I’ve been thinking recently about the difference between the principles of an idea and how that idea functions in the classroom; the difference between theory and practice. Conversations about education, especially those in popular media, tend to make broad generalizations on principle without ground-truthing to figure out how an idea plays out with real live teachers and students.

The principles of personalized learning, that one-size-fits-all education does not meet the needs of every student, are undoubtedly true. But in practice, that idea often functions to put students in front of computers for long periods of time, creating lifeless classroom where learning is reduced to spreadsheets and joy is sucked from the room.

The principles of Understanding by Design are useful to organize purposeful curriculum. But in practice, that idea often functions to require teachers to write an objective or enduring understanding on the board, without actually engaging with backwards design or creating a more meaningfully sequenced curriculum.

The principles of constructivism, that students must create their own meaning of new ideas, communicate something true about human cognition. But in practice, that idea often functions to ask students to figure everything out themselves and withhold necessary supports in the name of inquiry, a pedagogy that exacerbates inequities by hurting previously low-performing students the most.

The principles of mindset research, that growth or fixed mindsets have a large influence on future learning, are sound. But in practice, that idea often functions to reduce mindset thinking to platitudes about praising effort rather than ability, platitudes that are often hollow and certainly insufficient to the challenging task of changing mindsets.

These are just a few examples; I could go on. My point is that I have worked to practice this type of thinking; when I hear an idea in education, I try to stay just as curious about the broader principles as about how it functions in classrooms with the imperfections of teachers and the fickle nature of learning.

I first started doing number talks (also called math talks) to start class when I taught 8th grade. If you’re unfamiliar with number talks, this site by Fawn Nguyen has some great stuff to get started. My first year teaching high school (Algebra II, Precalculus, and Calculus) I stopped, opting for Visual Patterns, Open Middle, Which One Doesn’t Belong?, and a few other rotating warm-up routines. I thought that the skills involved in number talks, while useful for middle school students, were less relevant for upper high school.

I came back to number talks at the start of this school year, and I’ve been happy with the results. When I wrote a problem up on the board one day toward the end of the year, a student blurted out, “oh, I love these”. That’s just one student, but engagement was usually high. Efficient strategies for these problems often did not come easily to students, which suggests that there’s potential for learning from them. More importantly, as they became comfortable with the routine, many students who were rarely willing to share started to speak up and take more risks.

Here’s a selection of my favorite problems, many courtesy of Fawn’s site.

Picking two numbers and an operation is often insufficient for a great number talk; I’ve found that careful selection of the numbers involved to ensure a variety of strategies is worth the effort every time.

I have one lingering question for next year. Engagement during number talks seems high, and seems to engage both high-performing and low-performing students. There is clearly a need for the skills that number talks are targeting. At the same time, I don’t have any real evidence that students are learning these mental math skills. I think they are, but that’s based on my intuition and a few anecdotes. One challenge is that I tend to cycle through a variety of types of number talk problems that require different strategies. One goal for next year is to reorder the number talks I use so that I expose students to one type of problem 2 or 3 times, lead a discussion that attempts to consolidate understanding of relevant strategies and when they may be useful in the future, and then revisit that problem a few weeks later. Hopefully this sequencing will provide more robust evidence as to whether or not students are actually learning.

I really enjoyed a series of blog posts I recently discovered summarizing how cognitive science can be applied to education. The section on attention in particular caught my eye. It’s worth noting that attention doesn’t just mean students are sitting up straight and looking at the front of the room. Instead, attention is about thinking. It’s about asking ourselves what students are thinking about, and how we can influence that thinking.

