Teacher Learning Laboratory 2021 Round Up

My lab had another great year, despite the chaos of the pandemic. We had a wide-range of publications from several projects that wrapped up recently. We explored issues of teacher learning, of course, but also issues of identity and math learning, instructional coaching, and more. Below, I am including journal articles, chapters, as well as some podcast episodes. Without further ado, here is the roundup: Grace A, Chen Samantha A. Marshall, and Ilana S. Horn. “‘How do I choose?’: mathematics teachers’ sensemaking about pedagogical responsibility.” Pedagogy, Culture & Society 29, no. 3 (2021): 379-396. Teachers’ decisions are often undergirded by their sense of pedagogical responsibility: whom and what they feel beholden to. However, research on teacher sensemaking has rarely examined how teachers reason about their pedagogical responsibilities. The study analyzed an emotional conversation among urban mathematics teachers about what they teach mathematics for, given the many non-mathematical challenges they and their students face. The familiarity and simplicity of love and life skills narratives deployed to describe what it means to be a good teacher and to do good teaching may be comforting, but limit teachers’ engagement with other authentic forms of pedagogical reasoning about their pedagogical responsibility in complex sociopolitical contexts. The findings reveal the importance of opportunities to explore alternate possibilities ‘for what,’ especially within structured and supportive teacher collaborative groups. Lara Jasien & Melissa Gresalfi (2021) The role of participatory identity in learners’ hybridization of activity across contexts, Journal of the Learning Sciences, 30:4-5, 676-706. Background: We explore how school-based mathematical experiences shape out-of-school mathematical experiences, developing the idea that learners hybridize norms and practices around authority and evaluation across these two contexts. To situate our study, we build on constructs of participatory identity and framing. Methods: Drawing from a large corpus of video records capturing children’s point-of-view, we present a case study of hybridization with two purposively sampled 12-year-old friends—Aimee and Dia—interacting in an out-of-school mathematics playspace. We use interaction analysis to articulate grounded theories of hybridization. Findings: We present a thick description of how children hybridize their activity in out-of-school spaces and how such hybridization is consequential for engagement. Dia’s case illustrates how traditional norms and practices around authority and evaluation can lead to uncertainty and dissatisfaction, while Aimee’s illustrates how playful norms and practices can lead to exploration and pleasure in making. We argue that their school-based mathematics experiences and identities influenced these differences. Contribution: This report strengthens theoretical and methodological tools for understanding how activity and identity development in one context become relevant and shape activity in another by connecting analytic constructs of identity, framing, and hybridizing. Samantha A. Marshall, and Patricia M. Buenrostro. “What Makes Mathematics Teacher Coaching Effective? A Call for a Justice-Oriented Perspective.” Journal of Teacher Education, vol. 72, no. 5, Nov. 2021, pp. 594–60. Mathematics teacher coaching is a promising but largely overlooked form of professional development (PD) for supporting mathematics teachers’ learning of justice-oriented teaching. In this article, we critically review the literature to illuminate what we currently know about mathematics teacher coaching and to highlight studies’ contributions and limitations to inform future work. Broadly, we find that four programs of research have developed, investigating: (a) coaches’ activities and relationships, (b) the effects of coaching on student assessment scores, (c) the effects of coaching on teachers’ practices or behaviors, and (d) the effects of coaching on teachers’ knowledge or beliefs. From this analysis, we argue that justice-oriented perspectives of teaching, in tandem with sociocultural theories of teachers’ learning, could allow for more nuanced investigations of coaching and could support design of learning experiences for teachers that bring us closer to educational justice. Ilana Seidel Horn and Melissa Gresalfi. “Broadening Participation in Mathematical Inquiry: A Problem of Instructional Design.” In R.G. Duncan and C.A. Chinn (Eds.) International Handbook of Inquiry and Learning. Routledge. Cultural myths about mathematics as a set of known facts pose unique obstacles for inquiry instruction. What is there to discover if everything is already known? At the same time, decades of mathematics education research shows the potential for inquiry instruction to broaden participation in the discipline. Taking a classroom ecology perspective, this chapter uncovers common obstacles to inquiry in school mathematics and identifies three leverage points for redesigning instruction toward this goal. These include: teachers’ knowledge for inquiry mathematics, curricular connections to other contexts, and classroom norms and practices. The chapter proposes that design thinking around these leverage points holds promise for wider-spread implementation of inquiry instruction in mathematics classrooms. Emma Gargroetzi, Ilana Seidel Horn, Rosa Chavez and Sunghwan Byun. “Institution-Identities in the Neoliberal Era: Challenging Differential Opportunities for Mathematics Learning.” In J. Langer-Osuna and N. Shah (Eds.) Making Visible the Invisible: The Promise and Challenges of Identity Research in Mathematics National Council of Teachers of Mathematics. Schools exert powerful forces on people’s lives. As society’s formal setting for learning, schools-or, more precisely, the people in authority there-certify the learning of the next generation. Contradictions between learning and the bureaucratized systems of schooling are particularly keen in mathematics classrooms, where students are constantly subjected to tools that measure, rank, sort, and label them and their learning. The use of technical instruments as the tools of measurement gives results a veneer of scientific truth such that shifting life trajectories get both rationalized and made invisible. We refer to the mathematical identities that come from such processes as institution-identities (Gee, 2000), exploring how policy language makes available and naturalizes certain positions for students within schools. In other words, we examine how policy language and practices shape and constrain possibilities for young people’s mathematical identities in school-based interactions. All four authors of this chapter taught in U.S. schools. As such, we all have been actors in processes that took full, complex human beings and sorted, labeled, and set them on different paths. In doing so, we co-constructed students’ mathematical institution-identities, giving credence to (or shedding doubt on) stories about their capabilities and future possibilities. In this chapter, we use thickly described examples from four research projects to examine and illuminate how policy language and practices shape and constrain possibilities for young people’s mathematical identities in school-based interactions. On the basis of this analysis, we develop a theory of how policies and neoliberal logics operate together to provide institution-identities that become consequential in children’s mathematical identities and learning. We argue that mathematics educators concerned with issues of access, equity, and inclusion should attend to institution-identities rooted in neoliberal policies that naturalize processes contributing to social stratification. We furthermore demonstrate that policy and its enactment can serve as a site for research into the discursive nature of mathematical identities. Rebuilding after 2020-2021 on the Human Restoration Project Podcast In this conversation, we discuss how teachers can wrap up the 2020-2021 school year through reflection. How can we build a better system after seeing the inequities, problems, and challenges that this school year has highlighted? And, how do we build a classroom in spite of a system that often demotivates and disenfranchises educators? Motivated” Summer Readaloud Series on the Heinemann Podcast Motivated is a guidebook for teachers unsatisfied with questions met by silence. By examining what works in other classrooms and following the example of been-there teachers, you’ll start changing slumped shoulders and blank stares into energetic, engaged learners. In this preview, Ilana digs into some common teaching strategies and explores the “how” and “why” behind them. ––––––––– Our lab has a lot more in store for you –– more articles coming out in Educational Researcher, Journal of Research in Mathematics Education, and Review of Educational Research, just to name a few. We are probably most excited about the monograph we have coming out this spring, Teacher Learning about Ambitious and Equitable Mathematics Instruction: A Sociocultural Approach. Authored by me and Brette Garner, my whole Project SIGMa team contributed to individual chapters. We are really looking forward to conversations about these ideas in the coming months and beyond.

