Hi, my name is Mike… and I love using Desmos with students.

This is not a bad thing at all.  I’m not giving up time with my family to spend on Desmos. It’s just that whenever I think I’ve exhausted all of the ways to use this fantastic tool with students, the Desmos team adds a new activity or game that I can and want to use right away!  These people know how to keep us wanting more!

Here you can find out what Desmos is all about!

Now, for all of you teachers out there that haven’t engaged your students in this amazing math tool, let me move from a user to a pusher.  4 reasons why you should use this amazing tool with your students:

1. It’s completely free!  (not just this first time – all the time)
2. It’s a graphing calculator that works beautifully online or as an app for students to Model with Mathematics – SMP 4.

This is a screenshot of how my son, Connor, used the Desmos Calculator to make sense of transforming quadratic functions.

3.  When you sign up as a teacher (again, for free) you can assign activities and games (yep, they’re all free to use, too) to your students and you can check their progress from your teacher page.

So, beyond the graphing calculator – which is amazing on its own – as a teacher you can assign an activity to your students based on the content they are investigating. Try Central Park  – it’s my favorite activity.  (If you like, you can go to the student page and type in the code qqbm.  I set this up for anyone reading this post. Feel free to use an alias if you like).

And as far as games go, check out Polygraphs.  It’s like the Guess Who? game for math class. Trust me, your students will love it and there are polygraphs for elementary as well as secondary. The polygraphs are all partner games, so students will need to work in pairs.  I’ve even made a few:

Polygraph: Teen Numbers

Polygraph: Inequalities on a Number Line

Polygraph: Geometric Transformations

4.  As you get sucked in to this tool, you may begin to think to yourself, “Boy, I really wish there was an activity for ______.  If only knew how to create an activity for my students to use on Desmos.” That’s taken care of, too, with Activity Builder and Custom Polygraph (and, yep, you guessed it – they’re free to use, too)

And before you begin to doubt whether you can create an online activity or polygraph, the Desmos team has already taken steps to make this extremely teacher friendly.  Before you know it, you’ll have your own Desmos activity published!

Finally, as a great end of year gift, Dan Meyer blogged about the latest from Desmos – Marbleslides.  If this doesn’t get you to use Desmos with your students. . . well, I’m sure they will think of something else, soon. But seriously, try this out.  I have re-learned and deepened my own understandings of mathematics by trying and reflecting on many of these activities and games, and then having my own kids do them (and then they ask me why their teachers aren’t using them – “Can you talk to them, Dad?”).  The conversations will be happening this semester for sure!

But the best part about all of this is that students get to use the calculator to investigate graphs and compare graphs and equations/functions.  They get to notice and wonder about what matters and what changes a graph’s slope, and y-intercept for linear functions and what changes the vertex and roots of parabolas.  They get to investigate periodics and exponentials and rationals and so much more.  They get to engage in activities and games that have components that ask them to reflect on what they’ve learned in the games and activities themselves.  The students are doing the mathematics.

Then, in class, we get engage students in talking about the math they’ve investigated!  How sweet is that?

You see, as great as Desmos is, it can’t take the place of great teaching.  It’s a tool that can help us become better at our craft and help our students gain a deeper understanding of mathematics!  Sounds like a win-win!

So, I guess I don’t have a Desmos math addiction.  Addictions have adverse consequences and I see none of that here!  I just have – as we all do thanks to Desmos – access to a powerful mathematical learning tool!  Thank you Desmos.  I can’t wait to see what’s coming next!

# Filling Gaps: Buy a Program or Help Teachers Grow?

This post actually started as a rant as I was sitting through meeting after meeting with really nice people trying to sell products to “Fill the Gaps.”  So, if it has a rant-y feeling, just know where I’m coming from.  If no one really likes this, that’s ok.  At least it’s out of my system for now.  You see, when you’re “invited” to attend meetings to raise student achievement, you really need to show up, or who knows what will  happen.  So, in the effort to stand up for teachers and students, I attended all of them.

These were really nice people presenting to us, and they were very passionate about their products.  I even largely agree with several of them on their basic philosophy.

At least one of the people listening with us in the room was sold on many of the ideas before we even started these meetings.  Every slide or picture shown was met with a “That’s good!” or a “That’s really good!”  I think if they showed us a shiny, new penny, this person would have said, “This is what our students need!” with the same reaction!  The pictures of bulletin boards showing concept maps and vocabulary word walls and even students working may be good – or may not.   Really, there’s no way to tell – especially with the picture of the students working.  What were the students saying?  Were they discussing mathematics?  Were they using the vocabulary on the bulletin board?  Were they making connections to the concept maps?  Did they give and receive feedback about their work?  Let’s see some video, so I can see how this is really working.

