Classroom Discussions

I Didn’t Know What I Didn’t Know…

Let me just start with this. If you live in Georgia, say within a 2 hour drive to the UGA Griffin campus, seriously consider joining the Masters’ or EdS program. I’m in my first semester. It’s amazing! ‘Nuff said.

Maybe it’s just me… I thought I understood everything I needed to know about fraction equivalence… until this week. If you get to the end and think, “Oh, I already knew that!” I apologize. This is post is really for me to reflect a bit. If it helps anyone else make sense of fractions…well that’s just gravy!

It all started with an assignment for one of my graduate classes. The assignment was to read Chapter 3 from Number Talks Fractions, Decimals, and Percents by Sherry Parrish and reflect on one of the big ideas and the common misconceptions connected to those big ideas. I chose to reflect on fraction equivalence.

In the section on equivalence, Dr. Parrish talks about how students want to take fractions like 1/4 and multiply by two to get an equivalent fraction of 2/8. This misconception may be fostered by teachers who wish to make equivalent fractions easy for their students to remember. This is never a good idea! Because really… if you multiply 1/4 by two, that means you have 2 groups of 1/4. And 2 groups of 1/4 gives you 2/4 and 1/4 can’t be the same as 2/4.

What I learned next came from a phone conversation I had with Graham Fletcher about 15 seconds after I finished reading the chapter. Sometimes I just think he knows when I’m learning some math and gives me a call. He had a question about equivalent fractions. Over the course of about 45 minutes talking on the phone, I think we both deepened our understandings about what makes two fractions equivalent.

Take the rule of multiplying the numerator and the denominator both by the same number to make an equivalent fraction. If we look at 1/4 and multiply the numerator and denominator by two to get 2/8, we get an equivalent fraction, but this isn’t necessarily the whole story. To really understand fraction equivalence, I had to be asked to dive a little deeper. Graham asked me to dive deeper. As we talked, multiplying by one came up, then the multiplicative identity. These ideas definitely strengthened my understanding of fraction equivalence.

I thought I now had a deep understanding of fraction equivalence. But wait, there’s more. This is the best part. I went to class this past Saturday and Dr. Robyn Ovrick gave us this:

We were asked to fold the paper as many times as we wanted as long as all of the sections were the same size. Some of us folded once (guilty – I hate folding almost as much as I hate cutting). We shared our folds and Robyn recorded what several of us did on the smart board. Then she asked what we noticed. This is where everything came together for me. I tried to share my thoughts but I don’t think I was very successful. I was really excited about this. Here is my (1 fold) representation of an equivalent fraction for 1/4:

For my example, someone said the number of pieces doubled, and at this point (my eyes probably almost shot out of my head) I thought, but the size of the pieces are half as big. I’m usually pretty reserved and quiet, but this was too much. So, with a lot of help from colleagues in class who know me a bit better than the others it all came clear to me. We visually made equivalent fractions, but connected the visual to the multiplicative identity and even explained it in the context of paper folding.

Here it is.

The original paper shows 1/4. When we fold it in half horizontally, we get 2 times as many pieces and the pieces are half the size. This can be represented here:

The 1/4 represents the original fraction. The 2 shows that we got twice as many pieces, and the 1/2 shows that each of those pieces is half the size. With a little multiplication and the commutative property we can get something that looks like this:

Knowing that two halves is one whole is definitely part of this understanding, but seeing where it can come from in the context of paper folding allows an opportunity for a much deeper understanding. The numerator tells that there are twice as many sections as before and the denominator (really the fraction 1/2) says that the pieces are now half the size. We looked at another example of how someone folded 1/4 (someone who folded 8 times!) and noticed that it worked the similarly – we got 8 times as many pieces and the pieces were each 1/8 the size of the original. I don’t think anyone thought it wouldn’t work similarly, but it sure is nice to see your ideas validate something you thought you really understood before waking up that morning!

I’m still thinking about this and I keep making more connections. This morning, in a place where I think I do my best thinking (the shower!), I realized that this is connected to the strategy of doubling and halving for multiplication. I’ll leave you with that. Time for you to chew.

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.

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:

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.”

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.

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.

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.”

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

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

“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.

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…

Storytime_logo

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:

No whole group instruction – ever
Students should be on a self-paced computer program for personalized learning
Teachers have to create new groups of students every day/week to make sure learning is personalized
Teachers should do project based learning several times per unit to engage learners
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:

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.

Screen Shot 2016-02-12 at 3.31.02 PM

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!

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.

Shared Student Work

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.
Where’s the answer…

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.
I like how you explained your answer.
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.

Re-viewed: Children’s Mathematics. . . It’s a Beautiful Thing

About a month ago, I was asked to preview the new edition of Children’s Mathematics and write about it on this blog. I was more than happy to oblige! Children’s Mathematics is one of a select few books that I’ve read in the past decade that have really had an impact on how I teach mathematics.

Let me begin by saying that no matter which edition you read, it’s worth it. If you’re a teacher (or parent) and have an unread edition of Children’s Mathematics sitting on your shelf (for whatever reason), do yourself and your students (or children) a huge favor and read it. Then do exactly what it says to do!

