Thursday, July 30, 2015

Goals for 2015-2016

I find these goal-setting types of posts pretty helpful. Ideally, I will actually go back to this and reread it at some point into the school year. Maybe when we do the #1TMCthing check-in at the end of October? I just reread some of the goals that I had set in previous years and it's pretty hit and miss in terms of how much I actually accomplished. My main concern is that I have too many goals and it's maybe unrealistic that I will be able to accomplish all, but what's that terrible saying that I've seen in a bunch of classrooms that doesn't actually make any sense? Oh yes...


That literally makes no sense. The stars are farther.


Anyway, without further ado, my goals for the 2015-2016 school year!!

Grading/feedback

  • Give back quizzes with feedback only, share the grade after class (we will have an SBG online gradebook this year that will hopefully make that easier), as described in this post by @mythagon.
  • Students must correct original quiz and demonstrate evidence of work/learning done in order to reassess. They may not reassess on the same day. Still debating whether I want to put a limit on the number of attempts. I thought about making all of the attempts count (with a weight making the later attempts count more), but decided that this is not in the spirit of SBG.
  • Teach students how to use feedback effectively. This was from @pegcagle's session at #TMC15 and is my official #1TMCthing that I have publically promised to follow up on this year. One way to do this is to give back feedback mixed up and not attached to assignments and have students in each group try to figure out which feedback should go to which person. Another idea that I want to try is to have students exchange papers and coach each other on what to do with the feedback they received. Last year, I did a bit of peer feedback prior to turning work in to me, with some success. I need to make this a more consistent part of my class.
  • Incorporate homework and classwork into students' self-assessment of their practices. Last year, I asked students to do this for projects, but I would really like a portfolio each unit with self-assessment that is more global and includes homework and classwork with linked examples of their work as evidence. This idea is based on a post by @jacehan.
  • I really need to get homework and projects graded more quickly so that students are getting feedback at a time when it's still useful to them. I got bogged down with grading big time last year, and I need to be better about staying on top of it. Update: after reading "Creating Cultures of Thinking" a few weeks ago, I would like to set up individual meeting times with each of my students outside of class every 2 weeks (we'll see if the schedule supports this) in order to discuss their progress in the class and go over their projects and reflections with them. One of the ideas in the book that I found really fascinating was that instead of thinking about time as the most limiting constraint, as we usually do, we can instead think about energy... what feels energizing and what feels draining. It might make sense to change something that takes less time, but is draining with something that takes longer, but energizes you. The example given was grading papers at home, which can be exchanged for in-person meetings with students with the paper graded in real time and written feedback accompanied by in-person interaction. I would like to try this model, knowing that it will mean trading off some time, but hopefully, will feel less painful than grading projects late at night.
Class culture
  • Continue using daily random groupings and whiteboarding, as described by @AlexOverwijk to increase student participation and engagement. Based on a Twitter conversation with @fnoschese about gender balance in groups, in which he discussed the research that groups in which there are as many or more girls as boys have higher performance outcomes for girls than groups in which there are fewer girls than boys (single gender groups are okay), I will be tweaking the gender ratios in my random groups to help make them more balanced, if needed. But not always. It depends. Basically, it's on my radar, but I'm not 100% sure that gender always trumps other status issues and I really do believe in the overall benefit of visibly random groups.
  • Continue my policy of having students volunteer to participate in class discussions, with the caveat that each person must participate at least once or they must start the next day's discussion. In addition, when groups report out, I can call on any member of the group.
  • Introduce talking points and exploratory talk ideas into class discussions, as described by @cheesemonkeysf here and as described in a similar Visible Thinking routine called Micro Lab. Teach language of argumentation and mathematical discourse, as described in the Claim-Support-Question Visible Thinking routine.
  • Continue assigning bi-weekly reflections as a way for students to reflect on their learning and also to build community and feelings of connection. Move from reflections that are only about learning and affect to reflections that also dig deeper into mathematical concepts and connections. Incorporate reflection questions into daily homework or exit tickets to have more formal processing opportunities and feedback on thinking routines and classroom structures and how they are impacting student learning. 
  • Build student-student connections to strengthen class culture. I really enjoyed the activities that @sophgermain demonstrated in her session on restorative justice, which facilitate student-student connections in the classroom. Some examples:
    • There are two circles of students facing each other, a question is asked and each person speaks for 1 minute on that topic (examples of questions and topics below), then one circle rotates to a new partner and another question/topic is asked.
    • Making one circle for the class and popcorn or going around the whole circle sharing on a particular topic or to give appreciation to someone.
    • Have students write their names on a piece of paper, then distribute randomly and each student must write something nice anonymously about the person whose paper they received. Can repeat this multiple times and then return to the original person.
    • Ask students to share at the end of a task who was helpful to their learning and how. 


