Author: mwiernicki

I am a husband and father of two quickly growing children. I have taught elementary, middle and high school math and a math course for preservice teachers at Mercer University. I am currently a math teacher on special assignment and a consultant in Henry County, GA. In my spare time, I study (and perform a little) magic, swim, and love to spend time with my wife and kids!

Revisiting 3-Act Tasks

My lessons never stay the same.  They’re always evolving.  Recently, I’ve taken a look at some 3-Act Tasks I created and I noticed:

  • Some of the tasks are lacking an act.
  • Others have resources that no students ask for (at least students that I’ve worked with).
  • The quality is low (shaky camera, point of changes, etc.)

So, I finally had a minute (read 2 days) and revisited each.  Below, you’ll see the tasks I’ve chosen to revisit.  An explanation of the original, what I changed, and why I changed it follows.  If you’d like to skip this and get to the revisited tasks, click here.

Revisited #1 – The Candy Bowl

My very first attempt at a 3-act task was the Candy Bowl task.  I was working in an elementary school at the time and Graham Fletcher had created problem to get 2nd and 3rd grade students reasoning about subtraction by removing the numbers from the problem context.  His context involved the lunchroom and numbers of students in three classes.  We talked on the phone about this for a while and though I liked the problem, I wasn’t crazy about the context.  I sat in my room trying to think of a context that would be a bit more engaging for students to think about.  And the Candy Bowl was created.

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It was a good problem, but it really lacked one of the most basic parts of a 3-Act Task… The third act.  The reveal was weak, because it relied on the teacher to give students validation.  The updated version, which had to be done from scratch (apparently whoppers candies are no where to be found anywhere near Valentine’s day), can be found here with all new updated resources for Act 2 and new video including two reveals, depending on which question students decide to tackle.

Revisited #2 – Sweet Tart Hearts

Another one of my early tasks was Sweet Tart Hearts.  I really liked this one from the beginning. There is a huge focus on estimation which allows for students to obtain solutions that are close, but not exact in most cases.  This also allows for the teacher to facilitate a discussion about why answers may not be exact for a variety of reasons.  But again, it really lacked that third act.  The task was good, but the closing of the lesson was weak due to the fact that the students were relying on the “all knowing” teacher to give them affirmation.

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Apparently Sweet Tart Hearts are a hot commodity a few days before Valentine’s day.  I went out the other day for a quick run to pick up a bag.  I had to go to 4 stores and finally found a bag (the last one).  I thought it would take about 10 minutes to do this revisit.  Surely the  numbers for the colors would be similar to the last time.  Not only was that not true, but Sweet Tarts changed the orange hearts to yellow!  But, the revisit is all done and I’m very pleased with the new reveal which allows the video to reveal the answer and the teacher to focus students on the reasonableness of their solutions.

Revisited #3 – The Penny Cube

My final revisit is the Penny Cube.  It is probably my favorite task.  I’ve certainly heard more from teachers about this task than any of the others.  I think I got the reveal right on this one.  The problem I found with this task was that I thought students would ask for things that I would want.  The first time I did this task with students, I guided them to the information I had ready for them.  They didn’t care anything about the dimensions of a penny.  They just wanted some pennies and a ruler.  It’s amazing what you learn when you listen to students, rather than try to tell them everything you think they need to know. So, to all of the students out there, Thank you for making your voices heard!

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So, this was the quickest fix.  I just updated the Penny Cube page (all of the coin specifications are still there – in case anyone wants them).

Note:  In this post I share how I changed my approach to teaching the Penny Cube task. 

So, it took a few days, but I’ve revisited some tasks that have been bugging me for a while and I hope it’s for the best.  I know I’ll probably give these another look in the future.  I’ll just need to start in early January to make sure I get the candy I need.

