Monthly Archives: August 2012
Without further ado, I present you the second post from 10 different new math bloggers as part of the Math Blogger Initiation.
Kristen Silverman | Numbers
Kirsten Silverman @klsilverman has a blog named Numbers. The second post for the Blogging Initiation is titled “Today“ and the author sums it up as follows: “Even when there are crummy days, now that I’ve started blogging, I can read about the better days. I can go back and read about the days I have been inspired.” A memorable quotation from the post is: “I’m trying to be more positive.”
–> My take: We all have those moments where we are like gaaah, why is everyone else making my life so hard!?!? Kristen brings up a really cool point about blogs being a place where you can go back and revisit positive moments on a particularly challenging day. Cool idea!
That Math Lady | That Math Lady’s Blog
That Math Lady @thatmathlady has a blog named That Math Lady’s Blog. The second post for the Blogging Initiation is titled “A Teacher’s Legacy” and the author sums it up as follows: “My post is a futuristic account of what my students would say about me and my class at their 10 year reunion. I hope that my real legacy is similar to this one! “ A memorable quotation from the post is: “I would trade all the okra-powered hovering hybrid carriers in the world for that legacy to become true.”
–> My take: In 20 years, will kids remember how to divide polynomials? Probably not. What things will they remember then? I think passion for mathematics is as good an answer as any, so I approve of this futuristic vision.
Maggie Acree | pitoinfinity
Maggie Acree @pitoinfinity8 has a blog named pitoinfinity. The second post for the Blogging Initiation is titled “Two Things Are Better Than One” and the author sums it up as follows: “I am looking to implement more investigative practice as I go throughout this school year. I did find two ‘things’ I have done in my classroom that is just that and is motivating me to do more. There are always concepts that just have to be done, but when the opportunity is there, it is important to do some sort of activity or investigation to get the students thinking.” A memorable quotation from the post is: “Coming back to the prompt I decided on, it was important for me to find a few things I have put together myself to help my students with investigative practices or constructing their own reasoning.”
–> My take: Piecewise functions are what I use for review for my Calculus course because they seem to hit a ton of great things from lower math classes. I really like this idea of cutting out graphs, and I think I might try it in my own class! I wonder if there is a way to do this though without such step by step instructions. And I think it might be useful to have some part of the activity go the other way, i.e. to write down an equation from a piecewise function. Great idea!
Duth Math | duthmath
DuthMath @DuthMath has a blog named duthmath. The second post for the Blogging Initiation is titled “I wish I had heard about the Ladder of Abstraction in Teacher Training” and the author sums it up as follows: “My post is about trying to start using the concept of Ladder of Abstraction with my developmental college math students.” A memorable quotation from the post is: “Only one week to go before I meet them for the first time!”
–> My take: Dan Meyer has been such an inspirational figure for a lot of us teaching math. It’s no surprise that he is continuing to do so!
Nathan Kraft | Out Rockin’ Constantly
Nathan Kraft @nathankraft1 has a blog named Out Rockin’ Constantly. The second post for the Blogging Initiation is titled “Working for the Man: A Cautionary Tale” and the author sums it up as follows: “I did some work writing math-tasks for a company this summer. However, once restrictions were placed on how I could design the lessons, it no longer became something I enjoyed doing. Having creative freedom allows me to do my best work. Although getting paid sounded like a great motivator, it had the opposite effect.” A memorable quotation from the post is: “Once I was getting paid to write math lessons and had restrictions placed on how that should be done, what used to be fun became work.”
–> My take: I helped work on our school’s salary scale through a committee this past year and one of the coolest facts I learned is that most teachers don’t know their exact salary (do you? I don’t) unlike some other professions where money is much more of a motivator. Nathan captures this sentiment well in talking about getting paid to write math lessons. I don’t know if I could do that! I get super stressed out by planning, so the idea of doing just that part as a job kind of freaks me out.
Lee Ann Smith | Expanding Horizons Through Education
Lea Ann Smith @SmithTeach has a blog named Expanding Horizons Through Education. The second post for the Blogging Initiation is titled “Useful teaching technique” and the author sums it up as follows: “I like to keep my algebra students engaged using a wide variety of activities. This post describes a way to use a little bit of stand up comedy to keep them attentive.” A memorable quotation from the post is: “If I have a concept that I really want my students (high school algebra 1) to remember, I will occasionally deliver the key point while standing on my desk.”
–> My take: This is a funny idea. I love making my classroom a little more whimsical, so I’ll have to try this out.