Again, philosophically I agree with their framework of instruction.   However, the product is not really necessary if the PL these companies are willing to provide is effective.

Now, on to the PL.  Lots of good strategies offered here.  And more pictures of students “engaged.”   My question:  what are the students engaged in?”  Are they engaged in the mathematics or the product?  My initial response to this self-posed question was:  Does it really matter?  The students are working.  After  thinking about this for just a few seconds, though, I can say without a doubt that it does matter!

Engaging students can be tricky.   A passerby, seeing students working silently in their seats, might conclude student engagement in a task.  A passerby, seeing and hearing students discussing a task, may conclude non-engagement in a task as well as lack of classroom management.  Really it’s hard to tell, in either case, whether there was any engagement or what kind of engagement there was.

So, what does engagement mean?  It depends on what you want.  One of my goals year after year is to engage students in the mathematics they’re studying.  When I first started teaching, I wanted students to just be engaged, no matter what.  As I think back, they were engaged – probably in my educational “performance.”  I was the “fun” teacher that did crazy math lessons.  As I grew professionally, my lesson focus evolved to take the students’ engagement away from me and toward the mathematical content.  So, why is it so important?  If students are engaged in creating the product (creating a poster, making a presentation, etc.)  they may be learning mathematics, but how do we know.  I’ve seen students engaged in creating beautiful products and walk away with little mathematical understanding.  I’ve also seen students engaged in mathematics and creating not so beautiful products, but beautiful understandings and mathematical connections.

So, for all of the professionals in the room thinking this (or any of the other presentations we’ve seen) is the silver bullet. . . It’s not.  The only silver bullet out there that’s going to raise student achievement is teacher PL grounded in  understanding mathematics conceptually and building teachers’ pedagogical understandings and strategies.  If we want high achieving students, we have to help teachers achieve their greatest potential.  No program out there will do that, but if you really want to become a better math teacher, Twitter and the #MTBoS are a great place to start!

# Change vs. What Worked in the Past

So, I’m at this Standards Setting meeting in Atlanta this week.  I’m working with people I’ve never met before.  As we settle into our assigned seats, we begin the small talk:

I introduce myself (since I seem to be one of the last ones to arrive).

Others at the table introduce themselves as well and before long we’ve found some common ground (many of us are in a coaching role) and start building a professional relationship.  I love this part of attending professional learning sessions at a state (and national) level.  “All of us are smarter than one of us.”  By day two of our work, you would think we worked at the same school.  Our conversations, while still mostly professional, are much more relaxed.

During one of our breaks, we begin talking about some of the teachers we work with who are “stuck in their ways.”  The question bouncing around (at least in my head) is “Why?”  Why are they so stuck?  As we talked at our table, the “reason” that seemed to dominate the conversation was one that many of us have heard before:

The teachers “reason” is that teaching this way has worked for the past “y” years so why should I change now?

Our conversation then takes an interesting turn.  A “what if” turn.

What if Apple thought the same way.  How would our world be different?  Would we still have iPads, iPhones, Apple TV, etc.?  It’s unlikely.  We’d probably have something that looks like this:

because what worked in the past should be good enough now, right?

What if Ford Motor Company thought the same way.  How would our driving experience be the same?  I doubt we’d have radios, or even seat belts.   Our new ride may look like this:

because what worked in the past should be good enough.

What if we wanted cataract surgery?  How would that look, if the surgeons of today had the same attitude about what works best?  Did you know that cataract surgery goes back to ancient Egypt?  Would you rather have Lasik or have a surgeon come at you with one of these?

If the only thing keeping you from changing is because it’s the way you’ve always done it, then it can’t be the best.  We’ve been growing and changing the way we do things because we are always searching for the most efficient way, or the more cost-effective way, or the safer way, or the way that will improve our lives.  Have we done that for students.  Is the way you’re teaching mathematics what’s best for your students?  Is your pedagogy guided by what’s proven through research or just what you’ve done for years?

Our conversation ended abruptly because we had to get back to work on standards, but as I met and reconnected with others at the workshop, this same conversation came up multiple times.  My thoughts on this are below, but I hope others chime in here with their own thoughts on this.

Steven Leinwand wrote something several years ago that I think relates well to this.  What he wrote was (and I’m totally butchering this, I’m sure) that we shouldn’t expect more than 10% growth/change per teacher per year.  On the flip side of that, he also said that teachers should strive for more than 10% growth/change per year.

