Sunday, May 7, 2017

Contemplate then Calculate: Progression of Systems Tasks

After playing with the Contemplate then Calculate routine a little bit at the beginning of the year, a workshop with Grace Kelemanik and Amy Lucenta inspired me to really commit to planning a series of CthenC tasks as part of my Systems of Equations unit. I started by looking at New Visions’ Algebra I systems tasks, which made me decide that my focus would be on using structure to solve systems represented visually, with the hope that they would translate those structural moves to solving systems algebraically. So I sought out a whole bunch of visual systems and thought about how I would solve them, which led me to the following list of big ideas.

Big Ideas:

  • Any time you can get an equation with one variable, you can find the value of the variable. You can make an equation with one variable…
    •  through substitution of either a value or another variable
    • through elimination when you have two equations that have all but one of the variables the same
  • The equations given to you don’t always lend themselves to creating equations with one variable. You can create new equivalent equations by…
    • scaling an equation
    • adding two equations together
    • finding the difference between two equations
    • adding/subtracting the same thing from each side of the equation

Phase 1 of the Unit

 I have spent the first two weeks of my unit on understanding what a system of linear equations is, understanding what a solution is, solving systems graphically, introducing standard form linear equations, and revisiting slope-intercept form equations. During this time, I have been using the CthenC routine with a variety of algebraic systems represented visually, with the goal ofing prompt most (if not all) of the big ideas above. I wanted each task to be able to be solved more than one way, so I intentionally gave more equations than the bare minimum to solve. Here's the progression of tasks I have made so far:

Task #1 (Most basic substitution and elimination, see what they do):

Potential noticings that lead to structural thinking:

  • We noticed that the second and third equations were almost the same, but the second equation had one more star and a greater value. Therefore, we found the difference between the two equations and since 20 – 16 = 4, the red star has to equal 4.
  • We noticed that two stars were equal to a square, so we replaced the square in the 3rd equation with two stars. Then we had an equation with all stars, so we could divide the 16 equally between the 4 stars.
  • We noticed that two stars were equal to a square, so we replaced the square in the 2rd equation with two stars. Then we had an equation with all stars, so we could divide the 20 equally between the 5 stars.
  • We noticed that two squares equal a star, so we replaced the two stars in the 3rd equation with a square. Then we had an equation with all squares, so we could divide the 16 equally between the two squares.

Annotations of actual student strategies:

Task #2 (have a couple of different elimination options):

Potential noticings that lead to structural thinking:

  • We noticed that the last equation had one more triangle than the first equation/We noticed that the first and last equations both have a chunk of two hearts and a square
  • We noticed that second equation has two more triangles than the first equation/We noticed that the first and second equations both have a chunk of two hearts and a square
  • We noticed that the second equation has one more triangle than the third equation/We noticed that the second and third equations both have a chunk of two hearts and a square

Annotations of actual student strategies:

Task #3 (keep an elimination possibilities, prompt substitution again):

Potential noticings that lead to structural thinking:

  • We noticed that a pen costs $3 more than a notebook, so in the 2nd equation we can replace the pen with $3 and a notebook
  • We noticed that the third equation has one notebook and one more pen than the second equation/We noticed that the second and third equations both have a chunk of two notebooks and a pen

Annotations of actual student strategies:

Task #4 (substitution hopefully more appealing than elimination):

Potential noticings that lead to structural thinking:

  • We noticed that a slice of cake costs $2 more than an ice cream cone, so we can replace the slice of cake in the 1st equation with $2 and an ice cream cone.
  • We noticed that the 2nd equation has 2 more pieces of cake than the first equation and that the items in the second equation cost $7 more than the ones in the first equation/ We noticed that the first and second equations both have 3 ice cream cones and a piece of cake

Phase 2 of the Unit (up next):

After the first two weeks, we go into solving systems algebraically using elimination and then using substitution. I am planning for my CthenC tasks to have only one method for solving and for that method to directly link to the algebraic manipulations students will be doing that day or the next day.

