Intro to Math: Circles and Squares

School is back in session! One of the students I tutor apparently has no homework tonight, otherwise I don’t know when I would have had time to put this into writing – things got busy in a hurry.

I’m doing the whole #teach180 thing so if you’re desperate to see more frequent updates from the classroom you can follow me on Instagram, @multiplefactorsi. I’ll be posting every day, held accountable by a reminder on my phone. I hate tiny little red numbers on that screen, so you know I’ll be following the rules.

Anyway! The first week of math class I always do “intro to math.” We started with some talk about what topics in math we were going to do over the year, and then we spent some time on an arts integration project inspired by these tasks for small children. This would have been awesome if we had more time, or maybe they would have gotten old – who can say? A couple students finished them beautifully at home, and others have kept their unfinished projects in case of momentary boredom. I’ll attach that whole assignment.

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PDF of the assignment: day-1-arts

The second day was all about seeing things from multiple perspectives and integrating multiple viewpoints into collaborative work. We started with a number talk, something I loved when I first saw it but just never used. I took the one from YouCubed’s “Week of Inspirational Math,” week 2 day 1. I had been very nervous that our conversation would be nothing like the thought-provoking and joyful example video, but this went amazingly. Here’s our board:

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Inspired by Sarah at Math = Love, who was in turn inspired by other bloggers, we then launched Broken Circles. That was great! I love hearing students talk about math, but I also loved the no talking rule. It’s also an inspired touch that one circle completes itself. In one group, the person with the “A” pieces sat there, self-satisfied while the rest of the group struggled, and it was kind of glorious. It’s almost like the point of this task was to show that working together and paying attention is crucial.

Between that and our next collaborative task, we went over our group norms. I translated them into Spanish for an extra touch, and perhaps that will inspire me to do more group work in Spanish class? Time will tell.

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The end of Wednesday we started the Pentomino task – I blogged about Pentominoes last time I did it, SO long ago! This year I didn’t leave as much time and consequently they didn’t come up with quite as many combinations, but I do still think it was a useful exercise in visualization and pattern recognition. I’m kind of in love with my independent reflection for that task, attached. Why indeed can you not build a 6×6 square?

Independent Reflection PDF: pentomino-ir

Things are shaping up, but I already know I won’t have nearly enough time to blog as I’d like : 0

Rose Curves, Tattoos, and Weekly Journals

Last year and the year before, I stared every Thursday’s class with a journal entry. The students wrote to me in response to the questions “how was your week?,” and “what did we do?,” as well as some third thing. This third thing was almost always a writing prompt about the math that we had done that week, or that we were about to do. Then I wrote back to them on little post-it notes. It created this nice sense of ownership of their learning, offered them a chance to reflect on and write about their week of mathematics, and provided a space to give me feedback about the class. I would always dread reading these on weeks I had deemed bad, but nobody ever wrote anything terribly negative.

Towards the end of last year I stopped doing this. I felt that it was taking too much time out of the day Thursday, which at our school is only scheduled to last 30 minutes (impossibly short!). I also felt that writing only one day out of the week wasn’t organic since the most prompt-worthy topics might have been covered on Monday. I told myself I would have them stop and write every week whenever it seemed like a good time to put math into sentences, but since the schedule wasn’t forcing me to do it, I hadn’t been since.

NEW SOLUTION: weekly journals will still be due each week, but there is no specific class time carved out to do them. They are expected by class on Monday, and students can do them after school, in the morning, over the weekend, and whenever they get done early in class. I will see if it works and let you know. I already got a “good luck with that” from one fellow teacher.

I polled the math 4 students about journals (yes or no about bringing them back) on an assignment from Monday involving plenty of purposeful writing. I will likely do a whole separate post about that because my polar unit got really excellent this year, but basically they had to discuss the visual and numeric patterns of the number of petals on rose curves with fractional coefficients. This is the best visual that I have found for that. I summarized the results on excel:

tats roses

In addition to the more mathematical questions, I asked them about the journal (the aye’s have it! good thing I was going to re-institute it anyway!) and which rose curve I should get as a tattoo. Yet another thing to stay tuned for – new math tattoo!

My major conclusion right now is that I should do more polls. When I was in high school we would do this thing we called “carpool car poll” where anyone could just declare a poll and we’d all have to say our opinions on a topic. I want a catchy phrase like that and more chances to give opinions.