Happy Accidents

When I was growing up in (rural-ish) central New York, we had one TV.  We received 5 local stations through the antenna on the roof (abc, nbc, occasionally cbs if the wind was blowing just right, then Fox came along, and a pbs station).  This was a time when TV programming on the major networks actually ended at about 1:00 a.m. with a video of the American flag waving in the wind and the national anthem playing.  When that was over, there was nothing on TV but static.  This is something my kids can’t imagine. Not that they watch regular TV that often anyway (YouTube, Vimeo, etc.), but every time they turn it on, there are at least 100 shows to choose from on 4 TVs.

This wasn’t the case for my siblings and me.  Usually, the first person in the living room got dibs on what show was on or there had to be a “discussion” to figure out what everyone would watch.  Sometimes this ended in the TV being turned off by Mom or Dad with a “suggestion” that we go outside and get some fresh air.  Other times, we would decide to figure it out on our own and end up on the local PBS station watching a man with a huge perm (this was the 1980s) paint beautiful scenes in about 25 minutes.

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We (my 5 siblings and I) were all in awe while we watched Bob Ross paint wonderful paintings while talking to us (the viewers) about everything from his pet squirrels to painting techniques.  And at the end of every episode I felt like I could paint just like Bob Ross!  I never tried, but I felt like I could!

Recently, my kids have discovered the talent and wonder of Bob Ross through YouTube and Netflix. They love his words of wisdom:

  • “Just go out and talk to a tree.  Make friends with it.”
  • “There’s nothing wrong with having a tree as a friend.”
  • “How do you make a round circle with a square knife?  That’s your challenge for the day.”
  • “Any time ya learn, ya gain.”
  • “You can do anything you want to do. This is your world.”

And I love that they love these words of wisdom.  You can find more here.

For Christmas this year, my son and I received Bob Ross T-shirts.  Connor’s has just an image, while mine has a quote as well:

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Bob Ross was referring to painting when he said these words; “In painting there are no mistakes, just happy accidents.”  In other words, when you paint your mountain the wrong shape, treat it as a happy accident.  It can still be a mountain, there may just end up being a happy tree or a happy cloud that takes care of your happy accident.

I think it works for math class, too.  Recently, I modeled a Desmos lesson for a 7th grade teacher.  The students had been working with expressions and equations but were struggling with the abstract ideas associated with expressions and equations.  The teacher and I planned for me to model Desmos using Central Park to see how students reacted to the platform (this was their first time using Desmos) and how I managed the class using the teacher dashboard.

During the lesson, there was a lot of productive struggle.  Students were working in pairs and making mistakes happy accidents.  They were happy accidents!  Because students kept going back for more.  At times there was some frustration involved and I stepped in to ask questions like:

  • What are you trying to figure out?
  • Where did the numbers you used in your expression come from?
  • What do each of the numbers you used represent?
  • Before you click the “try it” button, how confident are you that the cars will all park?

The last question was incredibly informative.  Many students who answered this question were not confident at all that their cars would all park, but as they moved through the lesson, their confidence grew.

One of the best take-aways the teacher mentioned during our post-conference was  when she mentioned a certain boy and girl who she paired together so the (high performing) girl could help the (low performing) boy.  The exact opposite happened.  The girl was trying to crunch numbers on screen 5 with little success.  The boy just needed a nudge to think about the image and to go back to some previous screens to settle some ideas in his mind before moving ahead with his idea that the answer is 8.  Then, he got to expain how he knew it was 8 with the picture, conceptually, to his partner.  The teacher’s mistake happy accident was in believing her students would always perform a certain way.  When students are engaged in tasks that are meaningful, they tend to perform differently than when they’re given a worksheet with 30 meaningless problems on it (the norm for this class before Desmos).  Ah-has all around and the “low student” shows that he knows more than the teacher thinks.

The icing on the cake?  Several students walking out of the classroom could be heard saying, “That was cool.” or “That was fun.”

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Let’s treat math mistakes as happy accidents, something to learn from and problem solve our way through.  When students (all humans) make a mistake, synapses fire.  The brain grows (More on this from Jo Boaler here).  What we do as teachers from this point, determines how much more the brain will grow.  If we treat student mistakes as happy accidents, perhaps their brains will grow a bit more than if we continue to treat mistakes in the traditional manner.

