#myfavfriday Kuta Software Presentation View

I discovered this week that my Kuta software has a presentation view. You can show anywhere from 1-4 problems, with or without lines, and with or without answers. You can even zoom in!

This is so perfect. Now anytime a lesson runs short, I can just throw up some problems and use my ZAP! review game or math poker for extra practice. I don't need to make fancy powerpoints (although I'm sure I still will) because this works just as well with no help from me. Excitation!

Also, I found some cute folders at Wal-Mart, 3 for $1.00. I labeled them Algebra I, Algebra II, and Geometry. I'm keeping my originals and answer keys for the day in order to train myself to put them into my binder at the end of the day. Kinda sad that I need a folder to remind me to put it in a binder, but hey, whatever works. I had papers flying around like crazy and clutter makes me crazy.

I also used two folders for retakes; one for the forms the students fill out and one for the actual retake, which I grade, show to the student, and keep in my own files.

I'm happy to report that I've had 6 students do retakes which is about 10% of my total student population. One student's grade went down, one stayed the same, and the other four went up.

Interesting...

Put the Common Core into Practice in Your Classroom

Live Blogging
Conference by Sue and Randy Pippen

What makes a good problem?

• Multiple answers with justification
• Correctness lies in the mathematical argument
• Don't stop with just an answer
• Multiple entry points
• We're not questioning the student's answer, we're questioning the thinking behind it
• Ask "Is that the only way?"
• If you tell the answer first, students quit caring about the understanding.
• Richer, requires more reasoning, a directed solution instead of just a single answer

Student yellow pages- students write down a problem they missed and the correct solution, making a note of where they went wrong or what they didn't know how to do; write down new strategies they observe from others during class

Teaching isn't telling- it's listening. (Be Less Talkative...yeah!)

When students ask for help and we stand and watch them work, we are sending the message that we don't think they can do it right on their own. Offer a suggestion and then walk away.

The mantra of Common Core is fewer, higher, more focused. It's a 3-legged stool: understanding, applications, and skills. It's highly visual and connected with multiple representations. Three shifts: focus, coherence, and rigor.

Common Core is about different representations: fractions will focus on equivalent fractions, so three-fourths is just as good as sixth-eighths. You won't see simplest form in the Common Core.

Fluency means fast and accurate- not memorize.

Testing questions:

• Single answer multiple choice
• Multiple answer multiple choice
• Extended response
• Short answer
• Drag and drop
• Fill in the blank
• Constructed response

Mimic the test in your own assessments by repeating the question stem but asking a different question rather than one stem with a four part question.

In error analysis, we focus on the mistake. Force students to decide if it is right or wrong, always asking for justification.

The goal is to prevent guessing.

Create problems where students find data from words, graphs, pictures combined- not a paragraph. Break up reading into bullets.

Focus on structure rather than procedure.

Rather than teaching in pieces, delve into multiple parts at once, especially through multiple answer multiple choice questions.

Can you take the numbers away and focus on how to do it? (Problems Without Figures, Gillan, 1909)

CC starts visually and graphically. (Noticing and wondering. Yay Max!) When graphical methods don't work, then there is a hook for teaching the algebraic method. Is one method better than another depending on what you're given? Explain. Graph first every time so you can see if you need a different method.

Give the problem and the answer so students have to justify/explain/prove. Take the focus off of the end result.

New testing starts in spring 2015.

Teach conceptually- don't teach rules!

Algebra I takes Algebra II concepts and introduces them graphically, asking for differences in order to make connections.

Give specific problems that lead students to the strategy and listen to other students' strategies.

Mathematical power and mathematical strategies through reasonable problems that are properly structured. (Makes me thinks of Exeter!)

Standards now include the verb 'understand'  which was never used before because it couldn't easily be assessed through a multiple choice test.

Each standard is not a new event, but an extension of previous learning. Use previous standards to launch later standards and build coherence.

If you value mental math, you can't force students to show work on every problem. It doesn't have to be all or nothing.

More in-depth mastery of a smaller set of things pay off. If our students were problem solvers, we could give them anything and they could attack it. Exposure doesn't work in math.

Mastery doesn't mean memorize- its knowing because you have worked with it SO MUCH.

Teach context first to create curiosity. Make sense of situations.

Standards with plus signs are not for ALL students. Consumer statistics would be more useful for students who are not STEM-career bound. A star means it is a modeling standard.

