Friday, August 4, 2017

Clothesline Math - Relational Thinking


So I’ve dabbled in the idea of clothesline math (using a clothesline as an open number line) before--read a lot of blog posts about it, done a square root clothesline activity with my students, and experienced it at a couple of workshops-- but Chris Shore’s TMC session got me excited about it in a new way. Although this was partially because of his charisma, and the enthusiasm of people around me, I think ultimately it was because I had some mathematical realizations that I didn’t anticipate. 

So I like that clothesline math is hands-on, gets kids out of their seats, and feels different than a lot of the math that we do in my classroom. But I want to investigate a little bit more in what it has to offer mathematically. In thinking about geometry, we started with this problem:

With benchmarks of 0 and 180, we are able to place x clothespinned to a and b & c clothespinned to each other. A & x were the same distance from 0 as b & c were from 180, and we moved x, there we a line of symmetry at 90 degrees. This was cool, but the realization that b & c needed to be x units away from 180 degrees wasn’t mind-blowing for me. We are essentially taking the equation x + b = 180 and then turning it into b = 180 – x. Certainly an important understanding for kids, but something that could be figured out algebraically on paper and have almost the same weight as on the clothesline.

But then we got to this clothesline prompt:

At first, I thought ok, fine. This was going to be basically the same. 2x + 9 and 3x – 6 would be clothespinned together, and 7a + 1 as their reflection over the “line” at 90 degrees. But then Chris took out 2x and 3x to be placed after 2x + 9 and 3x – 6, and that’s when I started to get excited.

Let’s say that kids are given the equation 2x + 9 = 3x – 6 to solve. Here’s one possibility of what I might expect from my eighth grade students (they might deal with the variables first, or choose to eliminate different terms, but something along these lines).


At best, this demonstrates an understanding of how to isolate a variable. In my class we talk about what “moves” we can make to simplify an equation—in this case we’re looking for terms that we can “turn into 0’s” using the additive inverse and coefficients we can “scale to 1’s” using the multiplicative inverse.

And yet, with essentially the same prompt on the clothesline (we have 2x + 9 equivalent to 3x – 6), a whole new world of thinking opens up. Here’s a representation on paper of the thinking that my group did in placing 3x, 2x, and then x onto the clothesline.


In this progression of thought we had to do a lot of relational thinking, which I would argue is absent in the solving of the equation above. 2x + 9 and 3x – 6 is somewhere between 0 and 180 degrees—more specifically, it’s definitely between 2x and 3x. And in fact, the value of 2x + 9 and 3x – 6 is 9 more than 2x and 6 less than 3x. And then this is the amazing part. That distance is 15 and it’s also x. I want to think more about how to prompt this realization (either through a series of clothesline tasks that build this thinking before we get to something this complicated, or in the moment with this take), because this type of relational thinking isn’t something I’ve given my students a lot of experience with. And then once we know x is 15 degrees, then we can find the value of 2x, 3x, and the two original expressions.

Placing these expressions and solving on the open number line is dependent on the idea that all expressions we deal with represent a value (the location on the number line) that can be seen in relation to other expressions (the distances on the number line). This is something that we spend a lot of time on with numerical values in elementary, but that I haven’t leveraged very much with my students when they start to deal with algebraic expressions and equations. So it’s for this reason that I’m sold on doing some algebraic work with expressions and equations on the clothesline this year.

Some other clothesline math tips from Chris:
  • Have benchmarks on another color to establish scale, but only put the number of benchmarks necessary to communicate your idea
  • Have blank cards of another color for any other values or expressions that might need to be added
  • First place the cards in the proper order, then proportionally space
  • Students can give a thumb up or down for correct placement, then point the direction it should be moved, and clap when they think the card is in the correct place
  • In order to have whole class engagement, students participate at their desks on whiteboards while 1 group is at the board. Then record final answer on paper when done with discussion, deductions, and decisions
  • If kids estimate the value of a variable, can put an equivalent (approximate) value on with a clothespin
  • Equivalency is better on the single line using a clothespin to attach the equivalent expressions. Save a double number line for rates

Thursday, August 3, 2017

Morning Session: MPs and Equity


For my morning session at Twitter Math Camp (TMC), I participated in What is the Relationship Between Standards for Mathematical Practice and Equity? (all materials here), which was led by Grace Chen, Brette Garner, and Sammie Marshall. I really appreciated there being a designated space to talk about equity at TMC. This also became a space for me where I felt comfortable in taking risks and getting personal—the trust that we built was essential for having real conversation.

