October 31, 2012

A Day in the Life of a Math Educator

Long ago, when I first started putting together the #matheme page, I thought a fun theme would be for each of us to record a day in our lives. I wonder how everyone accomplishes the million tasks required of a teacher, who has routines that might work for me and the many unique approaches we all take to grading, lesson planning, email and that other thing- working with students. Sam suggested we do this in October but the start of school year insanity lasted a lot longer for many of us than usual so we are opting for November instead. The idea is that everyone picks a day during the week of November 12, and records everything they do that day. I'm planning to start with the alarm going off in the morning and end with falling asleep that night. You can provide as much or as little detail as you like, but the goal is to get some insight into what it's like to be you. If you have a blog, write your day up in a post and let us know so we can link back to you (the plan is to have a fancy form, but leaving a link in the comments works great too). If you don't have a blog I would love to have you write a guest post!

My original intention was for us to share our experiences within the online math teacher community. But after reading Lisa's post over the weekend I really want our 'day in the life' theme to be bigger than just us sharing with each other what we do each day. Then I saw this article, and Fouss' post. Peg Cagle made a big deal about teachers as professionals this year at PCMI (for good reason) and I think that public misconceptions of what educators do are only getting worse. I would love to see this turn into something big, a "Dear world, this is what teaching really entails." I'm not yet sure how to turn this public, so I'm hoping you have some ideas. In the meantime, I'm going to ask my department to participate (none of them blog) and hope that after the elections are over some type of media will be interested in what we have to say.

If you'd like to include a logo with your blog post or as you publicize this project, feel free to nab whichever one of these fits your color scheme.  Sam also made some fancier banners. We absolutely welcome non-math teachers; if you'd like one of these banners tailored to you just let me know. Or you could make your own!  We will be tweeting about it using the hashtag #DITLife.  I look forward to hearing about your day.

October 27, 2012

NCTM Adventures, Day 2

I attended the regional NCTM conference in Hartford this week, these are the ideas and activities I got out of day 2.  Read about Day 1.

Incorporate Interactive, Colorful Activities into Your Mathematics Classroom by Tom Reardon

I thought colorful activities meant crayons, but it actually meant TI-nSpires in color.  However, I was still really happy to have attended this session.  Tom was a great speaker and he was really practical - he knows that tech should supplement the lesson, not be the lesson.  Most of the applets he showed us were meant to be quick explorations or models.  Best of all, there's a TI-nSpire player which is free!  Students can interact with files that other people made, but it only works with the mouse (no keyboard inputs) and you can't edit the files without buying the software.  One really nice example was a visual for summing a geometric series.  The geometric interpretation worked especially well for this limits problems as students could quickly see that we were filling up the box, but would never finish.  Check out his website, the link to "all my stuff" gets you to a dropbox folder, choose the folder Player Files which is filled with files he hand selected for the player (no keyboard inputs required), the word document at the bottom gives you a list sorted by topic.  I need to take some time to really go through that list and see which ones are worth reserving the laptop cart for, and which other ones I can use on the smart board.

"Green" Geometric Modeling by Sharon McCrone and May Chaar

This session was about packaging and Sharon made an interesting point about how we teach Geometry.  So much of the course (sometime all of the course!) is spent on two dimensional objects, but we live in a three dimensional world.  These projects start with physical objects, but then students examine them in ways that allow them to practice skills in 2-D.  For example, if you consider sports drinks, they have to fit onto the shelf at the store.  So you examine a rectangular shelf surface and the footprint of the bottle.  Students can create layouts, calculate areas of various shapes and determine how efficiently they used the space.  Then if you consider packaging there are a variety of ways to look at it, but examining surface area and drawing nets lets us continue to practice area while including visualization skills and continuing to discuss a "real world" object.  The bonus question was to look at a bike helmet; they assumed that the highest and widest part was a semi-circle and then asked us to design packages that would fit the helmet with some wiggle room.  It would be interesting to ask students to compare the assumption to a real helmet, and of course this is a great time to make a joke about physicists: "assume a spherical cow."

