Monday, July 28, 2014

Chin Up, You

We're getting a post-TMC rush of folks posting about how interacting with our awesome community is making them feel a tad inadequate. I appreciate the honesty and the willingness to be vulnerable. Blogs are a great place for processing feelings, aren't they? And this is from a person who barely has any feelings, so you know it's true.

I have some perspective now on being well-known by a certain small niche of the internet, and I want to tell you, every time you think "I am not as cool as that cool blogger person" that person is likely thinking "I am not nearly as cool as this person thinks I am." At least, that is what I am thinking.

It is natural to read about someone's practice and compare yourself to them. But you have to remember that when you are reading about awesome things on blogs, you are reading about that person's best day. Their best day that week or month. Maybe even their best day that school year or in their career. When you attend a presentation at TMC, you are hearing about one aspect of a person's practice that they have been thinking about for a while.

And you're comparing it to the totality of you, because you are stuck inside your head 24/7.

In the fantastic morning session organized by Elizabeth Statmore, which touched on collaborative group problem solving, restorative practices, classroom circles, and so many other things, I had an opportunity to present a quick task and ask the participants to... participate. I'm really grateful that they were game and willing to work through a few things with me.

There were a few consequences of the way I structured the task that I didn't anticipate, and I wasn't super thrilled with how it turned out. I felt for a hot second like, "Oh no, now they know I'm not that good a teacher and will think I'm a fraud." But of course, it would have been delusional for anyone to think that someone with only a few years' classroom experience, who has been out of the game for a while, could plan something that would be awesome right out of the gate. So it was dumb of me to assign them that expectation. And if they came out of it thinking "Kate is not nearly as awesome as I thought she was," then GOOD. Because that is the truth.

So, I'd just like to say, everybody chill the &^%$ out. We are all good at some things and suck at other things. One thing we all share is the recognition that we all have work to do, and that we can all get better, and that focusing on that is worth our time. There was an adorable tweet yesterday from a math teacher asking how they could "join the MTBoS," with a perfect response from Jed Butler. If you're asking the question, you're already in. Show up. Learn. Teach. Get better. We're all right there with you.

Monday, July 14, 2014

Essential Questions for Algebra 2's what we have so far.

Sequences and Series
  • What kinds of patterns commonly arise in our world? 
  • Why is it sometimes desirable to describe a pattern mathematically? 
  • When we notice a real-world or mathematical pattern, what are some different ways in which we can describe it? 
  • How is it possible to keep getting closer and closer to something, but never actually touch it? 
Probability and Statistics
  • How can I use probability and statistics to make predictions and decisions that will benefit me in life?
  • How should I interpret statistical information about myself and that I see in the news?
  • What is the bell curve, why does it appear in many aspects of society, why is understanding it so important to our society?
  • What are are some more sophisticated ways of counting, and when are they useful?
Intro to Functions
  • How are functions used to represent/simulate the world we live in, and why are they so important?
  • How do functions help us to make the best decision?
  • What are some different kinds of functions, and what sorts of real-world situations can they model?
  • Why is the idea of "inverse" so important in mathematics?
  • How are quadratic functions used to understand/represent the Universe we live in?
  • How can writing a mathematical statement in different but equivalent ways highlight its various features?
  • Often, solving problems involves making choices. How can we make smart choices for any problem?
  • How are polynomial functions used to understand/represent the Universe we live in?
  • How are all the different representations of a polynomial function related?
  • How are rational functions and different types of variation used to understand/represent the Universe we live in?
  • How is it possible to keep getting closer and closer to something, but never actually touch it?
  • How are radical functions used to understand/represent the Universe we live in?
  • How can something that "doesn't exist" affect our world?
  • How can we make sense of exponents that are not integers?
Exponentials and Logs
  • How are exponential and logarithmic functions used to understand/represent the Universe we live in?
  • Why does the graph of an exponential function have its shape? How is it possible to get closer and closer to something and never touch it?
  • Why is the idea of "inverse" important in mathematics?
Modeling with Data
  • How do you decide if a mathematical model is "good"?
  • How can we use existing measurements to make predictions?
  • What are some possible pitfalls of using mathematical models to make predictions?

