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Realism and Idealism

Math Curmudgeon juxtaposes two arguments I’ve seen on Twitter several times in the last few weeks:

“If kids in your class are more engaged by a fidget spinner than they are by your lesson, the spinner isn’t the problem. Your lesson is.”


“Learning is hard. Kids fidget. Fads come, then go. Your lesson doesn’t suck simply because two kids out of 25 are fiddling with this thing.”

I find the tension between these two perspectives one of the most important interior monologues I have as a teacher. It’s a tension between the idealism of teaching engaging, meaningful, purposefully-structured lesson every day and the realism of the challenges and imperfections of classroom teaching and doing the best with what I have to work with. I need to be able to see both perspectives. I need to ask myself every day if I can be more engaging, ask better questions, be more responsive to student needs, build stronger relationships, and structure more meaningful curriculum. I also need to acknowledge that much of what happens in my classroom is at least partially outside of my control, and a lesson that doesn’t go well or a disengaged student doesn’t mean I’m a terrible teacher.

Honoring the cognitive dissonance of realism and idealism applies beyond the recent craze of fidget spinners.

  • Teaching the last class of the day or the last day before vacation, I need to bring my best and believe that students can learn every minute while acknowledging that those classes are going to look different than mid-morning classes on a typical Tuesday.
  • Working with students with math anxiety, I need to believe that every student can learn and achieve at high levels while acknowledging that the most practical way forward may be to adjust my expectations in the short term.
  • When I look at a set of exit slips and see that my students all have the misconception I tried to address in class that day, I need to ask myself how to improve that lesson next time while acknowledging that human brains are mysterious, learning is hard, and even the best lesson plans often come up short.

Every moment of teaching is useful feedback on what works and what can be better next time; every moment of teaching is an imperfect teacher with imperfect students in an imperfect institution trying to do what’s best in that moment.

Transfer: The Low Road and the High Road

I often justify my existence as a math teacher by arguing that math is worth learning because it teaches humans to think clearly and reason abstractly. Or, in the words of Underwood Dudley:

What mathematics education is for is not for jobs. It is to teach the race to reason. It does not, heaven knows, always succeed, but it is the best method that we have. It is not the only road to the goal, but there is none better.

Is this true? Does a mathematical education teach transferable skills that can be applied beyond the classroom?

Thanks to Michael Pershan for sharing an informative research paper on this topic that gave me a new perspective on teaching for transfer: Are Cognitive Skills Context-Bound?, by D. N. Perkins and Gavriel Salomon.

Half a century ago, many psychologists would argue that effective thinking is a function of intelligence and general strategies for problem solving and critical thinking. Polya’s work in problem solving was particularly influential, as he identified a number of heuristics such as breaking a problem into subproblems or examining extreme cases that could be applied to a wide variety of problems in different domains. Many thought that expertise consisted largely of general strategies like these that helped humans to reason across a range of contexts.

In the following decades, a growing body of research provided evidence that seemed to contradict this hypothesis. Studies of experts in various fields showed that their knowledge did not transfer readily outside of the domain in which it had been learnt, suggesting they had developed a specialized skill rather than a broad array of general reasoning strategies. Chess players could excel at chess, but not broader strategic thinking. Doctors who were expert diagnosticians in one field were no better than chance in another. Research looking specifically for transfer found that it rarely happened spontaneously and seemed much more elusive than had been previously thought.

For many today, this is a dominant paradigm of psychology: knowledge gained in one domain is unlikely to support thinking in another domain. However, there are plenty of contradictory results suggesting there is more to the story. That transfer exists is self-evident; under some conditions humans are capable of solving problems they haven’t seen before. This has been replicated in some studies but not in others; the question is, what are the conditions to make transfer more likely?

