## Saturday, September 29, 2012

### Kindle Book: Let's Play Math!, by Denise Gaskins

I love Denise's blog, and now I'm loving her book. She wants people to understand that this is a beta version, which means that it will be even better when her official first edition comes out. She hopes to have it out in print next year.

But if you don't mind reading electronic books, it's only \$3.99 for the Kindle version now. (You don't need a Kindle; you can read these online or on your computer.) Once she gets it perfected, I'm sure the price will go up.

Here's a peek at part of her chapter on home-made manipulatives:
Or put the number line at a slant, like a hill on which you can run up and down. In many applications of math and physics, “zero” is an arbitrary location, based on whatever makes our problem easy to solve.
In this case, you might invent a peasant who lives in a small hut at 0, partway up the hill, and dreams of becoming a knight. At the top of the hill is a castle, and at the bottom there is a cave where a terrible dragon lives. Take turns making up story problem adventures.
I don't know when I'll have time to savor the rest of this book, but I'm definitely looking forward to it. It'll be even nicer to have the print copy in my hands one day. I'll be waiting...

## Tuesday, September 25, 2012

### Math Stories: Special Triangles

Two weeks ago I was introducing the special triangles (with angles of 45-45-90 and 30-60-90) in my pre-calc class, and on a whim, I asked them to make up stories with those two triangles as characters. Only about a dozen students did the assignment, but what they created was really fun to read. A few students have given me permission to share their stories here. I'm sharing those 3 stories below. I may get to share a few more later.

This post is my birthday present to myself. Anyone else want to share some math stories?

Different Views of Shapes, by Aldrich Pablo
Once upon a time, there was a square name Geo. Geo was a hard working square who worked in the slaughterhouse. Geo loved his work. He loved his work because his wife, Tri, an equilateral triangle, also worked at the slaughterhouse alongside of him.  Like any other day, both Geo and Tri cut fellow square and triangular shapes vertically and diagonally, making sure they cut them into 2 different types of triangles, a 45®-45®-90® triangle, derived from the squares, and a 30®-60®-90® triangle, derived from the equilateral triangles.
One day however, Geo got accidentally pushed into the cutting machines by his enemy, Bre, the circle. With the affectionate love Tri had for Geo, Tri jumped in and tried to save Geo. But it was too late. Both Geo and Tri have been sliced diagonally and vertically. Geo became a 45®-45®-90® triangle, and Tri became a 30®-60®-90® triangle.
Even though Geo and Tri got sliced in half, they both wanted to do the same to Bre. With all the mixed emotions both triangles have, they were able to learn something new about each other. Geo, the 45®-45®-90® triangle, learned that he still has the same leg lengths, but half of the original shape. Tri, the 30®-60®-90® triangle, learned that she is just half of the original shape, forming different angles.
With previous shapes that have been sliced by the workers, Geo and Tri were also able to understand that with their new shapes, multiple of the same shapes together form a circle. Bre was horrified. Once Geo and Tri were able to understand their new shapes, they too, pushed Bre into the slaughter machines. Because of a malfunction when Bre went into the machines, Bre got stuck, and the machine exploded.
When the machine exploded, both Geo and Tri saw him fly into the air. When both triangles went to go look for him however, Bre was nowhere to be found. They believe that Bre became the sun, and he was never to be seen again. So every time when Geo and Tri go to work at the factory, they will always remember Bre, as they look towards the sun.
The End.

~  *  ~  *  ~  *   ~  *  ~  *  ~  *   ~  *  ~  *  ~  *   ~  *  ~  *  ~  *   ~  *  ~  *  ~  *

