Exponents & Error Analysis

Screen Shot 2014-01-13 at 7.50.49 PMThere aren’t many strategies that tell me more about student thinking than “error analysis” opportunities.  When students “grade” another hypothetical student’s work, there’s something special about the experience – students’ senses are heightened.  They’re on a quest for truth.  They feel exceptionally “safe” speaking about the mathematics because the work belongs to “somebody else”.

Many textbooks do a nice job of incorporating the occasional error-analysis opportunity somewhere toward the end of a problem set.  I find error-analysis becomes a more engaging experience if you “set the stage” so-to-speak.  For instance, as my algebra students entered class, I handed each of them this sheet —> QRCodeFindNFixErrorAnalysis before they crossed the classroom threshold.   Then I said, nonchalantly, “Will you go ahead and grade this for me?”  Some students were eager to please me, smiled, sat down, and got to work.  Others looked at me as though I’d said something very strange!  One girl giggled and said, “Mrs. Yenca, I actually thought you just said you wanted me to grade this!”  My response?  “Yep!  That’s exactly what I just said!  Thanks in advance!”

They were on to my gimmick 😉 but that’s okay – each student “graded” the paper silently and individually first, red pen in hand, then paired up after a few minutes to compare with a peer.  Finally, students scanned the QR-codes to see if the revealed text message confirmed or denied their work.

But I never stop there – asking random students to take their work to the document camera to present to the class is where the true meat-and-potatoes of this experience happens.  As students explain the way they “graded” the paper, they reveal the darndest things about their *own* thinking.  MANY students confirm correct work, but for the *wrong* reasons.  MANY students struggle with using appropriate academic vocabulary, and need a little encouragement to “talk the talk”.  It’s enlightening to have students with different opinions present their work and allow discussion and debate to ensue.  Letting them talk is the best part.

How do you use error analysis to enrich classroom discussion?  Confirm mathematical understanding? Reinforce vocabulary?  Reveal misconceptions?

Andrew Stadel has some rocking resources for error analysis too – check them out here.

Here are a few more I’ve used as well.

 

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Nearpod “Best Practices” For Math Class

Screen Shot 2013-12-28 at 7.30.08 PMI kid you not – when I say, “Get out your iPads and go to Nearpod” my 8th graders literally say, “YESSSSS!” and give a little fist-pump… every time.  It’s not necessarily Nearpod that they love, but the peer interactions and instant feedback it facilitates.  In a nutshell, making your Nearpods about “them” and not about “you” is the way to go.  Think about what the students will be doing versus what you, as the teacher, will be doing.  Students should be engaged and have ample opportunity to show what they know during any NPP (Nearpod presentation).

Perhaps having a few days away from the classroom is getting those creative juices going in the minds of math teachers.  This week I’ve been contacted several times by different math folks all aspiring to use Nearpod in their mathematics classrooms.  Even more specifically, mathy people are wanting to use Nearpod more effectively, and have asked me for ideas regarding “Nearpod Best Practices” in the mathematics classroom.

Here are some excerpts from e-mails I’ve been sending to individuals – I hope these first-hand experiences and tips will help as you Nearpod in the New Year!

Yes, I just used Nearpod as a verb.

 

Nearpod ≠ PowerPoint

It’s tempting to just refresh those digital presentations you might already have by dumping them into Nearpod.  This idea in and of itself isn’t a bad one, but if there are more static slides than opportunities for student input, your Nearpod may fall flat.  I’ve had success in limiting the number of “static” slides in a NPP and adding lots of assessment tools that constantly gauge student thinking.

In my humble opinion, Nearpod is best used as a segment of a lesson, not the ENTIRE lesson.  Consider a Nearpod at the start of class to pre-assess or review homework.  Consider a Nearpod to introduce a topic.  Or, use Nearpod to review and assess after a lesson has taken place.  Just as a PowerPoint or Keynote shouldn’t last 50 minutes, a NPP shouldn’t either.

 

Nearpod “Draw” = Instant Error Analysis

Work SamplesIt’s really not about the NPP – it’s about the peer relations Nearpod facilitates with ease.  “Draw” doesn’t have to mean “draw a picture” – mathematical work can be “drawn” too!  Present some content, then ask the students to solve a problem and “draw” what they know.  (AKA “show your work!”)  This allows you to virtually send every student to the board!  Think of the paper “exit ticket” of the past, but administered when needed during a lesson (rather than only at the end) with instant digital work samples from every student (rather than annoying little piles of paper to sort in hopes of helping your students tomorrow.)  Nearpod redefines the “exit ticket” concept completely.