Radian Protractor by Jennifer Silverman

Did you read that title?  Radian. Protractors.  Why haven't they existed for as long as degree protractors have?  I have no idea, but Jennifer decided to remedy this problem and the first run just arrived on Tuesday.  I feel so cutting edge!  They are sleekly designed with measurements only going in one direction (so much more logical than the confusing 2 directional ones) and they come in both fractions of π and straight numbers.  She has also provided us with some carefully scaled drawings that are easy to measure using the new protractors.  Check out the website for the lessons and to buy your own radian protractors ($3.14 each, so funny).

The original presentation is now available as a pdf.

Reasoning, Sense Making and Proof by Fred Dillon

I went to PCMI with Fred and he's great fun.  We got to do the locker problem, which is one of my favorites.  Then we talked about limits in terms of stocking fish in a pond.  Here Fred shared the 5 different representations he uses; I've seen what some people call the rule of four (graph, table, equation, words) but he also includes an area model (like Tom's geometric series box).  Finally, we worked on a puzzling probability problem.  That was the most interesting one:  Two people take their lunch break from 12 to 1.  They plan to meet up for coffee during their hour, but don't plan a time.  Both are rather impatient though, so they will only wait 10 minutes until they give up and leave.  What's the probability they meet?

Making Sense of Similarity by Terry Coes

Terry has been teaching for a long time and he had some good, solid ideas to share.  Since I developed a 10 hour PD on proportional reasoning as part of my grad program the challenges he listed for studying similarity weren't a surprise to me.  However, I think many teachers think like he once did "Oh, it's February, that's why this isn't going well."  (That comment also made me question what other topics I blame on timing...)  Thinking multiplicatively is hard for students after so many years of being trained to think additively.  Terry made an excellent point that I am going to try to keep in mind at all times: complete sentences make sense.  So often we let kids (and ourselves) get away with phrases that are meaningless out of context.  The Pythagorean Theorem is not "a squared plus b squared equals c squared."  Instead, the theorem states "If you have a right triangle with legs a and b, hypotenuse c, then a squared plus b squared equals c squared."  The math practice Attend To Precision doesn't just mean be careful when you round, it also means to communicate precisely.

I hope you found something useful in my session summaries, I found it useful to summarize since now I have all of the ideas from two intensive days more organized (both online and in my brain).  If you haven't been to an NCTM conference I would recommend checking out the schedule and trying to make the next one near you.  You'll get a lot more out of it than the tidbits I managed to share here.

October 26, 2012

NCTM Adventures, Day 1

This week was the regional NCTM conference for the East Coast.  Since it was in Hartford, one town away from where I grew up and where my parents still live, it seemed like an excellent opportunity to finally attend.  Lately I have more big ideas, strategies and philosophies to try than time to implement them, so I chose sessions that were hands on and related to Geometry or PreCalculus specifically.  I came away with a lot of great activities that I'm excited to share with you, my department and my students.

Standards Based Instruction by Suzanne Mitchell, NCSM President

She showed us some cool sample problems which included a ferris wheel problem that I thought would be a nice addition to the one I currently use, but a search of mathedleadership.org leads me to believe such things are only available to members.  Maybe I will expand my investigation of the ferris wheel beyond the problem we currently do though (which I might blog about someday).