Saturday, June 21, 2014

There's This Book. You Probably Want It.

If you don't know Sue VanHattum, you're missing out. She's a community college math teacher and math circle leader, and just one of the warmest, most thoughtful people I know. (If you're reading this, you probably know her blog, Math Mama Writes.)

For the past few years, Sue has been assembling and editing an anthology called Playing with Math, featuring writing by people who like to play with math. (Stories from Math Circles, Homeschoolers, and Passionate Teachers is the subtitle.) I appear in it, as do many of my heroes. It's stories, but it's also games and puzzles and fun things to play around with.

The book is finally ready to go! If you like f(t) and others of the genre, this book is probably right up your alley. The initial print run will need some funding. Donate as little as you like, but $25 gets you a print copy. (Oh, and don't put it off! The campaign only runs for a month.) Better yet, get one for you and a friend! Maybe that friend who is a parent and wants their kids to like math, but is afraid of it. You know the one. This is how we change people's minds about learning mathematics. One positive interaction at a time.

Thursday, May 22, 2014

The AMS Published a Kids' Book, and It's Really Good

The American Mathematical Society does some pretty great things. Now, a kids' book! It's called Really Big Numbers, by Richard Evan Schwartz.

This trailer is a great intro:

Here are particular things I really dig about it:
  • The subject matter is prime kid-bait. This book will give them many ways to think about how big a million, trillion, etc really is. It starts small with grokkable quantities (ladder rungs, cutting a cube into 1000 little cubes) and the thread of making really big numbers concrete runs through the entire book. It does this with distances (a million people joining hands would stretch from Providence to Chicago), volumes (100 billion basketballs would fill New York City roughly to the height of a person), and arrangements (between 5 and 6 trillion 9-letter "words"), among other constructs.
  • The bright, humorous, borderline-psychedelic illustrations.
  • The conversational and non-threatening invitations to think about mathematics that go past the words on the page. "There are about 20,000 ways to color a tic-tac-toe board with three colors." (following page) "You know, when I said that there are about 20,000 patterns like this, I was hoping that you would try to figure out exactly how many patterns there are." There are many such pages that set up tantalizing problems, that could launch some great explorations and conversations, particularly in combinatorics, geometric sequences, and graph theory.
  • It introduces concepts and notation only as needed. Exponents come up organically, as do special names for them. (As a kid, I would have been particularly tickled to learn names for powers of ten past a decillion. I was a weird little nerd, but still.)
  • It invites kids to read as far as makes sense to them. Schwartz compares reading the book to a game of bucking bronco: hold on as long as you can, and when you get thrown off, come back any time. You might pick the book up for your third grader, and the concept of exponents (about 1/3 of the way through) might stretch her mind. But the ideas and problems beyond could entertain and challenge her if she picks it up again in middle school and then high school.
I can't wait until my nieces come at me with "Infinity!" for the first time. I'll be ready for them.

Monday, April 14, 2014

NCTM 2014 Presentation --- One of Us, Every Teacher a Blogging Teacher

Mine was one of several on Friday billed in the "Leveraging Technology" strand. All the presenters appeared on a panel discussion at the end of the day. I give the NCTM program committee huge props for putting the strand together. In previous years, there were complaints that NCTM had not tapped into the voices, experience, and community going on in MTBoS every day. My interpretation is, this criticism was heard, and this strand was their response.

My overall thesis was that in many places, the professional development opportunities offered to teachers are not good enough to result in consistent improvement in their practice over time. And that starting a blog about what is going on in their classes can be a really effective way for teachers to take responsibility for their own professional growth.

I owe a big public thank you to Ashli Black, Sadie Estrella, Chris Lusto, and Meg Lane for sitting through a rehearsal Thursday night and offering thoughtful criticism and feedback. And an extra thank you to Ashli for suggesting some very positive changes in the weeks leading up.

I didn't want to just post slides, even though that would have been way easier than writing this all out, because they wouldn't make any sense by themselves. So you're welcome.

