January 17, 2016

Solving One Variable Equations

I teach ninth graders who arrive to high school without a solid foundation in all the prerequisite math skills. Sometimes I have to correct firmly held misconceptions - students are sure their teacher last year did it that way. Other times students are equally sure they've never seen a topic before. So when I teach a concept that students may have seen before I aim to be as clear as possible. Many teachers think that the best way to be clear is to provide very detailed step by step directions on how to complete a process. But my version of clear is providing the bare minimum information for students to be successful. After we played with pennies, I boiled that experience down to two pieces of information to solve one variable equations:
  • Combine like terms
  • Use opposite operations

As I continued, I recognized that students needed a bit more clarity on when to use opposite operations (they were continuing to subtract from the same expression twice) so I modified it to:
  • Combine like terms
  • Use opposite operations when terms are in different expressions

Now when I encounter a student who is stuck when solving an equation, I ask them "where are your like terms?" If they're in the same expression - combine. If they're in different expressions - combine using opposite operations. If they're inside parentheses or absolute value - uh oh, they're trapped! Then students can either choose a different pair of like terms or apply knowledge of the distributive property, definition of absolute value, etc. 

Limiting my instructions to two items means students have a chance of remembering them. It means students learn to make choices: some kids like have the variable on the left side - go for it! Some kids want to combine all like terms within an expression first - awesome! Some kids don't think ahead and move terms back and forth across the equal sign several times - you're making progress! It's really important to me to value all my students ideas. Later in the unit I might stop a kid and say - you're moving all the numbers to the same expression as the variable, it's legal algebra but it's not the fastest way to get there. But at the beginning? Yes! That's a great idea! Well done recognizing like terms!

When we get to inequalities - same rules and one additional note on how to shade. When we get to equations in more than one variable - same rules and a note that an expression is a valid answer. By giving students the bare necessities I'm making it easier to see the connections. It still takes an awful lot of practice and some students struggle at first with paralysis of choice - it really doesn't matter whether I combine the numbers or variables first?? - but it's important for them to start making some decisions as early as possible so they aren't entirely paralyzed when they reach trig identities and the only way to solve them is just trying a substitution to see what works.

January 13, 2016

A Day in the Life, Teaching Math to Students with Disabilities

I arrived at my classroom at 7:12 on the dot this morning. That’s the official teacher start time. Sunrise was at 7:11 this morning and I firmly believe that I should not have to rise earlier than the sun so I'm obviously not one to arrive early for school. I smile to see the identity proofs still on the board that my students worked on after school yesterday. Turned on my computer and checked a few emails before the bell rang at 7:24. One of those emails was a request to meet first block today so I checked in with the Algebra team, discovered they were not quite ready to start and figured I could go find out exactly what this somewhat vague string of emails had been about.

It turned out that there is a student in our alternative program which runs in the building who is really struggling with geometry. He has some substantial learning disabilities and the special education teacher working with him needed help figuring out how to modify the curriculum for him. We discussed his goals and history (he’s a junior who hasn’t passed the sophomore state test which he needs to graduate) and then I was able to make some recommendations. I shared the materials that I used with the contained math class (students with substantial learning disabilities where the state test was a major hurdle) as well as the materials that I used with my inclusion geometry classes (students with moderate learning disabilities). Dropbox for work files is really the best thing ever, I can send anyone an entire year’s worth of materials organized into units with the push of a button! However, an entire year’s worth of materials is overwhelming so I got her started with some specific tasks to do and ended with an invitation to email any time and an offer to meet again. I enjoy being at the point where I’ve been in a school long enough that teachers in another program know that I would be a useful resource for the particular problem they are facing. I was especially glad they called because the poor special ed teacher was trying to write all the lessons herself – way too much work!

