- "I don't think this is going to work"
- "Why?"
- "Maybe we can try..."
- "Shouldn't that happen because..."
- "What does the graph look like?"
- "What if we..."
- "Can you verbally explain what you did?"
- "Can you tell us what we did wrong?" (one group to another group)
- "We were real mathematicians today"

# M^3 (Making Math Meaningful)

## Friday, 24 February 2017

### Why Should You Try a Thinking Classroom?

Overheard in calculus class this morning (day 2 of full thinking classroom):

## Thursday, 23 February 2017

### Thinking Classroom - Day 1

After last week's workshop with Peter Liljedahl I decided to go full-on thinking classroom in both my calculus classes. I told them that they wouldn't be taking notes today and that they would be working in groups at the boards around the room. We talked a little about what the derivative function is and how we find it, along with the issues that would arise if they tried to find the derivative of y = x^729 from first principles. Next they each chose a card to determine their group. Off they went to their whiteboards/chalkboards and they started on the first question. Even though many had already been told the power rule, I made them "convince me" (and themselves) by finding each derivative from first principles. Here is the order of the questions they did:

They noticed patterns in parts (a) and (b) and were able to explain why the derivatives of (c) and (d) were the same as (a). Part (e) went better than expected and generally confirmed their conjectures. The results from parts (f) and (g) were confusing for many and I found that it was helpful to rewrite the question and get them to write the answer in the same form in order for them to see the pattern still held. They got stuck trying to do part (h) from first principles so needed to find the derivative another way.

At this point we stated the power rule as a group and turned to proving it. In the past, I have gone through the proof with my classes and many students' eyes have glazed over as they completely tuned me out. This time I gave them the expansion of x^n - a^n, we talked about how many terms there would be in part of it and let them try the proof. At least one group in each class finished the proof on their own! And all groups made good headway with it which helped them stay engaged when I showed them the full thing. I think they thought it was kind of cool!

I gave them two more questions after the proof:

The first was no problem and the second was done incorrectly by almost 100% of groups. We stopped there for today and I asked them to write down a summary of what they had learned. I didn't do anything else to close the lesson as I felt like it wasn't needed.

Here is the sequence for tomorrow:

Overall I thought today went well. I have done enough of this type of work with students that I was very comfortable and my students were great. There were a few times when I took a marker (there was only one marker/piece of chalk for each group) and handed it to a particular student, but in general they took turns doing the questions. Those not writing the solutions were watching what was going on, looking for errors. There were some good discussions going on today, but I anticipate more tomorrow due to the nature of the questions. There were some groups that would call me over to check their work, but they got a lot of "What do you think?" and "Convince me" and "Are you sure?" so I suspect that will diminish as we continue. I had to ask a few students to put their phones away, but it was not really an issue. They all did math and were all thinking and even those who came in knowing the power rule learned something new.

They noticed patterns in parts (a) and (b) and were able to explain why the derivatives of (c) and (d) were the same as (a). Part (e) went better than expected and generally confirmed their conjectures. The results from parts (f) and (g) were confusing for many and I found that it was helpful to rewrite the question and get them to write the answer in the same form in order for them to see the pattern still held. They got stuck trying to do part (h) from first principles so needed to find the derivative another way.

At this point we stated the power rule as a group and turned to proving it. In the past, I have gone through the proof with my classes and many students' eyes have glazed over as they completely tuned me out. This time I gave them the expansion of x^n - a^n, we talked about how many terms there would be in part of it and let them try the proof. At least one group in each class finished the proof on their own! And all groups made good headway with it which helped them stay engaged when I showed them the full thing. I think they thought it was kind of cool!

I gave them two more questions after the proof:

The first was no problem and the second was done incorrectly by almost 100% of groups. We stopped there for today and I asked them to write down a summary of what they had learned. I didn't do anything else to close the lesson as I felt like it wasn't needed.

Here is the sequence for tomorrow:

Overall I thought today went well. I have done enough of this type of work with students that I was very comfortable and my students were great. There were a few times when I took a marker (there was only one marker/piece of chalk for each group) and handed it to a particular student, but in general they took turns doing the questions. Those not writing the solutions were watching what was going on, looking for errors. There were some good discussions going on today, but I anticipate more tomorrow due to the nature of the questions. There were some groups that would call me over to check their work, but they got a lot of "What do you think?" and "Convince me" and "Are you sure?" so I suspect that will diminish as we continue. I had to ask a few students to put their phones away, but it was not really an issue. They all did math and were all thinking and even those who came in knowing the power rule learned something new.