The last few years have seen a movement towards the discussion of “non-cognitive skills.”  But what these really get at are ways into attention:

• Motivation is really about the voluntary direction of attention.  When we are motivated to do something, we pursue it more often; we give it more attention.  
• Similarly, Carol Dweck’s research on mindsets–whether we believe our intelligence is fixed, or whether we think intelligence is malleable–her research really explores whether we sustain our attention in the face of adversity.  If we have a growth mindset, we believe that our work improves with effort, and so we direct our attention to it repeatedly.
• Roy Baumeister’s research on willpower explores the factors that influence whether we sustain attention.  Our attention and resolve are limited, but we can exercise and adjust the factors that marshall our limited attention.
• And out of Stanford, Clifford Nass’ research on multitasking (and our inability to do it) further informs how we channel, and lose, attention.

In all these–motivation, mindsets, willpower, and multi-tasking–we find we are really talking about attention, and that exploring these “non-cognitive skills” is really another way to understand how and why people direct their attention–or not.

I really like this interpretation of a number of “pop psychology” publications that are popular with educators. I want to add my own spin.

I think of attention in terms of working memory. We can only hold a few ideas in the mind at one time. The research cited above provides a useful window into whether students direct and sustain attention on what we want them to pay attention to in class. While creating environments where students direct and sustain their attention in school is important, perhaps more important is what they are paying attention to.

I see two more important questions, building off of the ideas above:

• Is attention focused on the right stuff?
• Is attention overwhelmed to the point that it’s hard to learn?

As I have grown as a teacher, I am better able to ask myself the question, “What are my students thinking about right now?” Motivation, mindset, willpower, and multitasking are one useful lens here. But a student who is effectively paying attention may still only be paying attention to surface features of the problem rather than its deeper structure, or to a calculation without considering why that tool is the appropriate choice in that situation. It’s not just whether a student pays attention; memory is the residue of thought, and what students think about is what they will learn. The more I am able to take students’ perspectives, the better I can design learning experiences that get their attention focused on the right stuff, the essential mathematics that I want them to learn, rather than surface features that fall short of my goals. Building this knowledge means pulling the right ideas into working memory, connecting them to larger ideas students already know, and making sure attention is laser focused on that thinking.

At the same time, even when student attention is focused on the right stuff, if the reasoning they are doing overwhelms their working memory, it’s unlikely that anything will be retained. Here, attention can be all in the right place, but there’s too much to pay attention to, and students lose the forest for the trees. The challenge of figuring out what they are trying to figure out prevents opportunities to step back, take a larger perspective, and consider how what they are doing is connected with other things they have learned and consider how they might use it in the future.

These constraints on attention provide some useful questions to ask. Are students paying attention? If not, how can I facilitate an environment that helps them direct and sustain attention? Are students paying attention to the right stuff? If not, how can I design activities that will help them do so? Are students overwhelmed by the demands on their attention? If they are, how can I reduce the cognitive load of the activity to help them learn effectively?

I’m still unpacking Lani Horn’s awesome talk on an asset-orientation that you can watch here, and I began exploring a few days ago here.

Ambitious Instruction

I want to explore the idea of ambitious instruction and the teacher actions connected to this idea. For a more academic read, this paper outlines the term ambitious instruction. Lani uses a much simpler representation to get across the key ideas:

I really like this contrast. Ambitious instruction takes typical practice and sets higher goals that are focused on student thinking and an expansive view of what it means to do mathematics.

Slipping Away From Ambitious Instruction

Lani talks about a result from the MIST study where many teachers were aiming for the ambitious instruction, but slipped back to the left side of the chart when students struggled. A number of teachers viewed the struggles as intractable because they focused on students’ deficits and shortcomings. This is a clear problem; if teachers’ conceptions of students cause us to think, implicitly or explicitly, that they aren’t capable of engaging with meaningful mathematics, we’re stuck.

Even tougher was that a larger group of teachers, even if they didn’t use deficit language to characterize students, still moved away from ambitious instruction when students struggled. They were trying, but when things got hard they slipped back and reduced the cognitive demands for students.