At 8:10 I make it back to the team meeting. They’ve been planning out the rest of our unit on linear functions. We were problem solving how to help students who are struggling with substituting values into equations (to fill in a table or find intercepts) and I suggested Elizabeth’s star method. We joked/bemoaned that students would end up with two digit numbers if x has a coefficient (2x where x=4 becomes 24) and another teacher said that she requires them to always substitute using parentheses. I had this moment of shock – I did that when we were studying function notation but somehow dropped the parentheses in the intervening months and forgot about them entirely. This is why team time is so great, we can bring something we’re struggling with in our own classroom, share ideas and get new ones or a reminder of something we’ve used in the past. Conversation about the unit wraps up around 8:40 when my co-teacher arrives and I take the opportunity to check in with her for a bit. Then I overhear another coworker mention his upcoming wedding – he got engaged over the summer without telling anyone! It’s January and we have classrooms with an adjoining door and talk every day and I somehow didn’t know this. So I demanded that story and we were all properly excited for him.

8:55 the bell rings. First block is over. A special ed teacher on the algebra team asks me about one of my students on his case load. We discuss his behavior and disability and what to do about his low grades. He's showing some initiative coming after school lately so hopefully that's a trend which will continue! Second block there’s a class in my room (I have a set of iPads in there so they use my room rather than their own room down the hall) so I grab my bags and head down to the library. I pick up a form from the office on the way (professional day request so I can attend an NCTM resources committee meeting in February), make some copies and finish processing my email. At 9:30 I switch over to planning. At 9:55 I transition to grading. I love block scheduling because I can have a solid span of uninterrupted work time to get things done! Some students in my precalculus class know interval notation for domain and range while another group didn’t even fill in the domain and range questions at all. That should be interesting. The bell rings at 10:26 but I sacrifice a few minutes of my 25 minute lunch to finish the pile of assignments I’m grading.

As I leave the library I see my principal talking to someone. I need his signature on that form I just picked up, what timing! Head up to lunch in the second floor faculty lunch. Yup, we still call it lunch even though it’s 10:30. One year I brought oatmeal every day but this year I’m pretending it’s a normal time to eat and have a sandwich. I feel bad for kids who have first lunch on one day but second lunch (at 12:00) on the other day of our alternating block. I can adjust to eating at 10:30 and then having a snack when I get home, but switching off would just be weird. Lunch time depends on the department, so kids in math class during third block have first lunch while kids in English class during third block have second lunch. I chat with a few other teachers in my department about students and parenting and puppies.

Lunch ends and I scurry up the stairs and across the building to my classroom. Most of my students beat me there but I’ve finally got them trained in appropriate waiting behavior (today one kid was pretending to kick the door – that’s major progress since September when it wasn’t pretend). Students pick up their binders and I turn on the projector and pull up the slides. My coteacher greets students and reminds them to get their materials. This is my Algebra Support class, 13 students who are all behind in math upon arriving to high school. We start every class with a couple minutes of skill practice. Today they are given shaded ten frames: they have to write the fraction, simplify the fraction and convert to a decimal. I wasn’t sure if giving them all ten frames would make the decimals too easy but it wasn’t – it was a great opportunity for them to use their calculators and then recognize a pattern. We talked about place value and equivalent representations. This was a great intro to our lesson on slope. I put a few tables on the board and students determined the slope. Then a pair of points. Then a graph with two points. Then I am so confident that I’ve said “y-distance over x-distance” so many times that they’re masters of slope and ready to conquer anything! I set them free to do a scavenger hunt where there are 17 papers taped around the room. Students answer the question at the bottom of the sheet and then find the matching answer on the top of a new sheet, repeating until they’ve made a complete loop. Except they’re all stuck? Uh oh. First mistake: I said rate of change, pattern, y-distance over x-distance… pretty much every word except slope. And this activity only says slope. Second mistake: we didn’t address horizontal or vertical lines. My co-teacher and I move around the room prompting and prodding and having kids draw examples and calculate. I think the scavenger hunt would have been great if the bottom half of the paper just had tables, graphs and pairs of points. However, I used a premade thing and the phrasing of the questions threw off more kids than it helped. I should have looked at it more carefully and kept my population in mind – several of them are working on phonics with the reading teacher, the vocab has to wait until after they’ve had some more experience. Next time I’ll make it myself… Despite the challenges they got some work done and I’ll be more prepared tomorrow.