## Saturday, 18 February 2017

### Here Goes!

As I wrote in my last post, I am jumping into the full VNPS-VRG-thinking-classroom in my calculus classes. I have two classes - one in the morning, the other in the afternoon - so I will be able to tweak my plan in between and hopefully really make progress with both my planning and implementation. We have finished our first unit on limits and introducing the derivative function. On Wednesday we will start with the derivative rules (power, product, quotient, chain) so that's what I'm trying to plan out. Sheri Walker and I thought through this progression together yesterday so I've tried to tie things together and include some of what I expect to see. The intro will be to the entire class, the sequence are the questions I will give each group of 3 at their whiteboards/blackboard. Groups should progress at different rates so my job is to circulate, observe and help keep them in flow. The last question (part i) may be for the speed demons or for everyone - we shall see! I am also trying to keep in mind what comes next.

You may notice that I have a number of unanswered questions. If you can help me think through those, I would be most appreciative.

My plan is to edit my document after I have done both classes and then post the new file in case it might be useful to others.

You may notice that I have a number of unanswered questions. If you can help me think through those, I would be most appreciative.

My plan is to edit my document after I have done both classes and then post the new file in case it might be useful to others.

### A Thinking Classroom

I was fortunate enough to be able to attend a half-day workshop on Thursday with Peter Liljedahl. I first heard about Peter's work a few years ago after he had spoken at a conference in Ottawa. There was much buzz from those who attended about VNPS and VRG, most notably from Alex Overwijk. Al was happy (!) to share all he had learned about vertical non-permanent surfaces and visible random groups (he may or may not have stopped strangers on the street to tell them about it). He has become Peter's #1 fan, even entitling the Ignite session he did last year "Things Peter Says". Although I had heard the vast majority of what Peter shared with us, there were a couple of important puzzle pieces that got filled in which make me believe that I can create what he refers to as a thinking classroom. This post is not intended to explain it all to you - for that you should visit Peter's website

I have used VNPS in my classroom for a few years now, but not every day and not

VNPS is a means to a goal, not the goal itself. It is, according to Peter's research, the most effective vehicle to creating a thinking classroom. One where all students are engaged in meaningful mathematics - doing the math, not watching someone else do it. They are learning to because autonomous, to look for the next question and persevere when they get stuck. The teacher's role in this is hard to nail down - you need to adjust to what you see constantly. It's structured chaos at its best. And I don't think it can be successful without really good planning (and I would highly recommend the book "5 Practices for Orchestrating Productive Mathematics Discussions" to help). Thursday's experience helped me see how all the pieces fit.

The structure looks like this:

**here**or read**this**post from Alex.I have used VNPS in my classroom for a few years now, but not every day and not

__instead__of teaching/facilitating lessons. I use them for review stations and for tasks. I do visible random groups whenever I do group work, so that's not new for me (best group size = 3). I also spiral some of my courses with lots of activities, so I think that in many respects I already have a thinking classroom. My students tell me that even when I teach the same lesson as other teachers, I do it differently - I make them do the math, I don't just give it to them. But one can always do better... What convinced me was hearing about an actual curricular example of how to structure VNPS. Peter only spoke about students learning how to factor for a couple of minutes, but with enough detail to make it all click for me. I will attempt to share how I see it all working, but apologize in advance if my thoughts have still not gelled.VNPS is a means to a goal, not the goal itself. It is, according to Peter's research, the most effective vehicle to creating a thinking classroom. One where all students are engaged in meaningful mathematics - doing the math, not watching someone else do it. They are learning to because autonomous, to look for the next question and persevere when they get stuck. The teacher's role in this is hard to nail down - you need to adjust to what you see constantly. It's structured chaos at its best. And I don't think it can be successful without really good planning (and I would highly recommend the book "5 Practices for Orchestrating Productive Mathematics Discussions" to help). Thursday's experience helped me see how all the pieces fit.

The structure looks like this:

- Start with a quick (~2 minute) lesson or prompt to activate prior learning or give students enough to build upon. Instructions should be oral as much as possible. If you give instructions in writing students have to decode them individually, whereas if they are oral instructions, students immediately start talking to each other.
- The questions they are working on must be sequenced in a logical way to develop the skills while keeping all students in "flow". These don't have to be incredible task questions - they can be everyday textbook questions. The importance of proper selection and sequencing is huge!
- There must be a lesson close that will level the class to the bottom. This could be a full-class debrief (going through a different example) or a gallery walk that is also thoughtfully sequenced.