Homework
  • Be more intentional about homework. I blogged about this here already, but I'd like to assign fewer homework problems, spiral it in more intentional ways, and always provide answers in advance and worked solutions from students' own work after we discuss homework in class. Homework will be organized into three sections: Review (questions/problems relating to old material), Reflect (processing questions relating to new material, connections between content), and Reach (deeper/harder questions from mostly old material).
  • From @bowmanimal's post on homework, I'm going to ask students to give themselves feedback on their homework and emphasize it as a learning tool. I normally tell students to limit themselves to 45 minutes per assignment and then provide a day every 2 weeks that is a designated catch-up day when they are expected to go back to old assignments that were not finished and put more work into them. I will do this more explicitly this year.
  • I am still playing around with how I want to handle homework questions. I tried using a Google form, as described by @z_cress, last year, and it was a bit hit or miss. Quite a few students simply didn't do it and of those that did, many filled it out too late for it to be useful in my planning of the next day. In @pegcagle's workshop, she discussed using entrance tickets as a way to assess which homework questions are worthy of discussion and to anchor that day's learning, with questions like, "Which homework problem was the hardest for you? Which homework problem was the most interesting?" I definitely like giving groups a few minutes to discuss homework questions together and asking students to present solutions to problems that many are wondering about or that are especially important, but am still working on how to do this efficiently and in a way that maximizes everyone's learning. I definitely treat homework as a vital part of class, with questions that bring out connections between topics and that preview new content or skills that will be useful in that day's learning.
  • I will continue having students turn in pictures of their homework digitally while keeping an organized notebook. I really liked that students had access to all of their work and that I didn't have to track papers last year.
Other
  • I would like to emphasize organization a bit more. Ideally (if I can make myself do this consistently), I would like students to create a table of contents at the front and number pages in their notebook. I was reminded of Magdalene Lampert's structures for student notebook writing in @sgnagni's post describing the sections that she had students create (Date, Problem of the Day, Experiments, and Reasoning). For my high school students, I am thinking something like:
    • Date
    • Questions being investigated
    • Tasks/Mathematical Work
    • Summary and Reasons
  • I would like to continue giving students 5 or so minutes at the end of each class to organize their written work and complete the summary section of their notes. I started doing this halfway through the year last year and students felt that this was very helpful in solidifying their learning, especially if much of their work had been done on whiteboards and they wanted a record of their thinking for that day.
  • I will spend more time planning tasks in anticipating student responses and how I will treat them. I did too much of this on the fly this past year, and while much of the time, it went okay, I am starting to see the benefit of spending more time on advance planning. I would also like to provide rubrics for all projects in advance. I did this sometimes and always got positive feedback when I was able to do it.

Wednesday, July 29, 2015

#TMC15 Takeaways


This is the first year of TMC that I felt like part of the crowd and not a fangirl on the sidelines.

I still had to push myself out of my default setting of wanting to be alone. I hugged a bunch of people, hung out with my virtual-Kansas-math-department, and asked a friend to lunch that I really wanted to get to know better. But as soon as it was lunch time, I literally had to stop myself from running to my car and going to eat alone. And when people talked to me, I asked questions instead of just answering theirs and walking away. It was a stretch.

I went to Huntington Beach with Amy and her Kansas people but the other nights I spent alone on purpose. I have to have alone time to recharge. Also I love to shop and haven't found anyone who can hang with me yet. And I like to eat bad food and stay up late so I'm kind of strange like that.






This is also the first year of TMC that I left feeling like I had a lot of practical, low-risk, high impact changes I could make right away that didn't involve redoing my entire curriculum (although I still feel that way).

So this year my TMC post is not touchy-feel-warm-fuzzy (although I experienced many of those moments) but practical.

Here are some of things I plan to do this year:

  • High Fives- Glenn Waddell high-fived every student every day this past school year. He said it was one of the best years he's ever had and he attributed a lot of it to the high-fives; it built a culture of trust quickly by having fun and laughing together. How do you high-five someone and not smile? I need to work on my mood and attitude in the classroom and this is a great way to start.
  • Music Cues- Matt Vaudrey claims you can save 23 hours a year by using music cues for transitions in the classroom. I hate to say type this aloud but I'm not all obsessed with music like a lot of people are. I mean I like it and I listen to it some but I never listen to it in the car, or well I don't know when I do. But music is incredibly important to teenagers and using it for cues can save me time and voice while also connecting better with the students.
  • "Ask Me Questions!"- This comes from Rachel Kernodle and helps build the expectation that I WANT them to ask questions. Christopher Danielson had another suggestion of saying "What new questions can you ask?" Kate Nowak also used "Would you explain your knowledge of his/her solution?" All of these are replacements for the classic "Any questions?" which I vow to never utter again! Chris Shore also used brain stickers to reward students for good questions. Rachel challenged us to take note of two things: What's the best question *I* asked today and what's the best question a *student* asked today?
  • "That's Not a Choice"- From my #fawncrush, this is a way to set and enforce boundaries and structure that both I and my students crave. It's also way better than just saying no and helps to refocus students to what the choices actually are.It keeps me focused on the things that are within my control so that I never give up on taking action.
  • "Shut It Down!"- I just finished watching 30 Rock so this is a Liz Lemon classic that I have the perfect tone of voice and facial expression to deliver with enough fear to consider it a classroom management tool. (Along with 'What the what?', 'Blergh!', "High-fiving a million angels', 'I want to go to there', and 'Dealbreaker!')
  • Show Your Thinking- Students are so used to hearing 'Show your work' that they just tune that right out. Asking them to show their thinking makes me feel like that opens up more room for students to express their thinking other than calculations. Inspired by this tweet:
  • #onegoodthing blog- I'm challenging myself  to post one good thing every school day this year as another way to keep myself focused on positive things and being happy in the classroom. Last year a lot of weirdbadcrazy things happened and I'm attributing that to a "sixth year slump" and being proactive about making this year better.
  • 180blog- Megan wrote a helpful blog post about automatically importing Instagram photos with a certain hashtag to a blog post. I'd like to also post a photo a day for this year too. I know, I know, I'm probably over-committing myself but the worst that can happen is that I don't do it. 
  • Error Analysis- I am a big fan of Andrew Stadel and I used his estimation180.com once a week this whole year. He talked about presenting concepts by giving wrong answers (that are common misconceptions) and having students try to decide the correct answers. He used exponents as an example and it seemed so elegant. I'm really intrigued to try this with more concepts like multiplying polynomials, solving equations, and logarithms.
  • Which One Doesn't Belong?- I'm using these as a warm-up one day a week this year. They are so simply complex and easy to extend- give three examples and have students create the fourth, have students create their own, have students write a justification for why each doesn't belong. So rich! 
  • Showing Student Work- Now I can remember who I talked with about this (Sadie maybe?) but it never occurred to me to have students show their work to the class on the document camera. What a great skill- to present and defend your thinking. I just thought students would compare with people around them and that would be good enough. False! A simple change that I can easily implement.