Talking Points

One of the ways that I felt like our facilitators built trust was by starting with talking points on the following prompts. If you are unfamiliar with this structure, check out Elizabeth Statmore’s post on talking points here.

The talking points structure and the statements themselves were a great way for us to get to know each other and build some of that trust. These are common statements in the education world, but with talking points, we were able to each give our own response to them and then revise/extend our thinking. By the end of each round, my group had a majority of disagrees and sometimes a couple unsures. I particularly want to call out the statement “math is math is math, regardless of race or culture or context.” I think that this philosophy is so harmful in classrooms because it is used to erase the identities of the students within the classroom.

Equity Eyes

Dylan Kane wrote about this in more detail here, but another part of our work each day was to develop and practice a way of viewing other TMC sessions, teaching decisions, or classroom structures through “equity eyes.” We started by looking at the following sets of group roles:

I found examining these group roles fascinating, particularly since I have used a version of the ones on the left. When I first looked at them, the main difference that jumped out at me was what type of actions they emphasized students doing. I saw the roles on the left as focusing on process and having the groupwork run smoothly. I saw the roles on the right as focusing on the type of thinking we are expecting of students—representing math multiple ways, asking questions in order to extend thinking, everyone understanding, and making connections. Someone else, I believe it was Brette, pointed out that based on different goals, each of these sets of group roles might be more valuable. If you have a group that’s really good at going deep, they might need more reminders about actually recording their thinking. If you have a group that works really well together, but often stays at a surface level, the roles on the right might be a better fit.

From this, my first set of “equity eyes” questions emerged:
  • What are we expecting all kids to do? What type of thinking/actions?
  • How do we support each kid in reaching those goals?
  • Does this promote a hierarchy or collaboration (or some other interpersonal dynamic)?
Over the next two days, based on what other people shared, I added to this list:
  • Do I have the trust of each learner in the classroom? Do I trust each learner in the classroom?
  • Who is valued? How are they valued?
  • What assumptions are we making about students?
  • Whose voices are we hearing?
  • Where is race/culture in this thing?
  • Who am I thinking about when I am planning?

 

Equitable Mathematics Teaching Practices

In January’s issue of NCTM’s Journal for Reasearch in Mathaematics, many authors put out an article called Toward a Framework for Research Linking Equitable Teaching With the Standards for Mathematical Practice. Here are the practices:
  1. Draw on students’ funds of knowledge
  2. Establish classroom norms for participation
  3. Position students as capable
  4. Monitor how students position each other
  5. Attend explicitly to race and culture
  6. Recognize multiple forms of discourse and language as a resource
  7. Press for academic success
  8. Attend to students mathematical thinking
  9. Support development of a sociopolitical disposition
We spent some time trying to understand what these practices meant and give some examples of how to do them. I have chosen the 5th practice to be my #1tmcthing. I will be thinking more about what this will look like for me and my students in our classrooms before the beginning of the year.

I got a chance to start thinking about this more when we responded to the following prompt “…our pedagogical responsibility as math teachers, not just to our students but to society at large…”,  and then silently sent our paper around the circle for people to read and respond.


Other Take-Aways

Here are some things that I want to remember from the rich discussions that we had in this session:
  • Listen to learn, not to respond (I'm sorry, I don't remember who offered up this norm)
  • Questioning and problematizing can lead to lack of action, so we need to be okay with making mistakes and slowly trying to get better and more automatic in making decisions from an equity lens. (--Grace)
  • When someone says something problematic—“I want you to think about what other people are hearing when you say this” (--Glenn)
  • Racism as a codified set of double standards (source: Racecraft)
  • Specifically in the context of “accents,” the burden of understanding is put on the speaker instead of the listener. How can we make sure that the listener shares some of this responsibility as well? (--Grace)
  • Depending on our own personal identify, there are some equity issues that are more personal or hit closer to home for us than others. These are the ones that we may be playing “defense” on. But there may be some issues that we can play “offense” on and we can share and distribute that responsibility within a community (--Grace)
  • Go beyond “we want diversity because it’s important/we value it” to examine why diversity is important to us (is it optics, missing perspectives, not wanting to contribute to a problem, building a microcosm, etc.) (--Marian)
  • When inviting new people into a community, we want to and must embrace the things they can’t offer us, as well as the things they can (--Grace)