Wednesday, April 19, 2017

NCTM 2017: Day 1

Where We Disagree on the SMPsRaymond Johnson

  • Group of Algebra I teachers met over the course of  two years where they individually identified which math practice(s) were encouraged by different IM tasks, and then talked about where they disagreed
  • Over time their agreement increased—this process a potentially powerful tool for coming to common understanding of the math practices
  • Are some math practices more likely to occur together in a single task? Here was the correlations for this group

6 x 2/3 or 2/3 x 6: Using Structure & Precision to Build Understanding of Fraction Multiplication – Ryan Casey

  • Structure across the table: each row goes from iterating --> partitioning --> associative property --> distributive property
  • “Students must be taught structure explicitly”—planning and being aware of it yourself isn’t enough, but annotations can be used to help students look for and make use of structure
  • Because students are already in the mode of partitioning/iterating, 2 1/3 x 15 is easier for students than 2/3 x 15 (and 16÷3 is easier to evaluate than 2÷3)


Let’s Be Detectives: The Search for Rules, Patterns, And Understanding with SMP 7 & 8 in the Early YearsSusan Looney

  • The number line is already too abstract for some students. A beaded number line is more concrete and really helps the students see and understand what value mean. Can go from beaded number line (concrete) --> number line (pictorial) --> computations (abstract)
  • “Structure: When I look at 7 is there a doubles fact hiding in there? What do I notice about all of these numbers?”
  • “Repeated Reasoning: Does this always work? And why or why not? Are there patterns to the way we say and write our numbers?”

Get Strategic: A Thoughtful Progression of Addition  & Subtraction Strategies – Susan Jensen

  • Computation Strategy: purposeful manipulations that may be chosen for specific problems, may not have a fixed order, and may be aimed at converting one problem into another vs. Computation Algorithm: a set of predefined steps applicable to a class of problems that gives the correct result in every case when the steps are carried out correctly
  • Activities to do with kids: counting center (estimate the # of something then put it into ten frames), 10-frame build it (flash a quick image and then have students recreate on 10 frame, use some sort of interesting progression ex: all have a value of 6), shake and spill (5 red/yellow counters in a cup, shake and spill, how many red and how many yellow?), macaroni squeeze (10 noodles in a bag, line in the middle, what are all the possible combinations)
  • Progressions Document for single digit addition and subtraction

Early Number Operations: Important Understandings for All K-2 Students – Kathleen Lynch-Davis and Chrystal Dean

  • 65-36 alternative algorithms: subtracting in parts, counting on by 10s, and compensation
  • Disrupting people’s conceptions about standard algorithms: when dividing a fraction by another fraction, it does work to divide the numerators to get the numerator of the quotient and divide the denominators to get the denominator of the quotient (ex: 6/21 ÷ 2/3 = 3/7)

Seeing Students Who HideCathy Yenca

  • In regards to only a subset of students participating, went from feeling indifferent --> insulted --> inspired (maybe not a linear progression)
  • How can you access the students who hide?
    • Ask everyone (anonymize and project answers)
    • Include everyone (everyone works a problem and puts it up)
    • Make insulting moments inspiring
    • Let them talk
    • Let them create
  • Students care what their peers think and they compare what they think to their peers

Mathematical Heart – Anarupa Ganguly

  • K-12 50/50 math achievement M/F, but post-secondary 75/25 or worse
  • Maybe this is because we are missing an explicit effort to humanize math, so try:
    • Pose a pursue questions that catalyze emotional resonance (not just about the past and present, but future)
    • Empower our students to make and explore conjectures

Math Play: A Few ThoughtsKassia Omohundro Wedekind

  • Play is characterized by diminished consciousness of self, improvisational potential, continuation desire
  • Goals of play: ownership and identity in math

The Art of Mathematical Anthropology  - Geoff Krall

  • Growth mindset tells us that effort leads to ability, but it’s hard for students to see that when they are always being compared to a changing standard. Students need an opportunity to zoom out and see how they have grown over time
  • Students can see this through the following iterative process: assign complex tasks that produce complex work, written reflection, conversation

Sunday, January 29, 2017

The Great-Granddaughter of Immigrants

This is an excerpt from Julia Holmes’ 100 New Yorkers, describing my great-grandfather, Kong Chow Chun, who was a shop owner and community leader in early- to mid-19th-century NYC Chinatown.