Look for ways to use previous mathematics in service of new ideas rather than re-teaching.

Use application problems to introduce a topic.

Our books give pieces and then ask them to put it together at the end. The brain works opposite- need to see the big picture in order to make connections. Learning is making connections to what you already know

Wrong answers are part of the process too. What was the student thinking?

What Math Do All Students Need?

• Understanding math
• Doing math
• Using math

CC constantly makes us go back to number sense to understand that algebra works because numbers work.

Write answers to word problems as a sentence so students think about their answers in context.

Let students choose the tools they need rather than handing it to them.

Use precise mathematical vocabulary, symbols, and notation. Constantly connect to properties. The language comes after the concept. Use it enough so that it needs to be named.  Stop trying too make it easy- make it accurate. You have to accept student language in the development of their ideas and thinking. But then go back and refine with precise vocabulary in order to make their ideas mathematically accurate.

Be ready to extend problems. When you have scaffolded questions prepared, give the next extension as students are ready- that's differentiation. Not a new problem, not more problems, but an extension of the problem they are already working on.

3 Part Lesson Plan

• Introduction
• Investigation
• Discussion and Processing- Notes may be created as a result of discussion, practice may result from methods presented

Here is the powerpoint if you're still reading. It has some prototype questions of what the new test is supposed to look like and some ideas for rich tasks, plus some stuff I mentioned here and stuff I didn't.



Also I have a handout of links with REALLY good resources, I tweeted two of them earlier but I just can't sum up the will to retype all of them. So, here's a pdf that I scanned in. It's worth it to go through and save the links, and maybe I will type them up. Or maybe not.



I was really happy to see that a lot of things mentioned in this conference were things that we all have been blogging and tweeting about and doing in our own classroom. I actually left feeling less overwhelmed even though I had about 1000 things going through my mind. I felt like this helped me to focus on what I'm doing right and how to do more of it. It made Common Core seem more accessible to me and that's thanks to all of you.

Go us!

P.S. Thanks to @Fouss for helping me embed my box documents into my post. Aren't they cute?

A Nod to Dan Meyer

I detest teaching the intro stuff in geometry....segments, lines, planes, angle addition, midpoint formula...it just seems so boring and obvious that I can't stir up much joy in teaching it.

My fourth year of teaching and I've done it all four different ways. Here is what I've done this year for midpoint formula.

Back in August, I loved the post Dan Meyer wrote "How Technology Can Help" where he mentioned having students guess the lengths of sides of triangles before calculating them with trig ratios. I've been itching to use this idea and it dawned on me that this would be a perfect way to introduce the midpoint formula.

We've just finished segment bisectors and students are familiar with the midpoint. I made up a set of 6 graphs with segments graphed on it. Then I asked students to write the ordered pair of each end point and finally to guess the ordered pair of the midpoint. We did #1 together and then I asked them to go through and do #2-6 on their own.

As I walked around I heard some good questions: "Does it matter which point we write down first?"(Umm no since I forgot to label them A or B in the first place! lol), "What do we do if it's not exactly on a square? Can we just make up a number?" Students seemed to feel as if they were breaking the rules by making a guess. I've realized we never give them permission to not know the answer. Or as Dan would put it, we abstract it away.

Next we went through each problem and I would ask them to share their guess before revealing the true answer. There were a lot of right and wrong answers but I felt that students really were more engaged by the way they were shouting out the answers and leaning forward in their seat or by the way they shouted Yes!!! when their guess was correct. I guess that deep down inside, we all want to know that we have what it takes to be a guesser- or rather to be able to figure things out on our own.

From there we went to a table of points where the midpoint was given. I wanted students to develop a pattern rather than a formula. They quickly realized that the midpoint coordinates were 'in between' the coordinates of the endpoints. But as numbers got farther apart, for example 10 and -5, it wasn't quite as easy to see. It didn't take long for them to figure out what to do actually, although I definitely have some stragglers who really miss out on anything that happens through discussion rather than explicit written examples.

I used the last two empty boxes as an opportunity for students who 'got' the pattern to do on their own and then I worked it out so those who were not getting it could see the actual work.

We concluded the lesson by writing, in words, how to find the midpoint. I was really proud of us for not needing a formula. Actually, I don't think I am going to mention the formula yet. I may have to mention it when we work backwards to find the other endpoint, but I will ponder that and see if I can avoid it.

