Don’t ask if it’s “good” without adding “for whom?”

Once upon a time, a friend and I were talking about a math teacher our daughters shared. She said, “You know, his lessons weren’t so exciting, but at least he’s got the content knowledge, so the kids could work it out.”

I grew quiet, and an awkward pause ensued.
This had not been my daughter’s experience at all.

My friend’s daughter is the quintessential math kid, the kind you can provide with some math content, and she will eagerly work through it. She loves patterns and puzzles, sticks with new things until she gleans key insights, with or without a teacher. She is a bright and thoughtful kid, and any teacher would be glad to have her in class.

My kid? The social world is her main interest in school. Honestly, she is a fabulous little social scientist, drawing me sociograms of the lunch tables to explain the different friend groups, designing “crush scales” to help her friends quantify the extent of their infatuations, providing me with details of the social markers of popularity at her school. She finds school fascinating, but the school curriculum does not always engage her. Her academic interests depend largely on her relationship with a given teacher.

With the math teacher in question, she had a strong strike against her. She is a doodler, and this same math teacher was squarely anti-doodle.

mr math man

In a parent conference, he told me that he viewed her doodling as a lack of seriousness. I tried to push back and help him see her more clearly, to little avail. In the end, the same perceptiveness that led my kid to understand the subtle workings of her school’s social dynamics led her to the conclusion that this teacher did not think very highly of her as a potential learner in his class. Not surprisingly, her engagement reflected that.

The differences between my friend’s and my experiences with the same teacher has me thinking about how limited we are when we talk about the quality of education. Often our discourse focuses on whether things are “good” ––  whether it is teachers, classes, or schools –– as if it is an essential property of the thing.  In this case, my friend would probably say that this teacher was “good enough,” because that was true –– for her daughter. As you may surmise, I would say otherwise.

Our impoverished way of sorting the educational world into good/not good has consequences for the systems we have created. I would go so far as to say that it contributes to inequality. Many middle class parents are probably familiar with the conversations about “good schools.” Parents seldom actually go and visit the schools, but there are signifiers that stand in for goodness, many of which are problematic: test scores (which correlate to parent income and education), student populations that skew White/Asian/affluent. As the consensus grows about where “goodness” lies, in our crazy U.S. system, property values respond accordingly, reinforcing the demographic factors that underlie those assessments.

The thing is, I go to a lot of schools. I have been to schools that are deemed “good” by powerful parents and have found myself distressed at the socio-emotional climate or underwhelmed by the substance of the academics. I have been to schools that are deemed “bad” and have conversely been bowled over by the widespread care and wowed by thoughtful teaching and learning.

Our discourse around “goodness” in education cuts off the essential qualifier –– for whom? In doing so, it reinforces goodness as inextricable and erases important questions about whose learning is being supported.

Auditing Your Classrooms for Competence and Status

This past weekend, I had the great pleasure of giving a keynote address at the Mathematics Council of Alberta Teachers (MCATA) Conference.

First things first: @minaclark did sketch notes of my talk!  I am delighted because I have always wanted somebody to do that. She did a fantastic job too.

During the breakout session afterwards, I talked about how we can audit our classrooms to support better interactions. In particular, we need to pay attention to issues of mathematical competence and student status. (I have written a lot on these topics since they are critical to fostering positive relationships between students and the subject. You can read earlier posts here, here, and here.)

Here are my audit questions.

Competence audit:

  • What kinds of competencies are valued in your classroom? Where do students have a chance to show them?
  • Consider the last few activities you have done in your class. Did they provide multiple entry points toward a rich mathematical idea? If not, can you use the table below to adapt them to become a low ceiling/high floor question?
  • When you look at your class roster, can you identify at least one way that every student is mathematically smart?
  • When you think of students who struggle, do they have competencies that you might better support by redesigning some of your class activities?
  • When you think of students who have a history of high achievement, do they value other ways to be smart aside from quick and accurate calculation? Do they value other competencies in themselves? In others?
table

Some low floor-high ceiling question types. (Adapted from Will Stafford’s “Create Debate” Handout)

Status audit:

  • When you think of the students you worry about, how much of their challenge stems from lack of confidence?
  • How much do students recognize the value and contributions of their peers?
  • What small changes could you make to address status problems and support more students in experiencing a sense of competence?