I can see myself in both of these examples. I’ve been guilty of using a deficit framing of struggling students, and I’ve been guilty of lowering the cognitive demand of tasks when the going gets tough. Both actions can seem benign on the surface, whether I’m describing a student as unmotivated or making a choice that a certain task isn’t appropriate for that class that day. But in practice, these actions functioned in a way that lowered expectations and denied opportunities to learners.

Moving Forward

One solution Lani offers is teacher education and ongoing professional development that focus on ability, bias, and an asset-orientation to counter deficit thinking. I want to continue thinking about how to build this habit: to catch myself in instances of deficit thinking, to educate myself in ways of seeing strengths in all students, and to surface and address my own biases.

At the same time, I think there’s an important instructional piece. I can enact high expectations for students by challenging them with high cognitive demand tasks and having scaffolds ready if they are necessary. I can practice the course corrections I need when I realize a class is not ready for a demanding task, step back to build the foundation, and return to an opportunity to challenge students with meaningful mathematics.

I see these as two different skills I can work to improve to support my practice:

• An asset-oriented approach to framing and talking about students that frames challenges as solvable and values students for what they bring to the classroom
• A focus on adjusting the scaffolds and supports rather than the rigor and expectations of demanding tasks that students struggle with

This still feels a little fuzzy to me, and I’m left with the same question Lani ends her talk with: what structures help teachers sustain this work and this practice on a day-to-day and a year-to-year basis?

Lani Horn gave a great talk at the University of Utah earlier this year that I just stumbled across a video of on Twitter. The title of the talk is, “An Asset-Orientation is Everything: How Strengths-Based Approaches to Math Teaching Help Teachers and Students. The heart of the talk is about 40 minutes and it is absolutely worth watching. I want to pick on one small element of what she said that resonated for me, and also poke at it a little bit.

In her talk, Lani focuses on the challenges of ambitious teaching in math and the influence an asset-orientation has on teachers’ ability to improve their practice. More specifically, she makes this claim:

An asset-orientation is necessary (but not sufficient) for math teachers to improve their practice.

If you’d like to see her reasoning and the breadth of research she cites to back up this claim, go ahead and watch the talk. In this post I want to focus on exploring what an asset-orientation is and also what I think it is not.

Asset-Orientation 

Lani defines an asset-orientation in her talk:

When teachers take an asset-orientation toward students, they seek to understand their strengths and value them as whole people.

She then offers a contrast between two ways of talking about a student:

It’s worth unpacking the first image, because those phrases and similar phrases can seem benign on the surface, but function in ways that perpetuate deficit thinking. Things that seem descriptive, like “a C student” can actually essentialize that student’s capabilities. Even when we mean well, phrases like “at-risk” can surface a deficit framing for a student that assumes they are less capable, assumptions that are likely to play out in practice.

An asset-orientation focuses on strengths and values students as whole people. Here’s another quote from Lani:

Doesn’t mean I’ve given up on her as a student, though. Doesn’t mean that I’ve excused her or written her off. I am going to work with her. to figure out how she’s going to develop her student skills, despite the challenges that she has, building off of the strengths I see in her.

Lani addresses a misconception I had about deficit thinking. Lani is talking about a shift in language from looking backwards — at prior performance, at demographics, at other things that are currently out of that student’s control — to looking forwards at the work that is necessary to help that student reach their potential. The purpose is not to pretend that deficits don’t exist. Instead, the purpose is to frame problems of teaching practice in ways that are solution-oriented and value what students bring to our classrooms.  Deficits do exist, but focusing on deficits and framing problems around deficits makes them suddenly intractable. An asset-orientation is solution-oriented and focuses on where the work forward begins — building off of student strengths and capabilities.

In the past I’ve felt a bit queasy about some of the language I hear teachers use when they chide others for using deficit framing. If deficit framing is being replaced with fluffy language that just replaces a deficit with a euphemism and offers no path forward, that’s not a useful change. If avoiding deficit framing is only focused on language and not what teachers do next, it’s just semantics. That’s why I think Lani’s conception of an asset-orientation is so important. It’s focused less on eliminating deficit framing — necessary, but not sufficient — and more on the language we should be using and the influence that language has on everyday practice.