Factoring Amazing Race

Sometimes I have a strong temptation to use my blog solely for successes and never mention anything that isn’t great within my classroom. But I recently had a realization so simple that I don’t know how my teaching could possibly have been effective before it! Realization: students won’t work together effectively until they are together on a team.

kids amazing racing

We had this excellent all-school camping trip last week, but as a consequence class did not take place for an entire week. I was concerned that they would have forgotten everything, so for my Math 2 class I created an Amazing Race game to review and practice binomial factoring. All along I had been encouraging them to work together and ask each other questions, but everyone was either lost and totally un-focused, or focused on doing their own work, not motivated to help others.

amazing race cards

I had them in 3 groups for the Amazing Race. I put three students who were on the verge of full understanding with an extremely competent but extremely quiet student. My thinking was that the quiet student could shine and the on-the-verge students would quietly listen in order to gain full understanding. This group was fairly effective although never fully cohesive – I mainly saw them interacting one-one-one rather than as a full group. I put four on-the-verge but potentially less-motivated students together so that they would all have to participate fully in order to understand. This went well – some of the best conversations came out of this group. Then I put a brilliant and patient student with a very silly but very smart student and two students on the struggle bus. I think that at least one of the strugglers is doing really well after that, but the other one is still struggling. I’m feeling sad about this but I’ll catch up with her on Make Up Work Day.

A Conceptual intro to Imaginaries

I feel like every single year I put in my first-draft sketchy lesson plans “conceptual intro to imaginary numbers.” Then I google that phrase and find a bunch of boring-seeming articles and dumbed-down explanations, stare at them really hard, and then end up just doing what I did the year before.

Ugh.

This year I actually came up with something, finally! We had a nice long discussion about the history of number systems, guided by this powerpoint:

Number Systems

This just goes through the different number systems in the order they were devised. It starts with counting numbers, and I asked them why people originally needed this type of number.

The next set of numbers is the natural numbers, which are the same just with 0 added in. Here I just told them that adding zero was a huge deal because humans used to think of it as useless but modern mathematics doesn’t function without it. We do a ton with zero later on for asymptotes, and my [totally realistic] hope is they will be remembering this moment in suspense!

The students correctly predicted that the next set was the integers, with the edition of negative numbers. I asked them why we have negative numbers and several of them independently came up with debt. I told them that one of the first uses was for pyramids, because they had structure above and below ground – this is a spoiler for that Crest of the Peacock chapter on 0. Here we started talking about infinity – the natural numbers are already infinite, because theoretically one can count forever. With negatives, the numbers are now infinite in both directions. Whoa.

They also saw rational numbers coming. I like to tie these into Mitch Hedberg’s joke about the 2-in-one shampoo/conditioner bottles, which he didn’t get since 2 could never fit in 1. Before rational numbers, there was no way to represent 1 divided by 2, and humans’ ancient minds were as boggled as Mitch Hedberg’s. I pointed out that with rational numbers, there are now an infinite amount of numbers BETWEEN all the whole numbers – embedded infinity! My brilliant and autistic student piped up here that of course, there are multiple levels of infinity, he already fully understood that concept. This did nothing to draw in my two math anxiety girls.

For irrational numbers, a term they remembered, I tied in Pythagoras’ cult, and the original discovery that the square root of 2 is irrational. I forget where I read this story but I assume somewhere reliable. Pythagoras, and therefore all of his followers, believed that every number could be expressed as a fraction. This one time, they were all on a boat, and one of the math guys was like “Pythagoras, I reallllllly don’t think that the square root of 2 can be expressed rationally,” demonstrating using a right triangle and Pythagoras’ own Theorem. This was unacceptable, and the cult threw him overboard. But now we all believe in irrational numbers (for the most part), and most of the students remembered pi as being labeled as one.

Oh also, we’d been talking about how the natural numbers are closed under addition, but not subtraction, so we had to add the integers to account for stuff like 4 minus 7. Then the integers are closed under multiplication but not division, so we had to have rational numbers to account for stuff like 1 divided by 2. The first example of an irrational number was the square root of a non-square number. I was like “so is there anything we can’t take the square root of even still?” and they were like “uhhhhhhhhhhhhh” and I was like “OK I’ll just tell you! It’s negative numbers! i is what you get when you take the square root of something negative!” I guess I could have handled that better?