This is something I’ve really tried to work on in my coaching role with teachers.  When learning something that seems daunting to a new or veteran teacher (moving toward a standards-based, student-centered approach to teaching mathematics for example), I suggest teachers choose one thing, one piece of what we’ve discussed that they think they can become really good at over several months, rather than trying to make everything fit at once.

Letting teachers know they are not expected to become experts all at once is great, but following through is even more important.  Without constructive feedback, teachers will likely fall back to their comfortable habits.  Just like teachers need to really listen to students, coaches need to listen to teachers.  We need to model what we expect.

If we don’t, we may end up with this 20 years from now:

# The Best Part about Blogging

This is super exciting!  I love it when teachers keep thinking – especially when I stop!  What you’re about to read is truly the best part of blogging!

Readers of Under the Dome have been terrific commenters and questioners of my posts over the last 2 years and you all just keep getting better.  Recently, Sharon Wagner, a teacher I met during a three-day summer institute in June visited my blog and reached out to share her ideas about the Olympic Cola Display 3-act task.

Sharon’s words:

Through the course of a few emails over the summer and a lot of my time spent doing things outside of the MTBoS (my lovely wife got some of her honey-do’s completed and I got some of my Mike-do’s finished) I have Sharon’s extension and am now posting it with her blessing!  Please take a look.  Her idea is a natural extension and allows students to design their own display using the colors of Coca-Cola twelve packs (which she most helpfully added to her document).  Any Pepsi fans out there?

Sharon’s idea also ups the rigor by providing an audience (the merchant).  This, again, is a part of that natural extension (of course someone designs these displays for the merchants).  As for the Standards for Mathematical Practice . . . let’s just say your students will be engaging in multiple SMPs.

Again, this is super exciting.  I love to share my ideas here, but when someone else takes it and makes it better – in this case by adding to it – everyone wins.  Especially the students in our classrooms.

Thank you Sharon.

Sharon’s Display Extension:

coca cola display project extension

# A Further Discussion of “Funny Math”

Georgia’s new state school superintendent, Richard Woods, recently wrote a column about teaching mathematics. “Funny math methods” was the catch-phrase taken from the article and sent out through the media.  This was not unexpected.  Frankly, I’m surprised it took this long.  This was part of his campaign platform.

Though his column has prompted some emotional responses from math educators, it is imperative that this significant dialogue he has opened, continue.  The best thing we can do for the students of Georgia is to keep this discussion going in order to come to a common understanding about the mathematical terms, strategies and ideas presented by Mr. Woods in his column.  We can truly help the students of Georgia by making sure we are all speaking the same language.

Since I am unable to respond directly to Mr. Woods’ column, I would like to continue the dialogue here.  I welcome any and all comments that keep this discussion moving forward in a positive light.  I encourage all viewpoints, since one-sided dialogues don’t tend to be very productive.

Mr. Woods talks about hearing from parents unable to help their children with their math homework.  I, too, have heard this from parents.  My response to this is:  If students are not able to do their homework independently, perhaps it should not have been assigned.  This is difficult for many to hear.  If you think about it, though, it really makes sense.  If we want students to build their understanding of mathematics based on what they have learned, we have to make sure they have learned it before they can build on it.  That said, I look at homework as falling into one of three categories:

1. Practice – students use understandings learned in class to practice and build a more solid understanding at home.
2. Preview – students are given a few problems to get them thinking about a new concept that is related to what they already know.
3. Extension – students take a problem or task they worked on in class and are asked to extend their understanding. For example, in middle school, students may discover a growth pattern and as an extension, they may be asked to create a growth pattern that grows twice as fast.

Notice that each of these types requires students to have an understanding before they begin.  Understanding in mathematics, as in reading, is crucial for student success.

Mr. Woods also mentions the need for students to have a firm understanding of the fundamentals of mathematics.  He goes on to say that basic algorithms, fact fluency, and standard processes for addition, subtraction, multiplication, and division contribute to building that strong foundation for student achievement.

This is interesting.  First, algorithms have gotten a bad rap.  But, there is a place for algorithms in the big picture of how students learn mathematics.  An algorithm is just a mathematical term for a series of steps that can be followed to determine a solution to a mathematical computation.  Problems occur when algorithms are taught just as a series of steps to memorize, rather than facilitating an understanding of the computation(s) first.  Without understanding, the steps often don’t make sense and one or more of three things may happen next:

• Students may complete algorithmic steps out of order.
• Students may skip one or more steps of the attempted algorithm.
• Students may confuse the steps of one algorithm with another.