If you’re interested in another thought-provoking talk from Lani, check out this video on what it means to have the knowledge one needs to be a teacher.

I have found that improving my understanding of human cognition — how thinking happens, what learning looks like in the human brain — helps me to better understand the learning that is happening or not happening in my classroom. The brain is complicated, and cognitive science has far more questions than answers. At the same time, there are things that cognitive science does understand. In communicating those ideas, metaphors are useful in illuminating how a theory plays out in practice.

I’m going to explore four metaphors for memory. Each is imperfect: each illustrates some principles and comes up short with others. I think that, together, they get at some core elements of thinking were not intuitive for me and can help to paint a rich picture of what is happening in students’ minds during classroom instruction. The first and third of these metaphors come from Dan Willingham in this blog post, which is well worth a read.

Flowchart

This model sets up the distinction between working memory and long-term memory. Working memory is where thinking happens, and long term memory is where knowledge and skills are accumulated. For information to enter long-term memory, it needs to go through working memory.

There are several shortcomings to this approach. One essential feature of cognition is that working memory is finite — we can only think about a few things at a time — while long-term memory is, as far as psychological science knows, infinite. At the same time, long-term memory is not just a database of information. The organization of information in long-term memory is essential, and knowledge that is chunked together effectively increases the capacity of long-term memory. Finally, working memory influences long-term memory. Every time we think about something, we influence that knowledge, which is not captured in the flowchart.

Library 
Another way to think of memory is to imagine the brain as a giant library, and thinking as a few flashlights shining on a certain areas of the library.

All knowledge is not created equal in this model, as the better organized the library is, the more information can be captured by the flashlights’ beams. It’s possible to know something and not be able to think about it if you’re unsure of where to shine the flashlight. It can also illustrate the limitations of what our minds are capable of. There are only a few flashlights, and if too many of those flashlights are searching for information or busy shining on something at one time, the mind gets overloaded. A central metaphor here is that thinking and memory are inextricably linked; they don’t happen in different places through different processes but are systems that are interconnected and working in parallel.

But this metaphor also falls short in important ways. It still does not get at the idea that thinking changes memory; thinking is not a passive flashlight shining but actually creates new knowledge and consolidates old knowledge. The library metaphor also further perpetuates the idea of memory as a static, inert database.

Hill of Sand 

Think of a hill of sand—that’s your mind. You pour water on it—water is thought. The water coursing over the sand creates gullies and rivulets. That’s memory. It’s a representation of where the water (thought) has been in the past and if the water moves through those same channels they will become a little deeper. The next time you think (pour water) it will likely happen in the channels it’s followed before….but not necessarily.  The new water also has the potential to change the gullies on the hill.

Long-term memory is what has been left behind by working memory; memory is the residue of thought. Every time you think about something, you both consolidate that knowledge further and change the nature of that knowledge. This metaphor emphasizes the connections between thinking and memory, and I think it says something powerful about the influence long-term memory has on thought. Those gullies that have been created by long-term memory guide all of future thinking, and can be thought of as our collection of beliefs, habits, and biases.

But this metaphor comes up short as well. Working memory capacity is finite; thinking of it as a stream of water can miss that point. A metaphor of cognition as a stream can miss the nuance of the connected networks of knowledge that make up useful and transferable skills. This metaphor also does not emphasize the difference between storage and retrieval; it’s possible to know something but not remember it at the time.

Stones in a Forest 
A final metaphor I’d like to entertain is of memory as stones in a forest. As I walk around arranging and building memories with stones, I also wear paths from one place to another creating links between different memories or connecting new memories to old ones.