The Low Road and the High Road to Transfer
The authors describe two conditions under which transfer seems more likely. The “low road” to transfer involves “much practice, in a large variety of situations, leading to a high level of mastery and near-automaticity” (22). Practice and fluency in a domain makes it more likely that those skills will be drawn upon in a novel situation. The “high road” to transfer “depends on learners’ deliberate mindful abstraction of a principle” (22). Knowledge that is contextualized and connected with other ideas or broader principles is better primed to apply to a new situation. In short, there are two ways I can teach students to increase the odds they will be able to flexibly apply their knowledge in the future: effective practice and mindful abstraction.

Neither of these “roads” is certain, but they also provide a blueprint for learning that is unlikely to transfer: if there is insufficient practice and learning only takes place in one context without explicit abstraction, transfer seems all but impossible.

Where To From Here?
I often get frustrated with the arguments between the inquiry-oriented “progressive” folks and the explicit instruction “traditionalist” crowd. From my perspective, they’re both right. Learning needs to focus on connections to different elements of prior knowledge, to examine the ways that mathematical content can apply in other domains, and to focus on depth and flexibility of knowledge, all arguments of progressive educators. At the same time, purposeful practice spaced over time leads to fluency and automaticity with key ideas, making it more likely that they can be applied and synthesized with other knowledge in the future, as emphasized by traditionalists.

I think both sides have a point. I’ve seen too many students struggle with a challenging problem because they lack prior knowledge I wish they had — whether that’s addition, fraction operations, integers, or reasoning about the structure of functions. With more practice and better fluency and automaticity, they could be more successful. And I’ve seen too many students struggle with a challenging problem because their prior knowledge is totally context-bound — they’ve only solved problems from one perspective, and are unable to see how their knowledge applies to a novel problem at hand. Prior mindful abstraction of the principles they need would support this thinking by making explicit connections they may be able to make use of in the future.

It’s not an either-or, it’s a both-and. I need to be teaching students so that they have access to both the low road and the high road to transfer. And doing that depends on the content, the students in the room, and where we are in the broader curriculum. There are no easy answers. But I think that considering these two pathways to transfer is a useful touch point for my pedagogical priorities in the classroom.

It’s Not How You Learn, It’s What You Do With It

One mistake we make in the school system is we emphasize understanding. But if you don’t build those neural circuits with practice, it’ll all slip away. You can understand out the wazoo, but it’ll just disappear if you’re not practicing with it.

-Barb Oakley, source

I stumbled across the above quote in a recent interview in the Wall Street Journal, and it struck me as a useful way to think about my teaching.

When I first started teaching, I spent most of my planning time thinking about how I wanted to introduce new topics to my students. I was always looking for clever ways of explaining ideas and interesting new perspectives and hooks relating content to prior knowledge or student interests. I designed inquiry lessons carefully leading students to the big mathematical ideas I wanted them to grapple with.

Now, I spend much more of my time thinking about practice. Not that how I introduce a topic is irrelevant, just less important than what students actually do with the knowledge they’ve gained. I think about how to space that practice and interleave different topics, how to build toward more rigorous applications, how to ensure students engage with a topic in multiple contexts and use multiple representations over time. I work to create collaborative structures that will support students in doing challenging math while still providing individual accountability. I design sequences of activities that move between whiteboarding, technological manipulatives, and pencil-and-paper to keep students engaged for a full class.

The core principle of my teaching is that students are active in their learning. Students learn math by doing math. Practice can have a negative connotation among teachers, and research suggests repetitive practice on low-level tasks is ineffective for learning. But focused, purposeful practice that pushes students outside their comfort zone, is designed to move toward meaningful goals, and involves useful feedback is absolutely necessary for deep, durable learning.

There’s a constant balance here. John Sweller’s Cognitive Load Theory suggests that if the demands of problem solving are too great, students may not retain what we want them to learn even if they are successful in solving the problem. I am partial to Ben Blum-Smith’s summary: “any thoughtful teacher with any experience has seen students get overwhelmed by the demands of a problem and lose the forest for the trees”. At the same time, Robert Bjork’s work on desirable difficulties suggests that if students don’t experience any difficulties in the learning process, what they learn is unlikely to be retained in long term memory or transfer to new contexts. Meaningful learning is hard; if it feels easy it’s likely a missed opportunity.