I got many kind compliments afterwards, but I think if we are all being honest, this talk was a 6, maybe a 7. I was not as nervous as I feared I would be, and I was well-prepared with what I had prepared, but I think I could have made my points better if I had more experience planning this sort of thing. Here are the conclusions I've come to about the very surreal scenario of speaking to a big room with a microphone and a slideshow:
  1. I have the capacity to do this well.
  2. Learning to do it really well would require focused, sustained, intentional preparation and practice.
  3. I am undecided whether this skill is important enough to me to take the time to develop it. But that's where I am on that.

This is a snapshot of why people came to see me. My Polleverywhere account only goes up to 50 responses, but here is the sample made up of the first 50 to respond:

I was glad that the second and third options were the most frequent, because that was what I was prepared to talk about. Hooray!

And here is a snapshot of some of the rock stars who came to see me and sat in the front row. I am a lucky one, indeed.

Photo Credit: Avery Pickford
Part 1: Why I Started Reading Blogs

I started with a little story about how terrible I was at teachin' school in my first few years. The intention was mostly to be funny and get the audience to like me. But every word was true.

In order to do something about that problem, I started poking around the internet.

I found that there were people sharing their lessons -- like, what was actually going on in their classes -- with narratives and media and supporting documents included. And they were good. I said an especial thank you to Dan for being a pioneer in this area, except I didn't address it directly at him, even though he was in the audience, because I spazzed. Also mentioned: Jackie and Sam.

Part 2: Why I Started Writing a Blog

I started writing because I had to. I was coming up with some good stuff, or had ideas for some good stuff, and I needed to share them.

I'm not sure if this itch can be learned, but I tried to get the audience to experience it. First, I said, think of something that went really well this year in your class (or a class you're involved in). I put up a 30-second timer to give people a chance to think of something good. Then, turn to a neighbor for two minutes and tell them about it.

People were game. It got satisfyingly loud and animated. After four minutes (two for each partner), we regrouped, and I asked a volunteer to share what she talked about with the room. Then, I asked her how that felt. The point I was trying to make was, it's satisfying when you try something that goes well in your classroom, but it's deeply satisfying when you know that your insight and planning was put to good use by hundreds of people all over the world.

Part 3: What Blogging is Good For

I gave a few examples of some of my favorite posts:

We know this sort of reflective summary of our work is a valuable activity for learning. We know it because we assign portfolio assessments to students. We know it as a profession, because National Board Certification for teachers requires four portfolios. Four!

Also important: most teachers do not spend their whole career in the same place. Having a record of your professional growth and a collection of your best work is very helpful when you apply for new jobs.

Then I cited one example of posting a lesson that I wasn't happy with, soliciting feedback, and then posting the update. I had initially planned on using the example of the introduction to right triangle trig (original) (revised). However, I was afraid it would take too long to explain the gist of the lesson, so instead I went with a problem set used to help Algebra 2 students review what they know about equations of lines and extend it to point slope form (original) (revised). In hindsight, I should have stuck with the trig stuff; it was meatier content and had better visuals. The upshot: once you have a critical mass of readers who are active in this community, you can tap into a hive mind of experienced, generous, knowledgable folks who will help you work out the kinks in a lesson.

Part 4: Typical PD Offerings are Not That Effective, but Blogging Is

A poll showed that, for this audience, professional development offerings skew unhelpful:

Then I asked them to choose one or two of these statements to best characterize the PD they've participated in this year, and discuss them with a neighbor:

Then I revealed that these statements were characteristics of effective and ineffective PD, because research:

And this was the thing I wanted to communicate most loudly and most clearly. Dear Teacher: if the professional development offered by your school or district is not helping you improve your practice in clear, consistent, measurable ways, then it's up to you to take responsibility for your professional growth. Blogging isn't the only mechanism for that, but it is fun, and it does exhibit the three characteristics of effective PD outlined by Linda Darling-Hammond and Nikole Richardson. It's sustained over time because you're doing it at least a few times a month, it's linked to curricula and applied to practice because you are reflecting on your planning and what went down in your classroom, and it is done in collaboration with commenters and other bloggers.

