Author Archives: dkane47

Reasoning through Content

I love writing because as I write, I understand things a little better. I’ve written and changed my mind about why math might be worth learning more times than I can count. Each time, I get a little sharper in my thinking. Long time readers will probably find this repetitive, but I enjoyed writing it, and hopefully there’s a bit more insight than before.

Math class has lots of purposes. Learning math keeps future doors open, allows people to better understand the world around them, helps students to experience beauty and wonder, and empowers learners by showing them what they are capable of. But the most important argument to me is that math teaches reasoning. It’s not the only way to get there, but I think math class has enormous potential to teach students to notice and generalize patterns, shift representations and see a problem from a new perspective, balance attention to the big picture with attention to the details, connect ideas that seem different on the surface, and move flexibly between abstractions. These are skills that could serve students in the future within and beyond math classrooms.

But I hesitate to argue for reasoning as my primary goal in teaching math. Reasoning is the end goal, but it’s not the means of getting there. I don’t think that reasoning is a skill that one can practice; students can’t go to class, do some hard problems, and assume they are better at reasoning. I can’t confidently say that I’ve seen my students reasoning more effectively or consistently at the end of a year. All humans can reason, and all humans struggle to reason consistently. It’s a gradual learning process, and I’m lucky if I have a small part in that as a teacher.

More than my feeling that teaching reasoning directly doesn’t work, it’s also inequitable. Reasoning always happens in context, whether we choose the context of factors and multiples, interpreting statistics, or making decisions about finances. Students who already have knowledge of that context are likely to learn, and students who don’t, won’t. Pretending that reasoning is divorced from context advantages some students while leaving others floundering.

The Road to Reasoning Goes Through Content

Reasoning is a fuzzy goal, with a long and uncertain path to reach it. But content is what I know how to teach. I don’t have much confidence that I can teach a student to notice patterns and create generalizations in a given class. I do think I can do a decent job of teaching students to solve problems with quadratic functions. So I start with content. If I do my job and give each student a foundation in the content we’re learning, they all have a chance to apply their knowledge in new ways, to stretch their ways of thinking, and to reason. And I believe that, as students practice reasoning in more contexts, it becomes a little more likely that they will be able to apply those skills somewhere new. At first it’s small jumps — learning some things about quadratics and polynomials that can then be applied to rational functions. But, over time, I think math education sets students up with skills they can transfer outside of the math classroom. I don’t claim to know exactly how to get there, but that’s the goal.

This might seem like a dull perspective. We teach math to teach reasoning, but all I want to do is focus on content? But there are so many rich questions left unanswered. What content is most useful to teach? I don’t love what I’m asked to teach right now. Functions and algebraic manipulation are great, and should be in the curriculum, but what could we cut to make room for the math of gerrymandering, statistical analysis for social good, linear and exponential modeling in the world, and more? And once we’ve decided on the content, how can I teach that content in ways that help students reason? What are the teacher moves to link students’ experiences in math class to future decisions? What experiences will help students see math as a tool that empowers them in their lives? I don’t have any more answers at the moment, but I think these are great questions, and they are even better questions to ask about the breadth of content I teach each year.

Teaching Reasoning, or Teaching Math

Why teach math? One reason I hear often is to teach reasoning. I’d love to teach students how to reason, but I’m skeptical. I have enough trouble teaching them to complete the square or graph a rational function. Is it even possible to teach someone to reason?

I wonder if the idea that math teaches some broader skill like reasoning is a kind of reaction to the experience of many teachers that it’s really hard to teach content. Lots of students leave class without the mathematical knowledge we want them to have, and we might like to believe they have. But if content isn’t the point, then it’s just fine that students forget everything they learned about logarithms and polynomials within a month or two. It’s a nice idea!

So why teach math?

Mostly, for me, to learn math. I’m biased, I really like math. And primary math is really useful! I wouldn’t mind throwing out a few things, but facility with arithmetic, fractions, decimals, proportions, the basics of geometry, and few principles of statistics and probability help people to live in the world, to work, and to fulfill their civic responsibilities.

Most high school math, I could take it or leave it. If higher education wasn’t so obsessed with calculus, high school might look very different. Who’s to say that learning about polynomials is more useful than graph theory, or projective geometry, or number theory?