Highlights of TMC:

  • Fawn and her telling me that I am one of the first blogs she started following *shock*
  • Being in a session with Andrew Stadel
  • The beach and Fred's with my Kansas people
  • Listening to Sadie speak
  • Watching Alex draw his perfect circle
  • Watching Lisa Henry be appreciated
  • Inside jokes that only us intronerds get
  • Lunch with Rachel
  • PANDA EXPRESS
  • Two Twitter ladies recognizing me in the airport
  • Meeting and following new tweeps
  • Jonathan telling me how my questions helped him fine tune some activities
  • Being remembered by people!
  • Reading funny math t-shirts
  • This awesome #needaredstamp
  • The gorgeous California weather and the super polite people
  • IN-N-OUT!
  • Teresa telling me that she really liked my site and activities
  • This amazeballs new teacher bag I bought
  • My People!








  • Reuniting with my people, feeling accepted and supported, and knowing that feeling will continue through the year

See you next year!

Tuesday, July 28, 2015

#TMC15 Teacher Woman (aka #fawncrush)

Teacher Woman
Fawn Nguyen

TMC15
Saturday Keynote
Claremont, CA

The Five F's

Fast
Fair
Friendly
Firm
Funny

Building relationships trumps content, pedagogy, common core, testing

Rita Pierson Ted Talk- Every Kid Needs a Champion

Relationships with Administrators

  • realize that administrators care the same but show it differently -Glenn Waddell
  • respect the position
  • put things in writing
  • invite them in the classroom
  • never bad mouth admin

Relationships with Colleagues

  • be willing to share ALL the lessons
  • imagine YOUR kid in their class
  • take care of each other
  • speak well of them to students

Relationships with Parents

  • the parent is always right
  • parents are sending us their best
  • do not judge

Relationships with Students

  • be honest with them
  • guarantee a safe environment
  • respect them and get to know them, laugh with them

Fawn's Sayings

  • 'That makes me fart'
  • 'Figure it out and make it happen'
    • Be solution oriented
    • Be a risk taker
    • Be powerful
  • 'Nobody cares'
    • Focus on what matters
    • Giving kids time to talk and do math is more important than the pacing guide
    • Family time is more important than homework time
    • Focus on the positives
    • Focus on YOU, let's stop looking for affirmations
  • 'That's not a choice' (for students)
    • Defining boundaries
    • Teaching respect and tolerance
    • Empowering students
  • 'That's not a choice' (for educators)
    • Providing equal access
    • Giving students our best
It's ALWAYS about the students! 

(insert crying emjois x 100 here)

I've twitter-known Fawn for a while and definitely read her blog- and sometimes not read it just to shield myself from the awesomeness. I used visualpatterns.org once a week this entire school year. And Friday at TMC Fawn-IRL came up to me, knew me IRL name, hugged me, and said it was nice to meet ME! And I said "Nice to meet you.....awkward silence....I don't know what to say." Smooth. But I recovered and asked her about sessions. So I'm already super pumped. On the last day, I ran to get a fangirl picture with her when she tells me that *I* was one of the first blogs she followed. *gasp* So anytime you feel like no one is reading your posts or tweets, just remember, you never know when and where a Fawn might be lurking!

Her keynote was super hilarious. I will let these tweets tell the story:



And then it went so fast to emotional that I didn't even have time to think about Kleenex. It was so touching to see a teacher of over 27 years that is still moved to tears over how much she cares about her students. 

And in that moment, I saw myself in her. 

I will never be bitter or angry. I will never stop caring. I will never stop changing and growing. I will never stop getting better. I will never stop loving my students.

That's what a teacher woman does.


#TMC15 Math Mistakes and Error Analysis: Diamonds in the Rough


Math Mistakes and Error Analysis: Diamonds in the Rough
Andrew Stadel
Estimation180.com/tmc15

Thursday 2:45-3:45
TMC15
Claremont, CA

We need a large window into student thinking.

Turn problems into play by pointing out that making predictions means everyone will be wrong. Who's going to be the least wrong?

Imagine the anxiety you feel when the copier isn't working. How do we clear that jam with our students?

How can we make student mistakes drive instruction, curb student misconceptions, and strengthen formative assessments?

Polygraph by Desmos for teachers; use teacher dashboard to analyze student data. Why did it go well? Why did everyone miss this one? Give students a cheat sheet of vocab words to support them.

Andrew Stadel's Survey: Goo.gl/fOdZKQ

Start teaching exponents by giving eight problems that aren't incorrect and asking students what the answers should be.



Research shows that trying something and then learning about your mistakes increases retention.

We don't need textbooks to give us mistakes, we have students for that.

We can use WWDB to improve students ability to see things that are out of place.

Pause and predict- if we know we're looking for errors, pause and predict what 'should' come next so students aren't just passively watching a video.

What are some other concepts we can teach through error analysis?


#TMC15 Feedback Quizzes


Feedback Quizzes
Using detailed, written formative assessments to promote low-stress learning
Julie Wright
sadarmadillo.blogspot.com

Thursday 4:00-4:30
TMC15
Claremont, CA

It's better to execute a plan, any plan, than to wait and do nothing.

Quizzes are designed with space built in for feedback.