Emily Allman | Algebra, Essentially
Emily Allman @allmanfiles has a blog named Algebra, Essentially. The second post for the Blogging Initiation is titled “Reflections and Transformations” and the author sums it up as follows: “I am not a new teacher and only a fairly new blogger, but this past year has been full of new experiences, new adventures, and new ambitions. Thanks to the mathtwitterblogosphere, it’s also full of new and wonderful ‘colleagues.’” A memorable quotation from the post is: “I spend lots of time thinking intensely about tiny details, which is a wonderful contrast to teaching – where you have teeny amounts of time to maneuver a plethora of calamities.”
–> My take: I love the reflective-ness of this post, and Emily’s ability to turn a situation that she could easily have just griped about into a great learning opportunity for her. I can’t wait to read some more of her thoughts.
gooberspeaks | Reflections from an Asymptote
gooberspeaks has a blog named Reflections from an Asymptote. The second post for the Blogging Initiation is titled “Staying Sane” and the author sums it up as follows: “I am going to provide some vainglorious advice about my ways of staying sane through the school year.” A memorable quotation from the post is: “I think my motto this year may just have to be Keep Calm and Carry On (thank you British government).”
–> My take: What a great idea to write down the things that will keep you sane during the school year. I really need to do this too. My most simple one is “No work on Friday” (our weekend is Friday-Saturday) but some of the other things that she does are simple and great!
Rachel Tabak | Writing to Learn to Teach
Rachel Tabak @ray_emily has a blog named Writing to Learn to Teach. The first post for the Blogging Initiation is titled ““Special Number” Project” and the author sums it up as follows: “This post is all about a project that I’ve been using for a few years. I like it because it gives kids an opportunity to show their knowledge of number theory, and also to be creative and silly.” A memorable quotation from the post is: “This activity holds tons of potential to get your kids geeking out to the max – showing you how goofy and creative they can be over topics like prime factorization, GCF, LCM, and a bunch of other stuff.“
–> My take: This is a nice idea! I love topics where students can be creative. My suggestions might be to take the [# pts] heading from each question. I think that takes away from the fun! Also, is there any way to make this more opened ended? Could the students figure out their OWN properties about the number they chose? This is a nice idea to do instead of toiling through a sub lesson!
Matt Moran | Maximize Interest
Matt Moran @mathewpmoran has a blog named Maximize Interest. The first post for the Blogging Initiation is titled “What I Wish I Knew Before I Started Teaching/Being the Worst First Year Teacher Ever” and the author sums it up as follows: “This post is something I have really wanted to write but struggled to find the words to say. In it I talk about how horrible my start in teaching was and what I wish I knew when I started teaching. Namely, what I wish I knew was something really obvious; I wish I better knew myself.” A memorable quotation from the post is: “To begin, I want to share a little humblebrag: I believe I set the all-time record for Greatest Disparity Between Expectations of What I Would Do As a First-Year Teacher and Actual Results of Teaching in Year One. Respect to all the other competitors for this accolade, but sorry guys this contest was never even close.”
–> My take: This is great (though perhaps a bit scary?) inspiration for a first year teacher – you don’t have to get it right the very first time you do it, and it takes a while until you feel really at home in the classroom. For me, the transition was all about learning to view mistakes as learning opportunities. You wont ever be perfect, but you can always be improving. Also, Matt sets a record for the number of times citing Dan Meyer in his posts. Also, for the record, I totally disagree about the not-smiling-til-Christmas thing!
As full as my Google Reader is, I am so excited about the math blogger initiation going on right now, run by everyone’s favorite fairy blog father – the wonderful, ever-present, inexhaustible Sam Shah (my co-star in Magic Mike 2). Here are fifteen great posts in no particular order from new bloggers. Support their foray into math blogging by getting on their blogs and commenting! Those first bits of feedback really get people hooked on blogging!