The research-based approaches to teaching mathematics you’ll learn from the contents in this book are invaluable. Cognitively Guided Instruction. That’s what it’s called. And it’s a beautiful thing to see in action. And it’s even easier to see in action as you read the book (more on that later). Empowering students to think, make sense of, and solve problems based on their own understanding. Why don’t we all teach this way? It makes so much sense.

To be clear, Cognitively Guided Instruction (CGI) is not a program or a curriculum. It’s an approach to teaching and it’s based on research on children’s mathematical thinking and how it develops. The idea that’s most intimidating to teachers (and parents) is that no direct instruction is used before giving students a problem. Many would argue that students won’t know how to solve the problem unless they are shown how first. This is so NOT TRUE. The contexts of the problems give the students all they need to jump in to the problem. Their pathways to solutions are defined by their own understandings. For example, students may be given a problem such as:

Luke had 7 toy cars. His friend gave him some more cars for his birthday. Now Luke has 12 cars. How many cars did his friend give him?

Students given this problem may solve it by counting down from 12 or up from seven, They may begin by choosing 7 objects to represent the cars, then counting some more to get to 12. They may even make two sets (one set of 12 and one set of 7). There are multiple ways students can represent the problem. All of them valid. Some are more efficient than others, but regardless of the strategies used, it’s a beautiful thing. It’s especially beautiful when students share their strategies and learn from each other. When we listen to students’ thinking we best know how to work with them in order to move them along their own mathematical journey.

Now this is all great, but you can get this and more from any of the earlier editions of this book.

Here’s some of the new goodness you get from the latest edition (out later this month):

A chapter dedicated to Base-Ten number concepts – this was nice to see, since base-10 understanding is a huge part of elementary mathematics.
Quotes from real teachers using CGI in the classroom. These can be found at the beginning of each chapter. Its a small part of the new edition, but really it’s one of the things I really enjoyed!
Video clips that you can watch as you read! No more CDs to have to load, or lose or break. When you’re reading and want to see the accompanying video, just scan the QR code in the book with your phone. It just pops right up!

Overall, this new edition has some updated content and makes it easier (thanks to technology) to see in action. As one teacher from the book put it:

The better I get at listening to children, the clearer I hear them tell me how to teach them.

Have I said this already? Beautiful. Absolutely beautiful!

What are you waiting for? Go out and get Children’s Mathematics and read it.

And then, go out and get the “sequels”:

Extending Children’s Mathematics: Fractions and Decimals

and

Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School

Check out the new edition here: http://heinemann.com/ChildrensMath

The Penny Cube

I just finished a 5th grade 3-Act task called Penny Cube that I created last spring. I tried it then, but just to get some feedback from students and see what I might need to change about how the task should be presented. Now, after completing this task with two groups of students (at two different points in the year), I’ve learned three things:

Students see a video and notice a bunch of things that teachers don’t even realize are there.
The curious questions students ask first are often “why” questions.
There’s no way to predict everything a group of students might wonder.

I’ll take this reflection from the beginning. First, I let students know that I was going to show them a video clip. I also told them that I was going to ask them what they noticed when it was finished playing. I gave students a chance to brainstorm ideas about what they could do to make sure they would be able to share what they noticed once the clip had finished playing. Their ideas were amazing:

We could look for expressions (on faces – I found out later that this student was thinking about context. A person’s facial expression can tell a lot) Unfortunately, there were no facial expressions in this video.
Listen carefully (they might be able to hear something that might give them a clue about what was going on – these students were already expecting a problem situation!)
Stay focused on the clip.
Take notes.
Try to remember as much as you can.

I had never done this before, but after hearing their ideas, I will be using this again.

After showing the video clip for Act 1, I immediately had them talk about what they noticed with their groups. Then, they were asked to share with the whole group. Here is what they noticed:

What’s missing from this picture is the wonderful reasoning given for some of these. For the last bullet, “container is open in the front,” the student told the class that it was open in front so the pennies could be placed in the container more easily (I never thought they’d see or think about that). They even began to wonder a bit here – “it might be an expression or it might be counting.” My favorite, though, is the estimation by the girl who said “it looks like 100 pennies in the stack ($1.00).” This was particularly interesting to me because of what happened when they were asked to estimate for the focus question.

The wonders were typical from what I usually get from students new to 3-Act tasks, but I handled it a bit differently this time. Here are their wonders (click here for a typed version of Penny Cube Notices&Wonders):

In my limited (yet growing) experience with teaching using 3-Act tasks, I’ve noticed that the wonders are initially “why” questions (as stated in number 2 above). I told the class that I noticed that the questions they were asking were mostly “why” questions. I asked them what other words could be used to begin questions. Rather than trying to steer students to a particular question, I decided to focus the students’ attention on the kinds of questions they were already asking, and guide them to other types of questions. It didn’t take long! Within about 5 minutes, students had gone from “why” questions to “how many . . .” and “how much . . .” questions which are much easier to answer mathematically.