Bell rings at 12:28 and one class leaves. My next class has 5 minutes to arrive. This one is the contained Algebra class. They all have a moderate learning disability in mathematics (and several other issues outside of mathematics but the math disability is a prereq for the course). We do the same lessons as in Algebra and Algebra Support but they all have the class every day (a few of my Algebra students don’t attend the support block) and there are only six of them so they get much more individual attention. As one kid comes into class he declares he wants to take Street Law. I explain how course selection works and we get into a discussion of the content of the class and if it’s a good idea to run from the cops. I’m silently appreciative for my Twitter feed, particularly the #educolor crew because while I emphasized that being respectful is important, we also talked about how important it is to know your rights. The students in that class come from a variety of ethnic backgrounds and with cognitive impairments they’re not going to come across as the most well educated bunch to the untrained eye (even though they’re amazingly smart kids! With really slow processing speeds). They’re definitely at risk for getting taken advantage of and assumed the worst of. Thanks tweeps for making sure that my message had more depth than “don’t do anything to get in trouble.”

We run the same lesson with this class but I’m much more explicit about using the word slope throughout the intro. I also add in a slide on horizontal and vertical slopes. I am thrilled when a kid (the same one who was curled up on her desk on Monday because the stress of a new coteacher replacing the sub was overwhelming) asks, “But why??” when we discover that 5/0 makes the calculator say “error.” We compare 0/5=0 and 5/0=error by looking at 5*0=0 and 0*?=5. One student is positive he can find a number that will make the second equation true. “Is it 5?” What’s 0*5? “Zero.” That’s not 5! “Is it 0?” What’s 0*0? “Zero.” That’s not 5! “Is it 1?” (repeat conversation) “Is it 5?” (repeat conversation) “Is it 0?” (repeat conversation) … “I don’t know.” Me neither! There isn’t a number we know. The calculator can’t think of a number either, that’s why it says Error. The girl who asked “But why??” originally says, “ooh!” and I already feel like whatever I did wrong last block I made up for it with this discussion. At least some kids are leaving today feeling enriched. They still need help with the scavenger hunt but with two teachers for five kids? We manage just fine. During the scavenger hunt one of my precalculus students is hovering outside the door. I go out and he asks what the course webpage is. I tell him (cardonmath.com) and he tells me that he was searching for it and found some other stuff about me “Apparently you’re really important! I had no idea.” It was rather adorable.

2:02 and school is over. One student from last block stays to do some make up work. I find him a packet that needs correcting. Several teachers stick their head into my room "Do we have a meeting today?" We always have meetings on Wednesdays but we got this afternoon off since we have a full professional development day on Friday. One of the teachers comes in to ask about adding a student to my contained class. He's a student I was concerned about earlier in the year (I was in his class covering an adoption leave in September) but he's just now reaching the point where he needs a lot of extra help. We'll look into amending his IEP. Two students from precalculus show up and announce that they’re going to work on their ferris wheels today. The Algebra student is confused and makes some awkwardly funny jokes and my precalculus students are awesome about being nice to him. Everyone gets some work done and finishes around 3:00. I realize that I never put attendance in. For the last several years I had the same coteacher and she always did attendance. I yell at my students for not knowing how to be ready for class when it’s January but I’m just as bad. I pack up and walk out the door at 3:12.

That makes today exactly an eight hour day! I usually leave around 3:45 and always do some work at home on Sundays but I make really efficient use of my prep and am in an awesome district that gives me 90 minutes of prep a day plus my duty is common planning time so I have a manageable amount of work. Tomorrow I will teach three blocks (honors precalculus, algebra and contained algebra) with one prep and 25 minute lunch in the middle. Then back to today’s schedule, alternating ad infinitum (well, for 180 days).

January 3, 2016

January Blogging Initiative

I, Tina, resolve to blog in 2016 in order to open my classroom up and share my thoughts with other teachers. I hope to accomplish this goal by participating in the January Blogging Initiation hosted by Explore MTBoS.

You, too, could join in on this exciting adventure. All you have to do is dust off your blog and get ready for the first prompt to arrive January 10th!