TJ Hitchman @ProfNoodlearms has a blog named **Circles and Tangents**. The first post for the Blogging Initiation is titled **What to do with linear algebra? Some Inquiry Based Learning!“** and the author sums it up as follows: **This is a “plan out loud” session for the basic structure of my linear algebra course. Writing is a good vehicle for finalizing decisions and making real commitment to change in the classroom.** A memorable quotation from the post is: **I have unreasonable expectations of the average undergraduate.**
Kyle Harlow @KBHarlow has a blog named **War and Piecewise Functions**. The first post for the Blogging Initiation is titled **New Blogger Initiative: Week One“** and the author sums it up as follows: **My AP Calculus students are weak in Precalculus skills, especially graphing functions and translations. They all have graphing calculators, which seems like a great aide in combating the problem. But so far, there is no evidence that the students are using them.** A memorable quotation from the post is: **I want to stand on my desk and loudly ask my Calculus students what they’ve been doing with those $120 paperweights they carry around.**
Pippi has a blog named **Pippi’s Adventures in Teaching**. The first post for the Blogging Initiation is titled **Literacy“** and the author sums it up as follows: **My ninth graders have trouble reading and understanding complicated (and sometimes not-so-complicated) problems. This kills some of them on their state tests, even if they know the concepts the question is supposed to be about. I’m trying to find ways to help them practice reading and get better at it.** A memorable quotation from the post is: **I’d like for the answer to be something they can draw as often as possible, since that forces them to really think and synthesize information, rather than just picking words semi-randomly out of the paragraph and writing them for their answer.**
Jennifer Wilson @jwilson828 has a blog named **Easing the Hurry Syndrome**. The first post for the Blogging Initiation is titled **The Title“** and the author sums it up as follows: **We often hear that technology speeds things up. In my classroom, however, I find that using technology actually slows down the pace. When students explore difficult concepts using dynamic technology, they ask questions that they haven’t thought of when thinking about the concept in a static environment. When I use a student response system to check for student understanding, I find out during class what misconceptions need to be addressed, instead of waiting to find out on a summative assessment. Using technology “eases the hurry syndrome” in my classroom.** A memorable quotation from the post is: **Using technology eases the hurry syndrome, forcing me to pay attention to the questions students have and allowing me to assess their progress in a timely manner.**
Meagan Bubulka has a blog named **variablesofmath**. The first post for the Blogging Initiation is titled **Hello World!“** and the author sums it up as follows: **About 2 goals I have for this year: more/better parent contact and making my flipped classroom more effective and useful!** A memorable quotation from the post is: **I want to make this process better for the students, streamline the videos and make them more my personality while making them even more valuable. **
Erin @ErinYBaker has a blog named **Math Lessons on the Loose!**. The first post for the Blogging Initiation is titled **Yay for Blogger Initiation: A Goal for the First Week of School“** and the author sums it up as follows: **I focused on the goal to start a robotics club team at my school by participating in a STEM program called FIRST Challenge. I also have an interesting link to free online Stanford classes.** A memorable quotation from the post is: **Through this robotics club, I am hoping to create an outlet for students who may not have had that moment/opportunity in school yet to let out some of the creative and explorative minds. **
Pamela Rawson has a blog named **rawsonmath**. The first post for the Blogging Initiation is titled **The Evolution of a Teacher“** and the author sums it up as follows: **It’s about how graphing calculators have changed over the years that I’ve been teaching, from non-existent to TI-Nspire. It’s also about how those changes affect how and what I teach. Well, more how than what.** A memorable quotation from the post is: **After all, it’s just a tool. If I can’t use it to teach something, then what’s the point?**
Pam Rissmann has a blog named **PPerfect Squares**. The first post for the Blogging Initiation is titled **PPerfect Squares — It’s a rap!“** and the author sums it up as follows: **Why did I name my blog PPerfect Squares? I wrote this math rap my first year teaching, and my classes love singing it in class, and I’ve been known to rap the song at school talent shows. Kind of corny, but fun. My blog post includes the lyrics and an avatar singing it.** A memorable quotation from the post is: **PPerfect Squares, perfect squares goes on and on, but this is the end of this song!**
haversine has a blog named **Bowditch’s Apprentice**. The first post for the Blogging Initiation is titled **Like a Butterfly’s Wings in China“** and the author sums it up as follows: **What an inspiration Nat Bowditch has been to me since 5th grade me read about him in Carry On, Mr. Bowditch. ** A memorable quotation from the post is: **Nat famously steered his ship home to Salem during a blinding snowstorm one Christmas, confident in his calculations.**
Tad Snaith @TadSnaith has a blog named **What Does Math Mean**. The first post for the Blogging Initiation is titled **Why are students “good” at math?“** and the author sums it up as follows: **An attribute that I believe helps students or anyone understand a math concept is the ability to play a movie in their head of the idea.** A memorable quotation from the post is: **From my experience talking to students about their inadequacies which impede these skills we take for granted I strongly believe that if students or anyone for that matter are able to play a movie in their head of the math concept at hand then that person will have success at deciphering tougher math problems.**
Erin Wade @ewade4 has a blog named **wadingthrumath**. The first post for the Blogging Initiation is titled **This year…“** and the author sums it up as follows: **While it’s my 8th year teaching, it is my second at my current school and I’m still adjusting. These are some of the areas I’d like to improve this year! ** A memorable quotation from the post is: **PS…any suggestions on how to best utilize 43 minute class periods?? **