The Story of the Special Right Triangles, by Miranda Barron
Once upon a time, there were two special triangle cousins, one named Mr. 45, full name 45-45-90 Triangle, and the other named Ms. 30, short for 30-60-90 Triangle. These two cousins both inherited their family’s 90° angle trait yet they each were very different and special in their own way. Mr. 45 had two legs that were equal lengths, meaning he also had the equal corresponding angles. If Mr. 45 could walk he’d walk like a human. Hard to believe, isn’t it?  Well Ms. 30 wasn’t so lucky as to have equal lengths. She had to constantly ask for help from her cousin. It was a hard life for her, but she never let it bring her down.
The reason she learned to live with it was when she had kids. Each kid had different sized legs just like her. They even had the same proportion. She was always able to find the sizes of pant legs for her kids without having to measure each child’s leg, since it made them feel self-conscious. Each child, like her, had a proportion that had to do with one leg being x inches, then the other leg was x√3 inches, to go with the rest of their body that was 2x inches. It was the magical method to go with their cursed life.
On the other hand, Mr. 45 and his kids weren’t so unique. He still found a way to make his children seem special with their normal equal length legs. He found that each of his kids were proportioned like him. Each of their legs would measure x inches to go with the rest of their body that was x√2.
This seems like a weird story but it shows how unique and special you can be when you’re different. Mr. 45’s kids never got their pants personally made, one reason being their dad refused to sew, while Ms. 30’s kids got to have personally made pants from their loving and caring mother. Being special is amazing and that’s how things should be for everyone and everything. :D

~  *  ~  *  ~  *   ~  *  ~  *  ~  *   ~  *  ~  *  ~  *   ~  *  ~  *  ~  *   ~  *  ~  *  ~  *

A Special Triangle, by Ayesha Saleem
Isis is a triangle and her friend, Trinity, is a triangle who is also a conjoined twin. She is conjoined with her brother and together they are an equilateral triangle, which has three congruent sides and angles. About two weeks ago, Trinity and her brother underwent a rigorous surgery to get separated from one another. They had to stay in the hospital for a little over a week so that the doctors could keep an eye on their recovery. The siblings were allowed to go home yesterday, and today Isis and Trinity went to the park to hang out. Isis wanted to see how her best friend was and how she looks now that she is not connected to her brother by one side.
They met at a local park where they used to go to a lot when they were younger. Isis was so shocked at how differently Trinity looked. She was so happy that she has been recovering well and that she is happy with the surgery. Trinity brought along a measuring tape and a protractor. She wanted Isis to help her measure her sides and angles since she didn’t know what their measurements were anymore.
Isis wanted to start by measuring her sides, so they started with Trinity’s base. They measured in across the bottom to be one foot long. Then they measured her hypotenuse, which was two feet long. Now they had to measure her height. They measured it and it came out to be a weird, decimal number. So, to be more exact, they decided to use the formula, a2+b2=c2, to find the last length. They found that Trinity’s height was √3 feet. They found this by doing the following:
a2+b2=c2

a2+12=22
a2+1-1=4-1

a2=3
a=√3
Next, they moved on to measuring her angles. They stared with the angle made between her height and base. They used the protractor and measured that it was a 90o angle, or a right angle. Then they measured her bottom right angle and found it to be 60o. Since Isis couldn’t reach up to the last angle at the top of Trinity, they decided to do it mathematically. They already knew that all triangles have three angles that will always add up to a total of 180o. They found that the last angle is 30o by doing the following:

90+60+x=180

150-150+x=180-150
x=30
After doing all the measuring, they discovered that Trinity is now a Special, 30-60-90, Triangle. Trinity was so surprised; she didn’t think that she deserved to be a Special Triangle! Isis congratulated her and she was happy for her best friend. Trinity suggested that they measure Isis and maybe that she is also a Special Triangle, but Isis said she already knew that she was an Isosceles Triangle, meaning she had to equal sides and two equal angles. Isis was fine with being an Isosceles Triangle and Trinity was happy to find out that she was a Special, 30-60-90, Triangle.

## Friday, September 21, 2012

### Apps: Dragon Box

I don't have an ipad or an ipod touch myself, so I don't know much about apps yet. But lots of the folks on Living Math Forum are talking about how much fun the Dragon Box App is, and how much algebra their kids are learning from it.

## Sunday, September 16, 2012

### Meet the New Bloggers (week 4)

The last installment... I hope you've subscribed to your favorites. Introducing Kyle, Maggie, Erin, Jillian, Kate, Nate, and (once again!) Algebrainiac.

Kyle Harlow (@KBHarlow), blogging at War and Piecewise Functions, wrote Just Some Cell Phone Photos From Denver. His summary: Spent last weekend in Denver, CO and went to the Broncos game.  Here are some pictures from my trip.
The hotel we stayed at had a restaurant called Pi Kitchen + Bar.  Its menu was a circle, and happy hour was from 3:14p to 6:28p.