January 1, 2016

Solving Inequalities with Learning Disabilites

Are you tired of the reminder that I teach Algebra for kids who arrive in high school not quite ready for Algebra 1? Or that one class is mostly students with learning disabilities and the other class is a contained class where all students have a math disability? I'm reminding you again because context is so, so important. While grading this morning I got frustrated that my students are still struggling with reading inequality symbols and organizing their work. I got a lot of great responses from people about all sorts of issues students face with inequalities, but it wasn't until an elementary teacher chimed in that I got a useful reply to help my students read the symbols. And the only person who even broached my issue with kids who struggle with organizing was an intervention teacher. I love this community because I can ask for help and get it from so many places! Today I really needed to hear from some people who have students like mine. Next time I'll type the hashtag correctly (#swdmathchat, not #swdchat) and get even more help!

Now on to an overview of the unit, followed by an overview of the test and where I plan to go from here.

We started by matching number lines to basic inequalities (one variable, one number, one symbol). We did a card sort and they gallery walked to see the categories other pairs used. They shared the categories they saw and I recorded them. I had them hold up an example of a card that might fit the category (to check if everyone knew the vocab and if the category is well defined). We discussed what open and closed circled might mean and also reviewed what the symbols meant. Everyone recorded the information in the box. Then they set off to match inequality to graph.

For one of my students who especially struggles with symbolic notation I color coded: I wrote the greater than (or equal) symbols in red and the less than (or equal) symbols in blue. I had him color code the symbol first, then check which number line would match. Interestingly, when I wrote the definitions next to the symbol I asked him what word to use - greater, bigger, larger...? He understood greater better than any of the other words. I'm not sure if it's because it's more similar to Spanish or more familiar because it's an academic word but I was glad I asked rather than assuming he'd prefer to use bigger!

In the past I've had plenty of students with misconceptions about inequality symbols, but this year I had students who had trouble identifying the symbols, even with the notes in front of them. Some of them will even turn their paper to copy the symbol, apparently ^ is easier for them to see and write than <. I haven't had any solutions for this until I asked today and Jen said:
I'm interested to see if drawing the dots on the endpoints and vertex can help my kids see the difference between <, = and > better.

After the card sort we did more practice with just matching. Then given basic inequalities (one variable, one number, one symbol) I had them graph on a number line. This was a nice change of pace since we'd just concluded our study of absolute value equations. They also needed it. This is really a many step process:
  1. Identify end point
  2. Determine if end point is shaded
  3. Determine which direction to shade from the end point
None of these steps are obvious for most of my students. We continued to practiced reading the inequality aloud, picking values to check and then shading as we moved into more complex inequalities.

    

While students noticed that dividing by a negative caused the direction of the inequality to change (and why! Those are student sentences! After some prompting and discussion of course, but students said them!) I really wanted them to be checking their work so I continued to emphasize how inequalities are exactly the same as equations until the final steps.

I have no idea what the IL was, though it appears to be my handwriting?
We practice, practice, practiced. We did a packet based off a CME problem set on the difference between an equation and an inequality. We did compound inequalities (basically double the practice) (check out my colleague's awesome intro via movie ticket prices). We did absolute value inequalities (a little extra work followed by double the practice). Several students continued to ask me what the symbols meant. :( A student had an aha moment about open vs. closed circles on a review day! :) The fifteenth time is the charm? Then we took the test a month after starting this unit and it was not awesome.

Common issues:
  1. Equation solving mistakes (integer operation errors, combining unlike terms, not using opposite operations).
  2. Terms jumping the inequality symbol (3=x and x=3 may be equivalent but 3<x and x<3 are not).
  3. Forgetting to change the direction of the inequality symbol after dividing by a negative.
  4. Shading in the wrong direction on the number line.
  5. Surprisingly, they made very few new mistakes on compound inequalities and absolute value inequalities. Some kids forgot to change the direction of the symbol for the negative inequality when splitting the absolute value.
Next steps:
  1. We need to address this. Mostly it has to do with sloppy work. Some kids will need individual intervention.
  2. Easy individual intervention when we address 1.
  3. I'm not at all worried about this. It's annoying and we'll talk about it but this is where the errors should be during this unit!
  4. I'll show all these kids Jen's dots on the symbol idea and hope that helps. I do think the issue is entirely in translating < and > into words, once they have words they're generally okay.
  5. Again, not at all worried. Normal mistakes that kids will understand when they get their test back.
So I made a sheet of equations, inequalities and equations with both x and y (leading into our linear unit coming up next). Each section has four problems using the exact same numbers. I gave them the answers and the goal is to show all work exceedingly neatly. Then to compare and contrast equations with inequalities (it's the exact same process! Except that pesky dividing by a negative thing and the extra work of graphing).