Tomorrow we will be doing the last three examples in class and then students will split into partners and do this row game. One partner will work algebraically and one will work graphically so that both partners end up with the same answer. Shoutout to @druinok for the idea. When they do the back, the two partners will switch roles. I've decided that since I've labeled the columns A and B, I will have students fold their paper hot dog style so they are truly focusing on their column only.

At the end I will give the answers and debrief- hopefully there will be some interesting conversation. If not, well at least I accomplished my mantra of "be less talkative" or as I like to call it....BLT!

;)

Made 4 Math #13 Idea Box

I've been working on this for a while and I have to admit that it's hard for me to publish it when it isn't done. But it will never be done so now is as good a time as any.

My 'project' for this week is my Idea Box page which you will now see at the top of my blog with it's own nifty little tab.

Really it is just a virtual filing cabinet like other great people have done before me. But I like to take an idea and make it my own; therefore, the idea box was born.

I've linked to topics in Algebra I, Geometry, and Algebra II. Eventually they will each have their own separate tab. I've also linked to games, projects, and valuable resources.

My next step is to link all of my own lessons I've blogged about as a more complete picture of what resources I have available.

I purely created this for myself, for the nights I don't want to do anything (which are happening more and more often) and can just browse, borrow, and steal. But I also know how valuable they are to other teachers since I myself have spent many a night stealing from others'.

Feel free to steal or link me to other resources that I might have overlooked.

#myfavfriday Math Poker

Math Poker is a game I learned from Heather Hart at a NCTM conference and I've used it before. But for some reason it has been more popular this year.

The way it works is that every student starts with $100. They make a bet based on if they think they can do the problem or not, before seeing the actual problem. If they get the problem correct, they add the money to their total. If they're wrong, they subtract it. Once a student reaches $0, they must borrow money from me, the International Bank of Miller. But they only get $10 at a time and to get it, they have to randomly draw one of my index cards, and do whatever the card says.

And of course that's the best part. For example: Do a cartwheel, Hold your left foot in your right hand and hop around the room, Go to the classroom next door and say "I have a gambling problem" and walk away, Say two nice things about the principal, etc.

My freshman absolutely loved it. I used it to teach compound inequalities. We start with simple one step qualities, moving up to multi-step, variables on both sides, and then compound inequalities. I would just do one example with them and then they were so eager to continue to playing that that's all I really needed to do.

I think we played it for 3 days in a row and they thought it was SO fun. It was hardly any work on my end and all I did during class was click to the next slide. They spent three full periods working problems non-stop and I can't really ask for much better than that.

You can also have students trade papers to check for accuracy. And I guarantee you will have at least one student who will bet everything every time and eventually lose. That's what keeps the motivation going, plus just asking who has the most money gets them riled up.

Here are some links:

Inequalities Poker (ppt, doc1, doc2)
Segment Bisector Poker (ppt, doc)

If you need ideas of funny things to do for your index cards, I would be happy to share some more of what I have used.

Made 4 Math #12 - Domain and Range Lesson

I am totally slacking.

Here's the best I have.

I spent over two hours creating this Algebra II lesson on domain and range. Thanks to @pamjwilson for reminding me to slow the pace down, give students a change to observe, explain, and do problems on their own.

I've been trying to build in as many different examples as possible. This lesson will take me at least 2 days if not longer but I feel like it is really solid.

Thanks to the twitter people who have helped remind me how to do set and interval notation and proofed my lesson.

Notes
Powerpoint

I could say more but I'm too tired. If you have any questions about how I structured the lesson, feel free to ask and I promise to give you an excellent answer as long as it does not require me to answer right now.

Thank you, come again.

#myfavfriday Favorite Moments

Some favorite moments from the school year:

My middle school students who told me today that I am their favorite teacher because I "let them do things other teachers won't".

When I lectured my middle school class about their behavior and a students came up afterward and apologized.

Last Friday when my students told me I had to dress older because they thought I was a student in hallways.

This week a student asked me how long I thought our newish Ag teacher would work there. When I thought he was going to complain, he instead said he was asking because he wanted to become an Ag teacher and come back and take her job. :)

When students who finished early in my Geometry class of 30 started to help other people because they could see I couldn't get to everyone on my own.