I’m uninterested in arguing about whether discovery or direct instruction is better. From my perspective, those terms have been overused and caricatured to become meaningless pejoratives. As Dan Willingham says, memory is the residue of thought. What are students thinking about? What does that thinking look like? Those are the key questions I’m interested in, and I think they lead conversations past surface features to the substance that has a real influence on learning.

So students learn math by doing math, and my job is to constantly monitor what that experience is like for students. To what extent are they challenged and thinking deeply about mathematics? To what extent are they overwhelmed and struggling to connect the dots? If I can find a balance between these two poles while keeping students doing substantive math that builds toward ambitious goals, it’s a good day for me.

Research to Practice: Feedback

The purpose of this post is to digest the research on feedback and explore results that I wish I had learned about earlier in my teaching career. This is not a formal research review; I am cherry-picking topics I find useful and ignoring areas where, from my perspective, researchers have had trouble agreeing or results have tenuous links to classroom practice. I’m also including my own extrapolations to how these ideas apply to my teaching. I’m not an expert in this field, so take things with a grain of salt.

By far the most useful source has been Valerie Shute’s review Focus on Formative Feedback. It is very readable and I highly recommend it. I’ve also learned a great deal from Kluger & DeNisi’s review, Dylan Wiliam’s chapter on feedback that moves learning forward in Embedding Formative Assessment, and John Hattie’s book Visible Learning. Collectively, these authors cite several thousand sources. My goal is not to provide an exhaustive bibliography, but to explore a small number of key ideas I find useful.

Use Feedback Sparingly 
One commonly cited result on feedback is that, in Kluger & DeNisi’s review of nearly a century of research, 38% of feedback interventions resulted in negative effects; that is, in 38% of experiments, feedback resulted in less learning than a control condition with no feedback. There’s reason to be skeptical of this number. It does not imply that 38% of teacher feedback is preventing student learning. Rather, it suggests that many things that a teacher might intuitively think could be useful feedback may be less useful than they think. My corollary here is that, in most cases, giving feedback takes a great deal of teacher time. There are lots of other things I can do in a day that I am fairly confident support student learning. I should consider alternatives before giving individual feedback and carefully evaluate what offers me the most value.

Sooner Isn’t Always Better
“Feedback should happen as soon as possible” is often treated as a truism in teaching. However, in a number of studies analyzed by Shute, research suggests that delayed feedback may be more effective than immediate feedback. It is difficult to sort through the various interactions here. One element is that feedback should not interrupt the learner during a task; this is one example of feedback that can be harmful for learning. A second is that, in some situations, immediate feedback causes the learner to rely on the feedback rather than their own thinking. Third, delayed feedback seems to be useful for simpler tasks. One possible mechanism is that, by revisiting a topic later through feedback and spacing learning, the student has an opportunity for more useful thinking than finishing a session with that feedback or relying on it rather than doing additional thinking. Immediate feedback seems more important for complex tasks. This is a tricky one, and there are no clear-cut answers, but it’s worth hesitating to consider the interaction between the learner and the feedback before giving it during or immediately after a task.