Don’t be selfish and keep all of those awesome work samples to yourself.  Launch correct and incorrect work to ALL students’ screens, or at the front of the class, so discussion can immediately ensue.  Consider framing each anonymous, launched work sample by saying, “Tell me something you like about this student’s work, and then tell me something you don’t like about it.”  Or, “Tell me something you notice about this problem, and then tell me something you’re still wondering about.”  Students are happy to comment on their peers’ work, especially in an error-analysis mode.  Being able to scan instant student work samples is powerful data for the teacher, but gains the MOST power when shared with students to encourage rich mathematical discussions.  Trust me, they are very happy to comment on their peers’ work.

 

Nearpod “Polls” and “Quizzes” = Positive Peer Pressure

QuizStudents also love polls and quizzes, but please don’t keep that pretty pie chart to yourself!  When launching a poll or quiz, I put my iPad under the document camera and zoom in so only the moving pie chart is visible on the screen at the front of the class.  They LOVE this – we even have weird sayings about these graphs during “quizzes”.  Since green sectors indicate correct submissions, and red sectors show incorrect submissions, my students yell for “key lime pie” (green) and get all worked up when a graph starts turning red (“No!  No more cherry pie!  C’mon guys!!!”)  This is a silly thing that has developed in my own classroom, and they really do like it… and own it.  Peer pressure plays a positive role in Nearpod “quizzes” and “polls”.

 

Do you feel like your Nearpods are boring?  Here are some questions to consider:

  • Do you have enough interactive features in your NPPs versus static slides of information?
  • Do you try using Nearpod for shorter spurts of the lesson rather than for an entire class period?
  • Do you encourage student dialogue as a key part of Nearpod experiences?
  • Do you display class data for everyone to see, versus keeping it only on your own screen?

I’d love to hear how you’re using Nearpod with success in the mathematics classroom! Feel free to comment below.

To hear me share more “Nearpod love” check out this interview.

To see Nearpod content I designed, click here.

Thanks for the mention on the Nearpod blog!

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Posted in Algebra 1, Pre-Algebra | Tagged , , , , , , , , , | 22 Comments

“Pittsburgh Protractors”

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While visiting my home town, I couldn’t pass through without locating a few “Pittsburgh Protractors”.  For several years, painted protractors have been mysteriously appearing all over the city.  While no one seems to know why the protractors keep popping up, or who is the culprit, blogger Eric Lidji has made it a mission to number and track the protractors via a Google Map.

 

I tested the map today and easily located three protractors… several were lodged behind dumpsters, but I was able to get close enough to pose with Protractor #153 in the Strip District.  A quirky math mystery indeed!

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Defending “The Worksheet”

In this era, “the worksheet” has gotten a pretty bad rap.  The seemingly popular push to go “paperless” doesn’t help the cause.  Are worksheets and paper inherently evil in the 21st-century classroom?

In the ed-tech realm, I sense a little hypocrisy in worksheet trash-talk.  I liken it to the excuses folks make when they blame technology for “distracting students” (your students were already distracted before the iPad showed up… they’re kids… they just passed notes when you weren’t looking… or doodled in their notebooks… or ignored you… but I digress…).  Maybe you’ve heard that worksheets are boring… worksheets don’t engage students… they’re “busy work”… friends shouldn’t let friends hand out worksheets… I think over-generalizations can damage any cause, and that’s why I’m here to defend the worksheet.

To defend it, we have to define it… or perhaps redefine it.  I think this is key to understanding do’s and don’ts of worksheet use.  In the spirit of SAMR, here’s my very simple hierarchy of the worksheet, ThingLink-style.

“Practice” – I think this level of worksheet use is the one that potentially gives all worksheets a bad name.  The worksheet is not evil, but its misuse can be.  Giving repetitious problems in mass quantity is bogus.  This is where phrases like “skill… drill… kill…” originate.  Copy these as-is, assign every problem every day?  Yep, you’re the target of the trash-talk.