Please feel free to add others or offer your thoughts in the comment section.

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

Who Belongs in our Math Classrooms?

Many students enter mathematics classrooms with a sense of trepidation.  For some, their discomfort reflects a larger sense of detachment from school. They may have not felt welcomed because of the gaps they experience navigating between their home language or culture and the expectations at school. The social milieu of school may make them feel like an outcast, as they see peers who seamlessly “fit in” while they remain on the outside. Unlike the sports field, their community center, or the stage, academics may make them feel untalented and incompetent. For other students, school itself is fine, but there is a distinct dread upon entering math class. Math has never made sense –– or perhaps it used to make sense when it was whole numbers and counting, but as soon as the variables showed up, all hope was lost. They may have been demoralized by a standardized test score that deemed them below grade level. They may get messages at home that “we’re not good at math.” For still other students, they love the subject, but must contend with people who do not see them as fitting their ideas of a person who is good at math. They have to combat stereotypes to be seen as legitimate participant in the classroom, as they defy expectations by holding forth with their smartness even as others look on in dismay.

For most students, alienation can be overcome by teachers who create a sense of belongingness. Belongingness comes about when students experience frequent, pleasant interactions with their peers and teacher. It also comes about with the sense that others are concerned for who they are and for their wellbeing.

Why Belongingness Matters
When I go and observe in mathematics classrooms, I can usually ascertain students’ general sense of belongingness. What is their affect as they walk through the door? How warmly and personally do they greet the teacher and each other? Are they represented –– through math work or other means –– on what is posted on the walls?
All too often, I see students enter their math classrooms with a sense of gloom. Smiles disappear as they cross the threshold of the doorway. Their posture slumps. They sit at the back of the room or put their heads on their desks.They may even groan or launch into a litany of complaints. When I observe these student behaviors as a teacher, it signals that I have work to do to make children feel more welcomed and excited about spending their time with me learning mathematics.

Teachers’ relationships with students are an important source of of belongingness, but peers are equally (if not more) important. Even if a teacher welcomes each student with a smile and takes an interest in who they are, frequent insults or intimidation from other students can create a negative classroom climate. To support belongingness, then, teachers need to do more than create strong relationships. In addition, they need to create norms and expectations about how students treat each other.

During adolescence, children face the enormous task of developing a strong and stable sense of themselves. Although this identity development happens over the course of a lifetime, adolescence is distinct because it is when children are first able to think abstractly enough to grapple with both their own emerging self-understandings as well as how society views them. This leads to both a delightful self-awareness as well as a sometimes painful sense of self-consciousness for many students, as they are more sensitive to others’ perspectives and feedback. Necessarily, then, inclusive and inviting classrooms provide a place for this crucial developmental work, particularly in relationship to school in general and mathematics in particular.

What Gets in the Way of Belongingness
Although I generally avoid absolutes when it comes to describing good teaching, I will highlight a few common instructional practices that feed a negative classroom climate, thus working against belongingness. First, many math classrooms emphasize competition. Whether this comes from formal races, timed tests, or just students’ constant comparison of grades, competition sends a strong message that some people are more mathematically able than others. This is problematic because there is typically one kind of smartness that leads students to “win” these competitions: quick and accurate calculation. To paraphrase mathematician John Allen Paulos, nobody tells you that you cannot be a writer because you are not a fast typist; yet we regularly communicate to students that they cannot be mathematicians because they do not compute quickly. While a competitive dynamic may be at play in other school subjects, it is especially toxic in math classrooms because students do not have other venues to explore and affirm their diverse mathematical talents.

Another contributor to negative classroom climate comes from devaluing who students are. This may come in many forms, some of which teachers may not realize. For instance, some teachers avoid using what is for them an unfamiliar (thus difficult-to-pronounce) name. Not only does this lead to fewer invitations to participate, it communicates to students that we are not comfortable with something that might make them different than us. Names are deeply personal, one of the first words students identify with: They often reflect home cultures and personal history. When teachers avoid them or change them without consent, they devalue something of who students are.

