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.

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

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

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

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

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

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

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

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

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

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

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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.

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

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

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

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

Katie Breedlove (@KatieBreedlove)

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

Jenise Sexton (@MrsJeniseSexton)

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

Karla Cwetna (@KCwetna)

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

Carla Bidwell (@carla_bidwell)

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

Brian Lack (@DrBrianLack)

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

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

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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.

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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.

- Building Multiplication Fluency in Middle School
- Building Multiplication Fluency in Middle School Part 2
- Building Multiplication Fluency in Middle School Part 3
- Building Multiplication Fluency in Middle School Part 4

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

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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.

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…

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.

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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:

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.”**No whole group instruction – ever**–

**2. S tudents should be on a self-paced computer program for personalized**

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. T eachers 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!

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Having attended and presented at conferences before, I have to say some conferences are good, and some not so good. NCTM Nashville 2015 was, in my opinion, the best I have attended – hands down!

Here’s why:

On Wednesday evening, in the opening session, Graham Fletcher, Robert Kaplinsky, Laila Nur, Andrew Stadel and Cathy Yenca set the tone for the conference. They spoke about their personal experiences of improving mathematics teaching and what they use to continuously improve their practice, they all spoke about how accessible and personalized PL for math teachers’ needs can be with a Math Blogs, Twitter, the #MTBoS (Math Twitter Blog-o-Sphere) that links them all together, and Web 2.0 tools that are not only changing the ways we think about teaching mathematics, but also the ways students engage in mathematics in their classrooms. One word: Powerful. And as I said before, it set the tone for the rest of the conference.

The rest of the sessions, at least the sessions I attended, all connected to the opening session. In the Desmos sessions I attended with Michael Fenton and Christopher Danielson, the presenters were able to take novices through the simplicity and beauty of this free graphing calculator (which is really much more – see my post on this here) and those of us who are just above the novices had plenty to learn as well. I even had a Desmos special tutoring session from Cathy Yenca and Julie Reulbach in the back of one of these sessions.

The twitter sessions I attended were always full and the session facilitators, as well as many attendees, lent a hand to those who wanted to get on board “this Twitter math train.” In addition, LOTS of people stopped by the MTBoS booth and were given some “small group” lessons on how to use Twitter, who to follow, and were given some general tips to make the whole experience low stress! Michael Fenton and John Mahlstedt were the facilitators of the Twitter sessions I attended. In each of these sessions, attendees were eager to learn more about Twitter and how it could help them become better math teachers. Even some not so eager people were asking questions near the end of these sessions!

The rest of the sessions I attended (I even co-presented one) had to do with modeling with mathematics – SMP 4. These sessions were probably the most valuable to me for two reasons:

- We got to really dig in to some math and have some great mathematical discussions!
- I got to experience more modeling in secondary mathematics which is great since I have just rejoined the secondary math world.

Ashli Black‘s session: Selecting and Using Tasks to Develop MP.4: Model with Mathematics was all about investigating characteristics of modeling tasks and working with pitfalls. I recommend following Ashli on twitter: @Mythagon. She really knows what modeling with mathematics should look like in the secondary math world, she’s a great presenter, and I’m thankful that she took the time to fill out the speaker form last year.

Michael Fenton‘s session on modeling provided a one-two punch – Modeling **WITH** Desmos! This was an incredible session. Michael’s presentation combining Desmos with mathetmatical modeling was. I was making sense of mathematics through the models created. I wish I had learned math this way, initially! While I can’t go back in time to learn this way for the first time, I can make sure that the students in my district have the opportunity. And it’s one of my goals for this year.

Andrew Stadel’s session: Model with Mathematics using Problem Solving Tasks. I have to admit, I’ve been using Andrew’s resources from his blog for a few years, but it was a real treat attending his session. He engaged us in a three-act task: Swing Wraps. This problem solving task engaged us in mathematical arguments, modeling, and sense making and a few other SMP’s. Mr. Stadel also did some modeling of his own through the types of questions he asked to the whole group and small groups, through his guiding of the discussion, and through his commentary about the importance of doing these types of problems.

So, in conclusion, here’s what this all boils down to:

- Join Twitter and become a part of the #MTBoS
- Allow
**students**to model the problems they solve with mathematics. - Take a look at Desmos – a long hard look – one that allows you to see it for more than just a free online graphing calculator that students can use to model with mathematics (that should be enough-but there’s oh-so much more to it!)

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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:

- It’s completely free! (not just this first time – all the time)
- 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: 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!

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