I spend part of Friday with Sheri Walker working on sequencing questions for calculus (she was gracious enough to work on topics that she has already covered). Although not finalized, I think it helped us both think through how to make this work.

I still have lingering questions/concerns.

I worry - maybe that's too strong of a word - about the introverts in my classes. Especially the shy introverts. Because I am one, and I know how exhausting working in groups where you may not be entirely comfortable with the material or people can be. I am already purposeful about making my classroom a safe space for learning, which includes making mistakes, but I still worry...

I wonder about the number of markers and who is using them. We worked through two problems on Thursday and each group only received one marker. There were no rules around who should do the writing, but Peter was going around the room taking the marker from some and handing it to others. I was very aware of how long I had the marker when I was in my first group and did my best to always put the marker down when I had finished with it. It is much easier to pick a marker up than to take it out of someone's hand. There were questions about all of this and suggestions of either using a timer so that each person had the marker for an equal-ish amount of time or that the person with the marker could only write others' ideas. This takes me back to the introverts issue - I would hate to be the one who had to write someone else's ideas if I didn't understand them. But I also know that I did not touch the marker in my second group, so it's not that hard to step back a little which is not what we want.

I don't know if I can make this work with my grade 10 applied class. MFM2P is generally made up of students where one half to two thirds have IEPs. Many require written instructions. Many cannot work in groups, only pairs. Many cannot work with certain other students in the class. Many (most?) hate math and are often very unwilling to do any work. There seem to be so many obstacles with that group, that I'm not certain this is the way to go. Spiralling with activities has really helped with engagement and success, so I think the VNPS may continue to be an every-so-often thing. If you can convince me otherwise, I would love to hear your thoughts!

Peter says we shouldn't

*give*students notes. I agree, however, I will still continue to post notes on Google Classroom in calculus because there would be a mutiny if I didn't. There are only so many battles that I will take on! I like the idea of finishing the "lesson" at around the 50-60 minute mark, doing the lesson close then leaving ~15 minutes for students to write down, in their own words, what they have learned. They will then be able to reference my posted lesson with examples as they need.
There is so much more to all of this, but this is where I am for now. I am not promising to blog every day, but I will write about how it's all going. I would love to hear your thoughts in the comments. Thanks.

## Wednesday, 8 February 2017

### Skyscrapers

I spent a lot of time thinking about what activity I should do with my grade 10 applied students on the first day of semester 2. I wanted them to be engaged in mathematical thinking, preferably with something hands-on (but nothing that would cause complete chaos!) and I wanted them to work with someone else in the class. What I ended up choosing was Skyscrapers from BrainBashers -

A completed board would look like this, where the numbers in the grid represent the height of each skyscraper:

I first learned about these puzzles from Alex Overwijk last year and I'm fairly certain that he heard about them from Peter Liljedahl (I spelled that correctly this first try!). Alex let us try them at our math PD day last year using linking cubes as the skyscrapers.

I set up the skyscrapers ahead of time for my class, using a different colour for each height so that it would be easy to see if they had more than one skyscraper of a particular height in the same row or column. It turned out that I found this feature more useful than them as I circulated and checked their work.

Each pair got their first puzzle and these:

Here is an example of one column. We reasoned through the fact that there is only one way to place the skyscrapers if you can see all 4.

Here are a couple of views of the completed puzzle:

I had printed out 6 different puzzles for them to work through, each on a different colour of paper so that I knew which one they were working. I also had the solutions printed on the same colour of paper to make checking their work faster. I checked the first two and then let them go. I should have printed some harder ones as some groups flew through these. I did have blank ones for them to make their own, but I really think this could have been a much richer experience if they had tried some of the harder ones, like these:

This is what I tweeted out:

What you may not realize is that very few grade 10 applied students ask for more "work", so it was awesome that they wanted more! Day 1 was definitely a success. I got to interact with all my students and see a little of how they think, whether they are able to follow directions easily, and how they work with others. It was a good day and a great start to the semester.

**here**is the link to their site. These are logic puzzles with only a few rules:A completed board would look like this, where the numbers in the grid represent the height of each skyscraper:

I first learned about these puzzles from Alex Overwijk last year and I'm fairly certain that he heard about them from Peter Liljedahl (I spelled that correctly this first try!). Alex let us try them at our math PD day last year using linking cubes as the skyscrapers.