Ann Gorsuch @AnnGorsuch has a blog named **anngorsuch**. The first post for the Blogging Initiation is titled **My Goals for Student Teaching“** and the author sums it up as follows: **Here are my six hopefully attainable goals for student teaching. ** A memorable quotation from the post is: **Goal 2: Teach students to justify (explain why) and challenge (ask why) their thinking**
Nick Gerhard @nickgerhard has a blog named **Gerhard’s World**. The first post for the Blogging Initiation is titled **New (School) Year’s Resolutions“** and the author sums it up as follows: **I have a lot of goals for this year that will hopefully improve my teaching and my students’ learning. If I can not procrastinate, change up my teaching styles, continue working with my PLN, and stay positive this will prove to be a great school year. ** A memorable quotation from the post is: **Most resolutions are stop smoking, lose weight, yada yada yada.**
Lori Ferrington @loferrington has a blog named **Shift(ed)ucator**. The first post for the Blogging Initiation is titled **Standardized Grading Setting the Standard?“** and the author sums it up as follows: **I’m reflecting on last year to see explore some changes to be made for the upcoming school year. Specifically, I look at the possibility of implementing standards based grading to more accurately reflect student achievement.** A memorable quotation from the post is: **I like the ideas and am just beginning to form the thought of how this will look in my classroom environment, but I’m excited about making changes for this next year and refuse to let a standardized grading policy be the measure of my students’ achievements.**
Dave Enrico @denrico1 has a blog named **Me Dot**. The first post for the Blogging Initiation is titled **New Blogger Initiation: A Star Is Born“** and the author sums it up as follows: **I used to want to become a “master teacher.” Now I prefer the ring of “improving teacher.” I think this could be a sign of growth.** A memorable quotation from the post is: **So I should be, what, a third-degree master now, right?**
My school uses Moodle as our platform for sharing course materials. I used it for two years, but it was just way too clunky for me – editing everything takes thrice as many clicks as it should. So last year I decided to upgrade to a Google Site for my class. I just redesigned it for the upcoming school year and it looks really pretty, so I wanted to share it:
Here’s the new site, which doesn’t have any content yet, but will aim to do the same thing as my old course website for the same class (if you want to see what content I put on there). On my site students…
- Check their homework assignments (I do not write homework on the board, just announce that there is homework)
- Check for upcoming tests and quizzes – I write which standards are included, so then students…
- Read the text of standards for the course
- Access materials to study for specific standards, whether for the first quiz, or for a reassessment
- Find daily materials including images, problems and links for students who take notes on the computer
- Leave anonymous feedback for me
- Check their grades (I make internet reports with EasyGrade Pro, host them in my Public Dropbox folder and link from this website)
- Access the grading scale for standards
- Request a Reassessment, through a Google Form, which goes to a spreadsheet I can see
- Access a virtual whiteboard through Scribblar where we can interact virtually after hours, or where students can interact with each other (experiment this year)
- Fold their laundry (there’s an app for that)
As you can tell, I use it for a lot of different things in my class, all aimed to INCREASE student accountability, which is why I spent time to make it look how I wanted it to. Some tweeps enjoyed the look (and the fact that I have some goofy elements in my Reassessment Request form, check them out), and they were wondering if I could post a template for the site, so I did! When you are making your Google site, if you are at the “Manage Site” interface, click on Themes at the bottom of the sidebar and then click on the Browse More Themes button at the top right.
Theme name: Math Class Portal – @bowmanimal
The pretty banner wont be there, but other than that everything else should. The only thing that you really have to change besides adding your own content are the forms embedded in the Reassessment Request page and the Anonymous Feedback page. Those are both Google Forms. You can either link your own existing forms … OR I posted templates of those in the Google Doc Templates which you can modify and use.
- Go to docs.google.com/templates
- Search for “Bowman Dickson“
- One is named “Anonymous Feedback Template” and one is named “Reassessment Request Template“
- Just click on “Use this template” to make your own. In the document, if you go to Form –> Edit Form, you can make your revisions.
- Then link the forms from the website to the one you created instead of the ones from my template.
PS. The background of my website is an origami crease pattern… if you do some origami and then unfold it so the paper is flat, you can color the creases and the folds different colors to reflect the 3D structure. Beautiful!
Day one of the school year is rapidly approaching so I’m continuing my mad scramble to outfit my classroom (complicated by the fact that I’m trying to stock up here in the US because random things are very difficult to find in Jordan). My classroom walls were pretty bare last year, so I decided to make some posters and get them printed in large format (18″ by 24″). I wanted some of them to reflect core aspects of my teaching philosophy so that I can more easily continue great conversations that start in the beginning of the year and seem to peter out. Here is what I got printed:
(I’m counting this as #made4math for this upcoming Monday – I’ll be on a plane then!)
Memorizing vs. Understanding
One of the big themes in my classroom is the difference between understanding and memorizing. I don’t need to rant about this now, but I think that students just are not aware that when they cram algorithms in place of problem solving, they aren’t really learning. Getting them to understand the difference between understanding and memorizing is one of the most important metacognitive lessons of my math class. I thought it would be nice to have a visual reminder I could point to with a sort of classroom meme.