Here’s the fun part. I teach math because I think math can open doors, can empower students to see what they are capable of, can inspire wonder and beauty. Those are powerful experiences for students in school, when we get them right. And there’s lots of great thinking that goes into learning math. And I think that maybe, just maybe, as students learn more math, and they think in new ways in different contexts over time, it becomes a little more likely that they can apply that thinking somewhere new, in or out of math class. I’m a skeptic here; I’m not interested in teaching a “SWBAT reason” lesson, and I don’t think I get there with very many students. But I think math has that potential. Bit by bit, over time, it becomes more likely that those ways of thinking about proportions and linear models and function transformations and circle theorems become useful outside of mathematics.

I’m happy to be humble here. I teach math because I think math is worth knowing, and because the experiences one can have in math class are worth having. But I like to dream that what I teach could help someone to live a richer life outside of math class.

On Perspective

I think I would identify myself as a social justice educator. I hesitate because it’s an ambitious title; there’s a lot that goes into educating toward social justice, and I’m not very good at most of it. I’m not sure I even know what most of it looks like.

But one piece I’ve thought a lot about is understanding the perspectives of those with different backgrounds from me.

I’m reading Carol Anderson’s White Rage as part of the #ClearTheAir chats, and it’s all the history I never learned but need to know. The erosion of progress during Reconstruction, the forces working against the Great Migration, the violent and defiant opposition to Brown v Board, the reversals of gains from the Civil Rights era, and finally, “How to Unelect a Black President.” This is the history that has created the world we live in today, the intentional discrimination and hostility toward the black community that has changed form but not impact over the last 150 years.

We often tell this history as a narrative of progress, as major victories that have moved our society forward. I think we tell those stories because they are comfortable, and because they fit into the narrative we want to tell ourselves about our country. What would it look like to value discomfort and truth-telling over progress and American exceptionalism when we talk about history? Whose stories do we tell? Whose narratives do we hear?

I’ve thought about this tweet often in the last few months. Which authors do I read? Whose stories do I hear?

I think about this in other contexts as well. Who do I retweet and amplify? Who do I reference and quote in my writing? Which students do I gravitate toward?

I am far from perfect; it would take me a very long time to retweet as many people of color as I have white people, or to link to as many white education writers as people of color on this blog. But White Rage is one example of a place where I’ve been pushed to expand my horizons. And White Rage is giving me new insights into the world around me, and how I can exist as an educator in it.

And here’s something I’ve noticed. As I’ve been more intentional about whose ideas I’m reading and sharing and referencing and amplifying, I’ve learned a lot that I wouldn’t have otherwise. I’m exposed to more culture, richer conversations, perspectives that push me to think in new ways, and ideas I wouldn’t have found without that active effort. I see the world around me through a new lens, and I am better for it.

I’ve read a lot recently about why diversity matters, in different contexts and from different perspectives. There’s not just one reason to seek out diverse perspectives. Take a sample here, here, here, and here. These are thoughtful folks and great arguments.

For me, one important driving force has been my personal experience that being intentional in who I read, reference, and amplify is leading me to a richer life than the comfortable, familiar perspectives I’m likely to learn from without that effort.

Active Caring and Patient Caring

I recently finished Geoff Krall’s book Necessary Conditions. It’s great! You should read it too. Here’s one of my takeaways:

Geoff writes about “active caring” and shares this great graphic in the book:

Active caring demands a two-way relationship independent of the student’s academic dispositions. Students who don’t demonstrate a preternatural appreciation for the subject receive the same level of personal and cultural care as those who do.

I think this distinction is really neat, and Geoff has great ideas for how active caring can play out with large class sizes and busy schedules. Check out his blog post on the topic.

But there’s another type of caring I think connects with active caring, and can be equally important. I’m going to call it “patient caring.” In the world, in particular when I read news about politics and prominent folks, there’s this assumption that humans are static, that the traits someone has today are the traits they’ll have tomorrow and next year and in ten years. I think this plays out in education; humans have a disposition to forget that other humans grow and change, yet that growth is a core part of our job as educators. And that growth doesn’t happen over time; it takes time and deliberate effort. I don’t have Geoff’s great visual designer, but here is my list:

Patient Caring

  • Teacher plans to revisit norms and routines throughout the year, knowing that classes need reminders and to hear expectations explicitly multiple times.
  • Teacher knows that students forget things they learned last week and last month and last year and plans proactively to reteach concepts and revisit key ideas over time.
  • Teacher recognizes when a lesson is going poorly, and finds ways to reframe expectations and adjust the goals to help students feel successful.
  • Teacher gives time for students to think and wrestle with ideas on their own terms, even when class is running behind.
  • Teacher listens to students rather than for right answers, and works to begin where students are rather than where one might wish they would be. (See Max Ray-Riek for listening to vs listening for.)