Supporting Instructional Growth in Mathematics (Project SIGMa)

Good news to share: another research grant has been funded by the National Science Foundation. Yay!

For this project, my research team and I will be working with Math for America in Los Angeles to design a video-based coaching method for their Master Teacher Fellow program.

sigma logo

This is what we pitched to the NSF:

This study addresses the need to develop processes for adequate and timely feedback to inform mathematics teachers’ instructional improvement goals. In this study, we propose using design-based implementation research to develop and investigate a process for documenting mathematics teachers’ instruction in a way that is close to classroom practice and contributes to teachers’ ongoing pedagogical sense making. The practical contribution will be a framework for formative feedback for mathematics teachers’ learning in and from practice. The intellectual contribution will be a theory of mathematics teachers’ learning, as they move from typical to more ambitious forms of teaching in the context of urban secondary schools. Both the practical and theoretical products can inform the design of professional development and boost other instructional improvement efforts.

In a recent Spencer study, my team and I investigated how teachers used standardized test data to inform their instruction. (That team was Mollie Appelgate, Jason Brasel, Brette Garner, Britnie Kane, and Jonee Wilson.)

Part of the theory of accountability policies like No Child Left Behind is that students fail to learn because teachers do not always know what they know. By providing teachers with better information, teachers can adjust instruction and reach more students. There are a few ways we saw that theory break down. First, the standardized test data did not always come back to teachers in a timely fashion. It doesn’t really help teachers adjust  instruction when the information arrives in September about students they taught last May. Second, the standardized test data took a lot of translation to apply to what teachers did in their classroom. Most of the time, teachers used data to identify frequently challenging topics and simply re-taught them. So students got basically the same instruction again, instead of instruction that had been modified to address central misunderstandings. We called this “more of the same,” which is not synonymous with better instruction. Finally, there were a lot of issues of alignment. Part of how schools and districts addressed the first problem on this list was by giving interim assessments –– basically mini versions of year end tests. Often, the instruments were designed in-house and thus not psychometrically validated, so they may have not always measured what they purported to measure. Other times, districts bought off-the-shelf interim assessments whose items had been developed in the traditional (and more expensive) manner. However, these tests seldom aligned to the curriculum. You can read the synopsis here.