To explain, this is a picture of my family dog Whiskey. Whiskey is awesome at learning how to do all sorts of tricks. I think he could easily be a commercial dog. But when he messes up, he has trouble getting back on track, because he has no idea what he is actually doing. He has memorized what gets him a treat (like how students perform mathematical algorithms for grades) but has no deeper level understanding to be able to tinker with the process or apply his knowledge to a new situation. My favorite Whiskey-isms:
- BANG, YOU’RE DEAD: When my mom puts her gun finger out, Whiskey responds by sticking his paws in the air innocently. Then, my mom yells “bang!” and Whiskey awkwardly flops to the floor, flips over and plays dead. He’s really good at this, but if, for some reason, he messes us up, he just tries to throw all the steps of the trick at my mom until she gives him a treat (like students who just try to write stuff on a test for points). I show my students the video in this post, which shows Whiskey messing up, to illustrate the problem with troubleshooting a process you don’t understand.
- OUR NEIGHBOR’S NEWSPAPER: Whiskey brings in the paper every morning for my parents, and loves life every time he does it. When he brings it in, he will chomp down on that paper until my parents give him a treat. The only problem with this has been that while on walks, he will sometimes see a neighbor’s newspaper and apply the same logic. He grabs it and the sprints home (however far away) and then wont let go until he gets a treat. I am going to use this example to talk about how you can misuse processes if you don’t understand them and try to apply them to different situations (like canceling out added terms in a rational expression).
If Whiskey could just ask “Why?” he could avoid these errors in his tricks! Good thing my students are capable of doing that.
The Growth Mindset
The Growth Mindset, a brainchild of Psychologist Carol Dweck, has become one of the lynch pins of my teaching philosophy. The philosophy espouses that believing intelligence is fixed hinders learning – “smart” kids will be scared to take risks and fail, and “dumb” kids will not see real results in their learning because they are comparing themselves to others instead of themselves. The growth mindset puts the emphasis on hard work leading to real learning, and normalizes (no, necessitates) mistakes as part of the learning process. I really like this image (which I did not make), despite how small it is, and I could see a student reading through this one day at the beginning or end of class. Even if not, I made the headers “Fixed” and “Growth” bigger so at the very least it will be a reminder to me while I teach about this important idea!
In the first week, I will give a survey that will get us talking about the Growth Mindset, as I did last year, but I hope to do a better job of continuing that conversation this time.
Math is Magical
I love the completely-not-subtle message that is so subtly expressed in this poster (which I also did not make). Yes, Math certainly is magical. In surveys, students always cite my enthusiasm for the subject as something that makes the class better, so I have embraced that as part of my teaching philosophy.
Foundations are Important
Okay, maybe this isn’t a huge part of my teaching philosophy, but I made this poster last year on regular printer paper and I really liked it, so I wanted a more durable poster. I like it especially because it worked as both a comforting thing for my Calculus students and an inspirational thing for the Algebra students. If you want a copy of your own, I have this image split into nine regular pieces of paper that you can print out and glue onto a poster.
As my students know, I am a bit of an efficiency freak (read: I’m kind of impatient and actively organize my classroom to avoid patience problems). I love having students work with students other than the ones that sit right around them ( I don’t have assigned seats), but I can’t stand the time it takes to reorganize into groups. So whenever we do this, I do it right from the beginning of class. Students walk into the classroom to find the whiteboards laid out on the tables like this:
I usually stand around the door just instructing them to sit where their names are (even though most figure it out anyway). What usually happens is about half of the kids show up a few minutes early and sit behind the boards twiddling their thumbs. But then the problem catches their eye… And you can see them looking back and forth, fiddling with the markers, trying to resist the urge to DO MATH! Inevitably someone asks: “Mr Bowman, can we get started?” I’m like the coyest math teacher in the world, so I always respond “Uh, yeah, um sure, I guess” while my mind is deviously tapping its fingers together like Mr. Burns. Then the other kids trickle in, and immediately engage because they see someone else working (and haven’t really missed anything). Every time I do this, every single student in class is engaged in math by 30 seconds into the period with little to no cajoling from me. Then after a few minutes, I might give a few more instructions (like, this is MISTAKE GAME!) I realize that this may seem trivial/common sense, but the thing that makes me happiest is using every minute of every class!
Variations: If you don’t want to assign groups (which I usually do to get specific students working together) you can do this randomly with a deck of cards, or something similar.
GOAL: develop frameworks and modes appropriate for MATH specific Whiteboarding.
I did a ton of experiments this year with whiteboarding and a lot of brainstorming, but here are my three favorite modes of math whiteboarding that I tried (some writing copied from previous posts). A good whiteboarding mode for me can be applied to many different topics and takes advantage of everything whiteboarding has to offer: collaborative, interactive, promotes risk taking and visually stimulating.