The purpose is to see their thinking and respond; scored for effort; can get EVERYTHING wrong and still get a perfect score; can impose time cutoffs

Alphabetize and scan to PDFs before grading (or have students take pictures of their work and email to you!), on Mac using Preview, you can annotate and add your feedback is text. Color code your feedback for your records even though students won't see it. Reuse your comments!




Give comments for correct answers so those students also have comments to process.  Spend a period revising. Students receive negative criticism better when points are not attached.

Removes shame of making mistakes and takes away the punishing relationship between teacher and student.

You can give hard problems without feeling guilty. 

When you've repeatedly shared all the feedback you can with students, it's on them to perform well on unit tests.


#TMC15 Planning an Assess-Respond-Instruct Cycle

Planning an Assess-Respond-Instruct Cycle
Michelle Naidu

Friday 4:00-5:00
TMC15
Claremont, CA

Decide what content you are responsible for teaching for. Then specifically list the skills  needed to master the content you are responsible for. Next cluster your pre-skills in a way that makes sense.

Use Popplet!




Pre-assessments are one skill per question, one page or less, and organized.




What do we do if they don't know?



What do we do if they do?



A good structure for stations is to use tri-fold science fair boards with pockets. Fill the pockets, easily change content, and fold away for later.

It can pay off to go slow now so you can go fast later.


Modeling tasks: what if there is no third act?

As part of a Modeling with Mathematics workshop that I ran for teachers with @zmill415 a few weeks ago, I had the chance to play with a few different formats for modeling activities and reflect on the types of thinking and representation that were done. We introduced a number of Three Act tasks, which the teachers really enjoyed. Generally, Three Act tasks have a specific question that students are trying to answer by creating a model... it's in the very structure of the format, in which the third act is the "reveal." I've always felt some tension between this format and the idea that the focus should be on process and not solution. Yes, we're valuing different paths and approaches, but ultimately, the one that gets us to the right answer is the one that's going to feel most rewarding for students. So for one of the modeling tasks with which we had teachers engage (a slinky lab in which we investigate the relationship between the weight attached to a slinkie and its length), I specifically did not ask the teachers any questions or set up a conflict that needed resolution. We simply discussed what they noticed and wondered about a slinky and then selected the relationship between weight and length as one that would be investigated in more detail. I was really struck by the richness of the conversations and representations that came out of this activity compared to other modeling tasks we had done together. The goal was to understand and represent the relationship and look for interesting connections between various representations, and the teachers really dug into this question.

Modeling teachers reflecting on their experience with the slinky lab

More views of their whiteboards


Obviously, I'm not saying that we should be throwing away the Three Act framework. Clearly, tasks with a fun and engaging hook that sets up a conflict or ones where a student might engage quickly with a guess are popular with students and provide excellent "needs" that prompt the development and application of mathematical tools. But, I do think that there is a place for tasks or activities in which the goal is just to tinker and think about how something works, where there is no resolution or ultimate reveal. I also make the bold claim that thinking of modeling only or primarily within the Three Act structure does a disservice to students in its focus on getting the right answer, especially if the purpose is to more closely mimic how actual mathematics is done.

I was recently reading a biography of Terry Tao, a leading mathematician and Fields medalist, and was struck by his description of doing mathematics:

From "The Singular Mind of Terry Tao" by Gareth Cook for the New York Times

His analogy to doing mathematics as being similar to being a jazz musician really struck me, as well as the notion that mathematicians are not handed problems to solve. I don't disregard the ability of Three Act tasks to hook and engage students, but I do hope that there is balance with the types of activities and tasks we ask students to do, and that there is also inclusion of more open investigations, an opening of our students' minds to curiosity and wonder about how things work and how we might describe and understand them better using mathematics, not just because it might give us correct answers, but because it's interesting and engaging to try to know and understand something beautiful.

Tuesday, July 21, 2015

Teacher Binder 2015

Each year, one of the highlights of my summer is printing off my Teacher Binder for the new year!  I love the crisp new, blank pages and filling in the dates of meetings, etc.

My Teacher Binder is my everything... it is my calendar for both my school and personal life, meetings, my lesson plan book, my gradebook, my to-do lists, and my notes for everything!  If I lose this planner, I am totally sunk!  I know a lot of people like to use digital resources for planning out their lives, but I get too scattered for that.  I need the tactile-ness of writing it into my agenda, having it on my desk at all times and seeing what all I have going on that day/week/month.


(For previous posts about my Teacher Binder:  2012-13, 2013-14, 2014-15)

To be honest, I was so pleased with the 2014-15 binder, that I just updated the dates and the subjects I'll be teaching, so if you are interested in the details or downloading my binder, please see the 2014-15 post above!

I did add a pacing calendar to the front of the binder, before my lesson plans.  I haven't starting using this year, but I really like the way it turned out:


But the BEST change this year was the binding!  In the past, I've tried several things, with varying degrees of success.  I love the look of the spiral binding that you can have done at Staples, but since I use my Teacher Binder for my gradebook, that wasn't an option since I wanted to be able to add pages easily.  I've tried 3-ring binders, but those don't allow you to fold them back for easy note-taking or for a smaller foot-print on your desk.  I've used the Mead Flex Binder, which is supposedly a mix between spiral and 3-ring, but pages did not turn easily and it was annoying to add gradebook pages, one ring at a time.  So, what to do?!?!?

This year, while having my Teacher Binder printed at Staples (I wanted thicker paper than I had at home), I was browsing around and found the ARC system.  The ARC system is a disc-bound system that I think I will really like!  Staples had a cute poly black & white cover for $2.49 and I got a set of 1" disc for $1.99.  The most expensive part of the system is the hole punch and my Staples was out (and I didn't want to drive across town to the other ones), so I ordered a hole punch on Amazon.  So far, I really love it! The pages turn smoothly on the discs and it is a breeze to add, remove, and move around pages in the binder.  I'm curious to see how well this system wears over the year, especially in terms of taking my binder back and forth from home to school.