Memory as a forest captures a distinction between retrieval and storage that I think is important. Once I’ve created a memory — arranged some stones in the forest — that memory will be very slow to degrade. The path to and from the memory will become overgrown quickly if it is not used (that’s forgetting), but an old path will reappear quickly when it starts to be walked again. These paths also do something to emphasize how, if knowledge is not richly connected with broader ideas, it is unlikely to be useful, and it will be challenging to synthesize that knowledge with other, new ideas.

Of course this metaphor has weaknesses as well. Thinking influences memory. While this model captures the importance of revisiting memories over time, it does not get at the idea that memories themselves are changed through continued use. It also focuses more on the nature of long-term memory than the process of working memory and the interaction between them beyond wearing paths between memories.

Metaphors 
Metaphors are useful, but they are also limited. I’m skeptical there exists an ideal metaphor for memory,, and each of these metaphors serves different functions and emphasizes different ideas while missing others. Together, they can create a useful understanding of how cognition occurs, but there’s plenty more left out. I’m sure some psychologists would say I’m missing the distinction between decision-making, auditory memory, and visual memory in my model of working memory. Others would argue that I’m oversimplifying chunking, the organization of knowledge, and the complexity of long-term memory. Many teachers would say that I am getting lost in theory without connecting it classroom actions. Others would say that I am focusing on memory at the expense of numerous other influences on students’ everyday learning. There are plenty more potential avenues for criticism.

But these metaphors serve a very specific purpose for me. As I think of memory through metaphors, I consolidate my understanding of my students’ thinking in new ways and organize it so the implications of that understanding are readily available as I teach. I think that these metaphors do contain some scientific accuracy, but my primary goal is for my knowledge to be useful for me in my teaching. I am constantly interpreting what happens in my classroom — the relationship between the learning experiences I design for students and the students’ learning, or lack thereof. If these metaphors can serve as a lens to understand why one strategy works or another strategy doesn’t, they have helped me to better understand my teaching and my students’ learning.

References 

Dan Willingham’s blog post was my primary influence on this post. I also found these sources useful:

Robert Bjork on storage and retrieval

Anna Sfard on metaphors

John Sweller on Cognitive Load Theory

Make It Stick

Why Don’t Students Like School

The claim that students will perform better when the teaching is matched to their preferred learning style is simply not supported by science.

Letter in The Guardian

The standard argument against learning styles is simple: experimental research seems to have definitively shown that matching instruction to students’ learning styles does not improve learning.

In my experience, this argument is also unconvincing for many teachers. I’d like to try on a different perspective.

Anecdotally, the most common learning style that people identify is being a visual learner. I’ve also met many people who identify “learning by doing” as their preferred style. Certainly other styles exist, but these are the two I have seen the most, drawn from the VAK (visual/auditory/kinesthetic) framework that is popular. These are people who are likely to speak up in learning situations by saying things like, “I’m a visual learner, I need to see a visual to understand this”, or “I really need a chance to do this to get it”.

One way to understand these requests is to acknowledge that effective visuals and active “learning by doing” strategies are underutilized, and all learners benefit from learning in a variety of ways rather than through a single modality. From this perspective, identifying with a certain learning style functions to influence instruction to improve learning. The learning style could be understood as a privilege — through prior experience, individuals have learned to advocate for better instruction through the lens of learning styles. Those who don’t identify with a certain learning style do not have these tools to advocate for themselves. Inevitably, this creates inequities between outspoken learners whose position allows them to advocate for more varied instruction to meet all learners’ needs, and those who, for whatever reason, do not.

While learning styles can create inequities that advantage those who advocate for themselves, they can also cement existing disparities by creating an avenue for learners to say, “well that’s not how I learn”. Labeling a student, or allowing a student to label themselves, as a certain type of learner is not very different from allowing them to label themselves a “math person” or “not a math person”. Whether or not the underlying label is valid, it functions to limit what the learner believes themselves capable of and steers them toward learning situations where they don’t need to push outside of their comfort zone or try new things.

Do These Land? 
The research evidence convinces me, but it doesn’t land for many other teachers. Is this perspective a useful one to change minds on the value of learning styles?