Got Something to Prove? by Ralph Pantozzi

First of all, Ralph was an awesome presenter.  He was funny, jumped up on chairs to reach people with the microphone and spoke passionately about his students.  Beyond that, I really like what he had to say.  His central theme was "Let them wonder about it first."  Not only did he use this phrase several times, but he modeled it - giving us plenty of wonder time - and shared that he types this sentence across the top of any page he will use when working with students, including the notes for his presentation that day.  The goal is to convince students (and remind yourself) that all of mathematics is something that someone had a question about.  We should ask those same questions, and use proof because we genuinely wonder what works, why and how.  Proof doesn't need to be formal (no mathematician will be coming in the door to say "That's not a real proof!") but it needs to convince the audience - in formal mathematics, theories are accepted by people talking to each other, the same should hold true in the classroom.  Ralph likes to use the phrase "convince me" with his students, I wonder if that's better than the phrase I use, "defend your answer."  My intention isn't to have students be defensive...  Sprinkled throughout the session were some fascinating puzzles:

1 + 2 =3
4 + 5 + 6 = 7 + 8
(no questions, just two equations that he put up on a slide while waiting for us to wonder)

Divide a square up into squares.  (No shapes other than squares are allowed.)

P points are connected with segments in pairs.
The segments can’t intersect.

Draw a closed doodle.  Place a point at each intersection, each arc/squiggle/segment(s) between two points is called a section.  Count points, sections, regions (include the space outside the doodle).

The original presentation is now available as a pdf.

Origamics by Michael Serra

I was feeling really spiffy because the first thing we folded was the project I just completed in class!  Then we got to the second construction and I was stumped.  Luckily the woman I sat next to was stumped as well, and we were able to work through part of it together before he gave us an overview of the million similar triangles we had missed.  I spoke to Peg Cagle about the session later and she shared that she has students write "always, sometimes, never" statements about these types of things, especially the 'homework' which heads students towards sophisticated generalizations.  The constructions we did (NCTM Hartford), along with other sessions, are available on Michael's website.

Hands on Activities for Geometry by Carol J. Bell

The focus of this session was on proof without words.  I really liked the proofs we did, but they lacked scaffolding which made them difficult to approach.  Some simply needed the addition of a diagram since we couldn't see from the back which angles she referred to, while others jumped directly to abstraction without any numerical exploration.  That said here are some cool proofs that need lessons built around them:

Hands on Geometry

Customized Web Homework by Stephen Kuhn, Sandy Watson

I attended this session out of curiosity but with no real intention of using web homework.  Internet access still isn't quite universal enough among my students for this to be fair, plus it's something else to do and I promised myself to only take away activities (at least for implementation in the near future).  That said, I think the system they're using is promising (multiple types of answers, unique problem sets for each student with multiple attempts allowed on auto-generated similar problems and built in math type communication with instructors).  But for right now, I'm really excited about one aspect- shared problem sets among teachers!  I haven't had a chance to figure things out yet, but at some point I will be searching for PreCalculus problem sets.  Googling "whs web homework system" gets you to the website, several articles on the research study and more. 

And that concludes Day 1, read about Day 2.

October 21, 2012

Geometry Proofs: Follow Up

My take on proof blocks was a success!  Last week I outlined my plan and asked for feedback.  Then on Friday I made some on the fly adjustments and was quite pleased with the result:

Before class:

I printed all the cards on yellow paper (with the intention of printing the triangle cards on a different color - thanks to Justin for that comment), cut them out, asked the librarian very nicely to laminate all 240 of them (16 per pair), asked some students to cut those and sorted them.

During class:

I asked students to come up with ways we know to proves lines are parallel, angles are congruent or angles are supplementary.  We made a list as a class that 'just happened to' match up really nicely with the cards (obviously I set this up, but so much of teaching seems magical from the student perspective).

I showed them the cards, with the hints on the back, and talked about the difference between a to:__, from:__ card and a to/from card (without using the word biconditional, but I might have said converse since we've used that term).