Have you used the ARC system?  What are your thoughts?

#SummerList Update

Almost two months ago, at the end of May, I blogged about my #SummerList.  Sadly, with less than one month still to go, I can't say that I've accomplished much from the list, other than most of the books have been read. :)  I find it very interesting to see how my #SummerList actually changes once Summer is officially here.

Books:
  • What's Math Got to Do with It by Jo Boaler - I bought this book over Memorial Day weekend and read it at the beginning of summer.  This is also going to be our next #EduRead book, so feel free to join in!  My big take-aways from this book include:
    • Making mistakes helps our brains to grow.  I need to incorporate more error analysis in my classroom and use those mistakes to provide feedback and growth.  I would like to learn how to use "My Favorite No" with AP Stat as well as more student grading using the AP Rubrics.
    • Students can grasp high-level concepts, but not if they are given low-level work.  I think I often suffer from the "Curse of Knowledge" and try to rescue my students before they really have the chance to grapple with the concepts.  After 15 years, I know where struggles will happen and I have structured some lessons to help students over those hurdles.  However, in trying to help, I think I have coddled too much and created a learned helplessness situation.
    • Talking and discussing mathematics with your peers is vital.  I totally agree with this and my students are often asked to discuss their methods.  However, I need to also add in more written explanations and reflections.
    • Self assessment and Peer assessment are tools that need to be used more.  I am going to try more self-assessment this year (see previous blog post) and would like to use the AP rubrics more this year for peer assessment. I want to use strategies like "Two Stars & a Wish" to provide students with peer feedback. (By the way, if you have a great method to organize all the paperwork that comes with AP Rubrics, please let me know! :)
  • Never Work Harder than Your Students by Robyn Jackson - We read this book in our Twitter Book Club several years ago, but I felt the need to read it again this summer.  It is a great book with lots of practical advice, although it can be very overwhelming.  
  • How to give Effective Feedback by Susan Brookhart - I have not re-read this one yet
  • Accessible Mathematics by Steve Leinwand - This was a very quick read and gives you 10 practical instructional methods that you can use in your classroom right away.  For me, the biggest take-away from this book was about the daily skills quiz.  This is something I plan to implement this year.
  • Make it Stick by Peter Brown, et all - This book was recommended at Best Practices Night by Daren Starnes and I'm very grateful that he did a brief talk on it!  This book started out strong, but did lag quite a bit for me in the middle.  However, Chapter 8 makes the laggy parts worth it!  The main idea is the science behind learning and what strategies have been proven effective and what has not.  Make It Stick was our #EduRead book this month and I really enjoyed these chats.  I've already blogged about my big take-away's but here are the big ones:
    • Retrieval Practice is necessary!  I plan to create retrieval opportunities through short 10-minute quizzes that spiral through the curriculum, exit tickets on one learning target to help students be more successful on chapter/unit assessments, and opportunities through class to stop what they are doing and answer a question without looking at their notes. It's important to note that retrieval practice can occur without a formal quiz structure... flashcards, Chalk Talk, writing down a list of things you learned (without looking at the text/notes), etc are all forms of retrieval.
    • Mix it up!  When we do massed practice of one skill, it feels effective, however, as any teacher can tell you, when it comes to the final exam and everything is mixed together, kids really struggle!  I need to develop activities that have that mixed practice throughout my course.
    • Reflection is powerful!  I already knew this because I spend a lot of time in reflection.  However, I don't think I ask my students to reflect as much as I should.  I need to explore the use of Learning Logs in my class.
    • Practice like you play and play like you practice!  This is something I did not do well this year.  My students spent a lot of time discussing and justifying their answers verbally, but since the AP exam is written, their practice did not reflect in their "game play".  This year, I want to do more written explanations.
    • You must practice!  One of the things that I really need to stress to my students this year is the importance of practice.  I think too many students look at a review worksheet or practice assessment and think, "Yup! I know that one!", but never actually sit down to practice answering the question.  Then when assessment day (or AP Exam) comes, they struggle to write a complete and concise response.  
  • Rethinking Grading by Cathy Vatterott - This book was the ASCD book of the month and I highly recommend it if you are looking at Standards Based Grading.  The book starts out with some history behind grades and grading, but then in Chapters 3 & 4, she really delves into the nitty-gritty details about how to change your classroom practice to incorporate SBG.  The book doesn't go into all of the theory like Marzano's work, but is a practical guide for teachers to dip their toe into the SBG waters.  
  • Rethinking Homework by Cathy Vatterott - Similar to the book above by the same author, this book also starts out with the history of homework, then gets into more practical strategies.  My favorite chapter was Chapter 4 about effective homework practices.  Homework has been a struggle for my entire teaching career and I don't think I've ever been completely happy with how I've handled it. 

Other Things on my List:
  • Better integration of the Chromebooks (1:1) - Yeah, so this hasn't happened yet.  I still want to explore the Chromebooks for formative assessment (Google Forms, Kahoot, etc), but after seeing my AP scores this year, my main focus will be "back to basics" as listed above.  I like technology but I don't want to integrate tech for the sake of tech.
  • Work on more Free Response writing - Still working out in my head how this is going to look this year :)

Stuff NOT on my List:
  • This was the first summer in many years where I did not travel or attend any conferences in July.  Don't get me wrong, I spent most of June traveling, but July has been pretty low-key.  However, even without traveling, I was able to get a lot done:
    • At the beginning of July, we rented a dumpster in order to do some major deep cleaning of our house.  After 10 days, the dumpster was full, several loads had been taken to Goodwill, and we again had an upstairs that we enjoyed spending time in!
    • Spent time with family and friends, including several days with my sister, who lives on the east coast.
    • House repairs... I really don't like being a home owner most days and this week is one where lots of repair people will be in and out of my house.  Today, the plumber is coming to run a water line to my fridge... so excited about having an ice maker! :)

How is your #SummerList progressing?