One particular student who has been 'difficult' over the past two years has really grown and matured over the summer. She is now working hard, being polite, having more patience, and is in overall more control of her behavior.

My first tutoring session today with middle schoolers who I had a chance to talk to more and who are so hilarious.

The support and encouragement I have received from people when talking about attempting Take One/ NBCT.

When I made time to work with an IEP student and realized he was more capable than I assumed. And just now when I graded his quiz and he did better than expected.

In fact, a much better batch of quizzes from all my classes compared to last week.

But mostly, the fact that we are quickly approaching October so I can finally hit my groove.

--

I did have one thing to share. I've been using my hanging file organizer for absent student work but then I had students absent on a test day and of course I wasn't going to stick the test in there. I just used a corner of the board and wrote Make Up Test and listed their names under it as a reminder to myself. Painfully obvious, but worked wonders for me.

Made 4 Math #11 - Diamond Foldable and EOC Review

Although this isn't the first foldable I've ever used, it's the first one I put any effort into designing. I stole the idea from Nora Oswald's Discriminant Foldable. I liked the way she folded hers so that it formed a diamond. I printed out a blank one to attempt writing examples in the boxes. I wanted two per box so I was trying to decide the best way to fit them into each box. I went ahead and folded it to make the diamond. When I unfolded it, I now had to even spaces to write in. Success!

I altered the original foldable by adding the diagonal lines as separators and as a guide to folding. They don't fold exactly right but it  works for me. I used this for solving equations: multi-step, variables on both sides, distributive property, and fractions. I did two examples each, one with integers and one with decimals and we used two different colors- one on the outer flaps and one on the inner.

Here's the results.

And here is the file.

Next up, I created a 50 question Powerpoint that serves as an end of course review for Algebra I. The first 40ish questions are straight from our EOC exam except with different numbers. The last 10 or so are some overarching themes from Algebra that I want students to pick up. I finished this when I was tired so it might not be the most excellent thing you've ever seen but I think it could be really helpful.

We are having a schoolwide movement to try out the L to J program which basically works like so. I use the powerpoint once a week and ask 2-3 questions randomly from it. As a class, we graph our results. No grades. Just graphs. We do this every week and it exposes students to things we will learn or have already learn continuously throughout the year in order to improve retention. We celebrate all time bests of the students as individuals and as a class. Students start to look forward to our 'weekly competitions' and attendance and test scores improve. Miraculous, right? I'm not really bitter toward this, I like the idea, a lot actually. I'm bitter that I procrastinated making these powerpoints this summer and now have it piled up on my never-ending-should-already-be-done list.

But at least I have accomplished one out of three.

Here are some cute things I bought at Wal-Mart- everything under $7 and nothing that I actually need or have a plan for.

And here are my binders with glittery stickers in my favorite colors that were on sale at Hobby Lobby.

Last but not least, I made this gift for my teacher friend by using a Sharpie to write on a Dollar Tree mug and then bake it at 350 degrees for 30 minutes. Hand-wash only! Then I filled it with suckers. Simple and sentimental. And cheap. :)

#myfavfriday What Experience Brings

The one thing that is most necessary and hardest to find is something you cannot rush or buy.

Experience.

I love noticing little changes and fixes that I have picked up just in my measly three years of experience.

For example, actually doing whatever I plan for the students ahead of time, including making the answer key before I ever pass it out.

Learning to be flexible at the drop of a hat, including when students are just not understanding or I am called to an 'emergency' meeting, or both.

Being able to come up with random filler activities when my oh so beautifully planned lesson runs short.

Looking for problems in the set-up or structure of my lesson when it feels like everyone is failing because of my terrible teaching.

Knowing that it's okay to quit doing something that isn't working.

Seeking out my students opinions first on how to fix or change things in their classroom.

Learning when it's just time to go to sleep already!

Noticing subtle hints and changes in students behavior that I was oblivious to during my first year.

In fact, just beginning to focus on student behavior rather than my own.

Knowing every idea won't work for me and that that is okay.

Thinking ahead to questions students might ask and building them into the lesson.

Picking out that moment where something has to change right now or I will lose everybody.

That's my summary of three years...

I hope the learning from experience rate is exponential!

Made 4 Math #10 - "Be Less Talkative"

I shared in my #myfavfriday post about teaching without lecturing or basically my new mantra 'be less talkative'. Today I'm going to post some activities that I've used so far.