Let’s hear it for Bob Ross.  He probably never thought his words of wisdom about painting would be translated to the math classroom.

Now, go make friends with a math problem.

 

 

 

Georgia Math Conference 2016

For the second year in a row, I had the privilege and honor to give an ignite talk at the Georgia Math Conference (Last year’s talk can be found here.)  What makes ignite talk sessions great is that you get a taste of what several speakers are passionate about and you get to walk away with at least one ember of at least one of those talks beginning to burn in you!

Special thanks to Graham Fletcher for putting this all together (in pre and post production!).  Graham is top notch, “for sure” (Must be a little of my inner Canadian there).

The featured speakers this year in the order of their talk:

Me (@mikewiernicki) – I didn’t ask to go first. 🙂


<p><a href=”https://vimeo.com/190360814″>Mike Wiernicki – Ignite Talk (GCTM 2016)</a> from <a href=”https://vimeo.com/user21534889″>Graham Fletcher</a> on <a href=”https://vimeo.com”>Vimeo</a&gt;.</p>

Katie Breedlove (@KatieBreedlove)


<p><a href=”https://vimeo.com/190362489″>Katie Breedlove – Ignite Talk (GCTM 2016)</a> from <a href=”https://vimeo.com/user21534889″>Graham Fletcher</a> on <a href=”https://vimeo.com”>Vimeo</a&gt;.</p>

Jenise Sexton (@MrsJeniseSexton)


<p><a href=”https://vimeo.com/190364708″>Jenise Sexton – Ignite Talk (GCTM 2016)</a> from <a href=”https://vimeo.com/user21534889″>Graham Fletcher</a> on <a href=”https://vimeo.com”>Vimeo</a&gt;.</p>

Karla Cwetna (@KCwetna)


<p><a href=”https://vimeo.com/190381786″>Karla Cwetna – Ignite Talk (GCTM 2016)</a> from <a href=”https://vimeo.com/user21534889″>Graham Fletcher</a> on <a href=”https://vimeo.com”>Vimeo</a&gt;.</p>

Carla Bidwell (@carla_bidwell)


<p><a href=”https://vimeo.com/190286621″>Carla Bidwell – Ignite Talk (GCTM 2016)</a> from <a href=”https://vimeo.com/user21534889″>Graham Fletcher</a> on <a href=”https://vimeo.com”>Vimeo</a&gt;.</p>

Brian Lack (@DrBrianLack)


<p><a href=”https://vimeo.com/190415942″>Brian Lack – Ignite Talk (GCTM 2016)</a> from <a href=”https://vimeo.com/user21534889″>Graham Fletcher</a> on <a href=”https://vimeo.com”>Vimeo</a&gt;.</p>

Graham Fletcher (@gfletchy) – The great Emcee’s talk is available elsewhere.  I’ll find it and link it asap.

 

 

Student Ideas Matter

Engaging students in math has always been a goal for me.  No… more than a goal…. a passion! And it’s not always easy to do.  For example, I used to hate teaching students how to find the sum of an arithmetic series.  I didn’t hate it because it was difficult to teach or because students had an overwhelming difficulty learning it.  I hated it because I was the only one that saw the beauty in it.  I was the only one who was passionate about it.

This lesson was “fun” (I use the quotes to denote that this was a fun lesson for me – not so much for my students).  But this all changed when I allowed my students the opportunity to think for themselves.

The task was very simple in concept:  Find the sum of the series of the numbers 1-20.

Before going any further, it may be useful to know about the

  • Class norms:
    • Estimate first,
    • the answer is never enough,
    • reasoning, explaining and looking for patterns are all expectations,
    • if you found one way, look again, you may find a more efficient way,
    • get out of your own head and talk about the math with your partner/group while you work

Several started adding 1 + 2 + 3 + 4 + . . .+ 19 + 20.  I noticed this and asked those groups for one word to describe their strategy.  Sample responses:  boring, lame, tedious (actually proud of that one), calculator worthy…

My reply to each of their descriptions:  If your strategy is [insert one: boring, lame, tedious, or just plain calculator worthy] why do you feel the need to use it?