I realized that the desks are dry-erase (awesome feature!) so I had kids working in pairs, desks cleared except the cards and a dry erase marker.  Everyone found their Given card and wrote down the given information next to it.  Then I suggested that they look for cards about angles since the proof was just about angles (another suggestion from Justin) and set them loose to figure out the rest.  Some students quickly noticed vertical angles and made the correct conclusion, other students saw the vertical angles but didn't know what to do - I was able to direct them toward the back of the card which made them realize they could now state the angles were congruent.  I cut this first proof a bit short (maybe half the class had the entire proof done) so that we could model on the board exactly how to set it up, and how to write QED (which I loosely translated as "I proved it!").  Then I put a new proof up and set students loose again.  Once a pair finished a proof to their satisfaction they wrote it down on paper and I put up another which the rest of the class got to at their own pace.  Repeat until the end of class.

After class:

I wish that the Given card was a different color to make it easy to find.  That might happen when I make the next set.

I love that students are willing to take more risks on dry-erase, but it's important for them to have something to refer back to while doing their homework.  Writing it down after wasn't ideal though since their desk was covered with information.  Maybe next time I'll just promise to post pictures on the website for them to refer to?  One of the students whose proof I photographed was really proud I wanted to record his work!

Edited 10/21 at 9 pm to add:
I had students write in their journals whether or not the cards were helpful and why, here are a few responses
Helpful to put a name to the [reason].
They confused me.
The cards helped me identify the angles.
Yes because we could flip through them and see which ones fit in.
The cards were helpful because I am a visual learner.
The cards were helpful but it would have been easier to just write it out.

October 17, 2012

Geometry Proofs: A New Approach

This year I have re-organized the Geometry curriculum.  I'll tell you more about how that's going later, but for now just know that we have studied patterns, lines and angles.  When we did angles formed by parallel lines and a transversal I introduced the word 'converse' but otherwise haven't done any logic.  Students have been asked to justify all of their answers, but Friday will be our first introduction to more formal proof.

In my class the goal of proof is to show a logical sequence and explain reasoning.  I hated proof when I was in Geometry in high school since they were all fill in the blank proofs- I could have proven the statement myself but could never follow the disjointed outline that the book offered.  However, I appreciate the scaffold that offers, which is where Proof Blocks come in.  People have mentioned them to me on a few occasions and I made a weak attempt at them last year, but this year I'm hoping to use this format faithfully.  The idea is that students have a set of cards which represent all the possible reasons they can use in a proof, then they arrange the cards in a logical order and fill in the information specific to the problem they are working on.  I didn't love that the provided cards tell students exactly how to use each theorem since I would like them to remember the definition of perpendicular on their own, so I reformatted them slightly to have hints on the back that students can check if they forget. (First column: front, Second column: back; which I just realized won't print double sided... this version will - I think - but it's not nice to look at.)

Proof Blocks 1
(On my computer all the words fit on one line and the perpendicular symbol shows up, hopefully that will be fixed if you download the file rather than viewing on scribd?)

On Friday I will give each pair of students a laminated set of card (and recommend that they print their own for use at home) and each student will have a page of proofs to work on.  I will pull the proofs from the book, but won't include any of the extra information (my book either makes them fill in the blank or offers a 'plan for proof' that eliminates all the thinking!).  Then I will offer a brief explanation of how the cards work and set them loose.  Hopefully this will be a nice way to bring together all the ideas we've studied on lines and angles.  I'm also hoping that proofs will be fun puzzles rather than onerous tasks.

So, have you done anything like this before?  Do you have any suggestions?  Did I miss any important reasons in my cards?  Thanks in advance.

Edited 10/21 to add: I tried it and was successful!  Read about the outcome.

October 13, 2012

Origami Angles

I have so many things that I want to blog about that I have idea lists going in several different locations!  However, I still just can't get caught up.  I'm sure it has nothing to do with restructuring geometry, figuring out pre-calc as I go or having a million things to do outside of school since I bought a house August 30 (best timing ever!).  So this isn't quite a real post, but it's better than nothing (or me whining about how busy I am, since I know you're all busy too).

Over the summer I took a course at the Education Development Center on the CCSS Standards for Mathematical Practice.  We have a follow up session on Monday and I had to write up a lesson plan and a discussion of what math practices it involves.  As long as I wrote it, I figured I'd share:

Course: Geometry (Grade 10)

Objectives: Students will determine angle measurements by reasoning.  Students will write sentences that justify their conclusions.