Part 5: Tips for Blogging Teachers

I used Bree's blog as an example:

  • Use your real name, and bonus points for including a picture. This is your professional work and should have your name on it. People resist trusting and interacting with a pseudonym. You have no reason to hide.
  • Put a creative commons badge on your work. It won't stop nefarious ne'er-do-wells from publishing your stuff under their own name, but it does give you some recourse should that ever happen.
  • Put pictures in your posts. For whatever reason, people are more likely to read and interact when there are pictures. I suppose readers are averse to an unbroken wall of text.  
  • The arrow at the top points to links to Bree's short stories. I'm not suggesting you should publish short works of fiction, but this message is more, you do you. Be your authentic self. Bree is a good writer and storyteller, and that talent comes through in all of her writing. I am kind of dry and practical and occasionally sarcastic. Hedge does this stream-of-consciousness thing that goes with her personality. Sam enthuses. Fawn swears. People respond to writers, in this genre, who are being themselves.

  • This is a comment left by Tina Cardone on Bree's blog. Tina's name is a link. If I clicked on it, it would take me to Tina's blog. If you would like more people to read your stuff, it helps to read their stuff, and let them know you did in this way. If your comments are quality, people will be curious about you and go check you out. You just include your blog's URL in a special spot when you leave a comment.
  • Also shown is Bree's response to Tina. This is a tip to engage commenters, answer questions, and have a dialog with them.
  • Final thing, for which there is no graphic, is that it's okay to reach out. If you have been writing for several months and feel like you're talking to yourself, shoot an email to a few people you read and admire. Ask them to consider sharing a link to your blog, if they like what you're doing. This is a gracious and generous community.

Part 6: Tips for Admins and Others Supporting Teachers in Blogging

This is a screenshot that a teacher took for me at her school. AT HER SCHOOL. WORDPRESS IS BLOCKED AT SCHOOL. Admins, you have to find a way to unblock everything you can for teachers' accounts. Same goes for Youtube, and Google Hangout, and just as much as you can. If your mission statement includes the phrase "21st century learning," and you are denying teachers access to the tools of 21st century learning, you're not really doing it. I know there are privacy concerns and bandwidth concerns, but this is a problem you need to solve.

This is a picture from the blog of Jonathan, a high school teacher in Texas. He posts photos from his classroom, but edits students' faces. This practice is, presumably, within the acceptable use guidelines of his district. Some teachers are reluctant to start blogging because they don't know what it's okay to share from their classrooms. Having a clear and well-publicized policy can help.

Final tip: find a way to make it count. If maintaining a blog is an additional thing a teacher has to find time for, on top of everything else she has to do, it's hard to keep up with. A decent blog post a few times a month is a time-consuming effort. If you can make that activity count in your district's existing inservice credit program, or find a way for blogging to replace another PD commitment in which the teacher already participates, it would be a huge encouragement. For practical reasons, but also to communicate that you recognize blogging as a worthwhile pursuit.

I ended on a link to the Welcome to MTBoS page, for people looking for blogs to start reading, and that was that. I had about fifteen minutes to spare, so I took some questions.

I'd tweak it a bit next time. First, it needed a better ending. The ending just kind of appeared abruptly with a thud. I need to end with an inspirational quote or a call to arms or somesuch.

Second, Sadie suggested offering some examples of easier onramps to blogging, like a 180 blog. If this is all too intimidating, just commit to seeking out, taking, and posting one photo a day from school. That is an excellent idea. Because, boom, you're blogging -- the task is easy and you can figure out all the technical stuff in the process. And more important, it gets you in the habit of paying attention to worthwhile things to share. I wish I had thought of that.

If you want to talk about any of this, you know the way.

Monday, April 7, 2014

Dear Reader

From the mailbag, a kind of question I hear a whole lot. I don't know that I'm all that qualified to respond, but, I want to state clearly my belief that the vast, vast majority of our teaching force is made up of smart, well-intentioned, hard-working people who want to do a good job, and are willing to entertain the adoption of CCSS-M as an opportunity to do better. I know that there are valid complaints about the way tests have been implemented and used for ridiculous purposes, and I don't want to have that argument. I do want to spend my energy on productive changes in classrooms.