Monday, July 20, 2015

Teaching Again

I've taken the last year off to have my son.
It's been quite a ride.
People have been asking me which is easier, parenting or teaching.  Since he's only 11 months old, I don't really know yet what parenting consists of but I can say that this has been the most relaxing year of my adult life.  What does that say about being a teacher?
I was able to take some classes for myself (abstract algebra, graph theory 1 and graph theory 2) and I reconnected with what I love and hate about being a student.  It was really helpful for me to remember what not knowing math feels like and what a different persona I adopt as a student (super quiet, shy and uncertain) vs. who I am as a teacher (gregarious, adventurous, unashamed of making mistakes.)  I spent a lot of time observing the other women in the classes (only about a quarter of the graduate students were women) and how much they participated compared to the men (about 90% of the comments made in class were by men.)  None of the students were black or Latino. I'm still processing how these observations should influence my teaching but for now, it's clear that I need to do more for my female and minority students.  Why don't women participate?  Why don't I participate?  My personal reasons are related to fear that at some point, I will hit a wall mathematically and just won't be able to understand something (even though I've overcome every wall so far), inherent shyness and introvertedness, fear of being wrong, math being so tied to my identity that I don't want to be revealed as a fraud (which I do feel like sometimes.  What right do I have to be telling other people how to do math when I'm unsure I could have pursued math seriously.)  I did have some sexist math teachers.  I never felt encouraged in math.  But these are my reasons.  Does every woman in math share these misgivings?  Or do we all have our own individual insecurities reinforced by our cultural context?  My sample size was really tiny.  And my shyness prevented me from sharing my observations with other women in the class.
Anyway.  I'm going back to teaching.  Algebra 1 and Japanese for next year.  I'm excited and scared to go back but I'm looking forward to catching up with what everyone's been doing on the MTBoS while I've been away.  I hope I can start contributing again and I'm so grateful I have this community to lean on when I'm in need of inspiration, which I always am!  I hope someday I can contribute something useful in exchange for all this community has given me.

Saturday, July 18, 2015

Some Thoughts from Make It Stick

Subtitled... Because Julie Said So :)

On Twitter, we've been reading the book Make It Stick (You can read the archived chat under the Book Chat tab above).  For the past 5 years or so, I've been very focused on assessment and grading and this summer is no exception.  What I really find interesting is how one book will lead to another book (or 3 or 4 books) and they all seem to tie together.

While reading Make It Stick, I've also read Rethinking Homework and Rethinking Grading, both by Cathy Vatterott.  While reading all of these books, I've been trying to think about how to create retrieval practice opportunities that aren't tied to a quiz or quiz-like structure.  I am more certain than ever that I want to use the Multiple Choice Mondays and the weekly skills check, but I've also been thinking about exit tickets and reflection tools as mentioned in Make It Stick.

As a reflection tool at the end of a chapter, I plan to hand out a self-assessment tool with the chapter objectives listed on it and ask students to tape it into their notebook:


Students will self assess for the objectives for that quiz, then do some sort of "brain dump" activity on the rest of the page.  Some ideas for a "brain dump" would be a One-Minute Paper, a Concept Map, etc just to see what they recall about the chapter.

What are some ways that you practice retrieval in your classroom?

The Perniciousness of Negative Numbers: Are Our Children Safe?

What follows is a paper I just wrote for a class called History of Mathematics for Middle School Teachers. The actual name of the paper was "The History of Negative Numbers in Mathematics and Education," but that's not exactly click bait, so I livened it up here. Most of the paper is about how mathematicians sort of pretended they were ignoring negative numbers for a while, then publicly freaked out about how ridiculous they were, then finally came to love them. I'm sure many of us have had an in-law relationship like that, so maybe you can relate. The end of the paper is an appendix with some thoughts about activities for students that reinforce the methods and philosophies described in the rest of the paper.


Why do we believe –5 is a number? To adult math teachers, the question seems silly or startling. We were told about negative numbers before our teenage years, and we’ve seen them used in budgets or economic reports, weather reports of temperatures, elevations below sea level, and, of course, math textbooks. We’ve added, subtracted, multiplied, and divided them just as we have with positive numbers. Why wouldn’t they be numbers? To our middle school students, however, understanding negative numbers and performing arithmetic with them are often far from natural, and in their confusion and skepticism they have much in common with mathematicians from previous centuries, who believed negative numbers were ridiculous or impossible. Considering the history of how mathematicians came to accept and embrace negative numbers provides some guidance for how we can help our students to understand them and work confidently with them.