I set up the skyscrapers ahead of time for my class, using a different colour for each height so that it would be easy to see if they had more than one skyscraper of a particular height in the same row or column. It turned out that I found this feature more useful than them as I circulated and checked their work.

Each pair got their first puzzle and these:

Here is an example of one column. We reasoned through the fact that there is only one way to place the skyscrapers if you can see all 4.

Here are a couple of views of the completed puzzle:

I had printed out 6 different puzzles for them to work through, each on a different colour of paper so that I knew which one they were working. I also had the solutions printed on the same colour of paper to make checking their work faster. I checked the first two and then let them go. I should have printed some harder ones as some groups flew through these. I did have blank ones for them to make their own, but I really think this could have been a much richer experience if they had tried some of the harder ones, like these:

This is what I tweeted out:

What you may not realize is that very few grade 10 applied students ask for more "work", so it was awesome that they wanted more! Day 1 was definitely a success. I got to interact with all my students and see a little of how they think, whether they are able to follow directions easily, and how they work with others. It was a good day and a great start to the semester.

## Wednesday, 25 January 2017

### Giving Thanks

This post is long overdue. I knew I needed to write it last May and am finally making the time.

Way, way, waaaayyyyy back when I was doing my education degree, my senior (grade 10-13, yes, grade 13 existed back then) math instructor was amazing. He drove about an hour each way to teach us twice a week. He brought in graphing calculators and taught us how to use them and how to teach with them. This was pretty incredible as it was 1994 (I told you it was as long time ago!). This laid the foundation for me to become a national instructor for TI a few years later (I have since resigned - my heart belongs to Desmos). This instructor also had us create "backward problems". Instead of just asking a question, we started from the answer and turned the question around. He helped me think in a different way and really turned around (pun intended) my idea of what assessment questions can look like.

So a public merci! goes out to Rodrigue St-Jean for helping me start my career in a positive way. I am grateful and a better teacher today thanks to you.

Way, way, waaaayyyyy back when I was doing my education degree, my senior (grade 10-13, yes, grade 13 existed back then) math instructor was amazing. He drove about an hour each way to teach us twice a week. He brought in graphing calculators and taught us how to use them and how to teach with them. This was pretty incredible as it was 1994 (I told you it was as long time ago!). This laid the foundation for me to become a national instructor for TI a few years later (I have since resigned - my heart belongs to Desmos). This instructor also had us create "backward problems". Instead of just asking a question, we started from the answer and turned the question around. He helped me think in a different way and really turned around (pun intended) my idea of what assessment questions can look like.

So a public merci! goes out to Rodrigue St-Jean for helping me start my career in a positive way. I am grateful and a better teacher today thanks to you.

### Math Minute

Back in September, I decided to try a new way of sharing some of the cool stuff I hear about online with the math department at my school. I call it "Math Minute" and this is part of the original email I sent them:

This was followed by a short description and the first link that was all about Desmos card sorts.

Here is the link to the Google doc I am using to keep track of what I have shared.

This is a sample (since it had no link):

"Week #8:

Mary"

Although I have had little feedback from these Math Minutes <insert sad face>, I thought I would share what I've done in case someone else is looking for a way of sharing ideas.

"Hello fellow mathies,

I thought I would share some of the great things I come across on Twitter and on the blogs I read. It might be a cool activity or link to an article or blog post, but should only take a minute (or so) to read - hence the Math Minute title. I'll do my best to send a Math Minute out once a week, however please feel free to let me know if you would prefer not to receive them."

This was followed by a short description and the first link that was all about Desmos card sorts.

Here is the link to the Google doc I am using to keep track of what I have shared.

This is a sample (since it had no link):

"Week #8:

I have been using the box (area model) method for multiplying and dividing polynomials for a while now. I like it because there are no tricks involved and students can see (and hopefully understand!) why they are doing what they are doing.

Below is a sample of factoring a non-monic trinomials. There would normally only be one box, but I tried to make my steps understandable for you. I have to say that I love algebra and decomposition and I have been friends for a long time, but I love the box for these. Try it out!

Cheers,

Although I have had little feedback from these Math Minutes <insert sad face>, I thought I would share what I've done in case someone else is looking for a way of sharing ideas.

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