Maggie Acree (@pitoinfinity8), blogging at pitoinfinity, wrote PreRequisite Knowledge/Rev. Her summary: This post is about reviewing concepts and what teachers do for reviewing. I used to spend a six weeks or so reviewing concepts from previous years, but it really did little to no good, so I have a solution I have found that has worked well for me.
No matter what, I am done reviewing concepts for the first six weeks.
What Maggie calls 'bell ringers' I call 'warmups'. I like her idea of using warmups to do the necessary review of concepts we wish our students already had down.

Erin Goddard (@ErinYBaker), blogging at Math Lessons on the Loose!!, wrote Thanks Blogger Community . Her summary: I related to mathemagicalmolly's blog. Only a teacher knows what a teacher's night sleep is like.
I always saw the real benefit of taking [from other blogs], but I learned the true benefit of reading, relating, learning, and also giving back hopefully as much as others have given me.
How nervous are you at the beginning of the school year?

Jillian Paulen (@jlpaulen), blogging at Laplace Transforms for Life, wrote My Math Autobiography (a week late). Her summary: I’ve always assigned a “Math Autobiography” to my Geometry students and I’ve really enjoyed reading them. But I’ve never written about myself! So here’s my (long) story.
My love for math has only grown since I started teaching, and I hope I can continue for a long, long time.
Her math ed course seemed too fluffy. I hear that. I wonder if there's a way to draw in the math talent in the math ed courses.

Kate (@fourkatie), blogging at Axis of Reflection, wrote Grade/Age Equivalents are NOT numbers!. Her summary: For my final post for the new blogger initiation I opted to write about what was on my mind. Today that was the use of age and grade equivalents by a special educator in a report. I stepped on my soap box to rant about why age/grade equivalents are NOT numbers and therefore should not be treated like they are.
Because they are not real numbers, so you can't do math with them like they are real numbers.
Do any one or two summary numbers really tell you anything important about a student?

Nate Gildersleeve, blogging at Hard Enough Problems, wrote Visual Multiplication. His summary: This post talks about a visual multiplication lesson I did, and what my rationale was for it.
It is that I want to use this as a way to practice and learn several things: the idea that there are multiple valid ways of doing something; if those ways do the same thing they are connected in some way; and by talking about these connections we gain a deeper understanding of whatever we're covering.
I'd like to hear more about the connections involved in doing the same thing two different ways.

Algebrainiac (@algebrainiac1), blogging at Algebrainiac, wrote Open House/Curriculum Night. AB's summary: I posted my plans for the set up of my room for 8th grade Open House, which was a new format for us this year. I included links to handouts and files I prepared.
The main difference I have seen so far is that the Open House format seems to require more front end work to prepare, but I don’t imagine I will leave as tired and drained as in past years.
I hope it went well. AB sure put in the preparation!

Maybe by next year, we'll have enough teachers at each level that we can split off. I'd love to review a half dozen new college math teachers' blogs.

_____
The round up of week four is at these blogs:: JulieFawnAnneMeganBowmanSamLisaJohnShelliTina, and Kate. And a roundup sorted by grade level taught (for those who responded to Julie's survey) is here.

## Saturday, September 15, 2012

### Get Ready: September 25 is Math Storytelling Day

Maria Droujkova made it up in 2009, as a birthday present to herself. September 25th is my birthday too, and I loved the idea. So now I'd love to get lots of presents - math stories you all write. Here's my post on Math Storytelling Day 2 years ago. (I must have been too busy last year?)

If you'd like some inspiration, read The Man Who Counted, The Number Devil,  or Math Curse.

Poems are good, too.

## Thursday, September 13, 2012

### In Seattle? Fun opportunity for Elementary Teachers

I love Dan's blog (Math for Love), and have met him in person. If you get a chance to do this free Math Teacher Circle, I think you'll love it.

## Wednesday, September 12, 2012

### Thanks for Reading Math Mama Writes ...

... I hit 100,000 page views late last night.

## Monday, September 10, 2012

### Planning Calculus: What Is My 2nd Unit?

A few weeks ago, just before the semester started, I had unit 2 as calculating the derivative, and unit 3 as using the derivative, much like Math Boelkins does. It seems like a logical division. But now I'm thinking it's not a good way to frame things pedagogically. We would spend way too much time coming up with 'rules' and learning to use them. That would get the students so stuck on the procedural that I'd never be able to get them to go back to trying to make sense out of these new ideas.