Sometimes students get stuck in their own thinking and just need to be made aware of it. To help nudge students to think in other ways, I had bowls of tiles with the numbers 1-20 written on them available for groups to use.

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It took several minutes before students began to grab tiles and began to notice things like:

  • “Hey, Mr. W., we can make a bunch of 20s.”

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  • “We got a bunch of 21s. 10 of them.  It can’t be that easy, right?”

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  • “We made 10s and 30s.  How did you make 21s?”

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  • “We did the 20s too.  That’s the easiest way for us.”

It was a bit chaotic, and I didn’t know it then, but there was a passion building.  This wasn’t just engaging, these students were ALL IN.  They were more than engaged and wanted to learn more about the strategies they came up with.  They wanted to share. Needed to know.  And the answer was almost irrelevant. The connections between all of their strategies became the focus.

From here, getting to the algebra made sense.  How would you find the sum of the numbers 1-50?  1-90?  1-100?  What about 5-50?  Some saw their ideas with the tiles transfer easily to an algebraic expression and equation.  Others not so much. So, more time to talk and share.  More time to find a strategy that is more convenient to generalize for a series of numbers of any range.  The success of the students’ mathematical ideas gave them power to reach further – to take another chance.

Teaching the lesson this way was a definite improvement on the original. In this version, the students’ ideas matter, so students matter.  In this version, students think for themselves and collaborate with others, and in turn get validation of their thinking, so students matter.  In this version, students built some passion.  They fed off of each other. And the content mattered because of the students’ interaction with it.

Is this lesson the best it can be? I’m not sure.  So, I’ll continue to try to improve on it.

Thoughts and comments welcome.

 

 

 

 

Blogarithm Posts

Last year I had the honor of being asked to write four posts for NCTM’s Math Teaching in the Middle School Blog: Blogarithm (one of the coolest math blog names out there).  They were posted every two weeks from November through the end of December (which just shows that I can post more frequently if someone is reminding me every other week that my next post is due (thanks Clayton).

Pythagorean Decanomial

The four posts are a reflection of a lesson I taught with a 6th grade teacher, in September of last year, who was worried (and rightfully so) that her students didn’t know their multiplication facts.  After a long conference, we decided to teach a lesson together.  I modeled some pedagogical ideas and she supported students by asking questions (certain restrictions may have applied).

Links to the four posts are below.

  1. Building Multiplication Fluency in Middle School
  2. Building Multiplication Fluency in Middle School Part 2
  3. Building Multiplication Fluency in Middle School Part 3
  4. Building Multiplication Fluency in Middle School Part 4

While you’re at the Blogarithm site check out some other guest bloggers’ posts.  Cathy Yenca has some great posts on Formative Feedback, Vertical Value Part 1 and Part 2, and 3-Act Tasks

 

About Strategy Development (and Algorithms)

So there’s this thing going around about algorithms being a bad.

They’re not.  What’s bad is when students learn an algorithm – any algorithm from anyone – without making sense of it on their own.

Enter (what is considered by some) the buzz  word: “Strategy” (Guess what, the strategies being taught now are all algorithms).

I often hear teachers talking about teaching students several different strategies for (insert operation here).  Good, right?

Not so much.  Here’s the thing.  If teachers teach all of these different strategies, without student understanding at the forefront, they may as well teach the standard algorithm.  The worst part here is that students can actually be worse off being taught these multiple strategies without understanding than one algorithm without understanding.

arrrgh

Essentially, students are being force-fed strategies (aka algorithms) that they don’t understand and they feel like they need to memorize all of these steps for all of these strategies.  We’re going down the wrong path here.  Our destination was right, but we took a wrong turn somewhere.

It’s time to stop the madness!

How?  you ask.