Hi Kate!

When you have time, can you give me an idea of how your class ran when you taught? I guess I should start with some explanation…

I am a fairly traditional teacher. My students come into class. I have some sort of warm up on the SMART Board and I check their homework. We go over the homework from the previous day – I give them the answers and ask for questions. Then I go into the lesson for the day and give them time to work on the assignment when possible. Some days we do stations or other stuff in my class to practice, but by in large, many of my classroom days are spent “teaching” or “lecturing.”

Now, I know there is a time and place for that, but I also know and am really coming to realize that I need and want to make changes in my classroom next year. I want my students to not be so dependent on me. I know that this will be a tough thing – it will move my students out of their comfort zone and it will move me out of mine. However, I have seen the Field Test (we are a PARCC state) and I can also see that there is a larger emphasis on being able to read the problem and apply the concepts rather than problems like “solve this system of equations.” I am trying to figure out how to get there. What I mean is, I am trying to figure out what I need to do differently in my classroom to not only “cover” the material but to prepare my students to better think on their own. Prepare them so they can read a problem and say “well, I haven’t seen this before exactly but I do know I can do or try x, y, or z” and then they’ll (hopefully) do that. I am fairly certain that if I were to give my students an assessment that did not look like their review sheets and their practice problems, they would pretty much freak out. I know and realize that it is not a change I can make abruptly, but what I would really like is that at this point in the year (well, maybe a lot earlier), that I would feel like my students are capable of looking at an unfamiliar problem and knowing how to begin. That they can think and apply on their own without totally freaking out in the process.

I was hoping that you would be so kind to share with me how your class was structured and if you had any great words of wisdom to get me pointed in the right direction. I know it will be a huge shift for me and my students, but in the end, I also know that it is the best thing to prepare them for life beyond school. 

Hi Reader!

Here we go:

1. Your class doesn't sound that different from the way mine ran. The details that may have been different were:

1a. When you "go into the lesson for the day" what that looks like. If we could watch a time-lapse of my classes over the years, we'd see too much of me talking to silent kids at the beginning, and much more of kids talking to each other at the end. It was a slow process for me, but whenever I discovered or thought up a way to replace me-telling with them-experiencing, I replaced a lesson with a better lesson. For example, for log laws, traditionally some of the most bewildering of the Algebra 2 content, I wrote up a series of questions that started kids with things they already knew and prompted them to look for patterns. I didn't do it "with them," I said, okay, read carefully, work together, I'd like you to try to get through question 8 before we stop and discuss, ten minutes from now.

An old lesson plan would look like the notes I would write and the things I would say. A new lesson plan would look like either a series of questions, or 2-3 tasks that kids were meant to work though independently or in small groups and my anticipation of their responses, and then how I planned to tie it all together. In practice, I'd only let them work in 5-10 minute chunks before I interrupted them to share what they found and re-cap. (See also: the 5 Practices book. If I were teaching right now, I'd also be paying attention to Assessment for Learning.)

The most important change is the shift to giving them questions and problems unlike ones they'd seen before. Whether they're working through sort-of familiar or unfamiliar tasks to develop a whole new concept, or you're asking them to apply something previously-learned in a different way, I think the principle is unavoidable if you want them to be able to think for themselves. The only way to learn to do it is to do it. It's less scary and more do-able if you build off something they already know, or start with something concrete that they can count or measure. Count stuff, make a table, look for a pattern, generalize, solve new problems is as good a lesson flow as any.

Alot of this work was just reading blogs and journal articles and saving useful things in a digital filing cabinet. Then, when I went to plan a unit, I would go look at everything I had saved for that unit, to see if I could replace any of the mathematical/pedagogical flow of the unit with something better. Unfortunately I don't have my filing cabinet organized in a sharable way, but lots of other people do. Start slow -- see if you can replace one lesson every unit -- if you try to change everything all at once it will be overwhelming and you'll give up on it.