I want to introduce the 'rules' in the context of real problems. I don't have the facility with equipment that Shawn Cornally does. If I could spend a year as an intern in his classroom, I think I'd take lots of his ideas back home with me. But every time I think about using lab equipment, I get nervous. (We used a lab thermometer in pre-calc to measure the temperature of a hot cup of coffee during our murder mystery. And we built inclinometers with a protractor, tape, a straw, thread and a weight. That's about the extent of 'lab equipment' for me so far.)

To rethink this, I started with the sections of our textbook (Briggs Calculus) that I'm supposed to 'cover':
3.2: rules (constant, sum, power, constant multiple, e^x!)
3.3 product rule
3.4 trig function derivatives
3.5: derivative as rate of change
3.6: chain rule
3.7 implicit
3.8 derivative of log and exp fcns
3.9 derivative of inverse trig fcns
3.10 related rates

4.1 maxima and minima
4.2 what derivatives tell us
4.3 graphing
4.4 optimization
4.5 linear approximation and differentials
4.6 mean value theorem
[3.1 was part of the first unit, and the last two sections of chapter 4 will be part of the last unit.]

I played around with separating the 'rules' and the 'applications', and it seemed like there were only 4 applications (highlighted, with 4.1 to 4.3 as one). I'm not sure why the text splits things up the way it does. Graphing could come earlier, and seems like a simple way to begin seeing what derivatives can do for you.

In the first week of class I had the students read some history from Morris Kline's (pages 1 to 6). He gives 4 major problems that needed calculus (no wonder two people invented it at once):
1. Motion:  planetary & projectile,
2. Tangents to curves: for projectiles (will it hit head on?) & for lenses (telescopes and microscopes),
3. Optimizing (best angle to shoot a cannon, when will a planet be closest and farthest from sun?), and
4. Lengths of curves, areas, volumes, also center of gravity (for example, the volume of the earth, which is an oblate spheroid; these topics are mostly delayed to calc II).
I'm wondering if any of these historical applications are useful to bring into the mix.

What I have for now is:

App1: Rate of change & Graphs
Rate of change:
3.5 derivative as rate of change, velocity, rate of growth, cost (we mostly did this and this can be review)

In pre-calc, we factored to find x-intercepts of polynomials. What we couldn’t find was the maxes and mins, which are often pretty important.
3.2: rules (constant, sum, power, constant multiple, e^x! skip e^x for now)

4.1: maxima and minima
4.2: what derivatives tell us
4.3: graphing

App2: Sound!
3.4 trig function derivatives
3.3 product and quotient rule  (Shawn C uses 1/x * sin x to model guitar string)

App3: Differential Equations & Growth, Related Rates & Optimization
Infection (logistic application by Bowman D)
e^x
3.6: chain rule
3.7: implicit
3.10: related rates
3.8 derivative of log and exp fcns,
3.9 derivative of inverse trig
4.4: optimization
I still need a home for:
4.5 linear approximation and differentials
4.6 mean value theorem

This would make 3 units instead of two. That might be fine. I have to decide today, and get a unit sheet made by this evening. I keep wishing I could teach for a week and prep for a week. (Dream on!) But as hectic as this is, I'm having fun working on getting it right. And I know it will go more smoothly next semester.

### Time Out (We Interrupt This Blog for a Public Service Announcement)

The last time I did this was over 2 years ago, when I recommended Dropbox as a way to do backups and to get files between different computers. I am still loving Dropbox.

I've been getting some physical therapy lately because I get too many headaches, and my neck is often tight. My physical therapist asked lots of questions about my computer habits: Where do I sit, how do I type, do I take breaks? Nope, I told him, I don't take breaks, mainly because I forget to. When I'm writing or powering through my google reader, I forget about anything else.

But as we talked I remembered that Maria Andersen had recommended some software that would make you stop working. She recommended WorkRave, free for the PC (not available for the Mac). I tried finding something for the Mac, and didn't see anything good. I asked on her blog, and Maria suggested Time Out. I've been using it for over a month, and I love it. It's freeware, but of course they want donations. (I've paid them \$10.)

Every 15 minutes, Time Out comes on top of what I'm doing, and tells me it's time for a 15 second 'micro-break'. Every hour, it has me take a 1 minute break (which I can skip). All the times can be changed, and there are other settings you can fiddle with. Whenever I'm on Skype, I have to tell it to 'pause breaks', which takes 3 quick clicks.