Accountability theory’s central idea  ––  giving teachers feedback –– seemed important. We saw where that version broke down, so we wanted to figure out a way to give feedback that was closer to what happens in the classroom and doesn’t require so much translation to improve instruction. Data-informed action is a good idea, we just wanted to think about better kinds of data. We plan to use a dual video coaching system — yet to be developed — to help teachers make sharper interpretations of what is happening in their classrooms.

Why did we partner MfA LA? When I reviewed the literature on teachers’ professional learning, they seemed to be hitting all the marks of what we know to be effective professional development. They focus on content knowledge; organize their work around materials that can be used in the classroom; focus on specific instructional practices; they have a coherent and multifaceted professional development program; and they garner the support of teacher communities. Despite hitting all of these marks, the program knows it can do more to support teachers.

This is where I, as a researcher, get to make conjectures. I looked at the professional development literature and compared it to what we know about teacher learning. MfA may hit all the marks in the PD literature, but when we look at what we know about learning, we can start to see some gaps.

*Conjecture 1 Professional learning activities need to address teachers’ existing concepts about and practices for teaching.

 

Conjecture 2 Professional learning activities need to align with teachers’ personal goals for their learning.

 

Conjecture 3 Professional learning activities need to draw on knowledge of accomplished teaching.

 

*Conjecture 4 Professional learning activities need to respond to issues that come up in teachers’ ongoing instruction

 

*Conjecture 5 Professional learning activities need to provide adequate and timely feedback on teachers’ attempts to improve their instructional practice to support their ongoing efforts.

 

Conjecture 6 Professional learning activities should provide teachers with a community of like-minded colleagues to learn with and garner support from as they work through the challenges inevitable in transformative learning.

 

*Conjecture 7 Professional learning activities should provide teachers with rich images of their own classroom teaching.

 

The conjectures with * are the ones we will use to design our two camera coaching method.

We need to work out the details (that’s the research!) but  teacher’s instruction will be recorded with two cameras, one to capture their perspective on significant teaching moments and a second to capture an entire class session. The first self-archiving, point-of-view camera will be mounted on the teacher’s head. When the teacher decides that a moment of classroom discourse illustrates work toward her learning goal, she will press a button on a remote worn around her wrist that will archive video of that interaction, starting 30 seconds prior to her noticing the event. (As weird as it sounds, it has been used successfully by Elizabeth Dyer and Miriam Sherin!)  The act of archiving encodes the moment as significant and worthy of reflection. For example, if a teacher’s learning goal is to incorporate the CCSSM practice of justification into her classroom discourse, she will archive moments that she thinks illustrate her efforts to get students to justify their reasoning. Simultaneously, a second tablet-based camera would record the entire class session using Swivl®. Swivl® is a capture app installed in the tablet. It works with a robot tripod and tracks the teacher as she moves around the room, allowing for a teacher-centered recording of the whole class session. Extending the prior example, the tablet-based recording will allow project team members to review the class session to identify moments where the teacher might support students’ justifying their reasoning but did not do so. The second recording also captures the overall lesson, capturing some of the lesson tone and classroom dynamics that are a critical context for the archived interactions. Through a discussion and comparison of what the teachers capture and what the project team notices, teachers will receive feedback on their work toward their learning goals. We will design this coaching system to address the starred conjectures in the table

Anyway, I am super excited about this project. I am working with amazing graduate students: Grace Chen, Brette Garner, and Samantha Marshall. Plus, my partners at MfA LA: Darryl Yong and Pam Mason.

I will keep you posted!

 

 

 

Renegotiating Classroom Treaties

Many classrooms are governed by tacitly negotiated treaties. That is, students trade in their compliance and cooperation –– student behaviors that alleviate the challenges of crowded classrooms ––  for minimal demands for engagement by the teacher. When I have worked with teachers trying out open-ended tasks for the first time, I will often hear about “pushback” or “resistance” from the students: “I tried using this activity but the kids balked. They complained the whole time and refused to engage.”

These student responses indicate that teachers are violating their part of the treaty by going beyond minimal demands for engagement and increasing intellectual press. Put differently, by using an open-ended task, teachers raise the social risk, leaving students open to judgment since they can not rely on the usual rituals of math class to hide their uncertainty. Treaties may, as their name suggests, keep the peace, but they reflect norms of minimal engagement that interfere with deeper learning.