Likewise, when teachers problematically differentiate their treatment of students based on cultural styles, they can devalue who students are. For instance, educational researcher Ebony McGee studies successful students of color in STEM fields. She interviewed a Black chemistry major at a primarily White institution who reported that a White instructor avoided her when she dressed in a way often perceived by middle class teachers as “ghetto.” When she changed her clothing and hair style, he told her, “Now you actually look presentable. I bet you are making better grades too.” Similarly, in a research project I conducted, a female high school student concluded that her math teacher “didn’t like” her after the teacher emailed her mother that her skirts were “too short.” Adolescents use clothing to express themselves and their culture as a part of the identity work they engage in. Avoiding or rejecting them because of these forms of self-expression can further estrange them from the classroom or school. If concerns need to be raised, they should be done in a way that respects students’ self-expression.

Finally, teachers may alienate students by correcting the inconsequential. Although our job is to help students become educated people, when we correct the inconsequential, we may work against other goals of engagement and inclusion. Deciding what is inconsequential is, of course, a judgment call: context is everything. For instance, our standards for speech and language differ when students try to explain an idea they are in the midst of grappling with versus when they are preparing for a job interview. In the former situation, correct grammar is not the point, while in the second, it may matter a lot. If our students are learning English as a second language, speaking a pidgin or African American Vernacular English (AAVE), our focus on correct grammar in situations where it is inconsequential may disinvite their participation.

Professional Development is Broken, but Be Careful How We Fix It

This morning, Jal Mehta tagged me on a tweet to linking to his recent Education Week blog post, entitled “Let’s End Professional Development as We Know It.”

The following exchange ensued:

He then asked if I could share some of my research to back my perspective. I sent him an email with journal articles and such, but I thought I would share my ideas with y’all too.

Here is my argument about why putting professional development (PD) back in schools may be necessary but not sufficient to improving its impact on teachers’ instruction.

Unlike medicine and other scientific fields, where problems are taken-as-shared and protocols for addressing problems are roughly agreed upon, teaching problems are locally defined. What needs attention in one school may not need attention in another. For instance, some schools’ “best practices” may center on adapting instruction to English learners, while other schools’ might center on the mental health ailments that have become prevalent among affluent teens. Likewise, other professions share language, representations, and goals for critical aspects of their work — these all important resources for learning together. In teaching, we see repeatedly that terms acquire the meaning of their setting more often than they bring new meanings to these places. Take, for instance, Carol Dweck’s ideas about mindset. The various ways that her construct has taken hold in education led her to explain why what she means by mindset is not how the idea is being used. If we leave professional development entirely up to individual school sites, this means that “doing PD” on Topic X probably looks fairly different from place to place, so radically localized professional development will exacerbate this problem.

Leaving professional development to local sites also limits teachers’ access to expertise. When my colleagues and I have studied teachers’ collaborative learning, we found that the learning opportunities are not equally distributed across all teacher groups. Some of this has to do with how teachers spend their time (e.g., focused on logistics or deeper analysis of teaching). But some of it has to do with who is sitting around the table and what they have been tasked to do.

Teachers’ collaborative learning can be described as an accumulated advantage phenomenon, where the rich get richer. That is, teachers who have sophisticated notions of practice are able to identify teaching problems in complex ways and deploy more sophisticated strategies for addressing them. This follows from my previous points, since problem definition is an important part of teachers’ on-the-job learning. For instance, if we have a lot of students failing a course, how do we get to the bottom of this issue? In many places, high failure rates are interpreted as a student quality problem. In others, they are taken as a teaching quality problem. Interpretations depend on how practitioners think this whole teaching and learning business goes down. In other words, problem definition is rooted in teachers’ existing conceptions of their work, which in other professions, are codified and disseminated through standardized use of language and representations.

Unequal access to expertise is only one of many reasons the optimistic premise of teacher community often does not pan out. There is a tendency to valorize practicing teachers’ knowledge, and, no doubt, there is something to be learned in the wisdom of practice. That being said, professions and professionals have blind spots, and with the large-scale patterns of unequal achievement we have in the United States, we can infer that students from historically marginalized groups frequently live in these professional blind spots. For reasons of equity alone, it is imperative to develop even our best practitioners beyond their current level by giving them access to more expert others.