In past centuries, many mathematicians, like our students, accepted the idea of unitless or abstract positive numbers, although with some limited exceptions, they avoided or openly scorned negative numbers in their methods and solutions. Philip E. B. Jourdain described abstract numbers in The Nature of Mathematics in 1913 [Newman p. 24]:
In arithmetic we use symbols of number. A symbol is any sign for a quantity, which is not the quantity itself. […] When we shake off all idea of “1,” “2,” &c., meaning one, two, &c., of anything in particular […] then the numbers are called abstract numbers.
Early Egyptian, Mesopotamian, and Greek mathematicians came to use abstract positive numbers to varying degrees, but they did not use negative numbers. In about 300 A.D., Diophantus, says Herbert Westren Turnbull in The Great Mathematicians, referred to “the impossible solution of the absurd equation 4 = 4x + 20” [Newman p. 115], although he accepted fractional solutions without difficulty.
Early Chinese and Indian mathematicians began using negative numbers in a limited way, often as tools for intermediate steps of problem solving. Early Chinese mathematicians used negative numbers as coefficients in intermediate steps of solving systems of equations. [Berlinghoff p. 81-82] In the 600s, Brahmagupta of India used and calculated with negative numbers to represent debts, and other Indian mathematicians later continued to use them and develop arithmetic rules for them. In the twelfth century, in his text the Lilavati, Bhaskara II found a negative distance for the position of a triangle’s altitude from a vertex, and he correctly interpreted it as “in the contrary direction,” producing an obtuse triangle. [Mumford, p. 126-127] Still, the Indian mathematicians were somewhat dubious about negative numbers, rejecting them as solutions to quadratic equations, for instance. [Berlinghoff p. 82]
In any case, Chinese and Indian mathematicians’ work with negative numbers did not end up being passed to other cultures. Arabic mathematicians, like Muhammad Ibn Musa Al-Khwarizmi in the 800s and Umar Al-Khayammi (Omar Khayyam) in about 1100 AD, avoided negative numbers in their work on complicated algebraic equations. Al-Khwarizmi expressed algebraic problems in words; Al-Khayammi placed each term on the side of the equation for which the coefficient would be positive. The limitations in how they wrote equations hindered them from seeing all quadratics or cubics as particular examples of a single type of equation. [Berlinghoff p. 82, 109-110]
In Europe from the time of the Renaissance onward, many types of mathematics flourished, including methods to solve increasingly complicated algebraic equations and the development of the coordinate plane. The focus on higher-order equations seems to have been part of why negative numbers seemed so unreasonable to European mathematicians, because if they accepted that negative numbers existed, they also felt obliged to accept taking their roots, and complex and imaginary numbers seemed completely over the top to them. (As David Mumford puts it, “It was because of negatives that square roots had a problem, so maybe it was best to consider them both as second class citizens of the world of numbers. […] The fate of –1 and i were inseparable.” [Mumford p. 140 & 142]) European mathematicians were finding negative numbers increasingly useful, yet they still shied away from embracing them fully. In his work Arithmetica integra in 1544, Michael Stifel represented all quadratic equations as a general form by using negative coefficients, but called them numeri absurdi and would not accept them as solutions. [Hollingdale p. 109] Turnbull says that in the 1500s, Girolamo Cardan (sometimes known as Cardano), “surmised the need” for negative, imaginary, or complex roots to cubic and quartic equations in accord with his ideas of how many roots these equations should have [Newman p. 119], but he seems to have avoided such solutions in his publications.
Before the late 1600s, many mathematicians were also confused about the size of negative numbers and their physical meaning. Mumford attributes European discomfort with negative numbers largely to “the overwhelming importance of Euclid,” with his focus on geometry and positive quantities, in the development of math in Europe. [Mumford p. 140-143] Antoine Arnauld felt, not unreasonably, that if -1 were truly smaller than 1, then it was not reasonable for their ratio to be the same no matter which came first (since this would never happen with positive numbers of different sizes). John Wallis believed that dividing by a negative number was essentially an even more extreme case of dividing by zero, and that therefore all such quotients would yield infinity. [Berlinghoff p. 84] Wallis was, however, a pioneer in the 1600s in using negative coordinates in the coordinate plane, which had been originally developed by Descartes with only positive coordinates. [Berlinghoff p. 140] This alteration allowed Europeans to think about positions of points more flexibly, in a way similar to that used by Bhaskara II centuries earlier in his writing about the obtuse triangle.
In his Treatise on Algebra in 1685, Wallis developed clear explanations of the meaning and laws of arithmetic of negative numbers, despite his confusion about division with them. He explained that multiplying by a negative number could mean taking away that many times. He also described multiplying a negative number by a positive number, and even explained the produce of two negative numbers: “[T]here may well be a Double Deficit as a Double Magnitude; and−2A is as much the Double of –A as+2A is the Double of A. . . But to Multiply –A by −2 is twice to take away a Defect or Negative. Now to take away a Defect is the same as to supply it; and twice to take away the Defect of A is the same as twice to add A or to put 2A .” [Mumford p. 137]

In another breakthrough, Wallis explained and illustrated a number line with negatives on the left and positives on the right, explaining that
 [W]hen it comes to a Physical Application, [a negative number] denotes as Real a Quantity as if the Sign were +; but to be interpreted in a contrary sense. […] [If a man] having Advanced 5 Yards […] thence retreat 8 Yards […] and it then be asked, How much is he Advanced […]: I say –3 Yards […] That is to say, he is advanced 3 Yards less than nothing […] (Which) is but what we should say (in ordinary form of Speech), he is Retreated 3 Yards […] [Mumford p. 138]
Very soon thereafter, Isaac Newton was thinking about negative numbers in a similar way:
[I]n local motion, progression may be called affirmative motion, and regression negative motion; because the first augments, and the other diminishes the length of the way made. And after the same manner in geometry, if a line drawn in a certain way be reckoned for affirmative, then a line drawn the contrary way may be taken for negative. [Mumford p. 138]
It is easy to picture that Newton’s acceptance of negative numbers helped pave the way for his tremendous advances in calculus and physics.
Other scientists of the Enlightenment era in Europe were probably less comfortable than Newton was with negative numbers. The development of temperature scales gives some interesting evidence of a pronounced desire to avoid them. Early scales of the 1700s, including Fahrenheit’s, set 0° temperatures low enough that lab scientists of the time could describe lab temperatures with non-negative numbers; cold weather temperatures were not yet an area of interest. Anders Celsius developed a scale using the endpoints that are still familiar to us, of water’s freezing and boiling temperatures, which he determined precisely through experiments. Negative temperatures on the Celsius scale are, of course, much more likely to occur than with the Fahrenheit and related scales; for instance, the freezing temperature of brine used to set 0° Fahrenheit is negative on the modern Celsius scale. But Celsius decided to avoid negative temperatures for lab scientists by reversing the direction of the scale, making his 100° temperature water’s freezing point, and 0° the boiling point! Scientists used this reversed direction for a few decades before settling on the modern Celsius scale in the mid-1700s. [Beckman]