In my own observations, I see teachers struggle to move students past their initial reluctance to participate and make it clear that active involvement is required in their classrooms. Renegotiating classroom treaties requires a clear vision for what student participation can look like, structures to support that vision, along with the determination to see it through. The teachers I interviewed for my forthcoming book all emphasize how critical the first days are for setting these expectations for their students, particularly since their expectations may differ from what students are used to in math class. “It’s entirely intentional that I begin setting norms and structures on the first day of school,” Fawn explains. By launching the new school year by showing students what it means to do math in her class, Fawn renegotiates the classroom treaty through norms and structures, introducing the Visual Pattern and other discourse routines from the start. She says, “I need to provide students with ample opportunities to experience the culture that we have set up. We need to establish and maintain a culture that’s safe for sharing and discussing mathematics, safe for making mistakes, and a culture that honors each person’s right to contribute. There needs to be a firm belief among everyone that mathematics is a vital social endeavor. Building this culture takes time.”
Starting the school year with clear expectations is important, but guiding individual students’ participation is an ongoing project. The teachers I interviewed have numerous strategies for monitoring and building positive participation throughout the year. Students students who hide or students who dominate make for uneven participation. The teachers describe how they contend with these inevitable situations.
When figuring out how to respond to quiet students, the teachers try to understand the nature of students’ limited participation. Not all quiet students are quiet for the same reasons. At times, quietness is rooted in temperament: some students inclined to hang back until they feel confident about what is going on, but they are tracking everything in class. These students do not contribute frequently, but, when they do, their contributions add a lot to conversations. This kind of quiet is less of a concern and can even be acknowledged: “Raymond, you don’t talk a lot, but when you do, I always love hearing what you have to say.”
Other times, quietness signals students’ lack confidence. That is, students indicate some understanding in their work or small group conversations, but they do not have the confidence to participate in public conversations. With these students, the teachers seek out individual conversations. Chris calls these doorway talks, while Peg calls them sidebars. (“Trying to deal with calculators and rulers at the end of class, I couldn’t make it to the doorway!” Peg tells me when I note the different names.) “I might say to a kid, ‘You know, you had really good ideas today, and I would have loved to have heard more of them in the conversation we had a the end. I think you have a lot more to contribute than you give yourself credit for.’” Sometimes, there are ways of encouraging good ideas to become public that do not directly address the student. Chris explains that he might say something like, “I haven’t heard from this corner of the room.” He then asks other students to hold their ideas while waiting for a contribution from the quiet group.

Of course, some students are quiet because they really do not know what is going on. This could be due to a language issue, in which case, the teacher needs to modify instruction to give them more access to the ideas. If there are other learning issues going on, this might suggest the need to check in with colleagues about the students performance in previous years or in other subjects.

eager-students
Talkative students pose another kind of challenge to the expectation that everyone participates.  On the one hand, they can provide wonderful models of sharing their thinking. They can be the “brave volunteers” who explore their thinking publicly, and teachers can lean on them to get conversations started. On the other hand, they can be domineering, making it difficult for other students to get a word in. The quiet students who temperamentally need to think before they speak have their counterparts in some talkative students: these are the students who think by talking. Asking for their silence sometimes gets heard as asking them not to think. When I have had students like that in my own classes, I make sure to assure them that I value their engagement but that I need them to find other strategies for processing so that other students can be heard. Sometimes, students with impaired executive functioning, like those with ADD, have a hard time with the turn-taking aspect of classroom dialogue, so not only do they talk a lot sometimes, they struggle to take turns. Again, teachers can respond by valuing students’ ideas while helping them participate more effectively: “I know you get excited, but we need to take turns so that we can hear each other.” Finally, domineering behavior can get expressed through a lot of talking: students who are highly confident in their understanding and want to explain to others. Teachers need to judge the extent to which this is altruistic, a sense of trying to share knowledge, and the extent to which it shuts conversations down. In the first case, students can be coached towards asking questions of their classmates, channeling their impulse to talk into something constructive. In the second case, the dominance can be corrosive to the classroom culture and the students might need stronger redirection. For all of this feedback, similar strategies of direct address (via sidebars or doorway talks) and indirect address (“Let’s hear from somebody else”) can help teachers manage participation.

Structure Can Change Agency

One great privilege of the work I do are the many opportunities I get to share the things I care about with different groups of people. If you do it enough, you get a chance to clarify your own ideas, learn from others, and notice connections.

This past weekend, I had the honor to give a keynote talk at the Carnegie Math Pathways Forum. If you don’t know about their work, it is worth checking out. Briefly, their work addresses the enormous blockage in the math pipeline as students transition from secondary to post-secondary. A staggering number of students get placed in developmental math classes, and often, these courses become a holding bin students cannot get out of. The Carnegie folks have worked primarily with community college instructors to re-think developmental math curricularly and pedagogically. It’s fascinating and important work.

My talk was about the relationship between structure and agency, how both contribute to inequalities in mathematics education. When we are teaching in a classroom, it is easy to see problems of inequality as they look locally: high enrollments in developmental math, over-representation of students coming from poverty and students of color, a sense of student apathy. To make progress, however, instructors can learn by linking the local to broader social processes: the maldistribution of qualified math teachers, STEM classrooms that are hostile environments to minoritized students, a K-12 curriculum that often reflects the institution of schooling more than what it means to do meaningful mathematics. I argued that if we frame these problems through what we see locally, we give ourselves, as teachers, less leverage to make progress on them. I shared two key concepts for linking these social processes to what we see in our classrooms: social risk and status. I have written about both of these (click the links if you are curious), but briefly, social risk refers to the threats people feel are posed to their status in a community while status describes the perception of students’ academic capability and social desirability. Both of these ideas link the social process explanations for inequality to what teachers see in their classrooms locally.