December 6, 2015

Scaling the Teaching Curve: Sharing Session

Last night, after a full day of doing math and talking about teaching, nearly everyone still had more to give! We gathered at St. Mark's lovely Choate House for dinner and a sharing session.


The sharing session was like an Ignite without slides - there were ten presenters and they all went fast! Too fast for me to tweet in between introducing each of them. Instead I took notes and will share them with you now. Hopefully we will get links for everyone's materials soon if they aren't included here.

Nick L: Using Voice Memos for Student Feedback
After looking at a student assignment, Nick uses his phone to record comments in a voice memo as an mp4 file which he then emails to the student. A voice recording allows him to convey tone, seems more personalized and encourages him to highlight student successes in addition to areas for revision. His students have also taken to asking questions via voice memo which is easier than attempting to type math notation into an email.

Kate H: Jigsaw
Students sit in groups and the whole group solves one problem (but different groups solve different problems). Then the teacher gives explicit instruction on how to be a good teacher - not telling the answer or showing the strategy. Once they are ready to be helpful group members, students shuffle their groupings so that each table has one expert on each problem. They work through all the problems, with a different student playing the role of teacher for each problem.

Dan H: 36 Questions to Fall in Love (with your students)
Do you have any students that you struggle to like? Do you ever have students who you struggle to like even by the end of the year? The NYTimes article is intended for romantic partners, but Dan discovered that it is a great way to get to know your students in a deeper way. He gave students 5 minutes of quiet writing time at the beginning of class, then 2 minutes to share with a neighbor. He shared his own answer to the question and then collected student responses to read later. By the end of the year he was truly sorry to see his students go!

Seth B: Team Based Learning
Seth's PreCalculus classes have students work in groups of four, groups that are assigned in September and they stay in for the entire year. He'd read research stating that social experiences help with retention and collaboration helps with understand so he thought he would hit both of those by doing group work. Students are assigned group pre-tests and group problem sets. Then anyone in the group can be called on to present to the class. They spend time explicitly talking about how to function in a group and students find their groove when they stay with the same people, to the point that they were aghast when he asked if they wanted to change groups mid-year. Students give each other a grade on how much they contributed to the problem set in an interesting way - each student has 100 points to allot to their three team members in any way they see fit. The most balanced groups collaborate on this aspect as well so each person gets a 34 once!

Karen B: Modeling in PreCalculus
Students do some sort of modeling project every week in this class. At the end of the year instead of an exam, students choose a modeling project. They present their final product to a panel of teachers.

Jennifer F: Activity Builder in Desmos
Jennifer started students on a polygraph then had them explore rational functions by building and describing them. She put two students at each computer which encouraged discussion.

Dianna S: Graphing Videos
With her class of students who are English language learners, who have learning disabilities, and who feed into her Algebra class from several schools, Dianna needs activities that are accessible. She started by showing students http://graphingstories.com/ and then tasked each student with coming up with a unique context for a piecewise linear function (including a horizontal segment). For example, one group had a dog going down the stairs, stop to eat a treat, then continue down the stairs. Interestingly, another group used a student moving around a track - the video shows elliptical motion but the function is linear - a common misconception! To aid students in making quality projects she has each group conference with her after making a plan.

Wendy M: Problem Solving Elective
For juniors and seniors who aren't on the traditional track, Wendy's school offers a problem solving course. She uses the book Crossing the River with Dogs. It's filled with chapters on problem solving strategies and the problems don't necessarily require Algebra or Geometry background. She found her students felt really successful with the problems in this book which was not the case in their past math classes.

Heather K: Interview Grid
Students answer an open question (Always, Sometimes, Never or wodb.ca) and record their answer. Then they interview two classmates before having another opportunity to answer the question (possibly changing their answer based on what they heard from other students). This strategy is adapted from an ELL strategy - Level 1 or 2 students would copy their partner's answer word for word while higher level students would paraphrase or summarize.

Finally, a group of us shared about twittermathcamp.com and ExploreMTBoS.wordpress.com