These are small changes in my attitudes and responses in the moment, and there are lots of times that the practicalities of classrooms get in the way of patient caring. But, at its core, I think patient caring is recognizing that students are works in progress, that learning is messy, and that things often don’t happen on my schedule. My response in these moments should be patience, and flexibility, and the perspective to consider the broader goals of math class.

Valuing Calculation

G was a sixth grader in my first full-time student-teaching stint in summer school. She didn’t conform well to typical classroom norms. She struggled to sit still for extended periods of time or raise her hand when she had something to say. And G had never felt successful in math class. I remember her saying, “I can do addition, but I haaate subtraction.” She would quickly become frustrated with any math that wasn’t easy for her, or just scribble down the first thing that came to mind. But she was good at addition; she could add multi-digit numbers incredibly quickly and accurately, and she loved doing so. She would have been happy to hang out and solve addition problems all day if I left her to it. Unfortunately, it was my job to teach her subtraction.

I thought of G reading Michael Pershan’s recent post on ancient Greek mathematics. He explores the two ways of practicing mathematics in ancient Greece: the theoretical, abstract, and logic-oriented branch, and the practical, social, procedural branch. We often associate Greece with the culture of theoretical mathematics, yet there was a large and thriving group largely interested in solving practical problems and finding new ways to calculate, measure, and count. Michael writes:

Looking at the ancient Greek example makes me think that we really ought to respect practical mathematics — which by definition is mathematics that is not concerned with the “why.”

And yet there is so often disdain among some teachers for “mindless” calculation or “thoughtless” problem-solving. That seems unfair to me.

Here’s a quote from a recent piece of mine that seems to argue the opposite perspective:

Lani Horn writes that “Schooling favors one type of mathematical competence: quick and accurate calculation” (Motivated, p. 61). Horn argues that we can value broader mathematical competencies — making astute connections, seeing and describing patterns, developing clear representations, being systematic, extending ideas, and more.

I’ve been interested in finding ways to broaden what we think of as school mathematics, finding new ways for students to recognize their mathematical competence. But the key word there is broaden: it seems silly to find new ways of being mathematically smart if at the same time I devalue others.

I was a pretty inept teacher when I met G, and I missed opportunities to build from her competencies toward new ways of mathematical thinking. But if I only saw mathematical competence as making connections, seeing and describing patterns, and the rest of Lani’s list, I would have missed the most important asset G brought to class, and the one she identified as her greatest strength: quick and accurate calculation. I don’t see any reason to diminish G’s self-efficacy by telling her that math is really about understanding why or explaining ideas or making connections, and not calculating things. She didn’t see those other competencies as her strengths. And students in math class desperately want to feel smart and successful in their learning; G needed to find ways to build from the skills she had toward those I wanted her to learn.

There is a tricky balance to walk here. Quick and accurate calculation is the dominant competence that is valued, implicitly and explicitly, in many classrooms. Given its prevalence, I see value in putting extra effort toward seeking out different ways of being mathematically smart that broaden students’ conceptions of what it means to practice mathematics and helps more students feel valued in math classrooms. But its prevalence also means that there are lots of students, like G, who come to class with strong skills in computation. And if I choose to tell them that those skills aren’t “real math” and are less important than other skills I prefer, I am reframing those strengths as deficits exactly as I am trying to find broader ways to see students’ strengths in math class.

Learning versus Performance

I hate using turn-by-turn directions. I often refuse to turn them on even in unfamiliar places. I look at a map and try to find major landmarks or make inferences based on what I know about an area. And I often get a little lost, but getting lost helps me learn where I am and I can find my way around more easily the next time.