Teachers can then work to design classrooms that reduce social risk by, in part, attending to status dynamics. In other words, to connect structure and agency, we need ways to think across scale and look at the social origins of problems too often narrated as individual issues. Instead of, for example, blaming students for being apathetic about mathematics learning, we need to recognize what their history has likely been in our current system and accept their apparent apathy as a reasonable response. Our task shifts from finger pointing (“My students just aren’t motivated!“) to having the productive challenge of honoring their experience while trying to change their ideas about math and learning.

In the end, then, structure can help us change agency in two ways. First, by recognizing that it is there, along with the social processes it holds in place, we can arrive at more productive framings of the problems we face locally. Second, we can leverage the structural designs in our classroom to invite students’ agency.

I have written about designing structures to promote agency before. If you don’t feel like reading that (I realize it’s summer!), maybe watch this video instead. It is quite a joy.

And don’t we all need more of that right now?

 

Why Meaningful Math Learning Matters

What Meaningfulness Means

Learning and schooling are not the same thing. There are children who are great learners but terrible students. These young people are full of ideas and questions, but they have not managed to connect their innate curiosity with their experiences in school. There are many possible reasons for this. Children may find school to be a hard place to inhabit, due to invisible expectations that leave them feeling alienated. Sometimes, school curriculum just seems irrelevant: their personal questions about the world do not find inroads in the work they are asked to do.
Although many parenting books extol children’s natural curiosity and emphasize its importance in their learning and development, schooling too often emphasizes compliance over curiosity. Thus, it is not surprising that children who are great learners and weak students have their antithesis: children who are great students but who are less invested in learning and sense making. Make no mistake: these students hit every mark of good organization, compliance, diligence, and timely work production, but they do not seek deep engagement with ideas. Given the freedom to develop a question or explore an idea, they balk and ask for more explicit directions. I have heard teachers refer to these children as “teacher-dependent.”
Too often, meaningfulness falls through this gap between learning and schooling. There is a fundamental contradiction at play: meaningfulness arises from and connects to children’s curiosity, yet “curious children” is not entirely synonymous with “successful students.” Meaningfulness comes about when students develop an appreciation for mathematical ideas. Rich and meaningful learning happens when students draw on prior knowledge and experiences to make sense of ideas and explore problems, invoke their own strategies, get to ask “what if…?”  In short, meaningful learning happens when students’ activity connects to their own curiosity. To make meaningfulness central to math teaching, then, teachers need to narrow the gap between being curious and being a good student.

___________________________________________________

Meaningfulness: When students connect their own curiosity and experience to ideas, thereby developing an interest in and appreciation for mathematical content.
___________________________________________________

Why Meaningfulness Matters

Every math teacher, at one one time or another, has been asked the question, “When are we going to use this?” While this question often gets cast as students’ resistance to learning, it can be productively reinterpreted as a plea for meaningfulness. When the hidden curriculum of math class –– the messages that are inadvertently relayed through classroom organization and activity –– consistently communicates that meaning does not matter, we end up with hordes of students who no longer reason when they are doing math. They instead focus on rituals, such as following the worked example, and cues, such as applying the last learned procedure to the current problem.
As researcher Sheila Tobias explained in her classic exploration of math anxiety, a lack of meaning exacerbates many students’ negative experiences learning mathematics. When math class emphasizes rituals and cues that rely on memorization over sense making, students’ own interpretations become worthless.

For instance, they memorize multiplication facts, and, in a search for meaning, they decide that multiplication makes things bigger. Then, they learn how to multiply numbers between 0 and 1. Their prior understanding of multiplication no longer works, so they might settle on the idea that mulitiplication intensifies numbers since it makes these fractional quantities even smaller. Finally, when they learn how to multiply negative numbers, all their ideas about multiplication become meaningless, leaving them completely at sea in their sense making. The inability to make meaning out of procedures leaves students grasping and anxious, as the procedures seem ever more arbitrary.
In contrast, when classrooms are geared toward supporting mathematical sense making, they reap multiple motivational benefits. First, students’ sense of ownership over their learning increases. Students see that multiplication can be thought of as repeated addition, the dimensions of a rectangle as related to its area, or the inverse of division. When they learn new types of multiplication, the procedures have a conceptual basis to expand on. Relatedly, their learning is more durable. Because they understand the meaning behind the mathematics they are learning, they are more likely to connect it to their own experiences. This, in turn, provides openings for their curiosity and questions. Beyond giving students opportunities for sense making, meaningful mathematics classrooms provide students chances to identify and explore their own problems. Indeed, in a systematic comparison of teacher-guided and student-driven problem solving, educational researchers Tesha Sengupta-Irving and Noel Enyedy found that the ownership, relevance, and opportunities to engage curiosity in student-driven problem solving supported stronger outcomes in student affect and engagement.[1]

The challenge, then, for teachers is how to help students engage in meaningful mathematical learning within the structures of schooling. I would love to hear your ideas about how to achieve this.