As we were working with imaginary operations like multiplication and addition, I kept finding myself drawing out some ghosts to represent these imaginary numbers:

imaginary numbers

I started doing this last year. This is such an abstract concept that they sometimes feel like all bets are off when it came to things like how addition works. The imaginary number i was so associated with negatives that they would assume all imaginary parts should be written negative. They also sometimes want 2 imaginary numbers multiplied to still have i. I tried to come up with a metaphor – ghosts are pretty transparent, but when 2 of them multiply/ overlap you can see them, they’re just negative. This doesn’t really need a gimmick, though, since they just need to know that i * i = -1, the central concept. Drawing out the ghosts helped them to see that 2i + 3i would just be 5i, since you have 2 imaginary things and then 3 more imaginary things.

For the journal this week I had them write about imaginary/ totally constructed things that serve a purpose. They came up with a huge variety of things, from Santa Claus to the value of money to “basically everything.” These [obviously] got really philosophical, which I appreciated. One student, actually the one who talked about levels of infinity, said “imaginary numbers are extremely real, they just can’t be physically counted,” which I like a lot. I wanted to push back against the idea that these constructed things are purposeless since they aren’t “real.” Like what even IS real?!

Launch – Explore – Discuss

Happy Saturday, blogosphere! I’m currently located at school, about to do my second round of ACT/SAT prep of the day, and furthermore drink my third cup of coffee of the day. In between, my colleagues and I were tabling at a conference for twice-exceptional students. Tabling typically involves me excitedly explaining our school to anyone willing to listen, and ramping up imaginary rivalries with our fellow small schools. I did try to tone that down this time, though, because I suddenly want to do some classroom observations (aka spying) at other small schools.

The congregation of the church we rent our space to is slowly arriving, and that is the only thing weirder than being at school by myself. There are also two cats in the office building instead of just one like usual,

None of this is what I had intended to discuss! I have been implementing and blogging about collaborative learning for years, explaining that we had learned all of these techniques and ideas in college and I had wanted to try it. All this time I had been ignoring what I have recently discovered is an important element: the discussion. What you’re supposed to do is launch the task by explaining the directions and process, have the kids explore the material on their own, and then have a discussion as a full class to bring together their ideas. Since I was so focused on having the kids prove to me and themselves that they knew the material as an individual, I’ve been skipping the last step.

Such a poor choice! Wednesday we spend most of class on a discussion of the Rectangle Pattern Task I discovered last year. I love this task so very much. It’s the physical embodiment of “look for and make use of structure.” As I had learned to do in college, I structured the discussion around which groups had discovered what in a nicely-phrased way, making sure that everyone would have a chance to speak. Here are my notes:

notes

The groups shared the patterns they had recognized. They predicted step 0 as just 2 blue squares, then determined that each time you add 4. To find the pattern, I wrote that out like this:

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This made it easier for them to see that the nth iteration would have 2 + 4n blue squares. Love it!

To discover the green and red patterns, which are quadratic, we went more visually. That’s the squares you can make out in my notebook. The students described that the pattern of green squares is that on each side, you add another row, and that row will include 2 more than the previous. I drew this out as they described it:

structure A

Then I asked them to find the largest square in the first one. They said 1. I asked for the largest square in the second. They said 4 at first, then clarified that that was 2 by 2. For the last one, someone pointed out that there are many different 2 by 2 squares in there, which is true. I asked them if they could re-arrange the boxes to create a larger square than that (should I have waited to pose this?). One poetically described that process as “you take the wings off of the bird, and you put them on the bottom.” I illustrated it slightly differently, and then gave everyone paper to cut in order to create squares. This was the end result here:

structure B

The pattern for one side is then n squared plus n!

The next day the room was entirely silent and productive as brilliant math learners completed their independent reflections and then determined the patterns on yet another set of box designs. Next week we’ll discuss those, then I want to give them an assessment on this [which I have yet to write], and then we’ll work on factoring in a hopefully visual / non-terrible way.

Group Norms

Working collaboratively is an incredibly important skill, and increasingly so. So in addition to teaching math, I’ve been working on strategies for teaching communication and collaboration skills.