In Geometry, although I haven't lectured per se I've still been leading a lot, even if that means being in the front of the room and controlling the powerpoint.

I didn't do it this way but here's what I would recommend:

Start with the Geometry Basics Graphic Organizer (modified from this post by @msrubinteach). Here is the answer key. I numbered the answer key, cut it up into squares, and then passed out one square to each student as they came in the door. When I gave students the GO (printed on bright colored cardstock), they had to copy their square down and then trade over and over again until all 30 blanks in the GO was full. (There are 30 squares and I have 29 students so I just projected the last one for everyone to copy)

Next, try the hands-on naming review. This requires some prep work but it can be used again and again. I used envelopes that the school gave me and then I cut pipe cleaners in half for segments, used fuzzy pom poms for points, cut up pipe cleaners into sixths and folded them into arrows, and wrote and cut out letters on construction paper for labeling. Each student has construction paper on their desk representing a plane and their desk represents a second plane (to practice coplanar). I display the directions and walk around as the class arranges. Then I click to show the answer. I ask questions throughout that are more like reminders, "How do we label a plane?", "What goes on the ends of a line?", "What do arrows represent?" etc. The students can use their graphic organizer as a guide.

Next I would use this Foundations of Geometry handout (doc, ppt) to reinforce labeling and properties of basic geometric figures. The last portion of this worksheet has students draw or label different figures, similar to the hands on naming review. I recommend this one last because it involves writing. Students take more risks with the pipe cleaners and pom poms (similar to dry erase boards) because mistakes are easier to fix without others noticing. Then they are better prepared to draw and can still use the GO as a guide.

Last but not least, I used @msrubinteach's Geometry Sketch game. I drew and labeled 10 drawings with geometric figures, labeling two sets of #1-5. I tried to label from easiest to hardest so each student would have equally complex drawings. Copy on to card stock, cut apart, put two sets into a sandwich baggie. Students sat in partners facing each other with their binders open and standing up as a divider. Each student has five cards. One student described the figure while the other drew the picture then traded roles. If their picture was close, they gave themselves a point on their worksheet. This is where I left off on Friday. To be continued...

In Algebra II, we started with matrices. Matrices in the Common Core don't appear until the fourth course, which would be trig/precalc here, but it does show up on the ACT, and most of my Alg II classes are juniors. (I just used a LOT of commas!)

I introduced matrices using an Algebra I station activity from the book Algebra I Station Activities for Common Core State Standards, which you can get if you request the free sample. I totally changed the fourth station and made a handout to go along with it so it is a little bit of my own creation. Before we got started, I wrote a matrix on the board with a scalar and labeled both. I asked them what they knew about the matrix and they mentioned the numbers in the movie on their own so I built on that and just said yes, the matrix is a way of organizing numbers.

Here's the setup:

I cut index cards in half to save paper.  Station 4  I changed to be word problems using matrices and I got too tired to think of how to use index cards so there are none for station four. Here is the handout. Students rotated through stations (although they hated actually getting out of their seats) in different orders. I found that students who did station four first needed more help because they just wanted to add all numbers together and get 535 rather than add the two matrices since they hadn't seen that station yet. This went pretty smoothly but did bleed over into the next day.

We finished that up and as a summary, I asked students how a matrix is like a jewelry box. Then I wanted to reinforce the skills we had already learned so they worked on a handout connecting matrix operations to geometric transformations. (You need to label the vertices of each shape before copying. It takes too long to do on Word.) I started them out on #1 by asking them the ordered pairs for each vertex and writing them into the matrix as a model and emphasized that all four problems started out this way so DON'T ASK ME HOW TO START. My students kind of sit in groups of 2, 3, and 4 so they casually worked in groups but basically just talked to the people around them. This went pretty good unless students forgot to graph the new ordered pairs. In part b. I asked them what happened to each figure and I was looking for a description like "it moved right, up, down," etc but students were more technical and used the words I was alluding to like translate, rotate, reflect, but dilate was the one they couldn't remember. Some students even went so far as to tell me which quadrant it started in and moved to. That's promising. At the end we debriefed as a class and filled in (or corrected) the words in part c with math words.