Guess and Check with a Partner
Students try to solve problems that take a certain amount of intuition or guesswork (like antiderivatives or factoring) by having one person write down a guess, and the other person check if it is correct.They would then keep doing this until they get a correct answer. After a certain number of problems solved, the two students switch roles. For example, above the students are looking for the antiderivative of – the guesser writes down and the checker takes its derivative to see if that is correct. Since does not equal , the guesser tries again. They continue this process until they finally get that back again. This mode is great for showing students that a great way to do math (at first) is to just try things and adjust their answer; it’s great for getting students to converse together about how to get a solution; and it’s great to get them in the habit of always checking their answers. I had a really hard time getting some students to follow the procedure for this one, but the ones that stuck to their roles got a lot out of it.
Color Coding Problems
Before solving a problem, students rewrite it using different colors to help them understand its important parts. For example, above is a whiteboarding exercise I did with the Chain Rule. Students were in groups of threes – for each problem, one person had to rewrite the problem in different colors to indicate which was the outside and which was the inside function, the next person had to differentiate it still using the colors to point out where each part of the new expression came from, and then the last person had to rewrite the expression in a simplified form. This was perfect because the hardest parts of the chain rule are recognizing when you need, seeing inside vs. outside and then seeing where the parts of the new expression come from.
The Mistake Game
Groups present solutions to semi-complicated/involved problems on whiteboards, but while presenting their solution, they purposely make a mistake (and not an silly arithmetic mistake like - a real hardcore-misconception-style mistake). Then, they present their work to the other students in the class, trying to sell their mistake as having been made for real. Other students ask thoughtful questions about the presenting group’s solution to try to help everyone find the mistake. This is always great with a quick class followup at the end collecting the most common mistakes. Check out the Guide to the Mistake Game from Kelly O’Shea, who introduced me to this game.
P.S. I’m realizing now that the example above actually isn’t a great example of a time to use this game… Some topics that it worked well for this year were graph sketching, solving for limits algebraically, using the quotient rule, implicit differentiation, related rates and using infinite limits in graphing exponential functions.
I make it standard practice of mine to steal as much as I can from science teachers out there. One of the best things I have borrowed from Physics Teachers has been whiteboarding, popularized by the rise of Modeling Instruction. I feel like lots of math teachers have these smaller whiteboards in their classroom, either individual ones or a good size for groupwork, but they maybe aren’t all that sure how to use them. Physics teachers have figured out a way to use them effectively that makes sense for them. From what I gather, most of these Physics teachers (at a basic level) have students whiteboard problems that they have previously worked out on homework assignments or worksheets and then present those to each other in “board meetings” as part of the Modeling process (sorry if I totally butchered that). It makes sense with Physics and other teachers can join in on the fun because they can see immediately how to use them.
But as great as stealing is,we need a whiteboarding framework for math, so that math teachers can see how to use them immediately and also join in on the fun. We need non-content specific techniques that teachers can use day to day so that whiteboards don’t just get added to that pile of crap in the back of the classroom. There are math teachers in the blogotwittersphere - like Anna (@borschtwithanna) and Timon (@MrPicc112) – who are working on this too, and I hope we can get more collaborators. I am going to devote the next few posts to math whiteboarding, but please join in on the fun if you have done something super cool that might be helpful for others (i.e. me).
but first, some observations from this past year when I started using whiteboards…
WHITEBOARDS PROMOTE COLLABORATION. My whiteboards are about 2 x 3 feet, which gives plenty of room for at least 3-4 students to work together on a problem. There is something about working on the same surface (as opposed to working on the same problem in individual notebooks) that gets students talking way more. Whatever written is owned by the whole group, so there is more of a natural desire for everyone to explain to each other and help each other understand, and to healthily debate various aspects of problems.
STUDENTS ARE MORE LIKELY TO TAKE RISKS. The magic of a whiteboard marker is that it erases easily. It’s really not a big deal to make a mistake on the board because it is so easy to change. Students will try things that they never would with a pen and paper.
WHITEBOARDS ARE A NICE CHANGE OF ROUTINE. My students loved using whiteboards just because it was something different. They would work with different groups of students, we would sit in a different spot in the classroom, and the onus of the learning would be on them instead of on me. I don’t think that the idea of changing up the routine is trivial.
ALL TYPES OF LEARNERS ARE TARGETED. With different color markers, verbal exchanges between students, lots of time where students are being active and opportunities for creativity, almost every type of learner is engaged by whiteboarding.