I got a jump on this during bonding week this year. Every year we divide the school into small groups of maybe 5 or 6 and task them with creating boats. We have this rotation – year 1 these boats are strictly cardboard and duct tape, year 2 strictly trash bags and duct tape, and year 3 cardboard, trash bags, and duct tape. It’s year 3 but this picture is from last year:

boats for blog

Chaos, right? But super fun chaos, and we get more buy-in than you’d expect. Even still, we had noticed this tendency for boat group communication to be subpar. The groups would usually defer to the ideas of whoever was loudest, kids sometimes got mad that their idea didn’t feel listened to, and without a teacher consistently in the room (which would ruin it in other ways), it was totally OK to not participate for kids who didn’t feel like it.

So this year we had them do limited-resource challenges in small groups, and then reflect on these group dynamics upon entering their boat groups. We orchestrated the groups so that nobody was in a group with someone they just worked with, so they could talk openly about their experience. I wrote up a set of reflection questions:

Communication Discussion Questions:
1. What was the most challenging part of communicating with your previous group?
2. Did you feel that your ideas were listened to?
3. Did you always feel comfortable sharing your ideas?
4. Did you feel that you listened to other ideas? Why/ why not?
5. What qualities in someone else make them easy to communicate/collaborate with?
6. What qualities in someone else make them difficult to communicate/collaborate with?
7. Do you find it helpful when someone steps up to be the leader, or does that make you more reluctant to share?
8. What can you do as a group member to make sure nobody is dominating the discussion?
9. What can you do as a group member to make sure quieter people are joining in?
10. What can you do as a group member to make the decision-making process easier?

So now in my math 2 class we’ve started our first group task, the rectangle pattern task I discovered last year. Before breaking into groups we went over the group norms, based on concepts we talked about in my college classes and the book Designing Groupwork by Elizabeth Cohen. Group norms:

For a productive group…
1) Stay in your group
2) Ask the other group members first
3) Everyone is accountable for what the group is doing
4) You are responsible to ask for and offer help

I like the last one best although the first one is a close second.

Best question ever: “wait – are we doing this to practice working in groups or to learn the math?” Answer – both!!! I think I exclaimed that with three exclamation points in class as well.

Tomorrow we finish the task in math 2 and launch the vector fields task in math 4.

Intro to Math Week Year 3: Arts Integration Edition

Just finished my last academic day of the first academic week of the school year. As it has been for the last couple of years, this week is “intro to math week,” and this year I focused specifically on arts integration, keeping the mathematical standards of practice front and center.

On day 1 we spent some time on the syllabus, focusing on class procedures and previewing the upcoming units. Thanks to keeping this blog I have plenty of pictures of previous classes doing the work they’ll be doing, so it’s not just a line about functions or data or whatnot on a sheet of paper. Then I had them each take a Math Attitude Survey. Somewhat unsurprisingly, most of the new ones had some pretty negative associations with math. Kids also weirdly LOVE sharing bad experiences in math classrooms, so it’s a nice first day icebreaker. I took everyone’s three words and put them into a word cloud:

math word cloud 2015

I’ve always been too scared to re-administer the survey at the end of the year, but I think I should do it, or at least have them give me three words again.

The next day was all about arts integration. I had them re-examine the mathematical practices and compare them to the National Core Arts Standards, since I had personally noticed so many great links. Some of them were a little unsure at first, or had trouble understanding what individual standards meant, but they didn’t shy away from asking each other or me for clarification, and the resulting conversations were pretty cool.

art and math

Then we watched this TED talk by Daina Taimina, or at least most of it – it’s super long [for my kids’ attention spans & non-university math levels] and the important parts are at the beginning and end. I love this talk so much though. It inspires me to see her construct her own understanding, especially given her initial troubles with the subject, I enjoy seeing stereotypes smashed, and is the perfect example of arts integration. The art form makes the mathematics more concrete and comprehensible, and the mathematics provides structure and context for some interesting-looking pieces.

Typically I have some major emphasis on group norms, but I didn’t want to insert something else random into this week, so instead today we just had an extended journal reflection and then dove right into our first topics. I’ll teach them the group norms as we start doing our first group tasks. I realized earlier that I’ve never actually blogged the group norms, so I’m working on a separate post about that.

So far things are going well BUT hey it’s only the first week.