Wednesday, August 2, 2017

Teaching is Political


Yesterday, I re-watched Grace Chen’s keynote that she gave last week at Twitter Math Camp (TMC). If you haven’t watched it yet, check it out here and here—it is totally worth 50 minutes of your day. Her talk left me almost in tears at TMC, so I wanted to watch to take it all in again and to try to figure out what about the talk I found so powerful. After re-watching (and re-playing to transcribe so many moments that I wanted a written record of), I was about to sit down to write this post. However, I was soon interrupted when a non-math-teacher friend of mine showed up for dinner. So instead of writing, I tried to describe to my friend what Grace’s talk was about and its impact on me.

I told her that it was about how teaching is necessarily political. That there are many narratives and that the people who are in power choose the ones that are told over and over again. That as individuals we shouldn’t just accept these stories, these policies, these ways of living as the way it is. Instead we need to be making conscious and communicable choices about what narratives we are perpetuating. Even as I was saying this, I was frustrated, because I felt like I wasn’t conveying the power of Grace’s words.

So today, I want to step back, and try again to figure out why this talk made me feel a sense of clarity, understanding, hope, shame, frustration, belonging, pride, purpose, overwhelm, power, relief, and camaraderie.

1. It’s Personal

One reason was its deeply personal nature.

In the context of her father giving up his Taiwanese citizenship and emigrating to the U.S., Grace told us that she sometimes asks people what would have to happen for them to give up their American citizenship. She then shared with us her own answer:  
“I think there’s not much that would convince me to give up my American citizenship because as screwed as this country is and as terrible of decisions I think our government has made over years and years and years, it’s still a pretty incredible place and it has the potential to be the beacon of freedom my dad thought it was. And this is my home.” 
I had the privilege of growing up in a town in MA that did a lot of questioning and criticism of America—both its history and its current policies.  Along with the need to search for the counter-narratives and erased narratives, this also taught me to have a sense of shame and embarrassment about being an American. I appreciate Grace’s example here of being able to “hold multiple truths simultaneously.” America is screwed up and she has hope for its future and it is her home. This is a more complex view than the one that I grew up with, but it is also one that seems more true/illustrative of the whole story and more empowering to live with.

Grace also became extremely personal with her message by telling the story of her grandmothers and the political history of Taiwan in how it relates to her and her family. Grace told us that:
All history is political history because what gets told depends on who is in power, what gets allowed to be told, and what gets whispered about behind closed doors. That doesn’t mean any history is more or less true than another, but to function in this reality we have to learn to hold multiple truths simultaneously. But that doesn’t mean every truth is equally valid.” 
This is not a new concept to me. However, it meant so much more to me when it was being said in the context of the different Taiwanese political history stories she shared with us. So when Grace then said:
The reason I don’t believe that story [that the history of Taiwan starts in 1949] is because I think it ignores, disrespects, and dehumanizes the lives and labor of millions of people over century. And I believe that because I see myself, my family, and people like me being ignored and [something I can’t hear]. But whether you believe my story or the 1949 story, is a choice that people make, either by default or deliberately, based on their values and experiences. I argue that it is our ethical responsibility to make those choices consciously, rather than defaulting to whatever we hear loudest, or whatever we hear more of, or whatever we hear people in power say, and to be able to communicate the reasoning and values behind our choices.”  
she made me care about her, her family, and people like her. And that further motivates me to make conscious choices about what stories to believe and value.

