November 13, 2017

Matrices - the Basics

I've been writing a lot of material lately. Why are matrix lessons so hard to find? I understand that they're plus standards in CCSS but even if you don't teach them in Algebra 2 someone should be teaching them somewhere...?? Anyway, it's been great in that I have total control over the difficulty level, language and content focus. Not so great in that it takes a ton of time. But since I put all this time into things, I'm going to share so that the time was worth it!

I was out (at ATMNE) for the first day of the unit so I made an introduction packet from my textbook and David's lovely post on matrix multiplication. The students who were willing to read and take some baby risks did great. The students who were scared to try informed me that they couldn't do it since I didn't teach them how. I need constant reminders that these are learned behaviors. Given enough time and consistency I can teach them new behaviors. But omg is it hard some days!

To determine what students had learned from the packet as well as see what details they would attend to, we did a WODB the next class. After my last post I got some more feedback and decided on this set. Kids found reasons for all four in every class! I was surprised that the non-square matrix didn't get the most votes since it's the one that jumps out to me.

Then we practiced matrix multiplication by comparing matrix multiplication to the multiplication of numbers. For some students this sequence flowed really nicely. We did a problem from the homework and someone asked if we always had to bring down the matrix on the left. Cue a couple problems that follow a*b vs. b*a. Lovely counterexample of the commutative property for matrix multiplication. The next pair of problems has students multiply by the identity. For students who were paying attention they caught on to the pattern quickly and realized that this matrix was special. Students who weren't paying attention got extra practice multiplying matrices. I let them in on the 'secret' that mathematicians are lazy! They look for patterns to make their work easier and we should be like mathematicians as we work. It's a tricky balance to structure a class so that students are always looking for patterns but to make sure that they aren't just assuming there's a pattern because there always is one. That was one benefit of starting with testing the commutative property - in that case the thing to notice was the lack of a pattern! Finally we multiplied some inverses. Another benefit to the pattern finding structure is that students compared work with their group when they weren't seeing a pattern, they found mistakes faster than in a random set of practice problems. I chose my inverses very carefully which allowed students to recognize two aspects of inverses (opposite signs and switching a with d). We finished off with some homework on all the operations they've learned so far.

A few students asked during all of that rather tedious matrix multiplication practice what the point of these things was, when would anyone ever use this?? So I told them about graphic design using a matrix to represent all of the points of a figure and operations with other matrices to transform them on the screen. I could've sworn I saw a video on them using this process for the fur in Monsters Inc. at the Pixar exhibit at the Boston Museum of Science but I can't find the video online. I could, however, make up a problem set to demonstrate the process by 'animating' a triangle. This might have been my best lesson all year! It was last block before a long weekend but yet students found this lesson very approachable and got some self checking practice applying matrix operations. Bonus- I enjoy activities that blur the (non-existent) lines between algebra and geometry.

Most students finished their transformations with time to spare (in an 80 minute period) so I had another practice sheet ready to start in class and finish for homework. My colleague and I like my textbook's problems with variables and expressions as entries in matrices. Some of the entries are self checking, others make simple equations and students get some practice solving one variable equations (which, yes, my algebra 2 students do still need). However, no one on the internet has these lovely problems, including our Kuta Software. So I searched class zone (I still don't understand the structure of that site but I'm learning, kinda) and compiled all the ones they wrote on an additional equation practice sheet.

Next class we'll do some other applications of matrices and row games to review the basics before heading into systems.

What else do you have for matrices? Where on the internet are these lessons hiding?

November 11, 2017

Explore Math Project and DREAM

Last week was both the end of the quarter and ATMNE. I got the chance to see Tracy Zager talk (both her keynote and her workshop) and reflect on how first quarter went. Tracy's keynote was about mathematical inquiry. One method of allowing kids to ask the questions is to provide students time to play with math. Her daughter wondered why we DEAR (drop everything and read) but never DREAM (drop everything and math). After the talk, some of us who teach high school were discussing what this might look like at our level. I suggested that my version of Explore Math is one option, after all one of the columns is titled play!

This year's version of Explore Math asks students to explore something from one column per quarter and then share briefly with the class. I assigned it in all three sections of Algebra 2 as well as my honors precalculus. I've really struggled with my Algebra 2 classes and part of it is that they don't enjoy math, at all. They see every aspect of the subject as painful and it is taking every teacher tactic in my bag to convince them that math might be interesting to think about. I hoped that they would find something to grab their interest when completing this project. However, students continued to resist - I'm too shy to present, none of those 17+ options appeal to me, I don't know how to [fill in the blank]. When the first person presented on a mathematician they realized that I wasn't asking for a giant research project and almost all decided to also look up a mathematician, but there wasn't much exploration of math happening. Don't get me wrong, it was awesome to see my diverse students represented in the diverse mathematicians they chose but most didn't even look up the kind of math they studied. Maybe step one needed to be seeing that someone like them did math before they could be convinced to take another tentative step toward enjoying math? Here are the statistics from the first round:

Precalc:
61% mathematician
0% didn't do
2 recent articles, 2 recent topics, 1 unsolved problem

Alg 2 (3 sections):
64% mathematician
32% didn't do
1 brilliant.org problem

59% mathematician
36% didn't do
1 grad level math

59% mathematician
41% didn't do

I'm thankful I made this a year long project so that now for the remaining three quarters students will have to choose other categories. It seems wrong to be forcing my students to play, but after many years of learning to resent math it's going to take a strong shove toward playfulness to get them to consider it as something that they could engage with independently. (These results have a little bit to do with the project being homework, but I provided time in class on several occasions to work on this or other make up work.)

