them. On the mastery tests marked (RA), you may do the retake any time (in my office, after showing me your retake packet). The other retakes will be scheduled for once before class, on a day we arrange as a group. Use this sheet to record your grades.

Problem-Solving

You will have multiple chances to solve one problem that requires problem-solving skills. (Each test will have one of these, and there will be a few available during the final.)

Unit 1: Exploring the Idea of the Derivative

·            Definition (RA, as oral exam)

·            Tangent (RA)

·            Graphs

·            Rate of change (from a table of data)

Unit 2: Derivatives of Polynomials, Limits, Trig functions, Products & Quotients; Graphing; Optimization

·            Derivatives (RA)

·            Limits

·            Graphing Polynomials (RA)

·            Optimization

Unit 3: Composition, Exponential Growth, Implicit Functions, Related Rates

·            Basics (RA)

·            Graphing with Limits (RA)

·            Implicit   

·            Related rates 

Unit 4: Anti-Derivatives, Area, Volume

·            Anti-derivatives  (RA)

·            Area

·            Graphs

·            Position, Velocity & Acceleration

Unit 5: Volume

(No mastery test, but this will appear on the final exam)

Notice that graphing appears repeatedly. Graphing is a great way to visualize functions. Each graphing mini-test will cover different skills.

Retake Packets

For the mini-test or quiz you wish to retake, you will do the following for each problem you got wrong:

1.     Re-do the problem you got wrong. If you aren’t sure how to do it correctly, get help (from me, a tutor, or a friend). Once you are sure you have it right, think about what mistakes you made on the test, and explain what you did wrong.

2.     Find 3 problems (in the textbook or one of our supplemental texts) like the one you got wrong, do them, and check your answers. If you needed help to get these problems right, do more. Keep working until you can get 3 right on your own.

3.     Email me to request a retake appointment. When you come, be ready to show me a packet with the test on top, including your revisions and practice problems for each problem you got wrong.

Linkfest, part 3

Making Math

• Simple Sonobe (Sonobe units are used for multi-piece origami constructions)
• Weaving the Three Times Table
• Fractal App

Interesting Questions from Classrooms

• Circles Rolling on Circles
  
• From Graph to Equation (See the last question on this post.)
• On 'wait time' in Calc II (Wait for it... he talks about words before he gets to the math. This guy is brave. I can't imagine waiting "a few minutes" for students to answer a question.)
• Games for Complex Numbers
• Analyzing a Sonic Boom in Calculus Class

Education Politics

• Who Gets to Graduate?
• Women in Science
• Does Math Teach Higher Order Thinking Skills? (Not necessarily...)

Other coolness

• Analyzing the Game of Go
  
• Statistical Noise in Reporting
• Links for Statistics Teachers
• Don Cohen, Math Man, Interview As He Ends his Career

Linkfest, part 2

(Saved from the drafts bin...)

• These gifs are making the rounds on Facebook. I'm putting the link here so I can find them later.
• Henry Segerman makes fascinating mathematical artwork using 3D printing. While honing the description of the piece we'll be featuring in Playing with Math, I can across his 100 prisoners and a lightbulb paper. The puzzle posed seems impossible, but...
• Here's a much easier puzzle from 9gag (found on Facebook).

Linkfest

Hour of Code
Skip Counting with Manipulatives
 

Puzzles and Games
Skyscraper Sum puzzles (from Tanya Khovanova)
Geometric Construction puzzles (from Henri Picciotto)
Chomp (I haven't beat it yet...)
Just Make 10 (harder than 2048 for me, easier for some of my friends)



 (Wondering about Playing with Math? The book will be out within a month or two. I sent in corrections to the page layout proofs yesterday. Another round of corrections, last round on the index, off to the printers, and we're done.)