-Origami paper (one per student, plus a few extras)
- Student handout (one per student)

Seat students in pairs or small groups.

Give each student one square of origami paper and a student handout.  

Instruct them to carefully fold their paper according to the instructions.  If a student gets stuck, ask them to read the step they are on, stopping them mid-sentence to have them point to the part of the origami paper the instructions are referring to.  (Note: for step 2 I find that it’s helpful if I keep a finger on the upper left vertex - holding the paper taut makes it clear when the crease will intersect that point.)  If a student makes an error give them a new origami paper since extraneous creases will be confusing later.

When everyone has folded their square, bring the class together to clarify instructions.  Make sure the students label the points on their origami paper as shown on the handout.  Emphasize that the goal is to come up with reasons that justify the angle measure, not just make a list of numbers.  Depending on the level of the class either set them to work or read the hints together and brainstorm methods of approach.

Circulate the room, encouraging students to work together rather than rely on the teacher for assistance.  Give strategic hints to groups that need them (start with right angles, show students how to fold one person’s paper and use that to find congruent angles on another paper, ask students what types of angles they have studied and where they see examples of those).

Typically students need more than one class period to finish, plan to assign the remainder for homework or allow them to finish on another day.

Note: Idea for lesson and Origami Instructions are taken from the workbook which accompanies the textbook Geometry published by McDougal Littell.

Make sense of problems and persevere in solving them.

One thing I like about this task is that the instructions are straightforward.  Students have some questions about folding the paper properly, but after that they all know exactly what they need to do.  However, it is not at all clear to them how to achieve the goal.  I continue to be surprised how many students struggle to complete this task independently.  As I am circulating the room I will stop by a pair, point to an angle and ask them if they can figure out the measure of it.  Before walking away I might say, “look for other angles like that one” but too often when I check back in with that same group they haven’t made progress.  I use this lesson in early October, and it reminds me that perseverance is a practice my students need to work on.

There are 31 angles to find in the diagram (excluding 180 degree angles but including sums of smaller angles - such as HAC - which students rarely include).  For Geometry CP they need to find at least 20 angles to receive full credit for the assignment.  For Fundamentals of Geometry they need to find at least 15 angles to receive full credit for the assignment.  (The Fundamentals of Geometry students are just as capable as the other students of finding 20 angles, but it takes them longer since they have learning disabilities or gaps in their math background.)  In each case students are asked to find more than the ‘easy’ angles.  They will need to continue working past the point where angle measures are obvious and this is where some students’ ingenuity shines through.

Construct viable arguments and critique the reasoning of others.

The main goal of this lesson is to introduce proof.  I repeatedly ask, “How do you know?”  Asking students to defend their claim that an angle is 90 degrees really highlights the nature of proof.  The conversation usually goes:

Teacher: How big is this angle? (pointing at an angle in a corner)
Student: 90 degrees.
Teacher: How do you know?
Student: It’s a right angle.
Teacher: How do you know?
Student: (getting frustrated) Because it looks like it!
Teacher: Where is it on the paper?
Student: The corner.
Teacher: That’s how you know it’s a right angle. You’re not guessing or assuming, you’re using what you know about origami paper.

When I am grading this assignment I give no credit for correct angle measurements if they do not have an accompanying reason.  So if a student submits their origami square with all the angle measures labeled (correctly or not), they will get back a note asking for their reasons.  To make this point clear I remind students that they need to hand in 20 (or 15) sentences.  In reality I will happily accept a sentence which says, “Angle AHD and Angle GHD are each 90 degrees because I folded a line (180 degrees) in half” so they won’t necessarily have as many sentences as they do angles, but focusing their attention on sentences rather than numbers has the effect I want.

Student Handout

Origami Angles