Issues to Work Out
STUDENTS DON’T HAVE ANYTHING TO REVIEW LATER. Lots of great learning happens during the whiteboarding, but for those that need to review later, they don’t have anything written in their notebooks. I tried taking pictures of the whiteboards and posting online, but I doubt that any students ever really looked at these. A solution that I heard from a colleague this summer was to give them 5 minutes at the end of class to copy down a problem that they would like to look at later. I like this solution, and would like to try it out this year.
SOMETIMES ONE STUDENT WILL TAKE OVER. I found that occasionally whiteboarding would turn into one student writing while a few others sat back and watched. This is a fallback with almost every type of group work. Though they felt a little forced, the best way I found to avoid this is to have specific roles, or structured ways for all students to get themselves involved. This is something I am going to try to document and work on
WHITEBOARDING TAKES A LOT MORE TIME. I’m sure some pros have managed to work whiteboarding into their curriculum without sacrificing pace, but I definitely did not. In fact, I sort of used it to slow things down. If you’re looking to power through material (like I was at times in my AP class), I don’t think whiteboarding is the solution, because its strengths are in allowing students to communicate, construct their own concepts, and spend more time exploring a concepts deeply.
IT IS TOUGH TO GRADE WORK DONE ON WHITEBOARDS. Some teachers have expressed this concern to me – they aren’t handing anything in and the work is done by everyone, which makes it difficult to grade. My solution? Don’t grade it. Not everything that happens in the classroom needs to be graded.
Getting the Materials
GETTING THE WHITEBOARDS: If you can, I think the easiest way to get whiteboards is to have your department order some, but you can also easily just head down to home depot, buy a huge piece of whiteboard and have them cut it for you. @borschtwithanna describes how she did this here, and @fnoschese describes a few different ways to get them in his classic post describing whiteboards here.
OTHER MATERIALS NEEDED: I found that for my whiteboards, the normal erasers just don’t work well. @mgolding suggests using black socks as erasers (white socks get gross) and @misscalcul8 suggests putting the markers right in the socks as an easy way to distribute the markers and a way to avoid students fighting over which color marker they get. With the whole class using markers instead of just you, I also found that you need tons of markers. Luckily, I don’t have a quota at my school, so I just always go nab some more from the supply closet, but @kellyoshea has a pretty good solution to this problem with refillable markers and marker buckets for each group.
Want to know what students think? I collect student feedback as often as possible, so I went into my most recent document and pulled out every comment having to do with whiteboards. Interestingly, most of the negative comments came from my AP class and very few negative comments came from my non-AP class. (emphasis below is mine)
I LOVE when we use the white boards because I get to see how my peers think and compare their thought process to mine. And then when we discuss it with the class its even more helpful. There is so much opportunity to lean with this exercise.
I really liked graph sketching on whiteboards, and then hiding a mistake in it. It was helpful because it helped me remember what common mistakes I should avoid.
You’re doing a great job with the white boards, it give us a chance to work together which is fun and helpful at the same time, we can talk a little but we generally do the work and its more of a competition to me instead of just work so that is an extra motivation
For me, graph sketching on whiteboards and working in groups on packets isn’t really helpful and it’s not your fault. Some kids in my class lack motivation and manners and a desire to improve; so I guess it’s really frustrating to work with them.
One learning activity I found particularly helpful is the activity in which we drew derivatives on whiteboards. I tend to find drawing graphs from functions to be difficult at times, especially when we are focusing on derivatives and so this activity helped me a lot. It allowed me to see common mistakes and different ways of drawing functions and the discussions we had while sketching graphs allowed me to realize my mistakes.
graph sketching on whiteboards [is helpful]. Since it was interactive I learnt much more
I really like graphing sketching on the whiteboards. I enjoy working in groups, then looking at what everyone did. It’s a good way of practice, and we learn from our mistakes.
when we use a white in groups [isn't helpful] because i get confused when were than one wants to solve it on the board rather than if we work in groups and each one with his/her own paper.
Drawing derivatives on whiteboards [isn't helpful], because I find that the diffusion of responsibility between the team members decreases their productivity in class.
I think the most helpful class activity that has been very helpful is sketching on the white boards leaving mistakes for others to pick, in this case we can learn where are the possible mistakes occurring and allow you to avoid them when dealing with your own.
I found that [it was helpful] when we graphed derivatives on white boards and slowly drilled each step aswell as common mistakes to avoid. It was perfect to clear any doubts both visually and algebraicly.
Drawing derivatives on white boards, because my classmates and I can discuss different methods of solving a question.
I like using the white boards because it’s nice when we all share our work and see everyone else’s work and compare it to ours and then we look at the mistake and fix it together.
The best activities which I felt that helped me a lot is the group work like the games and the white boards
I think that when we do goup work and work on answering a question on the white-board I feel that one student work and the rest just watch him working which is not as beneficial to everyone.