1b. You also said you give homework every day. I gave homework, but I gave less at the end than at the beginning, and I became more intentional about its purpose. A person can really learn the content of a high school math course in 45 minutes a day. If your class time is firing on all cylinders, you can give less homework. You need to stop thinking of it as an insurance policy.

And when you do, there should be a reason. It's okay if the reason is fluency with some specific thing -- "I want kids to be able to identify the correct triangle congruence theorem by looking at a diagram with congruent things marked" for example. But there can be other good reasons, too, like "Kids are going to estimate the coordinates of a bunch of points on the unit circle to two decimal places -- I don't want to suck up 15 minutes of class time with this exercise that they can complete independently, but it's a necessary part of tomorrow's unit circle introduction." I found more kids completing homework when it had a clearly articulated purpose.

Corollary to 1b: Don't flip. They won't watch the videos.

2. Try to have this conversation with your whole department. The best possible thing would be for your school to adopt, and get your colleagues committed to, a research-based, field-tested, high-quality curriculum with lots of instructional supports like CME or Core-Plus. You still may have to supplement; for example, I think both of these are light on authentic real-world applications. But developing a curriculum from scratch is not the work of teaching -- both roles are too complicated to be done well by one person.

But even if that kind of systemic change is a non-starter, still see if you can get others in your department to start making some changes, because it's hard to make big changes isolated to your classroom. Let's say you try more problem-based investigation in small groups, and expect kids to articulate their reasoning in writing, and make it count on assessments. Kids will perceive your class as harder than other teachers'. They will complain, which is annoying, but also they will resist trying what you are asking of them, because it's soooo not fair! Or let's say you implement some flavor of standards-based grading. No matter what you do, kids will interpret that as a million retakes and second chances -we don't have to study, woo! Your class will be perceived as easier, and guidance counselors will start shunting more of the struggling learners into your classes, because they are more likely to be able to pass. This isn't fair to you.

I hope that helps! You can do it. A little at a time.

Sunday, February 23, 2014

CCSS Geometry and Proof through Transformations

Congruence and similarity proofs through transformations are new to most teachers with CCSS-M. I have noticed instances of them making teachers freak out. But they are actually delightful, once you understand what is expected. I find transformations a much clearer way to show why figures must be congruent and similar than with the axiomatic approach most of us used to use.

In the two triangles pictured below m(A)=m(D) and m(B)=m(E)Using a sequence of translations, rotations, reflections, and/or dilations, show that ABC
is similar to DEF.

This is a task called Similar Triangles from Illustrative Mathematics. If you can do it, you've proven AA, so, hey, pretty darn useful, too. 

If you give it to teachers, they will mostly freak out. You'll hear things like "This would be so much easier on a coordinate grid." "I don't know what I can assume." "What definition of similarity are we using?" 

So, clarify that the only givens are the congruent angles. And that similarity is defined as one figure being the product of a sequence of transformations of the other figure. Still, they are freaking out because they don't know what to put. Here is a magical, yet still general, procedure that will unlock their willingness to engage with this task. It was shared with me by Dr. Kristin Umland from University of New Mexico, who is a badass.
  • state the transformation precisely
  • state the object you are applying the transformation to
  • draw the intermediate step 
Here is what it will look like. Colored pencils are very helpful.

1.  I'm going to do a translation by vector AD. I'm going to apply it to triangle ABC. Here is the intermediate step:

2. Then, I will do a rotation, about point D, clockwise, by the number of degrees in angle B'DE. The object I'll rotate is triangle A'B'C'. Here is the intermediate step:

As a result of the rotation, A''B'' will lie right on top of DE, and A''C'' will lie right on top of DF.  I know this because it was given that angles A and D are congruent, and angle measures are preserved in translations and rotations.

3. Finally, I will do a dilation, with center D, with scale factor that is equal to DE/AB.  The object I'll dilate is triangle A''B''C''.  Here is the final step:

I know this final transformation to triangle A'''B'''C''' will land precisely on triangle DEF.  The dilated segments will lie right on top of DE and DF, because the center of dilation was D, so dilated segments that include D will grow in their original direction. B''' will land right on top of E, because of the scale factor I chose, and because distance was preserved in the translation and rotation.  Since it was given that angles B and E are congruent, and angle measures were preserved in all the transformations, B'''C''' will be right on top of EF.  D''' will land right on F, because the two dilated segments have no choice but to intersect at F.