“Transformation” – Just because a worksheet begins its purpose as an 8.5″ by 11″ sheet of paper doesn’t mean it has to stay that way.  Start small – take a worksheet of the “Practice” nature, and cut the thing up.  Group students randomly, and give one problem to each group to solve and ultimately present to the class.  Encourage communication, provide a menu of presentation tools (the board, a document camera, a screencast using Explain Everything or Tellagami) and you’ve just transformed a worksheet.  That was easy!

Not only can paper be cut… but it can be folded too! 😉  Ever use “foldables” in your classroom?  Kids love them – they’re great for organized and interactive note-taking, and students take pride in them because they’re more like a product than yet another sheet of lined paper in a notebook.

I don’t think any teacher would poo-poo hands-on learning experiences.  Did you know paper can be cut, manipulated, and sorted?  Can encourage rich mathematical communication?  Can cause students to defend their mathematical point-of-view?

Okay, so I am getting very tongue-in-cheek here, but my point is this – mathematical communication, problem-solving and even hands-on learning can occur from paper worksheets if used correctly.  I like this action research review and the cited research on physical and virtual manipulatives – they both matter (and this teacher’s students seemed to prefer physical manipulatives to virtual ones – see pages 33-34).  Don’t throw the paper out  just yet.  Do your students a favor and transform a worksheet!

Need some ideas?  Explore the ThingLink to see what bloggers are up to in the “worksheet transformation” realm.

Posted in Algebra 1, Pre-Algebra | Tagged , , , , , , , , , , , , , , , | 2 Comments

Texas-Sized Snowman

Perhaps it’s the knowledge that the probability of snow here in Austin, Texas is close to zilch that inspires my neighbors to buy such a massive inflatable yard snowman.  When I bike past this house, I giggle every time at the sheer silliness of this fellow.  I’m standing next to it to help you think of questions to ask and investigate.

In the spirit of www.101qs.com what’s the first question that comes to your mind?

How will you attempt to answer it?

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2013 Edublog Awards

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There are amazing folks sharing about the wonderful things they’re doing through reflections and resources on blogs.  Getting globally connected and inspired has never been simpler.  If you’re not blogging… you should be!  🙂

Please take a moment and scan all the names of folks whose blogs and tweets have enriched your practice and your life, and give them a vote!

Edublogs is using List.ly for voting this year.   Check out all the categories here:

Voting has officially begun!  Thanks so much for the nominations!

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Sort-of-Real-World Math

Dan Meyer’s latest post got me thinking about what seems to motivate students as far as “real-world-ness” goes.  What’s timely about this post is, I think some things my students did in class today go along well with reflecting upon, as Dan puts it, “theories of engagement”.  How “real-world” does the math have to be to be worthwhile to kids?

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Take my algebra students, for example.  Today’s objective was to “Solve Linear Systems Using Substitution.”  I started class with a little novelty and my students ate it for lunch.  In small groups and before any attempt at instruction on my part, students solved “The Leg Problem” by guessing-and-checking, drawing all sorts of interesting pictures, or by writing an equation that would later end up being half of our system – namely, something like 2x + 4y = 74 where x represents the number of chickens and y represents the number of cows.  What’s fun for me is showing them the problem statement, which is novel but certainly not a “real-world scenario”, yet no student has EVER complained.  They’re motivated enough to want to see this thing through to a solution.  They rise to the challenge, start talking mathematics, and can’t wait to share and present their methods to their peers.  Check out two work samples below.

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My pre-algebra students started today’s lesson with a graphic I found on Facebook.  It made an immediate impression.  Our objective – “Finding the Percent of Change”.

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Before talking about how to calculate a percent of change, I asked students to guess what they thought the percent increase was in the scenario.  Their guesses are shown below.

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We veered from the scenario and did some typical textbook examples… and once they got the hang of it, they were all trying to sneak back and solve the t-shirt problem.  One by one, they couldn’t contain their reactions as they realized we were talking about around a 900% increase.  By the time *I* was “ready to return to the t-shirt problem”, they were busting at the seams.  I love acting like I didn’t know what they had been doing… I love that they couldn’t resist returning to the t-shirt problem to test their guesses.  Why was it so irresistible?  Maybe because it truly was “real-world” to them.  Maybe it was the urge to find out just how great or how horrible their individual estimates were at the start.