[1] Tesha Sengupta-Irving & Noel Enyedy (2014): Why Engaging in Mathematical Practices May Explain Stronger Outcomes in Affect and Engagement: Comparing Student-Driven With Highly Guided Inquiry, Journal of the Learning Sciences, DOI: 10.1080/10508406.2014.928214

The Best of the Math Teacher Blogs 2015

It’s never been easier to miss a great math blog post. The MathTwitterBlogoSphere –– known as #MTBoS around social media –– was once a small group of math teachers willing to make themselves vulnerable, putting their practice online. As the community has expanded, even the most dedicated readers struggle to keep up with the deluge of thoughtful commentary, engaging and interesting tasks, and stories that we can all learn from.

To help keep you from missing out, we have compiled some favorite posts from this past year, as nominated by #MTBoS folks on Twitter, into a book. These posts are as rich and varied as the educators who wrote them. Some delve into specific content. Some tell stories of change and growth. Others explore teaching practices, new or well established. We hope that you find some that provoke and push you, and others that make you smile. Most of all, we hope you make some new connections in the MTBoS community.

This book has another purpose as well. Since 2012, folks from the MTBoS have participated in an annual “tweet up,” a two-day math extravaganza called Twitter Math Camp (TMC). Unlike regular conferences, teachers come knowing who they want to meet. They come to continue conversations that have been taking place online, through blogs and twitter. TMC is a rich and personal learning environment. The grassroots nature of TMC means it is lively, personal, tailor-made, and unpredictable. However, most teachers have to pay their own way. We will use the money raised through sales of this book to start a fund to bring along some of the teachers who would not otherwise be able to participate. We think that TMC is a unique professional learning experience, and we hope to share it while we grow our community.

The book is nearly ready for publication, but we need assistance with a few tasks (we’d like to add an index and list embedded links at the bottom of each post so they’re accessible to anyone reading a paper copy). If you’re interested in assisting please email Tina (tina.cardone1 on gmail) and she’ll get you set up with a task.

Thank you for reading, and thank you for your support.

— Lani Horn & Tina Cardone
P.S. Sorry that we were super secret on this project! We didn’t decide to do this until after the #MTBoS2015 conversation started. We were so impressed by the quality of the nominated posts, it seemed like a great opportunity to do something for this amazing community. As long as we are confessing, we also didn’t announce it until now because we weren’t sure we’d be able to finish it! If people like the idea then we’ll have a more public and organized process for 2016.

Laying the Groundwork for Logarithms

s

Strangely, I have had occasion to do a few tutoring sessions with different kids recently around exponential and logarithmic functions.

This particular mistake set off a few alarm bells:

logblog

 

Do you see what the student is doing here? She is treating

log a

like a variable that is being divided instead of a function.

I looked at the student’s notes, and all the usual log laws were there. But she did not yet have the unshakable understanding that logs are functions. I realized that there are some foundational ideas that she needed before we could really make sense of all of this.

Here are a couple of essential ideas I want to communicate to students about logarithm functions.

First, functions can be described as actions, so I always make students explain what a function is doing.

The question you should ask about every function is: what are we doing to the input to get to the output? I call it “saying the function in English.”

Since we usually teach logarithms after exponential functions, let me start with them.

I ask, What do exponential functions do? They provide rules based on repeated multiplication. So the function

2x

tells us that “some number (y) equals 2 multiplied by itself any number of (x)  times” to get y. We can do this with different examples, talk about how the function grows, look at the graph, look at tables, compare the growth of exponential functions to linear and quadratic ones. My goal is to get kids to have a feel for what is happening with exponential growth so well that when somebody says, “It was growing exponentially!” they can decide whether that is an accurate statement or not.

This is the first part of the groundwork for understanding logarithms.

Second, remember that anything we do in mathematics, we always find ways to undo.

This is thematic in all of mathematics. It becomes a chant when I teach math.
I say to students:
“Since this is math, anything we learn to do, we need to ….?”
They soon learn to respond with:
UNDO!!!”

Doing-and-undoing is a good mathematical habit of mind to emphasize, because students start to anticipate that when we learn some new funky function or operation, an inverse is coming down the pike. They are not at all surprised to learn that trig functions have an inverse and so on.

In this case, since we have learned to exponentiate, they can guess we need to un-exponentiate.

shrug

That’s just how math works!

I like to show inverses of functions in all of the representations. The idea is the same in tables, graphs and equations: the x’s and y’s switch places.