If you work intensely on the computer, and need to take more breaks, Time Out works great.

And now, back to our regularly scheduled programming....

## Sunday, September 9, 2012

### Meet the New Bloggers (week 3)

This is the shortest bunch so far. Introducing Rachel, Emily, Kevin, Meagan, Algebrainiac, Nate, and Aaron...

Rachel Tabak (@ray_emily), blogging at Writing to Learn to Teach, wrote Moar River Crossing and Whiteboarding!!! Her summary: In this post, I share one of my (just-discovered) favorite problems, a river-crossing puzzle designed by Mark Driscoll to help kids develop algebraic thinking. First, I relate how we approached this problem in my classroom. Then, I reflect on what I noticed as I watched my students work with various models (all the while honing their perception of what clear mathematical communication entails).
Students independently discovered that they needed to seek out structure within this problem, and then they did just that - without my saying a word.
I love this problem! (I thought I had blogged about it long ago, but I never did.) We did this at the 'math salon' I hold at my home with families. Thanks for reminding me of it, Rachel.

Emily Allman (@allmanfiles), blogging at  Algebra, Essentially, wrote Parenthetically Speaking. Her summary: My aha moment surrounding the definition of parentheses.
I find a subtle beauty in tiny moments of enlightenment, even if it is only my own.
What are parentheses?  Emily knows...  I love lightbulb moments.

Kevin Laxton, blogging at A Beginners View of Math Education, wrote Favorite Math Quote (New Blogger Initiation Post 3). His summary: I love this quote. Man, do I love this quote!
"Mathematics is the language with which God has written the universe."  - Galileo
Is there a god? I'm not sure. Is there math? Yes!

Meagan Bubulka, blogging at variablesofmath, wrote Flipping and Common Core. Her summary: This blog is the start of my blogs on flipping my classroom.  I also discuss how that will play into Common Core.  It gives an intro and then I have taken the liberty of answering many of the questions I have heard for the past 8 months since I started Flipping!
This is what we all wanted – time to let our students get to mastery through activities, projects, labs, etc.
I can't yet imagine doing a flipped class. It seems like so much more prep to do video lessons (and have activities for classtime).

Algebrainiac (@algebrainiac1), blogging at Algebrainiac, wrote Why I teach Math. AB's summary: My post this week is all about why I chose teaching and why I teach math.  It was an interesting road for me, but I couldn't be happier that I ended up a math teacher.
I grew up playing school with my little sister, older cousin and sometimes my stuffed animals and even my barbies played school.
I played school when I was little, too. Then, as a high school feminist, I thought I should do something 'more important'. Hah! I love teaching math too.  Thanks for sharing your story, AB.

Nate Gildersleeve, blogging at Hard Enough Problems, wrote Alg 2 and Precalc. His summary: It's a short post, but I make the argument that Algebra 2 and Precalc are both centered around exploration of functions.
Algebra 2 and Precalc should be called Functions I and Functions II.
Agreed.

Aaron C. (@CarpGoesMoo), blogging at Random Teaching Tangents, wrote New Blogger Initiation 3. His summary: Relating how I like to introduce students to "real math."
I mean 2 + 2 = FISH is honestly just as valid if you know what you’re doing.
i heart fish?  ;^)

That was fun. Next week is our last chance to savor all these new blogs together. So get your reading in while you can.

_____
Roundup of all the week 3 posts: Julie, Fawn, Anne, Megan, Bowman, Sam, Lisa, John, Shelli, Tina, and Kate

## Saturday, September 8, 2012

### Calculus: Finishing Our First Unit

For the textbook, I told students they could get the official textbook (Briggs, pretty expensive) or one of the open source textbooks (free, under \$25 for print copies of both Boelkins and Guichard).

Unit 1: Exploring the Idea of the Derivative
I've worked with my students for 3 weeks on what is mostly contained in one section of the Briggs (3.1). We started working on average velocity activities in Boelkins and on slope of a secant line. We touched on limits just enough to make sense out of the definition of derivative. We use the definition with h -> 0 most of the time, but did look at a few problems using x -> a. There aren't enough really simple homework exercises available to help students see the basic concept. I'd like to have more to provide next semester.