As a merchant, Chun was a Section Six exception to the 1882 Exclusion Act, which banned all Chinese immigrants—save merchants, scholars, and students—from entering the United States, though even Section Six immigrants could not apply for US citizenship, making the Chinese the first nationality barred by law from becoming U.S. citizens… Chun devoted his retirement years to activism: he fought for the removal of immigration quotas, for the right of Chinese immigrants to attend New York public universities, and for the posting of street and subway signs in Chinatown in both English and Chinese. (pg. 51)

It is in honor of my great-grandfather, and all of my students, that I say that we as teachers become part of the problem if we carry on with teaching as if it can be completely separate from the current presidential administration’s discrimination and the resistance against it. Teaching math can be an act of social justice itself (see this rubric for culturally responsive math teaching), but first and foremost I teach students. I’m not yet sure what it looks like in the Trump administration for teachers and students to resist, practice self-care, learn productively, and maintain a safe and inclusive community. But as I head to school tomorrow, and each day after that, I know that I am going to start with conversations with both adults and students. People’s hopes, concerns, fears, and anger are going to affect our learning environment no matter what.  Power and strength come from supporting each other and acting together rather than being isolated and silent.

Wednesday, November 16, 2016

Teaching Through Tasks

So at this point I am 35 days into my first (and longest) unit, focusing on linear relationships. The overall goal of this unit is for students to have a deep understanding of slope and y-intercept and be able to build, represent, and compare general rules for linear relationships. So far, this has been my most successful attempt at structuring a whole unit around period-long tasks. Generally speaking, I have used tasks for two purposes-- to have students surface informal reasoning that lends itself to both the concepts and procedures that are part of the unit (which I generally do as a whiteboard task routine) and to continue to develop or apply a concept/procedure that we've started to formalize (which I generally do with a refine your strategy routine). Following a class-long task, we analyze and apply one or more strategies that were generated as students worked on the task. In between tasks, we formalize some of the work they've been doing, introduce vocabulary associated with it, and practice the associated procedure(s).

Here's the sequence of tasks we've done so far:

Day 4 - Refine Your Strategy
Day 5 - Follow-Up

Day 9 - Whiteboard Task (adapted from Mathalicious)

Day 10 - Refine Your Strategy (adapted from Mathalicious)

Day 18 - Whiteboard Task

Day 19 - Follow-Up

Day 26 - Whiteboard Task (Adapted from Mathalicious)
Day 27 - Follow Up

Day 30 - Refine Your Strategy

Day 35 - Refine Your Strategy (Adapted from the Shell Centre)

But now, with about 10 days to wrap up the unit, students are in very different places. I would say that at least half are pretty comfortable with identifying and understanding slope and y-intercept (in context) and writing equations with understanding for linear relationships. This has been a way of learning that really works for them. A smaller, but still substantial group of students, are pretty frustrated and confused. This group includes many, but not all, of my beginning English language learners and students with learning disabilities. Clearly, this is a big problem. So here are some things that I am thinking about trying in order to make sure that all of my students are learning:
  • Putting more structures in place to help students listen and learn from each other
  • Push our formalization of student-generated strategies farther (step-by-step directions?)
  • Use more targeted questions (than what did they do? how are these strategies similar/different) when analyzing worked examples
  • Modeling strategies more explicitly
  • Pausing in the middle of working on a problem for me to teach a new skill, rather than always waiting for one or more groups to figure out that skill, and then analyze it at the end/the next day
What have other people found to be successful?

Saturday, October 22, 2016

Year Four Goals

I wrote this back in August, but got taken over by preparations for school and then school actually starting. Here they are now, though

1) Something compelling every class
  • Compelling questions
  • Compelling problems
  • Compelling structure 
  • Longer instructional routines
    • Refine a Strategy
    • Whiteboard Task
    • Creating Homework Mistakes
    • Desmos Activity
    • Contemplate then Calculate
  • Shorter routines
    • WODB
    • Visual Patterns
    • Estimation 180
    • Graphing Stories  
3) Transparency and student feedback around my teaching actions

Monday, September 5, 2016

Notes from the Field: Doing Time in Education

This weekend I went to see Notes from the Field: Doing Time in Education. This is Anna Deavere Smith’s latest one-woman show, which focuses on the school-to-prison pipeline. In the first act and coda, Smith takes on the persona and words of many people she has interviewed, and in the second act the audience members are split into groups in order to have facilitated conversations about what they have seen so far. I found the performance incredibly powerful and thought provoking and I strongly recommend it to any educators (or people) in the Boston area.