These may seem like easy fixes -“just tell the students again”.  Telling them where their errors are and having them practice more problems does not work.  Without a conceptual understanding of what the computation means, students will continue to make these errors.  Though students may be able to show some success in the short term, over the long term they will revert back to one or a combination of the error patterns above.

Completing algorithms incorrectly doesn’t even compare to one of the worst side effects of this procedural teaching: students who don’t realize their answers are unreasonable.  For example, a teacher recently sent me the email below:

This student has some major misconceptions.  With a conceptual understanding, this student could have reasoned that 1/3 of a pound is less than a whole pound, so the answer should be less than \$5.25.  Without conceptual understanding, the student is attempting to recall and use procedures they do not understand, is confusing procedures, and is unable to determine whether or not the solution they have found is reasonable.  This is only one piece of numeracy that is lost in the procedural mathematics instruction that Mr. Woods seeks.

Fact fluency and the standard procedures for the four basic operations is next.  I don’t think there is a math teacher anywhere in the world that doesn’t think fluency is important.  In order to be clear though, memorization and fluency are not the same thing.  Not even close.  To keep this short and sweet, with the focus on students, I have copied the excerpt below from the GA DOE frameworks for mathematics.  I think this sums everything up nicely (no pun intended).  However, if you would like to learn more about fluency, click the links below.

Fluent students:

• flexibly use a combination of deep understanding, number sense, and memorization.
• are fluent in the necessary baseline functions in mathematics so that they are able to spend their thinking and processing time unpacking problems and making meaning from them.
• are able to articulate their reasoning.
• find solutions through a number of different paths.

The fundamentals that Mr. Woods should have mentioned are actually reasoning and sense making.  This is what it takes to learn and do mathematics.  So, again, understanding must take place. However, understanding cannot take place through the memorization of algorithmic steps alone.  This is not just what I think.  It’s what I know from years of teaching students mathematics. This is also backed by research, papers, & videos.  The building of understanding is also fostered through a passionate, grassroots movement of mathematics teachers #MTBoS (Math Twitter Blog-o-Sphere).  This is our place to collaborate, share, and work to improve our teaching of mathematics.

Next, Mr. Woods discusses teaching “funny math methods.”  He specifically mentions “the lattice method” and he correctly states that this method is not state mandated and not required for students to achieve on state tests.  Mr. Woods is absolutely right!  This is not state mandated because it is a ridiculous strategy for multiplying multi-digit whole numbers.  to be fair, it works – every time. If you follow the steps of this algorithm by making the grid correctly, and placing the digits of the numbers correctly in the grid, and placing the products of the digits in the right places, and drawing the diagonals correctly, and adding the digits along the diagonals correctly, and copying the product correctly from the grid written in standard form.  You think that’s ridiculous?  Here’s something even worse – it’s often used for those students who have trouble remembering the steps of the standard algorithm.  This method is the definition of “funny math” and math teachers should not use this since it is does not make sense to students (or teachers) and it does not align with any of the standards for multiplication.

Mr. Woods says that mathematics has become over complicated.  It hasn’t.  It is only as complicated as it has been for centuries and that complication is exacerbated through teaching without sense making.  We can teach students to think mathematically on their own.  We can support and help them grow through their own understandings of mathematics.  We can help students make sense of mathematics and learn to use this to make informed decisions, rather than listening to others make these decisions for them.  We can do this because what we know now is how students learn mathematics.  It is not through memorization. It is through sense making and reasoning.  What we know now is that teaching students to think mathematically, through problem solving by building conceptual understanding provides students experience and allows them to make connections to algorithms they create and those created by others.  What we know now is that this works best for all students.  Not just average students, or above average students, or below average students.  All students.

Finally, just to be clear:

• We (mathematics teachers) are most likely more current than most on research-based, best practices in the mathematics classroom.
• There is a place for the algorithms you wish to see in the classroom, and they are found in the appropriate grade level standards. However, using an algorithm is not the end-all, be-all for learning mathematics.  There is always a need for students to be flexible, efficient, and accurate in their computations.  Multiple strategies, based on student understanding, must be explored.

At the end of the column, Mr. Woods states that “Offering choices and clarification are some of the steps we are taking to address the concerns surrounding mathematics in our state.” I applaud these steps.  Choices are always a good idea.  Clarification is even better!  Let’s work together to clarify the misconceptions about best practices in mathematics instruction.  Armed with these common understandings, Georgia can lead the charge as a state united to raise student achievement in mathematics.

I look forward to all comments and continuing this dialogue to help build these common understandings.

# Unlikely Students in Unknown Places

I recently got back from Santa Fe.  I was attending a conference there for a few days last week and afterward, I drove to El Paso to visit my brother’s family (he’s currently stationed in the middle east so I didn’t get to see him – unless you count face time) for a day before flying home.