The students were then asked to figure out what they needed to solve the problem. From experience with this task, I knew that most students would want pennies, so I had some ready. I didn’t give them the Coin Specifications sheet, because no one asked for it. I did have it ready, just in case. Every group asked for pennies and rulers. I wasn’t sure how they would use them, but I was pleasantly surprised.

Here’s what they did:

How many pennies in 2 inches

How many pennies in an inch

How many pennies fit on a 6 inch edge of the base

How many pennies cover base

How many stacked pennies in 2 inches?

The students all started in a place that made sense to them. Some wanted to figure out how many in the stack, so they stacked pennies and quickly realized (as I did when filling the cube) that you can’t stack pennies very high before they start to wobble and fall. So, they measured smaller stacks and used that info to solve the problem. Others wanted to find number of pennies along an edge to find how many cover the base, then work on the stacks. Students were thoroughly engaged.

After three 1 hour classes, students were wrapping up their solutions. Some groups were still grappling with the number of pennies in a stack. Others were finished. A few were unsure about what to do with some of the numbers they generated. All of this told the classroom teacher and me that there were some misconceptions out there that needed to be addressed. Many of the misconceptions had to do with students disengaging from the context, rather than integrating their numbers into the context:

One group was unsure of whether to multiply the number of pennies in a stack by 12 (6 inches + 6 inches) or to use 64.
Another group found the number of pennies to cover the base and multiplied it by itself to get their solution.
A third group found 37 pennies in 2 1/2 inches and was having a difficult time handling that information.
A fourth group had come up with two different solutions and both thought they were correct. Only one could defend her solution.

Eventually, several groups arrived a solution that made sense to them.

Time to share!

I chose one group to share. This group had a reasonable solution, but their method and numbers were different from many of the other groups, so this is where we were hoping for some light bulbs to begin to glow a bit.

This group shared their work:

I asked the class what they liked about the work. The responses:

The math (computations) are written neatly and they’re easy to follow.
I know what their answer is because it has a bubble around it.
The question is on it.
It’s colorful.

All good. Now, for the best part:

What questions do you have for this group? The responses:

Where did you get 34?
What does the 102 mean?
How about the 64?

Any suggestions for this group to help them clarify their work to answer some of your questions?

Maybe they could label their numbers so we know what the numbers mean.
Maybe they could tell what the answer means too. Like put it in a sentence so it says something like “6,528 pennies will fit in the container.”
Maybe they could have a diagram to show how they got a number like 64 or 34. I know that would help me (this student had a diagram on his work and thought it was useful).

The light bulbs really started to glow as students began making suggestions. As soon as a suggestion was made, students began to check their own work to see if it was on their work. If it wasn’t, they added it. All of the suggestions were written on the board so they could modify their work one final time. The best part about this whole exchange was that students were suggesting to their peers to be more precise in their mathematics (SMP 6 – Attend to precision). And, they really wanted to know what 34 was because they didn’t have that number on their boards (which is why I chose this group).

Now for the reveal! When I asked the class if they wanted to know how many pennies were in the cube, they were surprised when I pulled up the reveal the video. I guess they thought I’d just tell them (that’s so 1980’s). They watched to see how close they were and when the total came up on the screen, many cheered because they were so close!

The students in this class were engaged in multiple content standards over the course of 3 days. They reasoned, critiqued, made sense, and persevered. It’s almost difficult to believe that this class was a “remedial” class!

Below, I’ve included a picture of each group’s final work.

Finally, one of the conversations witnessed in a group was between a girl and a boy and should have been caught on video, but wasn’t. This group had an incorrect solution, but they were convinced they were correct, so to keep them thinking about the problem, I asked them how many dollars would be equal to the number of pennies in their answer (3,616).

Girl: There are 100 pennies in a dollar. So 600 pennies is . . .
Boy in group: $6.00
Girl (after a long pause): 1,000 pennies equals $10.00
Boy: So that’s . . . um. . .
Me: How does knowing 1,000 pennies = $10.00 help you.
Girl: We have 3,000 pennies, so that’s $30.00.
Boy: $36.00
Me: Share with your group how you know it’s $36.00
Boy: Because $30.00 and $6.00 is $36.00
Girl: And the rest (16) are cents. $36.16!

And they didn’t even need a calculator!

Math really does make sense!

	mwiernicki on Relevant Decimals Lesson
	Ella on Relevant Decimals Lesson
	Dividing Decimals an… on Relevant Decimals Lesson
	mwiernicki on Real Math Homework and Real…
	Norah A on Real Math Homework and Real…

Under the Dome

Thoughts about Teaching

Classroom Discussions

I Didn’t Know What I Didn’t Know…

Happy Accidents

Student Ideas Matter

Blogarithm Posts

About Strategy Development (and Algorithms)

Personalized Learning Can’t Trump Content & Pedagogy

Connecting Percents and Fractions

Empowering Students with In-N-Out Burger

Re-viewed: Children’s Mathematics. . . It’s a Beautiful Thing

The Penny Cube

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