My school year with students starts on Thursday and my focus for the last two weeks has been taken over by logistical details and setting up my classroom—covering bulletin boards, mounting whiteboards, arranging desks, unpacking all of my stuff from the closet, and scrubbing everything. However, going to this play reminded me that while everything I just listed is necessary in order to be ready for school, it is the bigger picture that I truly need to ground myself in before getting started. What am I doing to make sure that students are empowered and not marginalized in my classroom and school? When am I prioritizing compliance and how can I find an alternative? How will I make sure my students’ voices are heard and respected? What am I doing to get to know and support my kids as people, not just math students? What will I do when I see injustices committed against my students? How will I respond when I am part of enacting an injustice? What’s my location on the school-to-prison pipeline?

This last question is one that we were all asked to reflect on at the end of the play’s act two breakout group. My answer? I was never going to be sent to prison from school, nor were any of my friends or family. I grew up with that privilege. However, I am now part of a system that enables the school-to-prison pipeline, which means that I also have the power to disrupt it.

When several people shared out their location on the school-to-prison pipeline, one member of our group expressed a concern. She was concerned that people would come to this play, feel the catharsis that theater is intended to elicit, encourage other people to come, but have that be the end of the experience. She pointed out that us talking today was important, but that it needed to be followed by action. The American Repertory Theatre, where I saw the play, seems to be trying to address that concern. They sent everyone who attended a list of ways to get involved in Boston. I left still trying to figure out what actions I will take. But as I start this school year, I am committing to becoming more aware of and trying to change the actions of mine that contribute to the school-to-prison pipeline and to speaking up about the injustices that I see/that are brought to me.

Monday, August 29, 2016

Instructional Routines

So I have become a little bit obsessed with instructional routines.  This is because I think they have great value to both teachers and students:

Benefits for students
  • knowing what to expect makes kids feel more comfortable and safe 
  • having a routine process enables kids to focus more on the math ideas and less about figuring out what the directions are
Benefits for teachers
  • planning is faster: can choose a routine that fits goal and then just slot in the particular problem
  • collaborating is easier: having commonalities in practice give a more narrow lens for focus

Inspired by Contemplate then Calculate, I am planning on using 9 routines in my classroom this year. Some of these are greatly inspired by others and some of them I developed. For each routine, I have a powerpoint template than I can adapt for each time I use the routine. For some of the routines, I have accompanying handouts that I will use, no modification necessary, each time we do the routine. All of these materials and a more in-depth description of each routine can be found in this google folder.

Longer Routines:

o   Goal: Build structural thinking in order for students to solve efficiently based on understanding rather than blindly following a procedure. Need tasks that have multiple shortcuts.
·      Desmos Activity
o   Goal: Various math goals. Also using a guiding question and technology to learn math and for me to talk as little as possible.
·      Homework Mistakes
o   Goal: Have students learn from each other in order to improve their understanding of the homework problems. Build the positive culture of mistakes in our classroom.
·      Refine A Strategy
o   Goal: To build students’ capacity to solve problems on their own through scaffolding by engaging in others’ ideas. This is for tasks where I expect students to already have a pretty well-developed strategy or strategies to solve the problem.
·      Whiteboard Task (4 versions, depending on launch choice)
o   Goal: For students to work collaboratively on a task that they probably couldn’t solve on their own. This is for tasks where I am not expecting them to have well-developed strategies, but instead to deepen or extend their conceptual understanding in order to develop new strategies. Also using Peter Liljedahl’s visible random groupings and vertical non-permanent surfaces to disrupt institutional norms.

Shorter Routines:

·      Estimation 180
o   What: Estimate a quantity
o   Why: Think like mathematicians by determining relevant information and a reasonable
·      Graphing Stories
o   What: Make a graph to match a video story
o   Why: Think like a mathematician by using math to represent something happening in the real world
·      Visual Patterns
o   What: Describe, continue, and generalize a pattern
o   Why: Think like a mathematician by looking at the structure of a pattern and using repeated reasoning
·      WODB
o   What: Figure out why each one doesn’t belong
o   Why: To “think like mathematicians” by comparing similarities and differences