For tables and graphs, it’s fairly easy for students to figure it out. But the algebra gets tricky. To find the inverse of the previous exponential, for example, we need to derive it from:

inverse 1

This immediately creates a mathematical need to “un-exponentiate.”

So when we want to solve that equation for y, let’s undo exponentiation with a function we call a logarithm. Logarithms undo exponentiation.

logging the inverse
Since the log undoes the exponentiation, we end up isolating the y.

this one!

I also tell them we read this as “log base two of x equals y.”

So when you see an equation like:

fixed

you are asking “2 to what power equals 8?” I have them practice explaining what different equations mean.

Now your students are ready to learn all the details of working with logs!

Tell me your ideas in the comments.

[Before I close, vaguely related Arrested Development reference:

bob loblaw

Because this is a log law blog. But I guess I don’t really want to talk about log laws. Anyways…]

Making Sense of Student Performance Data

Kim Marshall draws on his 44 years’ experience as a teacher, principal, central office administrator and writer to compile the Marshall Memo, a weekly summary of 64 publications that have articles of interest to busy educators. He shared one of my recent articles, co-authored with doctoral students Britnie Kane and Jonee Wilson, in his latest memo and gave me permission to post her succinct and useful summary.

In this American Educational Research Journal article, Ilana Seidel Horn, Britnie Delinger Kane, and Jonee Wilson (Vanderbilt University) report on their study of how seventh-grade math teams in two urban schools worked with their students’ interim assessment data. The teachers’ district, under pressure to improve test scores, paid teams of teachers and instructional coaches to write interim assessments. These tests, given every six weeks, were designed to measure student achievement and hold teachers accountable. The district also provided time for teacher teams to use the data to inform their instruction. Horn, Kane, and Wilson observed and videotaped seventh-grade data meetings in the two schools, visited classrooms, looked at a range of artifacts, and interviewed and surveyed teachers and district officials. They were struck by how different the team dynamics were in the two schools, which they called Creekside Middle School and Park Falls Middle School. Here’s some of what they found:

  • Creekside’s seventh-grade team operated under what the authors call an instructional management logic, focused primarily on improving the test scores of “bubble” students. The principal, who had been in the building for a number of years, was intensely involved at every level, attending team meetings and pushing hard for improvement on AYP proficiency targets. The school had a full-time data manager who produced displays of interim assessment and state test results. These were displayed (with students’ names) in classrooms and elsewhere around the school. The principal also organized Saturday Math Camps for students who needed improvement. He visited classrooms frequently and had the school’s full-time math coach work with teachers whose students needed improvement. Interestingly, the math coach had a more sophisticated knowledge of math instruction than the principal, but the principal dominated team meetings.

In one data meeting, the principal asked teachers to look at interim assessment data to predict how their African-American students (the school’s biggest subgroup in need of AYP improvement) would do on the upcoming state test. The main focus was on these “bubble” students. “I have 18% passing, 27% bubble, 55% growth,” reported one teacher. The team was urged to motivate the targeted students, especially quiet, borderline kids, to personalize instruction, get marginal students to tutorials, and send them to Math Camp. The meeting spent almost no time looking at item results to diagnose ways in which teaching was effective or ineffective. The outcome: providing attention and resources to identified students. A critique: the team didn’t have at its fingertips the kind of item-by-item analysis of student responses necessary to have a discussion about improving math instruction, and the principal’s priority of improving the scores of the “bubble” students prevented a broader discussion of improving teaching for all seventh graders. “The prospective work of engaging students,” conclude Horn, Kane, and Wilson, “predominantly addressed the problem of improving test scores without substantially re-thinking the work of teaching, thus providing teachers with learning opportunities about redirecting their attention – and very little about the instructional nature of that attention… The summative data scores simply represented whether students had passed: they did not point to troublesome topics… By excluding critical issues of mathematics learning, the majority of the conversation avoided some of the potentially richest sources of supporting African-American bubble kids – and all students… Finally, there was little attention to the underlying reasons that African-American students might be lagging in achievement scores or what it might mean for the mostly white teachers to build motivating rapport, marking this as a colorblind conversation.”

  • The Park Falls seventh-grade team, working in the same district with the same interim assessments and the same pressure to raise test scores, used what the authors call an instructional improvement logic. The school had a brand-new principal, who was rarely in classrooms and team meetings, and an unhelpful math coach who had conflicts with the principal. This meant that teachers were largely on their own when it came to interpreting the interim assessments. In one data meeting, teachers took a diagnostic approach to the test data, using a number of steps that were strikingly different from those at Creekside:
  • Teachers reviewed a spreadsheet of results from the latest interim assessment and identified items that many students missed.
  • One teacher took the test himself to understand what the test was asking of students mathematically.
  • In the meeting, teachers had three things in front of them: the actual test, a data display of students’ correct and incorrect responses, and the marked-up test the teacher had taken.
  • Teachers looked at the low-scoring items one at a time, examined students’ wrong answers, and tried to figure out what students might have been thinking and why they went for certain distractors.
  • The team moved briskly through 18 test items, discussing possible reasons students

missed each one – confusing notation, skipping lengthy questions, mixing up similar-sounding words, etc.