I will be testing them on Monday. The atmosphere in the classroom changed on Thursday from positive, seeming to be engaged, to tension and distress. I talked with one student, C, who seemed particularly unhappy (by his facial expression and body language). He said he was irritated, and I asked him to come to my office so I could help him. I did something that felt to me almost exactly like what I had done in class a dozen times - I talked him through the slope idea. But he suddenly got it, and was much happier. I don't think he'll know how to practice, though, so I'm guessing he'll need to re-take the test - along with many others. I want to have material he can use to get up to speed; more on that below.

Back to my interaction with C. We chatted until we got to my office, then I closed the door so he'd have more privacy and asked him how he felt about the course. He has to take it for business, and I have the impression it's just a hoop he has to jump through. He has felt like it's not making sense since day one. That is not surprising when the students are so used to algebra classes - find out the new procedure the teacher is pushing, learn how to follow it, practice a few times to get the kinks out from all the other algebraic procedures cluttering up your brain, done. He said, "We were doing one thing, and then switched to something totally unrelated." I see the relationships between the activities (all leading toward derivative, or working with it), but the students don't. They are all related by one basic concept. But the students are looking for things to be procedurally related.

As I tried to understand what the two unrelated things were, I started working with him from the beginning. Did he understand f'(x) = the limit as h goes to 0 of (f(x+h)-f(x)) / ((x+h)-x)? His response made me back up more.

"Do you know slopes for straight lines, from algebra?"

"Yeah. y1-y2 over x1-x2."

So I wrote down what he had said, and drew my picture of a curve with one point labelled (x,f(x)), which I call the stable point, and a second point with x-coordinate x+h, a distance h away from the first, labelled (x+h, f(x+h)), which I call the sliding point. I asked if he could see that these were two points on the curve, with their x and y coordinates labelled. Yep. Then I crossed out the y1 and replaced it with f(x+h), and did the same with the other 3 terms in his slope definition. All of a sudden the light went on for him. I think the notation (f(x+h) especially, see this post and comments) gets in the way for students with weak algebra skills; I also think their skills can be strengthened. I was excited that it suddenly clicked for C, and I see 3 possible factors:
1. we were working with his definition of slope
2. I crossed out and replaced things term by term, so it was clear I was just using different symbols for the same thing
3. we were able to zoom in on his stuck point because we were one-on-one
If #2 was the most important, a small tweak of my lecture will help a lot. But I suspect that each student will need slightly different things. I have 45 students (we started with 50, 5 have already dropped). It would be hard to have individual sessions with each one. But maybe there's a way. They're in groups of 4. I've changed the groups every week so far, so they'd meet lots of other students. Now they get to pick a group and stick with it (at least for the next unit, maybe for the rest of the course). If I talked with one person in each group, maybe they could then help each other. I think I could do that in a week. Hmm...

Today and tomorrow I'll be creating the test, framing the next unit, and thinking about how to help students who didn't yet pass the first test. For material that might help them, I think I need exercises they can do on their own. Problems that have less notational issues, but get them thinking in the right direction. Christopher Danielson proposed working with finite differences. I think that might help them see the light. I wonder if there's any already-made materials for this. (If you know of anything, please point me to it.) I'll be trying to come up with something myself.

Unit 2: Calculating the Derivative
We'll be working on all the usual 'rules' of derivatives. I am thinking about which applications to mix in so that the material includes some motivation. Here's where we need to be more interactive. A textbook can tell the 'rule' and show the proof. But that won't stick. Students need to find the rule from a pattern, then see the need for proving it will work in all cases, then be able to come up with parts of the proof, and be able to explain and reproduce other parts.

Right now I'm thinking about how to lead them to the power rule. The pattern is easy to see from a few examples.  They'll want to assume that pattern holds. Is there time (and are they ready for) an activity that shows that patterns you see aren't always the right one? (My favorite is the circle where you add a dot and lines to it from all other dots, and count the total regions. But the solution to that is way hard. Ben Blum-Smith has two posts collecting problems with patterns that break. I might try some of those.) I want to show them Pascal's Triangle (1653, but used 600 to 700 years earlier in India and China, by Halayudha and Jia Xian, respectively), and get them to describe the relationship of (x+h)^n with a row of the triangle.

More to come...