I think this is a useful illustration of the difference between learning and performance. If I am interested in performance — getting where I’m going on time — I would want to use turn-by-turn directions. If I am interested in learning — being able to navigate around an area in the future — then looking at a map and thinking about where I’m headed is likely to be more helpful. As Dan Willingham writes, “memory is the residue of thought.” If I’m thinking “ok now I turn left here,” I learn less than if I’m thinking, “ok after I pass I-25 I need to keep an eye out for the exit, get off, and turn north.” That thinking might mean I trade a short-term drop in performance for durable learning I can use in the future.

Nicholas Soderstrom and Robert Bjork write about this in their review of research on learning versus performance:

…learners often mistakenly conflate short-term performance with long-term learning, ostensibly thinking, “If it’s helping me now, it will help me later.”

…instructors and students need to appreciate the distinction between learning and performance and understand that expediting acquisition performance today does not necessarily translate into the type of learning that will be evident tomorrow. On the contrary, conditions that slow or induce more errors during instruction often lead to better long-term learning outcomes, and thus instructors and students, however disinclined to do so, should consider abandoning the path of least resistance with respect to their teaching and study strategies.

I think of learning and performance when teaching topics like end behavior of polynomials and rational functions. End behavior is determined by a small number of principles — larger powers get larger much faster than smaller powers, even powers are always positive, and fractions behave in certain ways. But students often want quick, easy rules, like “If the degree is even and the leading coefficient is positive, the end behavior is positive in both directions.” The latter rule will improve performance in the short term, but misses an opportunity to connect the problem to the students’ prior knowledge in ways that can be applied in different contexts in the future.

Elizabeth and Robert Bjork call these “desirable difficulties,” difficulties that reduce performance but create thinking that improves learning. But here’s the catch. I sometimes get lost trying to navigate without directions, but I’m an adult with half-decent patience and self-regulation skills. And I still sometimes drive people I’m with crazy. Lots of students have a much more fragile relationship with math class. In a lab experiment, desirable difficulties sound like a great idea. But outside of psychological studies, students are largely motivated by their perceptions of success in school. Desirable difficulties might cause students to give up or convince themselves they’re destined not to understand a concept, erasing any benefits to learning. Students want to feel successful in school, and deliberately reducing performance undermines those perceptions and can prevent them from recognizing the valuable ideas they bring to class.

I think this is a great example of the complex relationship between research and practice. I have been dogmatic in the past pushing students to think in certain ways or struggle through problems, swinging the pendulum too far in one direction. But desirable difficulties don’t offer easy and straightforward solutions for the classroom, and neither do other areas of research.

Instead, I think it’s worth thinking about the tension between learning and performance, and the balancing act between desirable difficulties and unproductive struggle. And more broadly, whenever I encounter an idea that seems simple on the surface, it’s worth probing for the tensions and contradictions that come up whenever principles of teaching make contact with the practicalities of classrooms.

The Limits of Language

And so I’ve been asking myself lately, is equity even a useful term for me anymore? And I’ve kindof come up with the decision that it’s not. Not because it no longer works for me personally; when I think of that word equity I know what I’m talking about. But equity is this word that has been used so often now, it’s like that word diversity, that it kindof doesn’t mean anything anymore, or at least it’s not clear what it means. I liken it to the organic and natural movement of the 1970s. Back in the 60s and 70s when somebody said that they had something that was natural or that was organic, there was a certain standard we held ourselves to, right? Now, Campbell’s soup is organic, you’ve got strawberries that are grown in fields that the organic ones are grown right next to the non-organic ones, it’s the same distributor, so you have to ask yourself, what is this thing we even say when we say equity? Because historically what has happened with that word, and how it has gotten taken up, I think that it fails to promote dialogue and visioning, because when we’re in a meeting, and somebody says “we really need to address equity in our math department” or “in this committee when we’re selecting a new person to come into our field we need to be thinking about equity issues.” Well we don’t tend to stop ourselves and ask, “what do you mean by equity?” We tend to all in our heads go, “yea, I agree, I want to address equity too,” but there’s many different versions out there.

-Rochelle Gutierrez

The answers to why we should be racially diverse often hover most around “because I know it’s the right thing to do”, which just doesn’t cut it. I wanted to ask:

Why is it the right thing to do?
Why do you want me here?
What do you miss by my absence?

All met with silence.