Let me preface this story by saying that we all probably have a story similar this, but how we handle it can be a possible game changer.

Somewhere on my long drive, I stopped at a fast food restaurant to grab a quick bite.  So, I went inside and got in line.  The following outlines the beginning of our interaction as I stepped to the counter:

Cashier:  May I take your order?

Me:  Yes, please.  I’d like a number 2.

Cashier:  Large or medium.

Cashier:  (after pushing more buttons than is conceivably necessary to enter my choice of “medium”): Your total will be \$6.05.

I dug through my wallet (receipts from the trip and everything) and found that all I had was a \$10 bill, so I handed it to her.  She entered \$10.00 correctly and the correct change of \$3.95 showed up on the little screen.  At just about that point, I remembered that I had a bunch of change in my pocket and said quite enthusiastically, “Oh, wait, I think I have a nickel.”  Who wants to carry around \$0.95 in change in their pocket.

The cashier didn’t miss a beat, and said, “So, your change will be \$4.00 even.”  I kind of smiled as I continued to look through my change, proud that she had a mental strategy to adjust to the situation and that she seemed quite confident and comfortable using it in this situation.

Unfortunately, I didn’t have a nickel, but I still didn’t want change falling out of my pocket into the depths of the rental car, never to be seen by me again.  So, I told her, “Oh, I’m sorry, I don’t have a nickel, but I do have a dime.”

As I handed her the dime, I saw her face morph from a confident smirk to a confused, almost terrified look of despair .  I had just taken her from a mathematical point of “Yeah, I can do this math stuff.  I may not use the computer for the rest of my shift” to “Holy \$#!+, what the #=|| just happened!”

I went into math teacher mode and waited patiently for her to begin breathing again.  And then I waited for her begin thinking.  She adjusted my change with my introduction of the idea of a nickel, why not a dime? After what seemed like 5 minutes (probably more to her), it was painfully obvious to all around that her anxiety in this situation was taking over her ability to tackle this problem. So, I tried to think of a “least helpful question” to ask.  Now I put myself on the spot.  If she only knew that we were both now feeling some of this pressure.

So, I finally asked her my question and she gave me the correct change within a few seconds.  She smiled as she gave me my change and my new “unknown” student and I parted ways.  I know I felt good about helping someone develop a strategy outside of the classroom.  I hope she had a similar feeling about learning to make sense (no pun here) of making change.

Being a math teacher is a 24-7 job sometimes and we can find our students anywhere – even in a fast food restaurant in New Mexico!

What you would have asked the cashier in this situation.  I’d love to hear what your “least helpful” question would have been. No pressure, take as long as you like.  No one is waiting in line behind you!

Feed the hungry!

Oh, here’s my question:  “If you could change the dime into some other coins, what would you change it for?”

# Math: A Fun After Homework Activity

All week long I’ve been asking Connor, my 9th grade son, what he has been working on in coordinate algebra.  Here’s a snippet of a recent conversation:

• Me:  So, Connor, what have you been working on in your coordinate algebra class?
• Connor:  We’ve been graphing.
• Me:  Graphing what?
• Connor:  Graphing different lines.
• Me:  What kinds of lines are you graphing?
• Connor:  Ummmm…
• Me:  Are they linear functions.
• Connor: Yeah, there are linear functions, but we also do curves…
• Me:  Like what kind of curves?
• Connor: Umm… exponents
• Me:  Ok.  Anything else?
• Connor: Umm…
• Me:  Hey, I want to show you something. . .

Versions of this conversation happened several times this week.  Due to soccer practices, games, homework, and Life in general, we never got much past Connor’s last “Umm…”

Until yesterday!  The conversation changed a bit:

• Connor:  We did something cool in class today.
• Me:  Oh, yeah?  What was it?
• Connor:  We had to build a picture using graphs of different lines.  We built a shamrock.
• Me:  That’s what I’ve been meaning to show you all week.  Go grab my laptop.
• Connor:  (playing game of war on an ipad) But I finished my homework.
• Me:  Just take a look at this for a few minutes and see what you think.
• Connor: (heavy sigh)

Enter Des-Man from Desmos.  Once he had gone through the tutorial, he was hooked. . . for a while!  He engaged in this for about 2 1/2 hours.  When he wanted to make something happen, but didn’t know how, he would come to me and ask.  We’d figure it out together.  The best part of this whole experience was when he realized he knew how to create something on his own and went to his math work from class as a reference.

Fast forward to 2 1/2 hours later, when Connor finished his Desman.