All I know is, when it comes to “theories of engagement” it’s not an exact science… but when you reach that point in a lesson where the mathematics becomes irresistible, you’re in a good place.  🙂

Posted in Algebra 1, Pre-Algebra | Tagged , , , | 1 Comment

Sunshine Award

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I’m honored to have been nominated for a Sunshine Award by Drew Frank – it’s great to connect with Drew via Twitter, and you’ll see from his blog that getting connected and using social media for learning are themes.  Check out Drew’s blog here.

The Sunshine award is an opportunity to learn more about me as well as 11 other bloggers, recommended by yours truly.  Here are the rules from Drew’s latest post:

  1. Acknowledge the nominating blogger. (That’s Drew!)
  2. Share 11 random facts about yourself.
  3. Answer the 11 questions the nominating blogger has created for you.
  4. List 11 bloggers. They should be bloggers you believe deserve some recognition and a little blogging love!
  5. Post 11 questions for the bloggers you nominate to answer and let all the bloggers know they have been nominated. (You cannot nominate the blogger who nominated you.)

 

11 Random Facts About Me

1.  I won the third grade spelling bee.  The winning word was “ankle”.

2.  I play piano by ear, but I can barely read music.

3.  I will sing in front of anyone, anywhere.  Ask anyone who knows me.

4.  I have a Vera Bradley… habit…

5.  I am a Barry Manilow fan.

6.  I find brain research fascinating and am a huge fan of Judy Willis.

7.  I grew up north of Pittsburgh, Pennsylvania.

8.  I married my college sweetheart, who was also a math major.

9.  Royito’s Hot Sauce is my weakness.

10.  My favorite pizza on the planet is the “bianca” at Backspace.

11.  My son has severe food allergies, and I wish the general public understood more about this topic.

 

Answer the 11 questions the nominating blogger has created for you.

  1. What is your favorite movie of all time?  Airplane!
  2. If you could have attended any concert anytime in history, what would it have been?  Not sure – not a big concert-goer typically
  3. What do you do for fun?  Hobby?  Walks and bike rides in Austin’s many amazing parks
  4. What two guests would make the best comedic pair as co-hosts for the Oscars?  Will Ferrell and Steve Carell
  5. Cat, Dog or Goldfish? Why?  Goldfish – my son is allergic to the other two
  6. How do you caffeinate?  Lipton black tea with a splash of milk
  7. Favorite twitter chat? #msmathchat
  8. Best place you ever vacationed? Disney World junkie
  9. Best book you’ve read in 2013? Brain Rules by John Medina
  10. Favorite television shows?  Parenthood and Grey’s Anatomy
  11. What is one thing you never/rarely share that you are exceptionally proud of?  Being chosen as an Apple Distinguished Educator

My Sunshine Award Blogger Nominees

1.  rndesigns.com

2.  mrvaudrey.com

3.  tapintoteenminds.com

4.  techchef4u.com

5.  hookedoninnovation.com

6.  Teaching Like It’s 2999

7.  hacktheclassroom.ca

8.  restructuringalgebra.blogspot.com

9.  mathequalslove.blogspot.com

10.  futura.edublogs.org

11.  fawnnguyen.com

Finally, according to the Sunshine Award rules, it’s my turn to ask 11 questions of my nominees:

1.  What’s your favorite thing about blogging?

2.  Favorite hobby?

3.  Favorite movie of all time?

4.  Favorite place to travel?

5.  Favorite Twitter chat?

6.  Favorite educational website?

7.  Least favorite food?

8.  Favorite book you’ve read in 2013?

9.  Your unique talent?

10.  Proudest moment?

11.  Funniest thing you ever said in front of a group of students/educators?

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Seriously? A Storify

Posted in Algebra 1 | 2 Comments

Happy ThingLinksGiving

If you haven’t heard, ThingLink is offering FREE premium access to teachers who sign up through the end of December 2013!

Here is a link to join!

I haven’t begun to take advantage of the “premium” features yet, but I aspire to.  Creating custom icons, image analytics, and the ability to create class “groups” all look promising.

I’ve used ThingLink to create content for my students, but would LOVE to shift more in the direction of STUDENTS creating content.  Using ThingLink as a student space for creating collaborative, interactive study guides, or for students to “annotate” images of their mathematical work would absolutely rock.

Here’s my latest!  Thanks also to Desmos.com for being awesome.



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Posted in Algebra 1 | Tagged , , | 1 Comment