From there, it's on to multiplying, inverses, and determinants. I don't go very far into this, basically just teaching them how to do it on the calculator. Most of the time ACT asks them to do scalar multiplication on two matrices and then add them together so I don't feel a need to go in depth. Last year I did and taught Cramer's rule and all that and I just think I can use my time better this year. So this handout leads them step-by-step through the process on a TI-84 Plus calculator. Students worked on their own on this one. I found my second section did better when I started the class by asking them to put on their big girl panties and big boy boxers. I told them it would be heavy on reading and light on math, that the steps will work, and that I spent a lot of time at home punching calculator buttons and typing steps.

By far the students were the most crybaby on this activity. They would tell me they did it five times and it wouldn't work. They would indignantly say "Watch, I'll do it again" and read the steps out loud to me, punching buttons as they went. It worked, I smiled, they were outraged. Quite amusing.

Once one student got the hang of it, it started to catch on, and they naturally began to help each other. In one class, three boys basically refused to read and just stared at the paper saying "I don't know what to do. If this is on the test then I will just guess." I had to stand behind them and repeatedly tell them that they had to read the steps. They were reluctant but that's the kind of laziness and unwillingness to read/try that I want to zap in the big boy boxers as soon as possible.

All in all I'm pretty satisfied with how it's went so far. I plan to start tomorrow with them working on algebraic matrices since we've only done numerical so far. We connected scalar multiplication to the distributive property so hopefully it will be easy to connect algebraic matrix operations to combining like terms. Then I plan to try out my new ZAP! review game on Wednesday and test on Thursday, but we'll see how it goes.

Sorry this post is so long but I am trying to blog as many of my creations as possible to keep me motivated and remind myself next year of little things that happened during the lesson that will be forgotten in the next 12 months.

Hope there was something that inspired you to make for math!

#myfavfriday 'Flipping' the Classroom

I've been thinking about what I wanted to write about and then trying to decide what to name it. I know flipping the classroom is such a buzz word right now and it has so many different meanings. For me, I'm kind of relating it to @cheesemonkeysf's presentation at #TMC12 about disrupting student expectations through alternative activity structures. Flipping the classroom for me has really meant flipping expectations. 

So far in Algebra II, I have not lectured once. I haven't even made any powerpoints. Everything we've done has been a handout I created that leads them step-by-step through what I want them to learn. I walk around and answer questions and we debrief together at the end. And I am loving it. It makes me feel like a good teacher that the students are doing all the work but it makes me feel like a bad teacher sometimes when I'm just sitting there or wandering around the room. Not only am I flipping the student's expectations of what math class looks like but their expectation and mine of what a good teacher looks like. I'm starting to redefine the role of a good teacher. While this may be commonplace to you, it's like a clarifying moment for me: A good teacher creates opportunities to learn but doesn't necessarily lead them. 

Plus the strengths, weaknesses, and personalities of the students come across much quicker this way. It's easy to see who is needy, independent, cry baby, willing to help others, efficient, easily distracted, a leader, etc.

In Geometry I've (thankfully) been able to use some lessons from last year. But even in my head I've been questioning how can I flip this around into something more engaging? For example, teaching the very very basics such as point, line, ray, segment, plane, etc. I stole a handout from @msrubinmath on points, lines, and planes and modified it for myself and my 'interactive students binders'. And then I wondered, how can I make this more interesting?

I numbered every box on my answer key and numbered each box on their graphic organizer.

Then I cut my answer key up into little squares and handed one to each student as they walked in the door.

Once I passed out the graphic organizer I told them to write what was on the square they had in their hand and then trade with someone else and keep trading until your chart is full. I could easily have just put this on the SMART board and make them copy it or literally hand out a copy. But by mixing things up, students could work at their own pace and not be rushed, have the chance to get up and out of their seats, have some freedom to chitchat, and focus on one piece at a time rather than a page filled with information.

Today's lesson built on that and made the graphic organizer useful. We did my hands-on naming review from last year and students used the GO as a guide for how to arrange their pieces. Again, I could have lectured on the graphic organizer and discussed each part. But I mixed things up by creating an activity that highlighted the usefulness of the GO so understanding it and using it was not a command from me but a demand of the activity. Ooh I like that. Quote, you just got bolded!

So no, I'm not making any videos to send home or anything like that but I am striving to flip everyone's expectations of what learning and teaching math looks and sounds like.

And that's my favorite.