-Marian Dingle

First, go watch Rochelle Gutierrez’s talk and read Marian Dingle’s blog post. They’re fantastic, and I learned a ton from both of them. I want to pick up on one idea I took from both pieces.

Language is tricky. When I use a word, I might mean a certain thing, but I don’t know if those I’m engaging with understand exactly what I mean.

The trickiness is compounded in equity work. First, because equity work is messy, especially in social media-oriented spaces that incentivize catchy sound bites and quick fixes. And second, because disrupting the status quo is uncomfortable, and humans don’t like being uncomfortable.

The word “equity” is a great example. When I first understood the distinction between equity and equality, that idea was really useful. Equality has a connotation of giving everyone the same thing, while equity means giving every individual what they need to thrive. Equity helped me to think more critically about what I want for my students. But there’s a lot more complexity under the surface. Similarly, diversity is something most folks in education seem to believe is a good thing. But does our collective belief in diversity mask hard questions that might force us to confront the places we’re coming up short, and think critically about why we want diversity? These are the questions Rochelle and Marian are asking, and they each pushed me to reconsider the ways I use language, and the ways that I might be able to use language to better say what I mean.

So certain words have useful meanings, but as they become more and more common that meaning becomes normalized, becomes comfortable, and stops dialogue and reflection rather than encouraging it.

I think this happens in the language we use around teaching math as well. I want to teach so that students develop “conceptual understanding” of math content. It’s the difference between teaching how to solve a single problem and teaching how that problem represents a larger concept, is connected to other problems, and how that knowledge might be applied in new contexts in the future. A nuanced perspective on conceptual understanding helps me to better make sense of times when students struggle to transfer their knowledge to new problems and inform what I emphasize when introducing a concept.

But when I talk to other teachers, conceptual understanding often ends conversation rather than starting it. Everyone agrees conceptual understanding is important. But when you say conceptual understanding, do you mean the same things I do? How, exactly, does one teach students in ways that promote conceptual understanding? In what ways has that idea changed your teaching practice? The language alone often serves to create agreement and consensus, rather than conversation and further thinking.

I see something similar with “misconceptions.” Students aren’t blank slates; they bring their ideas and prior knowledge to class, and sometimes the ideas they bring are counterproductive for what I want them to learn. Anticipating those misconceptions and addressing them is much more productive than pretending that each student is an empty vessel that I’m pouring knowledge into.

But some folks argue that labeling student thinking as a “misconception” is counterproductive. They argue that students are trying to make sense of new ideas based on what they know, and their ideas are really just conceptions — “misconception” is a judgment based on our perspective as teachers. I find this really useful too! Not that I think misconceptions are evil or don’t exist, but thinking of student conceptions instead of misconceptions helps me to seek out the valuable thinking that students bring and build off of it, rather than see their ideas as broken or unusable.

The language I use shapes the way I act. If I am in the habit of labeling student ideas as misconceptions, I am likely to emphasize correcting them over valuing their thinking, and to miss opportunities to see potential bridges between where students are and where I want them to get. If I push myself to think more about conceptions, I get in the habit of assuming students bring valuable ideas to class, and creating space for students to recognize their value as mathematical thinkers. This isn’t just semantics; language influences how I see the world, and changing my language helps me to see and act in new ways.

Some people would argue that the policing of political correctness has run amok, that liberals only want to tell people what they can and cannot say, and blame and shame those who don’t conform. That argument misses the point; the point is not to change language, but to use language as a lever for action. Being more thoughtful and specific and concrete in the language I use is a generative experience; it leads me to new ways of thinking, being, and doing. Rochelle articulates the ways that “equity” has become inadequate and offers “rehumanizing mathematics” as a framework for thinking more deeply about equity in math education. Her talk helped me to find new places to work on my practice, and a new lens through which to look at my classroom. Marian asks hard questions about the ways that “diversity” promotes agreement without pushing people to clarify why they want to be in diverse spaces. Her post helped me to interrogate my motives, to better understand why I believe what I believe, and to locate the places I need to do more internal work. I’m grateful for both perspectives. Both took ideas that might seem common-sense and comfortable, and unpacked the messiness and the places where folks are falling short. Both seek to hold folks accountable for following through in practice. And both will help me to find new places where my language is inadequate to a task, and push myself to find new language that holds me accountable for more productive action.