Even in highly collaborative, well-intentioned teacher communities, other institutional pressures (e.g., covering curriculum, planning lessons) pull teachers’ attention to the nuts-and-bolts of their work, rather than broader learning or improvement agendas. Add to this the norms of privacy and non-interference that characterize teachers’ work, you can see why deeper conversations around issues of teaching and learning are difficult to come by.

What about, you might say, bringing in expert coaches? Research shows that expert facilitators or coaches can make a difference. In fact, there is evidence that having expert coaches may matter more than expert colleagues when it comes to teacher development. At the same time, we suspect that expert facilitators are necessary but not sufficient, as coaches often get pulled into other tasks that do not fully utilize their expertise. In our current study, we see accomplished coaches filling in for missing substitute teachers, collating exams, or working on classroom management with struggling teachers. None of these tasks taps into their sophisticated instructional knowledge. Additionally, being an accomplished teacher does not guarantee you have the skill to communicate your teaching to others. In our data, we have numerous examples of really great teachers underexplaining their teaching to others.

Lee Shulman famously called out the missing paradigm of teacher knowledge, giving rise to a lot of research on pedagogical content knowledge (PCK). While PCK gave a very useful way to think about teachers’ specialized knowledge, little progress has been made on understanding how teachers develop this and other forms of knowledge, particularly in the institutional context of schools, which often presses teachers’ practice away from what might be deemed “good teaching.” As long as we don’t have strong frameworks for understanding how teachers learn, PD –– even localized, teacher-led PD –– risks being just another set of activities with little influence on practice.

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…]

Building Teaching as a Responsive Profession

Those of you who spend real or virtual time with me have heard me talk about how hard it is to talk about teaching.

One frequently mentioned issue is that, unlike other professions, teaching does not have its own technical language. Professions like aviation and medicine have common professional terms that highlight important features of critical situations and guide practice. In aviation, for instance, pilots identify wind patterns to aid in landing planes. Likewise, surgeons have cataloged human anatomy and surgical procedures so the protocol for appendectomies can be named and routinized, with appropriate modifications for anatomical variations such as hemophilia or obesity. But a strong headwind in China is similar to a strong headwind in Denmark; a hemophiliac in Brazil will require more or less the same modifications as a hemophiliac in Egypt.

In contrast, an urban school may not be the same as an urban school a few blocks away, nor an ADHD kid the same as an ADHD kid in the same classroom. Although such terms attempt to invite descriptions about particular teaching situations, the language often relies on stereotyped understandings. Everyday categories like an urban school, an honors class, or an ADHD kid seldom work to describe teaching situations adequately to help teachers address the challenges they face. Words characterizing social spaces and human traits are inherently ambiguous and situated in particular social, cultural and historical arrangements.

The variation teachers encounter cannot always be codified, as they often are in aviation and surgery. In fact, in the United States, when educational situations are codified, they often presume the “neutral” of White, English-speaking, and middle class culture. However, the widespread practice of glossing cultural particulars, or only seeing them as deviants from a norm, reduces teachers’ ability to teach well. From Shirley Brice Heath’s  seminal work comparing home literacy practices in White and African American communities to Annette Lareau’s identification of social class-specific parenting patterns, we see time and again that children from non-dominant groups frequently encounter schooling expectations that are incongruous with their home cultures, often to the detriment of their learning. Conversely, when instructional practices align with children’s home cultures, teachers more are more effective at cultivating students’ learning. (See, for a few well documented examples, this work by Kathryn Au and Alice Kawakami, Gloria Ladson-Billings, and Teresa McCarty.)

Culturally responsive pedagogies are, by definition, highly particular and have been documented to yield better student learning. To communicate sufficiently, professional language for teaching would need to encompass this complexity, avoiding simplistic –– perhaps common sense –– stereotypes about children, classrooms, schools, or communities.

How, then, can we develop shared professional language for teaching and build professionals responsive to the children they serve? I have some ideas I will share in another post.