To see the picture in detail along with the equations Connor used to create this graph, click Connor Face Graph.

It didn’t stop there.  I had some tabs open and clicked on one with the In-N-Out Burger task from Robert Kaplinsky.  He was curious enough to work through it even after all of the Des-man work.  So, I showed him more by clicking on the Open Middle tab (also from Robert Kaplinsky).  I selfishly pulled up the task that I wrote in collaboration with Graham Fletcher called The Greatest Difference of Two Rounded Numbers.  After making sense of the problem, and a lot of eye opening moments that led to phrases like “Oh, I can make it larger!” He got what he thought was the final answer and we validated his reasoning by clicking on the answer.  A slight smile!

So, we’re looking at close to 3 hours of after homework math investigation that ranged from rounding numbers to graphing equations, and solving problems.  Sounds like a great evening to me.  Great conversations and fun while learning and reinforcing mathematics understanding!  What could be better?  Talking Math With Your Kids – High School Edition.

Feed the hunger of all ages!

More with Connor:  Real Math Homework and Real Learning

# Connecting Percents and Fractions

Not understanding mathematics can be extremely frustrating for students.  As a teacher, figuring out how to help students understand mathematics can be just as frustrating.  My primary go-to resource for these situations is Teaching Student-Centered Mathematics, by John Van de Walle et. al.  because it’s all about focusing on big ideas and helping students make sense of the math they’re learning in a conceptual way.

Recently, I was asked to model a lesson for a 6th grade class who was having difficulty working with percents.  So, I turned to my go-to resource, and during planning, I realized that I didn’t know anything about these students other than that they were struggling with percents.  So, I couldn’t assume anything.  I ended up creating three separate lessons and combined them into 1.

First, I handed groups of students a set of Percent Cards and Circle Graph Cards.  Their task was to match the percent with the corresponding circle graph.  As students were working on this, I heard groups reasoning about how they were matching the cards.  Many started with benchmarks of 25%, 50%, and 75%, while others started with the smallest (10%) and matching it to the graph with the smallest wedge.  As groups finished, they were asked to find pairs, using the matches they made, that totaled 100%.  Once finished, a discussion about their process for completing these tasks revealed a solid understanding of percent as representing a part of a whole.

Now to shake their world up a bit.  I asked them to leave their cards because they would be using them again shortly.  I introduced these Percent circles and asked them what they were.  A brief discussion revealed some misconceptions.  Some students said they were fractions, others said they were wholes because nothing was shaded.  I altered my planned line of questioning to questions that eventually led to a common understanding of what fractions were and how the pictures of the fraction (percent) circles really showed wholes and parts (fractions).

Their next task was to match their cards with the equivalent fraction circle.  This was incredibly eye-opening. Groups began to notice that some percent card matches could fit with multiple fraction circles (50% could be matched with the halves, quarters, eighths, and tenths).   Thirds and eighths were the last to be matched.  But their reasoning didn’t disappoint.  One group noticed that the percents ending in .5 all belonged with the eighths because they were too small to be thirds (the other percents with decimals).

Students were eager to share their thoughts about what they learned about fraction circles and percents:

• Fractions and percents are the same because the pieces look the same.
• 1/4 is the same as 25% and 2/8
• I don’t get why the eighths end in .5.
• The percents all can be fractions.
• 1/8 is 12.5% because it’s half of 25%

Finally, I asked students to solve a percent problem (now that they’ve all realized that fractions and percents can be used interchangeably).   I gave them the m & m problem from this set of percent problem cards.  The only direction I gave was that they had to solve the problem using some representation of the percent in the problem before they wrote any numbers.

My bag of M&M’s had 30 candies inside.  40% of the candies were brown.  How many brown candies is that?

While this was problematic at first, students looked at their fraction circles and percent cards and realized they could use four of the tenths since each tenth was the same as 10%.  Most students needed just one “least helpful” question to get on the right track:  Where do the 30 m & m’s belong in your representation?

Most groups were able to make sense and persevere to solve the problem correctly, and explain why they “shared the 30 m & m’s equally among the ten tenths in the fraction circle” and why they “only looked at four of the tenths because that’s the same as 40%.”

My beliefs that were reinforced with this lesson:

• We can’t assume understanding from correct answers alone.  We need to listen to students reason through problematic situations.
• Students really want to share their thinking when they realize that someone is really interested in hearing it.
• Students crave understanding.  They really want to make sense.
• Procedures are important, but not at the expense of understanding.
• Empowering students by allowing them to build their own understanding and allowing them to make connections allows students to feel comfortable taking risks in problem solving.