Let me tell you a story…

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Back in early fall 2007, when I was still a toddler of a math coach, my beautiful wife’s grandmother passed away and the whole family went to her school on the weekend to help her get some lessons together for the few days she would be out.  Truthfully, I was the only one helping since the kids were 7 and 4 at the time.  Kim gave me jobs to do and I did them with precision and efficiency.  One of the tasks she gave me was to make a 18 copies of a few tasks for her students to complete during her absence.

To help her out, I took my son, Connor (the second grader), with me to the copy room so she’d only have 1 child to keep track of while she was trying to work.  When we got to the teacher work room, Connor watched as I placed the small stack of papers on the copy machine tray, typed in the number of copies (18) needed and then hit the copy button.  Within seconds he asked me (in the most exasperated voice he could muster) “How many copies is that going to make?”

I swear, when things like this happen, mathematicians in heaven play harmonious chords on harps using ratios.  I hear them and respond accordingly.  This time, I brought Connor over to the copy machine screen and showed him the numbers. 

Me: “Do you see that 5 right there?  That’s how many papers, the copy machine counted, and that 18 right there?  That’s how many copies of each piece of paper I asked the copy machine to make.”

Connor:  “Oh…”

Commercial break:  I didn’t really expect much more than an estimate.  This was September and Connor was a second grader.  He may have heard the word multiplication, but likely didn’t know what it meant.  

And we’re back!  His eyes looked up as he thought about this briefly and within seconds of his utterance of “Oh,” he said in a thinking kind of voice, “50…..”

Now, I’m not one to interrupt a student’s thought process – I work with teachers to keep them from doing it.  I actually remember having a mental argument with myself about whether I should ask him a question.  I was so excited in this moment, I couldn’t help myself.  I asked (with as much calm as I could), “Where did you get 50?”

I kid you not, he replied by pulling me over to the screen on the copier and said, “You see that 1 right there (in the 18), that’s a ten. And 5 tens is 50.”

I could hardly contain myself.  Naturally, since I had already interrupted him, I asked what he was going to do next.  I was floored when he said that he didn’t know how to do five eights.  I was floored because he knew how to multiply a 2 digit number, he just lacked the tools to do so.  In the context of this copy machine excursion, Connor made sense of the problem, reasoned quantitatively, showed a good degree of precision, and I’m sure if he had some tools, he would’ve come to a solution within minutes.

As we left the teacher work room, with copies in hand, I asked him to think about it for a bit and see what he could come up with.  When we got back to my wife’s room, I told her all about it.  When I got to the part where he didn’t know how to do five eights, I called across the room to him and asked him if he figured out what five eights was.  As he said, “No.” he paused and thought for a few seconds and said, “Can I do 8 fives?  ‘Cause that’s 40.”  Before I could ask him (thank God), “What about the other 50?”  He said, “40…50…90!”

This second grade boy (My Son!) who had never been taught multiplication, what it means, or any algorithm for it, created a strategy for finding a solution to a contextual problem that most of us would solve using multiplication.  He came up with the strategy.  It was based on his understanding of number and place value and he created it.  These are the strategies students need to use — the ones they develop.

I’ve told this story at least 50 times (I’ve even told it to myself while on the road).  Afterward, I often challenge teachers to take their students to the copy machine and watch this play out for themselves.  Some pushback does come out occasionally with comments like these (my responses follow each):

  • That’s because he’s probably gifted.  He is, but that’s not a reason to not do this with any group of kids.  Every student can and will do this when presented with contextual problems and access to familiar tools and where teaching through problem solving is the norm.
  • You probably worked with him on multiplication tables.  Yes, and no.  When Kim was pregnant with Connor and on the sonogram table with a full bladder, I leaned close to her stomach and started reciting multiplication facts to make her laugh (I’m cruel for a laugh sometimes) Other than the 4 or 5 facts I quickly rattled off that afternoon, I’ve never recited them since.  I doubt that did much, if anything, for his math achievement.
  • You must work with him a lot with math.  Not really!  Other than natural math wonders that have piqued my kids’ interests and sparked some discussion, no.  Questions they’ve had, like – “Dad, how many tickets do you think I have in this Dave & Busters cup?” are all we’ve spent any amount of quality time on.  That and puzzles.