Tracy asserted that allowing kids to ask the questions is 1) intellectually honest, 2) a good way to teach and 3) important for equity and access. I have been trying harder to make the structure of class transparent to students. For example, I've shared with them the 'secret' that mathematicians are lazy! So they look for patterns to make their work easier. Most lessons involve us playing with problems to test some ideas and then generalizing our results, but when students aren't driving the questions it doesn't feel like playing. A recent lesson on matrices flowed beautifully for some students but others were so caught up in the drudgery of multiplying matrices (because integer operations require significant brainpower) that they weren't seeing the overarching ideas on their own. Turns out it's hard to recognize the significance of the identity matrix when you forget whether 5*0 is 5 or 0. So we're working our way toward 1) and 2) with some students feeling like mathematicians each day. What about 3)?
"The person who poses the question is the person who frames the debate."
When Tracy said this sentence I stopped, wrote it down, tweeted it out and only partially heard her next several sentence as I grappled with this huge revelation. Politics can be decided based on who poses the question and how they frame it. How do I give my students this power? How do I make sure that my students recognize this so they can question the premise of others questions? And, honestly, how do I even consider doing any of this when I have two new preps this year? I look back on the things I did last year and regret how many of them I've let drop this year. And then I remember that I only had two preps last year and I'd taught both of them at least 3 times before. While I should cut myself some slack I'm not going to give up entirely. Up next is solving systems using matrices with technology which sounds like a good time to mix in some messy data, hopefully I can find some worthy data sets for students to play with. They can ask the questions. They can judge others' questions. We can do some aspect of this important work each class and hope that by the end of the year students see their relationship with mathematics with a bit more positivity.

October 31, 2017

Matrices WODB

I've managed to avoid teaching Algebra 2 for all but one year of my career (prior to this one). Between Algebra 1 and Precalculus I am familiar with most of the topics in the course, but matrices don't appear in either of those courses in our sequence. I forgot how much familiarity with content makes a difference when prepping! Pulling from our textbook and David's lovely post on matrix multiplication I made a packet to introduce matrices and their basic operations while I'm out. I wanted to start the next class with a Which One Doesn't Belong? but I couldn't find one. So I thought I'd make one. I started out thinking about dimensions- the top right is the only one without a dimension of 3, the bottom left is the only one without a dimension of 2. Then I thought we could play with the numbers to get something interesting.


But when I was chatting with my colleague we were struggling to come up with what other reasons we could set up. I know that when Christopher first started making these he asked people for four characteristics and then made each shape using just 3 of the four (so each one wouldn't belong due to the characteristic it was missing). We tried to think what other characteristics we could include and decided to use the scalar multiplication and equation solving from the packet to build other aspects. I also made the sum of the digits equal 10 for three of the four just for fun.


But when I tweeted this set I still wasn't getting the kind of responses I was hoping for. And I am confident this is because I'm not familiar with matrices. I'm going to have to spend some time familiarizing myself with inverses before I can teach them. By the end of the unit I'll have an entirely different idea about what important characteristics of matrices are. So until I get there, help me out? Describe a matrix using four characteristics and let's see if we can build a good WODB together!

October 2, 2017

Prepping System

This year I'm teaching three (substantially) different courses and my prep periods are consecutive* so at the beginning of the year I knew I'd have to find a good system for making sure I kept track of everything. It only took until the sixth day of school for me to have a close call:

Prepping has several stages. I usually complete them in this order:
1) decide what to teach
2) write/edit paper materials
3) make slides
4) copy paper materials

But when I have three different courses it's easy to get distracted with an idea for another class (or an email or a student showing up or...) and then forget what step I'm on. I had a detailed plan including handouts for precalc that day I had a close call, but nothing to project which would give students instructions and remind me of the plan!

Based on my #1TMCThing I'm using a google doc to plan. Here's the one for Algebra 2:


I am using this to keep track of steps one and two. Sometimes I get motivated to do some long term planning so I fill in the agenda with broad ideas of topics for each day. Then as I get to more detailed level planning I use the to do column to remind myself of what I need to find/edit/write. I have a paraprofessional in all three of my algebra 2 classes, which is awesome but new for me. I've had coteachers in the past and I know how to work with them since they also get prep periods and are expected to stay after school once a week. I'm still figuring out how to collaborate with someone when we don't have any time to chat except for 5 minutes between the two classes we teach in a row but we do take some teacher time outs in the middle of class to check in and game plan. Anyway, that column isn't getting much use yet but maybe eventually. The notes column I fill in after class in case I'm stuck teaching algebra 2 again next year (can you tell it's not my favorite course?)