Okay, I am going to anticipate some of your objections and pre-emptively respond to them:

Objection: "Aren't you just teaching another procedure? Isn't that what we are trying to get away from?"

Response: State the transformation, state the object, draw the intermediate step doesn't feel like an evil procedure in the evil sense of the word. It's not actually dictating what you'll write. It's more a framework. The shallowest of footholds - something that can be used to gain purchase, and free you up to talk about the details of the proof. I wouldn't feel evil for teaching it.

Objection: "This is so stupid! Why are we making kids learn this?! At no time in my adult life have I had to prove that two shapes are similar!"

Response: I believe you, and I don't think anyone is suggesting that writing Euclidean proofs, itself, is a necessary life skill. The thing we really want kids to learn is how to reason logically and communicate their reasoning clearly. Geometry is a context for that, and has been for over 2000 years, because it is a pure abstraction, and you don't have to account for the limitations of measuring devices or friction or any other real-world complicating factors, so we can focus on the argument itself without any distractions. If you don't think that learning to reason logically and communicate your reasoning is a desirable skill, then you and I aren't going to have a very productive conversation.

Objection: "The students I work with might be able to follow those three steps, but no way can they complete the argument like you did. The bit about preserving distances and the argument for why C''' and F are coincident."

Response: Yeah, maybe. But the three steps are a good chunk of the proof, and if all of my students could do that, I'd be about 75% happy. Let's think about what students need to be able to do to complete just the three steps: visualize and draw the outcome of a transformation, communicate the details of a transformation clearly, and understand which attributes are preserved under transformation (whether they actually state them or not -- but with practice, I think most could make progress stating the argument with less formal language). For test-taking purposes, I'd be willing to bet that communicating the three steps clearly will count for at least half the available credit. The rest is difficult, and comes with practice and intellectual maturity, I'll totally concede that. But, if we're being honest, all practicing teachers make our peace with the reality that not 100% of the kids learn 100% of what they are supposed to learn, so I don't see why this should be any different.

Objection: There is too much going on here! The kids have to remember all the details for describing transformations, and then they have to draw them accurately, on top of constructing this argument. Too hard!

Response: Yeah, it's a lot. You'll need to probably spend an entire unit before this where they just learn how to describe transformations with precision. However, technology can alleviate some of this burden. For example, Khan Academy is creating some interactive modules where the transformations can be performed through button clicks. I'd cite this as possibly the first example I've seen of KA doing something right. I'd definitely use these modules, selectively, while teaching this, especially for any students with motor-control difficulties, but also for all students to have opportunities to focus on the reasoning and not worry about drawing. I've heard rumors that Desmos is cooking up something similar, so stay tuned for that.

An exercise for you: if you'd like to practice a transformational proof with a task that's suitable for, say, advanced high school students and up, try this:  Show that when a rectangle is dilated by a factor of k, its perimeter changes by a factor of k, and its area changes by a factor of k2

Monday, January 13, 2014

Our Favorites... Tuesday, Tuesday, Tuesday!

Sometimes you read a MTBoS post and you're like, "Dag. I want to sit down and buy this person a burrito and get them to tell me all their secrets."  That happens to me pretty frequently, and I normally try to share my enthusiasm on Twitter, or whatever, because I don't live within burrito-sharing distance of most of these people, but I don't do this in any organized kind of way.

At the same time I'm thinking about this, I've been involved in some conversations about how Global Math Department is going through minor growing pains, as most successful endeavors do. Don't worry, we have some really smart and dedicated people who are ON IT.

These two things have converged! And resulted in yours very truly hosting an edition of Global Math tomorrow night, where I got to invite three of my favorite bloggers to come talk about three of my favorite posts from the past year. This one is geared toward high school content, but I have no doubt there will be good takeaways for middle school and post-secondary folks, as well. I hope you can join us! (Unfortunately you will have to provide your own burrito.)