No matter how well feedback is articulated, it is only useful if the learner engages with it. Wiliam explores three triggers that may cause a learner to reject feedback. Each of these triggers is dependent on the learner’s perception of the feedback, rather than the intention of the teacher. In other words, no matter how thoughtful feedback is, and even if that same feedback was successful for another student, if a student perceives it in certain ways they are unlikely to learn. Truth triggers occur when a learner gets feedback that they perceive as incorrect or unfair. Relationship triggers occur when a learner gets feedback from an individual they don’t trust or don’t think has their best interests in mind. Identity triggers occur when a learner interprets feedback as saying something about who they are as a person rather than communicating concrete ideas about their thinking or their work. To consider the flip side of each of these triggers, feedback needs to consider the student’s perspective, needs to be built on authentic relationships, and needs to communicate concrete ideas about the student’s work rather than giving grades that are value-laden for many students. Building off of the last idea, research suggests that comments are much more useful than grades for promoting learning. What is more surprising is that, when comments and grades are given together, there is little difference for learning than if the grade was given alone; the grade acts as an identity trigger that causes the learner to focus on themselves rather than making use of the feedback in the comments.

Verification and Elaboration 
Shute writes that effective feedback often combines two elements: verification — communicating the extent to which student ideas are right or wrong; and elaboration — explanatory information communicating analysis of the work or areas for further thinking. Verification should avoid potential triggers by communicating about specific features of the task rather than using grades or similar value-laden information that can distract from specific features of student work.

Feedback Should Cause Thinking 
Dan Willingham writes that memory is the residue of thought. Wiliam explores this idea as well, writing that if feedback does not cause the learner to do additional thinking, it is unlikely to lead to learning. The central idea of this principle is that feedback should be concretely connected to some student action that involves future learning. This can happen in a variety of ways, for instance when the feedback is presented in a way that requires thoughtful interpretation by the student, leads to revision or reassessment, or sets up a classroom activity engaging with the ideas presented in the feedback. If the feedback is more work for the donor than the recipient, or if the feedback focuses more on the past than the future, it may present a missed opportunity for additional learning.

Shute notes that many of the experiments informing the research results above were conducted in laboratory settings where motivation was largely controlled for. This should cause classroom teachers to take research on feedback with healthy skepticism. While a certain feedback strategy might be research-based and well-intended, if students are unmotivated it is likely to be ineffective. This leads me to two ideas. I need to communicate to students that I care about the quality of their work. This can happen through feedback; it can also happen through other means like student work analysis or targeted review. But one useful function of feedback is to improve motivation by communicating that student work matters. Whether through feedback or other means, I need to find ways to let students know that their work matters regularly. Second, I need to clearly articulate to students the feedback strategies I am using and why I am using them, so they understand how feedback connects to their learning and the purpose behind my classroom decisions.

Looking Back 
Considering these principles of effective feedback, I think they are fairly limited in their utility. I don’t think there’s enough to go on to really design research-based classroom feedback strategies that will work across a variety of contexts. That said, I spent the first few years of teaching without any benchmarks for what effective feedback looked like; I did what seemed right at the time and leaned heavily on what teachers around me were doing and what I had experienced as a student. I think that the ideas above offer a useful lens to move beyond those anecdotal experiences to more purposeful strategies focused on maximizing student learning.

Problem Solving and Creativity

I try not to put too much stock in the endless hole of people talking about Silicon Valley and startups and the tech world on the internet. But this quote caught my eye:

You need to know the things that you need to know to solve the problem. And you need to not believe things that will get in the way of solving the problem.

Sourced here, which credits the quote to Scott Klemmer, though I can’t find the original anywhere. He was talking about design and what research says about creativity, but I think it applies well to problem solving in math. Leads me to the question: is there anything else to problem solving?

Talking About Learning Styles

Screenshot 2017-04-29 at 7.59.31 AMScreenshot 2017-04-29 at 7.58.53 AM

These are the results of a completely unscientific survey I did of people who happened to see these tweets, with a sample size of 75 < n < 147.

My hypothesis, which remains unproven by these polls, is that many of the people who say “it’s important to address student learning styles” actually mean “it’s important to use a variety of modalities”.