Please take a look at Jenise Sexton’s recent blog about percents with 7th grade students for some fantastic ideas about students using number lines and double number lines to solve percent problems.  It’s SWEET!

# Empowering Students with In-N-Out Burger

The following is a reflection on a 3-Act task I modeled for an 8th grade teacher last week.  The 3-Act is In-N-Out Burger from Robert Kaplinsky and the plan I followed I completely stole from the amazing @approx_normal ‘s blog post on her work with the same 3-Act with administrators last year.

This past Thursday was the day we agreed on to model the lesson.  So, this group of 8th grade students, who have never even seen me before, are wondering who this guy is that’s about to teach their class.  And, just as planned, they were giving me weird looks when I showed them the first cheeseburger picture and asked them what they noticed.  I believe one of them even asked, “Are you a teacher?”

Fast forward through to the “What do you wonder?” piece and the questions were amazingly well thought:

• “How much weight would you gain if you at that whole thing (100×100 burger?)”
• “How much do the ingredients cost for it (100×100 burger)?”
• How much does it (100×100 burger) cost?
• “Why would someone order that (100×100 burger)?”
• “Did someone really order that (100×100 burger)?”
• “How long did it take to make the (100×100 burger?)”

There were just a couple more, and they all came up very quickly.  The students were curious from the moment we started the lesson.  They are still working on precision of language.  The parentheses in their questions above denote that this phrase was not used in the question, but was implied by the students.  We had to ask what “it” or “that” was periodically throughout the lesson as they worked and as time went on, they did become more consistent.

The focus question chosen was:

• How much does it (the 100×100 burger) cost?

Students made estimates that ranged from \$20 to \$150.  We discussed this briefly and decided that the cost of the 100 x 100 burger would be somewhere between \$20 and \$150, and many said it would be closer to \$150 because “Cheeseburgers cost like \$1.00, and double cheeseburgers cost like \$1.50, so it’s got to be close to \$150.”  That’s some pretty sound reasoning for an estimate by a “low” student.

As students began Act 2, they struggled a bit.  They weren’t used to seeking out information needed, but they persevered and decided that they needed to know how much a regular In-N-Out cheeseburger would cost, so I showed them the menu and they got to  work.

I sat down with one group consisting of 2 boys (who were tossing ideas back and forth) and 1 girl (Angel) who was staring at the menu projected at the front of the room.  She wasn’t lost.  She had that look that says “I think I’ve got something.”  So, I opened the door for her and asked her to share whatever idea she had that was in her head.  She said, “Well, I think we need to find out how much just one beef patty and one slice of cheese costs, because when we buy a double double we aren’t paying for all of that other stuff, like lettuce and tomato and everything.”  The boys chimed in: “Yeah.”  I asked them how they would figure it out.  Angel:  “I think we could subtract the double-double and the regular cheeseburger.  The boys, chimed in again:  “Yeah, because all you get extra for the double double is 1 cheese and 1 beef.”  “Well done, Angel!”  You helped yourself and your group make sense of the problem and you helped create a strategy to solve this problem!  Angel: (Proud Smile)!

We had to stop, since class time was over.  Other groups were also just making sense of the idea that they couldn’t just multiply the cost of a cheeseburger by 100, since they didn’t think they should have to pay for all of the lettuce, tomato, onion, etc.

They came back on Friday ready to go.  They picked up their white boards and markers and after a quick review of the previous day’s events and ah-ha moments, they got to work.  Here is a sample after about 15 minutes:

Many groups had a similar answer, but followed different solution pathways.  I wanted them to share, but I also wanted them to see the value in looking at other students’ work to learn from it.  So I showed this group’s work (below-it didn’t have the post-its on it then.  That’s next.).  I asked them to discuss what they like about the group’s work and what might make it clearer to understand for anyone who just walked in the classroom.

Here’s what they said:

• I like how they have everything one way (top to bottom).
• I like how they have some labels.
• I’m not sure where the 99 came from.  Maybe they could label that.

During this discussion, many groups did just what @approx_normal saw her administrators do when she did this lesson with them.  They began to make the improvements they were suggesting for the work at the front of the room. It was beautiful.  Students began to recognize that they could make their work better.  After about 5 minutes, I asked the class to please take some post-its on the table and do a gallery walk to take a close look at other groups’ work.  They were to look at the work and give the groups feedback on their final drafts of the work using these sentence starters (again, from @approx_normal – I’m a relentless thief!):

• I like how you. . .
• It would help me if you. . .
• Can you explain how you. . .