So, when it comes to strategy building, it all has to begin at the student level of understanding.  The best way to do that is to let students develop their own strategies, share them with each other, and build more powerful understanding from there.  Then, if they do get “taught” a standard algorithm somewhere down the road, it has a better chance of making sense.

 

 

 

 

Personalized Learning Can’t Trump Content & Pedagogy

The problem I’m seeing with personalized learning (overall and especially as it pertains to math instruction) is the common understandings about what it is, what it can look like, what it shouldn’t look like, and how it works as related to our own learning experiences are fragile at best.

Many school systems, including my own, are looking at personalized learning as a means to improve math instruction, raise math test scores, and increase student engagement. These goals are great and many systems have them in some form or another. However, when personalized learning forces teachers into using sweeping generalized practices that often trump solid content pedagogy, something is drastically wrong.

I don’t think this is necessarily the fault of personalized learning as a concept,  but I do think it is problematic when common understandings become compromised.  These compromised understandings lead to sweeping generalized practices like:

  1. No whole group instruction – ever
  2. Students should be on a self-paced computer program for personalized learning
  3. Teachers have to create new groups of students every day/week to make sure learning is personalized
  4. Teachers should do project based learning several times per unit to engage learners
  5. Teachers need to use choice boards for every standard they teach.

This is not a definitive list – just what I’ve heard from within my own district over the last few years.

I may not have a response to each of these, but I can point out a few sources in addition to my thoughts:

  1.  No whole group instruction – ever – Dan Meyer’s post: http://blog.mrmeyer.com/2014/dont-personalize-learning/  my favorite idea from this is from Mike Caufield: “if there is one thing that almost all disciplines benefit from, it’s structured discussion. It gets us out of our own head, pushes us to understand ideas better. It teaches us to talk like geologists, or mathematicians, or philosophers; over time that leads to us *thinking* like geologists, mathematicians, and philosophers. Structured discussion is how we externalize thought so that we can tinker with it, refactor it, and re-absorb it better than it was before.”

2.  Students should be on a self-paced computer program for personalized learning Personalized learning is not something you get get from the App Store or Google Play  or from any ed tech vendor.

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Some other comments from Dan Meyer:  Personalized Learning Software: Fun Like Choosing Your Own Ad Experience  and from Benjamin Riley:  “Effective instruction requires understanding the varying cognitive abilities of students and finding ways to impart knowledge in light of that variation. If you want to call that “personalization,” fine, but we might just also call it “good teaching.” And good teaching can be done in classroom with students sitting in desks in rows, holding pencil and paper, or it can also be done in a classroom with students sitting in beanbags holding iPads and Chromebooks. Whatever the learning environment, the teacher should be responsible for the core delivery of instruction.”

3.  Teachers have to create new groups of students every day/week to make sure learning is personalized – I’m not sure this is the case.  If teachers really know where their students are in their mathematical progressions (lots of ways to do this – portfolios, math journals, student interviews (GloSS and IKAN from New Zealand, etc.)  These types of data are much more effective that computer testing programs because teachers are able to see and hear students’ thinking as well as their answers.  In my opinion, you can’t get more personalized than that!

4.  Teachers should do project based learning several times per unit to engage learners – anyone who has had PBL training knows that 1 per year is a good start!  PBL takes time – to plan, and plan some more (most often with other content areas).  If anyone expects more than one per year or semester initially, it’s time to have some Crucial Conversations!

5.  Teachers need to use choice boards for every standard they teach – student voice and choice does not have to be a choice board.  And really, how much of a choice do students have if we’re giving them all possible choices with no input from them?

To sum up: In order to really improve those goals of improving math instruction, increasing student engagement, and raising math test scores one thing is certain – an investment to increase teacher content and pedagogy knowledge must be at the forefront.  There is no other initiative or math program that will help districts reach these goals more effectively than this!