As for the post-Enlightenment European mathematicians, even after advances like Wallis’s and Newton’s, many remained skeptical of negative numbers for more than a century. In 1843, Augustus De Morgan wrote in his article Negative and Impossible Quantities, “These creations of algebra retained their existence, in the face of the obvious deficiency of rational explanation which characterized every attempt at their theory.” [Mumford p. 113] Philip Jourdain, the twentieth century mathematician mentioned earlier, perceived that negative numbers were useful; he acknowledged that “‘generalisations of number’ and transference of methods to analogous cases” were useful tricks of the trade that had led mathematicians to have “arrived at the truth by a sort of instinct.” Nevertheless, he did not feel that the generalization of numbers to negative numbers had been on a sound logical footing historically:
For centuries mathematicians used “negative” and “positive” numbers […] without any scruple, just as they used fractionary and “irrational” numbers. And when logically-minded men objected to these wrong statements, mathematicians simply ignored them or said: “Go on; faith will come to you.” And the mathematicians were right, and merely could not give correct reasons—or at least always gave wrong ones—for what they did. [Newman p. 25-27]
Jourdain reconciled himself to the logic of negative numbers by concluding that a negative number is the “operation which is the fulfillment of the order: ‘Subtract,’” adding, “Mathematicians call it a ‘number’ […] because of analogy: the same rules for calculation hold […] when ‘addition,’ ‘subtraction,’ &c., are redefined for these operations.” [p. 27-28] He goes on to describe negative numbers on a number line in a way that would be familiar to any of us in 2015, including the idea of +a and –a being placed symmetrically about 0. Later, he summarizes “the essence of algebra” as describing generally the “exceedingly complicated relations in which abstract things stand to one another. The motive for studying such relations,” he continues, “was originally, and still is in many cases, the close analogy of relations between certain things we see, hear, and touch in the world of actuality round us,” but in general, he notes approvingly, “we have reduced the definitions of all ‘numbers’ to logical terms” [p. 65], which is obviously a relief to him. Modern algebraists took similar ideas much further, defining our number system as a ring in ways that require negative numbers to be included as inverse elements for addition. At long last, we have come to a time when mathematicians consider negative numbers on an equal footing with their positive counterparts, and teaching children about them is not only accepted, but encouraged.

Activities for Students

As mathematicians came to accept negative numbers, they were used more freely in various contexts, so they are more familiar to today’s middle school students than they would have been to math students before the 1900s. Debts are still frequently described as a negative amount of money. Now that outdoor temperatures are frequently measured, negative temperatures are common contexts. Negative elevations (locations below sea level) are rare for land, but make sense to students as a context for problems. Students might be familiar with scales that go from a negative to a positive number (for instance, rating mood on a scale of -10 to 10). Finally, and perhaps most importantly for their flexible use of math as they grow up, students are now exposed to a variety of changes described with negative numbers, such as stock market drops or weight loss, and in middle school science they start describing physical changes (in position, for example) in terms of negative numbers as well. These examples provide a rich source of contexts for teachers to help students make sense of negative numbers, and our cultural acceptance and familiarity with negative numbers give our students advantages in learning about them that mathematicians in past centuries did not have. Seeing the contrast between historical and current use of negative numbers in “regular life” helps me better appreciate how essential it is to build upon this familiarity in order to help my students achieve a greater degree of comfort with negative numbers than mathematicians of the past ever experienced. The Illustrative Mathematics tasks “Above and below sea level,” “Comparing temperatures,” and “Bookstore account” would each help students explore negative numbers in a meaningful and familiar context.
Nevertheless, negative numbers will always represent another level of abstraction for our students beyond what they have experienced with positive numbers or even zero, which are easier to “see”. Mathematical advances of the past can guide how we help students with this abstract thinking. Multiplication and division, in particular, often are difficult to understand, and don’t have a clear meaning for many contexts, such as temperatures in degrees Fahrenheit or Celsius. Reading about historical explanations of negative numbers and how mathematicians made sense of them has given me a renewed appreciation for the number line developed by Wallis and currently promoted within the Common Core Mathematical Content Standards. The Illustrative Mathematics tasks “Integers on the number line 1,” “Integers on the number line 2,” “Fractions on the number line,” and “Distances between homes” are representative student work I will consider to help them understand and use number lines.


References

  1. Olaf Beckman, “History of the Celsius temperature scale,” 2001.
  2. William P. Berlinghoff and Fernando Q. Gouvea, Math Through the Ages: A Gentle History for Teachers and Others, Oxton House Publishers, Farmington, Maine, 2002 [1st edition].
  3. Common Core State Standards Initiative: Mathematics Standards. 
  4. Stuart Hollingdale, Makers of Mathematics, Penguin Books, London, 1989.
  5. David Mumford, “What’s So Baffling About Negative Numbers? — a Cross-Cultural Comparison.”
  6. James R. Newman, The World of Mathematics (vol. 1), Simon & Schuster, New York, 1956.