This is important because, after a number of reviews of the research (lit review, letter in The Guardian, research summary), there seems to be convincing evidence that adapting instruction so that teaching meshes with each individual student’s learning style does not improve learning. That’s important for teachers to know. It’s also a pretty narrow claim. If I am working to provide visual learners visual instruction, auditory learners auditory instruction, and so on (or other variations on learning styles) then the research has something to offer me. If, when teachers talk about learning styles, they really mean that they try to use a variety of representations and activities in class, that is a separate pedagogical strategy, and one that many more educators agree with. My experience is that it’s actually fairly rare for a teacher to attempt to determine student learning styles and tailor instruction to those styles. It’s not a unicorn; it does exist, and the research suggests it’s an ineffective use of teacher time. But pretty rare.

I think this is worth noting because discussions between educators on learning styles can quickly become angry and bitter. I think some of those conversations would benefit from a pause and clarification of what is actually being discussed. I think those who believe the research that learning styles are a myth could be much more careful to ensure that they are being critical of learning styles theory, and not other, related ideas that a teacher is using learning styles as shorthand for. And I think all teachers could be much more precise in their language and what they actually mean when they talk about learning styles.

Meaning Making and Structure Building

Of all I saw and learned this past half year, one thing stands out. What goes on in the class is not what teachers think — certainly not what I had always thought. For years now I have worked with a picture in mind of what my class was like. This reality, which I felt I knew, was partly physical, partly mental or spiritual. In other words, I thought I knew, in general, what the students were doing, and also what they were thinking and feeling. I see now that my picture of reality was almost wholly false. Why didn’t I see this before?

-John Holt, How Children Fail

I was fooling around on Youtube and ended up watching Beyonce’s Irreplaceable. It’s a song I’ve liked for a while, though I’m only a casual fan. Watching it last night I realized I’d been both hearing and singing it wrong for years. I had been singing, “everything I own in a box to the left” when the actual lyric is “everything you own in a box to the left”. And that’s a pretty significant distinction. Here’s a context clue from the video to help:

Screenshot 2017-04-24 at 10.20.48 PM.png

I had been listening to and enjoying the song for a long time — but in all that time, I had managed not to change a significant misconception or probe beyond the surface of my understanding of what was happening.

I’m curious how many of my students experience my teaching in this way, spending their time in class thinking about surface features of the mathematics we are studying without putting significant cognitive work into the underlying meaning of the content.

At NCTM a few weeks ago, I attended a talk by Skip Fennell, Beth Kobett, and Jon Wray on formative assessment. You can check out their slides here. One strategy that stuck with me was using an interview to explore student thinking after a task, asking the student how they solved the problem, why they solved it that way, and what else they can tell me about their thinking. It’s obviously impractical to do this every day or with every student. But it’s also a strategy I’ve never used to explore student thinking in depth, and with the premise that students often know less than I think they do I’m sure I would get some great insights out of it. One more point the presenters made was that the interview doesn’t have to be with a student who is struggling; talking with a student who is effectively using certain strategies could be useful in figuring out what moved their thinking forward and how to help other students with that thinking.

Here’s a final thought:

There do appear to be cognitive differences in how we learn. … One of these differences is the idea … that psychologists call structure building: the act, as we encounter new material, of extracting the salient ideas and constructing a coherent mental framework out of them. These frameworks are sometimes called mental models or mental maps. High structure-builders learn new material better than low structure-builders. The latter have difficulty setting aside irrelevant or competing information, and as a result they tend to hang on to too many concepts to be condensed into a workable model (or overall structure) that can serve as a foundation for future learning.

-Brown, Roediger & McDaniel, Make It Stick

I see structure building as the biggest difference between successful students and students who struggle. The most important piece of the research that the authors present on structure building is that guidance toward the key elements of a problem that makes explicit the essential relationships can support all students in structure building and making sense of the mathematics.

I’m not sure how well these ideas are connected — Holt, Beyonce, formative assessment interviews, and structure building. But it’s been some good food for thought in probing more meaningfully into student thinking, and constantly asking myself whether students are actually doing the thinking that I hope they are doing.