Some of the feedback (because the picture clarity doesn’t show the student feedback well):

• I like how you showed your work and labeled everything.
• I like how you broke it down into broke it down into separate parts.
• It would help me if you spaced it out better.
• It would help me if it was neater.
• I like how you explain your prices.
• I like how you wrote your plan.
• I like how you explain your plan.
• I like how you told what you were going to do.
• Can you explain how you got your numbers.
• I like how you wrote it in different colors.
• It would help me if you wrote a little larger.

Some samples with student feedback:

Not only was the feedback helpful to groups as they returned to their seats, it was positive.  Students were excited to see what their peers wrote about their work.

Now for the best part!  Remember Angel?  As she was packing up to leave, I asked her if her brain hurt.  She said, “No.”  After a short pause she added, “I actually feel smart!”  As she turned the corner to head to class, there was a faint, proud smile on her face.  Score one for meaningful math lessons that empower students.

Please check out the websites I mentioned in this post.  These are smart people sharing smart teaching practices that are best for students.  We can all learn from them.

# Feed the Hungry

Kim, my beautiful bride of 16 ½ years, does not like for me to go grocery shopping on my own.  Recently she had a procedure and when I brought her home to rest I told her I would do the grocery shopping while she rested and no one would have to go the following day.  I was surprised by the fact that she was totally against this idea.  When I asked her why she wanted to go with me, she told me that I get hungry when I shop and buy a bunch of things that are not on the list, so if I was going shopping, she had to go too.  We both ended up going later that day.

While I was a bit hurt by her reasoning, I couldn’t deny it.  She was absolutely right.  I get hungry when I shop.  Lots of food, free samples, items I like on sale, items I’ve never tried not on sale, eye catching packaging. . . I can’ help myself!  And there’s no pattern to my binge shopping.  It just depends on the aisles, the samples, and my cravings. This got me to thinking if everyone does this.  I think so.

So what does this have to do with math teaching?

Bear with me for a bit.  I started a new position in January as Math Teacher on Special Assignment for our district.  My focus is working with middle and high schools.  I’ve taught elementary school, middle school, high school and even some college courses for pre-service teachers, so I’m comfortable working with students at all of these levels.  But when I started I just wasn’t sure how teachers would react to the support I was offering.  Would they want feedback?  Would they want support in their planning?  Would they want a model lesson or to co-teach a lesson?

As I began my work with these teachers I thought about the different kinds of teachers I would encounter as I move from school to school.  I determined, through my interactions with many that teachers seem to fit into one of three categories:

The bottomless pit.  These teachers are hungry all the time!  They ask for feedback, and resources, plan for co-teaching lessons, conference, and do just about anything asked of them.

The nibbler.  These teachers are willing to take a taste, if it’s not too spicy or too bland.  They want new strategies and will try something if they can immediately see how it will fit within their classroom without dropping something that’s “tried and true”.  If they don’t shop in that aisle, they sure won’t take a taste.

The Pepto-Bismols.  These teachers just ate a three course meal with desert and coffee.  The only thing these teachers might want is an antacid.

At first glance, you may choose one to work with over another, but read on.  There’s more to these categories than meets the eye.

• The bottomless pits are always eating, but they may be eating things that lead them away from the aisles containing the foods for best practice. So it’s my job to make sure to steer these teachers down the aisle for the food that they need and  They may be devouring number talks, but they may be giving speed tests.  They may be sitting down to the table for a 3-Act Task, but they may not be letting their students come to the table of wonder to eat some for themselves.  Tricky stuff here.  We all have our favorite junk foods, but if it’s all we’re eating we are going to have a lot of problems down the line.

• The nibblers are kind of tricky. They’re a bit pickier about the food they eat.  Most of the time, these teachers just need an alternative, something that might replace what they’re currently eating.  Like the bottomless pits, these teachers could be in the wrong aisle and nibbling just because it’s easier than walking to the next aisle – even when the food over there is SO much better!
• The Pepto-Bismols are my favorites. These are the ones who think they cannot possibly eat another bite.  They don’t think they’re still hungry, but deep down they are still craving.  They’ve eaten and their plates are still full.  It’s time to steer these teachers toward the pharmacy aisles.  If we can ease the bloating (often caused by lack of standards-based diet), maybe we can slip in a small piece of gourmet math food here and there (Open Middle anyone?).  When they realize their hunger it isn’t long before these teachers are feeding others!

Ultimately, we all need to help our peers in this global math grocery store realize that they are hungry, feed them a little bit of the math goodness, then teach them how to shop for themselves.  You see, all of us are hungry.  Some of us are just walking down the wrong grocery aisles.

I’m still not allowed to grocery shop alone and I’m ok with that – as long as I can be a Mathmart associate, it’s all good!