There are still two more steps - slides and planning. My school computer is a Mac so I'm using the Stickies program for those steps. There are 3 notes: slides, copies and to do. At the beginning of my prep I bold the whole slides note. Then as I finish a set of slides I unbold the course name and update the filename and date. Once all three sets of slides are complete I update my agenda board (which I never look at and I'm pretty sure the students don't either, but it's ready for any admin who would like to know what I'm up to). As I complete handouts I add them to my copies sticky, once I make copies I delete them from the note. The final to do sticky is general stuff like randomizing seats (weekly) or updating my google calendar with the staff meeting schedule they sent out.

I'm still overwhelmed trying to get everything done. My contained class is following the IM curriculum so I need to read the lessons, figure them out for myself and then consider modifications for my students. My algebra 2 classes are sort of following a textbook but we are supplementing a lot, luckily the teacher next door taught it last year so she has some materials. My honors precalc class is back on solid ground after we added a bonus unit, I've been teaching this class since 2012. It still takes some effort to prep though because I have a small class of quick thinking students so I have to be sure to adjust previous plans for the pace of this group. But at least with a system in place I'm pretty sure I won't be forgetting about a class anytime soon!

*We run an alternating day schedule. I have prep last block one day and first block the other day. This means that I teach all my classes in a row with no break between. I would rather have to prep algebra 2 one day and precalc the other day, but with this schedule I prep all three classes during the last block and then do all my copying during the first block the next day. I thought I'd have time to grade during the first block too but so far that hasn't happened. Maybe after the first month insanity is over?

September 30, 2017

Using IM 8th Grade with Students with Disabilities

Side note about the title: I wish my class had a more recognizable name so I could title this post "Learning Skills, One Month In" but I forget that "Learning Skills" only means something to a handful of teachers at my school. It's what we used to call the level above Life Skills. This year my schedule just says "Math" as the course name, so helpful right?

A few weeks ago I wrote about how I was trying to figure out what to teach my contained special ed class. I decided to try some activities and then determine if the 8th grade IM curriculum would be a good fit. The class rocked my word problems with tape diagrams/bar models handout. I was impressed with their language skills; this activity is great for testing if students can make sense of problems and my group absolutely can! They needed counting objects for most problems but they were able to take the concrete representations and turn them into number sentences (no one used variable equations but that was just fine) and word sentences. Buoyed by this success we decided to move into the IM curriculum.

We've continued to play how many every day. It's interesting how some students are still resisting. They ask "Again??" when they walk in but even on days where I plan a short discussion they bring up interesting ideas and want to share all the things they notice once we get started! I'm enjoying having an organized folder of images. Right now we are working on arrays. Not every student is using multiplication as a strategy reliably but we write down a row x column = total equation each day. We are also working on one to one correspondence (I think? Remember, I'm a high school teacher by training so this is non-native vocab). We talk about how to figure out how many stems there are if we already counted the number of peppers (the same) or how many eyes there are if we already counted the number of stuffed animals (double - this was surprisingly challenging for them to grasp!).

The first lesson of 8th grade IM started with the same response. I think it stems from a feeling of "this seems like it should be easy but I don't feel confident." So I didn't get many kids to dance with me but we got enough practice that we built a list of transformation vocab that was sufficient to define terms the next class. We tried to do the card sort from lesson 2 as a desmos activity but that was too abstract for my concrete thinkers. So I opted to replace the next couple IM lessons which relied on geogebra with paper handouts from my days teaching geometry. We did a variety of transformation practice activities starting with moving (and tracing) physical shapes and graduating to using wax paper for reflections and rotations. Some students had a much easier time seeing the transformations than others, but all felt successful using the wax paper. After two great classes of following transformation instructions I posed the question "Does order matter?" and asked students to generate some transformations of their own to test the hypothesis. This was far too abstract and I stopped them, apologized mid class, thanked Desmos for their amazing timing and opened up transformation golf to demonstrate completing the same transformations in different orders. Then they were eager to play!

Last class we returned to the IM activities to do an info gap. It was excellent and the structured conversation was perfect for my students to practice vocabulary. Then I let them play transformation golf and was surprised to find they had a hard time understanding how to use the arrows (I automatically put the endpoint on the purple figure I was trying to transform, this wasn't intuitive for my students). Even though they struggled some with the interface they were successful at completing several tasks and were eager to have me play their various solutions so the class could see how many different ways there were to solve a problem!

So, where does that leave us? I've learned that concrete representations are going to be essential for this group. I'm not going to be able to use the IM curriculum as it stands with my students but hopefully I can use most of the ideas and activities just supplementing with physical models wherever possible. I think I'm going to need to spend some time soon looking at a year long calendar and the list of topics. Since I'm getting through one lesson per block and we only have class every other day I'm going to have to cut a lot. It would be good to have some idea of how that will work since the Pythagorean theorem is at the end and its definitely worth spending time on with this group. A task for tomorrow perhaps. It would be cool if the OUR site had a built in planning tool, especially because it could link to each lesson (4 clicks isn't hard, but it's still 3 more clicks than ideal).