At Twitter Math Camp 2012, I gave a session about getting started with GeoGebra. Here are the resources from my session, including eight tutorials and links to pages with lots of other tutorials
TABLE OF CONTENTS
- Why Should you use GeoGebra?
- How do you use GeoGebra in your classroom?
- Why use GeoGebra instead of Geometer’s Sketchpad or another math visualization program?
- Where do I get GeoGebra?
- How can I learn how to use the program? <– TUTORIALS!
- How can I find ridiculously cool applets that are way above my skill level?
Why should you use GeoGebra?
The idea of learning a new technology and incorporating it into your teaching can sometimes very overwhelming. And you should never just use technology for technology’s sake, as some administrators seem to espouse. You have to have a real reason to use it. GeoGebra can improve math instruction in a million ways. The dynamic nature of the program gives you the ability to explain and explore concepts that simple pen and paper (or marker and whiteboard) can’t! I find myself using the program at least weekly, sometimes more.
How do you use GeoGebra in your classroom?
AS A DYNAMIC DEMONSTRATOR: To help students understand a tricky concept during direct instruction.
How can you get students to understand that the perpendicular bisectors of a triangle ALWAYS meet at one point? Construct a triangle with perpendicular bisectors in GeoGebra and move the vertices of the triangle around and let them observe that those lines always meet up at a point. (PS, sorry I didn’t upload these – this post took forever as it is and there are lots of examples of applets in the tutorials section below)
DYNAMIC WORKSHEET: To give students a chance to explore a concept at their own pace in small groups or individually.
One activity I do every year is let students “discover” derivative rules using a derivative tracer. They enter a function into a GeoGebra applet, which then traces out its derivative. With that, students try to guess what the equation of the derivative is. Once they collect a bunch of examples or correct derivative equations, they look for patterns to come up with a rule.
STUDENT EXPLORATIONS: To give students a powerful tool with which to complete their own investigations.
I have had students convert pictures to integrals, fit functions to data of really crazy things that they wanted to study, and calculate the volume of real world solids of revolution. Getting them comfortable with program with more guided activities earlier in the year gives them the skills to be able to do amazing things with it on their own later in the year.
CREATING WORKSHEETS/ASSESSMENTS: A tool for you to make your worksheets and assessments very professional looking.
You can copy and paste anything from GeoGebra into a Word Document, giving you the ability to put very good looking graphs and diagrams in your teaching materials.
Why use GeoGebra instead of Geometer’s Sketchpad or another math visualization program?
Well, first, it’s free. I mean, that should really be enough, but I’ll keep going. Because it’s free, you can install it on as many computers as you need (so students can use the program at home and at school). And you don’t actually need to install it – you can run GeoGebra right from a web browser, or host web applets that just require a student to have a browser with Java installed (i.e. 99% of people who own a computer and keep it even remotely up to date). Basically, no matter how annoying the tech department at your school is, GeoGebra is pretty easy to get going.
Additionally, because the program is free, it is developing quickly, and resources are easy to share and easy to come by. The community around GeoGebra is strong and constantly growing – check out GeoGebraTube, a ridiculously large repository of GeoGebra applets.
Where do I get GeoGebra?
Download it here. Click on “Webstart” to download the installer. You can also start a web applet here (in “Applet Start”) and download an offline installer for students without internet access.
How can I learn how to use the program?
Luckily, the program is incredibly intuitive. The best way to learn is to open up the program and experiment! But some people hate that and need a bit more of a push to get going (I had to teach my mom how to text with her new phone, so I think she is one of those people). That’s totally okay – my recommendation is to work through some tutorials that can show you how powerful you can be with the program. I wrote 8 tutorials that progress from GeoGebra basics to some cool intermediate to advanced things that will go a long way in creating your own applets.
GeoGebra Tutorials (written by me):
7. Uploading to GeoGebra Tube
(uploading your creations to the web to make sharable web applets)
(tutorial, finished product) <– With GeoGebra 4.0, this is even easier! There is a menu item in File–>Export–>Dynamic Worksheet as Webpage (.html), and then you can directly upload to GeoGebra Tube.
Other tutorials I have found:
- Math and Multimedia Tutorials – Over 50 GeoGebra Tutorials at all levels from the blog Mathematics and Multimedia.
- Lance Bledsoe’s Tutorials - A similar collection to mine above of basic tutorials to get you started with the program.
How can I find ridiculously cool applets that are way above my skill level?
If you aren’t all that interesting in making your own, you can still find tons and tons of great applets. Like this applet that helps derive the equation for the area of a circle…
Head to GeoGebra Tube, an official searchable database of GeoGebra applets for just about any topic imaginable. Feel free to be inspired by the amazing work that some people do with the program!!