And then she also made it personal for me in thinking about my students and classroom. I am a biracial Asian/white woman who came from an upper middle class background teaching predominantly black and Latino students, many of whom are living in poverty. 
Like my grandmothers, my students weren’t poor and hungry because they didn’t work hard, or because they had no ambitions, or because they made poor choices. They weren’t poor and hungry because of “cultural priorities,” a term that is often used to mask deficit assumptions about people who are not white and middle class… But the story that I am telling you instead, is that like my grandmothers, my students were poor and hungry for political reasons. This kind of story is a counter-story, or counter-narrative because it confronts and disrupts the prevailing view.”  
This is the story that I want my friends, my family, the mainstream media, and really everyone to value and believe. And one thing that I am taking from this talk is that it is part of my job to tell this story to people who might not otherwise hear it or to people who might unconsciously be choosing to believe one of the other narratives about why my students are poor (or more generally why people are poor in America).  And I need to do a better job in researching and articulating exactly what those political reasons are.


2. It’s a Math Classroom

I think the other reason that Grace’s talk was so powerful and made me feel so many things is because she talked about these ideas in the context of a classroom, and specifically a math classroom.

Here was one of Grace’s big points:


There’s a prevailing narrative that math is not influenced by people, cultural contexts, governmental policies, or anything else. There’s this idea that math has some sort of purity outside of people and that “math is math is math.” And yet here’s what Grace said to a room full of math teachers:
 “Likewise, my students’ stories didn’t start when they entered my classroom, or when they opened a math textbook, or even when they enrolled school. Their stories started generations ago, also influenced by people, cultural context, and governmental policies."
"Who you are as a math teacher doesn’t just start when you wake up and drive to school, or when you plan a lesson, or in your teacher prep program. Who you are as a teacher started generations ago, shaped by the people who raised you, who inspired you, the people who challenged you, people who may have tried to hold you back. Your story is shaped by cultural context, what foods and languages and mannerisms you consider normal and what you take for granted. And your story is shaped by governmental policies that control education, employment, housing, access to resources. But who you are is not wholly determined by all of that. And that not wholly determined is where my hopefulness lies.”  
And so this makes me want to better examine how the choices I am making in the classroom (consciously and unconsciously) are influenced by my experiences and history and what I can do differently than how I have been conditioned to think and act. It also makes me want to think about what are the narratives my students have been told about themselves and math and what are the counter-narratives I want us to explore together. And how can I invite and take advantage of their histories and stories in our math classroom.

And this is a conversation that I haven’t found there’s really space for or is a priority in the education world as a whole. I have pockets of people who I can talk with about how education, and math in particular, is political and what implications that has on us. But certainly I haven’t done this in an organized or sanctioned way. And so I want to finish by saying that I’m hoping to continue the conversation on these prompts that Grace gave us—with people at my school, with other educator friends in my district, and with educators friends online.

  1. Create a Microcosm
    “One of my academic heroes, Kevin Kumashiro, writes that stereotypes aren’t harmful just because they are untrue.  Even though they do generalize, they do make assumptions, they do erase complex and nuanced stories, they are harmful because every time they are repeated they recall and recite history [or oppression].” 
     
    I have made a place to start doing some of this work here.
     
  2.  Teach to Gray Areas
    “But when we teach the power of mathematics, we can also teach its limitations—what it cannot understand or interpret or predict on its own.”
  3. Explore Alternatives
    “Politics is a way of valuing our way of finger counting over their way of finger counting. And exploring the idea of alternate mathematics helps us recognize that what we take for granted is just one of multiple, possible stories.” 
    “The version of U.S. school mathematics that emphasizes universal, and generalizable, and abstract, I think of it almost as an analogy to the 1949 story of Taiwanese political history. Or maybe it’s the 1895 version. It’s hard to tell whether your history of mathematics is complete or if it’s true or something you want to stand behind, until you take the time to consider other possibilities.”
  • Where else have you seen different ways of doing mathematics, and what values do they communicate?
  • What do these ways of thinking mathematically offer?

And so I will just finish (this post, I am committed to continuing to think and act) with a couple more quotes from Grace

“So when I say that our choices should be conscious and communicable about the stories we are telling and the policies we are living out, our mathematics should be conscious and communicable as well. On top of that, I would encourage us to think about the values that underlie our conscious, and communicable mathematics.”
“The mathematics that we value depends on our context, and I think one thing we can do in our classrooms is to be really explicit about that when we can. Not to not teach it, but to teach the alternative, to teach different ways of thinking, and to say that in this context this makes a lot of sense, in this other context this might make more sense."

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?