## Wednesday, September 5, 2012

### New Routine: Compiling Mathematical Habits of Mind

Thanks to Avery, the phrase 'mathematical habits of mind' has been in my brain more lately. Now, every time I point out a mathematical habit of mind (MHOM) in class, I want students to add it to their growing list of MHOM. I ask them to turn to the last page of their notebooks and add it.

I've found myself doing this in each of my classes, and I think it's a great change to be making.

In my office, when I tell a student how to check their work, I have them write down: "Check the steps" on the last page of their notebook. And when I ask them to re-read the question when they're done, to see if they've answered it, I ask them to write: "Look back to see if I've answered the question I was trying to answer."

I've mentioned precision and organizing information. I think I've also mentioned trying a simpler problem first.

I've often mentioned these things in the past. (In fact, here's a blog post where I summarized some of these ideas.) What I'm doing differently is asking the students to compile a list of these ideas, to help them become more self-aware as they work on math.

Maybe a month or so into the semester, we can take some time out to discuss their lists and make a class list, with examples. I like where this is going.

## Monday, September 3, 2012

### What is a good textbook?

First of all, there are things a textbook cannot do well, as Dan Meyer points out. A textbook can't wait for student responses in order to find out how helpful it might need to be. Nor can it show you a video of a mathematically-relevant situation. Another sort of problem with textbooks is that inexperienced teachers follow them too closely, and students believe them too readily. 'Question authority' is a pretty good attitude to bring to mathematics, and textbook use does not promote that.

But perhaps, like me, you still want a textbook to offer your students as a one-stop resource for the material you'll be thinking together about in the course. For me, the most important function of the textbook is providing homework problems for my (college) students to use to practice. I also want the mathematical progression of ideas to be recorded somewhere that the students can easily access.

So what makes a good textbook?

At the least, it needs to:
• do the math properly,
• be written in good pedagogical style,
• provide problems that progress from simple (so the students who want to learn the ideas through the problems can) to deeply thought-provoking, including problems that address whatever twists and turns can come up with the material.
• ??   (What other criteria would you add?)

I told my pre-calc students I'd help them get print copies of one of the open source books I listed, with just the sections they'll need included. But I'm not particularly happy with either one. One thing I do like is that the personalities of the author teams shine through in both of them. (1 pedagogical point earned.)

Here are a few of the things I noticed:

Stitz and Zeager seem to be going for better mathematical reasoning than the commercial textbooks. On page 11:

... the distance formula. Mostly I like what they're doing here. (Many texts treat it as separate from the Pythagorean theorem, which makes me cringe.) But I would not ask 'Do you remember why ...' here. I'd ask if they know why, understand why, or can figure out why. Remembering is for factoids, and why questions should not be thought of as having that sort of answer. I'm reminded of my post on teaching for understanding. Just as the word 'understand' can have deeper and shallower meanings, so can the word 'why'.

Also the use of 'extract' is a bit archaic. It refers to the process used before calculators (bc?) for finding square roots. Nowadays we 'take' the square root of both sides.

Lippman & Rasmussen are probably less likely to include an archaic term like 'extract', but they have other problems. On page 158, in a list of 6 vocabulary words referring to polynomials, they include:
A term of the polynomial is any one piece of the sum, that is any aixi. Each individual term is a transformed power function.

Does 'piece' do anything to define 'term'? When I'm teaching, I sometimes say 'chunk' myself, but I think we do this because we see the terms as standing separately. Our students don't yet see that. So I try to always add in something more helpful like 'terms are separated by a plus or minus'. If you're going to define things in a math book, your definition must give more information than the original word did. 'Any aixi' does that, by example, but a definitional phrase would be helpful.

On the next page, they say:

For any polynomial, the long run behavior of the polynomial will match the long run behavior of the leading term.
They give no indication of why this is true. I do not want to encourage students to think of math as a bunch of  little bits and pieces strung together with no rhyme or reason. Really, it's just the opposite - one huge coherent edifice, full of beauty and mystery. I want students to try to figure out things like long run behavior, based on what they already know.

From the little bit of checking I've done, I'm guessing I prefer having students use the Stitz & Zeager, even though the Lippman & Rusmussen may have more cool problems. I like how Stitz and Zeager write, and the feel they give is usually one of reasoning things out. I'll go with it, and maybe I can send them my suggestions for improvements. (Yeay for open source.)

Thinking about all this makes me want to write my own textbook. That's probably way more work than I want to take on, but I do like writing about math. Maybe some day...