  • Teachers were quite critical of the quality of several test items – rightly so, say Horn, Kane, and Wilson – but this may have distracted them from the practical task of figuring out how to improve their students’ test-taking skills.

The outcome of the meeting: re-teaching topics with attention to sources of confusion. A critique: the team didn’t slow down and spend quality time on a few test items, followed by a more thoughtful discussion about successful and unsuccessful teaching approaches. “The tacit assumption,” conclude Horn, Kane, and Wilson, “seemed to be that understanding student thinking would support more-effective instruction… The Park Falls teachers’ conversation centered squarely on student thinking, with their analysis of frequently missed items and interpretations of student errors. This activity mobilized teachers to modify their instruction in response to identified confusion… Unlike the conversation at Creekside, then, this discussion uncovered many details of students’ mathematical thinking, from their limited grasp of certain topics to miscues resulting from the test’s format to misalignments with instruction.” However, the Park Falls teachers ran out of time and didn’t focus on next instruction steps. After a discussion about students’ confusion about the word “dimension,” for example, one teacher said, “Maybe we should hit that word.” [Creekside and Park Falls meetings each had their strong points, and an ideal team data-analysis process would combine elements from both: the principal providing overall leadership and direction but deferring to expert guidance from a math coach; facilitation to focus the team on a more-thorough analysis of a few items; and follow-up classroom observations and ongoing discussions of effective and less-effective instructional practices. In addition, it would be helpful to have higher-quality interim assessments and longer meetings to allow for fuller discussion. K.M.] “Making Sense of Student Performance Data: Data Use Logics and Mathematics Teachers’ Learning Opportunities” by Ilana Seidel Horn, Britnie Delinger Kane, and Jonee Wilson in American Educational Research Journal, April 2015 (Vol. 52, #2, p. 208-242

How Does School Culture Reflect Middle Class Culture?

Class is rarely talked about in the United States; nowhere is there a more intense silence about the reality of class differences than in educational settings.

bell hooks

One of the things teachers often hear in the course of teacher education is that school culture typically reflects middle class culture. For teachers who grew up middle class, this statement can be perplexing. It’s like trying to alert fishes to the unique presence of water: they are so immersed in it that alternatives cannot be fully imagined.

Yet class shapes everything from interactional styles to the kinds of competencies valued in the home. In her famous ethnography of class and American childhoods, Annette Lareau characterized working class and poor families as tending to promote natural growth in children. These parents tend to let children determine their leisure activities. When they interject authority, they tend to do so with directives.

lareau cover

In contrast, middle-class families tended to practice a form of parenting Lareau calls concerted cultivation. These parents tended to equate good parenting with deliberate development of their children’s talents, especially through organized leisure activities. They also used fewer directives, instead reasoning with their children when seeking to change their behavior.

(There are other contrasts between these approaches to parenting, as summarized in this table.)

Lareau’s point is not that one style is better than the other, but instead to point out that school often assumes middle class parenting, leaving poor and working class families with less of an institutional fit. In fact, as somebody who was raised in this manner, I personally see many strengths that come out of the accomplishment of natural growth. Children have more opportunities to develop autonomy and engage in more social problem solving than children whose leisure activities are organized and led by adults.

How do these middle class assumptions play out in school? Classrooms are crowded places, and teachers frequently need to direct children’s attention and activities. Many teachers tend toward the middle class style of suggesting a transition (“Would you like to join us on the rug?”) rather than directing it (“Please come to the rug now”). If you are used to the latter, the former can be understandably ambiguous and confusing.

What is more, middle class children, through their greater experience with formally organized leisure activities, usually come to school with tacit understandings about how to participate. They have more experience responding to the authority of a non-kin adult with whom they will likely form a superficial and transitory relationship. In contrast, if your early socialization has been primarily with family, taking directions from a stranger may seem like a strange and maybe not entirely wise endeavor.

There are also subject-specific ways that social class makes school more or less a fit with children. Valerie Walkerdine has documented the ways class can interact with mathematics education in particular. She points to the quantitative fictions common to math class, describing, for example, an elementary number game requiring the “purchase” of various items for 1 to 10 pence and then making change. The working class children she observed, whose lives were much more consequentially tied to actual prices of things, found the premise of the game absurd. As I often tell my pre-service teachers, which of your students knows where to find the best price on a gallon of milk, and which simply look to make sure it’s organic? How does that change your job in making sure the cost in your word problem is realistic?

To feel comfortable participating in classrooms, children need to have a reason to be there. They need to see a connection to their lives and experience a sense of belonging. Social class differences are sometimes the source of cultural barriers to feeling like you belong in school, that school is a place that matters, that things make sense. Teachers need to be thoughtful in how they bridge these differences with their students.