If you'd like to do some reading ahead, check out the posts we'll be discussing:

Shireen D, Math Teacher Mambo, Related Rates and Crowdsourcing in which frustrations with related rates problems are shared (including the dreaded cheese factor), and a plan to address them. Shireen's going to tell us how it went.

Mimi Yang, I Hope This Old Train Breaks Down, Graphically Analyzing Inequalities and Equations Flexibly
Here is a quote that will get your blood pumping: "On the quiz for inequalities, my kids had 100% accuracy on all the equations and inequality questions because they were asked to show their work two ways, one by hand and one by the graph, for each equation or inequality."

Breedeen Murray, The Space Between the Numbers, More Projects, Please
I got sincerely excited by the combination of accessibility and mathematical depth going on in Bree's projects. And she's going to share some student work! Woo!

Saturday, December 21, 2013

My Geogebra Fancy Pants

The #1 thing you should learn in Geogebra if you want to make cool-as-in-cool things is the Sequence command. If you're already familiar with the sequence command on TI's it works the same: Sequence[expression, variable, start, end, interval].

As a simple example, here is a tool I made to subdivide a segment with evenly-spaced points.

But you don't have to just make sequences of points, you can make sequences of anything! Segments, so you can make a grid with a variable number of gridlines. Or an ice cube chopped up into smaller ice cubes:

Sectors, so you can make a customizable circle graph.

And this thing for Pandemic that is just too fun:

Sunday, December 1, 2013

Catching Fire/Hell

Warning: cranky old lady rant coming. Avert your eyes if you don't like this sort of thing.

I saw Catching Fire last night, on its second weekend, at an 8 PM show on a Saturday. Normally, I go to matinees, because I am cheap. But, for complicated reasons, I was there. This is all to say, I haven't been to a crowded showing where the audience skews young in quite a while.

During the previews, I could see lots of phone screens. Maybe a dozen. They were the brightest things in the theater. Far brighter than the projected image on the screen. I thought, surely, everyone would put away his phone when the movie started.


The young man (age hard to tell...I put him at 16-20) sitting right next to me, in fact, was looking at his phone more than he was watching the movie. He was reading his Facebook feed, composing status updates, and tagging lots of people. I know because it kept distracting me from the film, so I read over his shoulder.


I noticed, though, that he kept logging out of Facebook, so when he went back to check it again, he had to log back in. I reasoned that he was trying to deter himself from checking his phone. Each time he finished, he thought, "I know, I'll log out. That way, it will be a pain to get back on, which will make me less likely to check it again." (I often use the same logic when polishing off a pint of ice cream.)

Before the film had started, he had to leave and come back twice, awkwardly stepping over my companion and me. Each time, he apologized for inconveniencing us and said thank you, and we were, of course, very polite and accommodating. This was not a rude kid.

So about twenty minutes into the film, when I couldn't take it anymore, I leaned to him and loud-whispered, "YOU KNOW. THAT SCREEN IS REALLY BRIGHT." He apologized and put his phone away. I thanked him. I was afraid that might not last very long, but no. He didn't turn it back on for the rest of the movie. There were a couple others in rows further down, and they were annoying, but they were too far away for me to yell at. And the one right next to me was the one really ruining the movie. Which cost eleven dollars.

The kids in the theater didn't make me mad. They're kids, and they need to be told. What made me mad, after the fact (I was not ruminating on this during the movie, mind you. It was great. You should go.) was remembering every dummy on the Internet, who inevitably is not someone who spends much time in a classroom, who suggests that if teachers' lessons were interesting enough, kids wouldn't be tempted to distract themselves with their phones. Therefore we shouldn't have to require kids to put away their phones sometimes during math class.

Um, Francis Lawrence can't keep kids from being distracted by their phones. With a crazy-good story about reluctant teen revolutionaries. And a $78 million budget.

Learning takes focus. Focus takes practice. Kids might never know how jaw-droppingly cool are the things we are trying to teach them, if their focus is interrupted by Snapchat every two minutes. There are some things they need to be told.